the Essential Standards
High School StudyText, Math A
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-890693-0 MHID: 0-07-890693-8 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 QWD 16 15 14 13 12 11 10
North Carolina StudyText, Math A
Using Your North Carolina StudyText North Carolina StudyText, Math A, is a practice workbook designed to help you master the North Carolina Essential Standards for High School Math A. By mastering the mathematics standards, you will be prepared to do well on your end-of-course (EOC) test. This StudyText is divided into two sections.
Chapter Resources • Each chapter contains four pages for each key lesson in your North Carolina Algebra 1 Student Edition. Your teacher may ask you to complete one or more of these worksheets as an assignment.
Mastering the EOC This section of StudyText is composed of three parts. Each part can help you study for your EOC test.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
• The Diagnostic Test can help you determine which standards you might need to review before taking the EOC test. Each question lists the standard that it is assessing. Your teacher may assign review pages based on the questions that you did not answer correctly. • Practice by Standard gives you more practice problems to help you become a better test-taker. The problems are organized by the North Carolina High School Math A Essential Standards. You can also use these pages as a general review before you take the EOC test. • The Practice Test can be used to simulate what an EOC test might be like so that you will be better prepared to take it in the spring.
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North Carolina StudyText, Math A
Contents in Brief Chapter Resources Chapter 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Chapter 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Chapter 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Chapter 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Chapter 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Concepts and Skills Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Mastering the EOC, Algebra I / Math A Diagnostic Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A1 Practice by Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A21 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A60
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North Carolina StudyText, Math A
NAME
DATE
0-9
PERIOD
Study Guide
SCS
MA.G.2.4
Volume Volume of Rectangular Prisms
To find the volume V of a rectangular prism, use the formula V = ℓ · w · h, where ℓ is the length, w is the width, and h is the height of the solid or the equivalent formula V = Bh where B is the area of the base. Example
Find the volume of the prism. V = ℓ · w ·h V = 3 · 6 ·4
4 cm
6 cm
V = 72
3 cm
The volume is 72 cm 3.
Exercises Find the volume of each figure. If necessary, round to the nearest tenth. 1.
2.
3.
2.5 in.
10 ft
10 mm
2.5 in.
12 ft 6 mm
25 ft
1.9 mm
Lesson 0-9
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2.5 in.
4. Rectangular prism: length 9 millimeters, width 8.2 millimeters, height 5 millimeters
5. Find the width of a rectangular prism with a length of 9 inches, a height of 6 inches, and a volume of 216 cubic inches.
6. FORTS Gina and her sister built a fort out of boxes. What is the volume of the fort? 3 ft 4 ft 2 ft 2.5 ft
Chapter 0
1
1.75 ft 3 ft
4 ft 2.5 ft
North Carolina StudyText, Math A
NAME
DATE
0-9
Study Guide
PERIOD
SCS
(continued)
MA.G.2.4
Volume Volumes of Cylinders Just as with prisms, the volume of a cylinder is based on finding the product of the area of the base and the height. The volume V of a cylinder with radius r is the area of the base, πr 2, times the height h, or V = πr 2h. Example 1
Find the volume of the cylinder.
V = Bh
2.2 ft
Volume of a cylinder.
2
V = πr h 2
= π 2.2 ·
4.5 ft
Replace B with πr 2. ·
4.5
Replace r with 2.2 and h with 4.5.
≈ 68.4 Use a calculator. The volume is about 68.4 cubic feet. Check: You can estimate to check your work. V = πr2h ≈ 3 · 22 · 5
Replace π with 3, r with 2, and h with 5.
≈ 60 Simplify. The estimate of 60 is close to the answer of 68.4. So, the answer is reasonable. Example 2 The volume of a cylinder is 150 cubic inches. Find the height of the cylinder. Round to the nearest whole number. V = πr2h
I
Volume of a cylinder. Replace V with 150 and r with 2.
150 ≈ 12.56h
Use a calculator.
4 in.
12 ≈ h Divide each side by 12.56. Round to the nearest whole number. The height is about 12 inches.
Exercises Find the volume of each cylinder. Round to the nearest tenth. 1.
6.5 cm
2.
5.4 in.
3. radius: 1.3 m height: 3 m
12 cm
4.
6.8 ft
2.3 in.
5.
6. diameter: 11 cm
2.1 m
height: 6 cm 9 ft
Chapter 0
3m
2
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
150 = π · 22 · h
NAME
DATE
0-9
PERIOD
Practice
SCS
MA.G.2.4
Volume Find the volume of each figure. If necessary, round to the nearest tenth. 1.
3.
2. 19.2 ft
5 cm
12 cm 30 ft
9 cm
4.
2.5 cm
8 cm
5.
8 mm
9 mm
6.
14 cm
9 in.
3 mm 30 mm
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7. Rectangular prism: length 22.5 feet, width 12.5 feet, height 1.2 feet
8. Cylinder: radius 2.6 m, height 8.4 m
9. Find the height of a rectangular prism with a length of 11 meters, a width of 0.5 meter, and a volume of 23.1 cubic meters.
10. Find the height of a cylinder with a radius of 15 feet and a volume of 60,052.5 cubic feet.
11. PLUMBING A pipe has a diameter of 2.5 inches and a length of 15 inches. To the nearest tenth, what is the volume of the pipe? Chapter 0
3
North Carolina StudyText, Math A
Lesson Lesson X-X 0-9
1.5 in.
19 mm
NAME
0-9
DATE
Word Problem Practice
PERIOD
SCS
MA.G.2.4
Volume 4. BARRELS Mrs. Washington has a rain barrel with a radius of 1.25 feet and a volume of 15.9 cubic feet. Find the height of the rain barrel to the nearest tenth.
2. GARDENS Paulo and his mother are building a rectangular compost bin for their garden. They want the bin to have a volume of 64 cubic feet. The width of the bin is 3.2 feet and the length is 5 feet. What should the height of the bin be?
5. SOUP CANS Mama Rosa’s soup is shipped in boxes with 12 cans to a box. Each can has a diameter of 3.5 inches and a height of 5 inches. Find the volume of 12 cans. Round to the nearest tenth.
3. PACKAGING The Fresh Chili Company is changing the size of their cans of chili. The new can needs to hold 500 cubic centimeters of chili. The height of the can is to be 11 centimeters. What must the radius of the new can be? Round your answer to the nearest tenth.
6. WASTE COLLECTION At Peter’s job, he needs to empty wastebaskets. The cylindrically shaped wastebaskets have a height of 3 feet and a diameter of
Chapter 0
1 feet. Peter empties the wastebaskets 1− 2
into a dumpster that is shaped like a rectangular prism. If the dumpster is 8 feet wide, 6 feet deep, and 5 feet tall, how many wastebaskets full of trash will fit in the dumpster?
4
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. CONSTRUCTION Johnson Construction Company is going to build a house on a concrete slab. The slab is to have dimensions 30 feet by 20 feet by 2 feet. How many cubic feet of concrete should the construction company order?
NAME
DATE
0-10
PERIOD
Study Guide
SCS
MA.G.2.4
Surface Area Surface Area of Prisms
The surface area of a figure is the sum of the areas of all the surfaces, or faces, of a solid. Surface area is measured in square units. To find the surface area S of a prism use the formula S = 2 ℓw + 2 ℓh + 2 wh. Example
Find the surface area of the rectangular prism.
Find the surface area.
2.8 ft 2.1 ft
5.8 ft
Surface area formula S = 2 ℓ w + 2 ℓh + 2wh S = 2(5.8)(2.1) + 2(5.8)(2.8) + 2(2.1)(2.8) ℓ = 5.8, w = 2.1, h = 2.8 Multiply. = 24.36 + 32.48 + 11.76 2 Add. = 68.6 ft
Exercises 1.
2. 14 in.
4.3 m
22 in.
2.8 m
13 in.
6.6 m
3. Cube: side length 8.3 centimeters
4. Rectangular prism: length 17 yards, width 4.5 yards, height 3 yards
5. Rectangular prism: length 16 feet, width 12 feet, height 42 feet
6. Rectangular prism: length 20.2 centimeters, width 10 centimeters, height 43 centimeters Chapter 0
5
North Carolina StudyText, Math A
Lesson 0-10
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the surface area of each prism. Round to the nearest tenth, if necessary.
NAME
DATE
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Study Guide
PERIOD
SCS
(continued)
MA.G.2.4
Surface Area Surface Area of Cylinders As with a prism, the surface area of a cylinder is the sum of the lateral area and the area of the two bases. If you unroll a cylinder, its net is a rectangle (lateral area) and two circles (bases).
r
C = 2πS
The lateral area L of a cylinder with radius r and height h is the product of the circumference of the base (2πr) and the height h. This can be expressed by the formula L = 2πrh.
I
r
The surface area S of a cylinder with a lateral area L and a base area B is the sum of the lateral area and the area of the two bases. This can be expressed by the formula S = L + 2B or S = 2πrh + 2πr2. Example
Find the surface area of the cylinder.
3.5 in.
Find the lateral area.
Find the surface area.
L = 2πrh
S = L + 2πr2
5 in. 2
= 2 π 3.5 5
= 35π + 2π(3.5)
2
= 35π in
= 59.5π in2
exact answer
≈ 109.9 in2
≈ 186.9 in2
approximate answer
Find the surface area of each cylinder. Round to the nearest tenth. 1.
4.6 m
2.
3.
8 mm
12.5 in.
12 mm 6 in. 6m
4.
5.
21 ft
6.
8.4 m 2.1 m
1.9 cm
18 cm
13 ft
7. diameter of 20 yards and a height of 22 yards 8. radius of 7.6 centimeters and a height of 10.2 centimeters Chapter 0
6
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
0-10
DATE
PERIOD
Practice
SCS
MA.G.2.4
Surface Area Find the surface area of each solid. Round to the nearest tenth. 1.
18 mm
2.
3. 5 cm
4 ft
10 mm
36 ft
24 cm 60 cm
12 cm
4.
5.
6.
38 cm
40 in. 12 in.
23 cm
32 in.
20 cm
7. Cylinder: radius 28 millimeters, height 32 millimeters
8. Rectangular prism: length 15.4 centimeters, width 14.9 centimeters, height 0.8 centimeters
9. Cylinder: diameter 25 inches, height 18 inches
PACKAGING For Exercises 10 and 11, use the following information. A cardboard shipping container is in the form of a cylinder, with a radius of 6 centimeters and a volume of 8595.4 cubic centimeters. 10. Find the height of the shipping container. Round to the nearest tenth.
11. Find the surface area of the shipping container. Round to the nearest tenth.
12. DECORATING A door that is 30 inches wide, 84 inches high, and 1.5 inches thick is to be decoratively wrapped in gift paper. How many square inches of gift paper are needed? Chapter 0
7
North Carolina StudyText, Math A
Lesson 0-10
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
20 cm
NAME
0-10
DATE
PERIOD
Word Problem Practice
SCS
MA.G.2.4
Surface Area 1. MANUFACTURING The Acme Canning Company produces cans for chicken soup. If each can has a diameter of
5. MUSEUMS The diagram below shows the floor plan of a museum.
4
65 ft
much aluminum is needed to make one can? Round to the nearest hundredth.
Hall B
1 inches, how 2 inches and a height of 3 −
105 ft Great Hall
40 ft
42 ft Hall A
85 ft
25 ft
The museum wants to repaint each of the three halls shown. The walls in each hall are 15 feet high. Each wall will receive a coat of primer and 2 coats of paint.
2. DECORATING Ms. Frank is going to wallpaper a living room with dimensions 24 feet long, 18 feet wide, and 8 feet high. What surface area does Ms. Frank plan to wallpaper?
a. What is the area that will be painted?
3. AGRICULTURE A farmer has a silo with a volume of 2491.6 cubic feet. The silo is 24 feet tall. Find the surface area of the silo.
c. The crew of painters can paint about 500 square feet an hour. About how many hours should it take them to completely paint each hall?
4. MUSEUM A museum curator needs to order a display case for a small artifact. The case needs to be a rectangular prism and made entirely of clear plastic. 1 feet by The bases must each measure 1− 2
3 1− feet and the sides each 3 feet high. 4
Find the cost of the case if the clear plastic costs $10 per square foot.
Chapter 0
8
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. One gallon covers about 400 square feet. About how many total gallons of primer and paint will need to be purchased?
NAME
DATE
0-12
PERIOD
Study Guide
SCS
MA.S.1.1
Mean, Median, Mode, Range, and Quartiles Measures of Central Tendency
When working with numerical data, it is often helpful to use one or more numbers to represent the whole set. These numbers are called the measures of central tendency. You will study the mean, median, and mode. Statistic
Definition
mean
sum of the data divided by the number of items in the data set
median
middle number of the ordered data, or the mean of the middle two numbers
mode
number or numbers that occur most often
Mon. 2
Tues. 3.5
Wed. 3
Thurs. 0
Fri. 2.5
Sat. 6
Sun. 4
sum of hours mean = − number of days
2 + 3.5 + 3 + . . . + 4 7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
= −− or 3 The mean is 3 hours. To find the median, order the numbers from least to greatest and locate the number in the middle. 0
2
2.5
3
3.5
4
6
The median is 3 hours.
There is no mode because each number occurs once in the set.
Exercises Find the mean, median, and mode for each set of data. 1. Maria’s test scores
2. Rainfall last week in inches
92, 86, 90, 74, 95, 100, 90, 50
Chapter 0
0, 0.3, 0, 0.1, 0, 0.5, 0.2
9
North Carolina StudyText, Math A
Lesson 0-12
Example Jason recorded the number of hours he spent watching television each day for a week. Find the mean, median, and mode for the number of hours.
NAME
DATE
0-12
Study Guide
PERIOD
SCS
(continued)
MA.S.1.1
Mean, Median, Mode, Range, and Quartiles The range and the interquartile range describe how a set of data varies. Term
Definition
range
The difference between the greatest and the least values of the set
median
The value that separates the data set in half
lower quartile
The median of the lower half of a set of data
upper quartile
The median of the upper half of a set of data
interquartile range
The difference between the upper quartile and the lower quartile
Example
Find the range and interquartile range for each set of data.
a. {3, 12, 17, 2, 21, 14, 14, 8} Step 1 List the data from least to greatest. The range is 21 - 2 or 19. Then find the median. 2
3
8 12 14 14 17 21 median = 12 + 14 or 13 2
2
3
8 12 14 14 17 21
LQ = 3 + 8 median 2 or 5.5
The data are displayed in order. The greatest value is 56. The least value is 22. So, the range is 56 - 22 or 34. There are 19 data values, so the median is the 10th value. The median is 42. The LQ is the 5th value or 31 and the UQ is the 24th value or 48. So, the interquartile range is 48 - 31 or 17.
UQ = 14 + 17 2 or 15.5
The interquartile range is 15.5 - 5.5 or 10.
Exercise WEATHER Use the data at the right. Find the range, median, upper quartile, lower quartile, and interquartile range for each set of data.
Chapter 0
Average Extreme July Temperatures in World Cities
10
Low Temperatures:
High Temperatures:
{80, 50, 64, 51, 59, 51, 70, 73, 70, 75, 74, 79, 78, 75, 76}
{90, 88, 79, 107, 69, 81, 64, 92, 67, 81, 83, 84, 83, 95, 91}
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 2 Find the upper and lower quartiles.
b. {22, 26, 29, 31, 31, 33, 34, 39, 40, 42, 45, 45, 47, 47, 48, 53, 54, 56, 56}
NAME
0-12
DATE
PERIOD
Practice
SCS
MA.S.1.1
Mean, Median, Mode, Range, and Quartiles Find the mean, median, and mode for each set of data. If necessary, round to the nearest tenth. 1. 4, 6, 12, 5, 8
2. 16, 18, 15, 16, 21, 16
3. 55, 46, 50, 42, 39
4. 17, 16, 13, 17, 17, 10, 10, 13, 10
5. 25, 25, 25, 20
6. 3.1, 4.5, 4.5, 4.3, 6.0, 3.2
Find the range and interquartile range for each set of data. 8. {8, 3, 9, 14, 12, 11, 20, 23, 5, 26}
9. {42, 50, 46, 47, 38, 41}
10. {10.3, 9.8, 10.1, 16.2, 18.0, 11.4, 16.0, 15.8}
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
11. {107, 82, 93, 112, 120, 95, 98, 56, 109, 110}
12. {106, 103, 112, 109, 115, 118, 113, 108}
13. TORNADOES The table below shows the number of tornadoes reported in the United States from 1997 –2007. Find the mean, median, and mode for the number of tornadoes. If necessary, round to the nearest tenth. Year
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Number of Tornadoes
1148
1417
1342
1071
1216
941
1367
1819
1264
1106
1074
14. SCHOOLS The following set of data shows the number of students per teacher at different elementary schools in one school district. Which measure of central tendency best represents the data? Justify your selection and then find the measure of central tendency. 13, 15, 11, 15, 20, 14, 16, 16, 13, 17
Chapter 0
11
North Carolina StudyText, Math A
Lesson 0-12
7. {3, 9, 11, 8, 6, 12, 5, 4}
NAME
DATE
0-12
PERIOD
Word Problem Practice
SCS
MA.S.1.1
Mean, Median, Mode, Range, and Quartiles 1. MARATHON Martin is training for a marathon. The table below shows the total number of miles he has run each week for the first 6 weeks of his training. What is the mode distance that Martin has run? Date
4. ACADEMICS The class average for the first social studies test in Molly’s class was 85%. 23 students took the test. When the 24th student joined the class and took the same test, the class average went up to 85.5%. What grade did the new student earn on the exam?
Miles Run
Week 1
47
Week 2
35
Week 3
53
Week 4
52
Week 5
47
Week 6
56
5. SPORTS Rodney researched the longestplaying professional baseball players. He made a table of the nine who have played professional baseball for 25 seasons or more. Years Played
Player
2. WAGES Each state in the U.S. has its own minimum wage. The table below shows the minimum wage for 6 states. What is the median minimum wage for these 6 states? State
Wage
Alaska
$7.15
California
$8.00
Florida
$6.79
Illinois
$7.50
New York
$7.15
Texas
$5.85
25 27 25 25 26 25 25 26 27
Source: Baseball Reference
a. Find the range of the data set.
b. What is the median of the data set?
c. Find the upper and lower quartile and the interquartile range.
Source: U.S. Department of Labor
3. EXERCISE Shown below is the number of minutes Yashika walked each day for two weeks. Find the upper and lower quartile of the data. Week 1 25
22
15
30
45
18
25
Week 2 35
42
30
25
20
15
10
Chapter 0
12
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Eddie Collins Cap Anson Jim Kaat Bobby Wallace Tommy John Charlie Hough Rickey Henderson Deacon McGuire Nolan Ryan
NAME
0-13
DATE
PERIOD
Study Guide
SCS
MA.S.1.1, MA.S.1.2
Representing Data Words
Stem-andLeaf Plot
One way to organize and display data is to use a stem-and-leaf plot. In a stem-and-leaf plot, numerical data are listed in ascending or descending order.
Model
Example
Leaf 0 1 1 2 0 3 3
1 2 4 7
2 3 8 =
3 5 5 6 7 9 8 37
ZOOS Display the data shown
The next greatest place value forms the leaves.
Size of U. S. Zoos
at the right in a stem-and-leaf plot.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 1 The least and the greatest numbers are 55 and 95. The greatest place value digit in each number is in the tens. Draw a vertical line and write the stems from 5 to 9 to the left of the line. Step 2 Write the leaves to the right of the line, with the corresponding stem. For example, for 85, write 5 to the right of 8. Step 3 Rearrange the leaves so they are ordered from least to greatest. Then include a key or an explanation. Include a title.
Stem 5 6 7 8 9
Zoo
Size (acres)
Audubon (New Orleans)
58
Cincinnati
85
Dallas
95
Denver
80
Houston
55
Los Angeles
80
Leaf
Oregon
64
5 8 4 5 0 0 5 0 2 5
St. Louis
90
San Francisco
75
Woodland Park (Seattle)
92
Leaf 8 5 4 5 5 0 0 5 0 2
U.S. Zoos Stem 5 6 7 8 9
8 | 5 = 85 acres
Exercises Display each set of data in a stem-and-leaf plot. 1. {27, 35, 39, 27, 24, 33, 18, 19}
2. {94, 83, 88, 77, 95, 99, 88, 87}
3. {108, 113, 127, 106, 115, 118, 109, 112}
4. {64, 71, 62, 68, 73, 67, 74, 60}
Chapter 0
13
North Carolina StudyText, Math A
Lesson 0-13
The greatest place value of the data is used for the stems.
Stem 2 3 4
NAME
DATE
0-13
Study Guide
PERIOD
SCS
(continued)
MA.S.1.1, MA.S.1.2
Representing Data Words A box-and-whisker plot divides a set of data into four parts using the median and quartiles. Each of these parts contains 25% of the data. Box-andWhisker Plot
Model
LQ
median
UQ upper extreme, or greatest value
lower extreme, or least value
Example FOOD The heat levels of popular chile peppers are shown in the table. Display the data in a box-and-whisker plot. Step 1
Find the least and greatest numbers. Then draw a number line that covers the range of the data.
Heat Level of Chile Peppers Name
0
25,000
Step 2
50,000
75,000
100,000 150,000 200,000 250,000 125,000 175,000 225,000
Mark the median, the extremes, and the quartiles. Mark these points above the number line. Check for outliers. lower extreme: 0 LQ: 4250
UQ: 67,500
0
upper extreme: 120,000
outlier: 210,000
50,000 100,000 150,000 200,000 250,000 25,000 75,000 125,000 175,000 225,000
Step 3 Draw a box and the whiskers. Heat Level of Chile Peppers
25,000
50,000
75,000
Bell
17,000 0
Cayenne
8000
Habañero
210,000
Jalapeño
25,000
Mulato
1000
New Mexico
4500
Pasilla
5500
Serrano
4000
Tabasco
120,000
Tepín
75,000
Thai hot
60,000
Source: Chile Pepper Institute *Scoville heat units
0
Aji escabeche
100,000 150,000 200,000 250,000 125,000 175,000 225,000
Exercises Construct a box-and-whisker plot for each set of data. 1. {17, 5, 28, 33, 25, 5, 12, 3, 16, 11, 22, 31, 9, 11} 2. {$21, $50, $78, $13, $45, $5, $12, $37, $61, $11, $77, $31, $19, $11, $29, $16}
Chapter 0
14
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
median: 12,500
Heat Level*
NAME
0-13
DATE
PERIOD
Practice
SCS
MA.S.1.1, MA.S.1.2
Representing Data Display each set of data in a stem-and-leaf plot. 2. {27, 32, 42, 31, 36, 37, 47, 23, 39, 31, 41, 38, 30, 34, 29, 42, 37}
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Construct a box-and-whisker plot for each set of data. 3. {14, 30, 35, 8, 29, 28, 31, 42, 20, 36, 32}
4. {$105, $98, $83, $127, $115, $114, $132, $93, $107, $101, $119}
5. {211, 229, 196, 230, 240, 212, 231, 233, 243, 214, 239, 238, 228, 237, 230, 234, 239, 240, 212, 232, 239, 240, 237}
6. {3.7, 6.2, 4.1, 2.4, 1.0, 1.5, 1.4, 2.1, 2.6, 3.0, 1.3, 1.7}
Display each set of data in a histogram. 7.
8.
Ages of Zoo Volunteers Age
Tally
Crossword Puzzle Solving Times
Frequency
Time (min)
Tally
Frequency
18–27
|||
3
0–4
|||
3
28–37
|||| |||
8
5–9
|
1
38–47
|||| |||| |||| |
16
10–14
|||| |
6
48–57
|||| |||| ||
12
15–19
|||| |||| ||||
58–67
||||
5
20–24
68–77
||
2
25–29
14 0
||
2
For Exercises 9–11, use the box-and-whisker plot shown. Major Peaks of the Hindu Kush (height in feet)
9. How tall is the highest peak of the Hindu Kush?
*
10. What is the median height of the major peaks?
19,000 20,000 21,000 22,000 23,000 24,000 25,000 26,000 Source: Peakware
11. Write a sentence describing what the box-and-whisker plot tells about the major peaks of the Hindu Kush.
Chapter 0
15
North Carolina StudyText, Math A
Lesson 0-13
1. {68, 63, 70, 59, 78, 64, 68, 73, 61, 66, 70}
NAME
DATE
0-13
PERIOD
Word Problem Practice
SCS
MA.S.1.1, MA.S.1.2
Representing Data 1. CUSTOMER SERVICE A restaurant owner recorded the average time in minutes customers waited to be seated each night. His data are shown in the table below. To organize the data into a stem-and-leaf plot, how many stems would you need? Week 1
15
8
10
5
20
35
45
Week 2
9
3
7
8
25
38
43
4. FOOD The table shows the recent top 10 ice cream-consuming countries. Make a box-and-whisker plot of the data. Consumption per Capita (pints) 36.8 27.8 27.5 23.8 22.2 20.6 20.2 19.4 16.9 16.7
Country Australia New Zealand USA Sweden Canada Ireland Norway Finland Denmark Germany
2. PHONE Allison’s mother makes a stemand-leaf plot to track the time in minutes that Allison spends talking on the phone each night. In which interval are most of Allison’s calls? Minutes on Phone 5 4 5 8 9 5 8 3 5
1| 5 = 15 minutes
3. MONEY A group of students were asked how much cash (in bills) is in your wallet right now? Construct a histogram to represent the data.
9
9
30 –3
9
Number of Students 54 20 16 5 4 1
50 40 30 20 10 0
0–
0–$9 $10–$19 $20–$29 $30–$39 $40–$49 $50–$59
Number of Teens
Amount
Time Spent Reading the Newspaper
Minutes
a. How many teens said they read a newspaper for less than 30 minutes?
b. How many teens were surveyed in all?
Chapter 0
16
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. NEWSPAPERS Teens ages 13 to 18 who read a newspaper at least once a week were asked, How many minutes a day, on average, do you spend reading the newspaper? The responses are displayed in the histogram.
20 –2
0 3 0 1
9
1 2 3 4
Leaf
10 –1
Stem
NAME
DATE
1-2
PERIOD
Study Guide
SCS
MA.N.2.1
Order of Operations Evaluate Numerical Expressions
Numerical expressions often contain more than one operation. To evaluate them, use the rules for order of operations shown below. Step Step Step Step
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 1
1 2 3 4
Evaluate expressions inside grouping symbols. Evaluate all powers. Do all multiplication and/or division from left to right. Do all addition and/or subtraction from left to right.
Evaluate each expression.
a. 34 34 = 3 ․ 3 ․ 3 ․ 3 = 81
Use 3 as a factor 4 times.
b. 63 63 = 6 ․ 6 ․ 6 = 216
Use 6 as a factor 3 times.
Example 2
Evaluate each expression.
a. 3[2 + (12 ÷ 3)2] 3[2 + (12 ÷ 3)2] = 3(2 + 42) Divide 12 by 3. = 3(2 + 16) Find 4 squared. = 3(18) Add 2 and 16. = 54 Multiply 3 and 18. 3 + 23 b. − 2
Multiply.
4 3 3 + 23 3+8 − =− 42 ․ 3 42 ․ 3 11 =− 42 ․ 3 11 =− 16 ․ 3 11 =− 48
Multiply.
Evaluate power in numerator. Add 3 and 8 in the numerator. Evaluate power in denominator. Multiply.
Exercises Evaluate each expression. 1. 52
2. 33
3. 104
4. 122
5. 83
6. 28
7. (8 - 4) ․ 2
8. (12 + 4) ․ 6
9. 10 + 8 ․ 1
10. 15 - 12 ÷ 4
11. 12(20 - 17) - 3 ․ 6
13. 32 ÷ 3 + 22 ․ 7 - 20 ÷ 5
14. −
․ 42 - 8 ÷ 2 16. 2− ․
17. − ․
(5 + 2) 2
Chapter 1
12. 24 ÷ 3 ․ 2 - 32
4 + 32 12 + 1
15. 250 ÷ [5(3 ․ 7 + 4)]
4(52) - 4 ․ 3 4(4 5 + 2)
5 -3 18. −
17
2
20(3) + 2(3)
North Carolina StudyText, Math A
Lesson 1-2
Order of Operations
NAME
DATE
1-2
PERIOD
Study Guide (continued)
SCS
MA.N.2.1
Order of Operations Evaluate Algebraic Expressions
Algebraic expressions may contain more than one operation. Algebraic expressions can be evaluated if the values of the variables are known. First, replace the variables with their values. Then use the order of operations to calculate the value of the resulting numerical expression.
Example x3 + 5( y - 3)
Evaluate x3 + 5( y - 3) if x = 2 and y = 12. = 23 + 5(12 - 3) Replace x with 2 and y with 12. = 8 + 5(12 - 3) Evaluate 23. = 8 + 5(9) Subtract 3 from 12. = 8 + 45 Multiply 5 and 9. = 53 Add 8 and 45.
The solution is 53.
Exercises 3 4 Evaluate each expression if x = 2, y = 3, z = 4, a = − , and b = − . 5
5
2. 3x - 5
3. x + y2
4. x3 + y + z2
5. 6a + 8b
6. 23 - (a + b)
8. 2xyz + 5
9. x(2y + 3z)
y2 x
7. −2
10. (10x)2 + 100a
z2 - y2 x
13. − 2
25ab + y
16. − xz
(y)
x 19. (− z) + − z 2
Chapter 1
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. x + 7
11. −
3xy - 4 7x
12. a2 + 2b
14. 6xz + 5xy
15. − x
5a 2b 17. − y
18. (z ÷ x)2 + ax
x+z 20. −
y÷x z÷x 21. − + − y z
2 (z - y)
(
y + 2z
18
) (
)
North Carolina StudyText, Math A
NAME
DATE
1-2
PERIOD
Practice
SCS
MA.N.2.1
Order of Operations Evaluate each expression. 1. 112
2. 83
3. 54
4. (15 - 5) ․ 2
5. 9 ․ (3 + 4)
6. 5 + 7 ․ 4
7. 4(3 + 5) - 5 ․ 4
8. 22 ÷ 11 ․ 9 - 32
9. 62 + 3 ․ 7 - 9
10. 3[10 - (27 ÷ 9)]
11. 2[52 + (36 ÷ 6)]
2 ․ 4 - 5 ․ 42 13. 5−
14. − 2
5(4)
12. 162 ÷ [6(7 - 4)2]
(2 ․ 5)2 + 4 3 -5
7 + 32 4 ·2
15. − 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
16. a2 + b - c2
17. b2 + 2a - c2
18. 2c(a + b)
19. 4a + 2b - c2
20. (a2 ÷ 4b) + c
21. c2 · (2b - a)
bc2 + a
22. − c
2c3 - ab 23. −
24. 2(a - b)2 - 5c
b2 - 2c2 25. −
Lesson 1-2
Evaluate each expression if a = 12, b = 9, and c = 4.
4
a+c-b
26. CAR RENTAL Ann Carlyle is planning a business trip for which she needs to rent a car. The car rental company charges $36 per day plus $0.50 per mile over 100 miles. Suppose Ms. Carlyle rents the car for 5 days and drives 180 miles. a. Write an expression for how much it will cost Ms. Carlyle to rent the car. b. Evaluate the expression to determine how much Ms. Carlyle must pay the car rental company. 27. GEOMETRY The length of a rectangle is 3n + 2 and its width is n - 1. The perimeter of the rectangle is twice the sum of its length and its width. a. Write an expression that represents the perimeter of the rectangle. b. Find the perimeter of the rectangle when n = 4 inches.
Chapter 1
19
North Carolina StudyText, Math A
NAME
1-2
DATE
Word Problem Practice
PERIOD
SCS
MA.N.2.1
Order of Operations 1. SCHOOLS Jefferson High School has 100 less than 5 times as many students as Taft High School. Write and evaluate an expression to find the number of students at Jefferson High School if Taft High School has 300 students.
5. BIOLOGY Lavania is studying the growth of a population of fruit flies in her laboratory. She notices that the number of fruit flies in her experiment is five times as large after any six-day period. She observes 20 fruit flies on October 1. Write and evaluate an expression to predict the population of fruit flies Lavania will observe on October 31.
2. GEOGRAPHY Guadalupe Peak in Texas has an altitude that is 671 feet more than double the altitude of Mount Sunflower in Kansas. Write and evaluate an expression for the altitude of Guadalupe Peak if Mount Sunflower has an altitude of 4039 feet.
6. CONSUMER SPENDING During a long weekend, Devon paid a total of x dollars for a rental car so he could visit his family. He rented the car for 4 days at a rate of $36 per day. There was an additional charge of $0.20 per mile after the first 200 miles driven.
b. Write an algebraic expression to represent the number of miles over 200 miles that Devon drove the rented car.
4. GEOMETRY The area of a circle is related to the radius of the circle such that the product of the square of the radius and a number π gives the area. Write and evaluate an expression for the area of a circular pizza below. Approximate π as 3.14.
c. How many miles did Devon drive overall if he paid a total of $174 for the car rental? 7 in.
Chapter 1
20
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. Write an algebraic expression to represent the amount Devon paid for additional mileage only.
3. TRANSPORTATION The Plaid Taxi Cab Company charges $1.75 per passenger plus $3.45 per mile for trips less than 10 miles. Write and evaluate an expression to find the cost for Max to take a Plaid taxi 8 miles to the airport.
NAME
DATE
1-6
PERIOD
Study Guide
SCS
MA.A.4.2
Represent a Relation
A relation is a set of ordered pairs. A relation can be represented by a set of ordered pairs, a table, a graph, or a mapping. A mapping illustrates how each element of the domain is paired with an element in the range. The set of first numbers of the ordered pairs is the domain. The set of second numbers of the ordered pairs is the range of the relation.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example mapping. x
y
1
1
0
2
3
-2
a. Express the relation {(1, 1), (0, 2), (3, -2)} as a table, a graph, and a y
O
X
Y
1 0 3
1 2 -2
x
b. Determine the domain and the range of the relation. The domain for this relation is {0, 1, 3}. The range for this relation is {-2, 1, 2}.
Exercises 1A. Express the relation {(-2, -1), (3, 3), (4, 3)} as a table, a graph, and a mapping.
x
X
y
Y
y
O
x
1B. Determine the domain and the range of the relation.
Chapter 1
21
North Carolina StudyText, Math A
Lesson 1-6
Relations
NAME
DATE
1-6
PERIOD
Study Guide (continued)
SCS
MA.A.4.2
Relations Graphs of a Relation
The value of the variable in a relation that is subject to choice is called the independent variable. The variable with a value that is dependent on the value of the independent variable is called the dependent variable. These relations can be graphed without a scale on either axis, and interpreted by analyzing the shape. Example 1 The graph below represents the height of a football after it is kicked downfield. Identify the independent and the dependent variable for the relation. Then describe what happens in the graph.
Example 2 The graph below represents the price of stock over time. Identify the independent and dependent variable for the relation. Then describe what happens in the graph. Price
Height
Time
Time
The independent variable is time, and the dependent variable is height. The football starts on the ground when it is kicked. It gains altitude until it reaches a maximum height, then it loses altitude until it falls to the ground.
The independent variable is time and the dependent variable is price. The price increases steadily, then it falls, then increases, then falls again.
Identify the independent and dependent variables for each relation. Then describe what is happening in each graph. 1. The graph represents the speed of a car as it travels to the grocery store.
Speed Time
2. The graph represents the balance of a savings account over time.
Account Balance (dollars) Time
3. The graph represents the height of a baseball after it is hit. Height Time
Chapter 1
22
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
1-6
PERIOD
Practice
SCS
MA.A.4.2
1. Express {(4, 3), (-1, 4), (3, -2), (-2, 1)} as a table, a graph, and a mapping. Then determine the domain and range. y
O
x
Describe what is happening in each graph. 2. The graph below represents the height of a tsunami (tidal wave) as it approaches shore.
3. The graph below represents a student taking an exam.
Number of Questions Answered
Height
Time
Express each relation shown as a set of ordered pairs. 4.
X
Y
0
9
-8
3
2
-6
1
4
5.
X
Y
9 -6 4 8
5 -5 3 7
y
6.
O
7. BASEBALL The graph shows the number of home runs hit by Andruw Jones of the Atlanta Braves. Express the relation as a set of ordered pairs. Then describe the domain and range.
x
Andruw Jones’ Home Runs 52 48 44 Home Runs
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Time
40 36 32 28 24 0
Chapter 1
23
’02 ’03 ’04 ’05 ’06 ’07 Year
North Carolina StudyText, Math A
Lesson 1-6
Relations
NAME
DATE
1-6
PERIOD
Word Problem Practice
SCS
MA.A.4.2
Relations
Maximum Heart Rate (beats per minute)
200
25 195
30 190
35 185
40 180
Graph B
y
Number of cookies
20
Graph A Number of cookies
Age (years)
3. BAKING Identify the graph that shows the relationship between the number of cookies and the equivalent number of dozens. Graph C
y
Number of cookies
1. HEALTH The American Heart Association recommends that your target heart rate during exercise should be between 50% and 75% of your maximum heart rate. Use the data in the table below to graph the approximate maximum heart rates for people of given ages.
y
x
x
x
Number of dozens
Number of dozens
Number of dozens
Source: American Heart Association
4. DATA COLLECTION Margaret collected
Maximum Heart Rate
Heart Rate
200
data to determine the number of books her schoolmates were bringing home each evening. She recorded her data as a set of ordered pairs. She let x be the number of textbooks brought home after school, and y be the number of students with x textbooks. The relation is shown in the mapping.
y
190 180 170 160 20 25 30 35 40 x Age
0
Gallons of Sap
0 1 2 3 4 5
8 11 12 23 28
a. Express the relation as a set of ordered pairs.
Maple Sap to Syrup 320
y
y
280 240
b. What is the domain of the relation?
200 160 120 80 0
1
2
3 4 5 6 7 Gallons of Syrup
8
9 x
c. What is the range of the relation?
Source: Vermont Maple Sugar Makers’ Association
Chapter 1
24
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. NATURE Maple syrup is made by collecting sap from sugar maple trees and boiling it down to remove excess water. The graph shows the number of gallons of tree sap required to make different quantities of maple syrup. Express the relation as a set of ordered pairs.
x
NAME
DATE
1-7
PERIOD
Study Guide
SCS
MA.A.4.1, MA.A.4.2
Functions Identify Functions Relations in which each element of the domain is paired with exactly one element of the range are called functions. Example 1
Example 2 Determine whether 3x - y = 6 is a function.
Since each element of the domain is paired with exactly one element of the range, this relation is a function.
y
Since the equation is in the form Ax + By = C, the graph of the equation will be a line, as shown at the right.
O
x
If you draw a vertical line through each value of x, the vertical line passes through just one point of the graph. Thus, the line represents a function.
Exercises Determine whether each relation is a function. y
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1.
x
O
4.
y
O
x
Chapter 1
3.
x
O
y
5.
7. {(4, 2), (2, 3), (6, 1)}
10. -2x + 4y = 0
y
2.
X
Y
-1 0 1 2
4 5 6 7
y
6.
x
O
8. {(-3, -3), (-3, 4), (-2, 4)}
11. x2 + y2 = 8
25
O
x
9. {(-1, 0), (1, 0)}
12. x = -4
North Carolina StudyText, Math A
Lesson 1-7
Determine whether the relation {(6, -3), (4, 1), (7, -2), (-3, 1)} is a function. Explain.
NAME
1-7
DATE
Study Guide (continued)
PERIOD
SCS
MA.A.4.1, MA.A.4.2
Functions Find Function Values Equations that are functions can be written in a form called function notation. For example, y = 2x -1 can be written as f(x) = 2x - 1. In the function, x represents the elements of the domain, and f(x) represents the elements of the range. Suppose you want to find the value in the range that corresponds to the element 2 in the domain. This is written f(2) and is read “f of 2.” The value of f(2) is found by substituting 2 for x in the equation. Example
If f(x) = 3x - 4, find each value.
a. f(3) f(3) = 3(3) - 4 =9-4 =5 b. f(-2) f (-2) = 3(-2) - 4 = -6 - 4 = -10
Replace x with 3. Multiply. Simplify.
Replace x with -2. Multiply. Simplify.
Exercises 1. f (4)
2. g(2)
3. f(-5)
4. g(-3)
5. f(0)
6. g(0)
7. f (3) - 1
1 8. f −
(4)
1 9. g −
10. f (a2)
11. f(k + 1)
12. g(2n)
13. f(3x)
14. f (2) + 3
15. g(-4)
Chapter 1
(4)
26
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
If f(x) = 2x - 4 and g(x) = x2 - 4x, find each value.
NAME
DATE
1-7
PERIOD
Practice
SCS
MA.A.4.1, MA.A.4.2
Functions Determine whether each relation is a function. Explain. X
Y
-3 -2 1 5
0 3 -2
2.
3.
X
Y
1
-5
-4
3
7
6
1
-2
y
O
x
4. {(1, 4), (2, -2), (3, -6), (-6, 3), (-3, 6)}
5. {(6, -4), (2, -4), (-4, 2), (4, 6), (2, 6)}
6. x = -2
7. y = 2
If f(x) = 2x - 6 and g(x) = x - 2x2, find each value.
( 2)
1 9. f - −
8. f(2)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
( 3)
10. g(-1)
1 11. g -−
12. f(7) - 9
13. g(-3) + 13
14. f(h + 9)
15. g(3y)
16. 2[g(b) + 1]
17. WAGES Martin earns $7.50 per hour proofreading ads at a local newspaper. His weekly wage w can be described by the equation w = 7.5h, where h is the number of hours worked. a. Write the equation in function notation. b. Find f(15), f(20), and f(25). 18. ELECTRICITY The table shows the relationship between resistance R and current I in a circuit. Resistance (ohms)
120
80
48
6
4
Current (amperes)
0.1
0.15
0.25
2
3
a. Is the relationship a function? Explain.
b. If the relation can be represented by the equation IR = 12, rewrite the equation in function notation so that the resistance R is a function of the current I. c. What is the resistance in a circuit when the current is 0.5 ampere? Chapter 1
27
North Carolina StudyText, Math A
Lesson 1-7
1.
NAME
DATE
1-7
Word Problem Practice
PERIOD
SCS
MA.A.4.1, MA.A.4.2
Functions 1. TRANSPORTATION The cost of riding in a cab is $3.00 plus $0.75 per mile. The equation that represents this relation is y = 0.75x + 3, where x is the number of miles traveled and y is the cost of the trip. Look at the graph of the equation and determine whether the relation is a function. 16
4. TRAVEL The cost for cars entering President George Bush Turnpike at Beltline road is given by the relation x = 0.75, where x is the dollar amount for entrance to the toll road and y is the number of passengers. Determine if this relation is a function. Explain.
y
14 Cost ($)
12
5. CONSUMER CHOICES Aisha just
10
received a $40 paycheck from her new job. She spends some of it buying music online and saves the rest in a bank account. Her savings is given by f(x) = 40 – 1.25x, where x is the number of songs she downloads at $1.25 per song.
8 6 4 2 0
1
2
3
4 5 6 7 8 Distance (miles)
9 10 x
a. Graph the function.
b. Find f(3), f(18), and f(36). What do these values represent?
3. GEOMETRY The area for any square is given by the function y = x2, where x is the length of a side of the square and y is the area of the square. Write the equation in function notation and find the area of a square with a side length of 3.5 inches.
c. How many songs can Aisha buy if she wants to save $30? Chapter 1
28
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. TEXT MESSAGING Many cell phones have a text messaging option in addition to regular cell phone service. The function for the monthly cost of text messaging service from Noline Wireless Company is f(x) = 0.10x + 2, where x is the number of text messages that are sent. Find f(10) and f (30), the cost of 10 text messages in a month and the cost of 30 text messages in a month.
NAME
2-2
DATE
PERIOD
Study Guide
SCS
MA.A.4.5
Solving One-Step Equations Solve Equations Using Addition and Subtraction
If the same number is added to each side of an equation, the resulting equation is equivalent to the original one. In general if the original equation involves subtraction, this property will help you solve the equation. Similarly, if the same number is subtracted from each side of an equation, the resulting equation is equivalent to the original one. This property will help you solve equations involving addition. Addition Property of Equality
For any numbers a, b, and c, if a = b, then a + c = b + c.
Subtraction Property of Equality
For any numbers a, b, and c, if a = b, then a - c = b - c.
Example 2
Solve m - 32 = 18.
m - 32 = 18 m - 32 + 32 = 18 + 32 m = 50
Original equation
Solve 22 + p = -12.
22 + p = -12 22 + p - 22 = -12 - 22 p = -34
Add 32 to each side. Simplify.
The solution is 50.
Original equation Subtract 22 from each side. Simplify.
Lesson 2-2
Example 1
The solution is -34.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation. Check your solution. 1. h - 3 = -2
2. m - 8 = -12
3. p - 5 = 15
4. 20 = y - 8
5. k - 0.5 = 2.3
1 5 =− 6. w - −
7. h - 18 = -17
8. -12 = -24 + k
9. j - 0.2 = 1.8
10. b - 40 = -40
11. m - (-12) = 10
3 1 =− 12. w - −
13. x + 12 = 6
14. w + 2 = -13
15. -17 = b + 4
16. k + (-9) = 7
17. -3.2 = + (-0.2)
3 5 +x=− 18. - −
19. 19 + h = -4
20. -12 = k + 24
21. j + 1.2 = 2.8
22. b + 80 = -80
23. m + (-8) = 2
3 5 = − 24. w + −
Chapter 2
29
2
2
8
8
4
8
2
8
North Carolina StudyText, Math A
NAME
DATE
2-2
PERIOD
Study Guide (continued)
SCS
MA.A.4.5
Solving One-Step Equations Solve Equations Using Multiplication and Division If each side of an equation is multiplied by the same number, the resulting equation is equivalent to the given one. You can use the property to solve equations involving multiplication and division. To solve equations with multiplication and division, you can also use the Division Property of Equality. If each side of an equation is divided by the same number, the resulting equation is true. Multiplication Property of Equality
For any numbers a, b, and c, if a = b, then ac = bc.
Division Property of Equality
a b For any numbers a, b, and c, with c ≠ 0, if a = b, then − c =− c.
Example 1
1 1 Solve 3 − p = 1− . 2
1 1 3− p = 1−
Original equation
2 2 7 3 −p = − 2 2
( )
()
Original equation
60 -5n = − −
Divide each side by
-5
n = -12
2 Multiply each side by − .
7 2 3 p=− 7
Solve -5n = 60.
-5n = 60 -5
Rewrite each mixed number as an improper fraction.
2 7 2 3 − −p = − − 7 2
Example 2
2
-5.
Simplify.
The solution is -12.
7
Simplify.
3 The solution is − . 7
Solve each equation. Check your solution. h 1. − = -2
1 2. − m=6
3 1 3. − p=−
4. 5 = −
1 5. -− k = -2.5
m 5 6. - − =−
1 7. -1− h=4
3 8. -12 = -− k
j 2 9. − = −
3
y 12
2
8
5
4
2
5
8
3
p 5
8
5
1 10. -3− b=5
7 m 11. − = 10
1 12. − = - −
13. 3h = -42
14. 8m = 16
15. -3t = 51
16. -3r = -24
17. 8k = -64
18. -2m = 16
19. 12h = 4
20. -2.4p = 7.2
21. 0.5j = 5
22. -25 = 5m
23. 6m = 15
24. -1.5p = -75
3
Chapter 2
10
30
4
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
2-2
PERIOD
Practice
SCS
MA.A.4.5
Solving One-Step Equations Solve each equation. Check your solution. 1. d - 8 = 17
2. v + 12 = -5
3. b - 2 = -11
4. -16 = m + 71
5. 29 = a - 76
6. -14 + y = -2
7. 8 - (-n) = 1
8. 78 + r = -15
9. f + (-3) = -9
10. 8j = 96 13. 243 = 27r a 4 16. − =− 15
5
11. -13z = -39 y 9 g 2 17. − = − 27 9
12. -180 = 15m j 12
14. − = -8
15. - − = -8 q 1 18. − = − 24
6
Lesson 2-2
Write an equation for each sentence. Then solve the equation. 19. Negative nine times a number equals -117. 3 20. Negative one eighth of a number is - − . 4
5 . 21. Five sixths of a number is - − 9 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
22. 2.7 times a number equals 8.37.
23. HURRICANES The day after a hurricane, the barometric pressure in a coastal town has risen to 29.7 inches of mercury, which is 2.9 inches of mercury higher than the pressure when the eye of the hurricane passed over. a. Write an addition equation to represent the situation. b. What was the barometric pressure when the eye passed over?
24. ROLLER COASTERS Kingda Ka in New Jersey is the tallest and fastest roller coaster in the world. Riders travel at an average speed of 61 feet per second for 3118 feet. They reach a maximum speed of 187 feet per second. a. If x represents the total time that the roller coaster is in motion for each ride, write an expression to represent the sitation. (Hint: Use the distance formula d = rt.) b. How long is the roller coaster in motion?
Chapter 2
31
North Carolina StudyText, Math A
NAME
2-2
DATE
PERIOD
Word Problem Practice
SCS
MA.A.4.5
Solving One-Step Equations 1. SUPREME COURT Chief Justice William Rehnquist served on the Supreme Court for 33 years until his death in 2005. Write and solve an equation to determine the year he was confirmed as a justice on the Supreme Court.
4. FARMING Mr. Hill’s farm is 126 1 the size of acres. Mr. Hill’s farm is − 4
Mr. Miller’s farm. How many acres is Mr. Miller’s farm?
5. NAUTICAL On the sea, distances are measured in nautical miles rather than miles. 1 nautical mile = 6080 feet
2. SALARY In 2007, the annual salary of the Governor of New York was $179,000. During the same year, the annual salary of the Governor of Tennessee was $94,000 less. Write and solve an equation it to find the annual salary of the Governor of Tennessee in 2007.
nautical mile 1 knot = 1− hour
a. If a boat travels 16 knots in 1 hour, how far will it have traveled in feet? Write and solve an equation.
3. WEATHER On a cold January day, Mavis noticed that the temperature dropped 21 degrees over the course of the day to –9ºC. Write and solve an equation to determine what the temperature was at the beginning of the day.
Chapter 2
32
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. About how fast was the boat traveling in miles per hour? Round your answer to the nearest hundredth.
NAME
2-3
DATE
PERIOD
Study Guide
SCS
MA.A.4.5, MA.A.5.1
Solving Multi-Step Equations Work Backward
Working backward is one of many problem-solving strategies that you can use to solve problems. To work backward, start with the result given at the end of a problem and undo each step to arrive at the beginning number. Example 1 A number is divided by 2, and then 8 is subtracted from the quotient. The result is 16. What is the number? Solve the problem by working backward. The final number is 16. Undo subtracting 8 by adding 8 to get 24. To undo dividing 24 by 2, multiply 24 by 2 to get 48. The original number is 48.
Example 2 A bacteria culture doubles each half hour. After 3 hours, there are 6400 bacteria. How many bacteria were there to begin with? Solve the problem by working backward. The bacteria have grown for 3 hours. Since there are 2 one-half hour periods in one hour, in 3 hours there are 6 one-half hour periods. Since the bacteria culture has grown for 6 time periods, it has doubled 6 times. Undo the doubling by halving the number of bacteria 6 times. 1 1 1 1 1 1 1 ×− ×− ×− ×− ×− = 6400 × − 6400 × − 2
2
2
2
2
2
64
= 100 There were 100 bacteria to begin with.
Solve each problem by working backward. 1. A number is divided by 3, and then 4 is added to the quotient. The result is 8. Find the number. 2. A number is multiplied by 5, and then 3 is subtracted from the product. The result is 12. Find the number. 3. Eight is subtracted from a number, and then the difference is multiplied by 2. The result is 24. Find the number. 4. Three times a number plus 3 is 24. Find the number. 5. CAR RENTAL Angela rented a car for $29.99 a day plus a one-time insurance cost of $5.00. Her bill was $124.96. For how many days did she rent the car? 6. MONEY Mike withdrew an amount of money from his bank account. He spent one fourth for gasoline and had $90 left. How much money did he withdraw?
Chapter 2
33
North Carolina StudyText, Math A
Lesson 2-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
2-3
PERIOD
Study Guide (continued)
SCS
MA.A.4.5, MA.A.5.1
Solving Multi-Step Equations Solve Multi-Step Equations
To solve equations with more than one operation, often called multi-step equations, undo operations by working backward. Reverse the usual order of operations as you work. Example
Solve 5x + 3 = 23.
5x + 3 = 23 5x + 3 - 3 = 23 - 3 5x = 20 5x 20 − =− 5
5
x=4
Original equation. Subtract 3 from each side. Simplify. Divide each side by 5. Simplify.
Exercises Solve each equation. Check your solution. 1. 5x + 2 = 27
2. 6x + 9 = 27
3. 5x + 16 = 51
4. 14n - 8 = 34
5. 0.6x - 1.5 = 1.8
7 p - 4 = 10 6. −
d - 12 7. 16 = −
3n 8. 8 + − = 13
9. − + 3 = -13
4b + 8 -2
10. − = 10 7x - (-1) -8
13. -4 = −
12
g -5
11. 0.2x - 8 = -2
12. 3.2y - 1.8 = 3
k 14. 8 = -12 + −
15. 0 = 10y - 40
-4
Write an equation and solve each problem. 16. Find three consecutive integers whose sum is 96.
17. Find two consecutive odd integers whose sum is 176.
18. Find three consecutive integers whose sum is -93.
Chapter 2
34
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
14
8
NAME
DATE
2-3
PERIOD
Practice
SCS
MA.A.4.5, MA.A.5.1
Solving Multi-Step Equations Solve each problem by working backward. 1. Three is added to a number, and then the sum is multiplied by 4. The result is 16. Find the number. 2. A number is divided by 4, and the quotient is added to 3. The result is 24. What is the number? 3. Two is subtracted from a number, and then the difference is multiplied by 5. The result is 30. Find the number. 4. BIRD WATCHING While Michelle sat observing birds at a bird feeder, one fourth of the birds flew away when they were startled by a noise. Two birds left the feeder to go to another stationed a few feet away. Three more birds flew into the branches of a nearby tree. Four birds remained at the feeder. How many birds were at the feeder initially? Solve each equation. Check your solution. 6. 17 + 3f = 14
u +6=2 8. −
d 9. − + 3 = 15
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5
-4
7 1 1 11. − y-− =−
7. 15t + 4 = 49 b 10. − - 6 = -2 3
3 12. -32 - − f = -17
3 13. 8 - − k = -4
14. − = 1
15 - a 15. − = -9
3k - 7 16. − = 16
x 17. − - 0.5 = 2.5
18. 2.5g + 0.45 = 0.95
19. 0.4m - 0.7 = 0.22
2
8
8
r + 13 12 7
5
3
8
5
Write an equation and solve each problem. 20. Seven less than four times a number equals 13. What is the number? 21. Find two consecutive odd integers whose sum is 116. 22. Find two consecutive even integers whose sum is 126. 23. Find three consecutive odd integers whose sum is 117. 24. COIN COLLECTING Jung has a total of 92 coins in his coin collection. This is 8 more than three times the number of quarters in the collection. How many quarters does Jung have in his collection?
Chapter 2
35
North Carolina StudyText, Math A
Lesson 2-3
5. -12n - 19 = 77
NAME
2-3
DATE
Word Problem Practice
PERIOD
SCS
MA.A.4.5, MA.A.5.1
Solving Multi-Step Equations 1. TEMPERATURE The formula for converting a Fahrenheit temperature to
4. NUMBER THEORY Write and solve an equation to find three consecutive odd integers whose sum is 3.
F - 32º a Celsius temperature is C = − . 1.8
Find the equivalent Celsius temperature for 68ºF.
2. HUMAN HEIGHT It is a commonly used guideline that for the average American child, their maximum adult height will be about twice their height at age 2. Suppose that Micah’s adult height fits the following equation a = 2c – 1, where a represents his adult height and c represents his height at age 2. At age 2 Micah was 35 inches tall. What is Micah’s adult height? Write and solve an equation.
5. GEOMETRY A rectangular swimming pool is surrounded by a concrete sidewalk that is 3 feet wide. The dimensions of the rectangle created by the sidewalk are 21 feet by 31 feet.
3 ft
pool
21 ft
a. Find the length and width of the pool.
b. Find the area of the pool.
c. Write and solve an equation to find the area of the sidewalk in square feet.
Chapter 2
36
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
31 ft
3. CHEMISTRY The half-life of a radioactive substance is the time required for half of a sample to undergo radioactive decay, or for the quantity to fall to half its original amount. Carbon 14 has a half-life of 5730 years. Suppose 5 of given samples of carbon 14 weigh − 8 7 a pound and − of a pound. What was the 8 total weight of the samples 11,460 years ago?
NAME
DATE
2-4
PERIOD
Study Guide
SCS
MA.A.4.5
Solving Equations with the Variable on Each Side Variables on Each Side
To solve an equation with the same variable on each side, first use the Addition or the Subtraction Property of Equality to write an equivalent equation that has the variable on just one side of the equation. Then solve the equation. Example 1 5y - 8 5y - 8 - 3y 2y - 8 2y - 8 + 8 2y
Example 2
Solve 5y - 8 = 3y + 12.
= = = = =
-11 - 3y = 8y + 1 -11 - 3y + 3y = 8y + 1 + 3y -11 = 11y + 1 -11 - 1 = 11y + 1 - 1 -12 = 11y
3y + 12 3y + 12 - 3y 12 12 + 8 20
2y 20 −=−
11y 11
-12 = − −
2
2
Solve -11 - 3y = 8y + 1.
11
1 -1− =y
y = 10
11
The solution is 10.
1 The solution is -1 − . 11
Exercises 1. 6 - b = 5b + 30
2. 5y - 2y = 3y + 2
3. 5x + 2 = 2x - 10
4. 4n - 8 = 3n + 2
5. 1.2x + 4.3 = 2.1 - x
6. 4.4m + 6.2 = 8.8m - 1.8
1 1 b+4=− b + 88 7. −
3 1 8. − k-5=− k-1
9. 8 - 5p = 4p - 1
2
8
4
4
10. 4b - 8 = 10 - 2b
11. 0.2x - 8 = -2 - x
12. 3y - 1.8 = 3y - 1.8
13. -4 - 3x = 7x - 6
14. 8 + 4k = -10 + k
15. 20 - a = 10a - 2
2 1 16. − n+8=− n+2
3 2 17. − y-8=9-− y
18. -4r + 5 = 5 - 4r
19. -4 - 3x = 6x - 6
20. 18 - 4k = -10 -4k
21. 12 + 2y = 10y - 12
3
Chapter 2
2
5
5
37
North Carolina StudyText, Math A
Lesson 2-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation. Check your solution.
NAME
DATE
2-4
PERIOD
Study Guide (continued)
SCS
MA.A.4.5
Solving Equations with the Variable on Each Side Grouping Symbols
When solving equations that contain grouping symbols, first use the Distributive Property to eliminate grouping symbols. Then solve. Example 4(2a - 1) 8a - 4 8a - 4 + 10a 18a - 4 18a - 4 + 4 18a
Solve 4(2a - 1) = -10(a - 5). = = = = = =
-10(a - 5) -10a + 50 -10a + 50 + 10a 50 50 + 4 54
54 18a =− − 18
18
a=3 The solution is 3.
Original equation Distributive Property Add 10a to each side. Simplify. Add 4 to each side. Simplify. Divide each side by 18. Simplify.
Exercises Solve each equation. Check your solution. 2. 2(7 + 3t) = -t
3. 3(a + 1) - 5 = 3a - 2
4. 75 - 9g = 5(-4 + 2g)
5. 5(f + 2) = 2(3 - f )
6. 4( p + 3) = 36
7. 18 = 3(2t + 2)
8. 3(d - 8) = 3d
9. 5(p + 3) + 9 = 3( p - 2) + 6
3+y 4
-y 8
10. 4(b - 2) = 2(5 - b)
11. 1.2(x - 2) = 2 - x
12. − = −
2a + 5 a-8 =− 13. −
14. 2(4 + 2k) + 10 = k
15. 2(w - 1) + 4 = 4(w + 1)
16. 6(n - 1) = 2(2n + 4)
17. 2[2 + 3( y - 1)] = 22
18. -4(r + 2) = 4(2 - 4r)
19. -3(x - 8) = 24
20. 4(4 - 4k) = -10 -16k
21. 6(2 - 2y) = 5(2y - 2)
12
Chapter 2
3
38
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. -3(x + 5) = 3(x - 1)
NAME
DATE
2-4
PERIOD
Practice
SCS
MA.A.4.5
Solving Equations with the Variable on Each Side Solve each equation. Check your solution. 1. 5x - 3 = 13 - 3x
2. -4r - 11 = 4r + 21
3. 1 - m = 6 - 6m
4. 14 + 5n = -4n + 17
3 1 k-3=2-− k 5. −
1 6. − (6 - y) = y
7. 3(-2 - 3x) = -9x - 4
8. 4(4 - w) = 3(2w + 2)
9. 9(4b - 1) = 2(9b + 3)
10. 3(6 + 5y) = 2(-5 + 4y)
11. -5x - 10 = 2 - (x + 4)
12. 6 + 2(3j - 2) = 4(1 + j)
5 3 t-t=3+− t 13. −
14. 1.4f + 1.1 = 8.3 - f
4
2
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5 2 1 1 x-− =− x+− 15. − 3
6
2
6
g 2
2
3 1 16. 2 - − k=− k+9 4
8
1 (3g - 2) = − 17. −
1 1 18. − (n + 1) = − (3n - 5)
h 1 (5 - 2h) = − 19. −
1 1 20. − (2m - 16) = − (2m + 4)
21. 3(d - 8) - 5 = 9(d + 2) + 1
22. 2(a - 8) + 7 = 5(a + 2) - 3a - 19
2
2
2
3
9
6
3
23. NUMBERS Two thirds of a number reduced by 11 is equal to 4 more than the number. Find the number. 24. NUMBERS Five times the sum of a number and 3 is the same as 3 multiplied by 1 less than twice the number. What is the number? 25. NUMBER THEORY Tripling the greater of two consecutive even integers gives the same result as subtracting 10 from the lesser even integer. What are the integers? 26. GEOMETRY The formula for the perimeter of a rectangle is P = 2, + 2w, where is the length and w is the width. A rectangle has a perimeter of 24 inches. Find its dimensions if its length is 3 inches greater than its width. Chapter 2
39
North Carolina StudyText, Math A
Lesson 2-4
2
NAME
2-4
DATE
PERIOD
Word Problem Practice
SCS
MA.A.4.5
Solving Equations with the Variable on Each Side 1. OLYMPICS In the 2006 Winter Olympic Games in Turin, Italy, the United States athletes won 3 more than 2 times the number of gold metals won by the French athletes. The United States won 6 more gold metals than the French. Solve the equation 6 + F = 2F + 3 to find the number of gold metals won by the French athletes.
4. NATURE The table shows the current heights and average growth rates of two different species of trees. How long will it take for the two trees to be the same height?
2. AGE Diego’s mother is twice as old as he is. She is also as old as the sum of the ages of Diego and both of his younger twin brothers. The twins are 11 years old. Solve the equation 2d = d + 11 + 11 to find the age of Diego.
Tree Species
Current Height
Annual growth
A
38 inches
4 inches
B
45.5 inches
2.5 inches
5. NUMBER THEORY Mrs. Simms told her class to find two consecutive even integers such that twice the lesser of two integers is 4 less than two times the greater integer.
3. GEOMETRY Supplementary angles are angles whose measures have a sum of 180º. Complementary angles are angles whose measures have a sum of 90º. Find the measure of an angle whose supplement is 10º more than twice its complement. Let 90 – x equal the degree measure of its complement and 180 – x equal the degree measure of its supplement. Write and solve an equation.
Chapter 2
b. Does the equation have one solution, no solutions, or is it an identity? Explain.
40
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. Write and solve an equation to find the integers.
NAME
DATE
2-6
PERIOD
Study Guide
SCS
MA.N.1.1, MA.N.1.2
Ratios and Proportions A ratio is a comparison of two numbers by division. The ratio x of x to y can be expressed as x to y, x:y or − y . Ratios are usually expressed in simplest form. An equation stating that two ratios are equal is called a proportion. To determine whether two ratios form a proportion, express both ratios in simplest form or check cross products. Example 1
Example 2
Determine whether the
10 25 determine whether − and − form a 18 45 proportion.
24 12 and − are equivalent ratios. ratios − 36
Use cross products to
18
Write yes or no. Justify your answer.
25 10 − − 18 45
24 2 − =− when expressed in simplest form. 36 3 12 2 − =− when expressed in simplest form. 18 3 24 12 The ratios − and − form a proportion 36 18
Write the proportion.
10(45) 18(25)
Cross products
450 = 450
Simplify.
10 25 The cross products are equal, so − =− . 18
45
Since the ratios are equal, they form a proportion.
because they are equal when expressed in simplest form.
Exercises Determine whether each pair of ratios are equivalent ratios. Write yes or no. 1 16 ,− 1. −
5 10 2. − ,−
10 25 3. − ,−
25 15 ,− 4. −
12 3 5. − ,−
4 12 6. − ,−
0.1 5 ,− 7. −
15 9 8. − ,−
14 20 9. − ,−
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2 32
8 15
36 20
2
100
20 49
32 16
9 27
20 12
12 30
2 20 ,− 10. −
5 25 11. − ,−
72 9 12. − ,−
5 30 ,− 13. −
18 50 14. − ,−
100 44 15. − ,−
1.5 6 17. − ,−
0.1 0.45 18. − ,−
3 30
5 20
Chapter 2
64 8
24 75
0.05 1 ,− 16. − 1
9 45
20
2
75
8
0.2
41
33
0.9
North Carolina StudyText, Math A
Lesson 2-6
Ratios and Proportions
NAME
DATE
2-6
Study Guide (continued)
PERIOD
SCS
MA.N.1.1, MA.N.1.2
Ratios and Proportions Solve Proportions If a proportion involves a variable, you can use cross products to solve x 10 =− , x and 13 are called extremes. They are the first the proportion. In the proportion − 5
13
and last terms of the proportion. 5 and 10 are called means. They are the middle terms of the proportion. In a proportion, the product of the extremes is equal to the product of the means. a c For any numbers a, b, c, and d, if − =− , then b d ad = bc.
Means-Extremes Property of Proportions x 10 Solve − =− .
Example
5
10 x =− − 13 5
Original proportion
13(x) = 5(10) 13x = 50
Cross products Simplify.
13x 50 − =− 13
13
Divide each side by 13.
13
11 x = 3−
Simplify.
13
Exercises Solve each proportion. If necessary, round to the nearest hundredth. 0.1 0.5 3. − =− x 2
x+1 3 4. − = −
8 4 5. − =− x 6
x 3 6. − =−
9 18 7. − =−
3 18 8. − =−
p 5 9. − =−
4 4 =− 10. −
1.5 12 11. − x =− x
12. − = −
a - 18 15 =− 13. −
12 24 14. − =−
2+w 12 15. − = −
t
8
4
y+1
b-2
12
4
54
d
12
3
k
3
21
3
63
8
24
3+y 4
6
k
-y 8
9
Use a proportion to solve each problem. 16. MODELS To make a model of the Guadeloupe River bed, Hermie used 1 inch of clay for 5 miles of the river’s actual length. His model river was 50 inches long. How long is the Guadeloupe River? 17. EDUCATION Josh finished 24 math problems in one hour. At that rate, how many hours will it take him to complete 72 problems?
