Chapter 7 Algebra: Graphs, Functions, and Linear Systems€¦ · Section 7.1 Graphing and Functions...

Chapter 7 Algebra: Graphs, Functions, and Linear Systems

Copyright © 2015 Pearson Education, Inc. 279

Check Points 7.1

1.

2. x 4y x ( , )x y

3 4 ( 3) 4 3 7y ( 3,7)

2 4 ( 2) 4 2 6y ( 2,6)

1 4 ( 1) 4 1 5y ( 1,5)

0 4 (0) 4 0 4y (0,4)

1 4 (1) 4 1 3y (1,3)

2 4 (2) 4 2 2y (2,2)

3 4 (3) 4 3 1y (3,1)

3. a. Without the discount pass With the discount pass

x 2y x ( , )x y x 10y x ( , )x y

0 2(0) 0y (0,0) 0 10 0 10y (0,10)

2 2(2) 4y (2,4) 2 10 2 12y (2,12)

4 2(4) 8y (4,8) 4 10 4 14y (4,14)

6 2(6) 12y (6,12) 6 10 6 16y (6,16)

8 2(8) 16y (8,16) 8 10 8 18y (8,18)

10 2(10) 20y (10,20) 10 10 10 20y (10,20)

12 2(12) 24y (12,24) 12 10 12 22y (12,22)

Chapter 7 Graphs, Functions, and Linear Systems

280 Copyright © 2015 Pearson Education, Inc.

b.

c. The graphs intersect at (10,20) . This means that if the bridge is used ten times in a month, the total monthly cost is

$20 with or without the discount pass.

4. a. ( ) 4 5

(6) 4(6) 5

29

f x x

f

b. 2

2

( ) 3 10

( 5) 3( 5) 10

65

g x x

g

c. 2

2

( ) 7 2

( 4) ( 4) 7( 4) 2

46

h r r r

h

5. a. A car’s required stopping distance at 40 miles an hour on dry pavement is about 190 feet.

b. 2

2

( ) 0.0875 0.4 66.6

(40) 0.0875(40) 0.4(40) 66.6 191

f x x x

f

6. x ( ) 2f x x

( , ) or

x,f(x)

x y

x ( ) 2 3g x x

( , ) or

x,f(x)

x y

2 ( 2) 2( 2) 4f ( 2, 4) 2 ( 2) 2( 2) 3 7g ( 2, 7)

1 ( 1) 2( 1) 2f ( 1, 2) 1 ( 1) 2( 1) 3 5g ( 1, 5)

0 (0) 2(0) 0f (0,0) 0 (0) 2(0) 3 3g (0, 3)

1 (1) 2(1) 2f (1,2) 1 (1) 2(1) 3 1g (1, 1)

2 (2) 2(2) 4f (2,4) 2 (2) 2(2) 3 1g (2,1)

The graph of g is the graph of f shifted vertically down 3 units.

Section 7.1 Graphing and Functions


7. a. y is a function of x.

b. y is a function of x.

c. y is not a function of x. Two values of y correspond to an x-value.

8. a. The concentration is increasing from 0 to 3 hours.

b. The concentration is decreasing from 3 to 13 hours.

c. The maximum concentration of 0.05 mg per 100 ml occurs after 3 hours.

d. None of the drug is left in the body.

e. The graph defines y as a function of x because no vertical line intersects the graph in more than one point.

Concept and Vocabulary Check 7.1

1. x-axis

2. y-axis

3. origin

4. quadrants; four

5. x-coordinate; y-coordinate

6. solution; satisfies

7. y; x; function

8. x; 6

9. more than once; function

Exercise Set 7.1

1.

2.

3.

4.

5.

6.

7.

8.



