CHAPTER Linear Functions 4 Solutions Key Functions Solutions Key Are you reAdy? 1. E 2. C 3. A 4. B 5 -12. 8 4 -8 x y -8 -4 0 4 8 F A G C E B H D 13. 2x + y = 8 _-2x -2x y

Linear FunctionsSolutions Key

Are you reAdy?

1. E 2. C

3. A 4. B

5-12. 8

4

-8

x

y

0 8 4 -8 -4

F

A G

C E

B

H D

13. 2x + y = 8 _______ -2x ____ -2x y = 8 - 2x

14. 5y = 5x - 10

5y

_ 5 = 5x - 10 _

5

y = x - 2

15. 2y = 6x - 8

2y

_ 2 = 6x - 8 _

2

y = 3x - 4

16. 10x + 25 = 5y

10x + 25 _ 5 =

5y _

5

2x + 5 = y

17. 4g - 3 = 4(-2) - 3 = -8 - 3 = -11

18. 8p - 12 = 8(4) - 12 = 32 - 12 = 20

19. 4x + 8 = 4(-2) + 8 = -8 + 8 = 0

20. -5t - 15 = -5(1) - 15 = -5 - 15 = -20

21. v = 0.05 + 0.01m

22. Possible answer: The amount of money in your bank account equals $100 minus the amount spent.

23. 322 miles _ 14 gallons

= 23 mi/gal 24. $14.25 _

3 pounds = $4.75/lb

25. 32 grams

_ 4 servings

= 8 g/serving

26. 120 pictures

__ 5 rolls

= 24 pictures/roll

IdentIFyIng LIneAr FunctIons

CHECK IT OUT!

1a. Yes; each domain value is paired with exactly one range value; yes

b. Yes; each domain value is paired with exactly one range value; yes

c. No; each domain value is not paired with exactly one range value; no, not a linear function.

2. Yes; a constant change of +2 in x corresponds to a constant change of -1 in y.

3a. y = 5x - 9 ____ -5x _______ -5x y - 5x = -9 -5x + y = -9 The equation can be written in standard form, so the

function is linear.

x y = 5x - 9 (x, y)

0 y = 5(0) - 9 = -9 (0, -9)

1 y = 5(1) - 9 = -4 (1, -4)

2 y = 5(2) - 9 = 1 (2, 1)

Plot the points and connect them with a straight line.

-4

-6

-8

-2

x y 0 2 -2

b. y = 12 0x + y = 12 The equation can be written in standard form, so the

function is linear.

x y = 12 (x, y)

-1 y = 12 (-1, 12)

0 y = 12 (0, 12)

1 y = 12 (1, 12)


8

4

x

y

0 2 -2

c. This is not linear, because x appears in an exponent.

103 Holt McDougal Algebra 1

4CHAPTER

4-1

CS10_A1_MESK710372_C04.indd 103 3/30/11 10:57:10 PM

4. x f(x) = 3x + 10 (x, f(x))

0 f(x) = 3(0) + 10 = 10 (0, 10)

1 f(x) = 3(1) + 10 = 13 (1, 13)

2 f(x) = 3(2) + 10 = 16 (2, 16)

3 f(x) = 3(3) + 10 = 19 (3, 19)

4 f(x) = 3(4) + 10 = 22 (4, 22)

5 f(x) = 3(5) + 10 = 25 (5, 25)

6 f(x) = 3(6) + 10 = 28 (6, 28)

7 f(x) = 3(7) + 10 = 31 (7, 31)

Rental Payment

0 2 4 6

8

16

24

Manicures

Rent

al p

aym

ent

($)

The number of manicures must be a whole number, so the domain is {0, 1, 2, 3, ...}. The range is {$10, $13, $16, $19, ...}.

THInK and dIsCUss

1. No; all the points of the function must form a line in order for it to be a linear function.

2. It is only possible to do a whole number of manicures, so the points whose x-coordinates are not whole numbers have no meaning in this situation.

3.

From its graph: All the points form a line.

From its equation: It can be written in standard form, Ax + By = C , where A and B are not both 0. Example: 6x + 2y = -2

From a list of ordered pairs: A constant change in x corresponds to a constant change in y.

Determining Whether a Function Is Linear

x y 0 6 1 3 2 0

+ 1

+ 1

- 3

- 3

x

y

ExErCIsEsguided practice

1. No; it is not in the form Ax + By = C.

2. Yes; each domain value is paired with exactly one range value; yes



5. Yes; a constant change of -1 in x corresponds to a constant change of +2 in y.

6. No; there is a constant change in y, but there is not a corresponding constant change in x.

7. Yes; a constant change of -2 in x corresponds to a constant change of -2 in y.

8. No; a constant change of -3 in x corresponds to different changes in y.

9. 2x + 3y = 5 The equation can be written in standard form, so the

function is linear. 2x + 3y = 5 ________ -2x ____ -2x 3y = 5 - 2x

3y

_ 3 = 5 - 2x _

3

y = 5 - 2x _ 3

x y = 5 - 2x _ 3 (x, y)

-2 y = 5 - 2(-2)

_ 3 = 3 (-2, 3)

1 y = 5 - 2(1)

_ 3 = 1 (1, 1)

4 y = 5 - 2(4)

_ 3 = -1 (4, -1)


-2

x

y

0 2 -2

10. 2y = 8 The equation can be

written in standard form, so the function is linear.