Chapter 2
42
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5 1 2. − =−
-3 2 1. − x =−
NAME
DATE
2-6
PERIOD
Practice
SCS
MA.N.1.1, MA.N.1.2
Determine whether each pair of ratios are equivalent ratios. Write yes or no. 7 52 ,− 1. −
3 15 2. − ,−
18 36 3. − ,−
12 108 4. − ,−
8 72 5. − ,−
1.5 1 6. − ,−
3.4 7.14 7. − ,−
1.7 2.9 8. − ,−
7.6 3.9 9. − ,−
6 48
11
11 66
99
24 48
9 81
5.2 10.92
9
1.2 2.4
6
1.8 0.9
Solve each proportion. If necessary, round to the nearest hundredth. 5 30 10. − a =−
v 34 11. − =−
40 k 12. − =−
28 4 13. − =− w
3 27 14. − u =−
y 48 15. − = −
10 2 16. − y =−
5 35 17. − =−
3 z 18. − =−
6 12 19. − =−
g 6 20. − = −
14 2 21. − =−
7 8 22. − =−
3 5 23. − q =−
m 5 24. − =−
54
49
46
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
16
h
7
3
x
11
t
9
56
162
60
61
23
9
51
4
17 a
49
6
6
8
v 7 25. − =−
3 12 26. − =−
6 3 27. − n =−
7 14 28. − =−
3 2 29. − =−
m-1 2 30. − =−
r+2 5 32. − = −
3 x-2 33. − =−
0.23
1.61
a-4
6
x+1 4
5 31. − =− 12
0.72 12
7
0.51
b
y+6 7
8
7
4
6
1 hours. 34. PAINTING Ysidra paints a room that has 400 square feet of wall space in 2 − 2
At this rate, how long will it take her to paint a room that has 720 square feet of wall space? 35. VACATION PLANS Walker is planning a summer vacation. He wants to visit Petrified National Forest and Meteor Crater, Arizona, the 50,000-year-old impact site of a large meteor. On a map with a scale where 2 inches equals 75 miles, the two areas are about 1 inches apart. What is the distance between Petrified National Forest and Meteor 1− 2
Crater?
Chapter 2
43
North Carolina StudyText, Math A
Lesson 2-6
Ratios and Proportions
NAME
2-6
DATE
PERIOD
Word Problem Practice
SCS
MA.N.1.1, MA.N.1.2
Ratios and Proportions 1. WATER A dripping faucet wastes 3 cups of water every 24 hours. How much water is wasted in a week?
5. MAPS A map of Waco, Texas and neighboring towns is shown below. 0
5 mi
Ross
Texas
35
China Spring
2. GASOLINE In November 2007 the average price of 5 gallons of regular unleaded gasoline in the United States was $15.50. What was the price for 16 gallons of gas?
Bellmead Waco
484
6
Neale
84
Robinson Hewitt
a. Use a metric ruler to measure the distances between Robinson and Neale on the map.
4. BUILDINGS The Sears Tower in Chicago is 1450 feet tall. The John Hancock Center in Chicago is 1127 feet tall. Suppose you are asked to build a smallscale replica of each. If you make the Sears Tower 3 meters tall, what would be the approximate height of the John Hancock replica? Round your answer to the nearest hundredth.
Chapter 2
44
b. Using the scale of the map, find the approximate actual distance by air (not by roads), between Robinson and Neale.
c. Approximately how many square miles are shown on this map?
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. SHOPPING Stevenson’s Market is selling 3 packs of toothpicks for $0.87. How much will 10 packs of toothpicks cost at this price? Round your answer to the nearest cent.
NAME
DATE
2-8
Study Guide
PERIOD
SCS
MA.A.2.2, MA.A.2.3, MA.G.2.4
Literal Equations and Dimensional Analysis Solve for Variables
Sometimes you may want to solve an equation such as V = lwh for one of its variables. For example, if you know the values of V, w, and h, then the equation V =− is more useful for finding the value of . If an equation that contains more than one wh
variable is to be solved for a specific variable, use the properties of equality to isolate the specified variable on one side of the equation. Solve 2x - 4y = 8, for y.
Solve 3m - n = km - 8, for m.
3m - n = km - 8 3m - n - km = km - 8 - km 3m - n - km = - 8 3m - n - km + n = - 8 + n 3m - km = -8 + n m(3 - k) = -8 + n
2x - 4y = 8 2x - 4y - 2x = 8 - 2x -4y = 8 - 2x -4y 8 - 2x −=− -4
Example 2
-4
8 - 2x 2x - 8 or − y=− 4
-4
2x - 8 . The value of y is − 4
m(3 - k)
-8 + n 3-k -8 + n n-8 m = − or − 3-k 3-k n-8 The value of m is − . Since division by 0 is 3-k
−=− 3 -k
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
undefined, 3 - k ≠ 0, or k ≠ 3.
Exercises Solve each equation or formula for the variable indicated. 1. ax - b = c, for x
2. 15x + 1 = y, for x
3. (x + f) + 2 = j, for x
4. xy + w = 9, for y
5. x(4 - k) = p, for k
6. 7x + 3y = m, for y
7. 4(r + 3) = t, for r
8. 2x + b = w, for x
9. x(1 + y) = z, for x
h(a + b) 2
10. 16w + 4x = y, for x
11. d = rt, for r
12. A = − , for h
5 13. C = − (F - 32), for F
14. P = 2 + 2w, for w
15. A = w, for
9
Chapter 2
45
North Carolina StudyText, Math A
Lesson 2-8
Example 1
NAME
2-8
DATE
Study Guide (continued)
PERIOD
SCS
MA.A.2.2, MA.A.2.3, MA.G.2.4
Literal Equations and Dimensional Analysis Use Formulas Many real-world problems require the use of formulas. Sometimes solving a formula for a specified variable will help solve the problem. Example
The formula C = πd represents the circumference of a circle, or the
distance around the circle, where d is the diameter. If an airplane could fly around Earth at the equator without stopping, it would have traveled about 24,900 miles. Find the diameter of Earth. C = πd
Given formula
C d=− π
Solve for d.
24,900 3.14
d=−
Use π = 3.14.
d ≈ 7930
Simplify.
The diameter of Earth is about 7930 miles.
Exercises 1. GEOMETRY The volume of a cylinder V is given by the formula V = πr2h, where r is the radius and h is the height. a. Solve the formula for h.
2. WATER PRESSURE The water pressure on a submerged object is given by P = 64d, where P is the pressure in pounds per square foot, and d is the depth of the object in feet. a. Solve the formula for d. b. Find the depth of a submerged object if the pressure is 672 pounds per square foot. 3. GRAPHS The equation of a line containing the points (a, 0) and (0, b) is given by the y x formula − a + − = 1. b
a. Solve the equation for y. b. Suppose the line contains the points (4, 0), and (0, -2). If x = 3, find y. 4. GEOMETRY The surface area of a rectangular solid is given by the formula x = 2w + 2h + 2wh, where = length, w = width, and h = height. a. Solve the formula for h. b. The surface area of a rectangular solid with length 6 centimeters and width 3 centimeters is 72 square centimeters. Find the height. Chapter 2
46
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. Find the height of a cylinder with volume 2500π feet and radius 10 feet.
NAME
DATE
2-8
Practice
PERIOD
SCS
MA.A.2.2, MA.A.2.3, MA.G.2.4
Literal Equations and Dimensional Analysis Solve each equation or formula for the variable indicated. 1. d = rt, for r
2. 6w - y = 2z, for w
3. mx + 4y = 3t, for x
4. 9s - 5g = -4u, for s
5. ab + 3c = 2x, for b
6. 2p = kx - t, for x
2 7. − m + a = a + r, for m
2 8. − h + g = d, for h
2 9. − y + v = x, for y 3
5
3 10. − a - q = k, for a 4
rx + 9 11. − = h, for x
3b - 4 12. − = c, for b
13. 2w - y = 7w - 2, for w
14. 3 + y = 5 + 5, for
5
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
15. ELECTRICITY The formula for Ohm’s Law is E = IR, where E represents voltage measured in volts, I represents current measured in amperes, and R represents resistance measured in ohms. a. Solve the formula for R. b. Suppose a current of 0.25 ampere flows through a resistor connected to a 12-volt battery. What is the resistance in the circuit? 16. MOTION In uniform circular motion, the speed v of a point on the edge of a spinning 2π r, where r is the radius of the disk and t is the time it takes the point to disk is v = − t travel once around the circle. a. Solve the formula for r. b. Suppose a merry-go-round is spinning once every 3 seconds. If a point on the outside edge has a speed of 12.56 feet per second, what is the radius of the merry-go-round? (Use 3.14 for π.) 17. HIGHWAYS Interstate 90 is the longest interstate highway in the United States, connecting the cities of Seattle, Washington and Boston, Massachusetts. The interstate is 4,987,000 meters in length. If 1 mile = 1.609 kilometers, how many miles long is Interstate 90?
Chapter 2
47
North Carolina StudyText, Math A
Lesson 2-8
3
NAME
2-8
DATE
Word Problem Practice
PERIOD
SCS
MA.A.2.2, MA.A.2.3, MA.G.2.4
Literal Equations and Dimensional Analysis 1. INTEREST Simple interest that you may earn on money in a savings account can be calculated with the formula I = prt. I is the amount of interest earned, p is the principal or initial amount invested, r is the interest rate, and t is the amount of time the money is invested for. Solve the formula for p.
4. PHYSICS The pressure exerted on an object is calculated by the formula
2. DISTANCE The distance d a car can travel is found by multiplying its rate of speed r by the amount of time t that it took to travel the distance. If a car has already traveled 5 miles, the total distance d is found by the formula d = rt + 5 . Solve the formula for r.
5. GEOMETRY The regular octagon is divided into 8 congruent triangles. Each triangle has an area of 21.7 square centimeters. The perimeter of the octagon is 48 centimeters.
F P=− , where P is the pressure, F is the A force, and A is the surface area of the object. Water shooting from a hose has a pressure of 75 pounds per square inch (psi). Suppose the surface area covered by the direct hose spray is 0.442 square inches. Solve the equation for F and find the force of the spray.
3. ENVIRONMENT The United States released 5.877 billion metric tons of carbon dioxide into the environment through the burning of fossil fuels in 2006. If 1 trillion pounds = 0.4536 billion metric tons, how many trillions of pounds of carbon dioxide did the United States release in 2006?
a. What is the length of each side of the octagon?
b. Solve the area of a triangle formula for h.
c. What is the height of each triangle? Round to the nearest tenth.
Chapter 2
48
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
h
NAME
DATE
3-1
Study Guide
PERIOD
SCS
MA.A.4.3, MA.A.4.4
Graphing Linear Equations Identify Linear Equations and Intercepts
A linear equation is an equation that can be written in the form Ax + By = C. This is called the standard form of a linear equation. Ax + By = C, where A ≥ 0, A and B are not both zero, and A, B, and C are integers with GCF of 1.
Example 1
Determine whether y = 6 - 3x is a linear equation. Write the equation in standard form. First rewrite the equation so both variables are on the same side of the equation. Original equation. y = 6 - 3x y + 3x = 6 - 3x + 3x Add 3x to each side. 3x + y = 6 Simplify. The equation is now in standard form, with A = 3, B = 1 and C = 6. This is a linear equation.
Example 2 Determine whether 3xy + y = 4 + 2x is a linear equation. Write the equation in standard form. Since the term 3xy has two variables, the equation cannot be written in the form Ax + By = C. Therefore, this is not a linear equation.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises Determine whether each equation is a linear equation. Write yes or no. If yes, write the equation in standard form. 1. 2x = 4y
2. 6 + y = 8
3. 4x - 2y = -1
4. 3xy + 8 = 4y
5. 3x - 4 = 12
6. y = x2 + 7
7. y - 4x = 9
8. x + 8 = 0
9. -2x + 3 = 4y
1 10. 2 + − x=y
1 11. − y = 12 - 4x
12. 3xy - y = 8
13. 6x + 4y - 3 = 0
14. yx - 2 = 8
15. 6x - 2y = 8 + y
1 16. − x - 12y = 1
17. 3 + x + x2 = 0
18. x2 = 2xy
2
4
Chapter 3
4
49
North Carolina StudyText, Math A
Lesson 3-1
Standard Form of a Linear Equation
NAME
3-1
DATE
PERIOD
Study Guide (continued)
SCS
MA.A.4.3, MA.A.4.4
Graphing Linear Equations Graph Linear Equations The graph of a linear equations represents all the solutions of the equation. An x-coordinate of the point at which a graph of an equation crosses the x-axis in an x-intercept. A y-coordinate of the point at which a graph crosses the y-axis is called a y-intercept. Example 1 Graph the equation 3x + 2y = 6 by using the x and y-intercepts. To find the x-intercept, let y = 0 and solve for x. The x-intercept is 2. The graph intersects the x-axis at (2, 0). To find the y-intercept, let x = 0 and solve for y. The y-intercept is 3. The graph intersects the y-axis at (0, 3). Plot the points (2, 0) and (0, 3) and draw the line through them. y (0, 3) (2, 0) O
Example 2 Graph the equation y - 2x = 1 by making a table. Solve the equation for y. Original equation. y - 2x = 1 y - 2x + 2x = 1 + 2x Add 2x to each side. y = 2x + 1 Simplify. Select five values for the domain and make a table. Then graph the ordered pairs and draw a line through the points. x
2x + 1
y
(x, y)
-2
2(-2) + 1
-3
(-2, -3)
-1
2(-1) + 1
-1
(-1, -1)
0
2(0) + 1
1
(0, 1)
1
2(1) + 1
3
(1, 3)
2
2(2) + 1
5
(2, 5)
y
x
O
x
Graph each equation by using the x- and y-intercepts. 1. 2x + y = -2
2. 3x - 6y = -3
y
O
3. -2x + y = -2 y
y
x
O
O
x
x
Graph each equation by making a table. 4. y = 2x
5. x - y = -1 y
y
O
6. x + 2y = 4
x O
Chapter 3
50
y
x
O
x
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
3-1
DATE
PERIOD
Practice
SCS
MA.A.4.3, MA.A.4.4
Graphing Linear Equations Determine whether each equation is a linear equation. Write yes or no. If yes, write the equation in standard form and determine the x- and y-intercepts. 1. 4xy + 2y = 9
2. 8x - 3y = 6 - 4x
4. 5 - 2y = 3x
5.
4
y 3
5 2 6. − x -− y =7
-−=1
Lesson 3-1
x
−
3. 7x + y + 3 = y
Graph each equation. 1 x-y=2 7. −
8. 5x - 2y = 7
y
y
2
O
x
O
9. 1.5x + 3y = 9 y x x
10. COMMUNICATIONS A telephone company charges $4.95 per month for long distance calls plus $0.05 per minute. The monthly cost c of long distance calls can be described by the equation c = 0.05m + 4.95, where m is the number of minutes. a. Find the y-intercept of the graph of the equation.
14
Long Distance
12 10 Cost ($)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
8 6 4 2
b. Graph the equation.
0
c. If you talk 140 minutes, what is the monthly cost?
40
80 120 Time (minutes)
160
11. MARINE BIOLOGY Killer whales usually swim at a rate of 3.2–9.7 kilometers per hour, though they can travel up to 48.4 kilometers per hour. Suppose a migrating killer whale is swimming at an average rate of 4.5 kilometers per hour. The distance d the whale has traveled in t hours can be predicted by the equation d = 4.5t. a. Graph the equation. b. Use the graph to predict the time it takes the killer whale to travel 30 kilometers.
Chapter 3
51
North Carolina StudyText, Math A
NAME
DATE
3-1
PERIOD
Word Problem Practice
SCS
MA.A.4.3, MA.A.4.4
Graphing Linear Equations 1. FOOTBALL One football season, the
4. BUSINESS The equation
Carolina Panthers won 4 more games than they lost. This can be represented by y = x + 4, where x is the number of games lost and y is the number of games won. Write this linear equation in standard form.
y = 1000x - 5000 represents the monthly profits of a start-up dry cleaning company. Time in months is x and profit in dollars is y. The first date of operation is when time is zero. However, preparation for opening the business began 3 months earlier with the purchase of equipment and supplies. Graph the linear function for x-values from -3 to 8.
2. TOWING Pick-M-Up Towing Company charges $40 to hook a car and $1.70 for each mile that it is towed. The equation y = 1.7x + 40 represents the total cost y for x miles towed. Determine the y-intercept. Describe what the value means in this context.
y 2000 O 2
4
6
8x
-2000 -4000
3. SHIPPING The OOCL Shenzhen, one of the world’s largest container ships, carries 8063 TEUs (1280 cubic feet containers). Workers can unload a ship at a rate of a TEU every minute. Using this rate, write and graph an equation to determine how many hours it will take the workers to unload half of the containers from the Shenzhen.
TEUs on Ship (thousands)
-8000
5. BONE GROWTH The height of a woman can be predicted by the equation h = 81.2 + 3.34r, where h is her height in centimeters and r is the length of her radius bone in centimeters. a. Is this is a linear function? Explain.
y
8 7 6
b. What are the r- and h-intercepts of the equation? Do they make sense in the situation? Explain.
5 4 3 2 1 0
10 20 30 40 50 60 70 80 x Time (hours)
c. Use the function to find the approximate height of a woman whose radius bone is 25 centimeters long.
Chapter 3
52
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9
-6000
NAME
DATE
3-2
PERIOD
Study Guide
SCS
MA.A.4.3, MA.A.4.5
Solving Linear Equations by Graphing Solve by Graphing
You can solve an equation by graphing the related function. The solution of the equation is the x-intercept of the function. Example
Solve the equation 2x - 2 = -4 by graphing.
First set the equation equal to 0. Then replace 0 with f(x). Make a table of ordered pair solutions. Graph the function and locate the x-intercept. 2x - 2 = -4 2x - 2 + 4 = -4 + 4 2x + 2 = 0 f(x) = 2x + 2
Original equation Add 4 to each side. Simplify. Replace 0 with f(x).
To graph the function, make a table. Graph the ordered pairs. 1
f(x) = 2x + 2
f(x)
y
[x, f(x)]
f(1) = 2(1) + 2
4
(1, 4)
-1
f(-1) = 2(-1) + 2
0
(-1, 0)
-2
f(-2) = 2(-2) + 2
-2
(-2, -2)
0
Lesson 3-2
x
x
The graph intersects the x-axis at (-1, 0). The solution to the equation is x = -1.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises Solve each equation. 1. 3x - 3 = 0
2. -2x + 1 = 5 - 2x
3. -x + 4 = 0
y
y
y
x x
0
4. 0 = 4x - 1
Chapter 3
x
5. 5x - 1 = 5x
y
0
0
6. -3x + 1 = 0 y
y
x
0
x
0
53
0
x
North Carolina StudyText, Math A
NAME
DATE
3-2
PERIOD
Study Guide (continued)
SCS
MA.A.4.3, MA.A.4.5
Solving Linear Equations by Graphing Estimate Solutions by Graphing Sometimes graphing does not provide an exact solution, but only an estimate. In these cases, solve algebraically to find the exact solution.
Make a table of values to graph the function. t
d = 7 - 3.2t
d
(t, d)
0
d = 7 - 3.2(0)
7
(0, 7)
1
d = 7 - 3.2(1)
3.8
(1, 3.8)
2
d = 7 - 3.2(2)
0.6
(2, 0.6)
The graph intersects the t–axis between t = 2 and t = 3, but closer to t = 2. It will take you and your cousin just over two hours to reach the ranger station.
Miles from Ranger Station
Example WALKING You and your cousin decide to walk the 7-mile trail at the state park to the ranger station. The function d = 7 – 3.2t represents your distance d from the ranger station after t hours. Find the zero of this function. Describe what this value means in this context. y 8 7 6 5 4 3 2 1 0
1
3 x
2
Time (hours)
You can check your estimate by solving the equation algebraically.
Exercises 90
Time Available (min)
80 70 60 50 40 30 20 10 0
5
10 15 20 25 30
2. GIFT CARDS Enrique uses a gift card to buy coffee at a coffee shop. The initial value of the gift card is $20. The function n = 20 – 2.75c represents the amount of money still left on the gift card n after purchasing c cups of coffee. Find the zero of this function. Describe what this value means in this context.
Value Left on Card ($)
Number of Songs
24 20 16 12 8 4 0
2
4
6
8
10 12
Coffees Bought
Chapter 3
54
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. MUSIC Jessica wants to record her favorite songs to one CD. The function C = 80 - 3.22n represents the recording time C available after n songs are recorded. Find the zero of this function. Describe what this value means in this context.
NAME
3-2
DATE
PERIOD
Practice
SCS
MA.A.4.3, MA.A.4.5
Solving Linear Equations by Graphing Solve each equation. 2. -3x + 2 = -1
2
y
y
x
0
1 1 4. − x+2=− x-1 3
3
4
4
y
x
y
x
0
x
0
3 3 6. − x+1=− x-7
2 5. − x+4=3
3
0
y
x
0
y
x
0
Solve each equation by graphing. Verify your answer algebraically 7. 13x + 2 = 11x - 1
8. -9x - 3 = -4x - 3
y
0
2 1 x+2=− x-1 9. -− 3
3
y
x
y
x
0
10. DISTANCE A bus is driving at 60 miles per hour toward a bus station that is 250 miles away. The function d = 250 – 60t represents the distance d from the bus station the bus is t hours after it has started driving. Find the zero of this function. Describe what this value means in this context.
x
0
Distance from Bus Station (miles)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. 4x - 2 = -2
Lesson 3-2
1 x-2=0 1. −
300 250 200 150 100 50 0
1
2
3
4
5
6
Time (hours)
Chapter 3
55
North Carolina StudyText, Math A
NAME
3-2
DATE
PERIOD
Word Problem Practice
SCS
MA.A.4.3, MA.A.4.5
Solving Linear Equations by Graphing 1. PET CARE You buy a 6.3-pound bag of dry cat food for your cat. The function c = 6.3 – 0.25p represents the amount of cat food c remaining in the bag when the cat is fed the same amount each day for p days. Find the zero of this function. Describe what this value means in this context.
5. DENTAL HYGIENE You are packing your suitcase to go away to a 14-day summer camp. The store carries three sizes of tubes of toothpaste.
2. SAVINGS Jessica is saving for college using a direct deposit from her paycheck into a savings account. The function m = 3045 - 52.50t represents the amount of money m still needed after t weeks. Find the zero of this function. What does this value mean in this context?
Size (ounces)
Size (grams)
A
0.75
21.26
B
0.9
25.52
C
3.0
85.04
Source: National Academy of Sciences
a. The function n = 21.26 - 0.8b represents the number of remaining brushings n using b grams per brushing using Tube A. Find the zero of this function. Describe what this value means in this context.
c. Write a function to represent the number of remaining brushings n using b grams per brushing using Tube C. Find the zero of this function. Describe what this value means in this context.
4. BAKE SALE Ashley has $15 in the Pep Club treasury to pay for supplies for a chocolate chip cookie bake sale. The function d = 15 – 0.08c represents the dollars d left in the club treasury after making c cookies. Find the zero of this function. What does this value represent in this context?
d. If you will brush your teeth twice each day while at camp, which is the smallest tube of toothpaste you can choose? Explain your reasoning.
56
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. The function n = 25.52 – 0.8b represents the number of remaining brushings n using b grams per brushing using Tube B. Find the zero of this function. Describe what this value means in this context.
3. FINANCE Michael borrows $100 from his dad. The function v = 100 - 4.75p represents the outstanding balance v after p weekly payments. Find the zero of this function. Describe what this value means in this context.
Chapter 3
Tube
NAME
DATE
3-3
Study Guide
SCS
PERIOD
MA.A.3.1, MA.A.3.2, MA.A.3.3, MA.G.1.1
Rate of Change and Slope Rate of Change
The rate of change tells, on average, how a quantity is changing
over time.
POPULATION The graph shows the population growth in China.
a. Find the rates of change for 1950–1975 and for 2000–2025. change in population 0.93 - 0.55 1950–1975: −− = − 1975 - 1950 change in time 0.38 =− or 0.0152 25 change in population 1.45 - 1.27 2000–2025: −− = − change in time
Population Growth in China People (billions)
Example
2.0 1.5
25
1.27
0.5 0
2025 - 2000
0.18 =− or 0.0072
0.93
1.0 0.55
1950
1.45
1975 2000 2025* Year *Estimated
Source: United Nations Population Division
b. Explain the meaning of the rate of change in each case. From 1950–1975, the growth was 0.0152 billion per year, or 15.2 million per year. From 2000–2025, the growth is expected to be 0.0072 billion per year, or 7.2 million per year.
1. LONGEVITY The graph shows the predicted life expectancy for men and women born in a given year.
Predicting Life Expectancy 100
a. Find the rates of change for women from 2000–2025 and 2025–2050.
95
b. Find the rates of change for men from 2000–2025 and 2025–2050.
85
c. Explain the meaning of your results in Exercises 1 and 2. d. What pattern do you see in the increase with each 25-year period?
90
84 80
80 75 70
87
78
81
74
65 2000 Women Men
2025* 2050* Year Born *Estimated
Source: USA TODAY
e. Make a prediction for the life expectancy for 2050–2075. Explain how you arrived at your prediction.
Chapter 3
57
North Carolina StudyText, Math A
Lesson 3-3
Exercises
Age
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
c. How are the different rates of change shown on the graph? There is a greater vertical change for 1950–1975 than for 2000–2025. Therefore, the section of the graph for 1950–1975 has a steeper slope.
NAME
DATE
3-3
Study Guide (continued)
PERIOD
SCS
MA.A.3.1, MA.A.3.2, MA.A.3.3, MA.G.1.1
Rate of Change and Slope Find Slope
The slope of a line is the ratio of change in the y- coordinates (rise) to the change in the x- coordinates (run) as you move in the positive direction. y -y
Slope of a Line
rise 2 1 m=− x 2 - x 1 , where (x1, y1) and (x2, y2) are the coordinates run or m = − of any two points on a nonvertical line
Example 1 Find the slope of the line that passes through (-3, 5) and (4, -2).
Example 2 Find the value of r so that the line through (10, r) and (3, 4) has a 2 . slope of - − 7 y2 - y1 m=− x2 - x1
Let (-3, 5) = (x1, y1) and (4, -2) = (x2, y2).
Slope formula
Slope formula
4-r 2 -− =−
2 m = -− , y2 = 4, y1 = r, x2 = 3, x1 = 10
-2 - 5 = −
y 2 = -2, y 1 = 5, x 2 = 4, x 1 = -3
4-r 2 =− -− 7 -7
Simplify.
-7 =− 7
Simplify.
y -y
2 1 m=− x2 - x1
4 - (-3)
= -1
3 - 10
7
-2(-7) = 7(4 - r) 14 = 28 - 7r -14 = -7r 2=r
7
Cross multiply. Distributive Property Subtract 28 from each side. Divide each side by -7.
Find the slope of the line that passes through each pair of points. 1. (4, 9), (1, 6)
2. (-4, -1), (-2, -5)
3. (-4, -1), (-4, -5)
4. (2, 1), (8, 9)
5. (14, -8), (7, -6)
6. (4, -3), (8, -3)
7. (1, -2), (6, 2)
8. (2, 5), (6, 2)
9. (4, 3.5), (-4, 3.5)
Find the value of r so the line that passes through each pair of points has the given slope. 10. (6, 8), (r, -2), m = 1
3 11. (-1, -3), (7, r), m = −
12. (2, 8), (r, -4) m = -3
13. (7, -5), (6, r), m = 0
3 14. (r, 4), (7, 1), m = −
15. (7, 5), (r, 9), m = 6
Chapter 3
4
4
58
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
3-3
Practice
SCS
PERIOD
MA.A.3.1, MA.A.3.2, MA.A.3.3, MA.G.1.1
Rate of Change and Slope Find the slope of the line that passes through each pair of points. y
2.
(–2, 3)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(–1, 0)
3.
(–2, 3)
y (3, 3)
(3, 1) O
x
O
x
O
(–2, –3)
4. (6, 3), (7, -4)
5. (-9, -3), (-7, -5)
6. (6, -2), (5, -4)
7. (7, -4), (4, 8)
8. (-7, 8), (-7, 5)
9. (5, 9), (3, 9)
10. (15, 2), (-6, 5)
11. (3, 9), (-2, 8)
12. (-2, -5), (7, 8)
13. (12, 10), (12, 5)
14. (0.2, -0.9), (0.5, -0.9)
7 4 1 2 , − , -− ,− 15. −
(3 3) (
3 3
x
)
Find the value of r so the line that passes through each pair of points has the given slope. 1 16. (-2, r), (6, 7), m = −
1 17. (-4, 3), (r, 5), m = −
2
4
9 18. (-3, -4), (-5, r), m = -−
7 19. (-5, r), (1, 3), m = −
20. (1, 4), (r, 5), m undefined
21. (-7, 2), (-8, r), m = -5
1 22. (r, 7), (11, 8), m = -−
23. (r, 2), (5, r), m = 0
2
5
6
24. ROOFING The pitch of a roof is the number of feet the roof rises for each 12 feet horizontally. If a roof has a pitch of 8, what is its slope expressed as a positive number?
25. SALES A daily newspaper had 12,125 subscribers when it began publication. Five years later it had 10,100 subscribers. What is the average yearly rate of change in the number of subscribers for the five-year period?
Chapter 3
59
North Carolina StudyText, Math A
Lesson 3-3
y
1.
NAME
DATE
3-3
PERIOD
Word Problem Practice
SCS
MA.A.3.1, MA.A.3.2, MA.A.3.3, MA.G.1.1
Rate of Change and Slope 1. HIGHWAYS Roadway signs such as the one below are used to warn drivers of an upcoming steep down grade that could lead to a dangerous situation. What is the grade, or slope, of the hill described on the sign?
4. REAL ESTATE The median price of an existing single-family home in the United States was $195,200 in 2004. The median price had risen to $221,900 by 2006. Find the average annual rate of change in median home price from 2004 to 2006.
5. COAL EXPORTS The graph shows the annual coal exports from U.S. mines in millions of short tons. 100 Million Short Tons
90
2. AMUSEMENT PARKS The SheiKra roller coaster at Busch Gardens in Tampa, Florida, features a 138-foot vertical drop. What is the slope of the coaster track at this part of the ride? Explain.
80 70 60 50
Total Exports
40 30
2006
2005
2004
2003
2002
2001
2000
0
Source: Energy Information Association
3. CENSUS The table shows the population density for the state of Texas in various years. Find the average annual rate of change in the population density from 2000 to 2006.
b. How does the rate of change in coal exports from 2005 to 2006 compare to that of 2001 to 2002?
Population Density Year
People Per Square Mile
1930
22.1
1960
36.4
1980
54.3
2000
79.6
2006
87.5
c. Explain the meaning of the part of the graph with a slope of zero.
Source: Bureau of the Census, U.S. Dept. of Commerce
Chapter 3
60
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. What was the rate of change in coal exports between 2001 and 2002?
NAME
DATE
3-4
Study Guide
PERIOD
SCS
MA.A.2.1, MA.A.2.3, MA.A.4.4
Direct Variation Direct Variation Equations
A direct variation is described by an equation of the form y = kx, where k ≠ 0. We say that y varies directly as x. In the equation y = kx, k is the constant of variation. Example 1 Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points.
Example 2 Suppose y varies directly as x, and y = 30 when x = 5. a. Write a direct variation equation that relates x and y. Find the value of k. y = kx Direct variation equation 30 = k(5) Replace y with 30 and x with 5. 6=k Divide each side by 5. Therefore, the equation is y = 6x. b. Use the direct variation equation to find x when y = 18. y = 6x Direct variation equation 18 = 6x Replace y with 18. 3=x Divide each side by 6. Therefore, x = 3 when y = 18.
y y = 1x 2
O
(0, 0)
(2, 1)
x
1 1 For y = − x, the constant of variation is − . 2 y2 - y1 m=− x2 - x1
Slope formula
1-0 =−
(x1, y1) = (0, 0), (x2, y2) = (2, 1)
1 =−
Simplify.
2-0
2
1 The slope is − . 2
Exercises Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points. 1. (–1, 2)
y
2.
y
y
3. y = 3x (1, 3)
(0, 0)
x
O y = –2x
2
O
O (0, 0)
y = 3x
(0, 0)
x
x
(–2, –3)
Suppose y varies directly as x. Write a direct variation equation that relates x to y. Then solve. 4. If y = 4 when x = 2, find y when x = 16. 5. If y = 9 when x = -3, find x when y = 6. 6. If y = -4.8 when x = -1.6, find x when y = -24. 3 1 1 when x = − , find x when y = − . 7. If y = − 4
Chapter 3
8
16
61
North Carolina StudyText, Math A
Lesson 3-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
NAME
3-4
DATE
Study Guide (continued)
PERIOD
SCS
MA.A.2.1, MA.A.2.3, MA.A.4.4
Direct Variation Direct Variation Problems
The distance formula d = rt is a direct variation equation. In the formula, distance d varies directly as time t, and the rate r is the constant of variation. Example
TRAVEL A family drove their car 225 miles in 5 hours.
a. Write a direct variation equation to find the distance traveled for any number of hours. Use given values for d and t to find r. Original equation d = rt 225 = r(5) d = 225 and t = 5 45 = r Divide each side by 5. Therefore, the direct variation equation is d = 45t.
45 m=− 1
rise − run
✔CHECK (5, 225) lies on the graph.
Distance (miles)
Automobile Trips
b. Graph the equation. The graph of d = 45t passes through the origin with slope 45.
d 360 270
d = 45t
180 90 0
1
c. Estimate how many hours it would take the family to drive 360 miles.
(1, 45) 2 3 4 5 6 Time (hours)
7
8
t
Original equation Replace d with 360. Divide each side by 45.
Therefore, it will take 8 hours to drive 360 miles.
Exercises
Cost of Jelly Beans
a. Write a direct variation equation that relates the variables.
Cost (dollars)
1. RETAIL The total cost C of bulk jelly beans is $4.49 times the number of pounds p.
C 18.00 13.50 9.00 4.50
b. Graph the equation on the grid at the right.
0
3 pound of jelly beans. c. Find the cost of − 4
a. Write a direct variation equation that relates the variables. b. Graph the equation on the grid at the right.
Charles’s Law Volume (cubic feet)
2. CHEMISTRY Charles’s Law states that, at a constant pressure, volume of a gas V varies directly as its temperature T. A volume of 4 cubic feet of a certain gas has a temperature of 200 degrees Kelvin.
p 2 4 Weight (pounds)
V 4 3 2 1 0
100 200 T Temperature (K)
c. Find the volume of the same gas at 250 degrees Kelvin. Chapter 3
62
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
d = 45t 360 = 45t t=8
(5, 225)
NAME
DATE
3-4
PERIOD
Practice
SCS
MA.A.2.1, MA.A.2.3, MA.A.4.4
Direct Variation Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points. 1.
y
2.
y=3x 4
(–2, 5)
y = - 5x 2
y=4x 3
(0, 0)
x
O
y
(3, 4)
(4, 3)
(0, 0)
3.
y
(0, 0)
x
O
O
x
Graph each equation. y
4. y = -2x
y
6 x 5. y = −
y
5 6. y = - − x
5
2
x
O
O
x
O
x
7. If y = 7.5 when x = 0.5, find y when x = -0.3. 8. If y = 80 when x = 32, find x when y = 100. 3 when x = 24, find y when x = 12. 9. If y = − 4
10. MEASURE The width W of a rectangle is two thirds of the length .
11. TICKETS The total cost C of tickets is $4.50 times the number of tickets t.
Rectangle Dimensions C
10
25
8
20
Cost ($)
W
6 4 2 0
Cost of Tickets
15 10 5
2
4
6 8 10 12 Length
0
1
2
3 4 5 Tickets
6
t
12. PRODUCE The cost of bananas varies directly with their weight. Miguel bought 1 pounds of bananas for $1.12. Write an equation that relates the cost of the bananas 3− 2
1 pounds of bananas. to their weight. Then find the cost of 4 − 4
Chapter 3
63
North Carolina StudyText, Math A
Lesson 3-4
Write a direct variation equation that relates the variables. Then graph the equation.
Width
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Suppose y varies directly as x. Write a direct variation equation that relates x and y. Then solve.
NAME
DATE
3-4
Word Problem Practice
PERIOD
SCS
MA.A.2.1, MA.A.2.3, MA.A.4.4
Direct Variation 1. ENGINES The engine of a chainsaw requires a mixture of engine oil and gasoline. According to the directions, oil and gasoline should be mixed as shown in the graph below. What is the constant of variation for the line graphed? 10
4. SALARY Henry started a new job in which he is paid $9.50 an hour. Write and solve an equation to determine Henry’s gross salary for a 40-hour work week. 5. SALES TAX Amelia received a gift card to a local music shop for her birthday. She plans to use the gift card to buy some new CDs.
y
9
Oil (fl oz)
8 7
a. Amelia chose 3 CDs that each cost $16. The sales tax on the three CDs is $3.96. Write a direct variation equation relating sales tax to the price.
6 5 4 3 2
b. Graph the equation you wrote in part a.
1 0
1
2 3 4 5 6 x Gasoline (gal)
c. What is the sales tax rate that Amelia is paying on the CDs?
3. CURRENCY The exchange rate from U.S. dollars to British pound sterling (£) was approximately $2.07 to £1 in 2007. Write and solve a direct variation equation to determine how many pounds sterling you would receive in exchange for $90 of U.S. currency.
Chapter 3
64
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. RACING In 2007, English driver Lewis Hamilton won the United States Grand Prix at the Indianapolis Motor Speedway. His speed during the race averaged 125.145 miles per hour. Write a direct variation equation for the distance d that Hamilton drove in h hours at that speed.
NAME
3-5
DATE
PERIOD
Study Guide
SCS
MA.A.4.3, MA.A.4.4
Arithmetic Sequences as Linear Functions Recognize Arithmetic Sequences A sequence is a set of numbers in a specific order. If the difference between successive terms is constant, then the sequence is called an arithmetic sequence. Arithmetic Sequence
a numerical pattern that increases or decreases at a constant rate or value called the common difference
Terms of an Arithmetic Sequence
If a1 is the first term of an arithmetic sequence with common difference d, then the sequence is a1, a1 + d, a1 + 2d, a1 + 3d, . . . .
nth Term of an Arithmetic Sequence
an = a1 + (n - 1)d
Example 1 Determine whether the sequence 1, 3, 5, 7, 9, 11, . . . is an arithmetic sequence. Justify your answer.
Example 2 Write an equation for the nth term of the sequence 12, 15, 18, 21, . . . . In this sequence, a1 is 12. Find the common difference. 12 15 18 21 +3
+3
+3
The common difference is 3. Use the formula for the nth term to write an equation. a n = a 1 + (n - 1)d Formula for the nth term a n = 12 + (n - 1)3 a 1 = 12, d = 3 a n = 12 + 3n - 3 Distributive Property a n = 3n + 9 Simplify. The equation for the nth term is a n = 3n + 9.
Exercises Determine whether each sequence is an arithmetic sequence. Write yes or no. Explain. 1. 1, 5, 9, 13, 17, . . .
2. 8, 4, 0, -4, -8, . . .
3. 1, 3, 9, 27, 81, . . .
Find the next three terms of each arithmetic sequence. 4. 9, 13, 17, 21, 25, . . .
5. 4, 0, -4, -8, -12, . . .
6. 29, 35, 41, 47, . . .
Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence. 7. 1, 3, 5, 7, . . .
Chapter 3
8. -1, -4, -7, -10, . . .
65
9. -4, -9, -14, -19, . . .
North Carolina StudyText, Math A
Lesson 3-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
If possible, find the common difference between the terms. Since 3 - 1 = 2, 5 - 3 = 2, and so on, the common difference is 2. Since the difference between the terms of 1, 3, 5, 7, 9, 11, . . . is constant, this is an arithmetic sequence.
NAME
DATE
3-5
Study Guide (continued)
PERIOD
SCS
MA.A.4.3, MA.A.4.4
Arithmetic Sequences as Linear Functions Arithmetic Sequences and Functions An arithmetic sequence is a linear function in which n is the independent variable, a n is the dependent variable, and the common difference d is the slope. The formula can be rewritten as the function a n = a 1 + (n-1)d, where n is a counting number. Example
SEATING There are 20 seats in the first row of the balcony of the auditorium. There are 22 seats in the second row, and 24 seats in the third row. a. Write a function to represent this sequence. The first term a 1 is 20. Find the common difference. 20
22 +2
b. Graph the function. The rate of change is 2. Make a table and plot points.
24 +2
The common difference is 2. a n = a 1 + (n - 1)d = 20 + (n - 1)2 = 20 + 2n - 2 = 18 + 2n
n
an
28
1
20
26
2
22
3
24
4
26
an
24 22
formula for the nth term 20
a 1 = 20 and d = 2 Distributive Property
2
1
0
3
4
n
Simplify.
Exercises 1. KNITTING Sarah learns to knit from her grandmother. Two days ago, she measured the length of the scarf she is knitting to be 13 inches. Yesterday, she measured the length of the scarf to be 15.5 inches. Today it measures 18 inches. Write a function to represent the arithmetic sequence. 2. REFRESHMENTS You agree to pour water into the cups for the Booster Club at a football game. The pitcher contains 64 ounces of water when you begin. After you have filled 8 cups, the pitcher is empty and must be refilled. a. Write a function to represent the arithmetic sequence. 72 64 56 48 40 32 24 16 8
b. Graph the function.
0
Chapter 3
66
an
1 2 3 4 5 6 7 8n
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The function is a n = 18 + 2n.
NAME
DATE
3-5
Practice
PERIOD
SCS
MA.A.4.3, MA.A.4.4
Arithmetic Sequences as Linear Functions Determine whether each sequence is an arithmetic sequence. Write yes or no. Explain. 1. 21, 13, 5, -3, . . .
2. -5, 12, 29, 46, . . .
3. -2.2, -1.1, 0.1, 1.3, . . .
4. 1, 4, 9, 16, . . .
5. 9, 16, 23, 30, . . .
6. -1.2, 0.6, 1.8, 3.0, . . .
Find the next three terms of each arithmetic sequence. 7. 82, 76, 70, 64, . . .
10. -10, -3, 4, 11 . . .
8. -49, -35, -21, -7, . . .
11. 12, 10, 8, 6, . . .
3 1 1 , −, −, 0, . . . 9. − 4 2 4
12. 12, 7, 2, -3, . . .
Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence.
30
an
14. -5, -2, 1, 4, . . .
8
20
4
10
O
O
2
4
6n
an
15. 19, 31, 43, 55, . . .
60
an
40 2
4
6n
-4
20 O
2
4
6n
16. BANKING Chem deposited $115.00 in a savings account. Each week thereafter, he deposits $35.00 into the account. a. Write a function to represent the total amount Chem has deposited for any particular number of weeks after his initial deposit. b. How much has Chem deposited 30 weeks after his initial deposit? 17. STORE DISPLAYS Tamika is stacking boxes of tissue for a store display. Each row of tissues has 2 fewer boxes than the row below. The first row has 23 boxes of tissues. a. Write a function to represent the arithmetic sequence. b. How many boxes will there be in the tenth row?
Chapter 3
67
North Carolina StudyText, Math A
Lesson 3-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
13. 9, 13, 17, 21, . . .
NAME
3-5
DATE
Word Problem Practice
PERIOD
SCS
MA.A.4.3, MA.A.4.4
Arithmetic Sequences as Linear Functions 1. POSTAGE In 2008, the price for first class mail was raised to 42 cents for the first ounce and 17 cents for each additional ounce. The table below shows the cost for weights up to 5 ounces. Weight (ounces)
1
2
3
4
5
Postage (cents)
42
59
76
93
110
4. NUMBER THEORY One of the most famous sequences in mathematics is the Fibonacci sequence. It is named after Leonardo de Pisa (1170–1250) or Filius Bonacci, alias Leonardo Fibonacci. The first several numbers in the Fibonacci sequence are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . . Does this represent an arithmetic sequence? Why or why not?
Source: United States Postal Servcie
How much did a letter weigh that cost $1.61 to send? 5. SAVINGS Inga’s grandfather decides to start a fund for her college education. He makes an initial contribution of $3000 and each month deposits an additional $500. After one month he will have contributed $3500.
2. SPORTS Wanda is the manager for the soccer team. One of her duties is to hand out cups of water at practice. Each cup of water is 4 ounces. She begins practice with a 128-ounce cooler of water. How much water is remaining after she hands out the 14th cup?
a. Write an equation for the nth term of the sequence.
3. THEATER A theater has 20 seats in the first row, 22 in the second row, 24 in the third row, and so on for 25 rows. How many seats are in the last row?
Chapter 3
68
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. How much money will Inga’s grandfather have contributed after 24 months?
NAME
3-6
DATE
PERIOD
Study Guide
SCS
MA.A.4.3
Proportional Relationships If the relationship between the domain and range of a relation is linear, the relationship can be described by a linear equation. If the equation passes through (0, 0) and is of the form y = kx, then the relationship is proportional. Example COMPACT DISCS Suppose you purchased a number of packages of blank compact discs. If each package contains 3 compact discs, you could make a chart to show the relationship between the number of packages of compact discs and the number of discs purchased. Use x for the number of packages and y for the number of compact discs. Make a table of ordered pairs for several points of the graph. Number of Packages
1
2
3
4
5
Number of CDs
3
6
9
12
15
The difference in the x values is 1, and the difference in the y values is 3. This pattern shows that y is always three times x. This suggests the relation y = 3x. Since the relation is also a function, we can write the equation in function notation as f(x) = 3x. The relation includes the point (0, 0) because if you buy 0 packages of compact disks, you will not have any compact discs. Therefore, the relationship is proportional.
1. NATURAL GAS Natural gas use is often measured in “therms.” The total amount a gas company will charge for natural gas use is based on how much natural gas a household uses. The table shows the relationship between natural gas use and the total cost. Gas Used (therms) Total Cost ($)
1
2
3
4
$1.30
$2.60
$3.90
$5.20 y
a. Graph the data. What can you deduce from the pattern about the relationship between the number of therms used and the total cost?
6
Total Cost ($)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
b. Write an equation to describe this relationship.
69
4 3 2 1
c. Use this equation to predict how much it will cost if a household uses 40 therms.
Chapter 3
5
0 1
2 3
4 x
Gas Used (therms)
North Carolina StudyText, Math A
Lesson 3-6
Proportional and Nonproportional Relationships.
NAME
DATE
3-6
PERIOD
Study Guide (continued)
SCS
MA.A.4.3
Proportional and Nonproportional Relationships Nonproportional Relationships
If the ratio of the value of x to the value of y is different for select ordered pairs on the line, the equation is nonproportional. Example Write an equation in functional notation for the relation shown in the graph. y
Select points from the graph and place them in a table. x
-1
0
1
2
3
y
4
2
0
-2
-4 x
0
The difference between the x–values is 1, while the difference between the y-values is –2. This suggests that y = –2x. If x = 1, then y = –2(1) or –2. But the y–value for x = 1 is 0. x
1
2
3
-2x
-2
-4
-6
y
0
-2
-4
y is always 2 more than -2x
This pattern shows that 2 should be added to one side of the equation. Thus, the equation is y = -2x + 2.
Write an equation in function notation for the relation shown in the table. Then complete the table. 1.
x
-1
0
1
y
-2
2
6
2
3
4
2.
x
-2
-1
0
y
10
7
4
1
2
3
Write an equation in function notation for each relation. 3.
O
Chapter 3
4.
y
x
y
O
70
x
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
3-6
PERIOD
Practice
SCS
MA.A.4.3
1. BIOLOGY Male fireflies flash in various patterns to signal location and perhaps to ward off predators. Different species of fireflies have different flash characteristics, such as the intensity of the flash, its rate, and its shape. The table below shows the rate at which a male firefly is flashing. Times (seconds)
1
2
3
4
5
Number of Flashes
2
4
6
8
10
a. Write an equation in function notation for the relation. b. How many times will the firefly flash in 20 seconds? 2. GEOMETRY The table shows the number of diagonals that can be drawn from one vertex in a polygon. Write an equation in function notation for the relation and find the number of diagonals that can be drawn from one vertex in a 12-sided polygon.
Sides
3
4
5
6
Diagonals
0
1
2
3
Write an equation in function notation for each relation.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3.
4.
y
O
O
5.
y
y
x
x O
x
For each arithmetic sequence, determine the related function. Then determine if the function is proportional or nonproportional. Explain. 6. 1, 3, 5, . . .
Chapter 3
7. 2, 7, 12, . . .
71
8. -3, -6, -9, . . .
North Carolina StudyText, Math A
Lesson 3-6
Proportional and Nonproportional Relationships
NAME
DATE
3-6
PERIOD
Word Problem Practice
SCS
MA.A.4.3
Proportional and Nonproportional Relationships 1. ONLINE SHOPPING Ricardo is buying computer cables from an online store. If he buys 4 cables, the total cost will be $24. If he buys 5 cables, the total cost will be $29. If the total cost can be represented by a linear function, will the function be proportional or nonproportional? Explain.
4. MUSIC A measure of music contains the same number of beats throughout the song. The table shows the relation for the number of beats counted after a certain number of measures have been played in the six-eight time. Write an equation to describe this relationship.
2. FOOD It takes about four pounds of grapes to produce one pound of raisins. The graph shows the relation for the number of pounds of grapes needed, x, to make y pounds of raisins. Write an equation in function notation for the relation shown.
1
2
3
4
5
6
Total Number of Beats (y)
6
12
18
24
30
36
Source: Sheet Music USA
5. GEOMETRY A fractal is a pattern containing parts which are identical to the overall pattern. The following geometric pattern is a fractal.
f(x)
3.5 3.0 2.5 2.0 1.5 1.0
a. Complete the table.
0.5 0
1
2 3 4 5 6 7 Pounds of Grapes
8 x
3. PARKING Palmer Township recently installed parking meters in their municipal lot. The cost to park for h hours is represented by the equation C = 0.25h.
Term
x
1
Number of Smaller Triangles
y
1
2
3
4
b. What are the next three numbers in the pattern?
a. Make a table of values that represents this relationship.
b. Describe the relationship between the time parked and the cost.
Chapter 3
c. Write an equation in function notation for the pattern.
72
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Pounds of Raisins
4.0
Measures Played (x)
NAME
DATE
4-5
PERIOD
Study Guide
SCS
MA.S.2.1, MA.S.2.2
Scatter Plots and Lines of Fit Investigate Relationships Using Scatter Plots
A scatter plot is a graph in which two sets of data are plotted as ordered pairs in a coordinate plane. If y increases as x increases, there is a positive correlation between x and y. If y decreases as x increases, there is a negative correlation between x and y. If x and y are not related, there is no correlation. Example
EARNINGS The graph at the right
Carmen’s Earnings and Savings
shows the amount of money Carmen earned each week and the amount she deposited in her savings account that same week. Determine whether the graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation.
35
Dollars Saved
30 25 20 15 10 5
The graph shows a positive correlation. The more Carmen earns, the more she saves.
0
40
80
120
Dollars Earned
Exercises
1.
2. Average Jogging Speed
Average Weekly Work Hours in U.S.
Miles per Hour
34.6
Hours
34.4 34.2 34.0 33.8 33.6
10
5
0 0
1
2
3
4
5
6
7
8
5
10 15 20 25
Minutes
9
Years Since 1995 Source: The World Almanac
4. U.S. Imports from Mexico Imports ($ billions)
Average U.S. Hourly Earnings 19 18 17 16 15 0
1
2
3
4
220 190 160 130 0
1
2
3
4
5
Years Since 2003 Source: U.S. Census Bureau
5
Years Since 2003 Source: U.S. Dept. of Labor
Chapter 4
73
North Carolina StudyText, Math A
Lesson 4-5
3. Hourly Earnings ($)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation.
NAME
DATE
4-5
PERIOD
Study Guide (continued)
SCS
MA.S.2.1, MA.S.2.2
Scatter Plots and Lines of Fit Use Lines of Fit Example
The table shows the number of students per computer in Easton High School for certain school years from 1996 to 2008.
Students per Computer
1996
1998
2000
2002
2004
2006
2008
22
18
14
10
6.1
5.4
4.9
a. Draw a scatter plot and determine what relationship exists, if any. Since y decreases as x increases, the correlation is negative. b. Draw a line of fit for the scatter plot. Draw a line that passes close to most of the points. A line of fit is shown. c. Write the slope-intercept form of an equation for the line of fit. The line of fit shown passes through (1999, 16) and (2005, 5.7). Find the slope.
Students per Computer
Year
Students per Computer in Easton High School 24 20 16 12 8 4 0
1996 1998 2000 2002 2004 2006 2008
Year
5.7 - 16 m=− 2005-1999
Exercises Refer to the table for Exercises 1–3. 1. Draw a scatter plot.
Movie Admission Prices 6.2
2. Draw a line of fit for the data. Admission ($)
3. Write the slope-intercept form of an equation for the line of fit.
6 5.8 5.6 5.4 5.2 5 1
2
3
4
Years Since 1999
Admission (dollars)
0
$5.08
1
$5.39
2
$5.66
3
$5.81
4
$6.03
5
Years Since 1999 Source: U.S. Census Bureau
Chapter 4
74
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
m = -1.7 Find b in y = -1.7x + b. 16 = -1.7 · 1993 + b 3404 = b Therefore, an equation of a line of fit is y = -1.7x + 3404.
NAME
DATE
4-5
PERIOD
Practice
SCS
MA.S.2.1, MA.S.2.2
Scatter Plots and Lines of Fit Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. 2.
64 60 56 52 0
State Elevations Highest Point (thousands of feet)
Temperature versus Rainfall Average Temperature (ºF)
1.
10 15 20 25 30 35 40 45
16 12 8 4 0
1000
2000
3000
Mean Elevation (feet)
Average Annual Rainfall (inches)
Source: U.S. Geological Survey
Source: National Oceanic and Atmospheric Administration
3. DISEASE The table shows the number of cases of Foodborne Botulism in the United States for the years 2001 to 2005. a. Draw a scatter plot and determine what relationship, if any, exists in the data.
U.S. Foodborne Botulism Cases Year
2001 2002 2003 2004 2005
Cases
39
28
20
16
18
Source: Centers for Disease Control
50
b. Draw a line of fit for the scatter plot. c. Write the slope-intercept form of an equation for the line of fit.
Cases
40 30 20 10 0
2001
2002
2003
2004
2005
Year
4. ZOOS The table shows the average and maximum longevity of various animals in captivity. a. Draw a scatter plot and determine what relationship, if any, exists in the data.
Longevity (years) Avg. 12 25 15
8
35 40 41 20
Max. 47 50 40 20 70 77 61 54 Source: Walker’s Mammals of the World
Animal Longevity (Years) 80
b. Draw a line of fit for the scatter plot.
70 60 50 40 30 20 10 0
5
10 15 20 25 30 35 40 45
Average Chapter 4
75
North Carolina StudyText, Math A
Lesson 4-5
c. Write the slope-intercept form of an equation for the line of fit. d. Predict the maximum longevity for an animal with an average longevity of 33 years.
Maximum
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
U.S. Foodborne Botulism Cases
NAME
DATE
4-5
PERIOD
Word Problem Practice
SCS
MA.S.2.1, MA.S.2.2
Scatter Plots and Lines of Fit 1. MUSIC The scatter plot shows the number of CDs (in millions) that were sold from 1999 to 2005. If the trend continued, about how many CDs were sold in 2006? 950
3. HOUSING The median price of an existing home was $160,000 in 2000 and $240,000 in 2007. If x represents the number of years since 2000, use these data points to determine a possible line of best fit for the trends in the price of existing homes. Write the equation in slope-intercept form.
y
Millions
900 850 800 750
4. BASEBALL The table shows the average length (in minutes) of professional baseball games in selected years.
700 650 ‘99 ‘00 ‘01 ‘02 ‘03 ‘04 ‘05 Year
x
Average Length of Major League Baseball Games
Source: RIAA
Year
‘92
‘94
‘96
‘98
‘00
‘02
‘04
Time (min) 170 174 171 168 178 172 167
2. FAMILY The table shows the predicted annual cost for a middle income family to raise a child from birth until adulthood. Draw a scatter plot and describe what relationship exists within the data.
Source: Elias Sports Bureau
Cost of Raising a Child Born in 2003 3
6
9
12
Length (minutes)
Child’s Age
180
15
Annual 10,700 11,700 12,600 15,000 16,700 Cost ($)
Annual Cost ($1000)
17
y
178 176 174 172 170 168 166 1990
1992
1994
1996
1988
2000
2002
2004
Year
16 15 14
b. Explain what the scatter plot shows.
13 12 11 10
c. Draw a line of fit for the scatter plot.
9 0
3
6
9 12 Age (years)
15
x
Source: The World Almanac
Chapter 4
76
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. Draw a scatter plot and determine what relationship, if any, exists in the data.
NAME
4-6
DATE
PERIOD
Study Guide
SCS
MA.S.2.1, MA.S.2.2
Equations of Best-Fit Lines Many graphing calculators utilize an algorithm called linear regression to find a precise line of fit called the best-fit line. The calculator computes the data, writes an equation, and gives you the correlation coefficent, a measure of how closely the equation models the data. Example
GAS PRICES The table shows the average price of a gallon of regular gasoline in Los Angeles, California. Year
2002
2003
2004
2005
2006
2007
Average Price
$1.47
$1.82
$2.15
$2.49
$2.83
$3.04
Source: U.S. Department of Energy
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. Use a graphing calculator to write an equation for the best-fit line for that data. Enter the data by pressing STAT and selecting the Edit option. Let the year 2002 be represented by 0. Enter the years since 2002 into List 1 (L1). Enter the average price into List 2 (L2). Then, perform the linear regression by pressing STAT and selecting the CALC option. Scroll down to LinReg (ax+b) and press ENTER . The best-fit equation for the regression is shown to be y = 0.321x + 1.499. b. Name the correlation coefficient. The correlation coefficient is the value shown for r on the calculator screen. The correlation coefficient is about 0.998.
Exercises Write an equation of the regression line for the data in each table below. Then find the correlation coefficient. 1. OLYMPICS Below is a table showing the number of gold medals won by the United States at the Winter Olympics during various years. Year Gold Medals
1988
1992
1994
1998
2002
2006
2
5
6
6
10
9
Source: International Olympic Committee
2. INTEREST RATES Below is a table showing the U.S. Federal Reserve’s prime interest rate on January 1 of various years. Year
2003
2004
2005
2006
2007
Prime Rate (percent)
4.25
4.00
5.25
7.25
8.25
Source: Federal Reserve Board
Chapter 4
77
North Carolina StudyText, Math A
Lesson 4-6
Regression and Median-Fit Lines
NAME
4-6
DATE
PERIOD
Study Guide (continued)
SCS
MA.S.2.1, MA.S.2.2
Regression and Median-Fit Lines Equations of Median-Fit Lines A graphing calculator can also find another type of best-fit line called the median-fit line, which is found using the medians of the coordinates of the data points. Example
ELECTIONS The table shows the total number of people in millions
who voted in the U.S. Presidential election in the given years. Year
1980
1984
1988
1992
1996
2004
Voters
86.5
92.7
91.6
104.4
96.3
122.3
Source: George Mason University
a. Find an equation for the median-fit line. Enter the data by pressing STAT and selecting the Edit option. Let the year 1980 be represented by 0. Enter the years since 1980 into List 1 (L1). Enter the number of voters into List 2 (L2). Then, press STAT and select the CALC option. Scroll down to Med-Med option and press ENTER . The value of a is the slope, and the value of b is the y-intercept. The equation for the median-fit line is y = 1.094x + 87.29. b. Estimate the number of people who voted in the 2000 U.S. Presidential election. Graph the best-fit line. Then use the TRACE feature and the arrow keys until you find a point where x = 20.
Exercises Write an equation of the regression line for the data in each table below. Then find the correlation coefficient. 1. POPULATION GROWTH Below is a table showing the estimated population of Arizona in millions on July 1st of various years. Year
2001
2002
2003
2004
2005
2006
Population
5.30
5.44
5.58
5.74
5.94
6.17
Source: U.S. Census Bureau
a. Find an equation for the median-fit line. b. Predict the population of Arizona in 2009. 2. ENROLLMENT Below is a table showing the number of students enrolled at Happy Days Preschool in the given years. Year
1999
2001
2003
2005
2007
Students
130
168
184
201
234
a. Find an equation for the median-fit line. b. Estimate how many students were enrolled in 2004. Chapter 4
78
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
When x = 20, y ≈ 109. Therefore, about 109 million people voted in the 2000 U.S. Presidential election.
NAME
4-6
DATE
PERIOD
Practice
SCS
MA.S.2.1, MA.S.2.2
Write an equation of the regression line for the data in each table below. Then find the correlation coefficient. 1. TURTLES The table shows the number of turtles hatched at a zoo each year since 2002. Year
2003
2004
2005
2006
2007
21
17
16
16
14
Turtles Hatched
2. SCHOOL LUNCHES The table shows the percentage of students receiving free or reduced price school lunches in Marin County, California each year since 2003. Year Percentage
2003
2004
2005
2006
2007
14.4%
15.8%
18.3%
18.6%
20.9%
Source: KidsData
3. SPORTS Below is a table showing the number of students signed up to play lacrosse after school in each age group. Age
13
14
15
16
17
Lacrosse Players
17
14
6
9
12
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. LANGUAGE The State of California keeps track of how many millions of students are learning English as a second language each year. Year
2003
2004
2005
2006
2007
English Learners
1.600
1.599
1.592
1.570
1.569
Source: California Department of Education
a. Find an equation for the median-fit line. b. Predict the number of students who were learning English in California in 2001. c. Predict the number of students who will be learning English in California in 2010.
5. POPULATION Detroit, Michigan, like a number of large cities, is losing population every year. Below is a table showing the population of Detroit each decade. Year
1960
1970
1980
1990
2000
Population (millions)
1.67
1.51
1.20
1.03
0.95
Source: U.S. Census Bureau
a. Find an equation for the regression line. b. Find the correlation coefficient and explain the meaning of its sign.
c. Estimate the population of Detroit in 2008. Chapter 4
79
North Carolina StudyText, Math A
Lesson 4-6
Regression and Median-Fit Lines
NAME
DATE
4-6
PERIOD
Word Problem Practice
SCS
MA.S.2.1, MA.S.2.2
Regression and Median-Fit Lines 1. FOOTBALL Rutgers University running back Ray Rice ran for 1732 total yards in the 2007 regular season. The table below shows his cumulative total number of yards ran after select games. Game Number Cumulative Yards
3. GOLF SCORES Emmanuel is practicing golf as part of his school’s golf team. Each week he plays a full round of golf and records his total score. His scorecard after five weeks is below. Week
1
3
6
9
12
184
431
818
1257
1732
Golf Score
Use a calculator to find an equation for the regression line showing the total yards y scored after x games. What is the real-world meaning of the value returned for a?
2004 $438.10
2005 $517.20
4
5
112
107
108
104
98
Hours Campaigning
1
3
4
6
8
Votes Received
9
22
24
46
78
a. Use a calculator to find an equation for the median-fit line.
2006 $636.30
b. Plot the data points and draw the median-fit line on the graph below.
Source: Global Financial Data
y
Use a calculator to find an equation for the regression line. Then predict the price of an ounce of gold on the last day of trading in 2009. Is this a reasonable prediction? Explain.
80
Votes Received
70 60 50 40 30 20 10 0
1
2 3
4
5
6
7
8
x
Campaign Time (h)
c. Suppose a sixth candidate spends 7 hours campaigning. Estimate how many votes that candidate could expect to receive. Chapter 4
80
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2003 $414.80
3
4. STUDENT ELECTIONS The vote totals for five of the candidates participating in Montvale High School’s student council elections and the number of hours each candidate spent campaigning are shown in the table below.
2. GOLD Ounces of gold are traded by large investment banks in commodity exchanges much the same way that shares of stock are traded. The table below shows the cost of a single ounce of gold on the last day of trading in given years. 2002 $346.70
2
Use a calculator to find an equation for the median-fit line. Then estimate how many games Emmanuel will have to play to get a score of 86.
Source: Rutgers University Athletics
Year Price
1
NAME
DATE
4-7
Study Guide
PERIOD
SCS
MA.A.4.2, MA.A.4.3
Special Functions Step Functions The graph of a step function is a series of disjointed line segments. Because each part of a step function is linear, this type of function is called a piecewise-linear function. One example of a step function is the greatest integer function, written as f(x) = x, where f(x) is the greatest integer not greater than x. Example
Graph f(x) = x + 3.
x
x+3
x + 3
0
3
3
0.5
3.5
3
1
4
4
1.5
4.5
4
2
5
5
2.5
5.5
5
Lesson 4-7
Make a table of values using noninteger values. On the graph, dots represent included points, and circles represent points that are excluded. f(x)
x
0
Because the dots and circles overlap, the domain is all real numbers. The range is all integers.
Exercises Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Graph each function. State the domain and range. 1. f(x) = x + 1
2. f(x) = –x
f (x)
4. f (x) = x + 4
Chapter 4
x
0
5. f (x) = x – 3
f (x)
0
f (x)
f(x)
x
0
3. f(x) = x – 1
6. f (x) = 2x f (x)
f (x)
x
0
81
x
0
x
0
x
North Carolina StudyText, Math A
NAME
DATE
4-7
Study Guide (continued)
PERIOD
SCS
MA.A.4.2, MA.A.4.3
Special Functions Absolute Value Functions
Another type of piecewise-linear function is the absolute value function. Recall that the absolute value of a number is always nonnegative. So in the absolute value function, written as f (x) = |x|, all of the values of the range are nonnegative. The absolute value function is called a piecewise-defined function because it can be written using two or more expressions. Example 1
Example 2
Graph x + 1 if x > 1 f(x) = . State the domain 3x if x ≤ 1 and range.
Graph f (x) = |x + 2|. State the domain and range.
{
f (x) cannot be negative, so the minimum point is f (x) = 0. f (x) = |x + 2| Original function 0=x+2 Replace f(x) with 0. -2 = x Subtract 2 from each side. Make a table. Include values for x > -2 and x < -2. f(x)
-5
3
-4
2
-3
1
-2
0
-1
1
0
2
1
3
2
4
f (x)
f(x)
0
x x
0
The domain is all real numbers. The range is all real numbers greater than or equal to 0.
The domain and range are both all real numbers.
Exercises Graph each function. State the domain and range. 1. f(x) = |x - 1|
2. f(x) = |-x + 2|
f (x)
0
Chapter 4
3. f(x) =
{-xx-+24 ifif xx >≤ 11 y
f(x)
x
x
0
82
0
x
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
Graph the first expression. When x > 1, f (x) = x + 1. Since x ≠ 1, place an open circle at (1, 2). Next, graph the second expression. When x ≤ 1, f(x) = 3x. Since x = 1, place a closed circle at (1, 3).
NAME
4-7
DATE
PERIOD
Practice
SCS
MA.A.4.2, MA.A.4.3
Special Functions Graph each function. State the domain and range.
f (x)
4. f(x) = |2x + 4| - 3
5. f(x) =
{x2 + 4 ififxx>≤--11
{
–2x + 3 if x > 0 1 − x - 1 if x ≤ 0 2
f (x)
x
0
x
0
Determine the domain and range of each function. 7.