9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21. 2

3 2 1 0 1 2 3

2 7 2 1 2 1 2 7

x

y x



22. 2

3 2 1 0 1 2 3

2 11 6 3 2 3 6 11

x

y x

23. 3 2 1 0 1 2 3

2 5 4 3 2 1 0 1

x

y x

24. 3 2 1 0 1 2 3

2 1 0 1 2 3 4 5

x

y x

25. 3 2 1 0 1 2 3

2 1 5 3 1 1 3 5 7

x

y x

26. 3 2 1 0 1 2 3

2 4 10 8 6 4 2 0 2

x

y x

27. 3 31 1 1

2 2 2 2 2

3 2 1 0 1 2 3

1 0 1

x

y x

28. 7 5 31 1

2 2 2 2 2

3 2 1 0 1 2 3

2 3 2 1

x

y x

29. 3

3 2 1 0 1 2 3

27 8 1 0 1 8 27

x

y x

30. 3

3 2 1 0 1 2 3

1 28 9 2 1 0 7 26

x

y x

31. 3 2 1 0 1 2 3

1 4 3 2 1 2 3 4

x

y x



32. 3 2 1 0 1 2 3

1 2 1 0 1 0 1 2

x

y x

33. f(x) = x – 4

a. f(8) = 8 – 4 = 4

b. f(1) = 1 – 4 = – 3

34. f(x) = x – 6

a. f(9) = 9 – 6 = 3

b. f(2) = 2 – 6 = –4

35. f(x) = 3x – 2

a. f(7) = 3(7) – 2 = 21 – 2 = 19

b. f(0) = 3(0) – 2 = 0 – 2 = –2

36. f(x) = 4x – 3

a. f(7) = 4(7) – 3 = 28 – 3 = 25

b. f(0) = 4(0) – 3 = 0 – 3 = –3

37. 2 1g x x

a. 22 2 1 4 1 5g

b. 22 2 1 4 1 5g

38. 2( ) 4g x x

a. 2(3) (3) 4 9 4 13g

b. 2( 3) ( 3) 4 9 4 13g

39. 2( ) 2g x x

a. 2(4) (4) 2 16 2 14g

b. 2( 3) ( 3) 2 9 2 7g

40. 2( ) 1g x x

a. 2(5) (5) 1 25 1 24g

b. 2( 4) ( 4) 1 16 1 15g

41. 23 5h r r

a.

24 3 4 5

3 16 5

48 5

53

h

b. 21 3 1 5 3 5 8h

42. 2( ) 2 4h r r

a. 2(5) 2(5) 4

2(25) 4

50 4

46

h

b. 2( 1) 2( 1) 4 2 4 2h

43. 22 3 1f x x x

a. 23 2 3 3 3 1

2(9) 9 1

18 9 1

26

f

b.

24 2 4 3 4 1

2 16 12 1

32 12 1

19

f

44. 2( ) 3 4 2f x x x

a. 2(2) 3(2) 4(2) 2

3(4) 8 2

12 8 2

18

f

b. 2( 1) 3( 1) 4( 1) 2

3 4 2

3

f



45. ( )x

f xx

a. 6

(6) 16

f

b. 6 6

( 6) 16 6

f

46. ( )x

f xx

a. 5

(5) 15

f

b. 5 5

( 5) 15 5

f

47. x 2 1f x x

–2 3

–1 0

0 –1

1 0

2 3

48. x 2( ) 1f x x

–2 5

–1 2

0 1

1 2

2 5

49. x 1f x x

–2 –3

–1 –2

0 –1

1 0

2 1

50. x ( ) 1f x x

–2 –1

–1 0

0 1

1 2

2 3

51. x 22f x x

0 4

1 1

2 0

3 1

4 4



52. x 2( ) ( 1)f x x

–3 4

–2 1

–1 0

0 1

1 4

53. x 3 1f x x

– 3 – 26

– 2 – 7

– 1 0

0 1

1 2

54. x 3( ) ( 1)f x x

–3 –8

–2 –1

–1 0

0 1

1 8

55. y is a function of x.



58. y is not a function of x. All values of y correspond to an x-value.

59. y is not a function of x. Two values of y correspond to an x-value.




63. 2

1 3 1 5 3 5 2

1 2 2 2 4

4 2 4 10

g

f g f

64. 2

1 3 1 5 3 5 8

1 8 8 8 4

64 8 4 76

g

f g f

65.

23 1 6 6 6 4

3 1 36 6 6 4

4 36 1 4

2 36 4

34 4

38

66. 24 1 3 3 3 6

4 1 9 3 3 6

3 9 1 6

3 9 6 6 6 0

67.