8.
y
x 0
9.
y
0
y
x
10. CELL PHONES Jacob’s cell phone service costs $5 each month plus $0.35 for each minute he uses. Every fraction of a minute is rounded up to the next minute. a. Draw a graph to represent the cost of using the cell phone. b. What is Jacob’s monthly bill if he uses 124.8 minutes?
x
0
Monthly Bill ($)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6. f(x) =
f(x)
x
x
0
x
0
f (x)
0
f (x)
f(x)
x
0
1 x| + 1 3. f(x) = -|− 2
2. f(x) = x + 3 - 2
7.50 7 6.50 6 5.50 5 0
1
2
3
4
5
6
Minutes Used Chapter 4
83
North Carolina StudyText, Math A
Lesson 4-7
1. f(x) = -2x + 1
NAME
DATE
4-7
PERIOD
Word Problem Practice
SCS
MA.A.4.2, MA.A.4.3
Special Functions 4. FUNDRAISING Students are selling boxes of cookies at a fund-raiser. The boxes of cookies can only be ordered by the case, with 12 boxes per case. Draw a graph to represent the number of cases needed y when x boxes of cookies are sold.
y 30 25 20 15
Cases Needed
Ariel’s Earnings ($)
1. BABYSITTING Ariel charges $8 per hour as a babysitter. She rounds every fraction of an hour up to the next half-hour. Draw a graph to represent Ariel’s total earnings y after x hours.
10 5 0
1
2
3 x
Hours Babysitting
1 2 3 4 5 6 7 8 9
17.40 19.30 22.40 25.50 28.60 31.70 34.80 37.90 41.00
3 2 12 24 36 48 60 72 x
Boxes Sold
5. WAGES Kelly earns $8 per hour the first 8 hours she works in a day and $11.50 per hour each hour thereafter. a. Organize the information into a table. Include a column for hours worked x, and a column for daily earnings f (x).
b. Write the piecewise equation describing Kelly’s daily earnings f (x) for x hours.
3. DISEASE PREVENTION Body Mass Index (BMI) is used by doctors to determine weight categories that may lead to health problems. According to the Centers for Disease Control and Prevention, 21.7 is a healthy BMI for adults. If the BMI differs from the desired 21.7 by more than x, there may be health risk involved. Write an equation that can be used to find the highest and lowest BMI for a healthy adult. Then solve if x = 3.2. Chapter 4
4
c. Draw a graph to represent Kelly’s daily earnings. Daily Earnings ($)
f x 144 120 96 72 48 24 0
2
4
6
8
10 12 x
Hours Worked
84
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Rate (dollars)
5
1 0
2. SHIPPING A package delivery service determines rates for express shipping by the weight of a package, with every fraction of a pound rounded up to the next pound. The table shows the cost of express shipping for packages between 1 and 9 pounds. Write a piecewise-linear function representing the cost to ship a package between 1 and 9 pounds. State the domain and range. Weight (pounds)
y 6
NAME
DATE
5-6
PERIOD
Study Guide
SCS
MA.A.5.3
Graph Linear Inequalities
The solution set of an inequality that involves two variables is graphed by graphing a related linear equation that forms a boundary of a half-plane. The graph of the ordered pairs that make up the solution set of the inequality fill a region of the coordinate plane on one side of the half-plane. Example
Graph y ≤ -3x - 2.
y Graph y = -3x - 2. Since y ≤ -3x - 2 is the same as y < -3x - 2 and y = -3x - 2, the boundary is included in the solution set and the graph should be O drawn as a solid line. Select a point in each half plane and test it. Choose (0, 0) and (-2, -2). y ≤ -3x - 2 y ≤ -3x - 2 0 ≤ -3(0) - 2 -2 ≤ -3(-2) - 2 0 ≤ -2 is false. -2 ≤ 6 - 2 -2 ≤ 4 is true. The half-plane that contains (-2, -2) contains the solution. Shade that half-plane.
x
Exercises Graph each inequality.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. y < 4
2. x ≥ 1
3. 3x ≤ y
y
y
y
x
O x
O
4. -x > y
5. x - y ≥ 1
6. 2x - 3y ≤ 6
y
y
x
O
1 x-3 7. y < - − 2
O
y
O
x
8. 4x - 3y < 6
y
O
x
9. 3x + 6y ≥ 12
y
y
x x
O
O
Chapter 5
x
O
85
x
North Carolina StudyText, Math A
Lesson 5-6
Graphing Inequalities in Two Variables
NAME
DATE
5-6
Study Guide
PERIOD
SCS
(continued)
MA.A.5.3
Graphing Inequalities in Two Variables Solve Linear Inequalities
We can use a coordinate plane to solve inequalities with
one variable. Example
Use a graph to solve 2x + 2 > -1.
Step 1 First graph the boundary, which is the related function. Replace the inequality sign with an equals sign, and get 0 on a side by itself. 2x + 2 > -1 Original inequality 2x + 2 = -1 Change < to = . 2x + 2 + 1 = -1 + 1 Add 1 to each side. 2x + 3 = 0 Simplify.
y
x
0
Graph 2x + 3 = y as a dashed line. Step 2 Choose (0, 0) as a test point, substituting these values into the original inequality give us 3 > -5. Step 3 Because this statement is true, shade the half plane containing the point (0, 0). 1 Notice that the x-intercept of the graph is at -1 − . Because the half-plane 2 1 to the right of the x-intercept is shaded, the solution is x > -1 − . 2
Exercises 1. x + 7 ≤ 5
2. x - 2 > 2
3. -x + 1 < -3 y
y
x
0
4. -x - 7 ≥ -6
Chapter 5
x
0
5. 3x - 20 < -17
x
0
86
x
0
6. -2x + 11 ≥ 15
y
y
0
y
y
x
0
x
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use a graph to solve each inequality.
NAME
DATE
5-6
PERIOD
Practice
SCS
MA.A.5.3
Determine which ordered pairs are part of the solution set for each inequality. 1. 3x + y ≥ 6, {(4, 3), (-2, 4), (-5, -3), (3, -3)} 2. y ≥ x + 3, {(6, 3), (-3, 2), (3, -2), (4, 3)} 3. 3x - 2y < 5, {(4, -4), (3, 5), (5, 2), (-3, 4)} Graph each inequality. 4. 2y - x < -4
5. 2x - 2y ≥ 8
y
y
6. 3y > 2x - 3 y x
O x
O
x
O
Use a graph to solve each inequality. 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
2
y
y
y O
7 1 9. − > -2 x + −
2 x+5 8. 6 > −
7. -5 ≤ x - 9
x x
O
0
x
10. MOVING A moving van has an interior height of 7 feet (84 inches). You have boxes in 12 inch and 15 inch heights, and want to stack them as high as possible to fit. Write an inequality that represents this situation. 11. BUDGETING Satchi found a used bookstore that sells pre-owned videos and CDs. Videos cost $9 each, and CDs cost $7 each. Satchi can spend no more than $35. a. Write an inequality that represents this situation. b. Does Satchi have enough money to buy 2 videos and 3 CDs?
Chapter 5
87
North Carolina StudyText, Math A
Lesson 5-6
Graphing Inequalities in Two Variables
NAME
DATE
5-6
PERIOD
Word Problem Practice
SCS
MA.A.5.3
Graphing Inequalities in Two Variables
2. FARMING The average value of U.S. farm cropland has steadily increased in recent years. In 2000, the average value was $1490 per acre. Since then, the value has increased at least an average of $77 per acre per year. Write an inequality to show land values above the average for farmland.
4. FUNDRAISING Troop 200 sold cider and donuts to raise money for charity. They sold small boxes of donut holes for $1.25 and cider for $2.50 a gallon. In order to cover their expenses, they needed to raise at least $100. Write and graph an inequality that represents this situation. c 90
Cider (gal)
1. FAMILY Tyrone said that the ages of his siblings are all part of the solution set of y > 2x, where x is the age of a sibling and y is Tyrone’s age. Which of the following ages is possible for Tyrone and a sibling? Tyrone is 23; Maxine is 14. Tyrone is 18; Camille is 8. Tyrone is 12; Francis is 4. Tyrone is 11; Martin is 6. Tyrone is 19; Paul is 9.
80 70 60 50 40 30 20 10 O
10 20 30 40 50 60 70 d Donut holes
5. INCOME In 2006 the median yearly family income was about $48,200 per year. Suppose the average annual rate of change since then is $1240 per year. a. Write and graph an inequality for the annual family incomes y that are less than the median for x years after 2006.
Income ($1000)
y h
lengt
girth
500
g
Girth
400 350
52,000 50,000 48,000 46,000
O
300
1
2 3 4 5 6 7 8 9 10 x Years since 2006
b. Which of the following points is part of the solution set? (2, 51,000) (8, 69,200) (5, 50,000) (10, 61,000)
250 200 150 100 50
Chapter 5
54,000
44,000
450
O
58,000 56,000
50
150 250 350 450 100 200 300 400 500 Length
88
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. SHIPPING An international shipping company has established size limits for packages with all their services. The total of the length of the longest side and the girth (distance completely around the package at its widest point perpendicular to the length) must be less than or equal to 419 centimeters. Write and graph an inequality that represents this situation.
NAME
6-1
DATE
PERIOD
Study Guide
SCS
MA.A.5.2
Graphing Systems of Equations Possible Number of Solutions
Two or more linear equations involving the same variables form a system of equations. A solution of the system of equations is an ordered pair of numbers that satisfies both equations. The table below summarizes information about systems of linear equations. intersecting lines
same line
O
parallel lines y
y
y x
O
x
O
x
Number of Solutions
exactly one solution
infinitely many solutions
no solution
Terminology
consistent and independent
consistent and dependent
inconsistent
Lesson 6-1
Graph of a System
Example
y
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use the graph at the right to determine whether each system is consistent or inconsistent and if it is independent or dependent.
y=x+1
a. y = -x + 2 y = -x - 1 y = -x + 2 y=x+1 x O Since the graphs of y = -x + 2 and y = x + 1 intersect, there is one solution. Therefore, the system is consistent 3x + 3y = -3 and independent. b. y = -x + 2 3x + 3y = -3 Since the graphs of y = -x + 2 and 3x + 3y = -3 are parallel, there are no solutions. Therefore, the system is inconsistent. c. 3x + 3y = -3 y = -x - 1 Since the graphs of 3x + 3y = -3 and y = -x - 1 coincide, there are infinitely many solutions. Therefore, the system is consistent and dependent.
Exercises
y
Use the graph at the right to determine whether each system is consistent or inconsistent and if it is independent or dependent. 1. y = -x - 3 y=x-1
2. 2x + 2y = -6 y = -x - 3
3x + y = 3 2x + 2y = 4 O
x
2x + 2y = -6 y=x-1 y = -x - 3
3. y = -x - 3 2x + 2y = 4
Chapter 6
4. 2x + 2y = -6 3x + y = 3
89
North Carolina StudyText, Math A
NAME
DATE
6-1
PERIOD
Study Guide (continued)
SCS
MA.A.5.2
Graphing Systems of Equations Solve by Graphing
One method of solving a system of equations is to graph the equations on the same coordinate plane. Example
Graph each system and determine the number of solutions that it has. If it has one solution, name it. y
a. x + y = 2 x-y=4 The graphs intersect. Therefore, there is one solution. The point (3, -1) seems to lie on both lines. Check this estimate by replacing x with 3 and y with -1 in each equation. x+y=2 3 + (-1) = 2 x-y=4 3 - (-1) = 3 + 1 or 4 The solution is (3, -1). b. y = 2x + 1 2y = 4x + 2 The graphs coincide. Therefore there are infinitely many solutions.
x+y=2 O
x
(3, –1) x-y=4
y y = 2x + 1
2y = 4x + 2 x
O
Exercises
1. y = -2
1 x 3. y = −
2. x = 2
3x - y = -1
2
2x + y = 1
y
x
O
x+y=3
y
y
x
O
x
O
4. 2x + y = 6 2x - y = -2
5. 3x + 2y = 6 3x + 2y = -4
y
y
O O
Chapter 6
6. 2y = -4x + 4 y = -2x + 2 y
x
O
x
x
90
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Graph each system and determine the number of solutions it has. If it has one solution, name it.
NAME
6-1
DATE
PERIOD
Practice
SCS
MA.A.5.2
Graphing Systems of Equations y
Use the graph at the right to determine whether each system is consistent or inconsistent and if it is independent or dependent. 2. 2x - y = -3 4x - 2y = -6
3. x + 3y = 3 x + y = -3
2x - y = -3 x+y=3 x
O 4x - 2y = -6
x + y = -3
4. x + 3y = 3 2x - y = -3
Graph each system and determine the number of solutions that it has. If it has one solution, name it. 5. 3x - y = -2 3x - y = 0
6. y = 2x - 3 4x = 2y + 6 y
y
y
x
O O
7. x + 2y = 3 3x - y = -5
x x
a. Graph the system of equations y = 0.5x + 20 and y = 1.5x to represent the situation.
40 30
a. Write a system of equations to represent the situation.
25 20 15 10 5
b. How many treats does Nick need to sell per week to break even?
9. SALES A used book store also started selling used CDs and videos. In the first week, the store sold 40 used CDs and videos, at $4.00 per CD and $6.00 per video. The sales for both CDs and videos totaled $180.00
0
40
CD and Video Sales
30 25 20 15 10 5
c. How many CDs and videos did the store sell in the first week?
0
91
5 10 15 20 25 30 35 40 45 Sales ($)
35
b. Graph the system of equations.
Chapter 6
Dog Treats
35
Cost ($)
8. BUSINESS Nick plans to start a home-based business producing and selling gourmet dog treats. He figures it will cost $20 in operating costs per week plus $0.50 to produce each treat. He plans to sell each treat for $1.50.
Video Sales ($)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
5 10 15 20 25 30 35 40 45 CD Sales ($)
North Carolina StudyText, Math A
Lesson 6-1
1. x + y = 3 x + y = -3
x + 3y = 3
NAME
DATE
6-1
PERIOD
Word Problem Practice
SCS
MA.A.5.2
Graphing Systems of Equations 1. BUSINESS The widget factory will sell a total of y widgets after x days according to the equation y = 200x + 300. The gadget factory will sell y gadgets after x days according to the equation y = 200x + 100. Look at the graph of the system of equations and determine whether it has no solution, one solution, or infinitely many solutions.
900
y
4. AVIATION Two planes are in flight near a local airport. One plane is at an altitude of 1000 meters and is ascending at a rate of 400 meters per minute. The second plane is at an altitude of 5900 meters and is descending at a rate of 300 meters per minute. a. Write a system of equations that represents the progress of each plane
y = 200x + 300 Widgets
800 Items sold
700 600 500 400
y = 200x + 100 Gadgets
300
b. Make a graph that represents the progress of each plane.
200 100 0
1
2
3 4 Days
5
y
6 x
Altitude (m)
x
0 Time (min.)
3. FITNESS Olivia and her brother William had a bicycle race. Olivia rode at a speed of 20 feet per second while William rode at a speed of 15 feet per second. To be fair, Olivia decided to give William a 150-foot head start. The race ended in a tie. How far away was the finish line from where Olivia started?
Chapter 6
92
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. ARCHITECTURE An office building has two elevators. One elevator starts out on the 4th floor, 35 feet above the ground, and is descending at a rate of 2.2 feet per second. The other elevator starts out at ground level and is rising at a rate of 1.7 feet per second. Write a system of equations to represent the situation.
NAME
6-2
DATE
PERIOD
Study Guide
SCS
MA.A.5.2
Substitution One method of solving systems of equations is substitution.
Example 1
Use substitution to solve the system of equations. y = 2x 4x - y = -4
Example 2 Solve for one variable, then substitute. x + 3y = 7 2x - 4y = -6
Substitute 2x for y in the second equation. 4x - y = -4 Second equation 4x - 2x = -4 y = 2x 2x = -4 Combine like terms. x = -2 Divide each side by 2
Solve the first equation for x since the coefficient of x is 1. First equation x + 3y = 7 x + 3y - 3y = 7 - 3y Subtract 3y from each side. x = 7 - 3y Simplify. Find the value of y by substituting 7 - 3y for x in the second equation. 2x - 4y = -6 Second equation 2(7 - 3y) - 4y = -6 x = 7 - 3y 14 - 6y - 4y = -6 Distributive Property 14 - 10y = -6 Combine like terms. 14 - 10y - 14 = -6 - 14 Subtract 14 from each side. -10y = -20 Simplify. y=2 Divide each side by -10
and simplify.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use y = 2x to find the value of y. y = 2x First equation y = 2(-2) x = -2 y = -4 Simplify. The solution is (-2, -4).
and simplify.
Use y = 2 to find the value of x. x = 7 - 3y x = 7 - 3(2) x=1 The solution is (1, 2).
Exercises Use substitution to solve each system of equations. 1. y = 4x 3x - y = 1
2. x = 2y y=x-2
3. x = 2y - 3 x = 2y + 4
4. x - 2y = -1 3y = x + 4
5. x - 4y = 1 2x - 8y = 2
6. x + 2y = 0 3x + 4y = 4
7. 2b = 6a - 14 3a - b = 7
8. x + y = 16 2y = -2x + 2
9. y = -x + 3 2y + 2x = 4
10. x = 2y 0.25x + 0.5y = 10
Chapter 6
11. x - 2y = -5 x + 2y = -1
93
12. -0.2x + y = 0.5 0.4x + y = 1.1
North Carolina StudyText, Math A
Lesson 6-2
Solve by Substitution
NAME
6-2
DATE
Study Guide (continued)
PERIOD
SCS
MA.A.5.2
Substitution Solve Real-World Problems
Substitution can also be used to solve real-world problems involving systems of equations. It may be helpful to use tables, charts, diagrams, or graphs to help you organize data.
Example CHEMISTRY How much of a 10% saline solution should be mixed with a 20% saline solution to obtain 1000 milliliters of a 12% saline solution? Let s = the number of milliliters of 10% saline solution. Let t = the number of milliliters of 20% saline solution. Use a table to organize the information. 10% saline Total milliliters Milliliters of saline
20% saline
12% saline
s
t
1000
0.10 s
0.20 t
0.12(1000)
0.10t 20 − =− 0.10 0.10
First equation Solve for s. Second equation s = 1000 - t Distributive Property Combine like terms. Simplify. Divide each side by 0.10.
t = 200 Simplify. s + t = 1000 First equation s + 200 = 1000 t = 200 s = 800 Solve for s. 800 milliliters of 10% solution and 200 milliliters of 20% solution should be used.
Exercises 1. SPORTS At the end of the 2007–2008 football season, 38 Super Bowl games had been played with the current two football leagues, the American Football Conference (AFC) and the National Football Conference (NFC). The NFC won two more games than the AFC. How many games did each conference win? 2. CHEMISTRY A lab needs to make 100 gallons of an 18% acid solution by mixing a 12% acid solution with a 20% solution. How many gallons of each solution are needed?
3. GEOMETRY The perimeter of a triangle is 24 inches. The longest side is 4 inches longer than the shortest side, and the shortest side is three-fourths the length of the middle side. Find the length of each side of the triangle.
Chapter 6
94
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write a system of equations. s + t = 1000 0.10s + 0.20t = 0.12(1000) Use substitution to solve this system. s + t = 1000 s = 1000 - t 0.10s + 0.20t = 0.12(1000) 0.10(1000 - t) + 0.20t = 0.12(1000) 100 - 0.10t + 0.20t = 0.12(1000) 100 + 0.10t = 0.12(1000) 0.10t = 20
NAME
6-2
DATE
PERIOD
Practice
SCS
MA.A.5.2
Substitution 1. y = 6x 2x + 3y = -20
2. x = 3y 3x - 5y = 12
3. x = 2y + 7 x=y+4
4. y = 2x - 2 y=x+2
5. y = 2x + 6 2x - y = 2
6. 3x + y = 12 y = -x - 2
7. x + 2y = 13 -2x - 3y = -18
8. x - 2y = 3 4x - 8y = 12
9. x - 5y = 36 2x + y = -16
10. 2x - 3y = -24 x + 6y = 18
11. x + 14y = 84 2x - 7y = -7
12. 0.3x - 0.2y = 0.5 x - 2y = -5
13. 0.5x + 4y = -1
14. 3x - 2y = 11
1 15. − x + 2y = 12
1 x-− y=4 2
x + 2.5y = 3.5 1 16. − x-y=3 3
17. 4x - 5y = -7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2x + y = 25
y = 5x
2
x - 2y = 6 18. x + 3y = -4 2x + 6y = 5
19. EMPLOYMENT Kenisha sells athletic shoes part-time at a department store. She can earn either $500 per month plus a 4% commission on her total sales, or $400 per month plus a 5% commission on total sales. a. Write a system of equations to represent the situation. b. What is the total price of the athletic shoes Kenisha needs to sell to earn the same income from each pay scale? c. Which is the better offer?
20. MOVIE TICKETS Tickets to a movie cost $7.25 for adults and $5.50 for students. A group of friends purchased 8 tickets for $52.75. a. Write a system of equations to represent the situation. b. How many adult tickets and student tickets were purchased?
Chapter 6
95
North Carolina StudyText, Math A
Lesson 6-2
Use substitution to solve each system of equations.
NAME
6-2
DATE
PERIOD
Word Problem Practice
SCS
MA.A.5.2
Substitution 1. BUSINESS Mr. Randolph finds that the supply and demand for gasoline at his station are generally given by the following equations. x - y = -2 x + y = 10
4. POPULATION Sanjay is researching population trends in South America. He found that experts expect the population of Ecuador to increase by 1,000,000 and the population of Chile to increase by 600,000 from 2004 to 2009. The table displays the information he found.
Use substitution to find the equilibrium point where the supply and demand lines intersect.
Country
2004 Population
Predicted 5-Year Change
Ecuador
13,000,000
+1,000,000
Chile
16,000,000
+600,000
Source: World Almanac
If the population growth for each country continues at the same rate, in what year are the populations of Ecuador and Chile predicted to be equal?
2. GEOMETRY The measures of complementary angles have a sum of 90 degrees. Angle A and angle B are complementary, and their measures have a difference of 20°. What are the measures of the angles?
A B
a. Write a system of equations that Shelby and Calvin could use to determine how many ounces they need to pour from each flask to make their solution.
3. MONEY Harvey has some $1 bills and some $5 bills. In all, he has 6 bills worth $22. Let x be the number of $1 bills and let y be the number of $5 bills. Write a system of equations to represent the information and use substitution to determine how many bills of each denomination Harvey has.
b. Solve your system of equations. How many ounces from each flask do Shelby and Calvin need?
Chapter 6
96
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. CHEMISTRY Shelby and Calvin are doing a chemistry experiment. They need 5 ounces of a solution that is 65% acid and 35% distilled water. There is no undiluted acid in the chemistry lab, but they do have two flasks of diluted acid: Flask A contains 70% acid and 30% distilled water. Flask B contains 20% acid and 80% distilled water.
NAME
DATE
6-3
PERIOD
Study Guide
SCS
MA.A.5.2
Elimination Using Addition and Subtraction Elimination Using Addition In systems of equations in which the coefficients of the x or y terms are additive inverses, solve the system by adding the equations. Because one of the variables is eliminated, this method is called elimination. Example 1 Use elimination to solve the system of equations. x - 3y = 7 3x + 3y = 9
Example 2 The sum of two numbers is 70 and their difference is 24. Find the numbers. Let x represent one number and y represent the other number. x + y = 70 (+) x - y = 24 2x = 94 2x 94 −=−
Write the equations in column form and add to eliminate y. x - 3y = 7 (+) 3x + 3y = 9 4x = 16 Solve for x. 4x 16 − =− 4
2
2
x = 47 Substitute 47 for x in either equation. 47 + y = 70 47 + y - 47 = 70 - 47 y = 23 The numbers are 47 and 23.
4
x=4 Substitute 4 for x in either equation and solve for y. 4 - 3y = 7 4 - 3y - 4 = 7 - 4 -3y = 3
Lesson 6-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-3y
3 −=− -3 -3 y = -1 The solution is (4, -1).
Exercises Use elimination to solve each system of equations. 1. x + y = -4 x-y=2
2. 2x - 3y = 14 x + 3y = -11
3. 3x - y = -9 -3x - 2y = 0
4. -3x - 4y = -1 3x - y = -4
5. 3x + y = 4 2x - y = 6
6. -2x + 2y = 9 2x - y = -6
7. 2x + 2y = -2 3x - 2y = 12
8. 4x - 2y = -1 -4x + 4y = -2
9. x - y = 2 x + y = -3
10. 2x - 3y = 12
11. -0.2x + y = 0.5
12. 0.1x + 0.3y = 0.9
4x + 3y = 24
0.2x + 2y = 1.6
0.1x - 0.3y = 0.2
13. Rema is older than Ken. The difference of their ages is 12 and the sum of their ages is 50. Find the age of each. 14. The sum of the digits of a two-digit number is 12. The difference of the digits is 2. Find the number if the units digit is larger than the tens digit. Chapter 6
97
North Carolina StudyText, Math A
NAME
DATE
6-3
PERIOD
Study Guide (continued)
SCS
MA.A.5.2
Elimination Using Addition and Subtraction Elimination Using Subtraction In systems of equations where the coefficients of the x or y terms are the same, solve the system by subtracting the equations. Example
Use elimination to solve the system of equations.
2x - 3y = 11 5x - 3y = 14 2x - 3y = 11 (-) 5x - 3y = 14 -3x = -3 -3x -3 −= − -3 -3 x=1 2(1) - 3y = 11 2 - 3y = 11 2 - 3y - 2 = 11 - 2 -3y = 9 -3y 9 −=− -3 -3
y = -3
Write the equations in column form and subtract.
Subtract the two equations. y is eliminated. Divide each side by -3. Simplify. Substitute 1 for x in either equation. Simplify. Subtract 2 from each side. Simplify. Divide each side by -3. Simplify.
The solution is (1, -3).
Exercises 1. 6x + 5y = 4 6x - 7y = -20
2. 3m - 4n = -14 3m + 2n = -2
3. 3a + b = 1 a+b=3
4. -3x - 4y = -23 -3x + y = 2
5. x - 3y = 11 2x - 3y = 16
6. x - 2y = 6 x+y=3
7. 2a - 3b = -13 2a + 2b = 7
8. 4x + 2y = 6 4x + 4y = 10
9. 5x - y = 6 5x + 2y = 3
10. 6x - 3y = 12 4x - 3y = 24
11. x + 2y = 3.5 x - 3y = -9
12. 0.2x + y = 0.7 0.2x + 2y = 1.2
13. The sum of two numbers is 70. One number is ten more than twice the other number. Find the numbers. 14. GEOMETRY Two angles are supplementary. The measure of one angle is 10° more than three times the other. Find the measure of each angle. Chapter 6
98
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use elimination to solve each system of equations.
NAME
6-3
DATE
PERIOD
Practice
SCS
MA.A.5.2
Elimination Using Addition and Subtraction 1. x - y = 1 x + y = -9
2. p + q = -2 p-q=8
3. 4x + y = 23 3x - y = 12
4. 2x + 5y = -3 2x + 2y = 6
5. 3x + 2y = -1 4x + 2y = -6
6. 5x + 3y = 22 5x - 2y = 2
7. 5x + 2y = 7 -2x + 2y = -14
8. 3x - 9y = -12 3x - 15y = -6
9. -4c - 2d = -2 2c - 2d = -14
10. 2x - 6y = 6 2x + 3y = 24
11. 7x + 2y = 2 7x - 2y = -30
12. 4.25x - 1.28y = -9.2 x + 1.28y = 17.6
13. 2x + 4y = 10 x - 4y = -2.5
14. 2.5x + y = 10.7 2.5x + 2y = 12.9
15. 6m - 8n = 3 2m - 8n = -3
16. 4a + b = 2
4 1 x-− y = -2 17. -−
3 1 18. − x-− y=8
4a + 3b = 10
3
3
1 2 − x-− y=4 3 3
4 2 3 1 −x + − y = 19 2 2
19. The sum of two numbers is 41 and their difference is 5. What are the numbers? 20. Four times one number added to another number is 36. Three times the first number minus the other number is 20. Find the numbers. 21. One number added to three times another number is 24. Five times the first number added to three times the other number is 36. Find the numbers. 22. LANGUAGES English is spoken as the first or primary language in 78 more countries than Farsi is spoken as the first language. Together, English and Farsi are spoken as a first language in 130 countries. In how many countries is English spoken as the first language? In how many countries is Farsi spoken as the first language?
23. DISCOUNTS At a sale on winter clothing, Cody bought two pairs of gloves and four hats for $43.00. Tori bought two pairs of gloves and two hats for $30.00. What were the prices for the gloves and hats? Chapter 6
99
North Carolina StudyText, Math A
Lesson 6-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use elimination to solve each system of equations.
NAME
6-3
DATE
PERIOD
Word Problem Practice
SCS
MA.A.5.2
Elimination Using Addition and Subtraction 1. NUMBER FUN Ms. Simms, the sixth grade math teacher, gave her students this challenge problem. Twice a number added to another number is 15. The sum of the two numbers is 11. Lorenzo, an algebra student who was Ms. Simms aide, realized he could solve the problem by writing the following system of equations. 2x + y = 15 x + y = 11
4. SPORTS As of 2007, the New York Yankees had won more Major League Baseball World Series than any other team. In fact, the Yankees had won 1 fewer than 3 times the number of World Series won by the Oakland A’s. The sum of the two teams’ World Series championships is 35. How many times has each team won the World Series?
Use the elimination method to solve the system and find the two numbers.
3. RESEARCH Melissa wondered how much it would cost to send a letter by mail in 1990, so she asked her father. Rather than answer directly, Melissa’s father gave her the following information. It would have cost $3.70 to send 13 postcards and 7 letters, and it would have cost $2.65 to send 6 postcards and 7 letters. Use a system of equations and elimination to find how much it cost to send a letter in 1990.
Chapter 6
100
Team
Average Ticket Price
Change in Price
Dallas
$53.60
$0.53
Boston
$55.93
-$1.08
Source: Team Marketing Report.
a. Assume that tickets continue to change at the same rate each year after 2005. Let x be the number of years after 2005, and y be the price of an average ticket. Write a system of equations to represent the information in the table.
b. In how many years will the average ticket price for Dallas approximately equal that of Boston?
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. GOVERNMENT The Texas State Legislature is comprised of state senators and state representatives. The sum of the number of senators and representatives is 181. There are 119 more representatives than senators. How many senators and how many representatives make up the Texas State Legislature?
5. BASKETBALL In 2005, the average ticket prices for Dallas Mavericks games and Boston Celtics games are shown in the table below. The change in price is from the 2004 season to the 2005 season.
NAME
DATE
6-4
PERIOD
Study Guide
SCS
MA.A.5.2
Elimination Using Multiplication Elimination Using Multiplication Some systems of equations cannot be solved simply by adding or subtracting the equations. In such cases, one or both equations must first be multiplied by a number before the system can be solved by elimination. Example 1 Use elimination to solve the system of equations. x + 10y = 3 4x + 5y = 5
Example 2 Use elimination to solve the system of equations. 3x - 2y = -7 2x - 5y = 10
If you multiply the second equation by -2, you can eliminate the y terms. x + 10y = 3 (+) -8x - 10y = -10 -7x = -7 -7x -7 −=−
If you multiply the first equation by 2 and the second equation by -3, you can eliminate the x terms. 6x - 4y = -14 (+) -6x + 15y = -30 11y = -44
-7
-7
x=1 Substitute 1 for x in either equation. 1 + 10y = 3 1 + 10y - 1 = 3 - 1 10y = 2 10
10
1 y=−
y = -4 Substitute -4 for y in either equation. 3x - 2(-4) = -7 3x + 8 = -7 3x + 8 -8 = -7 -8 3x = -15 3x 15 − = -−
5
( )
1 . The solution is 1, − 5
11
3
3
x = -5 The solution is (-5, -4).
Exercises Use elimination to solve each system of equations. 1. 2x + 3y = 6 x + 2y = 5
2. 2m + 3n = 4 -m + 2n = 5
3. 3a - b = 2 a + 2b = 3
4. 4x + 5y = 6 6x - 7y = -20
5. 4x - 3y = 22 2x - y = 10
6. 3x - 4y = -4 x + 3y = -10
7. 4x - y = 9 5x + 2y = 8
8. 4a - 3b = -8 2a + 2b = 3
9. 2x + 2y = 5 4x - 4y = 10
10. 6x - 4y = -8 4x + 2y = -3
11. 4x + 2y = -5 -2x - 4y = 1
12. 2x + y = 3.5 -x + 2y = 2.5
13. GARDENING The length of Sally’s garden is 4 meters greater than 3 times the width. The perimeter of her garden is 72 meters. What are the dimensions of Sally’s garden? Chapter 6
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North Carolina StudyText, Math A
Lesson 6-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10y 2 −=−
11y 11
44 − = -−
NAME
DATE
6-4
PERIOD
Study Guide (continued)
SCS
MA.A.5.2
Elimination Using Multiplication Solve Real-World Problems
Sometimes it is necessary to use multiplication before
elimination in real-world problems. Example
CANOEING During a canoeing trip, it takes Raymond 4 hours to paddle 12 miles upstream. It takes him 3 hours to make the return trip paddling downstream. Find the speed of the canoe in still water. Read
You are asked to find the speed of the canoe in still water.
Solve Let c = the rate of the canoe in still water. Let w = the rate of the water current. r
t
d
r·t=d
Against the Current
c–w
4
12
(c – w)4 = 12
With the Current
c+w
3
12
(c + w)3 = 12
24
24
c = 3.5 The rate of the canoe in still water is 3.5 miles per hour.
Simplify.
Exercises 1. FLIGHT An airplane traveling with the wind flies 450 miles in 2 hours. On the return trip, the plane takes 3 hours to travel the same distance. Find the speed of the airplane if the wind is still. 2. FUNDRAISING Benji and Joel are raising money for their class trip by selling gift wrapping paper. Benji raises $39 by selling 5 rolls of red wrapping paper and 2 rolls of foil wrapping paper. Joel raises $57 by selling 3 rolls of red wrapping paper and 6 rolls of foil wrapping paper. For how much are Benji and Joel selling each roll of red and foil wrapping paper?
Chapter 6
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North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
So, our two equations are 4c - 4w = 12 and 3c + 3w = 12. Use elimination with multiplication to solve the system. Since the problem asks for c, eliminate w. 4c - 4w = 12 ⇒ Multiply by 3 ⇒ 12c - 12w = 36 3c + 3w = 12 ⇒ Multiply by 4 ⇒ (+) 12c + 12w = 48 24c = 84 w is eliminated. 24c 84 −=− Divide by 24.
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6-4
DATE
Practice
PERIOD
SCS
MA.A.5.2
Elimination Using Multiplication 1. 2x - y = -1 3x - 2y = 1
2. 5x - 2y = -10 3x + 6y = 66
3. 7x + 4y = -4 5x + 8y = 28
4. 2x - 4y = -22 3x + 3y = 30
5. 3x + 2y = -9 5x - 3y = 4
6. 4x - 2y = 32 -3x - 5y = -11
7. 3x + 4y = 27
8. 0.5x + 0.5y = -2
3 y = -7 9. 2x - −
5x - 3y = 16
x - 0.25y = 6
10. 6x - 3y = 21 2x + 2y = 22
11. 3x + 2y = 11 2x + 6y = -2
4 1 x + −y = 0 2
12. -3x + 2y = -15 2x - 4y = 26
13. Eight times a number plus five times another number is -13. The sum of the two numbers is 1. What are the numbers? 14. Two times a number plus three times another number equals 4. Three times the first number plus four times the other number is 7. Find the numbers. 15. FINANCE Gunther invested $10,000 in two mutual funds. One of the funds rose 6% in one year, and the other rose 9% in one year. If Gunther’s investment rose a total of $684 in one year, how much did he invest in each mutual fund?
16. CANOEING Laura and Brent paddled a canoe 6 miles upstream in four hours. The return trip took three hours. Find the rate at which Laura and Brent paddled the canoe in still water. 17. NUMBER THEORY The sum of the digits of a two-digit number is 11. If the digits are reversed, the new number is 45 more than the original number. Find the number.
Chapter 6
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North Carolina StudyText, Math A
Lesson 6-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use elimination to solve each system of equations.
NAME
DATE
6-4
Word Problem Practice
PERIOD
SCS
MA.A.5.2
Elimination Using Multiplication 1. SOCCER Suppose a youth soccer field has a perimeter of 320 yards and its length measures 40 yards more than its width. Ms. Hughey asks her players to determine the length and width of their field. She gives them the following system of equations to represent the situation. Use elimination to solve the system to find the length and width of the field.
4. TRAVEL Antonio flies from Houston to Philadelphia, a distance of about 1340 miles. His plane travels with the wind and takes 2 hours and 20 minutes. At the same time, Paul is on a plane from Philadelphia to Houston. Since his plane is heading against the wind, Paul’s flight takes 2 hours and 50 minutes. What was the speed of the wind in miles per hour?
2L + 2W = 320 L – W = 40
3. ART Mr. Santos, the curator of the children’s museum, recently made two purchases of clay and wood for a visiting artist to sculpt. Use the table to find the cost of each product per kilogram. Clay (kg)
Wood (kg)
Total Cost
5
4
$35.50
3.5
6
$50.45
Chapter 6
a. Write and solve a system of equations representing the total costs and revenue of your business.
b. Describe what the solution means in terms of the situation.
c. Give an example of a reasonable number of scooters you could assemble and sell in order to make a profit, and find the profit you would make for that number of scooters.
104
North Carolina StudyText, Math A
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2. SPORTS The Fan Cost Index (FCI) tracks the average costs for attending sporting events, including tickets, drinks, food, parking, programs, and souvenirs. According to the FCI, a family of four would spend a total of $592.30 to attend two Major League Baseball (MLB) games and one National Basketball Association (NBA) game. The family would spend $691.31 to attend one MLB and two NBA games. Write and solve a system of equations to find the family’s costs for each kind of game according to the FCI.
5. BUSINESS Suppose you start a business assembling and selling motorized scooters. It costs you $1500 for tools and equipment to get started, and the materials cost $200 for each scooter. Your scooters sell for $300 each.
NAME
6-5
DATE
PERIOD
Study Guide
SCS
MA.A.5.2
Applying Systems of Linear Equations Determine The Best Method
You have learned five methods for solving systems of linear equations: graphing, substitution, elimination using addition, elimination using subtraction, and elimination using multiplication. For an exact solution, an algebraic method is best. Example At a baseball game, Henry bought 3 hotdogs and a bag of chips for $14. Scott bought 2 hotdogs and a bag of chips for $10. Each of the boys paid the same price for their hotdogs, and the same price for their chips. The following system of equations can be used to represent the situation. Determine the best method to solve the system of equations. Then solve the system. 3x + y = 14 2x + y = 10 • Since neither the coefficients of x nor the coefficients of y are additive inverses, you cannot use elimination using addition. • Since the coefficient of y in both equations is 1, you can use elimination using subtraction. You could also use the substitution method or elimination using multiplication
x 3(4) +
=4 y = 14 y=2
The variable y is eliminated. Substitute the value for x back into the first equation. Solve for y.
This means that hot dogs cost $4 each and a bag of chips costs $2.
Exercises Determine the best method to solve each system of equations. Then solve the system. 1. 5x + 3y = 16 3x - 5y = -4
2. 3x - 5y = 7 2x + 5y = 13
3. y + 3x = 24 5x - y = 8
4. -11x - 10y = 17 5x – 7y = 50
Lesson 6-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The following solution uses elimination by subtraction to solve this system. 3x + y = 14 Write the equations in column form and subtract. (-) 2x + (-) y = (-)10
Chapter 6
105
North Carolina StudyText, Math A
NAME
6-5
DATE
PERIOD
Study Guide (continued)
SCS
MA.A.5.2
Applying Systems of Linear Equations Apply Systems Of Linear Equations
When applying systems of linear equations to problem situations, it is important to analyze each solution in the context of the situation. Example
BUSINESS A T-shirt printing company sells
T-shirt Printing Cost
T-shirts for $15 each. The company has a fixed cost for the machine used to print the T-shirts and an additional cost per T-shirt. Use the table to estimate the number of T-shirts the company must sell in order for the income to equal expenses.
Printing machine
$3000.00
blank T-shirt
$5.00
Understand You know the initial income and the initial expense and the rates of change of each quantity with each T-shirt sold. Plan
Write an equation to represent the income and the expenses. Then solve to find how many T-shirts need to be sold for both values to be equal.
Solve
Let x = the number of T-shirts sold and let y = the total amount. total amount
initial amount
0
rate of change times number of T-shirts sold
income
y
=
expenses
y
= 3000 +
+
15x 5x
You can use substitution to solve this system. The first equation.
15x = 3000 + 5x
Substitute the value for y into the second equation.
10x = 3000
Subtract 10x from each side and simplify.
x = 300 Divide each side by 10 and simplify. This means that if 300 T-shirts are sold, the income and expenses of the T-shirt company are equal. Check
Does this solution make sense in the context of the problem? After selling 300 T-shirts, the income would be about 300 × $15 or $4500. The costs would be about $3000 + 300 × $5 or $4500.
Exercises Refer to the example above. If the costs of the T-shirt company change to the given values and the selling price remains the same, determine the number of T-shirts the company must sell in order for income to equal expenses. 1. printing machine: $5000.00; T-shirt: $10.00 each
2. printing machine: $2100.00; T-shirt: $8.00 each
3. printing machine: $8800.00; T-shirt: $4.00 each
4. printing machine: $1200.00; T-shirt: $12.00 each
Chapter 6
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North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y = 15x
NAME
6-5
DATE
PERIOD
Practice
SCS
MA.A.5.2
Applying Systems of Linear Equations Determine the best method to solve each system of equations. Then solve the system. 1. 1.5x – 1.9y = -29 x – 0.9y = 4.5
2. 1.2x – 0.8y = -6 4.8x + 2.4y = 60
3. 18x –16y = -312 78x –16y = 408
4. 14x + 7y = 217 14x + 3y = 189
5. x = 3.6y + 0.7 2x + 0.2y = 38.4
6. 5.3x – 4y = 43.5 x + 7y = 78
8. SCHOOL CLUBS The chess club has 16 members and gains a new member every month. The film club has 4 members and gains 4 new members every month. Write and solve a system of equations to find when the number of members in both clubs will be equal.
9. Tia and Ken each sold snack bars and magazine subscriptions for a school fund-raiser, as shown in the table. Tia earned $132 and Ken earned $190.
Item
Number Sold Tia Ken
snack bars
16
20
magazine subscriptions
4
6
a. Define variable and formulate a system of linear equation from this situation.
b. What was the price per snack bar? Determine the reasonableness of your solution.
Chapter 6
107
North Carolina StudyText, Math A
Lesson 6-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7. BOOKS A library contains 2000 books. There are 3 times as many non-fiction books as fiction books. Write and solve a system of equations to determine the number of nonfiction and fiction books.
NAME
DATE
6-5
PERIOD
Word Problem Practice
SCS
MA.A.5.2
Applying Systems of Linear Equations 1. MONEY Veronica has been saving dimes and quarters. She has 94 coins in all, and the total value is $19.30. How many dimes and how many quarters does she have?
5. SHOPPING Two stores are having a sale on T-shirts that normally sell for $20. Store S is advertising an s percent discount, and Store T is advertising a t dollar discount. Rose spends $63 for three T-shirts from Store S and one from Store T. Manny spends $140 on five Tshirts from Store S and four from Store T. Find the discount at each store.
2. CHEMISTRY How many liters of 15% acid and 33% acid should be mixed to make 40 liters of 21% acid solution? Concentration of Solution
Amount of Solution (L)
15%
x
33%
y
21%
40
Amount of Acid
a. How far will the train travel before catching up to the barge? b. Which shipment will reach New Orleans first? At what time?
4. PRODUCE Roger and Trevor went shopping for produce on the same day. They each bought some apples and some potatoes. The amount they bought and the total price they paid are listed in the table below. Apples (Ib)
c. If both shipments take an hour to unload before heading back to Baton Rouge, what is the earliest time that either one of the companies can begin to load grain to ship in Baton Rouge?
Potatoes Total Cost (Ib) ($)
Roger
8
7
18.85
Trevor
2
10
12.88
What was the price of apples and potatoes per pound?
Chapter 6
108
North Carolina StudyText, Math A
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3. BUILDINGS The Sears Tower in Chicago is the tallest building in North America. The total height of the tower t and the antenna that stands on top of it a is 1729 feet. The difference in heights between the building and the antenna is 279 feet. How tall is the Sears Tower?
6. TRANSPORTATION A Speedy River barge bound for New Orleans leaves Baton Rouge, Louisiana, at 9:00 A.M. and travels at a speed of 10 miles per hour. A Rail Transport freight train also bound for New Orleans leaves Baton Rouge at 1:30 P.M. the same day. The train travels at 25 miles per hour, and the river barge travels at 10 miles per hour. Both the barge and the train will travel 100 miles to reach New Orleans.
NAME
6-7
DATE
PERIOD
Study Guide
SCS
MA.A.5.2
Using Matrices to Solve Systems of Equations
Example 1 Write an augmented matrix for each system of equations.
Example 2 Write an augmented matrix for each system of equations.
2x + 3y = 8 -x + 4y = 7 Place the coefficients of the equations and the constant terms into a matrix. ⎡ 2 3 ⎢ 8⎤ 2x + 3y = 8 ⎢ ⎢ -x + 4y = 7 ⎣-1 4 ⎢ 7⎦
x + 3y = 6 2y = 6 Place the coefficients of the equations and the constant terms into a matrix. ⎡1 3 ⎢ 6⎤ x + 3y = 6 ⎢ ⎢ 2y = 6 ⎣0 2 ⎢ 6⎦
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises 1. 2x - 2y =10 -4x + y = -35
2. x - y = 3 x - 2y = -1
3.
- x = -4 3x - 2y = 2
4. -x + y = 9 4x - y = -6
5. -2x + 5y = 11 3x - y = 0
6.
2y = 8 -3x = 12
7. x - y = 3 2x = 12
8. 2x - y = 1 3x + y = 4
9. 2x - y = 4 10x - y = 24
10.
-2x = -4 x + 3y = 8
13. x + 4y = 10 x-y=0
Chapter 6
11. 5x - 2y = 7 x + 4y = 19
12.
x-y=5 2x + 3y = 15
14. 3x - y = 1 x + 2y = 19
15.
4y = -2 x + 2y = 3
109
North Carolina StudyText, Math A
Lesson 6-7
Augmented Matrices An augmented matrix consists of the coefficients and constant terms of a system of equations, separated by a dashed line.
NAME
6-7
DATE
PERIOD
Study Guide (continued)
SCS
MA.A.5.2
Using Matrices to Solve Systems of Equations Solve Systems of Equations
You can solve a system of equations using an augmented matrix. To do so, you must use row operations until the matrix takes the form ⎡1 0⎤ ⎢ , also known as the identity matrix. ⎣0 1⎦ Example
Use an augmented matrix to solve the system of equations.
2x - y = 5 x + 3y = 6
⎡2 -1 ⎢ 5⎤ Write the augmented matrix: ⎢ ⎢ . ⎣1 3 ⎢ 6⎦ Notice that the first element in the second row is 1. Interchange the rows so 1 can be in the upper left-hand corner.
⎡2 -1 ⎢ 5⎤ ⎡1 3 ⎢ 6⎤ ⎢ ⎢ ⎢ ⎢ ⎣1 3 ⎢ 6⎦ ⎣2 -1 ⎢ 5⎦ To make the first element in the second row 0, multiply the first row by –2 and add the result to row 2. ⎡1 3 ⎢ 6⎤ ⎡1 3 ⎢ 6⎤ ⎢ ⎢ ⎢ ⎢ ⎣2 -1 ⎢ 5⎦ ⎣0 -7 ⎢ -7⎦ 7
⎡1 3 ⎢ 6⎤ ⎡1 3 ⎢ 6⎤ ⎢ ⎢ ⎢ ⎢ ⎣0 -7 ⎢ -7⎦ ⎣0 1 ⎢ 1⎦ To make the second element in the first row a 0, multiply the second row by –3 and add the result to row 1.
⎡1 3 ⎢ 6⎤ ⎡1 0 ⎢ 3⎤ ⎢ ⎢ ⎢ ⎢ ⎣0 1 ⎢ 1⎦ ⎣0 1 ⎢ 1⎦ The solution is (3, 1).
Exercises Use an augmented matrix to solve each system of equations. 1. x + 5y = -7 2x - y = 8
2. 2x - 2y = 6 x - 2y = -1
4. -x + y = 9 -4x + y = 6
5. -2x + 2y = -8 3x - y = 4
Chapter 6
110
3.
6.
-x = -3 2x + 6y = 0 2y = 10 6x +y = -7
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 . To make the second element in the second row a 1, multiply the second row by – −
NAME
6-7
DATE
PERIOD
Practice
SCS
MA.A.5.2
Using Matrices to Solve Systems of Equations Exercises 1. 4x - 2y = 10 x + 8y = -22
2.
-12y = 6 3x + 2y = 11
3.
x + y = 10 2y - 3y = 0
4. -x + 2y = 8 3x - y = 5
5. 4x - y = 11 2x - 3y = 3
6.
2x = 9 x - 5y = -5.5
Write a system of equations for each augmented matrix.
⎡2 0 ⎢ 8⎤ 7. ⎢ ⎢ ⎣3 4 ⎢ -2⎦
⎡ 1 1 ⎢ 9⎤ 8. ⎢ ⎢ ⎣-2 3 ⎢ -3⎦
⎡2 3 ⎢ -6⎤ 9. ⎢ ⎢ ⎣1 -4 ⎢ -14⎦
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use an augmented matrix to solve each system of equations. 10. 4x - y = 4 3y = 12
11. 2x + 5y =1 -x - y = -2
12. 2x + 3y = 0 -x + 2y = 14
13. 2x - y = 3 7x + y = 24
14. 2x - y = 4 9x - 3y = 12
15.
4x - y = 7 -2x + 3y = -16
16. COMMUTER RAIL The cost of a commuter rail ticket varies with the distance traveled. This month, Marcelo bought 5 round-trip tickets to visit his grandmother and 3 roundtrip tickets to his friend’s house for $31.50. Last month, Marcelo bought 2 round-trip tickets to visit his grandmother and 6 round-trip tickets to visit his friend’s house for $27.00. a. Write a system of linear equations to represent the situations. b. Write the augmented matrix. c. What is the cost of each type of ticket?
Chapter 6
111
North Carolina StudyText, Math A
Lesson 6-7
Write an augmented matrix for each system of equations.
NAME
6-7
DATE
PERIOD
Word Problem Practice
SCS
MA.A.5.2
Using Matrices to Solve Systems of Equations 1. RENOVATIONS A contractor is renovating a pair of bathrooms. The contractor buys 20 marble tiles and a can of paint to renovate the first bathroom, spending $290. The contractor buys 35 marble tiles and 2 cans of paint for the second bathroom, spending $470. Write an augmented matrix to model the situation.
2. CATERING The Mitchell Family is spending $122 on 4 trays of ziti and 3 trays of grilled chicken for their family reunion. The Arroyo Family is spending $208 on 7 trays of ziti and 5 trays of chicken from the same caterer. Use an augmented matrix to find out how much the caterer charges for a tray of ziti and a tray of grilled chicken.
4. FOOTBALL During the 2006–2007 season, the New York Jets scored a total of 59 times by getting touchdowns or kicking field goals, for a combined total of 282 points. If each touchdown is worth 6 points and each field goal is worth 3 points, write an augmented matrix to model the situation. Then find the number of touchdowns scored by the Jets.
y feet
x x
3. DIAMONDS As of 2007, the most expensive stone ever sold at auction is The Star of the Season, a pear-shaped diamond, which was sold in May 1995. The second most expensive diamond is the Chloe Diamond, which was sold in November 2007. The two diamonds together sold for $32.7 million. The Chloe Diamond sold for $33.3 million less than 3 times the selling price of the Star of the Season. Write an augmented matrix to model the situation. Then find the selling price of each diamond.
16x
a. Write a system of linear equations to model the situation.
b. Write the augmented matrix.
c. What are the dimensions of the pool?
Chapter 6
112
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. POOLS The design plans for a swimming pool call for a cement walkway of width x feet to surround the entire pool. The pool will be y feet long and 16x feet wide. The perimeter of the pool is 560 feet, and the outer perimeter of the walkway is 600 feet.
NAME
DATE
6-8
PERIOD
Study Guide
SCS
MA.A.5.4
Systems of Inequalities Systems of Inequalities
The solution of a system of inequalities is the set of all ordered pairs that satisfy both inequalities. If you graph the inequalities in the same coordinate plane, the solution is the region where the graphs overlap. y
Solve the system of inequalities
by graphing. y>x+2 y ≤ -2x - 1
y=x+2
Example 2 by graphing. x+y>4 x + y < -1
x
O
The solution includes the ordered pairs in the intersection of the graphs. This region is shaded at the right. The graphs of y = x + 2 and y = -2x - 1 are boundaries of this region. The graph of y = x + 2 is dashed and is not included in the graph of y > x + 2.
y = -2x - 1
y
Solve the system of inequalities
x+y=4
The graphs of x + y = 4 and x + y = -1 are parallel. Because the two regions have no points in common, the system of inequalities has no solution.
x O x+y=1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises Solve each system of inequalities by graphing. 1. y > -1 x -2x + 2 y≤x+1 y
y
x
0
4. 2x + y ≥ 1 x - y ≥ -2
Chapter 6
y
x
O
5. y ≤ 2x + 3 y ≥ -1 + 2x
x
O
113
x
O
6. 5x - 2y < 6 y > -x + 1 y
y
y
O
3. y < x + 1 3x + 4y ≥ 12
x
O
x
North Carolina StudyText, Math A
Lesson 6-8
Example 1
NAME
DATE
6-8
PERIOD
Study Guide (continued)
SCS
MA.A.5.4
Systems of Inequalities Real-World Problems In real-world problems, sometimes only whole numbers make sense for the solution, and often only positive values of x and y make sense. Example
BUSINESS AAA Gem Company produces necklaces and bracelets. In a 40-hour week, the company has 400 gems to use. A necklace requires 40 gems and a bracelet requires 10 gems. It takes 2 hours to produce a necklace and a bracelet requires one hour. How many of each type can be produced in a week?
Bracelets
50
Let n = the number of necklaces that will be produced and b = the number of bracelets that will be produced. Neither n or b can be a negative number, so the following system of inequalities represents the conditions of the problems.
40 30 20
10b + 40n = 400 b + 2n = 40
10 0
10 20 30 40 50 Necklaces
n≥0 b≥0 b + 2n ≤ 40 10b + 40n ≤ 400 The solution is the set ordered pairs in the intersection of the graphs. This region is shaded at the right. Only whole-number solutions, such as (5, 20) make sense in this problem.
Exercises
2. RECREATION Maria had $150 in gift certificates to use at a record store. She bought fewer than 20 recordings. Each tape cost $5.95 and each CD cost $8.95. How many of each type of recording might she have bought?
60
30
50
25
Compact Discs
Fat Grams
1. HEALTH Mr. Flowers is on a restricted diet that allows him to have between 1600 and 2000 Calories per day. His daily fat intake is restricted to between 45 and 55 grams. What daily Calorie and fat intakes are acceptable?
40 30 20 10 0
Chapter 6
1000 2000 Calories
20 15 10 5 0
3000
114
5 10 15 20 25 30 Tapes
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
For each exercise, graph the solution set. List three possible solutions to the problem.
NAME
6-8
DATE
PERIOD
Practice
SCS
MA.A.5.4
Systems of Inequalities Solve each system of inequalities by graphing. 2. y ≥ x + 2 y > 2x + 3 y
y
O
O
x
4. y < 2x - 1 y>2-x
x
O
y
x
O
x
7. FITNESS Diego started an exercise program in which each week he works out at the gym between 4.5 and 6 hours and walks between 9 and 12 miles.
b. List three possible combinations of working out and walking that meet Diego’s goals.
14 12 10 8 6 4 2 0
a. Make a graph showing the numbers of each price of stone Emily can purchase. b. List three possible solutions.
2
3 4 5 6 Gym (hours)
7
8
Turquoise Stones
5 4 3 2 1 0
115
1
6 $6 Stones
8. SOUVENIRS Emily wants to buy turquoise stones on her trip to New Mexico to give to at least 4 of her friends. The gift shop sells stones for either $4 or $6 per stone. Emily has no more than $30 to spend.
x
Diego’s Routine
16
Walking (miles)
a. Make a graph to show the number of hours Diego works out at the gym and the number of miles he walks per week.
Chapter 6
x
6. 2x - y ≥ 2 x - 2y ≥ 2
y
O
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
5. y > x - 4 2x + y ≤ 2
y
O
3. x + y ≥ 1 x + 2y > 1
Lesson 6-8
1. y > x - 2 y≤x
1
2
3 4 5 6 $4 Stones
7
8
North Carolina StudyText, Math A
NAME
DATE
6-8
PERIOD
Word Problem Practice
SCS
MA.A.5.4
Systems of Inequalities 1. PETS Renée’s Pet Store never has more than a combined total of 20 cats and dogs and never more than 8 cats. This is represented by the inequalities x ≤ 8 and x + y ≤ 20. Solve the system of inequalities by graphing. 20 18
y
Dogs
16 14 12
3. FUND RAISING The Camp Courage Club plans to sell tins of popcorn and peanuts as a fundraiser. The Club members have $900 to spend on products to sell and want to order up to 200 tins in all. They also want to order at least as many tins of popcorn as tins of peanuts. Each tin of popcorn costs $3 and each tin of peanuts costs $4. Write a system of equations to represent the conditions of this problem.
10 8 6 4 2 O
2
4. BUSINESS For maximum efficiency, a factory must have at least 100 workers, but no more than 200 workers on a shift. The factory also must manufacture at least 30 units per worker.
4 6 8 10 12 14 16 18 20 x Cats
b. Graph the systems of inequalities.
y 300 275 250 225 200 175 150 125 100 75 50 25 0
Chapter 6
7 Units (1000)
Wages ($)
a. Let x be the number of workers and let y be the number of units. Write four inequalities expressing the conditions in the problem given above.
y
6 5 4 3 2 1 O
50
150 250 350 x 100 200 300 400 Workers
c. List at least three possible solutions.
5 10 15 20 25 30 35 40 45 50 x Hours
116
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. WAGES The minimum wage for one group of workers in Texas is $7.25 per hour effective Sept. 1, 2008. The graph below shows the possible weekly wages for a person who makes at least minimum wages and works at most 40 hours. Write the system of inequalities for the graph.
NAME
DATE
7-1
Study Guide
PERIOD
SCS
MA.N.2.1, MA.N.2.2
Multiplying Monomials Monomials
A monomial is a number, a variable, or the product of a number and one or more variables with nonnegative integer exponents. An expression of the form xn is called a power and represents the product you obtain when x is used as a factor n times. To multiply two powers that have the same base, add the exponents. For any number a and all integers m and n, am
Example 1 Simplify (3x6)(5x2). (3x6)(5x2) = (3)(5)(x6 ․ x2) Group the coefficients
Example 2 Simplify (-4a3b)(3a2b5). (-4a3b)(3a2b5) = (-4)(3)(a3 ․ a2)(b ․ b5) = -12(a3 + 2)(b1 + 5) = -12a5b6 The product is -12a5b6.
and the variables
= (3 ․ 5)(x6 + 2) = 15x8 The product is 15x8.
․ a n = a m + n.
Product of Powers Simplify.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Simplify each expression. 1. y(y5)
2. n2 ․ n7
3. (-7x2)(x4)
4. x(x2)(x4)
5. m ․ m5
6. (-x3)(-x4)
7. (2a2)(8a)
8. (rn)(rn3)(n2)
9. (x2y)(4xy3)
1 10. − (2a 3b)(6b 3) 3
(5
1 13. (5a 2bc 3) − abc 4
Chapter 7
)
11. (-4x3)(-5x7)
12. (-3j2k4)(2jk6)
14. (-5xy)(4x2)(y4)
15. (10x3yz2)(-2xy5z)
117
North Carolina StudyText, Math A
Lesson Lesson X-X 7-1
Product of Powers
NAME
DATE
7-1
Study Guide
PERIOD
SCS
(continued)
MA.N.2.1, MA.N.2.2
Multiplying Monomials Simplify Expressions
An expression of the form (xm)n is called a power of a power and represents the product you obtain when xm is used as a factor n times. To find the power of a power, multiply exponents. Power of a Power
For any number a and any integers m and p, (am)p = amp.
Power of a Product
For any numbers a and b and any integer m, (ab)m = ambm.
We can combine and use these properties to simplify expressions involving monomials. Example
Simplify (-2ab2)3(a2)4.
(-2ab2)3(a2)4 = (-2ab2)3(a8) = (-2)3(a3)(b2)3(a8) = (-2)3(a3)(a8)(b2)3 = (-2)3(a11)(b2)3 = -8a11b6 The product is -8a11b6.
Power of a Power Power of a Product Group the coefficients and the variables Product of Powers Power of a Power
Exercises Simplify each expression. 2. (n7)4
3. (x2)5(x3)
4. -3(ab4)3
5. (-3ab4)3
6. (4x2b)3
7. (4a2)2(b3)
8. (4x)2(b3)
9. (x2y4)5
10. (2a3b2)(b3)2
(5 )
1 13. (25a 2b) 3 − abf
2
16. (-2n6y5)(-6n3y2)(ny)3
Chapter 7
11. (-4xy)3(-2x2)3
12. (-3j2k3)2(2j2k)3
14. (2xy)2(-3x2)(4y4)
15. (2x3y2z2)3(x2z)4
17. (-3a3n4)(-3a3n)4
18. -3(2x)4(4x5y)2
118
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. (y5)2
NAME
DATE
7-1
PERIOD
Practice
SCS
MA.N.2.1, MA.N.2.2
Multiplying Monomials Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. 21a 2 1. − 7b b 3c 2 2. − 2
3. (-5x2y)(3x4)
4. (2ab2f 2)(4a3b2f 2)
5. (3ad4)(-2a2)
6. (4g3h)(-2g5)
(
)
1 3 xy 7. (-15xy 4) - − 3
(
8. (-xy)3(xz)
1 mn 2 9. (-18m 2n) 2 - −
(3 )
2 p 11. −
6
Lesson Lesson X-X 7-1
Simplify each expression.
)
10. (0.2a2b3)2
2
(4 )
1 12. − ad 3
13. (0.4k3)3
2
14. [(42)2]2
GEOMETRY Express the area of each figure as a monomial. 16.
15. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
17. 5x3
3ab2
6ab 3
6a2b4
4a2b
GEOMETRY Express the volume of each solid as a monomial. 19.
18.
n
3h2 m3n 3h2
mn3
20.
3g 7g2
3h2
21. COUNTING A panel of four light switches can be set in 24 ways. A panel of five light switches can set in twice this many ways. In how many ways can five light switches be set? 22. HOBBIES Tawa wants to increase her rock collection by a power of three this year and then increase it again by a power of two next year. If she has 2 rocks now, how many rocks will she have after the second year?
Chapter 7
119
North Carolina StudyText, Math A
NAME
7-1
DATE
Word Problem Practice
PERIOD
SCS
MA.N.2.1, MA.N.2.2
Multiplying Monomials 1. GRAVITY An egg that has been falling for x seconds has dropped at an average speed of 16x feet per second. If the egg is dropped from the top of a building, its total distance traveled is the product of the average rate times the time. Write a simplified expression to show the distance the egg has traveled after x seconds.
2. CIVIL ENGINEERING A developer is planning a sidewalk for a new development. The sidewalk can be installed in rectangular sections that have a fixed width of 3 feet and a length that can vary. Assuming that each section is the same length, express the area of a 4-section sidewalk as a monomial.
4. SPORTS The volume of a sphere is given 4 3 πr , where r is the by the formula V = − 3 radius of the sphere. Find the volume of air in three different basketballs. Use π = 3.14. Round your answers to the nearest whole number. Ball
Radius (in.)
Child’s
4
Women’s
4.5
Men’s
4.8
Volume (in3)
5. ELECTRICITY An electrician uses the formula W = I2R , where W is the power in watts, I is the current in amperes, and R is the resistance in ohms. a. Find the power in a household circuit that has 20 amperes of current and 5 ohms of resistance.
x 3 ft
3. PROBABILITY If you flip a coin 3 times in a row, there are 23 outcomes that can occur. Outcomes HHH
TTT
HTT
THH
HTH
TTH
HHT
THT
If you then flip the coin two more times, there are 23 × 22 outcomes that can occur. How many outcomes can occur if you flip the quarter as mentioned above plus four more times? Write your answer in the form 2x.
Chapter 7
120
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. If the current is reduced by one half, what happens to the power?
NAME
DATE
7-2
PERIOD
Study Guide
SCS
MA.N.2.2
Dividing Monomials Quotients of Monomials
To divide two powers with the same base, subtract the
exponents.
Quotient of Powers
am m-n For all integers m and n and any nonzero number a, − . n = a
Power of a Quotient
a For any integer m and any real numbers a and b, b ≠ 0, −
a
(b)
a 4b 7 Simplify − . Assume 2
Example 2
ab
that no denominator equals zero.
( )( )
a 4b 7 a4 b7 − − = a − ab 2 b2 4-1
am =− m . b
3 2a 3b 5 Simplify − . Assume 2
( 3b )
that no denominator equals zero.
Group powers with the same base.
7-2
= (a )(b ) Quotient of Powers = a3b5 Simplify. The quotient is a3b5 .
3
(2a 3b 5) 3 3b (3b ) 2 3(a 3) 3(b 5) 3 =− (3) 3(b 2) 3 8a 9b 15 =− 27b 6 8a 9b 9 =− 27 8a 9b 9 The quotient is − . 27
(
3
5
)
2a b − 2
=− 2 3
Power of a Quotient Power of a Product Power of a Power Quotient of Powers
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises Simplify each expression. Assume that no denominator equals zero. 55 1. − 2
m6 2. − 4
a2 4. − a
5. − 5 2
5
xy 6 yx
(rw )
Chapter 7
m
x 5y 3 xy
6. −5
(
-2y 7 14y
)
2a 2b 8. − a
7. − 4
2r 5w 3 10. − 4 3
p 5n 4 pn
3. − 2
4
( 2r n )
r 6n 3 11. 3− 5
( ) 4p 4r 4 3p r
3
3
9. − 2 2
4
7 7 2 nt 12. r− 3 3 2
nrt
121
North Carolina StudyText, Math A
Lesson 7-2
Example 1
m
NAME
DATE
7-2
Study Guide
PERIOD
SCS
(continued)
MA.N.2.2
Dividing Monomials Negative Exponents
Any nonzero number raised to the zero power is 1; for example, (-0.5)0 = 1. Any nonzero number raised to a negative power is equal to the reciprocal of the 1 . These definitions can be used to number raised to the opposite power; for example, 6 -3 = − 3 6 simplify expressions that have negative exponents. Zero Exponent
For any nonzero number a, a0 = 1.
Negative Exponent Property
1 1 n For any nonzero number a and any integer n, a -n = − n and − -n = a . a
a
The simplified form of an expression containing negative exponents must contain only positive exponents. Example
-3
6
4a b . Assume that no denominator equals zero. Simplify − 2 6 -5 16a b c
-3
( 16 )( a )( b )( c )
6
-3
6
4a b b 4 a 1 − − − − = − 2 6 -5 2 6 -5 16a b c
1 =− (a -3-2)(b 6-6)(c 5)
( )
Quotient of Powers and Negative Exponent Properties Simplify. Negative Exponent and Zero Exponent Properties Simplify.
Exercises Simplify each expression. Assume that no denominator equals zero. p -8 p
22 1. − -3
m 2. − -4
b -4 4. − -5 b
5. − -1 2
x 4y 0 x
8. − 2 4
2
7. − -2
-3 -5 t − 10. m 2 3 -1
(m t )
Chapter 7
3. − 3
m
(-x -1y) 0
(a 2b 3) 2 (ab)
6. − -2
4w y
(6a -1b) 2 (b )
( 8m )
4m 2n 2 11. − -1
(3rt) 2u -4 r tu
9. − -1 2 7
(-2mn 2) -3 4m n
0
12. − -6 4
122
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4 1 -5 0 5 =− a bc 4 1 1 − (1)c 5 =− 4 a5 c5 =− 4a 5 c5 The solution is − . 4a 5
Group powers with the same base.
NAME
DATE
7-2
PERIOD
Practice
SCS
MA.N.2.2
Dividing Monomials Simplify each expression. Assume that no denominator equals zero. 8
3. − xy
m 5np mp
5c 2d 3 5. − 2
6. − 6 5
ab
4. − 4
( ) 4f 3g 3h
3
10. x3(y-5)(x-8)
(7)
3 13. −
8y 7z 6 4y z
-4c d
7. − 6
-2
(
5
)
6w 8. − 6 3 7p r
2
-4x 2 9. − 5 24x
11. p(q-2)(r-3)
(3)
4 14. −
-15w 0u -1 16. − 3
-4
12. 12-2
22r 3s 2 15. − 2 -3 11r s
( )
8c 3d 2f 4 4c d f
x -3y 5 4
17. − -1 2 -3
18. − -3
19. − -2 -5 3
-12t -1u 5x -4 20. − -3 5
r 21. − 3
m -2n -5 22. − 4 3 -1
23. − 3 3
5u
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
xy 2
a 4b 6 2. − 3
6f -2g 3h 5 54f g h
(m n )
( ) q -1r 3 qr
25. − -2
-5
2t ux
( j -1k 3) -4 jk
( c dh )
7c -3d 3 26. − 5 -4
Lesson 7-2
8
8 1. − 4
0
4
(3r)
(2a -2b) -3 5a b
24. − 2 4
-1
(
2x 3y 2z 3x yz
27. − 4 -2
)
-2
28. BIOLOGY A lab technician draws a sample of blood. A cubic millimeter of the blood contains 223 white blood cells and 225 red blood cells. What is the ratio of white blood cells to red blood cells?
29. COUNTING The number of three-letter “words” that can be formed with the English alphabet is 263. The number of five-letter “words” that can be formed is 265. How many times more five-letter “words” can be formed than three-letter “words”? Chapter 7
123
North Carolina StudyText, Math A
NAME
7-2
DATE
PERIOD
Word Problem Practice
SCS
MA.N.2.2
Dividing Monomials 1. CHEMISTRY The nucleus of a certain atom is 10-13 centimeters across. If the nucleus of a different atom is 10-11 centimeters across, how many times as large is it as the first atom?
4. METRIC MEASUREMENT Consider a dust mite that measures 10-3 millimeters in length and a caterpillar that measures 10 centimeters long. How many times as long as the mite is the caterpillar?
2. SPACE The Moon is approximately 254 kilometers away from Earth on average. The Olympus Mons volcano on Mars stands 25 kilometers high. How many Olympus Mons volcanoes, stacked on top of one another, would fit between the surface of the Earth and the Moon?
5. COMPUTERS In 1995, standard capacity for a personal computer hard drive was 40 megabytes (MB). In 2010, a standard hard drive capacity was 500 gigabytes (GB or Gig). Refer to the table below.
Memory Capacity Approximate Conversions 8 bits = 1 byte 103 bytes = 1 kilobyte 103 kilobytes = 1 megabyte (meg) 103 megabytes = 1 gigabyte (gig) 103 terabytes = 1 petabyte
a. The newer hard drives have about how many times the capacity of the 1995 drives?
3. E-MAIL Spam (also known as junk e-mail) consists of identical messages sent to thousands of e-mail users. People often obtain anti-spam software to filter out the junk e-mail messages they receive. Suppose Yvonne’s anti-spam software filtered out 102 e-mails, and she received 104 e-mails last year. What fraction of her e-mails were filtered out? Write your answer as a monomial.
Chapter 7
124
b. Predict the hard drive capacity in the year 2025 if this rate of growth continues.
c. One kilobyte of memory is what fraction of one terabyte?
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
103 gigabytes = 1 terabyte
NAME
7-3
DATE
PERIOD
Study Guide
SCS
MA.N.2.2
Scientific Notation Scientific Notation Very large and very small numbers are often best represented using a method known as scientific notation. Numbers written in scientific notation take the form a × 10n, where 1 ≤ a < 10 and n is an integer. Any number can be written in scientific notation. Example 1 Express 34,020,000,000 in scientific notation. Step 1 Move the decimal point until it is to the right of the first nonzero digit. The result is a real number a. Here, a = 3.402.
Example 2 Express 4.11 × 10-6 in standard notation. Step 1 The exponent is –6, so n = –6. Step 2 Because n < 0, move the decimal point 6 places to the left. 4.11 × 10-6 ⇒ .00000411
Step 2 Note the number of places n and the direction that you moved the decimal point. The decimal point moved 10 places to the left, so n = 10.
Step 3 4.11 × 10-6 ⇒ 0.00000411 Rewrite; insert a 0 before the decimal point.
Step 3 Because the decimal moved to the left, write the number as a × 10n. 34,020,000,000 = 3.4020000000 × 1010 Step 4 Remove the extra zeros. 3.402 × 1010
Express each number in scientific notation. 1. 5,100,000
2. 80,300,000,000
3. 14,250,000
4. 68,070,000,000,000
5. 14,000
6. 901,050,000,000
7. 0.0049
8. 0.000301
9. 0.0000000519
10. 0.000000185
11. 0.002002
12. 0.00000771
Express each number in standard form. 13. 4.91 × 104
14. 3.2 × 10-5
15. 6.03 × 108
16. 2.001 × 10-6
17. 1.00024 × 1010
18. 5 × 105
19. 9.09 × 10-5
20. 3.5 × 10-2
21. 1.7087 × 107
Chapter 7
125
North Carolina StudyText, Math A
Lesson 7-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
7-3
Study Guide
PERIOD
SCS
(continued)
MA.N.2.2
Scientific Notation Products and Quotients in Scientific Notation
You can use scientific notation to simplify multiplying and dividing very large and very small numbers. Example 1
Evaluate (9.2 × 10-3) 8 (4 × 10 ). Express the result in both scientific notation and standard form. (9.2 × 10-3)(4 × 108) = (9.2 × 4)(10-3 × 108) = = = =
36.8 × 105 (3.68 × 101) × 105 3.68 × 106 3,680,000
Original Expression Commutative and Associative Properties
Example 2
(2.76 × 10 7) (6.9 × 10 )
Evaluate − . 5
Express the result in both scientific notation and standard form. (2.76 × 10 7) 2.76 10 7 −5 − = − 5 (6.9 × 10 )
( 6.9 )( 10 )
Product rule for fractions
= 0.4 × 102
Quotient of Powers
36.8 = 3.68 × 10
= 4.0 × 10-1 × 102
0.4 = 4.0 × 10-1
Product of Powers
= 4.0 × 101
Product of Powers
= 40
Standard form
Product of Powers
Standard Form
Exercises Evaluate each product. Express the results in both scientific notation and standard form. 2. (2.8 × 10-4)(1.9 × 107)
3. (6.7 × 10-7)(3 × 103)
4. (8.1 × 105)(2.3 × 10-3)
5. (1.2 × 10-11)(6 × 106)
6. (5.9 × 104)(7 × 10-8)
Evaluate each quotient. Express the results in both scientific notation and standard form. (4.9 × 10 -3) (2.5 × 10 )
5.8 × 10 4 8. − -2
(1.6 × 10 5) (4 × 10 )
8.6 × 10 6 10. − -3
(4.2 × 10 -2) (6 × 10 )
8.1 × 10 5 12. − 4
7. − -4
9. − -4
11. − -7
Chapter 7
5 × 10
1.6 × 10
2.7 × 10
126
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. (3.4 × 103)(5 × 104)
NAME
DATE
7-3
PERIOD
Practice
SCS
MA.N.2.2
Scientific Notation Express each number in scientific notation. 1. 1,900,000
2. 0.000704
3. 50,040,000,000
4. 0.0000000661
Express each number in standard form. 5. 5.3 × 107
6. 1.09 × 10-4
7. 9.13 × 103
8. 7.902 × 10-6
Evaluate each product. Express the results in both scientific notation and standard form. 10. (7.5 × 10-5)(3.2 × 107)
11. (2.06 × 104)(5.5 × 10-9)
12. (8.1 × 10-6)(1.96 × 1011)
13. (5.29 × 108)(9.7 × 104)
14. (1.45 × 10-6)(7.2 × 10-5)
Evaluate each quotient. Express the results in both scientific notation and standard form. (4.2 × 10 5) (3 × 10 )
16. − -5
(7.05 × 10 12) (9.4 × 10 )
18. − 5
15. − -3
17. − 7
(1.76 × 10 -11) (2.2 × 10 ) (2.04 × 10 -4) (3.4 × 10 )
19. GRAVITATION Issac Newton’s theory of universal gravitation states that the equation m1m2 F = G− can be used to calculate the amount of gravitational force in newtons 2 r
between two point masses m1 and m2 separated by a distance r. G is a constant equal to 6.67 × 10-11 N m2 kg–2. The mass of the earth m1 is equal to 5.97 × 1024 kg, the mass of the moon m2 is equal to 7.36 × 1022 kg, and the distance r between the two is 384,000,000 m. a. Express the distance r in scientific notation. b. Compute the amount of gravitational force between the earth and the moon. Express your answer in scientific notation. Chapter 7
127
North Carolina StudyText, Math A
Lesson 7-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9. (4.8 × 104)(6 × 106)
NAME
7-3
DATE
PERIOD
Word Problem Practice
SCS
MA.N.2.2
Scientific Notation 1. PLANETS Neptune’s mean distance from the sun is 4,500,000,000 kilometers. Uranus’ mean distance from the sun is 2,870,000,000 kilometers. Express these distances in scientific notation.
5. COAL RESERVES The table below shows the number of kilograms of coal select countries had in proven reserve at the end of 2006.
Coal Reserves, 2006 2. PATHOLOGY The common cold is caused by the rhinovirus, which commonly measures 2 × 10-8 m in diameter. The E. coli bacterium, which causes food poisoning, commonly measures 3 × 10-6 m in length. Express these measurements in standard form.
Country
Coal (kg)
United States
2.46 × 1014
Russia
1.57 × 1014
India
9.24 × 1013
Romania
4.94 × 1011
Source: British Petroleum
a. Express each country’s coal reserves in standard form.
b. How many times more coal does the United States have than Romania?
4. AVOGADRO’S NUMBER Avogadro’s number is an important concept in chemistry. It states that the number 6.022 × 1023 is approximately equal to the number of molecules in 12 grams of carbon 12. Use Avogadro’s number to determine the number of molecules in 5 × 10-7 grams of carbon 12.
Chapter 7
c. One kilogram of coal has an energy density of 2.4 × 107 joules. What is the total energy density of the United States’ coal reserve? Express your answer in scientific notation.
128
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. COMMERCIALS A 30-second commercial aired during the 2007 Super Bowl cost $2,600,000. A 30-second commercial aired during the 1967 Super Bowl cost $40,000. Express these values in scientific notation. How many times more expensive was it to air an advertisement during the 2007 Super Bowl than the 1967 Super Bowl?
NAME
DATE
7-4
PERIOD
Study Guide
SCS
MA.G.2.4
Polynomials Degree of a Polynomial
A polynomial is a monomial or a sum of monomials. A binomial is the sum of two monomials, and a trinomial is the sum of three monomials. Polynomials with more than three terms have no special name. The degree of a monomial is the sum of the exponents of all its variables. The degree of the polynomial is the same as the degree of the monomial term with the highest degree. Example
Determine whether each expression is a polynomial. If so, identify the polynomial as a monomial, binomial, or trinomial. Then find the degree of the polynomial. Expression 3x - 7xyz -25
Polynomial?
binomial
3
monomial
0
none of these
—
trinomial
3
Yes. -25 is a real number. 3 =− , which is not a n4
No. 3n monomial
Yes. The expression simplifies to 9x3 + 7x + 4, which is the sum of three monomials
Exercises Determine whether each expression is a polynomial. If so, identify the polynomial as a monomial, binomial, or trinomial. 1. 36
3 +5 2. − 2
3. 7x - x + 5
4. 8g2h - 7gh + 2
1 + 5y - 8 5. − 2
6. 6x + x2
q
4y
Find the degree of each polynomial. 7. 4x2y3z
9. 15m
8. -2abc
10. r + 5t
11. 22
12. 18x2 + 4yz - 10y
13. x4 - 6x2 - 2x3 - 10
14. 2x3y2 - 4xy3
15. -2r8x4 + 7r2x - 4r7x6
16. 9x2 + yz8
17. 8b + bc5
18. 4x4y - 8zx2 + 2x5
19. 4x2 - 1
20. 9abc + bc - n5
21. h3m + 6h4m2 - 7
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9x3 + 4x + x + 4 + 2x
Degree of the Polynomial
Yes. 3x - 7xyz = 3x + (-7xyz), which is the sum of two monomials
-4
7n3 + 3n-4
Monomial, Binomial, or Trinomial?
NAME
DATE
7-4
Study Guide
PERIOD
SCS
(continued)
MA.G.2.4
Polynomials Write Polynomials in Standard Form
The terms of a polynomial are usually arranged so that the terms are in order from greatest degree to least degree. This is called the standard form of a polynomial. Example
Write -4x2 + 9x4 - 2x in standard form. Identify the leading coefficient.
Step 1: Find the degree of each term. Polynomial: -4x2 + 9x4 - 2x Degree:
2
4
1
Step 2: Write the terms in descending order: 9x4 - 4x2 - 2x. The leading coefficient is 9.
Exercises Write each polynomial in standard form. Identify the leading coefficient. 2. 6x + 9 - 4x2
3. x4 + x3+ x2
4. 2x3 - x + 3x7
5. 2x + x2 - 5
6. 20x - 10x2 + 5x3
7. x3 + x5 - x2
8. x4 + 4x3 - 7x5 + 1
9. -3x6 - x5 + 2x8
10. 2x7 - x8
11. 3x + 5x4 - 2 - x2
12. -2x4 + x - 4x5 + 3
13. 2 - x12
14. 5x4 - 12x - 3x6
15. 9x9 - 9 + 3x3 - 6x6
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1. 5x + x2 + 6
NAME
7-4
DATE
PERIOD
Practice
SCS
MA.G.2.4
Polynomials Determine whether each expression is a polynomial. If so, identify the polynomial as a monomial, binomial, or trinomial. 1 3 2. − y + y2 - 9
1. 7a2b + 3b2 - a2b
3. 6g2h3k
5
Find the degree of each polynomial. 4. x + 3x4 - 21x2 + x3
5. 3g2h3 + g3h
6. -2x2y + 3xy3 + x2
7. 5n3m - 2m3 + n2m4 + n2
8. a3b2c + 2a5c + b3c2
9. 10r2t2 + 4rt2 - 5r3t2
10. 8x2 - 15 + 5x5
11. 10x - 7 + x4 + 4x3
12. 13x2 - 5 + 6x3 - x
13. 4x + 2x5 - 6x3 + 2
GEOMETRY Write a polynomial to represent the area of each shaded region. 14.
b
15.
Lesson 7-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write each polynomial in standard form. Identify the leading coefficient.
b d a
16. MONEY Write a polynomial to represent the value of t ten-dollar bills, f fifty-dollar bills, and h one-hundred-dollar bills. 17. GRAVITY The height above the ground of a ball thrown up with a velocity of 96 feet per second from a height of 6 feet is 6 + 96t - 16t2 feet, where t is the time in seconds. According to this model, how high is the ball after 7 seconds? Explain.
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NAME
7-4
DATE
PERIOD
Word Problem Practice
SCS
MA.G.2.4
Polynomials 1. PRIMES Mei is trying to list as many prime numbers as she can for a challenge problem for her math class. She finds that the polynomial expression n2 - n + 41 can be used to generate some, but not all, prime numbers. What is the degree of Mei’s polynomial?
2. PHONE CALLS A long-distance telephone company charges a $19.95 standard monthly service fee plus $0.05 per minute of long-distance use. Write a polynomial to express the monthly cost of the phone plan if x minutes of longdistance time are used per month. What is the degree of the polynomial?
4. ARCHITECTURE Graphing the polynomial function y = -x2 + 3 produces an accurate drawing of the shape of an archway inside a historical library, where x is the horizontal distance in meters from the base of the arch and y is the height of the arch. At x = 0, what is the height of the arch?
5. DRIVING A truck and a car leave an intersection. The truck travels south, and the car travels east. When the truck had gone 24 miles, the distance between the car and truck was four miles more than three times the distance traveled by the car heading east.
A
B
3. COSTUMES Jack’s mother is sewing the cape of his costume for a charity masked ball. The pattern for the cape (lying flat) is shown below. The radius of the neck hole is 6 inches. What is the area, in square feet, of the finished cape?
Fabric
North
C
a. Suppose the truck stops at point C and the car stops at point B. Write a polynomial in standard form to express the sum of the distances traveled by the car and the truck.
Neck hole 3 ft
b. Write a simplified polynomial to express the perimeter of triangle ABC.
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24 mi
x mi
NAME
7-5
DATE
PERIOD
Study Guide
SCS
MA.A.1.1, MA.A.1.2
Adding and Subtracting Polynomials Add Polynomials To add polynomials, you can group like terms horizontally or write them in column form, aligning like terms vertically. Like terms are monomial terms that are either identical or differ only in their coefficients, such as 3p and -5p or 2x2y and 8x2y. Example 1
Example 2
Find (2x2 + x - 8) + (3x - 4x2 + 2).
Horizontal Method Group like terms. (2x2 + x - 8) + (3x - 4x2 + 2) = [(2x2 + (-4x2)] + (x + 3x) + [(-8) + 2)] = -2x2 + 4x - 6. The sum is -2x2 + 4x - 6.
Find (3x2 + 5xy) + (xy + 2x2).
Vertical Method Align like terms in columns and add. 3x2 + 5xy (+) 2x2 + xy Put the terms in descending order. 2 5x + 6xy The sum is 5x2 + 6xy.
Exercises 1. (4a - 5) + (3a + 6)
2. (6x + 9) + (4x2 - 7)
3. (6xy + 2y + 6x) + (4xy - x)
4. (x2 + y2) + (-x2 + y2)
5. (3p2 - 2p + 3) + (p2 - 7p + 7)
6. (2x2 + 5xy + 4y2) + (-xy - 6x2 + 2y2)
7. (5p + 2q) + (2p2 - 8q + 1)
8. (4x2 - x + 4) + (5x + 2x2 + 2)
9. (6x2 + 3x) + (x2 - 4x - 3)
10. (x2 + 2xy + y2) + (x2 - xy - 2y2)
11. (2a - 4b - c) + (-2a - b - 4c)
12. (6xy2 + 4xy) + (2xy - 10xy2 + y2)
13. (2p - 5r) + (3p + 6r) + ( p - r)
14. (2x2 - 6) + (5x2 + 2) + (-x2 - 7)
15. (3z2 + 5z) + (z2 + 2z) + (z - 4)
16. (8x2 + 4x + 3y2 + y) + (6x2 - x + 4y)
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Find each sum.
NAME
7-5
DATE
Study Guide
PERIOD
SCS
(continued)
MA.A.1.1, MA.A.1.2
Adding and Subtracting Polynomials Subtract Polynomials
You can subtract a polynomial by adding its additive inverse. To find the additive inverse of a polynomial, replace each term with its additive inverse or opposite. Example
Find (3x2 + 2x - 6) - (2x + x2 + 3). Vertical Method Align like terms in columns and subtract by adding the additive inverse. 3x2 + 2x - 6 (-) x2 + 2x + 3 3x2 + 2x - 6 (+) -x2 - 2x - 3 2x2 -9 The difference is 2x2 - 9.
Horizontal Method Use additive inverses to rewrite as addition. Then group like terms. (3x2 + 2x - 6) - (2x + x2 + 3) = (3x2 + 2x - 6) + [(-2x)+ (-x2) + (-3)] = [3x2 + (-x2)] + [2x + (-2x)] + [-6 + (-3)] = 2x2 + (-9) = 2x2 - 9 The difference is 2x2 - 9.
Exercises Find each difference. 2. (9x + 2) - (-3x2 - 5)
3. (9xy + y - 2x) - (6xy - 2x)
4. (x2 + y2) - (-x2 + y2)
5. (6p2 + 4p + 5) - (2p2 - 5p + 1)
6. (6x2 + 5xy - 2y2) - (-xy - 2x2 - 4y2)
7. (8p - 5r) - (-6p2 + 6r - 3)
8. (8x2 - 4x - 3) - (-2x - x2 + 5)
9. (3x2 - 2x) - (3x2 + 5x - 1)
10. (4x2 + 6xy + 2y2) - (-x2 + 2xy - 5y2)
11. (2h - 6j - 2k) - (-7h - 5j - 4k)
12. (9xy2 + 5xy) - (-2xy - 8xy2)
13. (2a - 8b) - (-3a + 5b)
14. (2x2 - 8) - (-2x2 - 6)
15. (6z2 + 4z + 2) - (4z2 + z)
16. (6x2 - 5x + 1) - (-7x2 - 2x + 4)
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1. (3a - 5) - (5a + 1)
NAME
7-5
DATE
PERIOD
Practice
SCS
MA.A.1.1, MA.A.1.2
Adding and Subtracting Polynomials 1. (4y + 5) + (-7y - 1)
2. (-x2 + 3x) - (5x + 2x2)
3. (4k2 + 8k + 2) - (2k + 3)
4. (2m2 + 6m) + (m2 - 5m + 7)
5. (5a2 + 6a + 2) - (7a2 - 7a + 5)
6. (-4p2 - p + 9) + ( p2 + 3p - 1)
7. (x3 - 3x + 1) - (x3 + 7 - 12x)
8. (6x2 - x + 1) - (-4 + 2x2 + 8x)
9. (4y2 + 2y - 8) - (7y2 + 4 - y)
10. (w2 - 4w - 1) + (-5 + 5w2 - 3w)
11. (4u2 - 2u - 3) + (3u2 - u + 4)
12. (5b2 - 8 + 2b) - (b + 9b2 + 5)
13. (4d2 + 2d + 2) + (5d2 - 2 - d)
14. (8x2 + x - 6) - (-x2 + 2x - 3)
15. (3h2 + 7h - 1) - (4h + 8h2 + 1)
16. (4m2 - 3m + 10) + (m2 + m - 2)
17. (x2 + y2 - 6) - (5x2 - y2 - 5)
18. (7t2 + 2 - t) + (t2 - 7 - 2t)
19. (k3 - 2k2 + 4k + 6) - (-4k + k2 - 3)
20. (9j 2 + j + jk) + (-3j 2 - jk - 4j)
21. (2x + 6y - 3z) + (4x + 6z - 8y) + (x - 3y + z)
22. (6f 2 - 7f - 3) - (5f 2 - 1 + 2f) - (2f 2 - 3 + f)
23. BUSINESS The polynomial s3 - 70s2 + 1500s - 10,800 models the profit a company makes on selling an item at a price s. A second item sold at the same price brings in a profit of s3 - 30s2 + 450s - 5000. Write a polynomial that expresses the total profit from the sale of both items. 24. GEOMETRY The measures of two sides of a triangle are given. If P is the perimeter, and P = 10x + 5y, find the measure of the third side.
3x + 4y
5x - y
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Lesson 7-5
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Find each sum or difference.
NAME
7-5
DATE
Word Problem Practice
PERIOD
SCS
MA.A.1.1, MA.A.1.2
Adding and Subtracting Polynomials 1. BUILDING Find the simplest expression for the perimeter of the triangular roof truss. 3a + 4
2a - 7
2
5a + a
2. GEOMETRY Write a polynomial to show the area of the large square below.
5. INDUSTRY Two identical right cylindrical steel drums containing oil need to be covered with a fire-resistant sealant. In order to determine how much sealant to purchase, George must find the surface area of the two drums. The surface area (including the top and bottom bases) is given by the following formula. S = 2πrh + 2πr 2 r
h
b
a. Write a polynomial to represent the total surface area of the two drums. a
b
3. FIREWORKS Two bottle rockets are launched straight up into the air. The height, in feet, of each rocket at t seconds after launch is given by the polynomial equations below. Write an equation to show how much higher Rocket A traveled. Rocket A: H1 = -16t2 + 122t Rocket B: H1 = -16t2 + 84t
c. The fire resistant sealant must be applied while they are stacked vertically in groups of three. If h is the height of each drum and r is the radius, write a polynomial to represent the exposed surface area.
4. ENVELOPES An office supply company produces yellow document envelopes. The envelopes come in a variety of sizes, but the length is always 4 centimeters more than double the width. Write a polynomial expression to give the perimeter of any of the envelopes.
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a
b. Find the total surface area if the height of each drum is 2 meters and the radius of each is 0.5 meter. Let π = 3.14.
NAME
7-6
DATE
PERIOD
Study Guide
SCS
MA.A.1.1, MA.A.1.2
Polynomial Multiplied by Monomial
The Distributive Property can be used to multiply a polynomial by a monomial. You can multiply horizontally or vertically. Sometimes multiplying results in like terms. The products can be simplified by combining like terms. Example 1
Example 2
Find -3x2(4x2 + 6x - 8).
Horizontal Method -3x2(4x2 + 6x - 8) = -3x2(4x2) + (-3x2)(6x) - (-3x2)(8) = -12x4 + (-18x3) - (-24x2) = -12x4 - 18x3 + 24x2 Vertical Method 4x2 + 6x - 8 (×) -3x2 -12x4 - 18x3 + 24x2 The product is -12x4 - 18x3 + 24x2.
Simplify -2(4x2 + 5x) - x(x2 + 6x).
-2(4x2 + 5x) - x( x2 + 6x) = -2(4x2) + (-2)(5x) + (-x)(x2) + (-x)(6x) = -8x2 + (-10x) + (-x3) + (-6x2) = (-x3) + [-8x2 + (-6x2)] + (-10x) = -x3 - 14x2 - 10x
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find each product. 1. x(5x + x2)
2. x(4x2 + 3x + 2)
3. -2xy(2y + 4x2)
4. -2g(g2 - 2g + 2)
5. 3x(x4 + x3+ x2)
6. -4x(2x3 - 2x + 3)
7. -4ax(10 + 3x)
8. 3y(-4x - 6x3- 2y)
9. 2x2y2(3xy + 2y + 5x)
Simplify each expression. 10. x(3x - 4) - 5x
11. -x(2x2 - 4x) - 6x2
12. 6a(2a - b) + 2a(-4a + 5b)
13. 4r(2r2 - 3r + 5) + 6r(4r2 + 2r + 8)
14. 4n(3n2 + n - 4) - n(3 - n)
15. 2b(b2 + 4b + 8) - 3b(3b2 + 9b - 18)
16. -2z(4z2 - 3z + 1) - z(3z2 + 2z - 1)
17. 2(4x2 - 2x) - 3(-6x2 + 4) + 2x(x - 1)
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Lesson 7-6
Multiplying a Polynomial by a Monomial
NAME
7-6
DATE
Study Guide
PERIOD
SCS
(continued)
MA.A.1.1, MA.A.1.2
Multiplying a Polynomial by a Monomial Solve Equations with Polynomial Expressions Many equations contain polynomials that must be added, subtracted, or multiplied before the equation can be solved. Example
Solve 4(n - 2) + 5n = 6(3 - n) + 19.
4(n - 2) + 5n = 6(3 - n) + 19 4n - 8 + 5n = 18 - 6n + 19 9n - 8 = 37 - 6n 15n - 8 = 37 15n = 45 n=3 The solution is 3.
Original equation Distributive Property Combine like terms. Add 6n to both sides. Add 8 to both sides. Divide each side by 15.
Exercises Solve each equation. 2. 3(x + 5) - 6 = 18
3. 3x(x - 5) - 3x2 = -30
4. 6(x2 + 2x) = 2(3x2 + 12)
5. 4(2p + 1) - 12p = 2(8p + 12)
6. 2(6x + 4) + 2 = 4(x - 4)
7. -2(4y - 3) - 8y + 6 = 4(y - 2)
8. x(x + 2) - x(x - 6) = 10x - 12
9. 3(x2 - 2x) = 3x2 + 5x - 11
10. 2(4x + 3) + 2 = -4(x + 1)
11. 3(2h - 6) - (2h + 1) = 9
12. 3(y + 5) - (4y - 8) = -2y + 10
13. 3(2a - 6) - (-3a - 1) = 4a - 2
14. 5(2x2 - 1) - (10x2 - 6) = -(x + 2)
15. 3(x + 2) + 2(x + 1) = -5(x - 3)
16. 4(3p2 + 2p) - 12p2 = 2(8p + 6)
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1. 2(a - 3) = 3(-2a + 6)
NAME
DATE
7-6
Practice
PERIOD
SCS
MA.A.1.1, MA.A.1.2
Multiplying a Polynomial by a Monomial Lesson 7-6
Find each product. 1. 2h(-7h2 - 4h)
2. 6pq(3p2 + 4q)
3. 5jk(3jk + 2k)
4. -3rt(-2t2 + 3r)
1 5. - − m(8m 2 + m - 7)
2 2 6. - − n (-9n 2 + 3n + 6)
4
3
Simplify each expression. 7. -2(3 - 4) + 7 9. 6t(2t - 3) - 5(2t2 + 9t - 3)
8. 5w(-7w + 3) + 2w(-2w2 + 19w + 2) 10. -2(3m3 + 5m + 6) + 3m(2m2 + 3m + 1)
11. -3g(7g - 2) + 3( g2 + 2g + 1) - 3g(-5g + 3)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation. 12. 5(2t - 1) + 3 = 3(3t + 2)
13. 3(3u + 2) + 5 = 2(2u - 2)
14. 4(8n + 3) - 5 = 2(6n + 8) + 1
15. 8(3b + 1) = 4(b + 3) - 9
16. t(t + 4) - 1 = t(t + 2) + 2
17. u(u - 5) + 8u = u(u + 2) - 4
18. NUMBER THEORY Let x be an integer. What is the product of twice the integer added to three times the next consecutive integer? 19. INVESTMENTS Kent invested $5000 in a retirement plan. He allocated x dollars of the money to a bond account that earns 4% interest per year and the rest to a traditional account that earns 5% interest per year. a. Write an expression that represents the amount of money invested in the traditional account. b. Write a polynomial model in simplest form for the total amount of money T Kent has invested after one year. (Hint: Each account has A + IA dollars, where A is the original amount in the account and I is its interest rate.) c. If Kent put $500 in the bond account, how much money does he have in his retirement plan after one year?
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NAME
7-6
DATE
Word Problem Practice
PERIOD
SCS
MA.A.1.1, MA.A.1.2
Multiplying a Polynomial by a Monomial 1. NUMBER THEORY The sum of the first n whole numbers is given by the 1 (n 2 + n). Expand the expression − 2 equation by multiplying, then find the sum of the first 12 whole numbers.
2. COLLEGE Troy’s boss gave him $700 to start his college savings account. Troy’s boss also gives him $40 each month to add to the account. Troy’s mother gives him $50 each month, but has been doing so for 4 fewer months than Troy’s boss. Write a simplified expression for the amount of money Troy has received from his boss and mother after m months.
5. GEOMETRY Some monuments are constructed as rectangular pyramids. The volume of a pyramid can be found by multiplying the area of its base B by one third of its height. The area of the rectangular base of a monument in a local park is given by the polynomial equation B = x2 - 4x - 12.
h
Circle of flags
a. Write a polynomial equation to represent V, the volume of a rectangular pyramid if its height is 10 centimeters.
r
Sidewalk
b. Find the volume of the pyramid if x = 12.
Write an equation that equates the outside circumference of the sidewalk to 1.10 times the circumference of the circle of flags. Solve the equation for the radius of the circle of flags. Recall that circumference of a circle is 2πr.
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3. LANDMARKS A circle of 50 flags surrounds the Washington Monument. Suppose a new sidewalk 12 feet wide is installed just around the outside of the circle of flags. The outside circumference of the sidewalk is 1.10 times the circumference of the circle of flags.
4. MARKET Sophia went to the farmers’ market to purchase some vegetables. She bought peppers and potatoes. The peppers were $0.39 each and the potatoes were $0.29 each. She spent $3.88 on vegetables, and bought 4 more potatoes than peppers. If x = the number of peppers, write and solve an equation to find out how many of each vegetable Sophia bought.
NAME
7-7
DATE
PERIOD
Study Guide
SCS
MA.A.1.1
Multiplying Polynomials Multiply Binomials To multiply two binomials, you can apply the Distributive Property twice. A useful way to keep track of terms in the product is to use the FOIL method as illustrated in Example 2. Find (x + 3)(x - 4).
Horizontal Method (x + 3)(x - 4) = x(x - 4) + 3(x - 4) = (x)(x) + x(-4) + 3(x)+ 3(-4) = x2 - 4x + 3x - 12 = x2 - x - 12 Vertical Method x+ 3 (×) x - 4 -4x - 12 2 x + 3x x2 - x - 12 The product is x2 - x - 12.
Example 2 Find (x - 2)(x + 5) using the FOIL method. (x - 2)(x + 5) First
Outer
Inner
Last
= (x)(x) + (x)(5) + (-2)(x) + (-2)(5) = x2 + 5x + (-2x) - 10 = x2 + 3x - 10 The product is x2 + 3x - 10.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises Find each product. 1. (x + 2)(x + 3)
2. (x - 4)(x + 1)
3. (x - 6)(x - 2)
4. ( p - 4)( p + 2)
5. ( y + 5)( y + 2)
6. (2x - 1)(x + 5)
7. (3n - 4)(3n - 4)
8. (8m - 2)(8m + 2)
9. (k + 4)(5k - 1)
10. (3x + 1)(4x + 3)
11. (x - 8)(-3x + 1)
12. (5t + 4)(2t - 6)
13. (5m - 3n)(4m - 2n)
14. (a - 3b)(2a - 5b)
15. (8x - 5)(8x + 5)
16. (2n - 4)(2n + 5)
17. (4m - 3)(5m - 5)
18. (7g - 4)(7g + 4)
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Lesson 7-7
Example 1
NAME
7-7
DATE
Study Guide
PERIOD
SCS
(continued)
MA.A.1.1
Multiplying Polynomials Multiply Polynomials
The Distributive Property can be used to multiply any
two polynomials. Example
Find (3x + 2)(2x2 - 4x + 5).
(3x + 2)(2x2 - 4x + 5) = 3x(2x2 - 4x + 5) + 2(2x2 - 4x + 5) = 6x3 - 12x2 + 15x + 4x2 - 8x + 10 = 6x3 - 8x2 + 7x + 10 The product is 6x3 - 8x2 + 7x + 10.
Distributive Property Distributive Property Combine like terms.
Exercises Find each product. 2. (x + 3)(2x2 + x - 3)
3. (2x - 1)(x2 - x + 2)
4. (p - 3)(p2 - 4p + 2)
5. (3k + 2)(k2 + k - 4)
6. (2t + 1)(10t2 - 2t - 4)
7. (3n - 4)(n2 + 5n - 4)
8. (8x - 2)(3x2 + 2x - 1)
9. (2a + 4)(2a2 - 8a + 3)
10. (3x - 4)(2x2 + 3x + 3)
11. (n2 + 2n - 1)(n2 + n + 2)
12. (t2 + 4t - 1)(2t2 - t - 3)
13. (y2 - 5y + 3)(2y2 + 7y - 4)
14. (3b2 - 2b + 1)(2b2 - 3b - 4)
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1. (x + 2)(x2 - 2x + 1)
NAME
7-7
DATE
PERIOD
Practice
SCS
MA.A.1.1
Multiplying Polynomials 1. (q + 6)(q + 5)
2. (x + 7)(x + 4)
3. (n - 4)(n - 6)
4. (a + 5)(a - 6)
5. (4b + 6)(b - 4)
6. (2x - 9)(2x + 4)
7. (6a - 3)(7a - 4)
8. (2x - 2)(5x - 4)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9. (3a - b)(2a - b)
10. (4g + 3h)(2g + 3h)
11. (m + 5)(m2 + 4m - 8)
12. (t + 3)(t2 + 4t + 7)
13. (2h + 3)(2h2 + 3h + 4)
14. (3d + 3)(2d2 + 5d - 2)
15. (3q + 2)(9q2 - 12q + 4)
16. (3r + 2)(9r2 + 6r + 4)
17. (3n2 + 2n - 1)(2n2 + n + 9)
18. (2t2 + t + 3)(4t2 + 2t - 2)
19. (2x2 - 2x - 3)(2x2 - 4x + 3)
20. (3y2 + 2y + 2)(3y2 - 4y - 5)
GEOMETRY Write an expression to represent the area of each figure. 5x - 4
22.
21. 2x - 2
x+1 4x + 2
3x + 2
23. NUMBER THEORY Let x be an even integer. What is the product of the next two consecutive even integers? 24. GEOMETRY The volume of a rectangular pyramid is one third the product of the area of its base and its height. Find an expression for the volume of a rectangular pyramid whose base has an area of 3x2 + 12x + 9 square feet and whose height is x + 3 feet.
Chapter 7
143
North Carolina StudyText, Math A
Lesson 7-7
Find each product.
NAME
7-7
DATE
PERIOD
Word Problem Practice
SCS
MA.A.1.1
Multiplying Polynomials 1. THEATER The Loft Theater has a center seating section with 3c + 8 rows and 4c – 1 seats in each row. Write an expression for the total number of seats in the center section.
2. CRAFTS Suppose a quilt made up of squares has a length-to-width ratio of 5 to 4. The length of the quilt is 5x inches. The quilt can be made slightly larger by adding a border of 1-inch squares all the way around the perimeter of the quilt. Write a polynomial expression for the area of the larger quilt.
5. ART The museum where Julia works plans to have a large wall mural replica of Vincent van Gogh’s The Starry Night painted in its lobby. First, Julia wants to paint a large frame around where the mural will be. The mural’s length will be 5 feet longer than its width, and the frame will be 2 feet wide on all sides. Julia has only enough paint to cover 100 square feet of wall surface. How large can the mural be?
Mural
Painted frame
a. Write an expression for the area of the mural.
b. Write an expression for the area of the frame.
c. Write and solve an equation to find how large the mural can be.
5 x-— 2
x+5 Source: American Flag Store
Chapter 7
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. SERVICE A folded United States flag is sometimes presented to individuals in recognition of outstanding service to the country. The flag is presented folded in a triangle. Often the recipient purchases a case designed to display the folded flag to protect it from wear. One such display case has dimensions (in inches) shown below. Write a polynomial expression that represents the area of wall space covered by the display case.
4. MATH FUN Think of a whole number. Subtract 2. Write down this number. Take the original number and add 2. Write down this number. Find the product of the numbers you wrote down. Subtract the square of the original number. The result is always –4. Use polynomials to show how this number trick works.
NAME
7-8
DATE
Study Guide
PERIOD
SCS
MA.A.1.1, MA.A.1.2
Special Products Squares of Sums and Differences
Some pairs of binomials have products that follow specific patterns. One such pattern is called the square of a sum. Another is called the square of a difference. Square of a Sum
(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2
Square of a Difference
(a - b)2 = (a - b)(a - b) = a2 - 2ab + b2
Example 1
Example 2
Find (3a + 4)(3a + 4).
Use the square of a sum pattern, with a = 3a and b = 4. (3a + 4)(3a + 4) = (3a)2 + 2(3a)(4) + (4)2 = 9a2 + 24a + 16 The product is 9a2 + 24a + 16.
Find (2z - 9)(2z - 9).
Use the square of a difference pattern with a = 2z and b = 9. (2z - 9)(2z - 9) = (2z)2 - 2(2z)(9) + (9)(9) = 4z2 - 36z + 81 The product is 4z2 - 36z + 81.
Lesson 7-8
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find each product. 1. (x - 6)2
2. (3p + 4)2
3. (4x - 5)2
4. (2x - 1)2
5. (2h + 3)2
6. (m + 5)2
7. (a + 3)2
8. (3 - p)2
9. (x - 5y)2
10. (8y + 4)2
11. (8 + x)2
12. (3a - 2b)2
13. (2x - 8)2
14. (x2 + 1)2
15. (m2 - 2)2
16. (x3 - 1)2
17. (2h2 - k2)2
1 18. − x+3
19. (x - 4y2)2
20. (2p + 4r)2
2 21. − x-2
Chapter 7
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(4
)
2
(3
)
2
North Carolina StudyText, Math A
NAME
7-8
DATE
Study Guide
PERIOD
SCS
(continued)
MA.A.1.1, MA.A.1.2
Special Products Product of a Sum and a Difference
There is also a pattern for the product of a sum and a difference of the same two terms, (a + b)(a - b). The product is called the difference of squares. Product of a Sum and a Difference
Example
(a
+ b)(a - b) = a2 - b2
Find (5x + 3y)(5x - 3y).
(a + b)(a - b) = a2 - b2 (5x + 3y)(5x - 3y) = (5x)2 - (3y)2 = 25x2 - 9y2 The product is 25x2 - 9y2.
Product of a Sum and a Difference a = 5x and b = 3y Simplify.
Exercises Find each product. 2. ( p + 2)( p - 2)
3. (4x - 5)(4x + 5)
4. (2x - 1)(2x + 1)
5. (h + 7)(h - 7)
6. (m - 5)(m + 5)
7. (2d - 3)(2d + 3)
8. (3 - 5q)(3 + 5q)
9. (x - y)(x + y)
10. ( y - 4x)( y + 4x)
11. (8 + 4x)(8 - 4x)
12. (3a - 2b)(3a + 2b)
13. (3y - 8)(3y + 8)
14. (x2 - 1)(x2 + 1)
15. (m2 - 5)(m2 + 5)
16. (x3 - 2)(x3 + 2)
17. (h2 - k2)(h2 + k2)
1 1 18. − x+2 − x-2
19. (3x - 2y2)(3x + 2y2)
20. (2p - 5r)(2p + 5r)
4 4 21. − x - 2y − x + 2y
Chapter 7
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)( 4
(3
)( 3
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1. (x - 4)(x + 4)
)
)
North Carolina StudyText, Math A
NAME
7-8
DATE
Practice
PERIOD
SCS
MA.A.1.1, MA.A.1.2
Special Products
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. (n + 9)2
2. (q + 8)2
3. (x - 10)2
4. (r - 11)2
5. ( p + 7)2
6. (b + 6)(b - 6)
7. (z + 13)(z - 13)
8. (4j + 2)2
9. (5w - 4)2
10. (6h - 1)2
11. (3m + 4)2
12. (7v - 2)2
13. (7k + 3)(7k - 3)
14. (4d - 7)(4d + 7)
15. (3g + 9h)(3g - 9h)
16. (4q + 5t)(4q - 5t)
17. (a + 6u)2
18. (5r + s)2
19. (6h - m)2
20. (k - 6y)2
21. (u - 7p)2
22. (4b - 7v)2
23. (6n + 4p)2
24. (5q + 6t)2
25. (6a - 7b)(6a + 7b)
26. (8h + 3d)(8h - 3d)
27. (9x + 2y2)2
28. (3p3 + 2m)2
29. (5a2 - 2b)2
30. (4m3 - 2t)2
31. (6b3 - g)2
32. (2b2 - g)(2b2 + g)
33. (2v2 + 3x2)(2v2 + 3x2)
34. GEOMETRY Janelle wants to enlarge a square graph that she has made so that a side of the new graph will be 1 inch more than twice the original side g. What trinomial represents the area of the enlarged graph? 35. GENETICS In a guinea pig, pure black hair coloring B is dominant over pure white coloring b. Suppose two hybrid Bb guinea pigs, with black hair coloring, are bred. a. Find an expression for the genetic make-up of the guinea pig offspring. b. What is the probability that two hybrid guinea pigs with black hair coloring will produce a guinea pig with white hair coloring? Chapter 7
147
North Carolina StudyText, Math A
Lesson 7-8
Find each product.
NAME
DATE
7-8
PERIOD
Word Problem Practice
SCS
MA.A.1.1, MA.A.1.2
Special Products 1. PROBABILITY The spinner below is divided into 2 equal sections. If you spin the spinner 2 times in a row, the possible outcomes are shown in the table below.
RED BLUE
red red
blue red
red blue
blue blue
What is the probability of spinning a blue and a red in two spins?
4. BUSINESS The Combo Lock Company finds that its profit data from 2005 to the present can be modeled by the function y = 4n2 + 44n + 121, where y is the profit n years since 2005. Which special product does this polynomial demonstrate? Explain.
5. STORAGE A cylindrical tank is placed along a wall. A cylindrical PVC pipe will be hidden in the corner behind the tank. See the side view diagram below. The radius of the tank is r inches and the radius of the PVC pipe is s inches.
2. GRAVITY The height of a penny t seconds after being dropped down a well is given by the product of (10 – 4t) and (10 + 4t). Find the product and simplify. What type of special product does this represent?
Wall
r+s r-s
r-s Floor
a. Use the Pythagorean Theorem to write an equation for the relationship between the two radii. Simplify your equation so that there is a zero on one side of the equals sign.
b. Write a polynomial equation you could solve to find the radius s of the PVC pipe if the radius of the tank is 20 inches.
Lincoln Memorial
Chapter 7
Pipe
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North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. TRAFFIC PLANNING The Lincoln Memorial in Washington, D.C., is surrounded by a circular drive called Lincoln Circle. Suppose the National Park Service wants to change the layout of Lincoln Circle so that there are two concentric circular roads. Write a polynomial equation for the area A of the space between the roads if the radius of the smaller road is 10 meters less than the radius of the larger road.
Tank
NAME
8-2
DATE
PERIOD
Study Guide
SCS
MA.A.1.3
Using the Distributive Property Use the Distributive Property to Factor
The Distributive Property has been used to multiply a polynomial by a monomial. It can also be used to express a polynomial in factored form. Compare the two columns in the table below. Multiplying
Factoring
3(a + b) = 3a + 3b
3a + 3b = 3(a + b)
x(y - z) = xy - xz
xy - xz = x(y - z)
6y(2x + 1) = 6y(2x) + 6y(1) = 12xy + 6y
12xy + 6y = 6y(2x) + 6y(1) = 6y(2x + 1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2
Use the Distributive Property to factor 12mp + 80m2.
Factor 6ax + 3ay + 2bx + by by grouping.
Find the GCF of 12mp and 80m2. 12mp = 2 2 3 m p 80m2 = 2 2 2 2 5 m m GCF = 2 2 m or 4m Write each term as the product of the GCF and its remaining factors. 12mp + 80m2 = 4m(3 p) + 4m(2 2 5 m) = 4m(3p) + 4m(20m) = 4m(3p + 20m) Thus, 12mp + 80m2 = 4m(3p + 20m).
6ax + 3ay + 2bx + by = (6ax + 3ay) + (2bx + by) = 3a(2x + y) + b(2x + y) = (3a + b)(2x + y) Check using the FOIL method. (3a + b)(2x + y) = 3a(2x) + (3a)(y) + (b)(2x) + (b)(y) = 6ax + 3ay + 2bx + by ✓
Exercises Factor each polynomial. 1. 24x + 48y
2. 30mp2 + m2p - 6p
3. q4 - 18q3 + 22q
4. 9x2 - 3x
5. 4m + 6p - 8mp
6. 45r3 - 15r2
7. 14t3 - 42t5 - 49t4
8. 55p2 - 11p4 + 44p5
9. 14y3 - 28y2 + y
10. 4x + 12x2 + 16x3
11. 4a2b + 28ab2 + 7ab
12. 6y + 12x - 8z
13. x2 + 2x + x + 2
14. 6y2 - 4y + 3y - 2
15. 4m2 + 4mp + 3mp + 3p2
16. 12ax + 3xz + 4ay + yz
17. 12a2 + 3a - 8a - 2
18. xa + ya + x + y
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North Carolina StudyText, Math A
Lesson 8-2
Example 1
NAME
DATE
8-2
Study Guide
PERIOD
SCS
(continued)
MA.A.1.3
Using the Distributive Property Solve Equations by Factoring
The following property, along with factoring, can be
used to solve certain equations. Zero Product Property
For any real numbers a and b, if ab = 0, then either a = 0, b = 0, or both a and b equal 0.
Solve 9x2 + x = 0. Then check the solutions.
Example
Write the equation so that it is of the form ab = 0. 9x2 + x = 0 Original equation x(9x + 1) = 0 Factor the GCF of 9x2 + x, which is x. x = 0 or 9x + 1 = 0 Zero Product Property 1 x = 0 x = -− Solve each equation. 9
{
}
1 The solution set is 0, - − . 9
Check Substitute 0 and - −19 for x in the original equation. 9x2 + x = 0 9(0)2 + 0 0 0=0✓
9x2 + x = 0 2
( 9 ) + (- −19 ) 0 1 1 − + (- − 0 9 9)
1 9 -−
0=0✓
Solve each equation. Check your solutions. 1. x(x + 3) = 0
2. 3m(m - 4) = 0
3. (r - 3)(r + 2) = 0
4. 3x(2x - 1) = 0
5. (4m + 8)(m - 3) = 0
6. 5t2 = 25t
7. (4c + 2)(2c - 7) = 0
8. 5p - 15p2 = 0
9. 4y2 = 28y
10. 12x2 = -6x
11. (4a + 3)(8a + 7) = 0
12. 8y = 12y2
13. x2 = -2x
14. (6y - 4)(y + 3) = 0
15. 4m2 = 4m
16. 12x = 3x2
17. 12a2 = -3a
18. (12a + 4)(3a - 1) = 0
Chapter 8
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
8-2
DATE
PERIOD
Practice
SCS
MA.A.1.3
Using the Distributive Property 1. 64 - 40ab
2. 4d2 + 16
3. 6r2t - 3rt2
4. 15ad + 30a2d2
5. 32a2 + 24b2
6. 36xy2 - 48x2y
7. 30x3y + 35x2y2
8. 9a3d2 - 6ad3
9. 75b2g3 + 60bg3
10. 8p2r2 - 24pr3 + 16pr
11. 5x3y2 + 10x2y + 25x
12. 9ax3 + 18bx2 + 24cx
13. x2 + 4x + 2x + 8
14. 2a2 + 3a + 6a + 9
15. 4b2 - 12b + 2b - 6
16. 6xy - 8x + 15y - 20
17. -6mp + 4m + 18p - 12
18. 12a2 - 15ab - 16a + 20b
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation. Check your solutions. 19. x(x - 32) = 0
20. 4b(b + 4) = 0
21. (y - 3)(y + 2) = 0
22. (a + 6)(3a - 7) = 0
23. (2y + 5)(y - 4) = 0
24. (4y + 8)(3y - 4) = 0
25. 2z2 + 20z = 0
26. 8p2 - 4p = 0
27. 9x2 = 27x
28. 18x2 = 15x
29. 14x2 = -21x
30. 8x2 = -26x
31. LANDSCAPING A landscaping company has been commissioned to design a triangular flower bed for a mall entrance. The final dimensions of the flower bed have not been determined, but the company knows that the height will be two feet less than the base. 1 2 b - b. The area of the flower bed can be represented by the equation A = − 2
a. Write this equation in factored form. b. Suppose the base of the flower bed is 16 feet. What will be its area? 32. PHYSICAL SCIENCE Mr. Alim’s science class launched a toy rocket from ground level with an initial upward velocity of 60 feet per second. The height h of the rocket in feet above the ground after t seconds is modeled by the equation h = 60t - 16t2. How long was the rocket in the air before it returned to the ground? Chapter 8
151
North Carolina StudyText, Math A
Lesson 8-2
Factor each polynomial.
NAME
8-2
DATE
Word Problem Practice
PERIOD
SCS
MA.A.1.3
Using the Distributive Property 1. PHYSICS According to legend, Galileo dropped objects of different weights from the so-called “leaning tower” of Pisa while developing his formula for free falling objects. The relationship that he discovered was that the distance d an object falls after t seconds is given by d = 16t2 (ignoring air resistance). This relationship can be found in the equation h = 4t - 16t2, where h is the height of an object thrown upward from ground level at a rate of 32 feet per second. Solve the equation for h = 0.
2. SWIMMING POOL The area A of a rectangular swimming pool is given by the equation A = 12w - w2, where w is the width of one side. Write an expression for the other side of the pool.
4. VERTICAL JUMP Your vertical jump height is measured by subtracting your standing reach height from the height of the highest point you can reach by jumping without taking a running start. Typically, NBA players have vertical jump heights of up to 34 inches. If an NBA player jumps this high, his height h in inches above his standing reach height after t seconds can be modeled by the equation h = 162t - 192t2. Solve the equation for h = 0 and interpret the solution. Round your answer to the nearest hundredth.
a. Assuming the dog doesn’t jump, after how many seconds does the dog catch the treat?
3. CONSTRUCTION Unique Building Company is constructing a triangular roof truss for a building. The workers assemble the truss with the dimensions shown on the diagram below. Using the Pythagorean Theorem, find the length of the sides of the truss.
b. The dog treat reaches its maximum height halfway between when it was thrown and when it was caught. What is its maximum height?
2x - 1 yd x yd
x + 1 yd
Chapter 8
c. How fast would Connor have to throw the dog treat in order to make it fly through the air for 6 seconds?
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5. PETS Conner tosses a dog treat upward with an initial velocity of 13.7 meters per second. The height of the treat above the dog’s mouth h in meters after t seconds is given by the equation. h = 13.7t - 4.9t2
NAME
DATE
8-3
PERIOD
Study Guide
SCS
MA.A.1.3
Quadratic Equations: x2 + bx + c = 0 Factor x2 + bx + c
To factor a trinomial of the form x2 + bx + c, find two integers, m and p, whose sum is equal to b and whose product is equal to c.
Example 1
x2 + bx + c = (x + m)(x + p), where m + p = b and mp = c.
Example 2
Factor each polynomial.
a. x2 + 7x + 10 In this trinomial, b = 7 and c = 10. Factors of 10
Sum of Factors
1, 10
11
2, 5
7
In this trinomial, b = 6 and c = -16. This means m + p is positive and mp is negative. Make a list of the factors of -16, where one factor of each pair is positive. Factors of -16
Since 2 + 5 = 7 and 2 5 = 10, let m = 2 and p = 5. x2 + 7x + 10 = (x + 5)(x + 2) b. x2 - 8x + 7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Factor x2 + 6x - 16.
Sum of Factors
1, -16
-15
-1, 16
15
2, -8
-6
-2, 8
6
Therefore, m = -2 and p = 8. x2 + 6x - 16 = (x - 2)(x + 8)
In this trinomial, b = -8 and c = 7. Notice that m + p is negative and mp is positive, so m and p are both negative. Since -7 + (-1) = -8 and (-7)(-1) = 7, m = -7 and p = -1. x2 - 8x + 7 = (x - 7)(x - 1)
Exercises Factor each polynomial. 1. x2 + 4x + 3
2. m2 + 12m + 32
3. r2 - 3r + 2
4. x2 - x - 6
5. x2 - 4x - 21
6. x2 - 22x + 121
7. t2 - 4t - 12
8. p2 - 16p + 64
9. 9 - 10x + x2
10. x2 + 6x + 5
11. a2 + 8a - 9
12. y2 - 7y - 8
13. x2 - 2x - 3
14. y2 + 14y + 13
15. m2 + 9m + 20
16. x2 + 12x + 20
17. a2 - 14a + 24
18. 18 + 11y + y2
19. x2 + 2xy + y2
20. a2 - 4ab + 4b2
21. x2 + 6xy - 7y2
Chapter 8
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North Carolina StudyText, Math A
Lesson 8-3
Factoring x2 + bx + c
NAME
8-3
DATE
Study Guide
PERIOD
SCS
(continued)
MA.A.1.3
Quadratic Equations: x2 + bx + c = 0 Solve Equations by Factoring
Factoring and the Zero Product Property can be used to solve many equations of the form x2 + bx + c = 0. Example 1
Solve x2 + 6x = 7. Check your solutions.
x2 + 6x = 7 Original equation 2 x + 6x - 7 = 0 Rewrite equation so that one side equals 0. (x - 1)(x + 7) = 0 Factor. x - 1 = 0 or x + 7 = 0 Zero Product Property x=1 x = -7 Solve each equation. The solution set is {1, -7}. Since 12 + 6 = 7 and (-7)2 + 6(-7) = 7, the solutions check. Example 2 ROCKET LAUNCH A rocket is fired with an initial velocity of 2288 feet per second. How many seconds will it take for the rocket to hit the ground?
Exercises Solve each equation. Check the solutions. 1. x2 - 4x + 3 = 0
2. y2 - 5y + 4 = 0
3. m2 + 10m + 9 = 0
4. x2 = x + 2
5. x2 - 4x = 5
6. x2 - 12x + 36 = 0
7. t2 - 8 = -7t
8. p2 = 9p - 14
9. -9 - 8x + x2 = 0
10. x2 + 6 = 5x
11. a2 = 11a - 18
12. y2 - 8y + 15 = 0
13. x2 = 24 - 10x
14. a2 - 18a = -72
15. b2 = 10b - 16
Use the formula h = vt - 16t2 to solve each problem. 16. FOOTBALL A punter can kick a football with an initial velocity of 48 feet per second. How many seconds will it take for the ball to return to the ground? 17. BASEBALL A ball is thrown up with an initial velocity of 32 feet per second. How many seconds will it take for the ball to return to the ground? 18. ROCKET LAUNCH If a rocket is launched with an initial velocity of 1600 feet per second, when will the rocket be 14,400 feet high? Chapter 8
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The formula h = vt - 16t2 gives the height h of the rocket after t seconds when the initial velocity v is given in feet per second. h = vt - 16t2 Formula 2 0 = 2288t - 16t Substitute. 0 = 16t(143 - t) Factor. 16t = 0 or 143 - t = 0 Zero Product Property t=0 t = 143 Solve each equation. The value t = 0 represents the time at launch. The rocket returns to the ground in 143 seconds, or a little less than 2.5 minutes after launch.
NAME
8-3
DATE
PERIOD
Practice
SCS
MA.A.1.3
Quadratic Equations: x2 + bx + c = 0 Factor each polynomial. 1. a2 + 10a + 24
2. h2 + 12h + 27
3. x2 + 14x + 33
4. g2 - 2g - 63
5. w2 + w - 56
6. y2 + 4y - 60
7. b2 + 4b - 32
8. n2 - 3n - 28
9. t2 + 4t - 45
10. z2 - 11z + 30
11. d2 - 16d + 63
12. x2 - 11x + 24
13. q2 - q - 56
14. x2 - 6x - 55
15. 32 + 18r + r2
16. 48 - 16g + g2
17. j 2 - 9jk - 10k2
18. m2 - mv - 56v2
19. x2 + 17x + 42 = 0
20. p2 + 5p - 84 = 0
21. k2 + 3k - 54 = 0
22. b2 - 12b - 64 = 0
23. n2 + 4n = 32
24. h2 - 17h = -60
25. t2 - 26t = 56
26. z2 - 14z = 72
27. y2 - 84 = 5y
28. 80 + a2 = 18a
29. u2 = 16u + 36
30. 17r + r2 = -52
31. Find all values of k so that the trinomial x2 + kx - 35 can be factored using integers. 32. CONSTRUCTION A construction company is planning to pour concrete for a driveway. The length of the driveway is 16 feet longer than its width w. a. Write an expression for the area of the driveway. b. Find the dimensions of the driveway if it has an area of 260 square feet. 33. WEB DESIGN Janeel has a 10-inch by 12-inch photograph. She wants to scan the photograph, then reduce the result by the same amount in each dimension to post on her Web site. Janeel wants the area of the image to be one eighth that of the original photograph. a. Write an equation to represent the area of the reduced image. b. Find the dimensions of the reduced image. Chapter 8
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Lesson 8-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation. Check the solutions.
NAME
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Word Problem Practice
PERIOD
SCS
MA.A.1.3
Quadratic Equations: x2 + bx + c = 0 1. COMPACT DISCS A standard jewel case for a compact disc has a width 2 cm greater than its length. The area for the front cover is 168 square centimeters. The first two steps to finding the value of x are shown below. Solve the equation and find the length of the case. Length × width = area x(x + 2) = 168 x2 + 2x - 168 = 0
2. MATH GAMES Fiona and Greg play a number guessing game. Greg gives Fiona this hint about his two secret numbers, “The product of the two consecutive positive integers that I am thinking of is 11 more than their sum.” What are Greg’s numbers?
5. MONUMENTS Susan is designing a pyramidal stone monument for a local park. The design specifications tell her that the height needs to be 9 feet, the width of the base must be 5 feet less than the length, and the volume should be 150 cubic feet. Recall that the 1 volume of a pyramid is given by V = − 3 Bh, where B is the area of the base and h is the height. a. Write and solve an equation to find the width of the base of the monument.
h d
If the driver of a car looks out at a height of 49 inches above the roadbed, at what distance(s) from the tower will the driver’s eyes be at the same height as the cable?
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b. Interpret each answer in terms of the situation.
North Carolina StudyText, Math A
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3. BRIDGE ENGINEERING A car driving over a suspension bridge is supported by a cable hanging between the ends of the bridge. Since its shape is parabolic, it can be modeled by a quadratic equation. The height above the road bed of a bridge’s cable h (in inches) measured at distance d (in yards) from the first tower is given by the equation h = d2 - 36d + 324.
4. PHYSICAL SCIENCE The boiling point of water depends on altitude. The following equation approximates the number of degrees D below 212ºF at which water will boil at altitude h. D2 + 520D = H In Denver, Colorado, the altitude is approximately 5300 feet above sea level. At approximately what temperature does water boil in Denver?
NAME
DATE
8-4
PERIOD
Study Guide
SCS
MA.A.1.3
Quadratic Equations: ax2 + bx + c = 0 Factor ax2 + bx + c
To factor a trinomial of the form ax2 + bx + c, find two integers, m and p whose product is equal to ac and whose sum is equal to b. If there are no integers that satisfy these requirements, the polynomial is called a prime polynomial.
Example 1
Example 2
Factor 2x2 + 15x + 18.
Note that the GCF of the terms 3x2, 3x, and 18 is 3. First factor out this GCF. 3x2 - 3x - 18 = 3(x2 - x - 6). Now factor x2 - x - 6. Since a = 1, find the two factors of -6 with a sum of -1.
In this example, a = 2, b = 15, and c = 18. You need to find two numbers that have a sum of 15 and a product of 2 18 or 36. Make a list of the factors of 36 and look for the pair of factors with a sum of 15. Factors of 36
Sum of Factors
Factors of -6
1, 36
37
1, -6
-5
2, 18
20
-1, 6
5
3, 12
15
-2, 3
1
2, -3
-1
Use the pattern ax2 + mx + px + c, with a = 2, m = 3, p = 12, and c = 18. 2x2 + 15x + 18 = 2x2 + 3x + 12x + 18 = (2x2 + 3x) + (12x + 18) = x(2x + 3) + 6(2x + 3) = (x + 6)(2x + 3) 2 Therefore, 2x + 15x + 18 = (x + 6)(2x + 3).
Sum of Factors
Now use the pattern (x + m)(x + p) with m = 2 and p = -3. x2 - x - 6 = (x + 2)(x - 3) The complete factorization is 3x2 - 3x - 18 = 3(x + 2)(x - 3).
Exercises Factor each polynomial, if possible. If the polynomial cannot be factored using integers, write prime. 1. 2x2 - 3x - 2
2. 3m2 - 8m - 3
3. 16r2 - 8r + 1
4. 6x2 + 5x - 6
5. 3x2 + 2x - 8
6. 18x2 - 27x - 5
7. 2a2 + 5a + 3
8. 18y2 + 9y - 5
9. -4t2 + 19t - 21
10. 8x2 - 4x - 24
11. 28p2 + 60p - 25
12. 48x2 + 22x - 15
13. 3y2 - 6y - 24
14. 4x2 + 26x - 48
15. 8m2 - 44m + 48
16. 6x2 - 7x + 18
17. 2a2 - 14a + 18
18. 18 + 11y + 2y2
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Lesson 8-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Factor 3x2 - 3x - 18.
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Study Guide
PERIOD
SCS
(continued)
MA.A.1.3
Quadratic Equations: ax2 + bx + c = 0 Solve Equations by Factoring
Factoring and the Zero Product Property can be used to solve some equations of the form ax2 + bx + c = 0. Example
Solve 12x2 + 3x = 2 - 2x. Check your solutions.
12x2 + 3x = 2 - 2x 12x2 + 5x - 2 = 0 (3x + 2)(4x - 1) = 0 3x + 2 = 0 or 4x - 1 = 0 1 2 x = -− x=− 3
4
Original equation Rewrite equation so that one side equals 0. Factor the left side. Zero Product Property Solve each equation.
{ 3 4} 1 2 2 2 + 3 (- − = 2 - 2(- − and 12 (− Since 12 (- − 4) 3) 3) 3) 2 1 The solution set is - − ,− . 2
2
(4)
(4)
1 1 +3 − =2-2 − , the solutions check.
Exercises Solve each equation. Check the solutions. 2. 3n2 - 2n - 5 = 0
3. 2d2 - 13d - 7 = 0
4. 4x2 = x + 3
5. 3x2 - 13x = 10
6. 6x2 - 11x - 10 = 0
7. 2k2 - 40 = -11k
8. 2p2 = -21p - 40
9. -7 - 18x + 9x2 = 0
10. 12x2 - 15 = -8x
11. 7a2 = -65a - 18
12. 16y2 - 2y - 3 = 0
13. 8x2 + 5x = 3 + 7x
14. 4a2 - 18a + 5 = 15
15. 3b2 - 18b = 10b - 49
16. The difference of the squares of two consecutive odd integers is 24. Find the integers.
17. GEOMETRY The length of a Charlotte, North Carolina, conservatory garden is 20 yards greater than its width. The area is 300 square yards. What are the dimensions? 8 in. x
18. GEOMETRY A rectangle with an area of 24 square inches is formed by cutting strips of equal width from a rectangular piece of paper. Find the dimensions of the new rectangle if the original rectangle measures 8 inches by 6 inches. Chapter 8
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x
x
6 in. x
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. 8x2 + 2x - 3 = 0
NAME
8-4
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PERIOD
Practice
SCS
MA.A.1.3
Quadratic Equations: ax2 + bx + c = 0 Factor each polynomial, if possible. If the polynomial cannot be factored using integers, write prime. 1. 2b2 + 10b + 12
2. 3g2 + 8g + 4
3. 4x2 + 4x - 3
4. 8b2 - 5b - 10
5. 6m2 + 7m - 3
6. 10d2 + 17d - 20
7. 6a2 - 17a + 12
8. 8w2 - 18w + 9
9. 10x2 - 9x + 6
10. 15n2 - n - 28
11. 10x2 + 21x - 10
12. 9r2 + 15r + 6
13. 12y2 - 4y - 5
14. 14k2 - 9k - 18
15. 8z2 + 20z - 48
16. 12q2 + 34q - 28
17. 18h2 + 15h - 18
18. 12p2 - 22p - 20
19. 3h2 + 2h - 16 = 0
20. 15n2 - n = 2
21. 8q2 - 10q + 3 = 0
22. 6b2 - 5b = 4
23. 10r2 - 21r = -4r + 6
24. 10g2 + 10 = 29g
25. 6y2 = -7y - 2
26. 9z2 = -6z + 15
27. 12k2 + 15k = 16k + 20
28. 12x2 - 1 = -x
29. 8a2 - 16a = 6a - 12
30. 18a2 + 10a = -11a + 4
31. DIVING Lauren dove into a swimming pool from a 15-foot-high diving board with an initial upward velocity of 8 feet per second. Find the time t in seconds it took Lauren to enter the water. Use the model for vertical motion given by the equation h = -16t2 + vt + s, where h is height in feet, t is time in seconds, v is the initial upward velocity in feet per second, and s is the initial height in feet. (Hint: Let h = 0 represent the surface of the pool.) 32. BASEBALL Brad tossed a baseball in the air from a height of 6 feet with an initial upward velocity of 14 feet per second. Enrique caught the ball on its way down at a point 4 feet above the ground. How long was the ball in the air before Enrique caught it? Use the model of vertical motion from Exercise 31. Chapter 8
159
North Carolina StudyText, Math A
Lesson 8-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation. Check the solutions.