68.

69.

70.

71. The coordinates of point A are (2,7). When the football is 2 yards from the quarterback, its height is 7 feet.

72. The coordinates of point B are (28,7). When the football is 28 yards from the quarterback, its height is 7 feet.

73. The coordinates of point C are approximately (6, 9.25).

74. The coordinates of point D are approximately (24, 9.5).

75. The football’s maximum height is 12 feet. It reaches this height when it is 15 yards from the quarterback.

76. The football’s height is 5 feet when it is caught by the receiver. The receiver is 30 yards from the quarterback when he catches the ball.

77. a. 2(30) 0.01(30) (30) 60 81G

In 2010, the wage gap was 81%. This is represented as (30,81) on the graph.

b. (30)G underestimates the actual data shown by

the bar graph by 2%.

78. a. 2(10) 0.01(10) (10) 60 69G

In 1990, the wage gap was 69%. This is represented as (10,69) on the graph.

b. (10)G underestimates the actual data shown by

the bar graph by 2%.

79.

2(20) 0.4 20 36 20 1000

0.4 400 720 1000

f

160 720 1000

560 1000 440

Twenty-year-old drivers have 440 accidents per 50 million miles driven. This is represented on the graph by point (20,440).

80.

2(50) 0.4 50 36 50 1000

0.4 2500 1800 1000

1000 1800 1000 200

f

Fifty-year-old drivers have 200 accidents per 50 million miles driven. This is represented on the graph by point (50,200).

81. The graph reaches its lowest point at 45.x

2(45) 0.4 45 36 45 1000

0.4 2025 1620 1000

810 1620 1000

810 1000

190

f

Drivers at age 45 have 190 accidents per 50 million miles driven. This is the least number of accidents for any driver between ages 16 and 74.

82. Answers will vary. One possible answer is age 16 and age 74.

2(16) 0.4 16 36 16 1000

0.4 256 576 1000

102.4 576 1000 526.4

f

2(74) 0.4 74 36 74 1000

0.4 5476 2664 1000

2190.4 2664 1000 526.4

f

Both 16-year-olds and 74-year-olds have approximately 526.4 accidents per 50 million miles driven.

89. makes sense



90. does not make sense; Explanations will vary. Sample explanation: The notation ( )f x does not

mean multiplication of f by x.

91. makes sense

92. makes sense

93. ( 1) ( 1) 1 ( 3) 2f g

94. (1) (1) 3 ( 5) 2f g

95. ( 1) ( 3) 1f g f

96. (1) ( 5) 3f g f

Check Points 7.2

1. Find the x-intercept by setting y = 0 2 3(0) 6

2 6

3; resulting point (3, 0)

x

x

x

Find the y-intercept by setting x = 0 2(0) 3 6

3 6


y

y

y

Find a checkpoint by substituting any value. 2(1) 3 6

2 3 6

3 4

4 4; resulting point 1,

3 3

y

y

y

y

2. a. 2 4 6

64 ( 3) 1

m

b. 5 ( 2) 7 7

1 4 5 5m

3. Step 1. Plot the y-intercept of (0, 1)

Step 2. Obtain a second point using the slope m. 3 Rise

5 Runm

Starting from the y-intercept move up 3 units and move 5 units to the right. This puts the second point at (3, 6).

Step 3. Draw the line through the two points.

4. Solve for y. 3 4 0

4 3 0

4 3 0

4 4 43

04

x y

y x

y x

y x

3

4m

and the y-intercept is (0, 0)

5. Draw horizontal line that intersects the y-axis at 3.

6. Draw vertical line that intersects the x-axis at 2 .

Section 7.2 Linear Functions and Their Graphs


7. The two points shown on the line segment for Medicare are (2007, 446) and (2016, 909).

11 45 340.85

2010 1970 40m

For the period from 1970 through 2010, the percentage of married men ages 20 to 24 decreased by 0.85 per year. The rate of change 0.85% is per year.

8. a. The y-intercept is 8 and the slope is Change in 24 8 16

0.32 Change in 50 0 50

ym

x

The equation is ( ) 0.32 8.C x x

b. ( ) 0.32 8

(60) 0.32(60) 8

27.2

C x x

C

The model projects that 27.2% of the U.S. population will be college graduates in 2020.

Concept and Vocabulary Check 7.2

1. x-intercept

2. y-intercept

3. 2 1

2 1

y y

x x

4. y mx b ; slope; y-intercept

5. 4 ; 3

6. 0,3 ; 2; 5

7. horizontal

8. vertical

Exercise Set 7.2

1. Find the x-intercept by setting y = 0 3

0 3


x y

x

x

Find the y-intercept by setting x = 0 0 3

3


y

y

y

2. Find the x-intercept by setting y = 0 4

0 4


x y

x

x

Find the y-intercept by setting x = 0 0 4


y

y


3 12


x

x

x


4 12


y

y

y




2 10


x

x

x


5 10


y

y

y

5. Find the x-intercept by setting y = 0 2 0 6

2 6


x

x

x

Find the y-intercept by setting x = 0 2(0) 6


y

y

6. Find the x-intercept by setting y = 0 3(0) 6


x

x

Find the y-intercept by setting x = 0 0 3 6

3 6


y

y

y


5 15

3; resulting point ( 3, 0)

x

x

x


0 3 15

3 15


y

y

y

y


3 6


x

x

x


0 2 6

2 6


y

y

y

y

9. 5 6 1

13 2 1

m

; line falls.

10. 4 2 2

23 4 1

m

; line falls.

11. 2 1 1

2 2 4m

; line rises.

12. 4 3 1

2 ( 1) 3m

; line rises.

13. 1 4 5

51 2 1

m

; line falls.



14. 2 ( 4) 2

14 6 2

m

; line falls.

15. 2 3 5

;5 5 0

m

Slope undefined. Line is vertical.

16. 5 ( 4) 9

3 3 0m

;

Slope undefined. Line is vertical.

17. 8 0 8

4;0 2 2

m

line falls.

18. 9 0 9

30 3 3

m

; line rises.

19. 1 1 0

0;2 5 7

m

line is horizontal.

20. 3 3 0

01 ( 2) 3

m

; line is horizontal.

21. y = 2x + 3 Slope: 2, y-intercept: 3 Plot point (0, 3) and second point using

2 rise

1 runm

22. y = 2x + 1 Slope: 2, y-intercept: 1 Plot point (0, 1) and second point using

2 rise

1 runm

23. y = – 2x + 4 Slope: – 2, y-intercept: 4 Plot point (0, 4) and second point using

2 rise

1 runm

24. y = –2x + 3 Slope: –2, y-intercept: 3 Plot point (0, 3) and second point using

2 rise

1 runm

25. 1

32

y x

Slope: 1

2, y-intercept: 3

Plot point (0, 3) and second point using 1 rise

.2 run

m



26. 1

22

y x

Slope: 1

2, y-intercept: 2


.2 run

m

27. 2

( ) 43

f x x

Slope: 2

3, y-intercept: – 4

Plot point (0, – 4) and second point using 2 rise

3 runm .

28. 3

( ) 54

f x x

Slope: 3

4, y-intercept: –5

Plot point (0, –5) and second point using 3 rise

.4 run

m

29. 3

44

y x

Slope: 3

4 , y-intercept: 4


.4 run

m

30. 2

53

y x

Slope: 2

3 , y-intercept: 5


.3 run

m

31. 5 5

( ) or ( ) 03 3

f x x f x x

Slope: 5

3 , y-intercept: 0


.3 run

m



32. 4

( )3

f x x or 4

( ) 03

f x x

Slope: 4

3 , y-intercept: 0


.3 run

m

33. a. 3 0x y

y = – 3x or y = – 3x + 0

b. Slope = – 3 y-intercept = 0

c.

34. a. 2x + y = 0 y = –2x or y = –2x + 0

b. Slope = –2 y-intercept = 0

c.

35. a. 3y = 4x 4 4

or 03 3

y x y x

b. Slope = 4

3

y-intercept = 0

c.

36. a. 4y = 5x 5

4y x or

50

4y x

b. Slope 5

4

y-intercept = 0

c.