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PERIOD
Word Problem Practice
SCS
MA.A.1.3
Quadratic Equations: ax2 + bx + c = 0 1. BREAK EVEN Breaking even occurs when the revenues for a business equal the cost. A local children’s museum studied their costs (wages, electricity, etc.) and revenues from paid admission. They found that their break-even time is given by the equation 2h2 - 2h - 24 = 0, where h is the number of hours the museum is open per day. How many hours must the museum be open per day to reach the break even point?
4. LADDERS A ladder is resting against a wall. The top of the ladder touches the wall at a height of 15 feet, and the length of the ladder is one foot more than twice its distance from the wall. Find the distance from the wall to the bottom of the ladder. (Hint: Use the Pythagorean Theorem to solve the problem.) Wall
15 ft.
Ladder
2. CARPENTRY Miko wants to build a toy box for her sister. It is to be 2 feet high, and the width is to be 3 feet less than its length. If it needs to hold a volume of 80 cubic feet, find the length and width of the box.
3. FURNITURE The student council wants to purchase a table for the school lobby. The table comes in a variety of dimensions, but for every table, the length is 1 meter greater than twice the width. The student council has budgeted for a table top with an area of exactly 3 square meters.
a. Write a quadratic equation (set equal to zero) to represent the information.
2w+1
b. Using 3 as an approximation for π, solve the equation for r.
w
Find the width and length of the table they can purchase.
Chapter 8
c. What radius should Mr. Hensley use for his tank?
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North Carolina StudyText, Math A
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5. FARMING Mr. Hensley has a total of 480 square feet of sheet metal with which he would like to construct a cylindrical tank for storing grain. The local zoning law limits the height of the tank to 13.5 feet. Recall that a formula for the surface area of a bottomless cylinder with radius r and height h is A = πr2 + 2πrh.
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PERIOD
Study Guide
SCS
MA.A.1.3
Quadratic Equations: Differences of Squares Factor Differences of Squares
The binomial expression a2 - b2 is called the difference of two squares. The following pattern shows how to factor the difference of squares. Difference of Squares
Example 1
a2 - b2 = (a - b)(a + b) = (a + b)(a - b).
Example 2
Factor each polynomial.
a. n2 - 64 n2 - 64 = n2 - 82 = (n + 8)(n - 8) b. 4m2 - 81n2
Write in the form a2
Factor each polynomial.
a. 50a2 - 72 50a2 - 72 = 2(25a2 - 36) Find the GCF. 2 2 = 2[(5a) - 6 )] 25a2 = 5a 5a and 36 = 6 6 =2(5a + 6)(5a - 6) Factor the difference of squares. b. 4x4 + 8x3 - 4x2 - 8x 4x4 + 8x3 - 4x2 - 8x Original polynomial 3 2 = 4x(x + 2x - x - 2) Find the GCF. 3 2 = 4x[(x + 2x ) - (x + 2)] Group terms. = 4x[x2(x + 2) - 1(x + 2)] Find the GCF. = 4x[(x2 - 1)(x + 2)] Factor by grouping. = 4x[(x - 1)(x + 1)(x + 2)] Factor the difference
- b2.
Factor.
4m2 - 81n2 = (2m)2 - (9n)2 Write in the form a2 - b2. = (2m - 9n)(2m + 9n) Factor.
Exercises Factor each polynomial. 1. x2 - 81
2. m2 - 100
3. 16n2 - 25
4. 36x2 - 100y2
5. 49x2 - 36
6. 16a2 - 9b2
7. 225b2 - a2
8. 72p2 - 50
9. -2 + 2x2
10. -81 + a4
11. 6 - 54a2
12. 8y2 - 200
13. 4x3 - 100x
14. 2y4 - 32y2
15. 8m3 - 128m
16. 4x2 - 25
17. 2a3 - 98ab2
18. 18y2 - 72y4
19. 169x3 - x
20. 3a4 - 3a2
21. 3x4 + 6x3 - 3x2 - 6x
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Lesson 8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
of squares.
NAME
DATE
8-5
Study Guide
PERIOD
SCS
(continued)
MA.A.1.3
Quadratic Equations: Differences of Squares Solve Equations by Factoring
Factoring and the Zero Product Property can be used to solve equations that can be written as the product of any number of factors set equal to 0. Example
Solve each equation. Check your solutions.
1 =0 a. x2 - − 25
1 =0 x2 - −
Original equation
25 1 2 1 1 1 − x2 - − =0 x2 = x · x and − = − 5 5 25 5 1 1 x+− x-− =0 Factor the difference of squares. 5 5 1 1 x+− = 0 or x - − =0 Zero Product Property 5 5 1 1 x = -− x=− Solve each equation. 5 5 1 1 1 1 1 2 The solution set is - − , − . Since - − -− = 0 and − 25 5 5 5 5
( )( )
() )( )
(
{
()
( )
}
b. 4x3 = 9x
4x3 = 9x 4x3 - 9x = 0 x(4x2 - 9) = 0 x[(2x)2 - 32] = 0 x[(2x)2 - 32] = x[(2x - 3)(2x + 3)] x = 0 or (2x - 3) = 0 or (2x + 3) = 0
2
1 -− = 0, the solutions check. 25
Original equation Subtract 9x from each side. Find the GCF. 4x2 = 2x 2x and 9 = 3 3
Zero Product Property
3 x=−
3 x = -− Solve each equation. 2 2 3 3 The solution set is 0, − , -− . 2 2 3 3 3 3 3 3 =9 − , and 4 - − = 9 -− , the solutions check. Since 4(0)3 = 9(0), 4 − 2 2 2 2
x=0
{ ()
} ()
( )
( )
Exercises Solve each equation by factoring. Check the solutions. 1. 81x2 = 49
2. 36n2 = 1
3. 25d2 - 100 = 0
1 2 x = 25 4. −
1 2 5. 36 = − x
49 6. − - x2 = 0
7. 9x3 = 25x
8. 7a3 = 175a
9. 2m3 = 32m
4
25
100
10. 16y3 = 25y
1 2 11. − x = 49
12. 4a3 - 64a = 0
13. 3b3 - 27b = 0
9 2 14. − m = 121
15. 48n3 = 147n
Chapter 8
64 25
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North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Factor the difference of squares.
NAME
8-5
DATE
PERIOD
Practice
SCS
MA.A.1.3
Quadratic Equations: Differences of Squares Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 1. k2 - 100
2. 81 - r2
3. 16p2 - 36
4. 4x2 + 25
5. 144 - 9f 2
6. 36g2 - 49h2
7. 121m2 - 144p2
8. 32 - 8y2
9. 24a2 - 54b2
10. 32t2 - 18u2
11. 9d2 - 32
12. 36z3 - 9z
13. 45q3 - 20q
14. 100b3 - 36b
15. 3t4 - 48t2
16. 4y2 = 81
17. 64p2 = 9
18. 98b2 - 50 = 0
19. 32 - 162k2 = 0
64 20. t2 - − =0
16 21. − - v2 = 0
1 2 x - 25 = 0 22. −
23. 27h3 = 48h
24. 75g3 = 147g
36
121
49
25. EROSION A rock breaks loose from a cliff and plunges toward the ground 400 feet below. The distance d that the rock falls in t seconds is given by the equation d = 16t2. How long does it take the rock to hit the ground? 26. FORENSICS Mr. Cooper contested a speeding ticket given to him after he applied his brakes and skidded to a halt to avoid hitting another car. In traffic court, he argued that the length of the skid marks on the pavement, 150 feet, proved that he was driving under the posted speed limit of 65 miles per hour. The ticket cited his speed at 70 miles 1 2 s = d, where s is the speed of the car and d is the length of per hour. Use the formula − 24
the skid marks, to determine Mr. Cooper’s speed when he applied the brakes. Was Mr. Cooper correct in claiming that he was not speeding when he applied the brakes?
Chapter 8
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North Carolina StudyText, Math A
Lesson 8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation by factoring. Check your solutions.
NAME
8-5
DATE
PERIOD
Word Problem Practice
SCS
MA.A.1.3
Quadratic Equations: Differences of Squares 1. LOTTERY A state lottery commission analyzes the ticket purchasing patterns of its citizens. The following expression is developed to help officials calculate the likely number of people who will buy tickets for a certain size jackpot. 81a2 - 36b2 Factor the expression completely.
4. BALLOONING The function f (t) = -16t2 + 576 represents the height of a freely falling ballast bag that starts from rest on a balloon 576 feet above the ground. After how many seconds t does the ballast bag hit the ground?
5. DECORATING Marvin wants to purchase a rectangular rug. It has an area of 80 square feet. He cannot remember the length and width, but he remembers that the length was 8 more than some number and the width was 8 less than that same number.
2. OPTICS A reflector on the inside of a certain flashlight is a parabola given by the equation y = x2 - 25. Find the points where the reflector meets the lens by finding the values of x when y = 0.
x+8 x-8
Area = 98m 2
a. Write a quadratic equation using the information given.
height
b. What are the length and width of the rug? base
Find the height of the truss.
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. ARCHITECTURE The drawing shows a triangular roof truss with a base measuring the same as its height. The area of the truss is 98 square meters.
NAME
DATE
8-6
PERIOD
Study Guide
SCS
MA.A.1.3
Factor Perfect Square Trinomials a trinomial of the form a2 + 2ab + b2 or a2 - 2ab + b2
Perfect Square Trinomial
The patterns shown below can be used to factor perfect square trinomials. Squaring a Binomial 2
2
2
Factoring a Perfect Square Trinomial
(a + 4) = a + 2(a)(4) + 4 = a2 + 8a + 16
a2 + 8a + 16 = a2 + 2(a)(4) + 42 = (a + 4)2
(2x - 3)2 = (2x)2 -2(2x)(3) + 32 = 4x2 - 12x + 9
4x2 - 12x + 9 = (2x)2 -2(2x)(3) + 32 = (2x - 3)2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 1 Determine whether 2 16n - 24n + 9 is a perfect square trinomial. If so, factor it. Since 16n2 = (4n)(4n), the first term is a perfect square. Since 9 = 3 3, the last term is a perfect square. The middle term is equal to 2(4n)(3). Therefore, 16n2 - 24n + 9 is a perfect square trinomial. 16n2 - 24n + 9 = (4n)2 - 2(4n)(3) + 32 = (4n - 3)2
Example 2
Factor 16x2 - 32x + 15.
Since 15 is not a perfect square, use a different factoring pattern. 16x2 - 32x + 15 Original trinomial 2 = 16x + mx + px + 15 Write the pattern. 2 = 16x - 12x - 20x + 15 m = -12 and p = -20 2 = (16x - 12x) - (20x - 15) Group terms. = 4x(4x - 3) - 5(4x - 3) Find the GCF. = (4x - 5)(4x - 3) Factor by grouping. 2 Therefore 16x - 32x + 15 = (4x - 5)(4x - 3).
Exercises Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1. x2 - 16x + 64
2. m2 + 10m + 25
3. p2 + 8p + 64
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 4. 98x2 - 200y2
5. x2 + 22x + 121
6. 81 + 18j + j2
7. 25c2 - 10c - 1
8. 169 - 26r + r2
9. 7x2 - 9x + 2
10. 16m2 + 48m + 36
11. 16 - 25a2
12. b2 - 16b + 256
13. 36x2 - 12x + 1
14. 16a2 - 40ab + 25b2
15. 8m3 - 64m
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Lesson 8-6
Quadratic Equations: Perfect Squares
NAME
8-6
DATE
Study Guide
PERIOD
SCS
(continued)
MA.A.1.3
Quadratic Equations: Perfect Squares Solve Equations with Perfect Squares Factoring and the Zero Product Property can be used to solve equations that involve repeated factors. The repeated factor gives just one solution to the equation. You may also be able to use the Square Root Property below to solve certain equations. Square Root Property
Example
For any number n > 0, if x2 = n, then x = ± √ n.
Solve each equation. Check your solutions.
2
a. x - 6x + 9 = 0 x2 - 6x + 9 = 0 Original equation 2 2 x - 2(3x) + 3 = 0 Recognize a perfect square trinomial. (x - 3)(x - 3) = 0 Factor the perfect square trinomial. x-3=0 Set repeated factor equal to 0. x=3 Solve. 2 The solution set is {3}. Since 3 - 6(3) + 9 = 0, the solution checks. b. (a - 5)2 = 64
Exercises Solve each equation. Check the solutions. 1. x2 + 4x + 4 = 0
2. 16n2 + 16n + 4 = 0
3. 25d2 - 10d + 1 = 0
4. x2 + 10x + 25 = 0
5. 9x2 - 6x + 1 = 0
1 6. x2 + x + − =0
7. 25k2 + 20k + 4 = 0
8. p2 + 2p + 1 = 49
9. x2 + 4x + 4 = 64
10. x2 - 6x + 9 = 25
11. a2 + 8a + 16 = 1
12. 16y2 + 8y + 1 = 0
13. (x + 3)2 = 49
14. (y + 6)2 = 1
15. (m - 7)2 = 49
16. (2x + 1)2 = 1
17. (4x + 3)2 = 25
18. (3h - 2)2 = 4
19. (x + 1)2 = 7
20. (y - 3)2 = 6
21. (m - 2)2 = 5
Chapter 8
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(a - 5)2 = 64 Original equation a - 5 = ± √ 64 Square Root Property a - 5 = ±8 64 = 8 8 a=5±8 Add 5 to each side. a = 5 + 8 or a = 5 - 8 Separate into 2 equations. a = 13 a = -3 Solve each equation. The solution set is {-3, 13}. Since (-3 - 5)2 = 64 and (13 - 5)2 = 64, the solutions check.
NAME
DATE
8-6
PERIOD
Practice
SCS
MA.A.1.3
Quadratic Equations: Perfect Squares
1. m2 + 16m + 64
2. 9r2 - 6r + 1
3. 4y2 - 20y + 25
4. 16p2 + 24p + 9
5. 25b2 - 4b + 16
6. 49k2 - 56k + 16
Lesson 8-6
Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it.
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7. 3p2 - 147
8. 6x2 + 11x - 35
9. 50q2 - 60q + 18
10. 6t3 - 14t2 - 12t
11. 6d2 - 18
12. 30k2 + 38k + 12
13. 15b2 - 24bf
14. 12h2 - 60h + 75
15. 9n2 - 30n - 25
16. 7u2 - 28m2
17. w4 - 8w2 - 9
18. 16a2 + 72ad + 81d2
Solve each equation. Check the solutions.
(2
)
2
19. 4k2 - 28k = -49
20. 50b2 + 20b + 2 = 0
1 21. − t-1
2 1 22. g2 + − g+− =0
6 9 23. p2 - − p+− =0
24. x2 + 12x + 36 = 25
25. y2 - 8y + 16 = 64
26. (h + 9)2 = 3
27. w2 - 6w + 9 = 13
3
9
5
25
=0
28. GEOMETRY The area of a circle is given by the formula A = πr2, where r is the radius. If increasing the radius of a circle by 1 inch gives the resulting circle an area of 100π square inches, what is the radius of the original circle? 10 29. PICTURE FRAMING Mikaela placed a frame around a print that measures 10 inches by 10 inches. The area of just the frame itself is 69 square inches. What is the width of the frame?
10 x x
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NAME
8-6
DATE
PERIOD
Word Problem Practice
SCS
MA.A.1.3
Quadratic Equations: Perfect Squares 1. CONSTRUCTION The area of Liberty Township’s square playground is represented by the trinomial x2 - 10x + 25. Write an expression using the variable x that represents the perimeter.
4. GEOMETRY Holly can make an opentopped box out of a square piece of cardboard by cutting 3-inch squares from the corners and folding up the sides to meet. The volume of the resulting box is V = 3x2 - 36x + 108, where x is the original length and width of the cardboard. x
2. AMUSEMENT PARKS Funtown Downtown wants to build a vertical motion ride where the passengers are launched straight upward from ground level with an initial velocity of 96 feet per second. The ride car’s height h in feet after t seconds is h = 96t - 16t2. How many seconds after launch would the car reach 144 feet?
Chapter 8
x
3 in
a. Factor the polynomial expression from the volume equation.
b. What is the volume of the box if the original length of each side of the cardboard was 14 inches?
c. What is the original side length of the cardboard when the volume of the box is 27 in3?
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3. BUSINESS Saini Sprinkler Company installs irrigation systems. To track monthly costs C and revenues R, they use the following functions, where x is the number of systems they install. R(x) = 8x2 + 12x + 4 C(x) = 7x2 + 20x - 12 The monthly profit can be found by subtracting cost from revenue. P(x) = R(x) - C(x) Find a function to project monthly profit and use it to find the break-even point where the profit is zero.
3 in
NAME
DATE
9-2
PERIOD
Study Guide
SCS
MA.A.4.7
Solving Quadratic Equations by Graphing Solve by Graphing Quadratic Equation
an equation of the form ax2 + bx + c = 0, where a ≠ 0
The solutions of a quadratic equation are called the roots of the equation. The roots of a quadratic equation can be found by graphing the related quadratic function f(x) = ax2 + bx + c and finding the x-intercepts or zeros of the function.
Example 2
Solve x2 + 4x + 3 = 0 by graphing.
Solve x2 - 6x + 9 = 0 by graphing.
Graph the related function f(x) = x2 + 4x + 3. Graph the related function f(x) = x2 - 6x + 9. The equation of the axis of symmetry is The equation of the axis of symmetry is 6 4 x=− or 3. The vertex is at (3, 0). Graph x=-− or -2. The vertex is at (-2, -1). 2(1) 2(1) the vertex and several other points on either Graph the vertex and several other points on side of the axis of symmetry. either side of the axis of symmetry. f(x)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
f(x)
O
x
O
x
To solve x2 - 6x + 9 = 0, you need to know where f(x) = 0. The vertex of the parabola is the x-intercept. Thus, the only solution is 3.
To solve x2 + 4x + 3 = 0, you need to know where the value of f(x) = 0. This occurs at the x-intercepts, -3 and -1. The solutions are -3 and -1.
Exercises Solve each equation by graphing. 1. x2 + 7x + 12 = 0
2. x2 - x - 12 = 0
f (x)
4 -8
-4
3. x2 - 4x + 5 = 0
f(x)
O
f (x) 4
8x
-4 -8 O
Chapter 9
x
-12
169
O
x
North Carolina StudyText, Math A
Lesson 9-2
Example 1
NAME
DATE
9-2
Study Guide
PERIOD
SCS
(continued)
MA.A.4.7
Solving Quadratic Equations by Graphing Estimate Solutions The roots of a quadratic equation may not be integers. If exact roots cannot be found, they can be estimated by finding the consecutive integers between which the roots lie. Example Solve x2 + 6x + 6 = 0 by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. Graph the related function f(x) = x2 + 6x + 6. x -5
f(x)
f(x) Notice that the value of the function changes 1 from negative to positive between the x-values of -5 and -4 and between -2 and -1.
-4
-2
-3
-3
-2
-2
-1
1
O
x
The x-intercepts of the graph are between -5 and -4 and between -2 and -1. So one root is between -5 and -4, and the other root is between -2 and -1.
Exercises Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. 2. x2 - x - 4 = 0
f (x)
O
3. x2 - 4x + 6 = 0 f(x)
f(x)
x
O
x
O
4. x2 - 4x - 1 = 0
5. 4x2 - 12x + 3 = 0
O
Chapter 9
6. x2 - 2x - 4 = 0
f(x)
f (x)
x
O
170
x
f(x) x
O
x
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. x2 + 7x + 9 = 0
NAME
DATE
9-2
PERIOD
Practice
SCS
MA.A.4.7
Solving Quadratic Equations by Graphing Solve each equation by graphing. 1. x2 - 5x + 6 = 0
2. w2 + 6w + 9 = 0
3. b2 - 3b + 4 = 0
f(w)
f (x)
O
O
x
f(b)
w
O
b
Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. 5. 2m2 + 5 = 10m
f (p) O
6. 2v2 + 8v = -7 f (v)
f(m) p
O
Lesson 9-2
4. p2 + 4p = 3
m
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
v
f (n)
7. NUMBER THEORY Two numbers have a sum of 2 and a product of -8. The quadratic equation -n2 + 2n + 8 = 0 can be used to determine the two numbers. a. Graph the related function f(n) = -n2 + 2n + 8 and determine its x-intercepts. O
b. What are the two numbers?
n
8. DESIGN A footbridge is suspended from a parabolic 1 2 support. The function h(x) = - − x + 9 represents 25
the height in feet of the support above the walkway, where x = 0 represents the midpoint of the bridge. a. Graph the function and determine its x-intercepts.
12
h (x)
6 -12 -6 O
6
12
x
-6
b. What is the length of the walkway between the two supports? Chapter 9
171
-12
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NAME
DATE
9-2
PERIOD
Word Problem Practice
SCS
MA.A.4.7
Solving Quadratic Equations by Graphing 1. FARMING In order for Ray to decide how much fertilizer to apply to his corn crop this year, he reviews records from previous years. He finds that his crop yield y depends on the amount of fertilizer he applies to his fields x according to the equation y = -x2 + 4x + 12. Graph the function, and find the point at which Ray gets the highest yield possible. 16 14 12 10 8 6 4 2 O
4. WRAPPING PAPER Can a rectangular piece of wrapping paper with an area of 81 square inches have a perimeter of 60 inches? (Hint: Let length = 30 – w.) Explain.
5. ENGINEERING The shape of a satellite dish is often parabolic because of the reflective qualities of parabolas. Suppose a particular satellite dish is modeled by the following equation. 0.5x2 = 2 + y
y
a. Approximate the solution by graphing.
x
1 2 3 4 5
4 3 2 1 -4-3-2
O
y
1 2 3 4x
-2 -3 -4
b. On the coordinate plane above, translate the parabola so that there is only one root. Label this curve A.
3. FRAMING A rectangular photograph is 7 inches long and 6 inches wide. The photograph is framed using a material that is x inches wide. If the area of the frame and photograph combined is 156 square inches, what is the width of the framing material?
c. Translate the parabola so that there are no roots. Label this curve B.
x 7 in.
x
Photograph 6 in. Frame
Chapter 9
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2. LIGHT Ayzha and Jeremy hold a flashlight so that the light falls on a piece of graph paper in the shape of a parabola. Ayzha and Jeremy sketch the shape of the parabola and find that the equation y = x2 - 3x - 10 matches the shape of the light beam. Determine the roots of the function.
NAME
DATE
9-3
PERIOD
Study Guide
SCS
MA.A.4.3
Transformations of Quadratic Functions Translations
A translation is a change in the position of a figure either up, down, or diagonal. When a constant c is added to or subtracted from the parent function, the resulting function f(x) ± c is a translation of the graph up or down. f(x)
The graph of f(x) = x2 + c translates the graph of f(x) = x2 vertically. If c > 0, the graph of f(x) = x2 is translated |c| units up. c >0
If c < 0, the graph of f(x) = x2 is translated |c|units down.
c =0 x
0 c 0. Therefore, the graph of g(x) = x2 + 4 is a translation of the graph of f(x) = x2 up 4 units.
b. h(x) = x2 - 3 The function can be written as f(x) = x2 + c. The value of c is –3, and –3 < 0. Therefore, the graph of g(x) = x2 – 3 is a translation of the graph of f(x) = x2 down 3 units. y
g(x)
f(x)
x
0
Lesson 9-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
f(x)
h(x) x
0
Exercises Describe how the graph of each function is related to the graph of f(x) = x2. 1. g(x) = x2 + 1
2. h(x) = x2 – 6
3. g(x) = x2 – 1
4. h(x) = 20 + x2
5. g(x) = –2 + x2
1 6. h(x) = - − + x2
8 7. g(x) = x2 + −
8. h(x) = x2 – 0.3
9. g(x) = x2 – 4
9
Chapter 9
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North Carolina StudyText, Math A
NAME
9-3
DATE
Study Guide
PERIOD
SCS
(continued)
MA.A.4.3
Transformations of Quadratic Functions Dilations and Reflections
A dilation is a transformation that makes the graph narrower or wider than the parent graph. A reflection flips a figure over the x- or y-axis.
a >1
a =1
The graph of f(x) = ax2 stretches or vertically compresses the graph of f(x) = x2. If |a| > 1, the graph of f(x) = x2 is stretched vertically. If 0 < |a| < 1, the graph of f(x) = x2 is compressed vertically.
0 0
two real roots
Example State the value of the discriminant for each equation. Then determine the number of real solutions of the equation. b. 2x2 + 3x = -4
a. 12x2 + 5x = 4 Write the equation in standard form. 12x2 + 5x = 4 Original equation 2 12x + 5x - 4 = 4 - 4 Subtract 4 from each side. 2 12x + 5x - 4 = 0 Simplify.
2x2 + 3x = -4 2x2 + 3x + 4 = -4 + 4 2x2 + 3x + 4 = 0
Now find the discriminant. b2 - 4ac = (5)2 - 4(12)(-4) = 217
b2 - 4ac = (3)2 - 4(2)(4) = -23
Original equation Add 4 to each side. Simplify.
Find the discriminant.
Exercises State the value of the discriminant for each equation. Then determine the number of real solutions of the equation. 1. 3x2 + 2x - 3 = 0
2. 3x2 - 7x - 8 = 0
3. 2x2 - 10x - 9 = 0
4. 4x2 = x + 4
5. 3x2 - 13x = 10
6. 6x2 - 10x + 10 = 0
7. 2x2 - 20 = -x
8. 6x2 = -11x - 40
9. 9 - 18x + 9x2 = 0
10. 12x2 + 9 = -6x
11. 9x2 = 81
12. 16x2 + 16x + 4 = 0
13. 8x2 + 9x = 2
14. 4x2 - 4x + 4 = 3
15. 3x2 - 18x = - 14
Chapter 9
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Since the discriminant is negative, the equation has no real roots.
Since the discriminant is positive, the equation has two real roots.
NAME
9-5
DATE
PERIOD
Practice
SCS
MA.A.4.7
Solving Quadratic Equations by Using the Quadratic Formula Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 1. x2 + 2x - 3 = 0
2. x2 + 8x + 7 = 0
3. x2 - 4x + 6 = 0
4. x2 - 6x + 7 = 0
5. 2x2 + 9x - 5 = 0
6. 2x2 + 12x + 10 = 0
7. 2x2 - 9x = -12
8. 2x2 - 5x = 12
9. 3x2 + x = 4
10. 3x2 - 1 = -8x
11. 4x2 + 7x = 15
12. 1.6x2 + 2x + 2.5 = 0
13. 4.5x2 + 4x - 1.5 = 0
3 1 2 14. − x + 2x + − =0
3 1 15. 3x2 - − x=−
2
2
4
2
16. x2 + 8x + 16 = 0
17. x2 + 3x + 12 = 0
18. 2x2 + 12x = -7
19. 2x2 + 15x = -30
20. 4x2 + 9 = 12x
21. 3x2 - 2x = 3.5
22. 2.5x2 + 3x - 0.5 = 0
3 2 23. − x - 3x = -4
1 2 24. − x = -x - 1
4
4
25. CONSTRUCTION A roofer tosses a piece of roofing tile from a roof onto the ground 30 feet below. He tosses the tile with an initial downward velocity of 10 feet per second. a. Write an equation to find how long it takes the tile to hit the ground. Use the model for vertical motion, H = -16t2 + vt + h, where H is the height of an object after t seconds, v is the initial velocity, and h is the initial height. (Hint: Since the object is thrown down, the initial velocity is negative.) b. How long does it take the tile to hit the ground? 26. PHYSICS Lupe tosses a ball up to Quyen, waiting at a third-story window, with an initial velocity of 30 feet per second. She releases the ball from a height of 6 feet. The equation h = -16t2 + 30t + 6 represents the height h of the ball after t seconds. If the ball must reach a height of 25 feet for Quyen to catch it, does the ball reach Quyen? Explain. (Hint: Substitute 25 for h and use the discriminant.)
Chapter 9
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North Carolina StudyText, Math A
Lesson 9-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
State the value of the discriminant for each equation. Then determine the number of real solutions of the equation.
NAME
9-5
DATE
PERIOD
Word Problem Practice
SCS
MA.A.4.7
Solving Quadratic Equations by Using the Quadratic Formula 1. BUSINESS Tanya runs a catering business. Based on her records, her weekly profit can be approximated by the function f(x) = x2 + 2x - 37, where x is the number of meals she caters. If f(x) is negative, it means that the business has lost money. What is the least number of meals that Tanya needs to cater in order to have a profit?
4. CRAFTS Madelyn cut a 60-inch pipe cleaner into two unequal pieces, and then she used each piece to make a square. The sum of the areas of the squares was 117 square inches. Let x = the length of one piece. Write and solve an equation to represent the situation and find the lengths of the two original pieces.
5. SITE DESIGN The town of Smallport plans to build a new water treatment plant on a rectangular piece of land 75 yards wide and 200 yards long. The buildings and facilities need to cover an area of 10,000 square yards. The town’s zoning board wants the site designer to allow as much room as possible between each edge of the site and the buildings and facilities. Let x represent the width of the border. 200 yd x x 75 yd
Buildings and Facilities
Border
3. ARCHITECTURE The Golden Ratio appears in the design of the Greek Parthenon because the width and height of the façade are related by the
b. Write the equation in standard quadratic form. c. What should be the width of the border? Round your answer to the nearest tenth.
H
model of the Parthenon is 16 inches, what is its width? Round your answer to the nearest tenth.
Chapter 9
x
a. Use an equation similar to A = × w to represent the situation.
W+H W . If the height of a equation − = − W
x
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North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. AERONAUTICS At liftoff, the space shuttle Discovery has a constant acceleration of 16.4 feet per second squared and an initial velocity of 1341 feet per second due to the rotation of Earth. The distance Discovery has traveled t seconds after liftoff is given by the equation d(t) = 1341t + 8.2t2. How long after liftoff has Discovery traveled 40,000 feet? Round your answer to the nearest tenth.
NAME
DATE
9-6
PERIOD
Study Guide
SCS
MA.A.5.1
Exponential Functions Exponential Function
Lesson 9-6
Graph Exponential Functions a function defined by an equation of the form y = a bx, where a ≠ 0, b > 0, and b ≠ 1
You can use values of x to find ordered pairs that satisfy an exponential function. Then you can use the ordered pairs to graph the function.
Graph y = 3x. Find the y-intercept and state the domain and range. x
y
-2
1 −
y
x
y
-2
16
-1
4
-1 0
1
0
1
(4)
1
3
1
1 −
-0.5
.
y
8
O
9
x
4 1 − 16
2
O
(4)
1 The value of −
The y-intercept is 1. The domain is all real numbers, and the range is all positive numbers. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
1 graph to approximate the value of −
9 1 − 3
2
(4)
1 Graph y = − . Use the
Example 2
Example 1
-0.5
x
2
is about 2.
Exercises y
1. Graph y = 0.3x. Find the y-intercept. Then use the graph to approximate the value of 0.3-1.5. Use a calculator to confirm the value.
2 O
1
x
Graph each function. Find the y-intercept and state the domain and range.
(3)
1 3. y = −
2. y = 3x + 1
x
1
x
O
-2 y
O
2
2
Chapter 9
x
y
y
O
(2)
1 4. y = −
+1
1
185
x
x
North Carolina StudyText, Math A
NAME
DATE
9-6
PERIOD
Study Guide (continued)
SCS
MA.A.5.1
Exponential Functions Identify Exponential Behavior It is sometimes useful to know if a set of data is exponential. One way to tell is to observe the shape of the graph. Another way is to observe the pattern in the set of data. Example Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. x
0
2
4
6
8
10
y
64
32
16
8
4
2
Method 1: Look for a Pattern The domain values increase by regular intervals of 2, while the range values have 1 . Since the domain a common factor of − 2 values increase by regular intervals and the range values have a common factor, the data are probably exponential.
Method 2: Graph the Data y The graph shows rapidly decreasing values of y as x increases. This is characteristic of exponential behavior. 8
O
x
2
Exercises
1.
3.
5.
x
0
1
2
3
y
5
10
15
20
x
-1
1
3
5
y
32
16
8
4
x
-5
y
1
Chapter 9
2.
4.
0
5
10
0.5
0.25
0.125
6.
186
x
0
1
2
3
y
3
9
27
81
x
-1
0
1
2
3
y
3
3
3
3
3
x
0
1
2
3
4
y
1 −
1 −
1 −
1 −
1 −
3
9
27
81
243
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not.
NAME
DATE
9-6
PERIOD
Practice
SCS
MA.A.5.1
Graph each function. Find the y-intercept and state the domain and range. Then use the graph to determine the approximate value of the given expression. Use a calculator to confirm the value. x
( 10 ) ( 10 )
1 1 1. y = − ; −
x
-0.5
(4) (4)
1 1 3. y = − ; −
2. y = 3x; 31.9
y
y
x
O
-1.4
y
x
O
x
O
Graph each function. Find the y-intercept, and state the domain and range. 4. y = 4(2x) + 1
5. y = 2(2x - 1)
y
y
y
x
O Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6. y = 0.5(3x - 3)
O
x
x
O
Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. 7.
x y
2
5
8
480 120
30
11
8.
7.5
x
21
18
15
12
y
30
23
16
9
9. LEARNING Ms. Klemperer told her English class that each week students tend to forget one sixth of the vocabulary words they learned the previous week. Suppose a student learns 60 words. The number of words remembered can be described by the function
()
x
5 , where x is the number of weeks that pass. How many words will the W(x) = 60 − 6 student remember after 3 weeks?
10. BIOLOGY Suppose a certain cell reproduces itself in four hours. If a lab researcher begins with 50 cells, how many cells will there be after one day, two days, and three days? (Hint: Use the exponential function y = 50(2x).)
Chapter 9
187
North Carolina StudyText, Math A
Lesson 9-6
Exponential Functions
NAME
9-6
DATE
Word Problem Practice
PERIOD
SCS
MA.A.5.1
Exponential Functions 1. WASTE Suppose the waste generated by nonrecycled paper and cardboard products is approximated by the following function. y = 1000(2)0.3x Sketch the exponential function on the coordinate grid below. 2450 2150 1850 1550 1250 950 650 350 -4-3-2-1O
4. DEPRECIATION The value of Royce Company’s computer equipment is decreasing in value according to the following function. y = 4000(0.87)x In the equation, x is the number of years that have elapsed since the equipment was purchased and y is in dollars. What was the value 5 years after it was purchased? Round your answer to the nearest dollar.
y
1 2 3 4x
3. PICTURE FRAMES Since a picture frame includes a border, the picture must be smaller in area than the entire frame. The table shows the relationship between picture area and frame length for a particular line of frames. Is this an exponential relationship? Explain.
Chapter 9
b. The McDonald Observatory in Texas is at an altitude of 2000 meters. What is the approximate atmospheric pressure there?
Side Picture Length Area (in.) (in2) 5
6
6
12
7
20
8
30
9
42
a. What is the pressure at sea level?
c. As altitude increases, what happens to atmospheric pressure?
188
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. MONEY Tatyana’s grandfather gave her one penny on the day she was born. He plans to double the amount he gives her every day. Estimate how much she will receive from her grandfather on the 12th day of her life.
5. METEOROLOGY The atmospheric pressure (in millibars) at a given altitude x, in meters, can be approximated by the following function. The function is valid for values of x between 0 and 10,000. f (x) = 1038(1.000134)-x
NAME
9-7
DATE
PERIOD
Study Guide
SCS
MA.A.4.6, MA.A.5.1
Growth and Decay Exponential Growth
Population increases and growth of monetary investments are examples of exponential growth. This means that an initial amount increases at a steady rate over time.
Example 1
Example 2
POPULATION The
population of Johnson City in 2000 was 25,000. Since then, the population has grown at an average rate of 3.2% each year. a. Write an equation to represent the population of Johnson City since 2000.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The rate 3.2% can be written as 0.032. y = a(1 + r)t y = 25,000(1 + 0.032)t y = 25,000(1.032)t b. According to the equation, what will the population of Johnson City be in the year 2010? In 2010, t will equal 2010 - 2000 or 10. Substitute 10 for t in the equation from part a. y = 25,000(1.032)10 t = 10 ≈ 34,256 In 2010, the population of Johnson City will be about 34,256.
INVESTMENT The Garcias have $12,000 in a savings account. The bank pays 3.5% interest on savings accounts, compounded monthly. Find the balance in 3 years. The rate 3.5% can be written as 0.035. The special equation for compound r nt interest is A = P(1 + − n ) , where A represents the balance, P is the initial amount, r represents the annual rate expressed as a decimal, n represents the number of times the interest is compounded each year, and t represents the number of years the money is invested. r nt A = P(1 + − n)
(
0.035 = 12,000 1 + − 12
)
36
≈ 13,326.49 In three years, the balance of the account will be $13,326.49.
Exercises 1. POPULATION The population of the United States has been increasing at an average annual rate of 0.91%. If the population of the United States was about 303,146,000 in the year 2008, predict the U.S. population in the year 2012.
2. INVESTMENT Determine the amount of an investment of $2500 if it is invested at an interest rate of 5.25% compounded monthly for 4 years.
4. INVESTMENT Determine the 3. POPULATION It is estimated that the population of the world is increasing at an amount of an investment of $100,000 average annual rate of 1.3%. If the population if it is invested at an interest rate of of the world was about 6,641,000,000 in the 5.2% compounded quarterly for year 2008, predict the world population in the 12 years. year 2015. Chapter 9
189
North Carolina StudyText, Math A
Lesson 9-7
Exponential Growth
The general equation for exponential growth is y = a(1 + r)t. • y represents the final amount. • a represents the initial amount. • r represents the rate of change expressed as a decimal. • t represents time.
NAME
9-7
DATE
PERIOD
Study Guide (continued)
SCS
MA.A.4.6, MA.A.5.1
Growth and Decay Exponential Decay
Radioactive decay and depreciation are examples of exponential decay. This means that an initial amount decreases at a steady rate over a period of time.
Exponential Decay
The general equation for exponential decay is y = a(1 - r)t . • y represents the final amount. • a represents the initial amount. • r represents the rate of decay expressed as a decimal. • t represents time.
Example
DEPRECIATION The original price of a tractor was $45,000. The value of the tractor decreases at a steady rate of 12% per year. a. Write an equation to represent the value of the tractor since it was purchased. The rate 12% can be written as 0.12. y = a(1 - r)t General equation for exponential decay t y = 45,000(1 - 0.12) a = 45,000 and r = 0.12 t y = 45,000(0.88) Simplify. b. What is the value of the tractor in 5 years? y = 45,000(0.88)t Equation for decay from part a 5 y = 45,000(0.88) t=5 y ≈ 23,747.94 Use a calculator. In 5 years, the tractor will be worth about $23,747.94.
1. POPULATION The population of Bulgaria has been decreasing at an annual rate of 0.89%. If the population of Bulgaria was about 7,450,349 in the year 2005, predict its population in the year 2015. 2. DEPRECIATION Mr. Gossell is a machinist. He bought some new machinery for about $125,000. He wants to calculate the value of the machinery over the next 10 years for tax purposes. If the machinery depreciates at the rate of 15% per year, what is the value of the machinery (to the nearest $100) at the end of 10 years? 3. ARCHAEOLOGY The half-life of a radioactive element is defined as the time that it takes for one-half a quantity of the element to decay. Radioactive carbon-14 is found in all living organisms and has a half-life of 5730 years. Consider a living organism with an original concentration of carbon-14 of 100 grams. a. If the organism lived 5730 years ago, what is the concentration of carbon-14 today? b. If the organism lived 11,460 years ago, determine the concentration of carbon-14 today. 4. DEPRECIATION A new car costs $32,000. It is expected to depreciate 12% each year for 4 years and then depreciate 8% each year thereafter. Find the value of the car in 6 years.
Chapter 9
190
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
9-7
DATE
Practice
PERIOD
SCS
MA.A.4.6, MA.A.5.1
Growth and Decay 1. COMMUNICATIONS Sports radio stations numbered 220 in 1996. The number of sports radio stations has since increased by approximately 14.3% per year. a. Write an equation for the number of sports radio stations for t years after 1996.
2. INVESTMENTS Determine the amount of an investment if $500 is invested at an interest rate of 4.25% compounded quarterly for 12 years.
3. INVESTMENTS Determine the amount of an investment if $300 is invested at an interest rate of 6.75% compounded semiannually for 20 years.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. HOUSING The Greens bought a condominium for $110,000 in 2010. If its value appreciates at an average rate of 6% per year, what will the value be in 2015?
5. DEFORESTATION During the 1990s, the forested area of Guatemala decreased at an average rate of 1.7%. a. If the forested area in Guatemala in 1990 was about 34,400 square kilometers, write an equation for the forested area for t years after 1990.
b. If this trend continues, predict the forested area in 2015.
6. BUSINESS A piece of machinery valued at $25,000 depreciates at a steady rate of 10% yearly. What will the value of the piece of machinery be after 7 years?
7. TRANSPORTATION A new car costs $18,000. It is expected to depreciate at an average rate of 12% per year. Find the value of the car in 8 years.
8. POPULATION The population of Osaka, Japan, declined at an average annual rate of 0.05% for the five years between 1995 and 2000. If the population of Osaka was 11,013,000 in 2000 and it continues to decline at the same rate, predict the population in 2050.
Chapter 9
191
North Carolina StudyText, Math A
Lesson 9-7
b. If the trend continues, predict the number of sports radio stations in this format for the year 2010.
NAME
9-7
DATE
Word Problem Practice
PERIOD
SCS
MA.A.4.6, MA.A.5.1
Growth and Decay 1. DEPRECIATION The value of a new plasma television depreciates by about 7% each year. Aeryn purchases a 50-inch plasma television for $3000. What is its value after 4 years? Round your answer to the nearest hundred. 2. MONEY Hans opens a savings account by depositing $1200 in an account that earns 3 percent interest compounded weekly. How much will his investment be worth in 10 years? Assume that there are exactly 52 weeks in a year and round your answer to the nearest cent. 3. HIGHER EDUCATION The table lists the average costs of attending a four-year college in the United States during the 2005–2006 school years.
Four-year Private
Tuition and Fees
Room and Board
$5941
$6636
$21,235
$7,791
5. MEDICINE When doctors prescribe medication, they have to consider the rate at which the body filters a drug from the bloodstream. Suppose it takes the human body 6 days to filter out half of the Flu-B-Gone vaccine. The amount of Flu-B-Gone vaccine remaining in the bloodstream x hours after an injection is x , given by the equation y = y0(0.5) − 6 where y0 is the initial amount. Suppose a doctor injects a patient with 20 μg (micrograms) of Flu-B-Gone. a. How much of the vaccine will remain after 1 day? Round your answer to the nearest tenth.
Source: College Board
Russ’s parents invested money in a savings account earning an average of 4.5 percent interest, compounded monthly. After 15 years, they have exactly the right amount to cover the tuition, fees, room and board for Russ’s first year at a public college. What was their initial investment? Round your answer to the nearest dollar.
Chapter 9
192
b. How much of the vaccine will remain after 12 days? Round your answer to the nearest tenth.
c. After how many days will the amount of vaccine be less than 1 μg?
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
College Sector Four-year Public
4. POPULATION In 2007 the U.S. Census Bureau estimated the population of the United States estimated at 301 million. The annual rate of growth is about 0.89%. At this rate, what is the expected population at the time of the 2020 census? Round your answer to the nearest ten million.
NAME
DATE
9-9
Study Guide
PERIOD
SCS
MA.A.3.1, MA.A.3.2, MA.A.3.3
Analyzing Functions with Successive Differences and Ratios Identify Functions Linear functions, quadratic functions, and exponential functions can all be used to model data. The general forms of the equations are listed at the right.
Linear Function
y = mx + b
Quadratic Function
y = ax2 + bx + c
Exponential Function
y = abx
You can also identify data as linear, quadratic, or exponential based on patterns of behavior of their y-values. Example 1
Graph the set of ordered pairs {(–3, 2), (–2, –1), (–1, –2), (0, –1), (1, 2)}. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function.
x
–2
–1
0
1
2
y
4
2
1
0.5
0.25
Start by comparing the first differences. 4 -2 2 -1 1 -0.5 0.5 -0.25 0.25
y
The ordered pairs appear to represent a quadratic function.
The table does not represent a quadratic function. Find the ratios of the y-values. 4 × 0.5 2 × 0.5 1 × 0.5 0.5 × 0.5 0.25 The ratios are equal. Therefore, the table can be modeled by an exponential function.
Exercises Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function. 1. (0, –1), (1, 1), (2, 3), (3, 5)
2. (–3, –1), (–2, –4), (–1, –5), (0, –4), (1, –1)
y
y 0 x x
0
Look for a pattern in each table to determine which model best describes the data. 3.
x
–2
–1
0
1
2
y
6
5
4
3
2
Chapter 9
4.
193
x
–2
–1
0
1
2
y
6.25
2.5
1
0.4
0.16
North Carolina StudyText, Math A
Lesson 9-9
The first differences are not all equal. The table does not represent a linear function. Find the second differences and compare. -2 + 1 -1 + 0.5 -0.5 + 0.25 -0.25
x
0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2 Look for a pattern in the table to determine which model best describes the data.
NAME
DATE
9-9
Study Guide
PERIOD
SCS
(continued)
MA.A.3.1, MA.A.3.2, MA.A.3.3
Analyzing Functions with Successive Differences and Ratios Write Equations Once you find the model that best describes the data, you can write an equation for the function.
Basic Forms
Linear Function
y = mx + b
Quadratic Function
y = ax2
Exponential Function
y = abx
Example
Determine which model best describes the data. Then write an equation for the function that models the data. x
0
1
2
3
4
y
3
6
12
24
48
Step 1 Determine whether the data is modeled by a linear, quadratic, or exponential function. 6 +6 12 + 12 24 + 24 48 First differences: 3 + 3 3
Second differences: 3
y-value ratios:
6
+3
6
×2
×2
12
+6
12
×2
+ 12
24
24 ×2
48
The ratios of successive y-values are equal. Therefore, the table of values can be modeled by an exponential function.
y = abx
Equation for exponential function
3 = a(2)0
x = 0, y = 3, and b = 2
3=a
Simplify.
An equation that models the data is y = 3 ․ 2x. To check the results, you can verify that the other ordered pairs satisfy the function.
Exercises Look for a pattern in each table of values to determine which model best describes the data. Then write an equation for the function that models the data. 1.
2.
3.
Chapter 9
x
–2
–1
0
1
2
y
12
3
0
3
12
x
–1
0
1
2
3
y
–2
1
4
7
10
x
–1
0
1
2
3
y
0.75
3
12
48
192
194
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 2 Write an equation for the function that models the data. The equation has the form y = abx. The y-value ratio is 2, so this is the value of the base.
NAME
DATE
9-9
Practice
PERIOD
SCS
MA.A.3.1, MA.A.3.2, MA.A.3.3
Analyzing Functions with Successive Differences and Ratios Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function.
(
) ( 3)
1 1 2. –1, − , 0, − , (1, 1), (2, 3)
1. (4, 0.5), (3, 1.5), (2, 2.5), (1, 3.5), (0, 4.5)
9
y
y
0
x
O
3. (–4, 4), (–2, 1), (0, 0), (2, 1), (4, 4)
4. (–4, 2), (–2, 1), (0, 0), (2, –1), (4, –2)
y
y
x
0
0
x
Look for a pattern in each table of values to determine which model best describes the data. Then write an equation for the function that models the data. 5. 6. 7. 8.
x
–3
–1
1
3
5
y
–5
–2
1
4
7
x
–2
–1
0
1
2
y
0.02
0.2
2
20
200
x
–1
0
1
2
3
y
6
0
6
24
54
x
–2
–1
0
1
2
y
18
9
0
–9
–18
Lesson 9-9
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
9. INSECTS The local zoo keeps track of the number of dragonflies breeding in their insect exhibit each day. Day
1
2
3
4
5
Dragonflies
9
18
36
72
144
a. Determine which function best models the data. b. Write an equation for the function that models the data. c. Use your equation to determine the number of dragonflies that will be breeding after 9 days.
Chapter 9
195
North Carolina StudyText, Math A
NAME
9-9
DATE
Word Problem Practice
PERIOD
SCS
MA.A.3.1, MA.A.3.2, MA.A.3.3
Analyzing Functions with Successive Differences and Ratios 1. WEATHER The San Mateo weather station records the amount of rainfall since the beginning of a thunderstorm. Data for a storm is recorded as a series of ordered pairs shown below, where the x value is the time in minutes since the start of the storm, and the y value is the amount of rain in inches that has fallen since the start of the storm. (2, 0.3), (4, 0.6), (6, 0.9), (8, 1.2), (10, 1.5) Graph the ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function.
3. BOATS The value of a boat typically depreciates over time. The table shows the value of a boat over a period of time. Years Boat Value ($)
Total Rainfall (in.)
1.2 1.0 0.8 0.6 0.4 0.2 4
6
8
Amount of Isotope Remaining (grams)
2. INVESTING The value of a certain parcel of land has been increasing in value ever since it was purchased. The table shows the value of the land parcel over time.
Land Value (thousands $)
0
1
2
$1.05
$2.10
$4.20
6930
3
3
4
5821.20 4889.81 4107.44
0
1
2
3
4
20
10
5
2.5
1.25
a. Is radioactive decay a linear decay, quadratic decay, or an exponential decay? b. Write an equation to determine how many grams y of a radioactive isotope will be remaining after x half-lives.
4
$8.40 $16.80
Look for a pattern in the table of values to determine which model best describes the data. Then write an equation for the function that models the data.
c. How many grams of the isotope will remain after 11 half-lives? d. Plutonium-238 is one of the most dangerous waste products of nuclear power plants. If the half-life of plutonium-238 is 87.7 years, how long would it take for a 20-gram sample of plutonium-238 to decay to 0.078 grams?
Chapter 9
196
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Time (min)
Year Since Purchasing
8250
Half-Lives Elapsed
10 12
2
4. NUCLEAR WASTE Radioactive material slowly decays over time. The amount of time needed for an amount of radioactive material to decay to half its initial quantity is known as its half-life. Consider a 20-gram sample of a radioactive isotope.
1.4
2
1
Write an equation for the function that models the data. Then use the equation to determine how much the boat is worth after 9 years.
1.6
0
0
NAME
DATE
10-1
PERIOD
Study Guide
SCS
MA.A.4.2
Square Root Functions Dilations of Radical Functions
A square root function contains the square root of a variable. Square root functions are a type of radical function. In order for a square root to be a real number, the radicand, or the expression under the radical sign, cannot be negative. Values that make the radicant negative are not included in the domain. y
Type of graph: curve
Lesson 10-1
Parent function: f(x) = √x y= x
Square Root Function Domain: {x|x ≥ 0} Range: {y|y ≥ 0}
Example
Graph y = 3 √ x . State the domain and range.
Step 1 Make a table. Choose nonnegative values for x. x 0 0.5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
0
Step 2 Plot points and draw a smooth curve. y
y 0 ≈ 2.12
1
3
2
≈ 4.24
4
6
6
≈ 7.35
y=3 x
x
0
The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
Exercises Graph each function, and compare to the parent graph. State the domain and range. 3√ x 1. y = − y
0
Chapter 10
5√ 3. y = − x
2. y = 4 √x
2
2
y
x
y
x
0
197
0
x
North Carolina StudyText, Math A
NAME
DATE
10-1
Study Guide
PERIOD
SCS
(continued)
MA.A.4.2
Square Root Functions Reflections and Translations of Radical Functions Radical functions, like quadratic functions, can be translated horizontally and vertically, as well as reflected across the x-axis. To draw the graph of y = a √ x + h + c , follow these steps. Step 1 Draw the graph of y = a √⎯⎯ x . The graph starts at the origin and passes through the point at (1, a). If a > 0, the graph is in the 1st quadrant. If a < 0, the graph is reflected across the x-axis and is in the 4th quadrant.
Graphs of Square Root Functions
Step 2 Translate the graph ⎪c⎥ units up if c is positive and down if c is negative. Step 3 Translate the graph ⎪h⎥ units left if h is positive and right if h is negative.
Example x + 1 and compare to the parent graph. State the Graph y = - √ domain and range. y
Step 1 Make a table of values. x
-1
0
y
0
-1
1 -1.41
3
8
-2
-3
y= x
x
0
y =- x+ 1
Exercises
Graph each function, and compare to the parent graph. State the domain and range. 1. y =
√x
2. y = √ x- 1
+ 3
y
3. y = - √ x- 1
y
y
0
0
Chapter 10
x
x
x
0
198
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 2 This is a horizontal translation 1 unit to the left of the parent function and reflected across the x-axis. The domain is {x | x ≥ -1 } and the range is {y | y ≤ 0 }.
NAME
DATE
10-1
PERIOD
Practice
SCS
MA.A.4.2
Square Root Functions Graph each function, and compare to the parent graph. State the domain and range. 2. y =
3
y
3. y = √ x-3
+2
y
x
0
√ x
y
x
0
5. y = 2 √ x-1 + 1
4. y = - √x + 1 y
x
0
6. y = - √ x-2 + 2 y
y
x x
0
x
0
7. OHM’S LAW In electrical engineering, the resistance of a circuit ⎯⎯ P , where I is the current in can be found by the equation I = −
√R
amperes, P is the power in watts, and R is the resistance of the circuit in ohms. Graph this function for a circuit with a resistance of 4 ohms.
Current (amperes)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
5 4 3 2 1 0
20 40 60 80 100
Power (watts)
Chapter 10
199
North Carolina StudyText, Math A
Lesson 10-1
4√ x 1. y = −
NAME
DATE
10-1
PERIOD
Word Problem Practice
SCS
MA.A.4.2
Square Root Functions 1. PENDULUM MOTION The period T of a pendulum in seconds, which is the time for the pendulum to return to the point of release, is given by the equation L . The length of the T = 1.11 √ pendulum in feet is given by L. Graph this function.
4. CAPACITORS A capacitor is a set of plates that can store energy in an electric field. The voltage V required to store E joules of energy in a capacitor with a capacitance of C farads is given ⎯⎯⎯ 2E by V = − C .
√
a. Rewrite and simplify the equation for the case of a 0.0002 farad capacitor. 5
Period (sec)
4
b. Graph the equation you found in part a.
3 2
350
1
300 0
4
8
12
16
20
Voltage (volts)
Pendulum Length (ft)
2. EMPIRE STATE BUILDING The roof of the Empire State Building is 1250 feet above the ground. The velocity of an object dropped from a height of h meters 2 gh , is given by the function V = √ where g is the gravitational constant, 32.2 feet per second squared. If an object is dropped from the roof of the building, how fast is it traveling when it hits the street below?
200 150 100
0
2
4
6
8
10
Energy (joules)
c. How would the graph differ if you wished to store E + 1 joules of energy in the capacitor instead?
200
d. How would the graph differ if you applied a voltage of V + 1 volts instead?
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
50
3. ERROR ANALYSIS Gregory is drawing the graph of y = -5 √ x + 1 . He describes the range and domain as {x x ≥ -1}, { y y ≥ 0}. Explain and correct the mistake that Gregory made.
Chapter 10
250
NAME
DATE
10-2
PERIOD
Study Guide
SCS
MA.N.2.3
Simplifying Radical Expressions Product Property of Square Roots
The Product Property of Square Roots and prime factorization can be used to simplify expressions involving irrational square roots. When you simplify radical expressions with variables, use absolute value to ensure nonnegative results. For any numbers a and b, where a ≥ 0 and b ≥ 0, √ ab = √ a √ b.
Product Property of Square Roots
Example 1
180 . Simplify √
√ 180 = √ 22335 = √ 22 √ 32 √5 = 2 3 √ 5 = 6 √ 5
Example 2
Prime factorization of 180 Product Property of Square Roots Simplify. Simplify.
Simplify √ 120a2 · b5 · c4 .
Lesson 10-2
√ 120a2 b5 c4 3 = √2 3 5 a2 b5 c4 √5 √ = √ 22 √ 2 √3 a2 √ b4 b √ c4 √ = 2 √ 2 √3 5 ⎪a⎥ b2 √ b c2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
= 2⎪a⎥b2c2 √ 30b
Exercises Simplify each expression. 28 1. √
2. √ 68
3. √ 60
4. √ 75
5. √162
6. √ 3 · √ 6
7. √ 2 · √ 5
8. √ 5 · √ 10
9. √ 4a2
13. 4 √ 10 3 √ 6
4 10. √9x
4 11. √300a
6 12. √128c
2 3 √ 14. √3x 3x4
2 4 15. √20a b
3 16. √100x y
24a4b2 17. √
4 2 18. √81x y
2 2 19. √150a bc
20. √ 72a6b3c2
2 5 8 21. √45x yz
4 6 2 22. √98x yz
Chapter 10
201
North Carolina StudyText, Math A
NAME
DATE
10-2
Study Guide
PERIOD
SCS
(continued)
MA.N.2.3
Simplifying Radical Expressions Quotient Property of Square Roots
A fraction containing radicals is in simplest form if no radicals are left in the denominator. The Quotient Property of Square Roots and rationalizing the denominator can be used to simplify radical expressions that involve division. When you rationalize the denominator, you multiply the numerator and denominator by a radical expression that gives a rational number in the denominator. Quotient Property of Square Roots
Example
Simplify
For any numbers a and b, where a ≥ 0 and b > 0,
√a
. √−ba = − √b
56 − . √ 45
56 4 14 − − = √ √ 45 95 2 √ 14 3 √ 5 5 2 √ 14 √ =−− √ √ 3 5 5
=−
2 √ 70 15
=−
Simplify the numerator and denominator. √ 5 Multiply by − to rationalize the denominator. √ 5
Product Property of Square Roots
Exercises Simplify each expression. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
√9
√8
1. −
2. − √24
√18
√100
√75
3. −
4. −
√121
√3
8 √2 2 √8
6.
√−52 √−56
7.
√−43 √−25
8.
√−75 √−52
9.
3a √− 10b
11.
100a √− 144b
5. −
2
6
4
8
√4
13. − 3 - √5 √5
15. − 5 + √5
Chapter 10
|
10.
√−yx
12.
75b c √− a
6
4
3 6
2
√8
14. − 2 + √3
√8
16. − + 4 √10 2 √7
202
North Carolina StudyText, Math A
NAME
DATE
10-2
PERIOD
Practice
SCS
MA.N.2.3
Simplifying Radical Expressions Simplify. 1. √24
2. √60
3. √108
√6 4. √8
√14 5. √7
5 √6 6. 3 √12
3 √18 7. 4 √3
3 8. √27tu
5 9. √50p
6 4 5 10. √108x yz
√8
2 4 5 11. √56m np
12. −
13.
2 √− 10
14.
5 √− 32
15.
√−43 √−54
16.
7 √−71 √− 11
√3k √8
18 √− x 9ab 20. √− 4ab
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
17. − 19.
Lesson 10-2
√6
18.
4y √− 3y 2
3
4
3 21. −
8 22. −
5 23. −
24. −
5 - √2
3 + √3 3 √7
√7 + √3
-1 - √27
25. SKY DIVING When a skydiver jumps from an airplane, the time t it takes to free fall a given distance can be estimated by the formula t =
2s − , where t is in seconds and s is √ 9.8
in meters. If Julie jumps from an airplane, how long will it take her to free fall 750 meters? 26. METEOROLOGY To estimate how long a thunderstorm will last, meteorologists can use d3 the formula t = − , where t is the time in hours and d is the diameter of the storm in 216 miles.
√
a. A thunderstorm is 8 miles in diameter. Estimate how long the storm will last. Give your answer in simplified form and as a decimal. b. Will a thunderstorm twice this diameter last twice as long? Explain.
Chapter 10
203
North Carolina StudyText, Math A
NAME
10-2
DATE
PERIOD
Word Problem Practice
SCS
MA.N.2.3
Simplifying Radical Expressions 1. SPORTS Jasmine calculated the height 15 of her team’s soccer goal to be − ⎯⎯ feet. √3 Simplify the expression.
4. PHYSICAL SCIENCE When a substance such as water vapor is in its gaseous state, the volume and the velocity of its molecules increase as temperature increases. The average velocity V of a molecule with mass m at temperature T 3 kT is given by the formula V = − m .
√
2. NATURE In 2004, an earthquake below the ocean floor initiated a devastating tsunami in the Indian Ocean. Scientists can approximate the velocity (in feet per second) of a tsunami in water of depth d 16d . (in feet) with the formula V = √ Determine the velocity of a tsunami in 300 feet of water. Write your answer in simplified radical form.
Solve the equation for k.
5. GEOMETRY Suppose Emeryville Hospital wants to build a new helipad on which medic rescue helicopters can land. The helipad will be circular and made of fire resistant rubber.
a. If the area of the helipad is A, write an equation for the radius r.
2PT − √ M
b. Write an expression in simplified radical form for the radius of a helipad with an area of 288 square meters.
V is the velocity (in meters per second). P is the car’s average power (in watts). M is the mass of the car (in kilograms). T is the time (in seconds). Find the time it takes for a 900-kilogram car with an average 60,000 watts of power to accelerate from stop to 26.82 meters per second (60 miles per hour). Round your answer to the nearest tenth.
Chapter 10
204
c. Using your calculator, find a decimal approximation for the radius. Round your answer to the nearest hundredth.
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. AUTOMOBILES The following formula can be used to find the “zero to sixty” time for a car, or the time it takes for a car to accelerate from a stop to sixty miles per hour. V=
r
NAME
DATE
10-3
PERIOD
Study Guide
SCS
MA.N.2.3
Operations with Radical Expressions Add or Subtract Radical Expressions
When adding or subtracting radical expressions, use the Associative and Distributive Properties to simplify the expressions. If radical expressions are not in simplest form, simplify them. Example 1
- 5 √ . 3 + 6 √ 3 - 4 √6 Simplify 10 √6
- 5 √ - 4 √ + (-5 + 6) √3 10 √6 3 + 6 √3 6 = (10 - 4) √6 = 6 √ 6 + √ 3 Example 2
Associative and Distributive Properties Simplify.
Simplify 3 √ 12 + 5 √ 75 .
3 √ 12 + 5 √ 75 = 3 √ 22 · 3 + 5 √ 52 · 3
Simplify.
= 3 · 2 √ 3 + 5 · 5 √3
Simplify.
+ 25 √ = 6 √3 3
Simplify.
= 31 √ 3
Distributive Property
Exercises + 4 √5 1. 2 √5
- 4 √6 2. √6
- √2 3. √8
+ 2 √5 4. 3 √75
+ 2 √5 - 3 √5 5. √20
+ √6 - 5 √3 6. 2 √3
+ 2 √3 - 5 √3 7. √12
+ 3 √2 - √50 + √24 8. 3 √6
- √2a + 5 √2a 9. √8a + 11. √3
√−31
13. √54
+ √24 10. √54 + 12. √12
√−61
√−31
- √20 + √180 14. √80
+ √18 - √75 + √27 15. √50
1 - 4 √45 +2 − 16. 2 √3
1 - 2 − + 17. √125
18.
√5 √−31
Chapter 10
√3
205
1 √−32 + 3√3 - 4√− 12
North Carolina StudyText, Math A
Lesson 10-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Simplify each expression.
NAME
DATE
10-3
Study Guide
PERIOD
SCS
(continued)
MA.N.2.3
Operations with Radical Expressions Multiply Radical Expressions Multiplying two radical expressions with different radicands is similar to multiplying binomials. Example
)(4 √ 2 - 2 √5 20 + √ 8 ). Multiply (3 √
Use the FOIL method. )(4 √ )(4 √ )( √8 ) (3 √2 - 2 √5 )(4 √ 20 + √ 8 ) = (3 √2 20 ) + (3 √ 2 )( √ 8 ) + (-2 √5 20 ) + (-2 √5 = 12 √ 40 + 3 √ 16 - 8 √ 100 - 2 √ 40
Multiply.
2 = 12 √2 · 10 + 3 · 4 - 8 · 10 - 2 √ 22 · 10
Simplify.
+ 12 - 80 - 4 √ = 24 √10 10
Simplify.
- 68 = 20 √10
Combine like terms.
Exercises Simplify each expression.
(
+ 4 √5 1. 2 √3
(
)
√3 - 2 √6 2. √6
(
)
3 √7 + 2 √5 4. √2
(
√5 - √2 3. √5
)(
(
)
)
(
2 + 4 √2 5. 2 - 4 √2
7. 2 - 2 √5
2
(
)
)
(
)(√2 + √6)
(
)(√5 + 3 √2)
(
)
- 2 √3 12. √2
(
- √2 13. √5
)(
)
+ √6 ) )(√10
- 2 √3 17. 2 √5
2
)(√3 + √6)
- √2 14. √8
- √18 7 √5 + √3 15. √5
Chapter 10
)
- 3 √2 10. √5
2
(
(
2
√8 + √24 8. 3 √2
(
(
)
6. 3 + √6
√2 + 5 √8 9. √8
+ √6 11. √3
)
(
+ 2 √6 ) )(√12
- √45 16. 2 √3
(
- 4 √8 ) )(√12
+ 3 √3 18. √2
206
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(
)
NAME
DATE
10-3
PERIOD
Practice
SCS
MA.N.2.3
Operations with Radical Expressions Simplify each expression. - 4 √30 1. 8 √30
- 7 √5 - 5 √5 2. 2 √5
- 14 √13x + 2 √13x 3. 7 √13x
+ 4 √20 4. 2 √45
- √10 + √90 5. √40
+ 3 √50 - 3 √18 6. 2 √32
+ √18 + √300 7. √27
+ 3 √20 - √32 8. 5 √8
√−72
+ √32 10. √50
+ 4 √28 - 8 √19 + √63 11. 5 √19
(
)
(
)
(
3 √12 + 5 √8 15. 2 √7 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
+ √75 - 2 √40 - 4 √12 12. 3 √10
)
5 √2 - 4 √8 14. √5
√10 + √15 13. √6
(
√−21
)
16. 5 - √15
(
- √18 ) )(√30
+ √12 18. √8
(
)(
- 2 √5 3 √10 + 5 √6 20. 4 √3
+ √6 17. √10
)
+ 2 √8 3 √6 - √5 19. √2
(
2
(
+ √18 ) )(√48
)(
)
21. SOUND The speed of sound V in meters per second near Earth’s surface is given by + 273 , where t is the surface temperature in degrees Celsius. V = 20 √t a. What is the speed of sound near Earth’s surface at 15°C and at 2°C in simplest form?
b. How much faster is the speed of sound at 15°C than at 2°C?