37. a. 2 3

2 3

x y

y x

b. Slope = – 2 y-intercept = 3

c.

38. a. 3x + y = 4 y = –3x + 4

b. Slope = –3 y-intercept = 4

c.



39. a. 7 2 14

2 7 14

77

2

x y

y x

y x

b. Slope = 7

2

y-intercept = 7

c.

40. a. 5 3 15

3 5 15

55

3

x y

y x

y x

b. Slope 5

3

y-intercept = 5

c.

41. y = 4

42. y = 2

43. y = – 2

44. y = –3

45. x = 2

46. x = 4



47. x + 1 = 0 or x = – 1

48. x + 5 = 0 or x = –5

49. 0

0

a a am

b b b

Since a and b are both positive, a

b is negative.

Therefore, the line falls.

50. 0

0

b b bm

a a a

Since a and b are both positive, b

a is negative.

Therefore, the line falls.

51.

0

b c b cm

a a

The slope is undefined. The line is vertical.

52.

a c c a

ma a b b

Since a and b are both positive, a

bis positive.

Therefore, the line rises.

53. Ax By C

By Ax C

A Cy x

B B

The slope is A

B and the y intercept is .

C

B

54. Ax By C

Ax C By

A Cx y

B B

The slope is A

Band the y intercept is .

C

B

55. 4

31 34

32

6 4

2

2

y

y

y

y

y

56. 1 4

3 4 2

y

1 4

3 4 21 4

3 66 3 4

6 12 3

18 3

6

y

y

y

y

y

y

57. 1 3 2 4, , ,m m m m

58. 2 1 4 3, , ,b b b b

59. a. Change in 29 49 20

0.5 Change in 62 22 40

ym

x

b. For each year of aging, the percentage of Americans reporting “a lot” of stress decreases by 0.5%. The rate of change is –0.5% per year of aging.

60. a. Change in 42 26 16

0.16 Change in 180 80 100

ym

x

b. For each minute of brisk walking, the percentage of patients with depression in remission increased by 0.16%. The rate of change is 0.16% per minute of brisk walking.



61. ( ) 0.725 18P x x

62. ( ) 0.45 23P x x

63. a. Find slope by using the endpoints of the line segment.

940 460 48016

2010 1980 30m

The value of b is the y-intercept, or 460. ( )

( ) 16 460

W x mx b

W x x

b. ( ) 16 460

( ) 16(40) 460

1100

W x x

W x

In the year 2020, the number of bachelor’s degrees that will be awarded to women will be about 1,100,000.

64. a. Find slope by using the endpoints of the line segment.

710 470 2408

2010 1980 30m

The value of b is the y-intercept, or 470. ( )

( ) 8 470

M x mx b

M x x

b. ( ) 8 470

( ) 8(40) 470

790

M x x

M x

In the year 2020, the number of bachelor’s degrees that will be awarded to men will be about 790,000.

74. does not make sense; Explanations will vary. Sample explanation: Either point can be considered

1 1,x y or 2 2, .x y

75. does not make sense; Explanations will vary. Sample explanation: Since college cost are going up, this function has a positive slope.

76. does not make sense; Explanations will vary. Sample explanation: The slope of lines whose equations are in this form can be determined in several ways. One such way is to rewrite the equation in slope-intercept form.

77. makes sense

78. false; Changes to make the statement true will vary. A sample change is: It is possible for .m b

79. false; Changes to make the statement true will vary. A sample change is: Vertical lines can not be expressed in slope-intercept forn.

80. true

81. false; Changes to make the statement true will vary. A sample change is: The line 2 3 7y x is

equivalent to 3 7

2 2y x which has a y-intercept of

7.

2

82. First, find the slope using the points 0,32 and 100,212 .

212 32 180 9

100 0 100 5m

The slope, m, is 9

5 and the y-intercept, b, is 32.

Instead of y mx b we will use F mC b .

932

5

F mC b

F C

Chapter 7 Algebra: Graphs, Functions, and Linear Systems€¦ · Section 7.1 Graphing and Functions...

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Transcript of Chapter 7 Algebra: Graphs, Functions, and Linear Systems€¦ · Section 7.1 Graphing and Functions...