+ 2 √ 22. GEOMETRY A rectangle is 5 √7 3 meters long and 6 √ 7 - 3 √ 3 meters wide. a. Find the perimeter of the rectangle in simplest form. b. Find the area of the rectangle in simplest form. Chapter 10
207
North Carolina StudyText, Math A
Lesson 10-3
9. √14
NAME
10-3
DATE
PERIOD
Word Problem Practice
SCS
MA.N.2.3
Operations with Radical Expressions 1. ARCHITECTURE The Pentagon is the building that houses the U.S. Department of Defense. Find the approximate perimeter of the building, which is a regular pentagon. Leave your answer as a radical expression.
4. RECREATION Carmen surveyed a ski slope using a digital device connected to a computer. The computer model assigned coordinates to the top and bottom points of the hill as shown in the diagram. Write a simplified radical expression that represents the slope of the hill.
23 √ 149 m
B (-2 √ 14 , 7 √ 7)
Ski
y
20
slop
e
15 ) A (2 √ 14 , 5 √7
2. EARTH The surface area of a sphere with radius r is given by the formula 4πr2. Assuming that Earth is close to spherical in shape and has a surface area of about 5.1 × 108 square kilometers, what is the radius of Earth to the nearest ten kilometers?
5
-10
-5
5
O
10
x
5. FREE FALL Suppose a ball is dropped from a building window 800 feet in the air. Another ball is dropped from a lower window 288 feet high. Both balls are released at the same time. Assume air resistance is not a factor and use the following formula to find how many seconds t it will take a ball to fall h feet. 1 √ h t=− 4
a. How much time will pass between when the first ball hits the ground and when the second ball hits the ground? Give your answer as a simplified radical expression.
3 ft 6 √
Deck h = 7 √ 5 ft
b. Which ball lands first?
12 √ 3 ft
House
c. Find a decimal approximation of the answer for part a. Round your answer to the nearest tenth.
Chapter 10
208
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. GEOMETRY The area of a trapezoid is found by multiplying its height by the average length of its bases. Find the area of the deck attached to Mr. Wilson’s house. Give your answer as a simplified radical expression.
10
NAME
DATE
10-4
PERIOD
Study Guide
SCS
MA.N.2.3
Radical Equations Radical Equations Equations containing radicals with variables in the radicand are called radical equations. These can be solved by first using the following steps. Isolate the radical on one side of the equation. Square each side of the equation to eliminate the radical.
Step 1 Step 2
Example 1
√x
Example 2
for x. Solve 16 = −
√x
16 = − 2
( ) √x
2(16) = 2 − 2
2
Solve √ 4x - 7 + 2 = 7.
√ 4x - 7 + 2 = 7 √4x -7+2-2=7-2 √4x -7=5 - 7 )2 = 52 ( √4x
Original equation Multiply each side by 2.
32 = √ x Simplify. 2 2 (32) = ( √ x) Square each side. 1024 = x Simplify. The solution is 1024, which checks in the original equation.
Original equation Subtract 2 from each side. Simplify. Square each side.
4x - 7 = 25 Simplify. 4x - 7 + 7 = 25 + 7 Add 7 to each side. 4x = 32 Simplify. x=8 Divide each side by 4. The solution is 8, which checks in the original equation.
Solve each equation. Check your solution. =8
2.
√ a
4. 7 = √ 26 - n
5.
√ -a
7. 2 √ 3 = √ y
8. 2 √ 3a - 2 = 7
1.
√a
+ 6 = 32
3. 2 √x = 8
6. √ 3r2 = 3
=6
9. √ x-4=6
10. √ 2m + 3 = 5
11. √ 3b - 2 + 19 = 24
13. √ 3r + 2 = 2 √3
14.
√−2x = −12
15.
2 + 5x = 2 16. √6x
17.
√−3x + 6 = 8
3x − 18. 2 + 3 = 11
Chapter 10
209
± √
12. √ 4x - 1 = 3
√−8x = 4
√5
North Carolina StudyText, Math A
Lesson 10-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
10-4
Study Guide
PERIOD
SCS
(continued)
MA.N.2.3
Radical Equations Extraneous Solutions
To solve a radical equation with a variable on both sides, you need to square each side of the equation. Squaring each side of an equation sometimes produces extraneous solutions, or solutions that are not solutions of the original equation. Therefore, it is very important that you check each solution. Example 1
x + 3 = x - 3. Solve √
√ x+3=x-3 Original equation 2 2 ( √ x + 3 ) = (x - 3) Square each side. 2 x + 3 = x - 6x + 9 Simplify. 2 0 = x - 7x + 6 Subtract x and 3 from each side. 0 = (x - 1)(x - 6) Factor. x - 1 = 0 or x - 6 = 0 Zero Product Property x=1 x=6 Solve. √ CHECK √ x+3 =x-3 x+3=x-3 √ √ 1+3 1-3 6+36-3 √ √9 3 4 -2 2 ≠ -2 3=3 Since x = 1 does not satisfy the original equation, x = 6 is the only solution.
Solve each equation. Check your solution. 1.
√a
=a
2. √ a+6=a
2-n 4. n = √
5.
y-1 =y-1 7. √
8. √ 3a - 2 = a
√-a
=a
3. 2 √x = x
6. √ 10 - 6k + 3 = k
9. √ x+2=x
2b + 5 = b - 5 10. √
11. √ 3b + 6 = b + 2
12. √ 4x - 4 = x
2-r =2 13. r + √
2 14. √x + 10x = x + 4
x − 15. -2 = 15
2 - 4x = x + 2 16. √6x
2 17. √2y - 64 = y
2 18. √3x + 12x + 1 = x + 5
Chapter 10
210
√8
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
10-4
DATE
PERIOD
Practice
SCS
MA.N.2.3
Radical Equations Solve each equation. Check your solution. = 8 1. √-b
= 2. 4 √3
+ 3 = 11 3. 2 √4r
= -2 4. 6 - √2y
5. √k +2-3=7
6. √m - 5 = 4 √3
7. √6t + 12 = 8 √6
8. √3j - 11 + 2 = 9
9. √2x + 15 + 5 = 18
3d −-4=2 √ 5
3x 11. 6 − -3=0
12. 6 +
13. y = √y +6
14. √15 - 2x = x
15. √w +4=w+4
16. √17 -k=k-5
17. √5m - 16 = m - 2
18. √24 + 8q = q + 3
19. √4t + 17 - t - 3 = 0
20. 4 - √3m + 28 = m
21. √10p + 61 - 7 = p
2 22. √2x -9=x
√3
5r = -2 √− 6
23. ELECTRICITY The voltage V in a circuit is given by V = √ PR , where P is the power in watts and R is the resistance in ohms. a. If the voltage in a circuit is 120 volts and the circuit produces 1500 watts of power, what is the resistance in the circuit? b. Suppose an electrician designs a circuit with 110 volts and a resistance of 10 ohms. How much power will the circuit produce? 24. FREE FALL Assuming no air resistance, the time t in seconds that it takes an object to √h 4
fall h feet can be determined by the equation t = − . a. If a skydiver jumps from an airplane and free falls for 10 seconds before opening the parachute, how many feet does the skydiver fall? b. Suppose a second skydiver jumps and free falls for 6 seconds. How many feet does the second skydiver fall?
Chapter 10
211
North Carolina StudyText, Math A
Lesson 10-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10.
√x
NAME
DATE
10-4
Word Problem Practice
PERIOD
SCS
MA.N.2.3
Radical Equations 1. SUBMARINES The distance in miles that the lookout of a submarine can see , where h is is approximately d = 1.22 √h the height in feet above the surface of the water. How far would a submarine periscope have to be above the water to locate a ship 6 miles away? Round your answer to the nearest tenth. 2. PETS Find the value of x if the perimeter of a triangular dog pen is 25 meters.
12 m
10 m
4. FIREFIGHTING Fire fighters calculate the flow rate of water out of a particular hydrant by using the following formula. F = 26.9d2 √p F is the flow rate (in gallons per minute), p is the nozzle pressure (in pounds per square inch), and d is the diameter of the hose (in inches). In order to effectively fight a fire, the combined flow rate of two hoses needs to be about 2430 gallons per minute. The diameter of each of the hoses is 3 inches, but the nozzle pressure of one hose is 4 times that of the second hose. What are the nozzle pressures for each hose? Round your answers to the nearest tenth.
x+1m
d-4 2 , where d is the log B=L −
(
4
)
diameter (in inches) and L is the log length (in feet). Suppose the truck carries 20 logs, each 25 feet long, and that the shipment yields a total of 6000 board feet of lumber. Estimate the diameter of the logs to the nearest inch. Assume that all the logs have uniform length and diameter.
5. GEOMETRY The lateral surface area s of a right circular cone, not including the base, is represented by the equation s = πr √ r2 + h2 , where r is the radius of the circular base and h is the height of the cone. a. If the lateral surface area of a funnel is 127.54 square centimeters and its radius is 3.5 centimeters, find its height to the nearest tenth of a centimeter.
b. What is the area of the opening (i.e., the base) of the funnel?
Chapter 10
212
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. LOGGING Doyle’s log rule estimates the amount of usable lumber (in board feet) that can be milled from a shipment of logs. It is represented by the equation
NAME
10-6
DATE
PERIOD
Study Guide
SCS
MA.G.1.1
Distance Formula
The Pythagorean Theorem can be used to derive the Distance Formula shown below. The Distance Formula can then be used to find the distance between any two points in the coordinate plane. Distance Formula
The distance between any two points with coordinates (x1, y1) and (x2, y2) is given by d = √ (x2 - x1)2 + (y2 - y1)2 .
Example 1 Find the distance between the points at (-5, 2) and (4, 5). d = √(x - x1)2 + (y 2 - y1)2 Distance Formula 2 = √ (4 - (-5))2 + (5 - 2)2 (x1, y1) = (-5, 2), (x2, y2) = (4, 5) = √ 92 + 32 Simplify. = √ 81 + 9 Evaluate squares and simplify. = √ 90 The distance is √ 90 , or about 9.49 units.
Example 2 Jill draws a line segment from point (1, 4) on her computer screen to point (98, 49). How long is the segment? (x2 - x1)2 + (y2 - y1)2 d = √ (98 - 1)2 + (49 - 4)2 = √ = √ 972 + 452 = √ 9409 + 2025 11,434 = √ The segment is about 106.93 units long.
Exercises Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the distance between the points with the given coordinates. 1. (1, 5), (3, 1)
2. (0, 0), (6, 8)
3. (-2, -8), (7, -3)
4. (6, -7), (-2, 8)
5. (1, 5), (-8, 4)
6. (3, -4), (-4, -4)
7. (-1, 4), (3, 2)
8. (0, 0), (-3, 5)
9. (2, -6), (-7, 1)
10. (-2, -5), (0, 8)
11. (3, 4), (0, 0)
12. (3, -4), (-4, -16)
Find the possible values of a if the points with the given coordinates are the indicated distance apart. 5 13. (1, a), (3, -2); d = √
14. (0, 0), (a, 4); d = 5
15. (2, -1), (a, 3); d = 5
16. (1, -3), (a, 21); d = 25
17. (1, a), (-2, 4); d = 3
18. (3, -4), (-4, a); d = √65
Chapter 10
213
North Carolina StudyText, Math A
Lesson 10-6
The Distance and Midpoint Formulas
NAME
DATE
10-6
Study Guide
PERIOD
SCS
(continued)
MA.G.1.1
The Distance and Midpoint Formulas Midpoint Formula The point that is equidistance from both of the endpoints is called the midpoint. You can find the coordinates of the midpoint by using the Midpoint Formula. The midpoint M of a line segment with endpoints at (x1, y1) and (x2, y2) is Midpoint Formula
(
x1 + x2 y1 + y2
)
given by M −, − . 2
2
Example 1
Find the coordinates of the midpoint of the segment with endpoints at (–2, 5) and (4, 9).
( x +2 x
y +y
)
1 2 1 2 ,− M −
(
2 -2 + 4 5 + 9
= M −, − 2
(2 2 )
2
Midpoint Formula
)
(x1, y1) = (-2, 5) and (x2, y2) = (4, 9)
2 14 =M − ,−
Simplify the numerators.
= M (1, 7)
Simplify.
Exercises Find the coordinates of the midpoint of the segment with the given endpoints. 2. (4, -2), (0, 6)
3. (7, 2), (13, -4)
4. (-1, 2), (1, 0)
5. (-3, -3), (5, -11)
6. (0, 8), (-6, 0)
7. (4, -3), (-2, 3)
8. (9, -1), (3, -7)
9. (2, -1), (8, 7)
10. (1, 4), (-3, 12)
11. (4, 0), (-2, 6)
12. (1, 9), (7, 1)
13. (12, 0), (2, -6)
14. (1, 1), (9, -9)
15. (4, 5), (-2, -1)
16. (1, -14), (-5, 0)
17. (2, 2), (6, 8)
18. (-7, 3), (5, -3)
Chapter 10
214
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. (1, 6), (3, 10)
NAME
DATE
10-6
PERIOD
Practice
SCS
MA.G.1.1
The Distance and Midpoint Formulas 1. (4, 7), (1, 3)
( 2)
Lesson 10-6
Find the distance between the points with the given coordinates. 2. (0, 9), (-7, -2)
(3 )
1 3. (6, 2), 4, −
1 4. (-1, 7), − ,6
, 3), (2 √3 , 5) 5. ( √3
, -1), (3 √2 , 3) 6. (2 √2
Find the possible values of a if the points with the given coordinates are the indicated distance apart. 7. (4, -1), (a, 5); d = 10 9. (6, -7), (a, -4); d = √18 11. (8, -5), (a, 4); d = √85
8. (2, -5), (a, 7); d = 15 10. (-4, 1), (a, 8); d = √50 12. (-9, 7), (a, 5); d = √29
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the coordinates of the midpoint of the segment with the given endpoints. 13. (4, -6), (3, -9)
14. (-3, -8), (-7, 2)
15. (0, -4), (3, 2)
16. (-13, -9), (-1, -5)
(
) ( 2)
1 1 , 1, − 17. 2, - − 2
(3 ) ( 3 )
2 1 18. − , -1 , 2, −
y
19. BASEBALL Three players are warming up for a baseball game. Player B stands 9 feet to the right and 18 feet in front of Player A. Player C stands 8 feet to the left and 13 feet in front of Player A.
16 12 8
a. Draw a model of the situation on the coordinate grid. Assume that Player A is located at (0, 0).
4
b. To the nearest tenth, what is the distance between Players A and B and between Players A and C?
-8
-4 O
4
8
x
c. What is the distance between Players B and C? 20. MAPS Maria and Jackson live in adjacent neighborhoods. If they superimpose a coordinate grid on the map of their neighborhoods, Maria lives at (-9, 1) and Jackson 1 mile, how far apart are Maria’s and lives at (5, -4). If each unit on the grid represents − 4 Jackson’s homes? Chapter 10
215
North Carolina StudyText, Math A
NAME
DATE
10-6
PERIOD
Word Problem Practice
SCS
MA.G.1.1
The Distance and Midpoint Formulas 1. CHESS Margaret’s last two remaining chess pieces are located at the centers of the squares at opposite corners of the board. If the chessboard is a square with 8-inch sides, about how far apart are the pieces? Round your answer to the nearest tenth.
4. UTILITIES The electric company is running some wires across an open field. The wire connects a utility pole at (2, 14) and a second utility pole at (7, -8). If the electric company wishes to place a third pole at the midpoint of the two poles, at what coordinates should the pole be placed? 5. MARCHING BAND The Ohio State University marching band performs a famous on-field spelling of O-H-I-O called “Script Ohio”. Sometimes they must adjust the usual dimensions of the word to fit it into the limited guest band performance area. The diagram below shows part of the adjusted drill chart. Each point represents one band member, and the coordinates are in yards.
2. ENGINEERING Todd has drawn a cul-de-sac for a residential development plan. He used a compass to draw the cul-de-sac so that it would be circular. On his blueprint, the center of the culde-sac has coordinates (-1, -1) and a point on the circle is (2, 3). What is the radius of the cul-de-sac?
(20, 13)
tuba player
(32, 10)
drum major
line property
9 8 7
(6, 8)
b. Carol is the band member at the top left of the first O in Ohio. She is located at (0, 26). How far away is Carol from the tuba player? Round your answer to the nearest tenth.
6 5 4 3
house
meters
s
ub
r sh
a. How far is the drum major from the tuba player who dots the “i”?
(2, 4) patio
2 1 0
Chapter 10
(5, 1)
tree 1
2
3
4
5 6 7 meters
8
9
216
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. LANDSCAPING Randy plotted a triangular patio on a landscape plan for a client. What is the length of fencing he will need along the patio edge that borders the property line? Round your answer to the nearest tenth.
NAME
DATE
11-1
PERIOD
Study Guide
SCS
MA.A.2.1, MA.A.2.2, MA.A.2.3
Inverse Variation Identify and Use Inverse Variations An inverse variation is an equation in the k form of y = − x or xy = k. If two points (x1, y1) and (x2, y2) are solutions of an inverse variation, then x1 ∙ y1 = k and x2 ∙ y2 = k. x1 ∙ y1 = x2 ∙ y2
Product Rule f or Inverse Variation
x
y
Example
If y varies inversely as x and y = 12 when x = 4, find x when y = 18. Method 2 Use a proportion.
Method 1 Use the product rule. x1 ∙ y1 = x2 ∙ y2 Product rule for inverse variation 4 ∙ 12 = x2 ∙ 18 x1 = 4, y1 = 12, y2 = 18 48 − = x2
x
Divide each side by 18.
18 8 − = x2 3
y
1 2 − x2 = − y1
Proportion for inverse variation
18 4 − x =−
x1 = 4, y1 = 12, y2 = 18
2
12
48 = 18x2 8 − = x2
Simplify.
3
Cross multiply. Simplify.
8 Both methods show that x2 = − when y = 18. 3
Exercises Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Determine whether each table or equation represents an inverse or a direct variation. Explain. 1.
x
y
3
6
5
10
8
16
12
24
2. y = 6x
3. xy = 15
Assume that y varies inversely as x. Write an inverse variation equation that relates x and y. Then solve. 4. If y = 10 when x = 5, find y when x = 2.
5. If y = 8 when x = -2, find y when x = 4.
6. If y = 100 when x = 120, find x when y = 20.
7. If y = -16 when x = 4, find x when y = 32.
8. If y = -7.5 when x = 25, find y when x = 5. 9. DRIVING The Gerardi family can travel to Oshkosh, Wisconsin, from Chicago, Illinois, in 4 hours if they drive an average of 45 miles per hour. How long would it take them if they increased their average speed to 50 miles per hour? 10. GEOMETRY For a rectangle with given area, the width of the rectangle varies inversely as the length. If the width of the rectangle is 40 meters when the length is 5 meters, find the width of the rectangle when the length is 20 meters. Chapter 11
217
North Carolina StudyText, Math A
Lesson 11-1
1 2 From the product rule, you can form the proportion − x2 = − y1 .
NAME
DATE
11-1
Study Guide
PERIOD
SCS
(continued)
MA.A.2.1, MA.A.2.2, MA.A.2.3
Inverse Variation Graph Inverse Variations Situations in which the values of y decrease as the values of x increase are examples of inverse variation. We say that y varies inversely as x, or y is inversely proportional to x. Inverse Variation Equation
an equation of the form xy = k, where k ≠ 0
Example 1 Suppose you drive 200 miles without stopping. The time it takes to travel a distance varies inversely as the rate at which you travel. Let x = speed in miles per hour and y = time in hours. Graph the variation.
Example 2 Graph an inverse variation in which y varies inversely as x and y = 3 when x = 12. Solve for k. xy = k Inverse variation equation 12(3) = k x = 12 and y = 3 36 = k Simplify. Choose values for x and y, which have a product of 36.
The equation xy = 200 can be used to represent the situation. Use various speeds to make a table. y
y
x
y
10
20
30
−6
−6
20
10
−3
−12
30
6.7
−2
−18
2
18
3
12
6
6
40
5
50
4
60
20 10 20
O
3.3
40
60
x
y 24 12 24 x
12
O
Exercises Graph each variation if y varies inversely as x. 1. y = 9 when x = -3 24
y
32
y
12
-24 -12 O
24 x
3. y = -25 when x = 5 100
16
12
16
-32 -16 O
32 x
-50
-24
-32
-100
20
36
10 -20 -10 O
5. y = -18 when x = -9 y
10
20 x
-36 -18 O
x 100
6. y = 4.8 when x = 5.4 7.2
18
50
-100 -50 O
-16
y
y
50
-12
4. y = 4 when x = 5
Chapter 11
2. y = 12 when x = 4
y
3.6 18
36 x
x -7.2 -3.6 O
-10
-18
-3.6
-20
-36
-7.2
218
3.6
7.2
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
NAME
DATE
11-1
Practice
PERIOD
SCS
MA.A.2.1, MA.A.2.2, MA.A.2.3
Inverse Variation Determine whether each table or equation represents an inverse or a direct variation. Explain. 2.
y
x
3. − x = -3
y
x
y
0.25
40
-2
0.5
20
0
0
2
5
2
-8
8
1.25
4
-16
7 4. y = − x
8
Asssume that y varies inversely as x. Write an inverse variation equation that relates x and y. Then graph the equation. 5. y = -2 when x = -12 16
y
24
8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-16 -8
O
6. y = -6 when x = -5
7. y = 2.5 when x = 2
y
y
12 8
16 x
-24 -12 O
-8
-12
-16
-24
12
24 x
O
x
Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then solve. 8. If y = 124 when x = 12, find y when x = -24. 9. If y = -8.5 when x = 6, find y when x = -2.5. 10. If y = 3.2 when x = -5.5, find y when x = 6.4. 11. If y = 0.6 when x = 7.5, find y when x = -1.25. 12. EMPLOYMENT The manager of a lumber store schedules 6 employees to take inventory in an 8-hour work period. The manager assumes all employees work at the same rate. a. Suppose 2 employees call in sick. How many hours will 4 employees need to take inventory? b. If the district supervisor calls in and says she needs the inventory finished in 6 hours, how many employees should the manager assign to take inventory? 13. TRAVEL Jesse and Joaquin can drive to their grandparents’ home in 3 hours if they average 50 miles per hour. Since the road between the homes is winding and mountainous, their parents prefer they average between 40 and 45 miles per hour. How long will it take to drive to the grandparents’ home at the reduced speed?
Chapter 11
219
North Carolina StudyText, Math A
Lesson 11-1
1.
NAME
11-1
DATE
PERIOD
Word Problem Practice
SCS
MA.A.2.1, MA.A.2.2, MA.A.2.3
Inverse Variation 1. PHYSICAL SCIENCE The illumination I produced by a light source varies inversely as the square of the distance d from the source. The illumination produced 5 feet from the light source is 80 foot-candles. Id2 = k 80(5)2 = k 2000 = k
4. BUSINESS In the manufacturing of a certain digital camera, the cost of producing the camera varies inversely as the number produced. If 15,000 cameras are produced, the cost is $80 per unit. Graph the relationship and label the point that represents the cost per unit to produce 25,000 cameras. 300 Price per Unit ($)
Find the illumination produced 8 feet from the same source.
4
14.4
5
12
6
10.29
7
x 30
a. Write an equation that represents the relationship between frequency f and length . Use k for the constant of variation. b. If you have two different length strings, which one vibrates more quickly (that is, which string has a greater frequency)?
3. ELECTRICITY The resistance, in ohms, of a certain length of electric wire varies inversely as the square of the diameter of the wire. If a wire 0.04 centimeter in diameter has a resistance of 0.60 ohm, what is the resistance of a wire of the same length and material that is 0.08 centimeters in diameter?
Chapter 11
10 20 Units Produced (thousands)
5. SOUND The sound produced by a string inside a piano depends on its length. The frequency of a vibrating string varies inversely as its length.
Annual Interest Rate (percent)
18
100
c. Suppose a piano string 2 feet long vibrates 300 cycles per second. What would be the frequency of a string 4 feet long?
220
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Years to Double Money
200
0
2. MONEY A formula called the Rule of 72 approximates how fast money will double in a savings account. It is based on the relation that the number of years it takes for money to double varies inversely as the annual interest rate. Use the information in the table to write the Rule of 72 formula.
y
NAME
DATE
11-2
PERIOD
Study Guide
SCS
MA.A.4.2
Rational Functions 10 The function y = − x is an example of a rational function. Because division by zero is undefined, any value of a variable that results in a denominator of zero must be excluded from the domain of that variable. These are called excluded values of the rational function.
Identify Excluded Values
Example
State the excluded value for each function.
3 a. y = − x The denominator cannot equal zero. The excluded value is x = 0.
4 b. y = − x-5
x-5=0 Set the denominator equal to 0. x=5 Add 5 to each side. The excluded value is x = 5.
State the excluded value for each function. 2 1. y = − x
1 2. y = −
x-3 3. y = −
4 4. y = −
x 5. y = −
5 6. y = - −
3x - 2 7. y = −
x-1 8. y = −
9. y = − x
x-2
2x - 4
x+3
3x
x+1
5x + 10
x-7 10. y = −
x-5 11. y = −
x-2 12. y = −
7 13. y = −
3x - 4 14. y = −
x 15. y = −
2x + 8
6x
3x + 21
x + 11
x+4
7x - 35
16. DINING Mya and her friends are eating at a restaurant. The total bill of $36 is split 36 among x friends. The amount each person pays y is given by y = − x , where x is the number of people. Graph the function. 36 32
Bill per Person ($)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x+1
x-4
28 24 20 16 12 8 4 0
1
2
3
4
5
6
7
8
Number of People Chapter 11
221
North Carolina StudyText, Math A
Lesson 11-2
Exercises
NAME
DATE
11-2
Study Guide
PERIOD
SCS
(continued)
MA.A.4.2
Rational Functions Identify and Use Asymptotes Because excluded vales are undefined, they affect the graph of the function. An asymptote is a line that the graph of a function approaches. a + c has a vertical asymptote at the x-value that A rational function in the form y = − x-b
makes the denominator equal zero, x = b. It has a horizontal asymptote at y = c. 1 Identify the asymptotes of y = − + 2 . Then graph the function.
Example
x-1
Step 1 Identify and graph the asymptotes using dashed lines. vertical asymptote: x = 1 horizontal asymptote: y = 2 Step 2 Make a table of values and plot the points. Then connect them. x
–1
0
2
3
y
1.5
1
3
2.5
y y =2
x
0
1
y = x-1 + 2 x =1
Exercises Identify the asymptotes of each function. Then graph the function. 3 1. y = − x
4 3. y = − x +1
-2 2. y = − x
0
x
2 4. y = − x -3
x
2 5. y = −
Chapter 11
-2 6. y = −
x+1
x-3
y
x
x
0
0
y
0
y
y
0
222
y
x
0
x
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
NAME
DATE
11-2
PERIOD
Practice
SCS
MA.A.4.2
Rational Functions State the excluded value for each function. 2x 3. y = −
-1 1. y = − x
3 2. y = −
x-1 4. y = −
5. y = −
x-5
x+5
x+1 2x + 3
12x + 36
1 6. y = − 5x - 2
Identify the asymptotes of each function. Then graph the function. 3 8. y = − x y
y
x
2 10. y = −
y
x
0
1 11. y = − +2
x+2
x+1
y
x
x
0
2 12. y = − -1
x-3
y
0
x-1
y
x
x
0
13. AIR TRAVEL Denver, Colorado, is located approximately 1000 miles from Indianapolis, Indiana. The average speed of a 1000 plane traveling between the two cities is given by y = − x ,
where x is the total flight time. Graph the function.
0
1000
Average Speed (mph)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
2 9. y = −
Lesson 11-2
1 7. y = − x
800 600 400 200 0
1
2
3
4
5
Total Flight Time
Chapter 11
223
North Carolina StudyText, Math A
NAME
DATE
11-2
Word Problem Practice
PERIOD
SCS
MA.A.4.2
Rational Functions
Average Speed (km/h)
1. BULLET TRAINS The Shinkansen, or Japanese bullet train network, provides high-speed ground transportation throughout Japan. Trains regularly operate at speeds in excess of 200 kilometers per hour. The average speed of a bullet train traveling between Tokyo 515 and Kyoto is given by y = − x , where x is the total travel time in hours. Graph the function.
4. USED CARS While researching cars to purchase online, Ms. Jacobs found that the value of a used car is inversely proportional to the age of the car. The average price of a used car is given by 17,900 x + 1.2
y = − + 100, where x is the age of
the car. What are the asymptotes of the function? Explain why x = 0 cannot be an asymptote.
300 250 200
5. FAMILY REUNION The Gaudet family is holding their annual reunion at Watkins Park. It costs $50 to get a permit to hold the reunion at the park, and the family is spending $8 per person on food. The Gaudets have agreed to split the cost of the event evenly among all those attending.
150 100 50 0
1
2
3
4
Time (hours)
2. DRIVING Peter is driving to his During the trip, Peter makes a 30-minute stop for lunch. The average speed of 40 Peter’s trip is given by y = − , where x + 0.5
x is the total time spent in the car. What
a. Write an equation showing the cost per person y if x people attend the reunion. b. What are the asymptotes of the equation?
are the asymptotes of the function? c. Now assume that the family wants to let a long-lost cousin attend for free. Rewrite the equation to find the new cost per paying person y.
3. ERROR ANALYSIS Nicolas is graphing 20 the equation y = − - 6 and draws a x+3
graph with asymptotes at y = 3 and x = – 6. Explain the error that Nicolas
d. What are the asymptotes for the new equation?
made in his graph.
Chapter 11
224
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
grandparents’ house 40 miles away.
NAME
DATE
11-3
PERIOD
Study Guide
SCS
MA.G.2.4
Simplifying Rational Expressions Identify Excluded Values an algebraic fraction with numerator and denominator that are polynomials
Rational Expression
x2 + 1 y
Example: − 2
Because a rational expression involves division, the denominator cannot equal zero. Any value of the denominator that results in division by zero is called an excluded value of the denominator. Example 1
Example 2
State the excluded
State the excluded
4m - 8 value of − . m+2
x2 + 1 values of − . x2 - 9
Exclude the values for which m + 2 = 0. m+2=0 The denominator cannot equal 0. m + 2 - 2 = 0 - 2 Subtract 2 from each side. m = -2 Simplify. Therefore, m cannot equal -2.
Exclude the values for which x2 - 9 = 0. x2 - 9 = 0 The denominator cannot equal 0. (x + 3)(x - 3) = 0 Factor. x + 3 = 0 or x - 3 = 0 Zero Product Property = -3 =3 Therefore, x cannot equal -3 or 3.
Exercises 2b 1. − 2
12 - a 2. −
x2 - 2 3. − 2
m2 - 4 4. − 2
2n - 12 5. − 2
6. − 2
32 + a
b -8
x +4
2m - 8
2x + 18 x - 16
n -4 2
x -4 7. − 2
x + 4x + 4
a-1 8. − 2
k 2 - 2k + 1 k + 4k + 3
m -1 10. − 2
9. − 2 2
a + 5a + 6 2
2m - m - 1 2x 2 + 5x + 1 x - 10x + 16
25 - n 11. − 2
12. − 2
2 - 2n - 3 − 13. n 2
n + 4n - 5
14. − 2
k 2 + 2k - 3 k - 20k + 64
16. − 2
n - 4n - 5
15. − 2
Chapter 11
Lesson 11-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
State the excluded values for each rational expression.
y2 - y - 2 3y - 12
x 2 + 4x + 4 4x + 11x - 3
225
North Carolina StudyText, Math A
NAME
DATE
11-3
Study Guide
PERIOD
SCS
(continued)
MA.G.2.4
Simplifying Rational Expressions Simplify Expressions
Factoring polynomials is a useful tool for simplifying rational expressions. To simplify a rational expression, first factor the numerator and denominator. Then divide each by the greatest common factor. Example 1
54z 3 . Simplify − 24yz
(6z)(9z 2) 54z 3 − =− 24yz (6z)(4y) (6z)(9z 2) (6z)(4y) 9z 2 =− 4y
The GCF of the numerator and the denominator is 6z.
1
=− 1
Example 2
Divide the numerator and denominator by 6z. Simplify.
3x - 9 Simplify − . State the excluded values of x. 2
x - 5x + 6 3(x 3) 3x - 9 − = − Factor. (x - 2)(x - 3) x 2- 5x + 6 3(x - 3)1 = − Divide by the GCF, x - 3. (x - 2)(x - 3)1 3 = − Simplify. x-2
Exercises Simplify each expression. State the excluded values of the variables.
ab
7n 3 2. − 8
3. − 2
x+2 x -4
m -4 4. − 2
2n - 8 5. − 2
6. − 2
12ab 1. − 2 2
n - 16 2
x -4 7. − 2
x + 4x + 4 2
k -1 9. − 2
k + 4k + 3 2
n - 25 11. − 2
n - 4n - 5
n 2 + 7n + 12 n + 2n - 8
13. − 2
Chapter 11
21n
2
m + 6m + 8
x 2 + 2x + 1 x -1
a 2 + 3a + 2 a + 5a + 6
8. − 2
m 2 - 2m + 1 2m - m - 1
10. − 2
x2 + x - 6 2x - 8 y2 - y - 2 14. − y 2 - 10y + 16
12. − 2
226
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exclude the values for which x2 - 5x + 6 = 0. x2 - 5x + 6 = 0 (x - 2)(x - 3) = 0 x = 2 or x = 3 Therefore, x ≠ 2 and x ≠ 3.
NAME
DATE
11-3
PERIOD
Practice
SCS
MA.G.2.4
Simplifying Rational Expressions State the excluded values for each rational expression. p 2 - 16 p - 13p + 36
4n - 28 1. − 2
2 - 2a - 15 − 3. a 2
2. − 2
n - 49
a + 8a + 15
Simplify each expression. State the excluded values of the variables. 6xyz 3 3x y z
48a
3
4
5c d 7. − 2 4 2 40cd + 5c d
2 - 4m - 12 − 9. m
m-6
2b - 14 11. − 2
6. − 2 5 p 2 - 8p + 12 p-2
8. − m+3 m -9
10. − 2 x 2 - 7x + 10 x - 2x - 15
b - 9b + 14
12. − 2
y 2 + 6y - 16 y - 4y + 4
14. − 2
13. − 2 2
t - 81 15. − 2
r 2 - 7r + 6 r + 6r - 7 r2 + r - 6 r + 4r - 12
t - 12t + 27
16. − 2
2x 2 + 18x + 36 3x - 3x - 36
18. − 2
17. − 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
36k 3np 2 20k np
5. − 2 2
2y 2 + 9y + 4 4y - 4y - 3
19. ENTERTAINMENT Fairfield High spent d dollars for refreshments, decorations, and advertising for a dance. In addition, they hired a band for $550. a. Write an expression that represents the cost of the band as a fraction of the total amount spent for the school dance. b. If d is $1650, what percent of the budget did the band account for? 20. PHYSICAL SCIENCE Mr. Kaminksi plans to dislodge a tree stump in his yard by using a 6-foot bar as a lever. He places the bar so that 0.5 foot extends from the fulcrum to the end of the bar under the tree stump. In the diagram, b represents the total length of the bar and t represents the portion of the bar beyond the fulcrum.
b
fulcrum
t
tree stump
a. Write an equation that can be used to calculate the mechanical advantage. b. What is the mechanical advantage? c. If a force of 200 pounds is applied to the end of the lever, what is the force placed on the tree stump?
Chapter 11
227
North Carolina StudyText, Math A
Lesson 11-3
12a 4. − 3
NAME
11-3
DATE
Word Problem Practice
PERIOD
SCS
MA.G.2.4
Simplifying Rational Expressions 1. PHYSICAL SCIENCE Pressure is equal to the magnitude of a force divided by the area over which the force acts. F P=− A
Gabe and Shelby each push open a door with one hand. In order to open, the door requires 20 pounds of force. The surface area of Gabe’s hand is 10 square inches, and the surface area of Shelby’s hand is 8 square inches. Whose hand feels the greater pressure?
2. GRAPHING Recall that the slope of a line is a ratio of the vertical change to the horizontal change in coordinates for two given points. Write a rational expression that represents the slope of the line containing the points at (p, r) and (7, -3).
4. PACKAGING In order to safely ship a new electronic device, the distribution manager at Data Products Company determines that the package must contain a certain amount of cushioning on each side of the device. The device is shaped like a cube with side length x, and some sides need more cushioning than others because of the device’s design. The volume of a shipping container is represented by the expression (x2 + 6x + 8)(x + 6). Find the polynomial that represents the area of the top of the box if the height of the box is x + 2.
x+2
a. Write a rational expression to express the ratio of public school students to x private school students.
0.0672ws 2 f= − r
b. How many students attended private school?
f = force in pounds w = weight in pounds s = speed in mph r = radius in feet
Chapter 11
228
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. AUTOMOBILES The force needed to keep a car from skidding out of a turn on a particular road is given by the formula below. What force is required to keep a 2000-pound car traveling at 50 miles per hour on a curve with radius of 750 feet on the road? What value of r is excluded?
5. SCHOOL CHOICE During a recent school year, the ratio of public school students to private school students in the United States was approximately 7.6 to 1. The students attending public school outnumbered those attending private schools by 42,240,000.
NAME
DATE
11-4
PERIOD
Study Guide
SCS
MA.A.1.1
Multiplying and Dividing Rational Expressions Multiply Rational Expressions
To multiply rational expressions, you multiply the numerators and multiply the denominators. Then simplify. 2
2c f a 2b . Find −2 · −
Example 1
3cf
5ab
2c 2f a 2b 2a 2bc 2f −2 · − = − 2 3cf
5ab
Multiply.
15ab cf 1
(abcf)(2ac) (abcf)(15b)
= − 1
Simplify.
2ac = −
Simplify.
15b
x+4 x 2 - 16 Find − · − . 2
Example 2
2x + 8
2
x + 8x + 16
(x - 4)(x + 4) 2(x + 4)
x+4 x + 8x + 16
x+4 (x + 4)(x + 4)
x - 16 − · − = − · − 2 2x + 8
(x - 4)(x + 4) 2(x + 4) 1
1
x+41 (x + 4)(x + 4)
Factor.
= −·−
Simplify.
1
x-4 =−
Multiply.
Exercises Find each product. mp 2
6ab a 2 · −2 1. − 2 2
4 2. − · − mp 3
x+2 x-4 3. − · −
m-5 16 4. − ·−
2n - 8 2n + 4 ·− 5. −
x+8 x 2 - 64 6. − · − 2
8x + 8 x-1 ·− 7. − 2
a 2 - 25 a2 - 4 8. − · −
ab
b
x-4
n+2
x-1
8
n-4
x - 2x + 1
2x + 16
2x + 2
x 2 + 6x + 8 x2 - x - 1 · 2− 9. − 2 2 2x + 9x + 4 2
x - 3x + 2 2
n -1 n - 25 ·− 11. − 2 2 n - 7n + 10
a 2 + 7a + 12 a + 2a - 8
n + 6n + 5 a 2 + 3a - 10 a + 2a - 8
· − 13. − 2 2
Chapter 11
m-5
a+2
x + 16x + 64 a-5
2
2m + 1 m - 2m + 1
m -1 10. − · − 2 2 2m - m - 1
3p - 3r 10pr
20p 2r 2 p-r
12. − · − 2 2 2 v 2 + 8v - 4v - 21 14. v− · − 2 2
3v + 6v
229
v + 11v + 24
North Carolina StudyText, Math A
Lesson 11-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2x + 8
NAME
DATE
11-4
Study Guide
PERIOD
SCS
(continued)
MA.A.1.1
Multiplying and Dividing Rational Expressions Divide Rational Expressions To divide rational expressions, multiply by the reciprocal of the divisor. Then simplify. 12c 2f 5a b
Example 1 2
c 2f 2 10ab
÷ −. Find − 2 2
2 2
2
12c f cf 12c f 10ab − ÷ −=− ×− 2 2 2 2 2 2 10ab
5a b
5a b cf 2 12 1c 2f 1 10 1ab 1 × − =− 2 2 5ab b c 2f 2 f 1 a 1
24 =− abf
Example 2 x 2 + 6x - 27 x + 11x + 18
x 2 + 6x - 27 x + 11x + 18
x-3 ÷ − . Find − 2 2 x +x-2
x 2 + 6x - 27 x + 11x + 18
x2 + x - 2 x-3
(x + 9)(x - 3) (x + 9)(x + 2)
(x + 2)(x - 1) x-3
x-3 − ÷− = − × − 2 2 2 x +x-2
= − × − 1
(x + 9)(x - 3) 1 1(x + 9)(x + 2) 1
1
(x + 2)(x - 1) x-31
= − × − =x-1
Find each quotient. 12ab b ÷− 1. − a a 2b 2
n n 2. − ÷− p 4
3xy 2 8
m-5 m-5 4. − ÷−
3. − ÷ 6xy
8
16
2
y - 36 y - 49
y+6 y+7
2n - 4 n2 - 4 5. − ÷− n 2n
6. − ÷− 2
x 2 - 5x + 6 x-3 7. − ÷ −
a 2b 3c 6a 2bc 8. − ÷− 2 2
5
2
x + 6x + 8 x + 4x + 4
15
x+4 x+2
9. − ÷ − 2 n 2 - 5n + 6 n + 3n
3-n 11. − ÷ − 2 4n + 12
2
a + 7a + 12 a2 - 9 13. − ÷ − 2 2 a + 3a - 10
Chapter 11
a - 25
3r t
8rt u
m 2 - 13m + 42 m 2 - 49 − 10. − ÷ m 3m 2 p 2 - 2pr + r 2
p2 - r2
12. − ÷ − p+r p+r 2
a+3 4a - 1
a -9 14. − ÷ − 2 2 2a + 13a - 7
230
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
11-4
PERIOD
Practice
SCS
MA.A.1.1
Multiplying and Dividing Rational Expressions Find each product or quotient. 3 18x 2 15y − · 1. − 2
10y
24rt 2 12r 3t 2 2. − ·− 4 3 2
24x
8r t
(x + 2)(x + 2) 8
4. − · −
(x + 2)(x - 2)
a+3 a-6
4x + 8 x
a-4 ·− 5. − 2 n 2 + 10n + 16 5n - 10
x 6. − · − 2 2
b 2 + 5b + 4 b - 36
t 2 + 6t + 9 t 2 - t - 20 10. − · − 2 2 t - 10t + 25
mn2p3 xy
t + 7t + 12
mnp2 xy
28a2 21a3 ÷− 11. − 2
12. − ÷− 4 2 3
2a ÷ (a + 1) 13. −
z2 - 16 14. − ÷ (z - 4)
7b
35b
a-1
4y + 20 y-3
3z
y+5 2y - 6
4x + 12 6x - 24
2x + 6 x+3
15. − ÷ −
16. − ÷ −
b2 + 2b - 8 2b - 8 ÷− 17. − 2
3x - 3 6x - 6 18. − ÷ − 2 2
2 a2 + 8a + 12 - 4a - 12 − ÷ a 19. − 2 2
20. − ÷ − 2 2
b - 11b + 18
a - 7a + 10
2b - 18
a + 3a - 10
x - 6x + 9
x - 5x + 6
y2 + 6y - 7 y + 8y - 9
y2 + 9y + 14 y + 7y - 18
21. BIOLOGY The heart of an average person pumps about 9000 liters of blood per day. How many quarts of blood does the heart pump per hour? (Hint: One quart is equal to 0.946 liter.) Round to the nearest whole number. 22. TRAFFIC On Saturday, it took Ms. Torres 24 minutes to drive 20 miles from her home to her office. During Friday’s rush hour, it took 75 minutes to drive the same distance. a. What was Ms. Torres’s average speed in miles per hour on Saturday?
Lesson 11-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y 2 - 8y + 16 y-3
8. − · − 2
n + 9n + 8
b 2 + 5b - 6 b + 2b - 8
x - 5x - 14
3y - 9 y - 9y + 20
n-2 7. − · − 2
· − 9. − 2 2
(m - 6)(m + 4) (m + 7)
m+7 (m - 6)(m + 2)
72 3. − · −
a - a - 12
36r t
b. What was her average speed in miles per hour on Friday?
Chapter 11
231
North Carolina StudyText, Math A
NAME
DATE
11-4
PERIOD
Word Problem Practice
SCS
MA.A.1.1
Multiplying and Dividing Rational Expressions 1. JOBS Rosa earned $26.25 for babysitting 1 hours. At this rate, how much will for 3 − 2 she earn babysitting for 5 hours?
5. MANUFACTURING India works in a metal shop and needs to drill equally spaced holes along a strip of metal. The centers of the holes on the ends of the strip must be exactly 1 inch from each end. The remaining holes will be equally spaced.
2. HOMEWORK Alejandro and Ander were working on the following homework problem.
d
d
d
d
d
n - 10 2n + 6 · −. Find − n+3
n+3
Alejandro’s Solution 2n + 6 n+3
n - 10 − ·− n+3
2(n - 10)(n + 3) 1 (n + 3)(n + 3) 1
1 inch
Ander’s Solution
a. If there are x equally spaced holes, write an expression for the number of equal spaces there are between holes.
2(n - 10)(n + 3) 1 n + 31
= −−
= −−
2n - 20 =−
= 2n - 20
n+3
2n + 6 n+3
n - 10 − ·− n+3
Is either of them correct? Explain.
c. Write a rational equation that represents the distance between the holes on a piece of metal that is inches long and must have x equally spaced holes.
5k2m3 expression − represents the area 2ab
2km of a section in a tiled floor and − a
represents the section’s length. Write a rational expression to represent the section’s width. 4. TRAVEL Helene travels 800 miles from Amarillo to Brownsville at an average speed of 40 miles per hour. She makes the return trip driving an average of 60 miles per hour. What is the average rate for the entire trip? (Hint: Recall that t = d ÷ r.)
232
d. How many holes will be drilled in a metal strip that is 6 feet long with a distance of 7 inches between the centers of each screw?
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. Write an expression for the distance between the end screws if the length is ℓ.
3. GEOMETRY Suppose the rational
Chapter 11
1 inch
NAME
11-5
DATE
PERIOD
Study Guide
SCS
MA.A.1.1, MA.A.1.2
Dividing Polynomials Divide Polynomials by Monomials
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Example 1
Example 2
Find (4r2 - 12r) ÷ (2r). 2
4r - 12r (4r2 - 12r) ÷ 2r = −
2r 4r2 12r =−-− 2r 2r 2r 4r2 12r 6 = − -− 2r 2r1 1
= 2r - 6
Find (3x2 - 8x + 4) ÷ (4x). 3x2 - 8x + 4 4x 3x2 2 8x 4 =−-−+− 4x 4x 1 4x 8x 4 3x2 - − +− = 3x4− 4x 4x 4x 3x 1 =−-2+− x 4
(3x2 - 8x + 4) ÷ 4x = − Divide each term. Simplify. Simplify.
Exercises
1. (x3 + 2x2 - x) ÷ x
2. (2x3 + 12x2 - 8x) ÷ (2x)
3. (x2 + 3x - 4) ÷ x
4. (4m2 + 6m - 8) ÷ (2m2)
5. (3x3 + 15x2 - 21x) ÷ (3x)
6. (8m2p2 + 4mp - 8p) ÷ p
7. (8y4 + 16y2 - 4) ÷ (4y2)
8. (16x4y2+ 24xy + 5) ÷ (xy)
15x2 - 25x + 30 5
10. −−
6x3 + 9x2 + 9 3x
12. − 2
m2p2 - 5mp + 6 mp
14. − pr
6a2b2 - 8ab + 12 2a
16. −−
9. − 11. − 13. − 2 2 15. −− 2 9x2y2z - 2xyz + 12x
17. −− xy
Chapter 11
10a2b + 12ab - 8b 2a
m2 - 12m + 42 3m p2 - 4pr + 6r2
2x2y3 - 4x2y2 - 8xy 2xy
2a3b3 + 8a2b2 - 10ab + 12 2a b
18. −− 2 2
233
North Carolina StudyText, Math A
Lesson 11-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find each quotient.
NAME
DATE
11-5
Study Guide
PERIOD
SCS
(continued)
MA.A.1.1, MA.A.1.2
Dividing Polynomials Divide Polynomials by Binomials To divide a polynomial by a binomial, factor the dividend if possible and divide both dividend and divisor by the GCF. If the polynomial cannot be factored, use long division. Example
Find (x2 + 7x + 10) ÷ (x + 3).
Step 1 Divide the first term of the dividend, x2 by the first term of the divisor, x. x x + 3 x2 + 7x + 10 (-) x2 + 3x Multiply x and x + 3. 4x Subtract. Step 2 Bring down the next term, 10. Divide the first term of 4x + 10 by x. x+4 2 x + 3 x + 7x + 10 x2 + 3x 4x + 10 (-) 4x + 12 Multiply 4 and x + 3. -2 Subtract. -2 The quotient is x + 4 with remainder -2. The quotient can be written as x + 4 + − . x+3
Find each quotient. 1. (b2 - 5b + 6) ÷ (b - 2)
2. (x2 - x - 6) ÷ (x - 3)
3. (x2 + 3x - 4) ÷ (x - 1)
4. (m2 + 2m - 8) ÷ (m + 4)
5. (x2 + 5x + 6) ÷ (x + 2)
6. (m2 + 4m + 4) ÷ (m + 2)
7. (2y2 + 5y + 2) ÷ ( y + 2)
8. (8y2 - 15y - 2) ÷ ( y - 2)
2 - 6x - 9 − 9. 8x
4x + 3
2 - 5m - 6 − 10. m
m-6
x3 + 1 x-2
12. −−
6a2 + 7a + 5 2a + 5
14. −
11. − 13. −
Chapter 11
6m3 + 11m2 + 4m + 35 2m + 5
8p3 + 27 2p + 3
234
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
11-5
DATE
PERIOD
Practice
SCS
MA.A.1.1, MA.A.1.2
Dividing Polynomials Find each quotient. 1. (6q2 - 18q - 9) ÷ 9q
3. −− 2
4. −− 3
5. (x2 - 3x - 40) ÷ (x + 5)
6. (3m2 - 20m + 12) ÷ (m - 6)
7. (a2 + 5a + 20) ÷ (a - 3)
8. (x2 - 3x - 2) ÷ (x + 7)
9. (t2 + 9t + 28) ÷ (t + 3)
2m3p2 + 56mp - 4m2p3 8m p
10. (n2 - 9n + 25) ÷ (n - 4)
13. (x3 + 2x2 - 16) ÷ (x - 2)
20w2 + 39w + 18 5w + 6
2 - 5r - 56 − 11. 6r
12. −−
3r + 8
14. (t3 - 11t - 6) ÷ (t + 3)
x3 + 6x2 + 3x + 1 x-2
16. −−
2k3 + k2 - 12k + 11 2k - 3
18. −
15. −−
17. −−
6d3 + d2 - 2d + 17 2d + 3
9y3 - y - 1 3y + 2
19. LANDSCAPING Jocelyn is designing a bed for cactus specimens at a botanical garden. The total area can be modeled by the expression 2x2 + 7x + 3, where x is in feet. a. Suppose in one design the length of the cactus bed is 4x, and in another, the length is 2x + 1. What are the widths of the two designs? b. If x = 3 feet, what will be the dimensions of the cactus bed in each of the designs? 1 20. FURNITURE Teri is upholstering the seats of four chairs and a bench. She needs − 4 1 square yard of fabric for each chair, and − square yard for the bench. If the fabric at 2 the store is 45 inches wide, how many yards of fabric will Teri need to cover the chairs and the bench if there is no waste?
Chapter 11
235
North Carolina StudyText, Math A
Lesson 11-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
12a2b - 3ab2 + 42ab 6a b
2. (y2 + 6y + 2) ÷ 3y
NAME
11-5
DATE
Word Problem Practice
PERIOD
SCS
MA.A.1.1, MA.A.1.2
Dividing Polynomials 1. TECHNOLOGY The surface area (in square millimeters) of a rectangular computer microchip is represented by the expression x2 - 12x + 35, where x is the number of circuits. If the width of the chip is x - 5 millimeters, write a polynomial that represents the length.
2. HOMEWORK Your classmate Ava writes her answer to a homework problem on the chalkboard. She has 6x2 - 12x simplified − as x2 - 12x. Is this 6 correct? If not, what is the correct simplification?
5. CIVIL ENGINEERING Greenshield’s Formula can be used to determine the amount of time a traffic light at an intersection should remain green. G = 2.1n + 3.7 G = green time in seconds n = average number of vehicles traveling in each lane per light cycle Write a simplified expression to represent the average green light time per vehicle.
6. SOLID GEOMETRY The surface area of a right cylinder is given by the formula S = 2πr2 + 2πrh.
4. SHIPPING The Overseas Shipping Company loads cargo into a container to be shipped around the world. The volume of their shipping containers is determined by the following equation.
a. Write a simplified rational expression that represents the ratio of the surface area to the circumference of the cylinder.
b. Write a simplified rational expression that represents the ratio of the surface area to the area of the base.
x3 + 21x2 + 99x + 135 The container’s height is x + 3. Write an expression that represents the area of the base of the shipping container.
Chapter 11
236
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. CIVIL ENGINEERING Suppose 5400 tons of concrete costs (500 + d) dollars. Write a formula that gives the cost C of t tons of concrete.
NAME
DATE
11-6
PERIOD
Study Guide
SCS
MA.A.1.1
Add and Subtract Rational Expressions with Like Denominators To add rational expressions with like denominators, add the numerators and then write the sum over the common denominator. To subtract fractions with like denominators, subtract the numerators. If possible, simplify the resulting rational expression. Example 1
5n 7n +− . Find − 15
5n + 7n 5n 7n − +− =− 15 15 15 12n =− 15 12n 4n = −5 15 4n =− 5
15
Add the numerators.
3x + 2 x-2 3x + 2 3x + 2 - 4x 4x −-−= − x-2 x-2 x-2
4x Find − - − .
Example 2
x-2
The common denominator is x - 2.
Simplify.
2-x =−
Divide by 3.
x-2 -1(x - 2) =− x-2 -1(x - 2) 1 =− x-2 1 -1 =− 1
Simplify.
Subtract. 2-x
= -1(x - 2)
Simplify.
=-1
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find each sum or difference. 3 4 1. − a +− a
x2 x 2. − +−
5x x -− 3. −
11x x 4. − -−
9
8
9
15y
2a - 4 -a +− 5. − a-4
y+7 y+6
a-4
1 7. − - − y+6
x+1 x-5 9. − + − x-2
x-2
x2 + x
x2 + 5x
3x + 2 x+2
x+6 x+2
-− 11. − x x 13. − + −
Chapter 11
8
15y
m+1 3m - 3 6. − + − 2m - 1
2m - 1
3y + 5 5
2y 5
8. − - − 5a 10a 10. − +− 2 2 3b
5a + 2 a
3b
4a + 2 a
12. − -− 2 2 a+6 a-4 14. − +− a+1
237
a+1
North Carolina StudyText, Math A
Lesson 11-6
Adding and Subtracting Rational Expressions
NAME
DATE
11-6
Study Guide
PERIOD
SCS
(continued)
MA.A.1.1
Adding and Subtracting Rational Expressions Add and Subtract Rational Expressions with Unlike Denominators Adding or subtracting rational expressions with unlike denominators is similar to adding and subtracting fractions with unlike denominators. Adding and Subtracting Rational Expressions
Example 1
Step 1 Find the LCD of the expressions. Step 2 Change each expression into an equivalent expression with the LCD as the denominator. Step 3 Add or subtract just as with expressions with like denominators. Step 4 Simplify if necessary.
n+3 8n - 4 Find − n + −.
Example 2
3x 1 -− . Find − 2 x - 4x
4n
Factor each denominator. n=n 4n = 4 . n LCD = 4n 8n - 4 Since the denominator of − is already 4n n+3 4n, only − n needs to be renamed.
x-4
3x 3x 1 1 − -− =− -− 2 x-4
x - 4x
x(x - 4)
x-4
3x x 1 -− ·− =− x-4 x x(x - 4) 3x x -− =− x(x - 4) 2x =− x(x - 4) 2 =− x-4
x(x - 4)
denominator. The LCD is x(x - 4).
1·x=x Subtract numerators. Simplify.
Exercises Find each sum or difference. 7 1 1. − a +−
3 1 2. − +−
5 1 -− 3. − 2
6 3 4. − -− 2 3
8 6 +− 5. − 2
4 2 6. − +−
3a
9x
6x
x
4a
y y-3
x
3a
8
x
h+1
h+2
y y-7
y+3 y - 4y - 21
3 7. − - − y+3
8. − - − 2
a 4 +− 9. −
6 2 10. − +−
a+4
a-4
3(m + 1)
4 2 -− 11. − x - 2y
x + 2y
y+2 y + 5y + 6
2-y y +y-6
3(m - 1)
7 a - 6b 12. − -− 2 2 a 2b 2a - 5ab + 2b q q - 16
q+1 q + 5q + 4
+− 13. − 2 2
14. − +− 2 2
Chapter 11
238
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4(n + 3) n+3 8n - 4 8n - 4 − =−+− n +− 4n 4n 4n 4n + 12 8n - 4 =−+− 4n 4n 12n + 8 =− 4n 3n + 2 =− n
Factor the
NAME
DATE
11-6
PERIOD
Practice
SCS
MA.A.1.1
Adding and Subtracting Rational Expressions n 3n +− 1. − 8
w+9 9
7u 5u 2. − +−
8
16
Lesson 11-6
Find each sum or difference. w+4 9
3. − + −
16
x-6 x-7 -− 4. −
n + 14 n - 14 5. − - −
6 -2 6. − -−
x-5 -2 +− 7. −
r+5 2r - 1 8. − + −
9. − + −
2
x+2
2
5
x+2
r-5
5
c-1
4p + 14 p+4
r-5
c-1
2p + 10 p+4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the LCM of each pair of polynomials. 10. 3a3b2, 18ab3
11. w - 4, w + 2
12. 5d - 20, d - 4
13. 6p + 1, p - 1
14. x2 + 5x + 4, (x + 1)2
15. m2 + 3m - 10, m2 - 4
Find each sum or difference. 6p 5x
2p 3x
m+4 m-3
2 17. − - −
16. −2 - − y+3 y - 16
3y - 2 y + 8y + 16
+ − 18. − 2 2 t+3 t - 3t - 10
4t - 8 - − 20. − 2 2
t - 10t + 25
m-6
p+1 p + 3p - 4
p p+4
19. − + − 2 4y y -y-6
3y + 3 y -4
21. − -− 2 2
22. SERVICE Members of the ninth grade class at Pine Ridge High School are organizing into service groups. What is the minimum number of students who must participate for all students to be divided into groups of 4, 6, or 9 students with no one left out? 23. GEOMETRY Find an expression for the perimeter of rectangle ABCD. Use the formula P = 2 + 2w.
A
5a + 4b 2a + b
B 3a + 2b 2a + b
D
Chapter 11
239
C
North Carolina StudyText, Math A
NAME
11-6
DATE
PERIOD
Word Problem Practice
SCS
MA.A.1.1
Rational Expressions with Unlike Denominators 1. TEXAS Of the 254 counties in Texas, 4 are larger than 6000 square miles. Another 21 counties are smaller than 300 square miles. What fraction of the counties are 300 to 6000 square miles in size?
4. INSURANCE For a hospital stay, Paul’s health insurance plan requires him to 2 the cost of the first day in the pay −
5 1 the cost of the second and hospital and − 5
third days. If Paul’s hospital stay is 3 days and cost him $420, what was the full daily cost?
2. SWIMMING Power Pools installs swimming pools. To determine the appropriate size of pool for a yard, they measure the length of the yard in meters and call that value x. The length and width of the pool are calculated with the diagram below. Write an expression in simplest form for the perimeter of a rectangular pool for the given variable dimensions. x
5. PACKAGE DELIVERY Fredricksburg Parcel Express delivered a total of 498 packages on Monday, Tuesday, and Wednesday. On Tuesday, they delivered 7 less than 2 times the number of packages delivered on Monday. On Wednesday, they delivered the average number delivered on Monday and Tuesday.
4m
3. EGYPTIAN FRACTIONS Ancient Egyptians used only unit fractions, which 1 are fractions in the form − n . Their mathematical notation only allowed for a numerator of 1. When they needed to express a fraction with a numerator other than 1, they wrote it as a sum of unit fractions. An example is shown below. 5 1 1 − =− +− 6
3
b. How many packages were delivered on Monday?
2
Simplify the following expression so it is a sum of unit fractions. 5x + 6 2x − +− 2 2 10x + 12x
Chapter 11
8x
240
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. Write a rational equation that represents the sum of the numbers of packages delivered on Monday, Tuesday, and Wednesday.
2x 5m
NAME
DATE
11-7
PERIOD
Study Guide
SCS
MA.A.1.1
Mixed Expressions and Complex Fractions x+y
b Algebraic expressions such as a + − are c and 5 + − x+3 called mixed expressions. Changing mixed expressions to rational expressions is similar to changing mixed numbers to improper fractions.
Simplify Mixed Expressions
2 Simplify 5 + − n.
5·n 2 2 5+− n =− n +− n 5n + 2
=− n
LCD is n. Add the numerators.
5n + 2
2 Therefore, 5 + − n =− n .
Example 2
3 . Simplify 2 + − n+3
2(n + 3) 3 3 =−+− 2+− n+3 n+3 n+3 2n + 6 3 =−+− n+3 n+3 2n + 6 + 3 = − n+3 2n + 9 =− n+3 2n + 9 3 = −. Therefore, 2 + − n+3 n+3
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write each mixed expression as a rational expression. 6 1. 4 + − a
1 2. − -3
1 3. 3x - − 2
4 4. − -2 2
60 5. 10 + −
h 6. − +2
9x
x
x+5
y y-2
7. − + y2 1 9. 1 + − x
x
h+4
4 8. 4 - − 2x + 1
4 10. − - 2m m-2
x+2 x-3
11. x2 + − 3p 2t
a-2 12. a - 3 + − a+3
q
13. 4m + −
14. 2q2 + − p+q
2 - 4y2 15. − 2
16. t2 + −
y -1
Chapter 11
p+t p-t
241
North Carolina StudyText, Math A
Lesson 11-7
Example 1
NAME
DATE
11-7
Study Guide
PERIOD
SCS
(continued)
MA.A.1.1
Mixed Expressions and Complex Fractions Simplify Complex Fractions
If a fraction has one or more fractions in the numerator or denominator, it is called a complex fraction. a −
Simplifying a Complex Fraction
ad b Any complex fraction − . c where b ≠ 0, c ≠ 0, and d ≠ 0, can be expressed as − bc − d
4 2+− a
Example
Simplify − . a+2 3
− 2a
4
4
2+− − a a +− a − =− a+2 3
−
Find the LCD for the numerator and rewrite as like fractions.
a+2 3 2a + 4 − a − a+2 − 3
−
=
Simplify the numerator.
2a + 4
3 =− ·− a a+2
2(a + 2)
Rewrite as the product of the numerator and the reciprocal of the denominator.
3 =− ·− a
Factor.
6 =− a
Divide and simplify.
a+2
Simplify each expression. 2 2−
1.
5 − 3 3− 4
1 1-− x
4. − 1 1+− x
7. − 3 2 x - 5x
Chapter 11
y x − y2
2. −
3. − 3
4 − y
1 −
1 1-− x
x-3 6. −
5. − 1 1-− x2
2 − 2 x -9
12 x-−
2
x - 25 − y
x − 3
3 − x
8.
x-1 − 8 x-− x-2
242
3 2 − -−
y+2 y-2 1 2 −-− y+2 y-2
9. −
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
DATE
11-7
PERIOD
Practice
SCS
MA.A.1.1
Mixed Expressions and Complex Fractions Write each mixed expression as a rational expression. 4d 2. 7d + − c
b+3 2b
5. 3 + − 2
a-1 6. 2a + −
p+1 p-3
n-1 8. 4n2 + − 2
4 9. (t + 1) + −
4. 5b - −
7. 2p + −
t+5 t -1
n -1
Simplify each expression. 5 10. − 5 2− 6
a-4 − 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a 13. − 2 a - 16 − a
b2 + b - 12 − b2 + 3b - 4 − b-3 − b2 - b
a+1
t+5
12.
x2 - y2 x − x+y − 3x
15.
k2 + 6k k + 4k - 5 − k-8 − k2 - 9k + 8
2
m −
2 3−
16.
6-n 3. 3n + − n
6p 3m − p2
11. − q2 - 7q + 12 − q2 - 16
14. − q-3
10 g-−
g+9 − 5 g-− g+4
17.
− 2
− 2
6 y+−
y-7 7 y-− y+6
18. −
1 19. TRAVEL Ray and Jan are on a 12− -hour drive from Springfield, Missouri, to Chicago, 2
1 Illinois. They stop for a break every 3 − hours. 4
a. Write an expression to model this situation. b. How many stops will Ray and Jan make before arriving in Chicago? 1 -inch wooden rods to reinforce the frame on a futon. 20. CARPENTRY Tai needs several 2 − 4
1 -inch dowel purchased from a hardware store. How many She can cut the rods from a 24 − 2
wooden rods can she cut from the dowel?
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Lesson 11-7
9 1. 14 - − u
NAME
11-7
DATE
PERIOD
Word Problem Practice
SCS
MA.A.1.1
Mixed Expressions and Complex Fractions 1. CYCLING Natalie rode in a bicycle event 2 of for charity on Saturday. It took her − 3
an hour to complete the 18-mile race. What was her average speed in miles per hour?
2. QUILTING Mrs. Tantora sews and sells 3 Amish baby quilts. She bought 42− yards 4 1 of backing fabric, and 2− yards are 4
needed for each quilt she sews. How many quilts can she make with the backing fabric she bought?
a. Find the illumination of the same 3 feet. Round light at a distance of 15− 4
your answer to the nearest hundredth. b. Is there enough illumination at this distance to meet OSHA requirements for lighting? c. In order to comply with OSHA, what is the maximum allowable working distance from this light source? Round your decimal answer to nearest tenth.
4. PHYSICAL SCIENCE The volume of a gas varies directly as the Kelvin temperature T and inversely as the pressure P, where k is the constant of variation. T V=k −
( )
P 13 If k = − , find the volume in liters of 157
13 helium gas at 273 degrees Kelvin and − 3 atmospheres of pressure. Round your answer to the nearest hundredth.
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3. TRAVEL The Franz family traveled from Galveston to Waco for a family reunion. Driving their van, they averaged 30 miles per hour on the way to Waco and 45 miles per hour on the return trip home to Galveston. What is their average rate for the entire trip? (Hint: Remember that average rate equals total distance divided by total time and that time can be represented as a ratio of distance x to rate.)
5. SAFETY The Occupational Safety and Health Administration provides safety standards in the workplace to keep workers free from dangerous working conditions. OSHA recommends that for general construction there be 5 footcandles of illumination in which to work. A foreman using a light meter finds that the illumination of a construction light on a surface 8 feet from the source is 11 foot-candles. The illumination produced by a light source varies inversely as the square of the distance from the source. I is illumination (in foot-candles). k I = −2 d is the distance from the source d (in feet). k is a constant.
NAME
DATE
12-2
Study Guide
PERIOD
SCS
MA.S.1.2
Analyzing Survey Results Summarize Survey Results To make survey data more useful, it can be summarized according to measures of central tendency: mean, median, and mode. Type
Description
Most Useful When
mean
the sum of the data divided by the number of items in the data set
The data sets have no outliers.
median
the middle number of the ordered data, or the mean of the middle two numbers
The data sets have no outliers, but there are no big gaps in the middle of the data.
mode
the number or numbers that occur the most often
The data set has many repeated numbers.
Example
Which measure of central tendency best represents the data? Justify your answer. Then find the measure.
List the values from least to greatest: 25, 26, 27, 28, 30, 32. The data set does not have any outliers, and does not have any repeated numbers. The mean would best represent the data.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
25 + 26 + 27 + 28 + 30 + 32 −− = 28 6
The mean of the data is 28. b. SOCCER A soccer team keeps a record of the number of points it scores in each game: {2, 3, 2, 1, 4, 3, 1, 3, 3, 4}. List the values from least to greatest: 1, 1, 2, 2, 3, 3, 3, 3, 4, 4. The data set has four sets of repeated numbers. The mode best represents the data. The mode is 3, the number that occurs the most often.
Exercises Which measure of central tendency best represents the data? Justify your answer. Then find the measure. 1. DEFECTS A furniture manufacturer keeps records of how many units are defective each day: {7, 12, 9, 8, 10, 14, 8}. 2. SCIENCE TESTS Mr. Wharton records his students’ scores on the last science test: {94, 88, 88, 94, 94, 84, 94, 88, 84, 94}. 3. PUPPIES A veterinarian keeps records of the weights of puppies in ounces: {4.1, 3.8, 5.0, 4.6, 5.6, 4.7, 11.6}. 4. COMMUTING The local newspaper conducted a telephone survey of commuters to see how they get to work each day. The responses were: commuter rail, 22; bus, 17; subway, 18; walking, 15; car, 224.
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Lesson 12-2
a. RESTAURANTS A restaurant records the number of people who order soup at lunch each day: {26, 25, 30, 32, 27, 28}.
NAME
12-2
DATE
Study Guide
PERIOD
SCS
(continued)
MA.S.1.2
Analyzing Survey Results Evaluate Survey Results After a survey’s data has been summarized and a report of the findings and conclusions has been made, it is important to be able to judge the reliability of the report. You can do this by verifying that the sample is truly random, that the sample is large enough to be an accurate representative of the population, and that the source of the data is a reliable one. Also check graphs accompanying surveys for misleading results. Example
MUSIC Given the following portion of a survey report, evaluate the validity of the information and conclusion. Results
Question: What is your favorite band? Sample: 100 concertgoers were randomly selected. Conclusion: America’s favorite band is October Hope.
Choice
Response
October Hope
40%
Rayne
20%
Weimar Republic
10%
Larry Blodgett Trio
30%
Source: October Hope Fan Club
The report says that concertgoers were chosen randomly, but there is no guarantee that a group of concertgoers is representative of America as a whole. In addition, a sample size of 100 may be too small to draw a conclusion from. Also, the report’s source is the “October Hope Fan Club,” which may be biased, considering that the report cites October Hope as America’s favorite band.
Given the following portion of a survey report, evaluate the validity of the information and conclusion. 1. SCHOOL UNIFORMS Survey USA polled 500 randomly selected adults in Cincinnati, Ohio, by telephone. Question: Should public school students wear uniforms? Results: should, 58%; should not, 36%; not sure, 6% Conclusion: Adults in Cincinnati believe students should wear uniforms to school.
2. ELECTIONS State Representative Beck commissioned a poll of 400 randomly selected adults visiting a mall in her district. Question: Do you approve of the job State Representative Beck is doing? Results: yes, 44%; no, 32%; undecided, 24% Conclusion: Senator Beck will win re-election.
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Exercises
NAME
12-2
DATE
PERIOD
Practice
SCS
MA.S.1.2
Analyzing Survey Results Which measure of central tendency best represents the data? Justify your answer. Then find the measure. 1. CALCULATORS The math department counts how many graphing calculators are in each classroom: {20, 19, 20, 20, 18, 19, 20, 18, 19}. 2. BUDGETING The Brady family keeps track of its monthly electric bills: {$134, $122, $128, $127, $136, $120, $129}. 3. AUTOMATED TELLERS A bank keeps track of how many customers use its ATM each hour: {39, 42, 44, 120, 54, 48, 43}.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. HOMEWORK Chris polled 16 of his friends during study hall. Question: Do teachers at Edison High School assign too much homework? Results: yes, 94%; no, 6% Conclusion: Teachers at Edison High School should assign less homework.
5. SMOKING SurveyUSA polled 500 randomly selected adults in Kentucky. Question: Do you want to see smoking banned from restaurants, bars, and most indoor public places in Kentucky? Results: banned, 58%; allowed, 41%; not sure, 1% Conclusion: The United States should ban smoking indoors.
Determine whether the display gives an accurate picture of the survey results. 6. REDEVELOPMENT A local news broadcast commissioned a poll of 600 randomly chosen Providence residents. Question: Do you support or oppose the redevelopment of the waterfront? Conclusion: Providence residents support redeveloping the waterfront.
Waterfront Redevelopment 4VQQPSU 0QQPTF
4USPOHMZ 0QQPTF
6OEFDJEFE 4USPOHMZ 4VQQPSU
7. PETS Ernesto took a poll of randomly selected students at his high school and asked them how many pets they owned. He recorded the results and made the graph shown at the right. Write a valid conclusion using data to support your answer.
Pets None One Two or More 0
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20
40
60
80
100 120
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Lesson 12-2
Given the following portion of a survey report, evaluate the validity of the information and conclusion.
NAME
12-2
DATE
PERIOD
Word Problem Practice
SCS
MA.S.1.2
Analyzing Survey Results 1. PROPERTY TAXES A landlord is keeping track of what he pays each month in property taxes so he can budget accordingly. For the first half of the year, the tax bills were $256, $256, $274, $256, $256, and $274. Which measure of central tendency best represents the data? Justify your answer. Then find the measure.
4. TRANSPORTATION The Ford Township School Board surveyed 86 randomly selected students to find out how students get to school each day. Question: What mode of transportation did you use to get to school today? Conclusion: Most students take the bus to school every day. School Transportation 50 40 30 20 10 0
Bus
Walk
Drive
a. Evaluate the validity of the information.
b. Evaluate the validity of the conclusion.
3. BODYBUILDING A bodybuilder keeps track of how many sets of each exercise he performs each day: {9, 8, 6, 5, 11, 7, 10}. Which measure of central tendency best represents the data? Justify your answer. Then find the measure.
Chapter 12
c. Write a valid conclusion of your own using data to support your answer.
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2. GAS PRICES Quinnipiac University surveyed 1534 randomly chosen registered voters nationwide and asked them, “As a result of the recent rise in gas prices, have you cut back significantly on how much you drive?” Among those who make less than $30,000 a year, 67% said they had cut back on how much they drive while 30% said they had not. Based on this information, a newspaper made the conclusion that “Americans are cutting back on their driving because of high gas prices.” Evaluate the validity of the information and conclusion.
NAME
DATE
PERIOD
CSB 1 Study Guide
SCS
MA.G.1.2, MA.G.2.2
Two-Dimensional Figures Polygons
A polygon is a closed figure formed by a finite number of coplanar segments called sides. The sides have a common endpoint, are noncollinear, and each side intersects exactly two other sides, but only at their endpoints. In general, a polygon is classified by its number of sides. The vertex of each angle is a vertex of the polygon. A polygon is named by the letters of its vertices, written in order of consecutive vertices. Polygons can be concave or convex. A convex polygon that is both equilateral (or has all sides congruent) and equiangular (or all angles congruent) is called a regular polygon. Example Name each polygon by its number of sides. Then classify it as convex or concave and regular or irregular. a. D
E
b.
F
H I
L
G J
The polygon has four sides, so it is a quadrilateral.
The polygon has five sides, so it is a pentagon.
Two of the lines containing the sides of the polygon will pass through the interior of the quadrilateral, so it is concave.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
K
No line containing any of the sides will pass through the interior of the pentagon, so it is convex. All of the sides are congruent, so it is equilateral. All of the angles are congruent, so it is equiangular.
Only convex polygons can be regular, so this is an irregular quadrilateral.
Since the polygon is convex, equilateral, and equiangular, it is regular. So this is a regular pentagon.
Exercises Name each polygon by its number of sides. Then classify it as convex or concave and regular or irregular. 1.
2.
3.
4.
5.
6.
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NAME
DATE
CSB 1 Study Guide
PERIOD
SCS
(continued)
MA.G.1.2, MA.G.2.2
Two-Dimensional Figures Perimeter, Circumference, and Area
The perimeter of a polygon is the sum of the lengths of all the sides of the polygon. The circumference of a circle is the distance around the circle. The area of a figure is the number of square units needed to cover a surface. Example Write an expression or formula for the perimeter and area of each. Find the perimeter and area. Round to the nearest tenth. a.
c
3 in. a
3 ft
b.
4 in. b
c.
2 ft w
5 in.
5 in. r
w
P=a+b+c =3+4+5 = 12 in. 1 A=− bh
C = 2πr = 2π(5) = 10π or about 31.4 in. A = πr2 = π(5)2 = 25π or about 78.5 in2
P = 2 + 2w = 2(3) + 2(2) = 10 ft A = w = (3)(2) = 6 ft2
2 1 =− (4)(3) 2
= 6 in2
Exercises
1.
2. 2.5 cm
3 cm
4 ft
2 cm 3.5 cm
3.
4.
6.25 cm
3.75 cm 14 yd
5 cm
COORDINATE GEOMETRY Graph each figure with the given vertices and identify the figure. Then find the perimeter and area of the figure. 5. A(−2, −4), B(1, 3), C(4, −4)
6. X(−3, −1), Y(−3, 3), Z(4, −1), P(4, 2) y
y
0
Concepts and Skills Bank
x
0
250
x
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the perimeter or circumference and area of each figure. Round to the nearest tenth.
NAME
DATE
PERIOD
CSB 1 Practice
SCS
MA.G.1.2, MA.G.2.2
Two-Dimensional Figures Name each polygon by its number of sides and then classify it as convex or concave and regular or irregular. 2.
1.
3.
Find the perimeter or circumference and area of each figure. Round to the nearest tenth. 17 ft
4.
5.
4 ft
6. 8.1 mm
7 mi
7 mm
8 mm
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
COORDINATE GEOMETRY Graph each figure with the given vertices and identify the figure. Then find the perimeter and area of the figure. 7. O(3, 2), P(1, 2), Q(1, -4), R(3, -4) 8. S(0, 0), T(3, -2), U(8, 0)
CHANGING DIMENSIONS Use the rectangle from Exercise 4. 9. Suppose the length and width of the rectangle are doubled. What effect would this have on the perimeter? Justify your answer.
10. Suppose the length and width of the rectangle are doubled. What effect would this have on the area? Justify your answer.
11. SEWING Jasmine plans to sew fringe around the circular pillow shown in the diagram. 8 in.
a. How many inches of fringe does she need to purchase? b. If Jasmine doubles the radius of the pillow, what is the new area of the top of the pillow? Concepts and Skills Bank
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North Carolina StudyText, Math A
NAME
DATE
CSB 1 Word Problem Practice
PERIOD
SCS
MA.G.1.2, MA.G.2.2
Two-Dimensional Figures 1. ARCHITECTURE In the Uffizi gallery in Florence, Italy, there is a room filled with paintings by Bronzino called the Tribune room (La Tribuna in Italian). The floor plan of the room is shown below.
4. ORIGAMI Jane takes a square piece of paper and folds it in half making a crease that connects the midpoints of two opposite sides. The original piece of paper was 8 inches on a side. What is the perimeter of the resulting rectangle?
La Tribuna
5. STICKS Amy has a box of teriyaki sticks. They are all 15 inches long. She creates rectangles using the sticks by placing them end to end like the rectangle shown in the figure.
What kind of polygon is the floor plan?
2. JOGGING Vassia decides to jog around a city park. The park is shaped like a circle with a diameter of 300 yards. If Vassia makes one loop around the park, approximately how far has she run?
b. What is the perimeter of each rectangle listed in part a?
3. PORTRAITS Around 1550, Agnolo Bronzino painted a portrait of Eleonore of Toledo and her son. The painting measures 115 centimeters by 96 centimeters. What is the area of the painting?
Concepts and Skills Bank
c. Which of the rectangles in part a has the largest area?
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North Carolina StudyText, Math A
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a. How many different rectangles can she make that use exactly 12 of the sticks? What are their dimensions?
300 yards
NAME
DATE
PERIOD
CSB 2 Study Guide
SCS
MA.G.2.1
Chords, Secants, and Tangents Chords and Secants
A circle consists of all points in a plane that are a given distance, called the radius, from a given point called the center. & A segment or line can intersect a circle in several ways. • A segment with endpoints that are at the center and on the circle is a radius.
" # '
% $
• A chord that passes through the circle’s center and made up of collinear radii is a diameter.
−− −−− chord: AE, BD −−− −− −− radius: FB, FC, FD −−− diameter: BD
• A secant is a line that intersects a circle in exactly two points.
secant: AE
• A segment with endpoints on the circle is a chord.
For a circle that has radius r and diameter d, the following are true d r=− 2
1 r=− d 2
d = 2r
Example a. Name the circle. The name of the circle is O. b. Name radii of the circle. −−− −−− −−− −−− AO, BO, CO, and DO are radii. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
O
A
C
B
D
c. Name chords of the circle. −− −−− AB and CD are chords.
Exercises For Exercises 1–5, refer to the figure shown.
"
9
1. Name the circle. 3
2. Name radii of the circle. :
#
3. Name chords of the circle. 4. Name diameters of the circle. 5. Identify a secant.
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NAME
DATE
PERIOD
CSB 2 Study Guide (continued)
SCS
MA.G.2.1
Chords, Secants, and Tangents Tangents
A tangent to a circle intersects the circle in exactly one point, called the point of tangency. There are important relationships involving tangents. A common tangent is a line, ray, or segment that is tangent to two circles in the same plane.
Point of tangency
3
4
1 Point of tangency
5
Example
Copy each figure and draw the common tangents. If no common tangent exists, state no common tangent. b.
a.
These circles have 3 common tangents.
These circles have 2 common tangents.
Copy each figure and draw the common tangents. If no common tangent exists, state no common tangent. 1.
2.
3.
4.
Concepts and Skills Bank
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North Carolina StudyText, Math A
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Exercises
NAME
DATE
PERIOD
CSB 2 Practice
SCS
MA.G.2.1
Chords, Secants, and Tangents For Exercises 1–7, refer to L.
S R
1. Name the circle.
L T
2. Name a radius. W
3. Name a chord. 4. Name a diameter. 5. Name a radius not drawn as part of a diameter. 6. Suppose the radius of the circle is 3.5 yards. Find the diameter. 7. If RT = 19 meters, find LW.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8. Copy the figure and draw the common tangents. If no common tangent exists, state no common tangents.
Identify a secant for each circle. 10.
9. 4
, 3 -
.
5
14. SUNDIALS Herman purchased a sundial to use as the centerpiece for a garden. The diameter of the sundial is 9.5 inches. Find the radius of the sundial.
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NAME
DATE
PERIOD
CSB 2 Word Problem Practice
SCS
MA.G.2.1
Chords, Secants, and Tangents 1. WHEELS Zack is designing wheels for a concept car. The diameter of the wheel is 18 inches. Zack wants to make spokes in the wheel that run from the center of the wheel to the rim. In other words, each spoke is a radius of the wheel. How long are these spokes?
4. PLAZAS A rectangular plaza has a surrounding circular fence. The diagonals of the rectangle pass from one point on the fence through the center of the circle to another point on the fence. 377 ft
245 ft
2. CAKE CUTTING Kathy slices through a circular cake. The cake has a diameter of 14 inches. The slice that Kathy made is straight and has a length of 11 inches. Based on the information in the figure, is the diagonal a radius, a diameter, a secant or a chord of the circle?
a. If he places them down as shown above, how many common tangents are there?
3. COINS Three identical circular coins are lined up in a row as shown.
10
10
10
b. If he places them down as shown below, how many common tangents are there?
Draw the common tangents.
Concepts and Skills Bank
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North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Did Kathy cut along a radius, a diameter, a secant or a chord of the circle?
5. EXERCISE HOOPS Taiga has two circular loops that he can twirl around his body for exercise.
NAME
DATE
PERIOD
CSB 3 Study Guide
SCS
MA.G.2.5
Surface Areas Lateral and Surface Area altitude
Lateral Area of a Prism
If a prism has a lateral area of L square units, a height of h units, and each base has a perimeter of P units, then L = Ph.
Surface Area of a Prism
If a prism has a surface area of S square units, a lateral area of L lateral edge square units, and each base has an area of B square units, then S = L + 2B or S = Ph + 2B
Lateral Area of a Cylinder Surface Area of a Cylinder
If a cylinder has a lateral area of L square units, a height of h units, and a base has a radius of r units, then L = 2πrh. If a cylinder has a surface area of S square units, a height of h units, and a base has a radius of r units, then S = L + 2B or 2πrh + 2πr2.
lateral face pentagonal prism base height
axis base
radius of base
Example 1 Find the lateral and surface area of the regular pentagonal prism above if each base has a perimeter of 75 centimeters and the height is 10 centimeters. L = Ph = 75(10) = 750
Lateral area of a prism P = 75, h = 10 Multiply.
S = L + 2B 1 = 750 + 2 − aP
36°
(2 )
= 750 +
(
B
)
7.5 − (75) tan 36°
15 cm
7.5 tan 36° = − a
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
≈ 1524.2
7.5 a=− tan 36°
The lateral area is 750 square centimeters and the surface area is about 1524.2 square centimeters. Example 2 Find the lateral and surface area of the cylinder. Round to the nearest tenth. If d = 12 cm, then r = 6 cm. L = 2πrh Lateral area of a cylinder 12 cm = 2π(6)(14) r = 6, h = 14 ≈ 527.8 Use a calculator. 14 cm S = 2πrh + 2πr2 Surface area of a cylinder 2 ≈ 527.8 + 2π(6) 2πrh ≈ 527.8, r = 6 ≈ 754.0 Use a calculator. The lateral area is about 527.8 square centimeters and the surface area is about 754.0 square centimeters.
Exercises Find the lateral area and surface area of each figure. Round to the nearest tenth if necessary. 2. 1. 3m
10 in.
10 m
6 in.
4m
Concepts and Skills Bank
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North Carolina StudyText, Math A
NAME
DATE
PERIOD
CSB 3 Study Guide (continued)
SCS
MA.G.2.5
Surface Area Lateral and Surface Area 1 The lateral area L of a regular pyramid is L = − Pℓ, where ℓ 2 is the slant height and P is the perimeter of the base.
Lateral Area of a Regular Pyramid
1 The surface area S of a regular pyramid is S = − Pℓ + B,
Surface Area of a Cone
height base
2
Surface Area of a Regular Pyramid
Lateral Area of a Cone
lateral edge slant height
where ℓ is the slant height, P is the perimeter of the base, and B is the area of the base. V
The lateral area L of a right circular cone is L = πr, where r is the radius and is the slant height.
V altitude
base
The surface area S of a right cone is S = πr + πr2, where r is the radius and is the slant height.
axis
oblique cone
slant height base
right cone
Example 1
Find the lateral area and surface area of a regular square pyramid if the length of a side of the base is 12 centimeters and the height is 8 centimeters. Find the slant height. ℓ2 = 62 + 82 Pythagorean Theorem 2 ℓ = 100 Simplify. ℓ = 10 cm Take the positive square root of each side. 1 Pℓ L=−
Lateral area of a regular pyramid
= 240 cm2
1 S=− Pℓ + B
Surface area of a regular pyramid
2
P = 4 12 or 48, ℓ = 10
= 240 + 144
1 − Pℓ = 240, B = 12 · 12 or 144 2
Simplify.
= 384 cm2
Simplify.
Example 2
Find the lateral area and surface area of a right cone if the radius is 6 inches and the height is 8 inches. Find the slant height. ℓ2 = 62 + 82 Pythagorean Theorem ℓ2 = 100 Simplify. ℓ = 10 in. Take the positive square root of each side. L = πrℓ = π(6)(10) ≈ 188.5 in2
Lateral area of a right cone r = 6, ℓ = 10 Simplify.
Exercises
S = πrℓ + πr2 ≈ 188.5 + π(62) ≈ 301.6 in2
Surface area of a right cone
πrℓ ≈ 188.4, r = 6 Simplify.
Find the lateral area and surface area of each figure. Round to the nearest tenth. 1.
2. 20 cm
9 cm
15 cm
Concepts and Skills Bank
12 cm
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North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2 1 =− (48)(10) 2
NAME
DATE
PERIOD
CSB 3 Practice
SCS
MA.G.2.5
Surface Areas Find the lateral and surface area of each figure. Round to the nearest tenth if necessary. 2. 1. 15 cm
32 cm
5 ft
10 ft 8 ft
15 cm
5 ft
3.
4. 4m
7 ft
8.5 m
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5.
6. 12 m 10 yd 7m
9 yd
7.
8. 5m
7 cm
4m
21 cm
9. GAZEBOS The roof of a gazebo is a regular octagonal pyramid. If the base of the pyramid has sides of 0.5 meter and the slant height of the roof is 1.9 meters, find the
Concepts and Skills Bank
259
North Carolina StudyText, Math A
NAME
DATE
CSB 3 Word Problem Practice
PERIOD
SCS
MA.G.2.5
Surface Area 1. PAPER MODELS Patrick is making a paper model of a castle. Part of the model involves 20 cm 20 cm 15 cm cutting out the net shown and folding it into a pyramid. The pyramid has a square base. What is the lateral area of the resulting pyramid?
3. CAKES A cake is a rectangular prism with height 4 inches and base 12 inches by 15 inches. Wallace wants to apply frosting to the sides and the top of the cake. What is the surface area of the part of the cake that will have frosting?
4. MEGAPHONES A megaphone is formed by taking a cone with a radius of 20 centimeters and an altitude of 60 centimeters and cutting off the tip. The cut is made along a plane that is perpendicular to the axis of the cone and intersects the axis 12 centimeters from the vertex. Round your answers to the nearest hundredth.
2. TOWERS A circular tower is made by placing one cylinder on top of another. Both cylinders have a height of 18 inches. The top cylinder has a radius of 18 inches and the bottom cylinder has a radius of 36 inches.
a. What is the lateral surface area of the original cone?
18 in.
b. What is the lateral surface area of the tip that is removed?
a. What is the total surface area of the tower? Round your answer to the nearest hundredth.
c. What is the lateral surface area of the megaphone? b. Another tower is constructed by placing the original tower on top of another cylinder with a height of 18 inches and a radius of 54 inches. What is the total surface area of the new tower? Round your answer to the nearest hundredth.
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North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
18 in.
NAME
DATE
PERIOD
CSB 4 Study Guide
SCS
MA.G.2.3
Volume Volume
The measure of the amount of space that a three-dimensional figure encloses is the volume of the figure. Volume of a Prism
If a prism has a volume of V cubic units, a height of h units, and each base has an area of B square units, then V = Bh.
Volume of a Cylinder
If a cylinder has a volume of V cubic units, a height of h units, and the bases have a radius of r units, then V = πr 2h.
cubic foot cubic yard 27 cubic feet = 1 cubic yard r
Example 1 of the prism.
Find the volume
h
Example 2 Find the volume of the cylinder. 3 cm
4 cm 4 cm 3 cm
7 cm
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
V = Bh Volume of a prism = (7)(3)(4) B = (7)(3), h = 4 = 84 Multiply. The volume of the prism is 84 cubic centimeters.
V = πr2h Volume of a cylinder 2 = π(3) (4) r = 3, h = 4 ≈ 113.1 Simplify. The volume is about 113.1 cubic centimeters.
Exercises Find the volume of each figure. Round to the nearest tenth. 1.
8 ft
3.
2.
8 ft
1.5 cm 4 cm
8 ft
3 cm
4. 2 cm
2 ft 1 ft
Concepts and Skills Bank
261
18 cm
North Carolina StudyText, Math A
NAME
DATE
PERIOD
CSB 4 Study Guide (continued)
SCS
MA.G.2.3
Volume Volume
This figure shows a prism and a pyramid that have the same base and the same height. It is clear that the volume of the pyramid is less than the volume of the prism. More specifically, the volume of the pyramid is one-third of the volume of the prism. Volume of a Pyramid
If a pyramid has a volume of V cubic units, a height of h units, 1 and a base with an area of B square units, then V = − Bh. 3
For a cone, the volume is one-third the product of the height and the area of the base. The base of a cone is a circle, so the area of the base is πr2. Volume of a Cone
r
If a cone has a volume of V cubic units, a height of h units, and 1 2 the bases have a radius of r units, then V = − πr h. 3
Example 1
Find the volume of the square pyramid.
1 V=− Bh
10 ft
Volume of a pyramid
3 1 = −(8)(8)10 3
B = (8)(8), h = 10
8 ft 8 ft
≈ 213.3 Multiply. The volume is about 213.3 cubic feet. Example 2
h
Find the volume of the cone.
5 cm
Volume of a cone 12 cm r = 5, h = 12
≈ 314.2 Simplify. The volume of the cone is about 314.2 cubic centimeters.
Exercises Find the volume of each figure. Round to the nearest tenth if necessary. 1.
2.
10 ft
15 ft
6 ft
8 ft 12 ft
10 ft
4.
3. 10 cm 6 cm
12 in. 30 in.
Concepts and Skills Bank
262
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 2 πr h V=− 3 1 =− π(5)2(12) 3
NAME
DATE
PERIOD
CSB 4 Practice
SCS
MA.G.2.3
Volume Find the volume of each figure. Round to the nearest tenth if necessary. 1.
2.
26 m
5 in.
10 m
17 m
5 in.
9 in. 5 in.
3.
4. 16 mm
25 ft
7 ft
17.5 mm
5.
6.
9 ft
13 yd 19 ft
9.2 yd
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9.2 yd
7. AQUARIUM Mr. Gutierrez purchased a cylindrical aquarium for his office. 1 inches and a radius of 21 inches. What is the The aquarium has a height of 25 − 2
volume of the aquarium in cubic feet?
8. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is 4 meters in diameter and the height of the shed is 3.8 meters. What is the volume of the shed?
9. HISTORY The start of the pyramid age began with King Zoser’s pyramid, erected in the 27th century B.C. In its original state, it stood 62 meters high with a rectangular base that measured 140 meters by 118 meters. Find the volume of the original pyramid.
Concepts and Skills Bank
263
North Carolina StudyText, Math A
NAME
DATE
PERIOD
CSB 4 Word Problem Practice
SCS
MA.G.2.3
Volume 1. TRASH CANS The Meyer family uses a kitchen trash can shaped like a cylinder. It has a height of 18 inches and a base diameter of 12 inches. What is the volume 18 in. of the trash can? Round your answer to the nearest tenth of a cubic inch.
4. ICE CREAM DISHES The part of a dish designed for ice cream is shaped like an upside-down cone. The base of the cone has a radius of 2 inches and the height is 1.2 inches.
12 in.
What is the volume of the cone? Round your answer to the nearest hundredth. 2. BENCH Inside a lobby, there is a piece of furniture for sitting. The furniture is shaped like a simple block with a square base 6 feet on each side and a height of 3 feet. 1−
5. STAGES A stage has the form of a square pyramid with the top sliced off along a plane parallel to the base. The side length of the top square is 12 feet and the side length of the bottom square is 16 feet. The height of the stage is 3 feet.
5
3
1 5 ft 6 ft
12 feet
What is the volume of the seat?
3 feet 16 feet
a. What is the volume of the entire square pyramid that the stage is part of?
3. FRAMES Margaret makes a square frame out of four pieces of wood. Each piece of wood is a rectangular prism with a length of 40 centimeters, a height of 4 centimeters, and a depth of 6 centimeters. What is the total volume of the wood used in the frame?
Concepts and Skills Bank
b. What is the volume of the top of the pyramid that is removed to get the stage?
c. What is the volume of the stage?
264
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6 ft
NAME
DATE
PERIOD
CSB 5 Study Guide
SCS
MA.D.1.1
Vertex-Edge Graphs In graph theory, a vertex-edge graph is a collection of points in which a pair of points called vertices, or nodes, are connected by a set of segments or arcs, called edges. In a simple graph, only one edge is allowed between any two vertices. Unlike a simple graph, a multigraph may have several edges connecting the same pair of vertices. These are called parallel edges. A multigraph may also contain edges that connect a vertex to itself. These are called loops. edge loop vertex parallel edges Multigraph
Simple Graph
One way to describe a graph is by the number of vertices and edges, notated G(n, m), where n is the number of vertices and m is the number of edges of a graph G. A more specific way to describe a graph is by listing the vertices and edges. Example 1
Use the notation G(n, m) to name the multigraph.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The multigraph has 6 vertices and 9 edges. Thus, the graph can be described as G(6, 9).
#
"
$
%
' &
Example 2
Draw a graph with V = {L, M, N} and E = {{L, M}, {M, N}}.
Two graphs that satisfy the given conditions are shown below. .
-
.
/
/
W
X
Z
EXERCISES 1. Use the notation G(n, m) to name the multigraph.
Y
Draw each graph described below. 2. V = {R, S, T, U, V}, E = {{R, T}, {S, U}, {U, V}, {S, V}} 3. V = {A, B, C, D}, E = ∅ Concepts and Skills Bank
265
North Carolina StudyText, Math A
NAME
DATE
CSB 5 Study Guide
PERIOD
SCS
(continued)
Complete Graphs
MA.D.1.1
"
The degree of a vertex C, denoted deg (C), is the number of edges connected to C. In the graph at the right, deg (C) = 3 and deg (B) = 1.
#
%
$
&
Describing Graphs For any graph G(n, m), where n is the number of vertices and m is the number of edges, the sum of the degrees of the vertices is twice the number of edges.
Example 1
Draw a graph with four vertices, where deg (A) = 1, deg (B) = 2,
deg (C) = 2, and deg (D) = 1. The sum of the degrees of the vertices is 1 + 2 + 2 + 1 or 6. Thus, the graph will have 6 ÷ 2 or 3 edges. Two graphs that satisfy the given conditions are shown below. "
% "
#
#
$
.
%
/
$
A complete graph with n vertices, denoted Kn, is a graph in which each pair of vertices is connected by exactly one edge. The graph at the right is an example of a complete graph. You can use a formula to find the number of edges in a complete graph.
0
2
,6
1
For any complete graph Kn,
(
n(n – 1)
)
n(n − 1) , and 1. there are − edges; that is, G(n, m) = G n, − 2 2
2. the maximum degree of every vertex is n − 1.
Example 2
Determine whether each graph could be complete. Explain.
a. G(3, 4) Find the total number of edges for a complete graph with 3 vertices or K3.
b. G(6, 15) Find the total number of edges for a complete graph with 6 vertices or K6.
n(n - 1) 3(2) − = − or 3 2 2
n(n - 1) 6(5) − = − or 15 2 2
Since 3 ≠ 4, the graph cannot be complete.
Since 15 = 15, the graph could be complete.
EXERCISES Draw each graph described below. 1. 5 vertices, each with degree 2
2. 6 vertices, each with degree 1
Determine whether each graph could be complete. Explain your reasoning. 3. G(14, 91) 4. G(100, 4900) Concepts and Skills Bank
266
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Complete Graphs
NAME
DATE
CSB 5 Practice
PERIOD
SCS
MA.D.1.1
Vertex-Edge Graphs Use the notation G(n, m) to name each multigraph. 1.
2.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Draw each graph described below. 3. G(4, 7)
4. G(5, 10)
5. V = {A, B, C, D}, E = {{A, B}, {A, C}, {C, D}}
6. V = {M, N, O, P}, E = {{M, N}, {M, P}, {N, O}, {N, P}, {O, P}}
7. 5 vertices; deg(A) = 1, deg(B) = 2, deg(C) = 1, deg(D) = 3, deg(E) = 4
8. 6 vertices; deg(A) = 2, deg(B) = 1, deg(C) = 3, deg(D) = 1, deg(E) = 1, deg(F) = 3
Determine whether each graph could be complete. Explain your reasoning. 9. G(2, 3)
Concepts and Skills Bank
10. G(9, 45)
267
North Carolina StudyText, Math A
NAME
DATE
CSB 5 Word Problem Practice
PERIOD
SCS
MA.D.1.1
Vertex-Edge Graphs 1. CAMPUS At Golden State Community College, the student union is located at the center of campus and walkways from the union lead to a dormitory, the recreation center, the business administration building, the math building, and the education building. Both the math and education buildings have walkways that lead to the business administration building. The dormitory has a walkway that leads to the recreation center. Draw a possible graph for this campus.
268
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. TRAINS For a class project, you have been named the director of transportation for a train company. The company wants to connect cities in Ohio, New Jersey, Illinois, South Carolina, Texas, and California. Let each city be represented by a vertex and let each city have a direct line with all other cities. How many train lines need to be built to accommodate these connections?
2. INTERSTATES State Route 181 goes from San Antonio to Floresville and Corpus Christi. Route 97 goes through Floresville, Pleasanton, Jourdanton, and Charlotte. Interstate 37 connects San Antonio, Pleasanton, and Corpus Christi. Draw a graph to model this situation.
Concepts and Skills Bank
3. AIRPLANES There is a roundtrip from New York to Los Angeles and from New York to Chicago. Two roundtrip flights leave Los Angeles. One flies to Chicago and the other to Dallas. Out of Atlanta, there are three round trips; Dallas, Chicago, and New York. The only other roundtrip is from Dallas to Chicago Draw a graph to model the flight paths of the airline. Let each round trip be represented by one edge.
NAME
DATE
PERIOD
CSB 7 Study Guide
SCS
MA.D.1.1, MA.D.1.2
Walks, Trails, and Paths • A walk is the course taken from one vertex to another vertex along edges of a graph. • A trail is a walk in which no edge is used more than once. • A path is a walk in which no vertex or edge is repeated. Example 1
Refer to the multigraph shown.
2
b
j
a. Find a L-Q walk. A walk can repeat vertices and edges. Thus, one L-Q walk is e, f, g, h, e, d, j.
/ a
c
d
b. Find a L-Q trail. A trail can repeat vertices but cannot repeat edges. Thus, one L-Q trail is e, f, g, h, a, j.
-
f
.
e
0 g
i
1
h
c. Find a L-Q path. A path cannot repeat vertices or edges. Thus, one L-Q path is a, j.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
d. Find a circuit for P. A circuit can repeat vertices but not edges. So, one circuit for P is g, f, d, b, a, e, i. e. Find a cycle for P. A cycle cannot repeat vertices or edges. Thus, one cycle for P is h, a, c, g. A graph is connected if there is a path between every two vertices. A trail that includes each edge exactly once and has the same starting and ending vertex is called an Euler circuit. A trail that includes each edge exactly once, but has different starting and ending vertices is called an Euler path. Euler Circuit Test A connected multigraph contains an Euler circuit if and only if the degree of each vertex is even.
Example 2
Refer to the multigraph shown.
a. Does the graph have an Euler circuit? Why or why not? No; each vertex does not have an even degree. b. Does the graph have an Euler path? Why or why not? No; it is not possible to include each edge exactly once. EXERCISES 1. Is the walk g, h, i, a, b a circuit, a path, or a trail? Use the most specific name. 2. If possible, find an Euler circuit.
b
a c
3. Is there an Euler path? Why or why not? i
Concepts and Skills Bank
269
e
d
h
f
g
North Carolina StudyText, Math A
NAME
DATE
CSB 7 Study Guide
(continued)
PERIOD
SCS
MA.D.1.1, MA.D.1.2
Algorithms An algorithm is a sequence of instructions that solves all cases of a certain type of problem. Fleury’s Algorithm If a connected multigraph contains an Euler circuit, the circuit can be located by the following process. Step 1 Choose a starting vertex and walk the edges. Mark off the edges walked as you go. Step 2 Do not choose a bridge, unless not choosing a bridge will disconnect the circuit.
Example 1
Find an Euler circuit in the connected multigraph.
Using Fleury’s algorithm begin at vertex Z c a and walk edge a. At vertex W you can choose V from 3 edges, none are bridges, let’s choose d. f At vertex X you can choose from 3 edges, h let’s choose e. Then choose f and j. Continue to check for bridges at each vertex. To complete the circuit, choose g, h, c, b, and i, respectively. Z So, an Euler circuit for this multigraph is a, d, e, f, j, g, h, c, b, i.
W d
e
X
b
g j
i
Y
A Hamilton path is a path that visits each vertex exactly once. If the path returns to the starting vertex, it is called a Hamilton cycle. In a weighted multigraph, a value is assigned to each edge. This value is called the weight of the edge. The weight of a path is the sum of the weights of the edges along the path. The efficient route is the path with the minimum weight. b
a c i
d
g
e
j
The path a, c, d, e, f, h, j is a Hamilton cycle because it contains all the vertices exactly once and returns to the starting vertex.
h
f
Determine the efficient route from A to Z and the weight of the Example 3 2 path for the graph. " Begin by tracing paths from A to Z. Three possible paths are shown. A, T, M, Z: 5 + 8 + 9 or 22 A, T, Z: 5 + 6 or 11 A, G, Z: 2 + 7 or 9 The efficient route is the path A, G, Z, with a weight of 9. .
(
5
5
7
6
8
;
9
EXERCISES 1. Draw a graph with 8 vertices that contains an Euler circuit.
$
7
2. Does the graph shown contain a Hamilton Path or a Hamilton cycle? If so, name the path or cycle.
" 2
*
3. Determine the efficient route from A to F and the weight of the path for the graph. Concepts and Skills Bank
270
#
5
5
3
2
3 )
4
1
&
( 3
% 3
'
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2 Determine whether the graph contains a Hamilton Path or a Hamilton cycle. If so, name the path or cycle. Use the most specific name.
NAME
DATE
PERIOD
CSB 7 Practice
SCS
MA.D.1.1, MA.D.1.2
Walks, Trails, and Paths Use the graph to determine whether each course is a circuit, cycle, path, trail, or walk. Use the most specific name. #
1. a, h, g
b
a
2. a, b, c, j, a
h
"
3. i, c, d
g
4. e, d, c, i
5. k
(
e
e
f a h
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
k
8. g
b
d
d
&
Determine whether each graph contains an Euler circuit or an Euler path. If so, name the circuit or path. 7.
%
l
'
c
i
f
6. f, g, h, a
$
j
a d
c
c
b
Determine whether each graph contains a Hamilton path or a Hamilton cycle. If so, name the path or cycle. Use the most specific name. 10.
9.
d a
h g
b
e f
c
c
a
b
d
Determine the efficient route from A to Z and the weight of the path for each graph. 11.
12. #
1
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6
4 1
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2
(
4
)
5
5
&
2 1 8
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1
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7
2 4
( +
5
) 3
,
271
2
&
2
Concepts and Skills Bank
$
4
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3
' 4
3 1 2
* 2
;
North Carolina StudyText, Math A
NAME
DATE
PERIOD
CSB 7 Word Problem Practice
SCS
MA.D.1.1, MA.D.1.2
Walks, Trails, and Paths 1. CHEMISTY A supplier needs to ship 6 chemicals from a refinery to a plant. The chemicals are U, V, W, X, Y, and Z. Shipping regulations require that certain chemicals must be shipped in separate tank cars due to the possibility of explosion. In the graph, chemicals linked by an edge may not be shipped together. What is the minimum number of tank cars needed to ship equal quantities of each chemical, and what chemicals will be in each car? 9 6
:
7
3. TRANSPORTATION A truck driver makes deliveries to the cities listed in the mileage table below. She wants to develop a schedule where no city-to-city trip exceeds 450 miles. nta
la At Atlanta
-
Birmingham 161
m
n
ha
ing
rm Bi
alo
ff Bu
nd
sto
rle
a Ch
le his lk e vil rfo bil mp sh No Na Mo Me 394 345 250 572
ela
ev
Cl 774
161
896
515
-
901
535
718
244
274
195
864
200
922
1246
721
657
256
609
808
395
407
789
Buffalo
896
901
-
Charleston
515
535
864
-
Cleveland
774
718
200
256
-
731
1030
530
571
Memphis
394
244
922
609
731
-
402
208
924
Mobile
345
274
1246
808
1030
402
-
467
893
Nashville
250
195
721
395
530
208
467
-
708
Norfolk
572
789
657
407
571
924
893
708
-
a. On the basis of the mileages given in the table, draw a graph with vertices representing cities and edges representing trips not exceeding 450 miles. If your initial graph has edge crossings, reposition the vertices and redraw the graph to eliminate edge crossings.
;
8
New York
Brooklyn
New Jersey
b. Using your graph, find the shortest path from Buffalo to Atlanta. Staten Island
c. Using your graph, find the minimal path and distance from Mobile to Nashville.
Atlantic Ocean
Concepts and Skills Bank
272
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. TOURISM Staten Island, New York, is connected to New Jersey by 3 bridges and to Brooklyn, New York, by one bridge. Can a tour bus from New Jersey cross all 4 bridges exactly once and end the tour back in New Jersey? If not, what is the least number of additional bridges needed to complete the tour, and where should they be located?
Name
Date
Diagnostic Test Wilbur and Orville Wright based their flight experiments at Kitty Hawk, North Carolina, on the principle that lift is directly proportional to the product of the area of the wings and the square of the velocity.
3
The scatter plot below shows the gas mileage of a car at various speeds. Gas Mileage at Various Speeds 40
MPG
1
Which of the following equations describes the relationship between lift L, wing area A, and velocity v? A B C
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
6.0 × 103 ft
What is an expression for the height h of the rectangle? 0.8 × 10 8 ft
C
8.0 × 10 8 ft
A
It is linear with a positive slope.
B
It is linear with a negative slope.
C
It is quadratic.
D It is exponential.
h
B
60
What does the graph suggest about the relationship between speed and gas mileage for the car?
The area of the rectangle shown below is 4.8 × 10 12 square feet.
0.8 × 10 4 ft
40
Speed (mi/h)
L = kA √v kA L=_ v2 LAv 2 = k
A
20
0
D L = kAv 2
2
20
The slopes of two sides of a quadrilateral are equal. The same two sides are also equal in length. What is the best description of the quadrilateral? A
square
B
rhombus
C
trapezoid
D parallelogram
D 8.0 × 10 9 ft
A1
North Carolina StudyText, Math A
Name
Date
Diagnostic Test 5
(continued)
The vertex-edge graph below represents a business park. The edges represent sidewalks. The vertices represent buildings.
7
& "
0
While traveling in northern Canada, Sasha hears a weather forecast for temperatures of 5° Celsius. The graph below shows the relationship between temperature in degrees Fahrenheit x and in degrees Celsius y.
'
20
y
#
$
°Celsius
10
%
0
10
20
30
40
50
60
x
−10
Which of the following is true of the route EOABCDFE? It is an Euler circuit because each building is visited once.
B
It is the most efficient route for traveling each sidewalk once.
C
It is a Hamilton cycle because each building is visited exactly once, and the path returns to the starting building.
°Fahrenheit
What is the approximate equivalent temperature in degrees Fahrenheit for 5° Celsius?
D It is an Euler circuit because no path is retraced.
6
Simplify
8
A
-15°F
C
B
-3°F
D 41°F
A landscaper wants to place a fence along the diagonal path of the rectangular area shown below. d
2 -3(-3) 4 _ .
2(3) 2 2 A - _4 (3) 1 B -_ 2 2(3) 4 1 C _ 4 2 (3) 4 22 D _ (3) 4
8°F
6 ft
8
12 ft d = √122 + 62
What is the length of the fence in simplest radical form with no perfect square in the radicand?
A2
A
20 ft
B
18 ft
√ 180 ft ft D 6 √5
C
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
Name
Date
Diagnostic Test 9
(continued)
Sarah read that the Wright Brothers flew 852 feet in 59 seconds at Kitty Hawk, North Carolina on December 17, 1903. She entered the keystrokes shown below in her calculator. 852
59
11
14.440678
Power measures how fast work is done. The formula for power is work P = _. seconds Which statement describes the relationship between the amount of power and time needed to do a specific amount of work?
Which unit should Sarah use for her answer?
A
If the time increases, the power needed to do the work increases.
A
feet
B
B
seconds
If the time decreases, the power needed to do the work increases.
C
seconds per foot
C
A change in time neither increases nor decreases the power needed to do the work.
D feet per second
D There is a linear relationship between power and time.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10
Which table shows a quadratic relationship between x and y? A
B
C
D
x
-2
-1
0
1
2
y
8
2
0
2
8
x
-2
-1
0
1
2
y
-2
-1
0
1
2
x
-2
-1
0
1
2
y
2
1
0
-1
-2
x
-2
-1
0
1
2
y
-8
-4
0
4
8
12
What is the volume of the triangular prism shown below?
CM CM CM 3
A
160 cm
B
240 cm 3
C
480 cm 3
D 520 cm 3
A3
North Carolina StudyText, Math A
Name
Date
Diagnostic Test 13
(continued)
Which expression represents the area of the figure shown below?
16
Which of the following is not the graph of a function? A
y
x
0 x+5
2x + 2
x
B
y
3x x
0
2
A
6x + 6x
B
5x 2 + x
C
5x 2 - x - 10 y
C
D 7x 2 + 11x
-2π
14
D
17 -3 -1
15
Simplify
2 3
3xyz 2
8 8
8 8
A B
3
(-2x ) (x y z) __ .
xyz -_
6 2x 5y 8z -_ 3
C
xyz _ 6
2x 5y 8z D _ 3
2π
x
The lift height of the Carolina Cobra at Carowinds is 125 feet. When Juan was at the highest point, a penny fell out of his pocket. Which function models the penny’s height h, in feet, as a function of the number of seconds t that the penny was falling? A
h(t) = -16t 2 - 125
B
h(t) = -16t 2 + 125
C
h(t) = 16t 2 - 125
D h(t) = 16t 2 + 125
A4
x
y
0
D 2
π
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
What is the slope of the line through the points (- 4, - 5) and (8, 3)? 1 A _ 2 2 B _ 3 3 C _ 2
0 -1
-π
Name
Date
Diagnostic Test 18
(continued)
Population density indicates how crowded or spread out a population may be in a given area. The formula for density is shown below.
20
In the figure below, what is the name of an angle that is formed by a tangent and a chord of the circle? 5
number of specimens Population = __ area density
1
A biologist counts 82 frogs in 100 square feet of a rain forest. Ten years later, double the area is needed to count the same number of frogs. What effect does this have on the population density?
A
∠RSP
A
The population density is doubled.
B
∠RST
B
The population density is halved.
C
∠PRS
C
The population density is unchanged.
D ∠PST
4 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D The population density is divided by four.
21
19
The speed required to travel a certain distance varies inversely with the available time. Cheryl can bike from home to school in 15 minutes, at an average speed of 10 miles per hour. If she had 12 minutes to make the trip, how fast would she have to travel? A
20 miles per hour
B
13 miles per hour
C
12.5 miles per hour
Which of the following relationships is linear? A
the relationship between the size of a rabbit population that doubles each year and the number of years
B
the relationship between the distance a car travels at a constant speed and the amount of time the car is traveling
C
the relationship between the area of a circle and its radius
D the relationship between the number of tomato plants that can be planted in a garden and the perimeter of the garden
D 8 miles per hour
A5
North Carolina StudyText, Math A
Name
Date
Diagnostic Test 22
(continued)
Maggie is delivering telephone books to every home in a neighborhood represented by the graph shown below. 1
24
Which table shows some solutions of the equation y = 4 x - 1? A
x
-1
0
1
2
y
0.25
1
4
16
2
B
x
-1
0
1
2
y
0
-1
1
3
3
6
C 5
4
D
The edges represent streets and the vertices represent intersections. Which is the most efficient route for Maggie to take so that she will travel every street and will also return to the starting point? QRSTUPQ
B
TUSRQPUT
C
PQRSTUQSUP
-1
0
1
2
y
0.0625
0.25
1
4
x
-1
0
1
2
y
-0.25
0
0.25
1
The height of the square prism shown below is 2 times the length of the side of the base.
D UPQUSTUQRSQU
23
2s
Jordan is training with a fully loaded backpack in preparation for hiking the Appalachian Trail. During his first training week, he hiked 50 miles. Each week, he hikes 10 more miles than he did the previous week. Which equation shows the relationship between the number of weeks w that he has trained and the distance d that he hiked that week?
The volume of the prism is given by the formula V = 2s 3. If the volume of the prism is 6000 cubic inches, how long is the side of the square base, in simplest radical form?
A
d = 10w
C
B
d = 50 + 10(w - 1)
C
d = 50(w - 1) + 10
s s
A B
3
in. 10 √3 in. 10 √3 3
10 √ 30 in. in. D 10 √30
D d = 10(50 + w)
A6
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
25
x
Name
Date
Diagnostic Test 26
(continued)
Maxine wants to find the surface area of the triangular prism shown below.
27
Which is the graph of a linear function? A
1.5
y
1 0.5 5 ft
8 ft
0
4 ft
B
6 ft
3x
2
4
6x
y
1
8 ft
0 6 ft
2
2
Maxine drew a net of the prism. Which of the following did she draw? A
3
1
4 ft
y
C 5 ft
B 5 ft
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0 8 ft 6 ft
4 ft
5 ft
C
x
0
4 ft
28
8 ft 6 ft
4 ft
4 ft
D
y
D
8 ft 5 ft
4 ft 6 ft
The distance from Asheville to Wilmington, North Carolina, is 360 miles. If you are traveling at 55 miles per hour, which equation can you use to determine how many miles m you must still travel after driving h hours? A
m = _h
B
m = 55h - 360
C
m = 360 + 55h
360 55
D m = 360 - 55h
A7
North Carolina StudyText, Math A
Name
Date
Diagnostic Test 29
(continued)
Which equation demonstrates the Commutative Property of Addition? A
4x(2x - 1) = 8x 2 - 4x
B
2x + 5x 2 - x = 5x 2 + 2x - x
C
1.5y 2(xz) + 1 = 1.5xy 2z + 1
32
D 18y + (3y - 4) = (18y + 3y) - 4
30
Elevator manufacturers must ensure that each elevator is safe for the number of people listed. For this reason, it is important that they predict the weight of the passengers. The table below shows some of the data that was collected by one elevator manufacturer.
The table below shows the coordinates of points on the graph of a line. x
y
-2
7
0
3
2
-1
4
-5
For what value of x does y = -4? 3
C
B
3.5
D 11
6
P(n) = 250 + 2(n + 1)
B
P(n) = 250(2n - 1)
C
P(n) = 250 2(n - 1)
1
152
2
346
2
384
3
485
3
562
4
605
4
626
4
798
5
942
What is the median line of best fit for the data?
Bonus points are added to a player’s score on a computer game after each level is completed. After the first level, players get 250 bonus points. The number of bonus points received for each of the following levels is double the number of bonus points they received for completing the previous level. Which equation represents the number of bonus points P that a player receives after completing the nth level? A
Total Weight
A
y = 226x
B
y = 226x - 102.6
C
y = 226x - 106
D y = 226x - 109.3
33
Divide (18x3y2 - 6x2y + 12xy2) ÷ (-4xy). A
-4.5x 2y + 1.5x - 3y
B
-4.5x 2y - 1.5x - 3y
C
-4.5xy - 1.5x - 3
D 4x 2y + 1.5x - 3y
D P(n) = 250(2 n - 1)
A8
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
31
A
Number of Passengers
Name
Date
Diagnostic Test 34
(continued)
Carlos maintains a blog on which he publishes data about the price of gasoline in his area. The graph below represents one week of data.
36
$PTUPG(BT5IJT8FFL
2.20
2.40
2.60
2.80
3.00
3.20
Just as he was about to post this graph on his blog, he was informed that a station increased the cost of gas from $2.85 to $3.20 per gallon. Which of the following best describes how Carlos will have to change his box-and-whisker plot because of this new information?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
The entire box on the graph will shift to the right.
C
The box will extend to the right and the right whisker will be longer.
A
It falls at a constant speed.
B
Its speed increases and decreases.
C
Its speed decreases.
D Its speed increases.
The graph will have a longer whisker on the right and the line for the median will move to the right.
B
While riding up a chair lift at Appalachian Ski Mountain, Swati accidentally dropped her container of lip balm. The height of the container in feet is modeled by the function h(t) = -16t 2 + 128 where t is the time in seconds after Swati dropped the container. Which of the following statements best describes the speed of the container as it falls?
37
Suppose the line shown below is extended. 4 3 2 1 -4 -3 -2
O
y
1 2 3 4x
-2 -3 -4
D The right whisker will be longer. Which of the following points will be on the line?
35
A
(-8, 6)
B
(-8, 8)
C
(6, -2)
Which of the following is an equation of the line through the point (-2, 5) with a slope of 0.75? A
y - 2 = 0.75(x + 5)
D (8, -3)
B
y + 2 = 0.75(x - 5)
C
y - 5 = 0.75(x - 2)
D y - 5 = 0.75(x + 2)
A9
North Carolina StudyText, Math A
Name
Date
Diagnostic Test 38
(continued)
Donovan sailed a distance of about 12 nautical miles along the North Carolina coast from Albermarle Sound to Elizabeth City. He knows that 1 nautical mile (nm) is equivalent to about 1.15 miles on land. Which of the following proportions can Donovan use to find x, the number of miles in 12 nautical miles? A
1 mi 1.15 nm _ =_
B
1.15 mi 12 nm _ =_
C
40
12 cm
x mi
12 nm
5 cm
x mi
1 nm
12 nm x mi _ =_ 1
1.15 mi
1 nm 12 nm D _ =_ 1.15 mi
x mi
39
What is the volume of the smallest cylinder into which the cone shown below can fit?
2
$140
3
$190
4
$240
5
$290
B
100π cm 3
C
120π cm 3
41
The graph shows the solution to which of the following systems of equations? y
0
Which equation can be used to find the price p to rent a personal watercraft for h hours? A
p = 50h
B
p = 70h
C
p = 50h - 40
x
⎧ x - 2y = -1 ⎨ ⎩ x + 2y = 5 ⎧x + 2y = -1 B ⎨ ⎩ x - 2y = 5 ⎧ 2x - y = -1 C ⎨ ⎩ 2x + y = 5 ⎧ 2x + y = -1 D ⎨ ⎩ 2x - y = 5
A
D p = 50h + 40
A10
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Price
60π cm 3
D 300π cm 3
A rental store posts the following chart to explain the cost of renting a personal watercraft. Hours
A
Name
Date
Diagnostic Test 42
(continued)
Which of the following statements is true about this graph? 1
45
Which graph shows the solution set to the following system of inequalities? y ≥ 5 - 2x
5
y ≤ 2x - 3 4
A
2
y
3
A
It is connected and is an Euler circuit.
B
T and R are the only vertices of even degree.
C
It can be drawn with a single line.
0
B
x
y
D There is only one path from P to T.
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
43
What is the surface area of the cylinder shown below to the nearest square centimeter? 14 cm
C
y
36 cm 0
44
A
1372 cm 2
C
B
3079 cm 2
D 4310 cm 2
x
3695 cm 2 D
Factor 3x 2 + 20x - 32.
y
0
A
(x + 4)(3x - 8)
C
B
(x - 4)(3x + 8)
D (x - 8)(3x + 4)
x
(x + 8)(3x - 4)
A11
North Carolina StudyText, Math A
Name
Date
Diagnostic Test 46
Each student in Mrs. Montino’s class read at least 85 books last year. Each read more than twice as many fiction books as nonfiction. Which table contains possible combinations of books that Mrs. Montino’s class read? A
B
C
D
nonfiction
10
20
30
40
50
fiction
19
42
62
82
102
nonfiction
10
20
30
40
50
fiction
80
70
60
50
40
nonfiction
10
20
25
30
32
fiction
85
75
70
65
65
nonfiction
10
20
25
30
32
fiction
75
65
60
55
50
48
Which of the following functions contains all of the points shown in the table below? x
-5
0
-2
y
0
-6
-3.6
A
-2x + y = _
B
-2x - _y = 10
C
5 x - 2y = 12 -_
2 5
5 3
3
5 D -_ x - 2y = 10 3
49
At what point will the diagonals of parallelogram LMNO, shown below intersect? 5 4 3 2 1
What is the range of the function shown by the table below?
−1 O
x
y
-5
-4
-2
2
1
8
A
(2, 3)
4
14
B
(2, 2) (3, 2)
y
-
0
.
/
1 2 3 4 5 6 7x
−2 −3
A
{-5, -2, 1, 4}
C
B
y ≥ -4
D (3, 3)
C
{-4, 2, 8, 14}
D y > -4
A12
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
47
(continued)
Name
Date
Diagnostic Test 50
(continued)
What is the volume of the cylinder shown below? Round to the nearest cubic foot.
52
5 ft
The Great Seal of the State of North Carolina is shown below. It depicts two figures, one representing Liberty, the other, Plenty.
4 ft
A
63 ft 3
B
126 ft 3
C
251 ft 3
D 314 ft 3
51
A computer screen saver shows the seal increasing in size. As the seal’s radius r increases, which of the following best describes the rate at which the circumference C of the seal changes?
The path of a golf ball is represented in the graph below. 40
y
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
30 20
A
C increases at a square of the rate of r.
B
C increases at π times the rate of r.
C
C increases at 2π times the rate of r.
10 0 40
80
120
x 160
Which of the following quadratic equations could represent the path of the ball? A B C D
D C increases at the same rate as r.
y = - _ x 2 - _x
1 1 320 2 1 2 1 _ _ x + x y=320 2 1 1 y = _ x 2 + _x 4 640 1 2 1 _ _ x + x y= 320 2
53
What is the shortest distance between 3 (3, -1) and the graph of y = _x - 4? 2
A B
A13
√ 13 _ 13 2 √ 13 _ 13
C D
3 √ 13 _ 13 √ 5 13 _ 13
North Carolina StudyText, Math A
Name
Date
Diagnostic Test 54
(continued)
Loggerhead turtles are an endangered species that lay their eggs at night along the North Carolina coast. For two consecutive years a biologist collected data about the number of eggs in loggerhead nests found on a beach. Analysis of the data is shown in the table below. Year 1
Year 2
Mean
98.4
85.6
Standard Deviation
10.9
12.8
43
56
Range
56
A
B
What do these figures indicate about the number of eggs in the nests of loggerhead turtles? The population of loggerhead turtles will remain about the same.
B
The average number of eggs in a nest did not change significantly.
C
The number of eggs in an average nest is decreasing and the data are more spread out.
C
D
57
D The number of eggs each turtle lays is less than the previous year, but the maximum and minimum nest sizes have not changed.
-6.25
B
-3.5
C
3.5
1
2
3
y
1
2
3
4
x
0
1
2
3
y
0
1
2
4
x
0
1
2
3
y
1
2
4
8
x
0
1
2
3
y
2
4
8
16
A
4 and 5
B
6 and 12 4 and 2 √5
and 4 √5 D 2 √5
What is the maximum value of the function f (x) = -x 2 + 7x - 6? A
0
How long are the diagonals of a parallelogram whose vertices are located at (0, 0), (5, 0), (8, 4) and (3, 4)?
C
55
x
D 6.25
A14
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
Damica folds a large sheet of paper in half repeatedly. When she unfolds the paper, the number of sections created by the folds is a function of the number of folds. The equation y = 2 x models the relationship between the number of folds and the number of sections formed. Which table shows that relationship?
Name
Date
Diagnostic Test 58
A school district compares student test scores on a standardized test with the previous year’s results for the same test. This year the first quartile increased by 6 points but the mean remained about the same. Which of the following best describes how the scores changed? A
B
C
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(continued) 60
More students earned scores in the upper fourth but others did not improve. There is no significant improvement or loss since the mean did not change significantly.
61
The lowest performing students improved their scores, but other scores are about the same because the mean stayed about the same.
" $
C
an angle formed by two diameters
C
x ≥ -3
A sculpture shaped like the pyramid shown below is to be packaged for shipping.
9.6 cm
A
210 cm 3
B
419 cm 3
C
629 cm 3
D 728 cm 3
&
an angle formed by two tangents
x≥3
The sculpture will be boxed in the smallest possible rectangular carton and the space around filled with biodegradable packing material. To the nearest cubic centimeter, what volume of packing material is needed?
#
B
B
7.8 cm
%
an angle formed by two chords
x≥0
8.4 cm
Which of the following describes angle A in relation to the circle?
A
A
D x ≤ -3
D The lowest performing students improved their scores, but other students had lower scores, since the mean did not change significantly.
59
The relationship between two variable quantities, x and y, is defined by the equation y = ⎪x + 3⎥. If x increases, for which values of x is y also increasing?
D an angle formed by two secants
A15
North Carolina StudyText, Math A
Name
Date
Diagnostic Test 62
(continued)
The ferry between Southport and Fort Fisher, North Carolina, charges pedestrians $1 and cyclists $2 for a one-way trip. The system of equations below models a trip in which $528 was collected from 366 passengers who were either pedestrians p or cyclists c.
64
Which table shows some possible solutions to the system of inequalities? y ≤ 3x y > -2x + 1 A
p + c = 366
B
p + 2c = 528 How many pedestrians were on that trip? A
162
B
183
C
204
C
D
x
-1
-2
y
0
-1
x
-1
0
y
4
5
x
-2
-3
y
2
3
x
2
2
y
0
-2
D 290
65 Solve the following equation for y. - 5(2x - 1) - 3y + x = _y - 4 2 5
A
y=0
B
9 9x + _ y=-_ 5 5
C
45 45 y = - _x + _ 17 17
A
square
B
rectangle
C
trapezoid
D quadrilateral
5 45 x - _ D y=- _ 17 17
A16
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
63
A quadrilateral is formed by connecting the points (0, 0), (4, 4), (8, 4), and (0, -4). What is the most specific description of the figure?
Name
Date
Diagnostic Test 66
(continued)
The field goals attempted and made by the Carolina Panthers from the 2000 to 2008 seasons are shown in the table below.
68
Which of the following tables represents a function? A
Field Goals
x
y
-1
5
0
4
Attempted
Made
25
20
-1
3
25
15
0
2
25
24
-1
1
28
24
28
24
x
y
31
28
0
2
34
26
1
2
35
31
2
2
38
32
3
2
4
2
B
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
What is the median line of best fit?
67
A
y = 1.6x - 23.6
B
y = 1.1x - 7.3
1
0
C
y = 1.1x - 7.5
1
1
D y = 1.1x - 6.8
1
2
1
3
1
4
C
D
Which system of equations could best solved using the substitution method? ⎧ 2x - y = 0 A ⎨ ⎩ 2x - 3y = -12 B C
x
x 0 -4
⎧ 2x - 5y = 13 ⎨ ⎩ 5x - 2y = -4 ⎧ 7x - 4y = 21 ⎨ ⎩ 3x + 5y = 12
⎧ 1.4x + 2.5y = 10 D ⎨ ⎩ 0.4x - 2.4y = 1.8
69
y
y 5 -3
8
6
-4
2
-1
11
Simplify 8a 2b - (3ab 2 + a) + 4a(ab - 2). A
12a 2b + ab 2 - 8a
B
12a 2b - 3ab 2 - 7a
C
12a 2b - 3ab 2 - 9a
D 12a 2b - 3ab 2 - a - 2
A17
North Carolina StudyText, Math A
Name
Date
Diagnostic Test 70
(continued)
Admission charges for a theater group’s performances are shown below.
72
Admission Charges Adults
$10
Seniors & Children
$8
The theater has 420 seats. To pay for costs, receipts for each performance must be at least $2000. If a is the number of adults and s is the number of seniors and children, which set of inequalities models this situation? ⎧ A ⎨ a + s ≤ 420 ⎩ a + s ≥ 2000 ⎧ B ⎨ a + s ≥ 420 ⎩ a + s ≤ 2000 ⎧ a + s ≥ 420 C ⎨ ⎩ 10a + 8s ≥ 2000 ⎧ a + s ≤ 420 D ⎨ ⎩ 10a + 8s ≥ 2000
$4.75
B
$5.20
C
$5.40
t ≤ 1.5
B
t ≥ 1.5
C
0 ≤ t ≤ 1.5
D 0≤t≤9
73
A bridge is designed for a maximum load of 2 tons. The surface of the bridge has an area of 640 square feet. Brent knows that a ton is equivalent to 2000 pounds. He enters the keystrokes shown below on his calculator. 2
2000
640
6.25
What unit of measure should he use for the answer?
When Maxine eats in a restaurant, she leaves a tip that varies directly with the cost of the meal. If she leaves a $3 tip for a meal that costs $15, how much should she leave for a meal that costs $27? A
A
A
square feet per pound
B
square feet per ton
C
tons per square foot
D pounds per square foot
D $5.75
A18
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
71
The height of a football after being kicked can be modeled by the function y = −16t 2+ 24t, where t is the number of seconds the ball is in the air and y is the height of the ball. What is the practical domain for this function?
Name
Date
Diagnostic Test 74
(continued)
Which graph represents the solutions to the system of inequalities?
75
y ≥ -x + 2
Glen drives from his home to the library. The graph below shows his travel speed s as a function of the time t since he left home. s
y ≤ -2x + 4 y
A
t
0 x
0
A
Glen leaves home, stays at the library, and then returns home.
B
Glen’s speed initially increases, then stays the same, and then decreases.
C
As Glen gets farther from home, his speed decreases.
y
B
x
0 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which statement best describes the data?
D Glen’s speed initially increases, then stays the same, and then increases again.
y
C
x
0
76
y
D
0
x
A19
Suppose y varies inversely as x, and y = 6 when x = -5. Which equation represents the relationship between x and y? A
6y = -5x
B
-5 y=_
C
-6 y=_
D
-30 y=_
6x
5x x
North Carolina StudyText, Math A
Name
Date
Diagnostic Test 77
Attendance figures for the Saturday matinee at a theater for three months are shown in the table below.
78
SATURDAY MATINEE ATTENDANCE Month 1
245
424
360
195
Month 2
234
302
288
145
Month 3
403
325
245
333
79
On the first Saturday of Month 4, the theater ran a blockbuster film and attendance jumped to 1064. Which statement below best describes the effect that this outlier will have on the standard deviation? A
C
A
-8
C
B
-3
D 8
3
The area of the rectangle shown below is expressed as a quadratic function. "(x) = 2x 2 - 7x + 3
?
?
One more data item means that the sum of the squares will be divided by 13 instead of by 12; this will have little effect on the standard deviation.
Which of the following are possible expressions for the length and the width?
The standard deviation is based upon the median, which will increase a little and cause the standard deviation to also increase slightly.
A
(x + 1) and (2x + 3)
B
(x - 1) and (2x - 3)
C
(x + 3) and (2x + 1)
D (x - 3) and (2x - 1)
The mean and median will both increase, so the standard deviation will also increase slightly.
80
D Standard deviation measures how spread out the data is. The new outlier increases the range and the mean, causing the standard deviation to increase significantly.
Which expression is equivalent to (x -3y 4z -1) -4? A
x-7 yz -5
B
x12y16z 4
C
x12 _
D
A20
y16z4 12 4 z x_ 16 y
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
B
What is the y-intercept of the function y = -3x 2 + 4x - 8?
Name
Date
Practice By Standard Clarifying Objective MA.A.3.1 1
In their first powered flight at Kitty Hawk, North Carolina, the Wright Brothers flew 120 feet in 12 seconds. Which table below describes a linear relationship between elapsed time t, in seconds, and distance d, in feet? A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
B
2
C
t
d
0
0
0
40
4
50 100
t
d
0 4 8
64
8
12
120
12
120
t
d
t
d
0
0
0
0
4
48
4
40
D
8
96
8
80
12
120
12
120
3
She folds the paper in half two more times, creating 8, and then 16, equal sections. What kind of pattern, if any, exists between the number of times the paper is folded and the number of sections into which the paper is divided?
Which table displays a quadratic relationship between x and y? A
B
C
x
y
x
y
2
4
2
4
3
1
3
8
4
0
4
16
5
1
5
32
6
4
6
64
x
y
D
x
y 0
2
5
2
3
3
3
9
4
0
4
16
5
1
5
25
6
5
6
36
A student folds a piece of paper in half, creating 2 equal sections. She folds this folded piece of paper in half again, parallel to the first fold. This creates 4 equal sections.
4
A
linear
C
B
quadratic
D no pattern
exponential
Which of the following relationships is linear? A
the relationship between the time and the speed of a car traveling a distance of 100 miles
B
the relationship between the height of a bouncing ball and the length of time it bounces
C
the relationship between the amount of money in an account earning simple interest and the age of the account
D the relationship between the price of a taxable good before and after tax is added
A21
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.A.3.2 1
2
The height of a football is given by the equation h(t) = -16t2 + 768t where h is the height of the ball in feet and t is the time in seconds after it is kicked. Over which interval of time is the ball’s height increasing? A
0 ≤ t ≤ 16
C
B
0 ≤ t ≤ 24
D 0 ≤ t ≤ 96
4
0 ≤ t ≤ 48
20
0
40
x≥0
B
x≥5
C
x≤5
5
d (feet) 0 20 40 60 80 100 120
50
0 ≤ x ≤ 50
C
B
-50 ≤ x ≤ 0
D 0 ≤ x ≤ 250
A The height of the rope decreases at a constant rate for 0 ≤ d ≤ 60. B The height of the rope varies directly with the distance from the metal pole.
- 50 ≤ x ≤ 50
C
3
The Space Club launches a newly designed rocket. Its height h in feet above ground level is given by the equation h(t) = -16t2 + 128t, where t is the number of seconds after the launch. As t increases, for which interval of time is h(t) decreasing? A
0≤t≤8
C
B
4≤t≤8
D 0≤t≤4
h (feet) 29.1 22.8 20.5 20 20.5 22.8 29.1
Which statement best describes the data?
As x is increasing, for which values of x is y decreasing? A
A rope is suspended between a metal pole and a wooden pole. The table below shows the height h of the rope at a distance d from the metal pole.
The height of the rope varies inversely with the distance from the wooden pole.
D As the distance from the metal pole increases, the rate at which the height increases is greatest for 100 ≤ d ≤ 120.
0 ≤ t ≤ 16
A22
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
−20
A
D all values of x
The outer edge of the semi-circular drive shown below can be described __________ by the equation y = √2500 - x2 .
−50 −40
The relationship between two variable quantities, x and y, is defined by the equation y = ⎪x - 5⎥. As x is increasing, for which values of x is y increasing?
Name
Date
Practice By Standard Clarifying Objective MA.A.3.3 1
A stone is dropped from a height of 64 feet. The function h = -16t2 + 64 represents the relationship between the height h of the stone in feet, and the number of seconds t after the stone is dropped. This function is shown in the graph below. 80
3
Which table shows y increasing at a faster rate than x? A
h
B
60 40 20 0
2
4
6
8t
x 0 1 2
y 10 8 6
3
4
4
2
x 0 1 2 3 4
y 1 3 6 10 15
C
x 0 1 2 3 4
y 5 10 15 20 25
D
x 0 1 2 3 4
y 20 18 15 11 8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which of the following statements best describes the speed of the stone? A
The stone’s rate of speed remains the same as it drops.
B
The stone’s rate of speed decreases as it drops.
C
The stone’s rate of speed varies as it drops.
4
The dimensions of the flag of the state of North Carolina are shown below. 1 x 2
1 x 2
D The stone’s rate of speed increases as it drops.
2
1 x 3
A computer screen saver shows the North Carolina flag shrinking in size. As the length x decreases, which of the following best describes the rate at which the area of the flag changes?
A software demo shows an expanding cube. If the side length s of the cube is growing at 1 mm/sec, which of the following statements best describes the rate at which the surface area, given by the formula A = 6s2, changes? A
It increases at the same rate as s.
B
It increases at a faster rate than s.
C
It increases at a slower rate than s.
x
A
It decreases at 2 times the rate of x.
B
It decreases at the same rate as x.
C
It decreases at a faster rate than x.
D It decreases at a slower rate than x.
D It increases at six times the rate of s.
A23
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.A.4.1 1
Is the relation shown below a function? Why or why not? {(5, 5), (4, 1), (3, 5), (0, 5), (4, 4)} A
4
Which of the following is a function? A
y
It is a function since every x-value is mapped to a single y-value.
B
It is a function since every y-value is mapped to a single x-value.
C
It is not a function because one y-value is mapped to more than one x-value.
x 0
B
D It is not a function because one x-value is mapped to more than one y-value.
y
x
0
C
2
Which relation is a function? {(0, 0), (1, -1), (-1, 0)}
B
{(0, 0), (0, -1), (-1, 0)}
C
{(0, 0), (0, 1), (-1, 1)}
x -π
B
x -1 0 -1 0 -1
y 4 5 6 7 8
C
x 0 1 2 3 4
y 1 1 1 1 1
D
π
−2
D none
Which of the following tables represents a function? A
0
5
x 0 0 0 0 0
y 0 1 2 3 4
x 0 25 25 36 36
y 0 5 -5 6 -6
Which of the following is not a function? A
height of a carousel rider as a function of distance from the ride center
B
height of a bouncing basketball as a function of time
C
cost of a train trip as a function of distance traveled
D number of tennis balls in a store as a function of the number of tennis ball canisters in the store
A24
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
2
D {(1, 0), (1, -1), (-1, 0)}
3
y
Name
Date
Practice By Standard Clarifying Objective MA.A.4.2 1
What is the domain and range of the function f(x) = x2 - x - 6? A
4
Domain: {all real numbers}
The path of a child’s ball thrown into the air from a height of 0 feet is modeled by the graph shown below. y
Range: {y ≥ -6.25} B
4
Domain: {all real numbers}
2
Range: {y ≤ -6.25} C
x
Domain: {-3 ≤ x ≤ 2}
0
Range: {y ≤ -6.25}
4
8
12
16
−2
D Domain: {-2 ≤ x ≤ 3} Range: {y ≥ 0.5}
2
What is the relevant domain and range for this function? A Domain: {all real numbers} Range: {y ≥ 5} B Domain: {all real numbers} Range: {y ≤ 5}
What is the domain and range of the following function? {(0, 1), (2, 3), (-1, 5), (0.4, 4), (4, -1)}
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
C
Domain: {-1, 1, 3, 4, 5}
D Domain: {0 ≤ x ≤ 16} Range: {0 ≤ y ≤ 5}
Range: {-1, 0, 0.4, 2, 4} B
Domain: {0 ≤ x ≤ 16} Range: {y ≤ 5}
Domain: {-1, 0, 0.4, 2, 4} Range: {1, 3, 5}
C
Domain: {-1, 0, 0.4, 2, 4}
5
Range: {-1, 1, 3, 4, 5} D Domain: {0, 0.4, 2}
A B
Range: {-1, 1, 3, 4, 5}
3
6
What are the x- and y-intercepts of the function f(x) = x2 + x - 12? A
(0, 4), (0, -3), (-12, 0)
B
(0, -4), (0, 3), (-12, 0)
C
(-4, 0), (3, 0), (0, -12)
What is the maximum value of the function f(x) = x2 - 2x + 15? C 15 D 16
-1 1
Which function has an x-intercept of (1, 0)? A
f(x) = -4x + 1
B
f(x) = 2x + 2
C
f(x) = 2x - 2
D f(x) = 3x - 1
D (4, 0), (-3, 0), (0, -12)
A25
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.A.4.3 1
2
The height of a football is modeled by the function h(t) = -16t2 + 24t where t is the number of seconds the ball is in the air. When does the ball reach its maximum height? A
0.75 second
C
B
1.5 seconds
D 9 seconds
4
8 seconds
A B
A function contains the points shown in the table. x
-4
-3
-2
y
-3
-5
-3
A stone is dropped from the Mile High Swinging Bridge in Linville, North Carolina. The function f(t) = -16t2 + 80 describes the stone’s height in feet relative to the time in seconds. How high is the bridge?
5
16 feet 64 feet
C 80 feet D 96 feet
Some roads are designed with slightly curved surfaces to prevent flooding. The graph below shows the shape of a cross-section of a curved surface.
Which function contains them? A
y = -2x - 11
B
y = 2(x + 3)2 - 5
C
y = 2x + 1
0.6
y
0.4 0.2 -80 -40
0
40
80
x
-0.2
3
Which of the following quadratic equations could be the function that describes the curve of this surface?
Which equation is graphed on the coordinate grid below? 4 3 2 1 -4-3-2-1 O
A y = 0.00004x2 - 0.004x - 0.5 B y = 0.00004x2 - 0.004x + 0.5
y
C
y = -0.00004x2 - 0.004x - 0.5
D y = -0.00004x2 - 0.004x + 0.5
1 2 3 4x
-2 -3 -4
6 A
y = -2x - 2 2 B y = -2x - _ 3 2x - 2 C y = -_ 3 2x - 2 D y=_ 3
A26
Which ordered pair is the maximum value of f(x) = -2x2 + 8x +1? A
(2, 9)
C
B
(8, 1)
D (-2, 0)
(-2, 8)
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D y = 2(x - 3)2 - 5
Name
Date
Practice By Standard Clarifying Objective MA.A.4.4 1
The cost of admission c to a museum exhibit for one teacher and s students is given by c = 10s + 15. Which of the following equations is equivalent? A
10s + c = 15
B
s = 0.1c - 1.5
C
c + 10s = -15
4
D s = 1.5c - 0.1
Johanna’s cell phone plan costs $35 per month for 500 minutes and $0.40 for each additional minute. Which equation represents the relationship between the total bill n and the number of extra minutes m Johanna used? A
n = 0.4m - 35
B
m + 0.4n = 35
C
n - 0.4m = 35
D n = 35 + 40m
2
Jasmine eats one less than two times the number of orange slices that Celia does. Which function can be used to determine the number of orange slices Celia ate g, if the number Jasmine ate h, is known?
5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
g = 2h - 1 1 1 B g=_ h+_ 2 2 C g = 2h + 1 1 1 D g=_ h+_ 2 2
Hima is paid 1.5 times her regular hourly rate when she works more than 40 hours per week. Her regular rate is $12 per hour. Which equation can Hima use to find her total pay p for a week in which she works h hours of overtime? A
p = 480 + 1.5h
B
p = 480 + 15h
C
p = 480 + 12h
D p = 480 + 18h
3
A rental store posts the following chart to explain the cost of renting a kayak. Hours
Price
2
$25
3
$30
4
$35
5
$40
6
Which equation can be used to find the price p to rent a kayak for h hours? A
p = 5h
B
p = 8h
C
p = 5h + 15
What is an equation for the line that has 7 slope _ and contains the point (-2, 3)? 5 19 7 A y=_ x+_ 5 5 7 B y = _x + 4 5 1 _ C y = 7x + _ 5 5 31 7 D y=_ x+_ 5 5
D p = 15h + 5
A27
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.A.4.5 1
Which point is a solution to the equation 8x + 12y = -32?
3
A
(0, -4) 2 B -2_ ,0 3 2 C 0, 2_ 3 D (-4, 0)
( (
A
(11, 3)
C
B
(3, 11)
D (-1, 4)
(4, -1)
)
4
In the graph below, for what value of x does y have a value of 2? 5 4 3 2 1
The relationship between miles x, and kilometers y, is represented in the graph below. 16
Kilometers
2
)
Which point lies on the line defined by the equation y = 4x - 1?
O
-5 -4 -3-2
y
y
1 2 3 4 5x
-2 -3 -4 -5
12 8
0
2
4
6
8x
A
1
C
B
2
D 5
3
Miles
Approximately how many kilometers did you travel if you drove 7 kilometers and 5 miles?
5
The table below shows the coordinates of some points on a line.
A
9 kilometers
B
10 kilometers
x
y
15 kilometers
0
-3
1
-1
2
1
C
D 16 kilometers
For what value of x does y = 0?
A28
A
-3
C
B
-1.5
D 2
1.5
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
Name
Date
Practice By Standard Clarifying Objective MA.A.4.6 1
Lee invests $500 at 4% annual interest, compounded monthly. The table below shows the value of his investment at the end of each year. Year
Value ($)
Year
Value ($)
1
520.37
5
610.50
2
541.57
6
635.37
3
563.64
7
661.26
4
586.60
8
688.20
3
The number of bacteria in a colony doubles every hour. The function A = 200(2x) describes the number of bacteria, after each hour, in a colony with an initial population of 200. The table below contains some solutions. Hour
Total
Hour
Total
1
400
7
25,600
2
800
8
51,200
3
1600
9
102,400
4
3200
10
204,800
5
6400
11
409,600
6
12,800
12
819,200
During what year will Lee’s investment exceed $600?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
A
Year 5
C
B
Year 6
D Year 10
Year 8
After how many hours is this population at least 100 times larger than it was initially?
Ana’s uncle puts money into her college fund for every “A” she earns. The dollar amount that he gives her is defined by the equation y = 1.5x where x is the number of As. Which table can Ana use to determine how many As she must earn to get $100? A B
x y
1 150
2 225
3 338
4 506
x y
5 75
6 90
7 105
8 120
C
x y
65 97.50
66 99
67 100.5
68 102
D
x y
10 57.67
11 86.50
4
A
6 hours
C
B
7 hours
D 10 hours
9 hours
The car that Carrie’s mother bought for $20,000 loses 12% of its value each year. The table below shows the value of the car at the end of each year. Year
Value ($)
Year
Value ($)
1
17,600
5
10,555
2
15,488
6
9288
3
13,629
7
8174
4
11,994
8
7193
Which function can be used to calculate the values in the table?
12 13 129.75 194.62
A
V = 20,000(2x)
B
V = 20,000(1.12x)
C
V = 20,000(0.88x)
D V = (20,000․0.88)x
A29
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.A.5.1 1
For Jason’s fourteenth birthday, his Uncle Matt put $200 into an education fund. Matt promised to double the amount in the fund each year until Jason is 21. Which equation represents the amount a in the fund as a function of the number of years n since Jason’s fourteenth birthday? A
4
A
A(n) = 200 + 2n
B
A(n) = 200(2n)
C
A(n) = 200 2n
For her sixteenth birthday, Maria’s family invests $2000 in an account with an interest rate that is equivalent to 6% compounded annually. Maria wants to make a table that shows the amount of interest earned each year and the total value of the investment. Which table should Maria use?
D A(n) = 200(2 n)
2
B
between 21 and 24
C
between 18 and 20
Piper starts strength training with 10-pound weights and plans to increase the weight by 1.5 pounds each week. Which equation shows the relationship between the number of weeks w, and the number of pounds p? A
p = 10 + 1.5w
B
p = 10w
C
p = 10w + 1.5
1
120.00
2120.00
2
120.00
2240.00
3
120.00
2360.00
4
120.00
2480.00
Year
Interest Earned
Total Value
1
120.00
2120.00
2
240.00
2360.00
3
360.00
2720.00
4
480.00
3200.00
Year
Interest Earned
Total Value
1
120.00
2120.00
2
127.00
2247.00
3
134.83
2381.83
4
142.91
2524.24
Year
Interest Earned
Total Value
1
120.00
2120.00
2
127.00
2127.00
3
134.83
2134.83
4
142.91
2142.91
C
D less than 17
3
Total Value
D
D p = 1.5(10 + w)
A30
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
at least 25
Interest Earned
B
Marianna has $50 in her savings account and saves $3 each week. Luis starts with no money in his savings account and saves $5.50 each week. After how many weeks will Luis have more money saved than Mariana? A
Year
Name
Date
Practice By Standard Clarifying Objective MA.A.5.2 1
Which graph shows the solution to the following system of equations?
2
2x + 3y = -3 4x - y = 8 A
5 4 3 2 1
y
1 2 3 4 5x
−3−2−10
3
−2 −3
B
3 2 1 −3−2−10
y
1 2 3 4 5x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
−2 −3 −4 −5
C
3 2 1 0
y
4 1 2 3 4 5 6 7x
−2 −3 −4 −5
D
y 5 4 3 2 1 −3−2−10
1 2 3 4 5x
−2 −3
A31
What is the solution to the system of equations? x+y=8 3x + 0.5y = 10 A
(-1, 11)
C
B
(5.6, 2.4)
D (2, 4)
(2.4, 5.6)
A baseball team’s manager buys 70 balls for $260. The balls used for practice cost $3.50 each, and the balls used in games cost $4.25 each. How many game balls did the manager buy? A
20
C
B
25
D 50
30
Which system of equations could best be solved using the substitution method? ⎧ A ⎨ 8x - 5y = 12 ⎩ 3x + 5y = -2 ⎧ 2 1 B _ x - _y = -2 3 ⎨ 2 _ 5 1 _ x + y = -2 6 ⎩2 ⎧ 4x - 5y = 10 C ⎨ ⎩ 8x - 10y = 20 ⎧ 3x - 2y = 8 D ⎨ ⎩ x + 7y = 12
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.A.5.3 1
A florist is preparing bunches of flowers to sell at a bazaar. Each bunch must have at least 8 flowers and cost no more than $10. Roses cost $3 each and daisies cost $0.50 each. Which set of inequalities models this situation? ⎧ x+y≥8 A ⎨ ⎩ 3x + 0.5y ≥ 10 ⎧ x+y≤8 B ⎨ 3x + 0.5y > 10 ⎩ ⎧ x+y>8 C ⎨ ⎩ 3x + 0.5y < 10 ⎧ x+y≥8 D ⎨ ⎩ 3x + 0.5y ≤ 10
3
Which graph represents the solution to the system of inequalities? 2y - x ≤ -5 -3y - 2x ≥ 4 A
3 2 1
y
1 2 3 4 5x
−3−2−10 −2 −3 −4 −5
B
3 2 1
y
1 2 3 4 5x
−3−2−10
2
−2 −3 −4 −5
A
B
C
D
C
3 2 1
nonfiction
1
2
3
4
5
−3−2−10
fiction
5
4
3
2
1
−2 −3 −4 −5
nonfiction
1
1
1
1
2
fiction
5
4
3
2
4
nonfiction
5
4
3
2
4
fiction
1
1
1
1
2
3 2 1
nonfiction
1
1
1
1
2
−3−2−10
fiction
4
3
2
1
3
−2 −3 −4 −5
D
A32
y
1 2 3 4 5x
y
1 2 3 4 5x
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Last month, Jemal read at least one fiction and one nonfiction book, up to a total of six books. He read twice as many fiction books as nonfiction. Which table contains the possible combinations of books?
Name
Date
Practice By Standard Clarifying Objective MA.A.5.4 1
2
Which is a possible graph of the solution set of the following system of inequalities? y≥-x+1 y ≤ 3x y
A
x
Castle.com is an Internet chat service. The site’s plan to increase the number of its users can be represented by the inequality y ≥ 6x + 8, shown on the graph below. A new competitor, Sassy.com, plans to at least double the number of its users every month. Sassy’s plan can be represented by the inequality y ≥ 2 x + 1, also shown below.
0 40 (4, 32)
Users (thousands)
32 y
B
x
24
Castle.com
16
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
1
2
3
4
Months
y
C
Sassy.com
8
0
Which of the following coordinates indicates that both companies are growing according to their plans?
x 0
A
(5, 40)
C
B
(2, 18)
D (3, 28)
(1, 12)
y
D
3
x 0
A33
Which point is a possible solution for this system of inequalities? x+y>4 y > 2x A
(0, 5)
C
B
(1, 3)
D (5, 5)
(3, 0)
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.D.1.1 1
The vertex-edge graph below represents some locations and the routes that connect them in the Cape May public transportation system. Downtown Station
Buck Spring Gap Mount Pisgah
UNCW Station Independence Station
Old Buck Spring Lodge Site
Pisgah Inn
Public Library
Turkey Spring Gap Little Bald Mountain
Pilot Rock Trail
A student at Lakeside High travels by public transportation to the public library. How many routes can he take without visiting any location more than once?
Which of the following statements is true about this graph? A
It is a connected Euler Circuit.
A
5
C
B
B
7
D 10
It is connected and can be drawn with a single line.
C
Old Buck Spring Lodge Site is the only vertex of odd degree.
9
D It can be drawn so that all vertices, except for the starting vertex, are visited once.
The vertex-edge graph below models the relationships among a group of high school students. Each vertex represents a person and each edge indicates a friendship. Jake
4
Ann Deval
What circuit(s) does the graph shown contain? p
n
Sophia
s
Lucy
Derrick Latoya
Li
v
How many friendships exist among these students? A
4
C
B
5
D 7
6
t
A
npvn, pnvtsp, pstvp
B
vtspn, nvpst, npvts
C
vtspvnp
D nvpn, tspvt
A34
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
The vertex-edge graph below represents some locations and the trails that connect them, in the Pisgah National Forest.
Central Station
Williston Middle
Lakeside High
3
Name
Date
Practice By Standard Clarifying Objective MA.D.1.2 1
Which of the vertex-edge graphs below is an Euler circuit?
3
A
The vertex-edge graph below represents a neighborhood. The edges represent streets. The vertices, labeled with letters, represent street intersections.
B
n
m
p
C
q
r
s
t
Which is the most efficient route that a snow plow can travel to remove snow from every street?
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
o
A
spnqsrpmnoqtsr
B
oqtsrpmno
C
mnoqtsrpsqnpm
D motrpnqs
2
The graph below represents a group of businesses. The edges represent the sidewalks between the businesses. Keys
Al’s Gas
Drug Store
Fruit
4
Laundry
A
Café
Which route forms a Hamilton cycle? A
Al’s, Gas, Drug Store, Keys, Laundry, Café, Fruit
B
Keys, Laundry, Café, Fruit, Drug Store, Gas, Al’s, Keys
C
Drug Store, Keys, Laundry, Café, Fruit, Drug Store, Gas, Al’s, Keys, Drug Store
Which vertex-edge graph is a Hamilton cycle?
B
C
D
D Laundry, Café, Fruit, Drug Store, Gas, Keys, Laundry
A35
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.G.1.1 1
An outline of the floor plan for the North Carolina State Capitol is placed on a coordinate grid as shown below. y
3
A parallelogram is represented on the coordinate plane as shown below. y
(30, 80)
(70, 20)
A
x
0
x C
What are the coordinates of the vertices of the parallelogram? B
(1, 2), (7, 2), (-3, -2), and (-3, -4)
Which coordinates best represent points A, B, and C?
B
(1, 2), (7, 2), (2, -3), and (-4, -3)
C
(2, 1), (2, 7), (-3, 2), and (-3, -4)
A
C A: (-30, 20) B: (-30, -80) C: (30, -20)
D (2, 1), (2, 7), (-2, -3), and (-4, -3)
B
D A: (-70, 20) A: (-30, 20) B: (-30, -80) B: (-80, -30) C: (-30, 20) C: (30, -20)
A: (-70, 20) B: (-80, -80) C: (-30, -20)
4
The vertices of a square are represented as coordinates in a plane. The coordinates (0, 1), (13, 1), and (0, -12) represent three vertices of the square. Which ordered pair represents the other vertex of the square? A
(-13, 12)
B
(-12, 12)
C
(12, -12)
The North Carolina Research Triangle is a triangular region with vertices formed by the cities of Raleigh, Durham, and Chapel Hill. Using a coordinate plane to represent the Research Triangle, the origin represents the city of Durham. Chapel Hill, located to the southwest of Durham, is represented by the coordinates (-3, -4). If Raleigh is located to the southeast of Durham, which coordinates best represents Raleigh? A
(8, 7)
C
B
(8, -7)
D (-8, -7)
(-8, 7)
D (13, -12)
A36
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
A
Name
Date
Practice By Standard Clarifying Objective MA.G.1.2 1
What is the slope of the line shown below? 5 4 3 2 1 -5 -4 -3 -2
O
4
y
1 2 3 4 5x
The points (0, 12) and (-16, 0) are connected to form a right triangle with the x- and y-axes. What are the coordinates of the midpoint of the hypotenuse? A
(6, -8)
C
B
(-6, -8)
D (-8, 6)
(-8, -6)
-2 -3 -4 -5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
3
5
C
B
_1
D -5
5
5
1 -_
A
5
Suppose the line shown below is extended. 5 4 3 2 1
A line is drawn through the point (3, 5) perpendicular to the x-axis. What is the least distance from the point (-3, -2) to the line? A
3
C
B
4
D 7
-5 -4 -3-2
O
y
1 2 3 4 5x
-2 -3 -4 -5
6
Which of the following points will be on the line? A
(-8, -10)
C
B
(-6, -7)
D (7, 11)
(6, 10)
What is the slope of a line that passes through the points (-2, 5) and (10, -4)? A
4 -_
6
3
3 B -_ 4 3 _ C
D
4 _4 3
A37
A line segment is drawn from P(-1, 2) to Q(3, 4). What −− is the distance from the midpoint of PQ to (-2, 7)? A
3
C
B
4
D 6
5
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.G.1.3 1
A quadrilateral has vertices at P(2, 2), Q(5, -2), R(1, -6), and S(-3, -3). What type of quadrilateral is PQRS? A
rhombus
B
parallelogram
C
trapezoid
4
D rectangle
2
Which statement best describes the difference between a line segment and a ray? A
C
A line segment has one distinct endpoint, and a ray has two distinct endpoints.
5
square
B
rectangle
C
rhombus
square
C
right triangle
Consider the figure shown below. 3 (3, 10)
4 (1, 6)
2 (5, 6)
1 (3, 2) x
O
Which statement best justifies the identification of the figure represented by the ordered pairs?
A figure is drawn in the coordinate plane with the vertices (-2, -3), (-4, -1), (-2, 1), and (0, -1). Which geometric figure do these points represent? A
B
y
D A line segment has two distinct endpoints, and a ray has one distinct endpoint.
3
trapezoid
A
The figure is a trapezoid because −− −− SR and PQ are parallel.
B
The figure is a parallelogram −− −− because SR and PQ are equal in length.
C
The figure is a rhombus because opposite sides are parallel, and all four sides are equal in length.
D The figure is a square because all four sides are equal in length.
D parallelogram
A38
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A line segment has two distinct endpoints, and a ray continues in both directions.
A
D isosceles triangle
A line segment continues in both directions, and a ray has two distinct endpoints.
B
Natasha draws a line segment with endpoints at (-1, -3) and (1, 1). She draws a second line segment between (1, 1) and (4, 7), a third line segment between (4, 7) and (9, -3), and a fourth line segment between (9, -3) and (-1, -3). Which geometric figure do the line segments form?
Name
Date
Practice By Standard Clarifying Objective MA.G.2.1 1
−− Which term best describes KL in the diagram below?
3
, ) +
.
Which of the statements is true? A
A diameter is a chord.
B
A chord is a secant.
C
A radius is a chord.
D A tangent is a secant.
-
A
tangent
C
B
secant
D chord
4
radius
Which is tangent to the circle below? " % (
2
$
and AD In the diagram below, AC intersect at point A.
#
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
$
A
.
#
B
−− BG −− D DB
−− AB −− AC
C
" %
5
Which statement best describes the , AD , and relationship between AC circle M? A
is a secant of circle M, and AC is tangent to circle M. AD
B
is a secant of circle M, and AD is tangent to circle M. AC
C
and AD are both secants of AC circle M.
Which line segment is a chord but not a diameter of circle R? # % "
$ 3
D AC and AD are both tangent to circle M.
A B
A39
−− RC −− AB
−−− CD −− D AC
C
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.D.2.2 1
A triangle has a base of 12 inches and an area of 36 square inches. What is the height of the triangle? A
3 inches
B
4 inches
C
5 inches
4
Some art students are painting a rectangular wall mural of the flag of North Carolina as shown below. 3 ft
D 6 inches
2
12 ft
What is the area of the figure below? " 5 in.
4 in.
#
4.5 in.
%
9 in.
The horizontal bars are congruent rectangles, and the length of each bar is equal to the height of the flag. What will be the area of the flag on the mural?
6 in.
$
A
24 square inches
A
36 ft 2
C
B
28.5 square inches
B
54 ft 2
D 144 ft 2
C
29.25 square inches
108 ft 2
5 3
A park is in the shape of the figure below.
The length of one leg of a right triangle is 12 centimeters, and the other leg is 21 centimeters. What is the area of the triangle? A
24.2 cm 2
B
63 cm 2
C
126 cm 2
D 252 cm
3.9 mi
2.7 mi
3.5 mi
3.2 mi
2
What is the area of the park?
A40
A
13.09 sq mi
C
B
13.65 sq mi
D 14.8 sq mi
14.21 sq mi
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D 40.5 square inches
Name
Date
Practice By Standard Clarifying Objective MA.G.2.3 1
A model of a square pyramid has the dimensions shown below.
3
The cylinder shown below is packaging for a sculpture that is in the shape of a right circular cone. 5 ft
6.4 cm
3 ft
11.5 cm
The sculpture is 3 feet high with a base diameter of 10 feet. What is the approximate volume of the padding that is needed to fill the space around the sculpture?
The model is packaged in a box that is a rectangular prism. If the box has the same height and base area as the pyramid, what is the approximate volume of the box?
A
52.3 ft 3
A
74 cm 3
C
471 cm 3
B
78.5 ft 3
B
132 cm 3
D 846 cm 3
C
157 ft 3
D 471 ft 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
Which of the following techniques can be used to find the volume of any right prism or cylinder? A
Find the area of each side and multiply by the height.
B
Multiply the length and the width and the height.
C
Double the area of each side and add the results together.
4
D Find the area of the base and multiply by the height.
3
A cylindrical container and a coneshaped container are both filled with water. If the containers have equal diameters and equal heights, how much more water can the cylindrical container hold than the cone-shaped container? A
2 times as much
B
3 times as much
C
4 times as much
D 5 times as much
A rectangular pyramid has a volume of 234 cubic centimeters. What is the volume of the smallest rectangular prism that can contain it? A
78 cm 3
C
702 cm 3
B
234 cm 3
D 1872 cm 3
A41
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.G.2.4 1
4
The shape of the lighthouse on Ocrakoke Island is almost a right circular cone.
A wooden fence post is in the shape of a right triangular prism. The volume of the prism is approximately 562.5 cubic inches. The base is a right isosceles triangle with legs that are each 5 inches long. What is the approximate height of the fence post? A
56 inches
B
45 inches
C
37 inches
D 22 inches If the height of the lighthouse is 75 feet and the diameter of the base is 25 feet, what is the approximate volume of the lighthouse?
3
2944 ft3
C
B
5888 ft3
D 36,797 ft3
12,266 ft3
A company sells pepper in cans that are cylinders. Each can is 14 centimeters tall and has a radius of 4 centimeters. What is the approximate volume of a can of pepper? A
56 cm3
C
B
224 cm3
D 2463 cm3
6
704 cm3
Cylinder X and cylinder Y have equal heights, but the diameter of cylinder Y 5 is _ the diameter of cylinder X. What 3 is the ratio of the volume of cylinder X to the volume of cylinder Y? A
3 to 5
C
B
5 to 3
D 25 to 9
A pyramid has a square base with each side measuring 6 meters. The distance from the center of the base to the top of the pyramid is 9 meters. What is the volume of the pyramid? A
18 m3
C
B
54 m3
D 324 m3
108 m3
Which of the following methods will give a reasonable estimate for the volume of a cone? A
Multiply the height by 9.
B
Multiply the square of the radius by the height.
C
Divide the square of the radius by 3.
D Multiply the radius by the height and divide by 3.
9 to 25
A42
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
A
5
Name
Date
Practice By Standard Clarifying Objective MA.G.2.5 1
3
The City Council is reviewing plans for a new monument to honor city heroes. One design is the cylinder in the diagram below.
What is the surface area of the box?
2 ft 10 in.
15 ft
3 in.
7.5 in.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The names of the heroes will be written on the lateral surface of the monument. What is the approximate lateral surface area of the monument? A
30.0 ft2
C
B
47.1 ft2
D 100.5 ft2
4
A
127.5 in2
C
B
210 in2
D 255 in2
240 in2
A box has the planar net shown below.
94.2 ft2 4 in.
4 in. 10 in. 4 in.
2
4 in.
A shed has the dimensions shown below. What is the surface area of the box?
3 in. 9 in.
A
160 in2
C
B
192 in2
D 272 in2
240 in2
10 in. 8 in.
If all surfaces except the base are painted, what is the total area that is painted? A
174 m2
C
B
324 m2
D 720 m2
448 m2
A43
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.S.1.1 1
The table below shows the 2009 annual salaries for six jobs in Raleigh. Job
Salary ($)
Administrative Assistant
35,427
Bank Teller
22,317
Office Manager
43,135
Pharmaceutical Sales Representative
65,182
Professional Sports Scout
64,272
Registered Nurse
52,304
3
1BOUIFSTWT'BMDPOTm
0
B
10
15
20
25
30
35
40
45
Mean: 46,064.60 Median: 47,719.50
A
The plot will not change.
Range: 16,877.00
B
The end of the left whisker will change.
C
The end of the right whisker will change.
Mean: 46,064.60 Median: 54,158.50
D The end of the left and right whiskers will both change.
Mean: 47,106.17 Median: 47,719.50 Range: 42,865.00
D Mean: 47,106.17
4
Median: 54,158.50
Daily low temperatures for one month are shown below.
Range: 16,877.00
2
Daily Low Temperatures (°F)
Rosa scores 75, 62, 91, 85, and 80 on her first five math quizzes. After she takes the sixth quiz, her median score is 82. What was her score on the sixth math quiz? A
82
C
B
84
D 99
4
6
5
6
8 9
6
2
2 3 3 3 3 5 5 5
5
6
6 8
7
0
1 2 2 2 2 3 6 6
7
7
8
8
9
Key 6|2 = 62°F
What is the median temperature if the highest and lowest temperatures are disregarded?
89
A44
A
66°F
C
B
66.5°F
D 67.5°F
67°F
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Range: 42,865.00 C
5
Only two games had point spreads greater than 30. Which statement best describes the effect on the boxand-whisker plot if these two outliers are removed?
What are the mean, median, and range, in dollars, for the data set? A
The Carolina Panthers played the Atlanta Falcons 28 times between 1995 and 2008. The point spreads for those games are represented by the box-and-whisker plot below.
Name
Date
Practice By Standard Clarifying Objective MA.S.1.2 The box-and-whisker plot below displays data for the highest elevations in all 50 states in the United States.
1
2
The number of available rooms at each of 10 large hotel chains in the United States is listed in the table. Number of Available Rooms
)JHIFTU&MFWBUJPOTJO4UBUFT (1.8) (4.6) (0.3)
Rooms
Rank
Rooms
1 2
209,765 186,410
6
97,007
7
94,738
3
156,376
8
83,276
4
109,205
9
77,519
5
101,628
10
60,454
(20.3)
0
2
4
6
8 10 12 14 16 Thousand Feet
18 20 22
The highest elevation in North Carolina is Mount Mitchell at 6684 feet. Which statement best describes the relationship between the highest elevation in North Carolina and the highest elevations in the other 49 states? A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Rank
(11.2)
B
C
To the nearest whole number, what is the standard deviation for the number of hotel rooms in these chains?
Exactly one fourth of the other highest elevations are greater than the highest elevation in North Carolina.
3
At least one fourth of the other highest elevations are greater than the highest elevation in North Carolina.
A
48,748
C
B
117,637
D 4,400,135,523
2,205,537,957
The heights of 14 members of the girls’ basketball team are shown. 65
Team Heights (inches) 66 63 62 58 66 67
64
64 61 64 63 69 65
Which statement best describes the spread of this data?
Exactly one half of the other highest elevations are greater than the highest elevation in North Carolina.
D At least one half of the other highest elevations are greater than the highest elevation in North Carolina.
A
About 68% of the team heights are between 62.8 and 65.4 inches.
B
About 68% of the team heights are between 61.5 and 66.7 inches.
C
About 95% of the team heights are between 62.8 and 65.4 inches.
D About 95% of the team heights are between 61.5 and 66.7 inches.
A45
North Carolina StudyText, Math A
Name
Date
Practice By Standard
(continued)
Clarifying Objective MA.S.1.2 Customers at a gasoline station purchase a mean of 16.4 gallons per transaction. The standard deviation is 4.8 gallons and 95% of purchases are within 2 standard deviations of the mean. If the price of gas is $2.80 per gallon, what is the range of the cost c, of purchases per transaction by 95% of customers? A
$19.04 ≤ c ≤ $72.80
B
$32.48 ≤ c ≤ $59.36
C
$39.20 ≤ c ≤ $52.64
6
Home Sales in June
0
Median
1.3
Maximum
4.2
Minimum
0.2
2nd Quartile
0.8
3rd Quartile
3.8
7
Which range best describes the growth g, in inches, of a student whose increase in height was in the 60th percentile? A
0.8 < g < 1.3
B
1.3 < g < 3.8
C
1.4 < g < 3.8
800
1000
A
The bell shape will increase in height.
B
The bell shape will decrease in height.
C
The bell shape will shift right and a longer tail will extend left.
A statewide math test was taken by 8000 students. The mean score was 82 and the standard deviation was 4. If the distribution of scores is normal, which of the following statements best describes the situation? A
All scores were between 78 and 86.
B
About 1000 students scored between 78 and 86.
C
About 5500 students scored between 78 and 86.
D About 7200 students scored between 78 and 86.
D 3.8 < g < 4.2
A46
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Increase in Height (inches) 1.4
600
D The bell shape will shift left and a longer tail will extend right.
Growth Study Results
Mean
400
After creating the graph, Mr. Montoya decides to add a sale for $2.5 million. Which statement best describes how the graph for June sales will change after the $2.5 million sale is added?
School nurses studied the growth of students by tracking the increase in height for 2000 middle school students for one year. The summary report for the study included the measures shown in the table below. Measure
200
Sale Price (thousands of dollars)
D $43.12 ≤ c ≤ $48.72
5
Mr. Montoya created the graph below to show home sales in June.
Number of Sales
4
Name
Date
Practice By Standard Clarifying Objective MA.S.2.1 1
At a field trip to the battleship North Carolina, students recorded data about the relationship between a person’s age and the time it takes to accomplish a task after a verbal command is issued. Their data is shown in the table below.
3
Number of People
Evacuation Time (minutes)
200
3.5
Age
Time (seconds)
400
4.5
8
3.4
600
6.0
9
3.6
800
6.5
10
3.8
1000
9.5
11
4.1
1200
10.5
12
4.5
13
4.6
Which equation represents the median line that best fits this data?
14
4.7
A
y = 0.0075x - 1.5
15
4.8
16
5.1
B
y = 0.0075x + 1.5
C
y = 0.75x - 1.75
Which of the following equations for a median line best fits this data? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The amount of time that it takes to evacuate a sports arena is related to the number of people in the arena. Data is shown in the table below.
A
y = 0.2x - 1.8
B
y = 0.2x + 1.9
C
y = 0.2x + 2.1
D y = 0.75x + 1.75
4
What is the median line of best fit for the data in the scatterplot below?
D y = 0.25x + 1.1 10 8
2
Which equation represents the leastsquares regression line that best fits the table below? x
6 4
y
5
-2
6
3
7
5
8
6
2
0
2
4
A
y = -x + 12.3
A
y = 2.6x + 13.9
B
y = -x - 12.3
B
y = 2.6x - 13.9
C
y = x + 12.3
C
y = 0.44x + 26.5
D y = x - 12.3
6
8
10
D y = 0.44x - 26.5
A47
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.S.2.2 1
The scatterplot below shows the relationship between a major league baseball team’s payroll and the number of games won.
3
Which scatterplot is best represented by a quadratic model? A
y
Annual Payroll vs. Games Won 2006-2009 100
Games Won
80 0
60
B
40
x
y
20
0
50
100
150
200
250
Average Annual Payroll ($1.0 million)
What type of model would best fit this situation? linear model with a negative slope
B
linear model with a positive slope
C
quadratic model
C
x
y
D exponential model 0
2
D
A communications company introduces new free software for cell phones and other mobile devices. The number of users increases each month as shown in the table below. Month
1
2
3
4
5
Users in Millions
0.5
1.1
4.2
64
2250
x
y
0
x
What type of model would best fit this situation? A
linear model with a negative slope
B
linear model with a positive slope
C
quadratic model
D exponential model
A48
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
0
Name
Date
Practice By Standard Clarifying Objective MA.N.1.1 1
The North Carolina State Capitol has the dimensions shown below.
4
160 ft West Wing
South Wing
North 140 ft Wing
5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
If a scale model is to measure 36 inches north to south, how wide should the model be east to west?
2
3
31.5 in.
C
B
41.14 in.
D 41.14 ft
31.5 ft
In December 2003, a Japanese JR-Maglev train reached a speed of about 360 miles per hour. What is this speed in feet per minute? A
60
C
B
88
D 31,680
6
A new concept car can accelerate at a steady rate from 0 to 80 miles per hour in 1 minute. How long will it take to accelerate from 50 miles per hour to 60 miles per hour? 7.5 sec
C
B
13.3 sec
D 48 sec
6
B
75
C
83
North Carolina has 13 delegates to the United States House of Representatives. During a recent year, the ratio of representatives to people living in North Carolina was about 1:700,000. About how many people lived in North Carolina that year? A
910,000
B
1,857,000
C
9,100,000
D 18,857,000
3600
A
A
D 120
East Wing
A
An inspector at a factory found 12 defective items among 160 that were checked. At this rate, how many defective items can be expected in a shipment of 1000 items?
A pump removed 120 gallons of water from a flooded area in 45 minutes. How long will it take that same pump to remove 800 gallons of water? A
3 hours
B
3.5 hours
C
5 hours
D 6.75 hours
37.5 sec
A49
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.N.1.2 1
Carlotta used the data below to solve a math problem about Wake County, North Carolina.
3
The table below shows the cost to rent a canoe for a given number of hours. RENTAL CHARGES
WAKE COUNTY, NC Population (approximate)
Number of Hours
Cost
1
$17.50
2
$25.00
3
$32.50
4
$40.00
850,000
Area (mi2)
832
Carlotta’s work is shown below.
Which of the following statements best describes the cost of renting a canoe? 850,000 x = 832 1 832x = 850,000 x = 1,021.634
A
The cost is $17.50 plus $5.00 for each hour.
B
The cost per hour increases as the number of hours increases.
C
The cost is $10.00 for each hour.
D The cost is $10.00 plus $7.50 for each hour.
A
The population of each square mile is 1021.634 people.
B
Each square mile in the county has more than 1021 residents.
C
An average of 1021.634 people live in each square mile.
4
D There are an average of 1021.634 people per mile.
2
A balloon is released from ground level and rises at a constant rate. Which unit best describes the rate that it climbs? A
feet
B
feet per minute D minutes per foot
C
A tire is inflated to a pressure of 32 pounds per square inch. Which statement describes the meaning of this measurement? A
The air inside the tire weighs 32 pounds.
B
The air inside the tire exerts 32 pounds of pressure on each square inch of the tire.
C
The surface area of the tire is 32 square inches.
D The air inside all four tires weighs 32 pounds.
minutes
A50
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which of the following best describes the meaning of her answer?
Name
Date
Practice By Standard Clarifying Objective MA.N.2.1 1
Multiply 4 3 · 4 2. A
45
B
46
C
86
5
The area of the rectangle shown below is 2.7 × 10 6 square feet.
D 16 5
2
h
23 · 53 Simplify _ . 5 -1
3 × 102 ft
2 ·5
A
22 · 54
B
2 -2 · 5 4
C
2
2 ·5
What is an expression for the height h, in feet, of the rectangle?
-2
D 2 -2 · 5 -2
A
9 × 10 3
B
9 × 10 4
C
8.1 × 10 4
D 8.1 × 10 8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
3.2 × 10 6 Divide _ . -4 1.6 × 10
A
0.2 × 10 2
B
1.6 × 10 10
C
2.0 × 10 2
6
D 2.0 × 10 10
Which expression is equivalent to (-2 3 × 5 -1) 2 ? A
-2 5 × 5
B
25 × 5
C
2 -_ 2 6
26 D _ 2
4
5
5
Which of the following has the same value as (-1.2 · 3 3) 2 ? A
1.44 · 3 6
B
1.44 · 3 5
C
-1.44 · 3 5
A
−40
D -1.44 · 3 6
B
0
C
8
7
What is the value of (2 5)(3 0)(2 -2)?
D 64
A51
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.N.2.2 1
Simplify (3x 4y 3) 3.
4
3x 7y 6
A
Simplify (5p 3q -4)(4p -2q). A
12 9
B
3x y
C
27x 7y 6
B
12 9
D 27x y
C
20p _ q3 -20p _ q3
20p -6q -4
D -20pq -3
2
Which expression is equivalent 1 to _ x -3y 5 ? 4
5
5
y A _3 x
4 B _ 3 5
D
xy
3
5
y C _3 4x 4y 5 _
1 Simplify _ ab 4c 2
(2
A
a 4b 16c 8 _
B
-a 4b 16c 8 _
C
16 _
x3
.
16
16
a 4b 16c 8
-16 D _ 4 16 8 ab c
4xy 2
6
5xy 2x
What is an expression for the volume of the solid? A
20x 2y 2
B
20x 2y 3
C
40x 2y 3
Which expression is equivalent -5 to (a -3b 4c -1) ? A
a -8b -1c -6
B
a 15b 20c 5
C
a 15 _ b 20c 5
a 15c 5 D _ 20 b
D 40x 3y 3
A52
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Expressions for the dimensions of a rectangular solid are shown below.
-4
)
Name
Date
Practice By Standard Clarifying Objective MA.N.2.3 1
What is an equivalent form of √ 56 with no perfect square in the radicand? A B
5
2 √ 7 2 √14
A B
4 √ 7 D 5 √ 6
C
C
2
D
3
What is an equivalent form of √ 72 with no perfect cube in the radicand? A
6
B
2 √ 3 3 2 √ 6
C
Which of the following is equivalent 3 to 5 √ 10 ?
6
3
D 2 √ 9
3 √ 500 3 √1250
The area of a square is 800 square feet. How long is each side of the square? B
4 √ 10 feet feet 20 √2
C
40 feet
A
3
3 √ 50 3 √250
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D 200 feet
3
3 What is an equivalent form of √8000 with no perfect cube in the radicand? 3
B
2 √ 100 3 10 √ 2
C
20
A
7
D 200
4
8
Which of the following is equivalent to 30? A B C D
√ 90 √ 300
The volume of a cube is 81,000 cubic centimeters. What is the length, in centimeters, of each edge? A
27
B
30
3
30 √ 3 3 D 30 √9 C
3
Simplify 3 √ 625 . A
75
B
15 √5
3
25 3 D 5 √25
C
√ 900 √ 270 3
A53
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.A.1.1 1
Simplify 5x 2y - (3xy 2 + x) + 5x(xy - 1). A
10x 2y - 3xy 2 - 6x
B
10x 2y - 3xy 2 - x - 1
C
5x 2y - 3xy 2 + x - 1 2
5
Expressions for the dimensions of a right triangle are shown in the diagram below.
3x + 1
2
D 5x y - 3xy - 4x 2x - 4
2
2
What is an expression for the area of the triangle?
2
Multiply -2a (a - b) . A
-2a 3 - b 2
A
3x 2 - 5x - 2
B
-2a 3 + 2a 2b 2
B
3x 2 - 7x - 2
C
-2a 4 - 4a 3b - 2a 2b 2
C
6x 2 - 10x - 4
D 6x 2 - 2x - 4
D -2a 4 + 4a 3b - 2a 2b 2
6
Simplify (6h 3p 2 - 4h 2p + 8hp 4) ÷ -2hp. A
-3h 2p - 2h - 4hp 3
B
-3h 2p + 2h - 4hp 3
C
-3h 2p + 2h - 4p 3
D -3h 2p - 4h + 4hp 3
4
-60x 9y 6
B
60x 9y 5
C
-120x 9y 6
A
rt + 2r
B
2r + t + 2
C
2rt + 4r + t + 2
D 2r 2t + 4r 2 + rt + 2r
Multiply 3x 2(-2xy 2) 3(5x 4). A
A cyclist pedals at rate of r miles per hour on a flat surface. If he cycles downhill, his rate increases to 2r + 1. What is an expression for the distance in miles that he cycles downhill in t + 2 hours?
D 120x 9y 6
A54
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
Name
Date
Practice By Standard Clarifying Objective MA.A.1.2 1
Solve the equation below for y.
4
4 -4(2x + 1) - 6y - x = _ y - 8
A
27 27 y = -_ x+_
B
y = -66x + _
C
y = - _x + 3
22
4 Rico solved the equation _x = 4 on 5 the paper below.
3
Rico
22
88 3
( ) ( )
5 4 5 x = (64) 4 5 4 5 4 x 5 • = (64) 4 5 4
45 4
1 • x = 80
6 27 D y = -_ x+_ 22
2
11
x = 80
Which property is demonstrated by
Which property did Rico use to solve the equation?
a 2b the equation _ _ = 1? 2b a A Associative Property of Multiplication B Commutative Property of Multiplication Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
( )
C
Multiplicative Inverse
A
Multiplicative Identity
B
Associative Property of Multiplication
C
Addition Property of Equality
D Distributive Property
D Multiplicative Identity
5 3
2 3x + _y + x - 5y = -8x + 6
Which equation demonstrates the Commutative Property of Addition? A
-6x(x - 2) = -6x 2 + 12x
B
3x + 4x 2 - x = 4x 2 + 3x - x
C
2
Solve the equation below for y. 3
2
36y xz + 1 = 36xy z + 1
D 3xy + (2xy + 5) = (3xy + 2xy) + 5
A
y = _x - _
B
y = - _x - _
C
y = _x - _
18 7
9 7
18 13
12 13
36 13
18 13
38 6 D y=_ x-_ 15
A55
5
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.A.1.3 1
Factor 6x 2 - 5x - 1.
5
What are the solutions of 2x 2 + 3x - 2 = 0 ?
A
(6x - 1)(x + 1)
B
(6x + 1)(x - 1)
A
_1 and -2
C
(3x + 1)(2x - 1)
B
- _ and 2
C
1 and -4
D (3x - 1)(2x + 1)
2
1 2
D -1 and 4
2
Factor 6x 2 + 5x - 4. A
(3x - 4)(2x + 1)
B
(3x + 2)(2x - 2)
C
(2x + 1)(3x - 4)
6
D (2x - 1)(3x + 4)
3
2
What are the zeros of y = x - x - 6? 2 and −3
B
−2 and 3
C
1 and −6
7
Which of the following quadratic expressions has 2x − 5 as a factor? A
4x 2 + 20x + 25
B
2x 2 - 5x - 25
C
-4x 2 + 25
0 and -1
B
1 and -1
C
x - 1 and x + 5
D x + 1 and x - 5
D −1 and 6
4
A
The expression 10x 2 - 13x - 3 represents the area of a rectangle. Which of the following are possible expressions for the length and width? A
(10x + 1) and (x - 3)
B
(10x - 1) and (x + 3)
C
(2x + 3) and (5x - 1)
D (2x - 3) and (5x + 1)
D 2x 2 + 7x + 5
A56
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
The points (-1, 0) and (5, 0) are zeros of a quadratic function. What are the factors of the quadratic function?
Name
Date
Practice By Standard Clarifying Objective MA.A.2.1 1
The time needed to travel a certain distance varies inversely with respect to the speed. Reza traveled an average of 60 miles per hour for 4 hours and 24 minutes. How long would it take him to travel the same distance if he travels at 55 miles per hour? A
4 hours and 30 minutes
B
4 hours and 44 minutes
C
4 hours and 48 minutes
4
5
D 4 hours and 55 minutes
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
Suppose h varies directly as g, and h = 12 when g = 5. What is the value of h when g = 8? A
3.3
B
7.5
C
15
A
-10
C
B
-2.5
D 10
2.5
The time it takes to paint a room varies inversely with the number of painters. In 6 hours, 4 painters can paint a room. If everyone works at the same speed, in how many hours can 3 painters paint the same room? A
4.5
B
7.5
C
8
D 9
6
D 19.2
3
Suppose s varies inversely as t, and s = 6 when t = −15. What is the value of s when t = -9?
When p is 5, q is 12. If q varies directly as p, what is the value of q when p = 3?
In North Carolina, a sales and use tax is charged on many items. In each county, the dollar value of the tax varies directly with the price of the item. When Carl bought a $25 shirt at a store in Alexander County, he paid a $1.75 sales and use tax. If Carl buys a $32 shirt from the same store, how much tax will he pay?
A
1.25
B
7.2
A
$1.73
C
10
B
$2.24
D 20
C
$5.60
D $22.24
A57
North Carolina StudyText, Math A
Name
Date
Practice By Standard Clarifying Objective MA.A.2.2 1
A recipe for fruit punch requires 3 cups of orange juice to make 8 servings of punch. If the number of cups of orange juice required by the recipe varies directly with the number of servings produced, which equation relates the number of cups c of orange juice, and the number of servings s of punch? A
c=s-5
C
B
c = _s
D
3 8
4
c = _s
8 3 s _ c= 24
5 2
Suppose h varies inversely as g, and h = -5 when g = 6. Which equation represents the relationship between h and g? A
3
C D
g h = -_ 30 _ h = - 30 g
6
When x is 15, y is 24. If y varies directly as x, which equation relates x and y? A
y = 1.6x
B
y=x+9
C
y = 0.625x
360 D y=_ x
A
s = 4P
C
B
s=_
D
4 P
s=_
P 4 1 s=_ 4P
When a is 9, b is 4. If b varies inversely as a, which equation relates a and b? b=_ a 36
A
b = 4 - 9a
C
B
b = 36a
a D b=_ 36
The total cost to rent a beach house for a week will be divided equally among several friends. The cost per person will vary inversely with the number of people renting the house. If 4 friends rent the house, the cost per person will be $435. Which equation represents the relationship between the cost per person c and the number of people n renting the house? A
c=_
B
c=_
C
c = 1740n
4n 435 435 4n
1740 D c=_ n
A58
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
B
5 h = -_ 6g 5g _ h=6
The perimeter of a square varies directly with the length of its side, as represented by the equation P = 4s, where P is the perimeter and s is the length of one side. Which equation solves P = 4s for s?
Name
Date
Practice By Standard Clarifying Objective MA.A.2.3 1
The equation t = _ s represents the relationship between the time t that is needed to travel the 100-mile distance between Charlotte and Greensboro and the travel speed s. What effect will a decrease in speed have on travel time between Charlotte and Greensboro? 100
A
It will decrease the travel time.
B
It will shorten the travel distance.
C
It will increase the travel time.
4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
q will be multiplied by 3.
B
q will be divided by 3.
C
q will be multiplied by 9.
The equation _ = k represents the T relationship between a gas’ pressure and temperature, where k is the constant of variation. What is the effect of an increase in gas pressure?
People of all ages enjoy riding on the recently restored 1912 Pullen Park Carousel in Raleigh, North Carolina. The equation s = 0.2πr approximates the relationship between the speed s, in feet per second, of a person on the carousel and her distance r, in feet, from the carousel’s center. If a person takes a ride and then moves 3 feet closer to the center for a second ride, what happens to her speed?
A
It will cause k to increase.
A
It increases by about 0.6 ft/sec.
B
Gas temperature will increase.
B
It decreases by about 1.9 ft/sec.
C
Gas temperature will decrease.
C
It increases by about 1.9 ft/sec.
5
P
D It decreases by about 4.4 ft/sec.
D It will cause k to decrease.
3
A
D q will be divided by 9.
D It will lengthen the travel distance.
2
Suppose p and q vary directly. If p is tripled, how will q change?
A
It will be divided by 4π.
A manufacturer is redesigning its boxes. The newly-designed box will be 2 times as long and 1.25 times as wide as the old box. The volume will not change. How will the height change?
B
It will be divided by 4.
A
It will be multiplied by 2.5.
C
It will be divided by 2π.
B
It will be multiplied by 1.6.
C
It will be multiplied by 0.5.
6
The circumference C of a circle varies directly with the radius r. If the radius is halved, how will C change?
D It will be divided by 2.
D It will be multiplied by 0.4.
A59
North Carolina StudyText, Math A
Name
Date
Practice Test 1
A person standing still in Raleigh travels around the center of Earth at the rate of about 844 miles per hour because of Earth’s rotation. Which expression equals that speed in feet per second? (1 mile = 5280 feet)
3
A ski trail map shows that a hill is 180 feet high. A ski trail, which runs straight down the hill, is 240 feet long. Kiara made the following table before skiing down the hill.
1 hr 1 min 844 mi _ A _ · ·_ 5280 ft
60 min
60 sec
240
180
5280 ft _ 1 hr 844 mi _ _ · ·
200
150
3600 sec
160
120
C
3600 sec 1 mi 844 mi _ _ · ·_
120
90
80
60
40
30
1 mi
1 hr
1 hr
5280 ft
1 hr
1 mi
1 hr
60 min
What can you conclude from Kiara’s table? A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Height (ft)
B
5280 ft _ 1 hr 844 mi _ D _ · ·
2
Distance (ft)
The slope of the hill is _. 3 4
Which table describes a quadratic relationship between x and y?
B
The ratio of the vertical distance to the horizontal distance is 4 : 3.
A
C
For every 4 feet that she skis, she will appear to go 3 feet on the map.
B
x
y
0
C
x
y
10
0
0
1
6
1
2
2
2
2
8
3
-2
3
18
4
-6
4
32
x
y
x
y
0
0
0
-4
1
2
1
-1
2
4
2
2
3
8
3
5
4
16
4
8
D
D For every 4 feet that she skis, her elevation will decrease 3 feet.
4
A series of canisters have a fixed height. The volume V of the canisters varies directly with the square of the radius r of the base. Which equation expresses this relationship? A
V = k + r2
B
V = kr
C
V = kr 2
D Vr 2 = k
A60
North Carolina StudyText, Math A
Name
Date
Practice Test 5
(continued)
A landscaper designs a rectangular garden. She splits the garden into two areas by placing a fence on one diagonal as in the diagram below.
7
20 ft
25 ft
45
B
15 √ 5 5 √41
C
A
lose 32 pounds
B
gain 2.28 pounds
C
gain 3.8 pounds
D gain 6 pounds
Which choice is the simplest expression for the fence length, in feet? A
A person’s weight on Mars varies directly with his weight on Earth. Someone who weighs 100 pounds on Earth would weigh 38 pounds on Mars. If Jason gains 6 pounds in one year, by how much would his weight change on Mars?
8
What is the slope of the line shown in the graph below?
-4 -3 -2
6
O
y
1 2 3 4x
-2 -3 -4
The volume of a rectangular prism can be expressed as 12x3 - 10x2 - 9x. If the height is 4x, what is an expression for the area of the base?
A
-2
A
3x2 - 6x - 5
B
-1
B
3x2 - 5x - 2.5
C
-_
C
3x2 - 10x - 2.5
1 D _
1 2
2
2
D 3x - 2.5x - 2.25
A61
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4 3 2 1
D 10 √ 10
Name
Date
Practice Test 9
(continued)
The volume of the reservoir formed by the hydroelectric Fontana Dam is 1,443,000 acres-feet3. The diagram below explains the meaning of the acre-foot unit of measurement.
10
1 acre
Which of the following situations describes a linear relationship? A
the height h of a basketball t seconds after it is thrown
B
the number of Celsius degrees m in n Fahrenheit degrees
C
the amount of money A in an account after x days if interest is compounded daily
1 ft
D the time t that it takes to drive a fixed distance at speed s
1 acre-foot = area of 1 acre with depth of 1 foot
The reservoir covers an area of 10,640 acres. Jesse made the following calculation.
11
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1,443,000 ÷ 10,640 = 135.62
What unit of measure should Jesse use for his answer, and what does his calculation mean? A
Towns; the reservoir provides water for 135 towns.
B
Feet above sea level; the height of the reservoir is about 136 feet above sea level.
C
The area of the base of a triangular prism can be expressed as 5x 2 + 1. If the height is 2x - 1, what is an expression for the volume of the prism? A
10x 3 - 1
B
10x 3 + 2x - 1
C
10x 3 - 3x 2 - 1
D 10x 3 - 5x 2 + 2x - 1
12
Feet; the average depth of the reservoir is about 136 feet.
D Acres; the reservoir has about 136 acres of water.
What is the distance between the points (-9, -3) and (15, -10)? A
5
B
25 5 √5
C
D 5 √ 53
A62
North Carolina StudyText, Math A
Name
Date
Practice Test 13
(continued)
Sliding Rock in Pisgah National Forest, North Carolina, is a natural water slide created by a flat, gradually sloping boulder. If the top of the slide is 30 feet high, and the horizontal length of the slide is 52 feet, which equation best models the slide? A
y = - 0.58x
B
y = -1.7x
C
y = -2x
16
Bacteria Growth Day
Number
0
10,000
1
20,000
2
40,000
3
80,000
Which of the following represents the number of bacteria on Day d?
D y = 3.7x
14
Bacteria in a culture are growing exponentially as shown in the table below.
What is the sum of the following expression?
A
2000 d
B
10,000 · 2d
C
10,000 · 2d
D 10,000 (2d +1)
2
2x + 5x - 8 + (-3x 2 - 4x + 8) _________________ -x2 - x + 6
B
-x2 + x - 6
C
x 2 + 9x - 10
17
D 5x 2 - x - 10
15
A square pyramid and square prism have congruent bases. Their heights are also equal. The combined volume of the two figures is 108 in3. What are the volumes of the pyramid and the prism? A
27 in3 and 81 in3
B
30 in3 and 78 in3
C
32 in3 and 76 in3
Moe has $6000 in a savings account. The amount of money A in the account doubles every 8 years. Which equation can he use to predict the amount of money he will have in the account after t years? A
A = (2 · 6000)t
B
A = 6000 (2) 8
C
A = 6000 + 2 8t
_t
D A = 6000 + (6000)(2)t
D 40 in3 and 68 in3
A63
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A
Name
Date
Practice Test 18
(continued)
The vertex-edge graph below is a model of connections among people at a party. Each vertex represents a person and each edge represents a handshake.
20
3
B
4
C
5
21
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
D 6
19
A
It is a parallelogram because two sides are parallel.
B
It is a parallelogram because the equal sides are also parallel.
C
It is not a parallelogram because the parallel sides must be equal.
D It is not a parallelogram because there are no right angles.
How many handshakes were there? A
A quadrilateral has one pair of parallel opposite sides and one pair of congruent sides. Is the quadrilateral a parallelogram? Why, or why not?
A square glass pyramid has the dimensions shown below.
30 cm
A volleyball is served and follows the path modeled in the graph below before the opposing team spikes it. 10 9 8 7 6 5 4 3 2 1 0
30 cm 30 cm
y
The pyramid will be packed in a cube with a side length of 30 centimeters. To the nearest cubic centimeter, what is the volume of the packing material that will be needed around the pyramid to fill the cube? 1 2 3 4 5 6 7 8 9 10 11 12 x
Over which interval is the ball’s height increasing? A
0≤x≤5
B
0≤y≤8
C
6≤y≤8
A
9000 cm3
B
18,000 cm3
C
24,000 cm3
D 27,000 cm3
D 5 ≤ x ≤ 13
A64
North Carolina StudyText, Math A
Name
Date
Practice Test 22
(continued)
ChatInc.com is a new social networking Web site. The company’s desire to increase the number of new customers y, can be modeled by y ≥ 1000(2) x. However, the number of new customers they can accommodate with their current technical capacity, can be modeled by y ≤ 4000 + 2000x. Both inequalities are graphed below. y
32
New Customers
24
A
It will decrease by about 3 minutes.
B
It will decrease by about 5 minutes.
C
It will increase by about 3 minutes.
D It will increase by about 5 minutes.
28 24 20 16 12
25
8 4
x
0
1
2 3
4
5
6
Months Since Startup
A
(1, 1000)
B
(2, 8000)
C
(3, 11,000)
What is the best description of the polygon formed by connecting the points (-3, 4), (1, 8), (7, 6) and (-2, -3)? A
parallelogram
B
rectangle
C
trapezoid
D rhombus
26
D (4, 13,000)
Which quadratic equation has roots of -2 and 5? A
y = x 2 - 3x - 10
B
y = x 2 + 3x - 10
What are the factors of 2x 2 - 11x - 6?
C
y = x 2 - 7x - 10
A
(2x - 3)(x + 2)
D y = x 2 + 7x - 10
B
(2x + 3)(x - 2)
C
(2x + 1)(x - 6)
D (2x - 1)(x + 6)
A65
North Carolina StudyText, Math A
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Which of the following coordinates meets both the company’s plan for growth and its current technical limitations?
23
The amount of time that it takes Carlie to drive to work varies inversely with the speed that she drives. If she drives at 50 miles per hour, it takes her 30 minutes. How will the amount of time change if she increases her speed by 10 miles per hour?
Name
Date
Practice Test 27
(continued)
According to a description of the current North Carolina State Capitol, the east and west porticoes have 1 columns that are 5 feet 2_ inches in 2 diameter. Approximately what is the volume of one of these 40-foot high cylindrical columns? A
271 ft3
B
852 ft3
C
3407 ft3
29
D 10,221 ft3
In 2003, Hurricane Isabel was one of the most powerful storms in a century, striking North Carolina with winds of 145 miles per hour. Out of 78 hurricanes from 1903 to 2003, only 2 had stronger winds. In 2003, if hurricane winds were ranked by percentiles, what would have been the percentile for Hurricane Isabel? Round to the nearest whole number. A
4th percentile
B
54th percentile
C
76th percentile
D 97th percentile
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
28
A child’s model of the Bodie Lighthouse in Nag’s Head, North Carolina is shaped like a cone.
30
The graph of an equation is shown below. y 5 4 3 2 (0, 2) 1
30 cm
-4 -2 O 1 2 3 4 5 x -2 (−2, −3) -3 -4 -5
The base of the model has a radius of 5 centimeters. To the nearest cubic centimeter, what is the volume of the smallest cylinder in which this model can be packaged?
For what value of x will y have a value of 7? A
2
A
B
3
C
4
B C
3
750 cm
3
2357 cm
3
14,130 cm
D 5
D 23,550 cm3
A66
North Carolina StudyText, Math A
Name
Date
Practice Test 31
(continued)
After a football is kicked, its height is modeled by the equation h(t) = -16t 2 + 288t, where h is the height of the ball and t is the time in seconds after it is kicked. Over what interval is the height of the ball increasing? A
0