2y = 8

2y

_ 2 = 8 _

2

y = 4

x y = 4 (x, y)

-1 y = 4 (-1, 4)

0 y = 4 (0, 4)

1 y = 4 (1, 4)


2

x

y

0 2 -2

11. This is not linear, because x has an exponent other than 1.


CS10_A1_MESK710372_C04.indd 104 3/30/11 10:57:13 PM

12. x _ 5 =

y _

3

3x = 5y ____ - 5y ____ -5y 3x - 5y = 0 The equation can be written in standard form, so the

function is linear. 3x - 5y = 0 ________ -3x ____ -3x -5y = -3x

-5y

_ -5

= -3x _ -5

y = 3 _ 5 x

x y = 3 _ 5 x (x, y)

-5 y = 3 _ 5 (-5) = -3 (-5, -3)

0 y = 3 _ 5 (0) = 0 (0, 0)

5 y = 3 _ 5 (5) = 3 (5, 3)


2

-2

x

y

0 2 -2

13. x f(x) = 75x (x, f(x))

0 f(x) = 75(0) = 0 (0, 0)

1 f(x) = 75(1) = 75 (1, 75)

2 f(x) = 75(2) = 150 (2, 150)

3 f(x) = 75(3) = 225 (3, 225)

4 f(x) = 75(4) = 300 (4, 300)

Train Travel

0 2 4

80

160

240

Time (h)

Dis

tanc

e (m

i) The number of hours does not need to be a whole number, so the domain is x ≥ 0. The range is y ≥ 0.

14. x f(x) = 2.50x + 6 (x, f(x))

0 f(x) = 2.50(0) + 6 = 6.00 (0, 6.00)

1 f(x) = 2.50(1) + 6 = 8.50 (1, 8.50)

2 f(x) = 2.50(2) + 6 = 11.00 (2, 11.00)

3 f(x) = 2.50(3) + 6 = 13.50 (3, 13.50)

4 f(x) = 2.50(4) + 6 = 16.00 (4, 16.00)

5 f(x) = 2.50(5) + 6 = 18.50 (5, 18.50)

Movie Rentals

0 2 4 6 8

4

8

12

16

Movies rented

Cost

($)

(0, 6.00)

(1, 8.50) (2, 11.00)

(3, 13.50) (4, 16.00)

(5, 18.50) The number of movies rented must be a whole number, so the domain is {0, 1, 2, 3, ...}. The range is {$6.00, $8.50, $11.00, $13.50, ...}.

practice and problem Solving

15. Yes; each domain value is paired with exactly one range value; no


17. Yes; each domain value is paired with exactly one range value; no

18. No; a constant change of +3 in x corresponds to different changes in y.

19. Yes; a constant change of +1 in x corresponds to a constant change of +1 in y.

20. Yes; a constant change of -3 in x corresponds to a constant change of -2 in y.

21. y = 5 0x + y = 5 The equation can be written in standard form, so the

function is linear.

x y = 5 (x, y)

-1 y = 5 (-1, 5)

0 y = 5 (0, 5)

1 y = 5 (1, 5)


4

2

x

y

0 2 -2


CS10_A1_MESK710372_C04.indd 105 3/30/11 10:57:15 PM

22. 4y - 2x = 0 -2x + 4y = 0 The equation can be written in standard form, so the

function is linear. -2x + 4y = 0 ________ +2x ____ +2x 4y = 2x

4y

_ 4 = 2x _

4

y = 1 _ 2 x

x y = 1 _ 2 x (x, y)

-2 y = 1 _ 2 (-2) = -1 (-2, -1)

0 y = 1 _ 2 (0) = 0 (0, 0)

2 y = 1 _ 2 (2) = 1 (2, 1)


2

-2

x

y

0 2

23. This is not linear, because x appears in the denominator of a fraction.

24. 5 + 3y = 8 _______ -5 ___ -5 3y = 3 0x + 3y = 3 The equation can be written in standard form, so the

function is linear. 3y = 3

3y

_ 3 = 3 _

3

y = 1

x y = 1 (x, y)

-1 y = 1 (-1, 1)

0 y = 1 (0, 1)

1 y = 1 (1, 1)


2

-2

x

y

0 2 -2

25.x f(x) = - 1 _

25 x + 15 (x, f(x))

0 f(x) = - 1 _ 25

(0) + 15 = 15 (0, 15)

25 f(x) = - 1 _ 25

(25) + 15 = 14 (25, 14)

50 f(x) = - 1 _ 25

(50) + 15 = 13 (50, 13)

75 f(x) = - 1 _ 25

(75) + 15 = 12 (75, 12)

100 f(x) = - 1 _ 25

(100) + 15 = 11 (100, 11)

Tony’s Drive

0 20 40 60 80

4

8

12

16

Distance driven (mi)

Gas

left

(gal

) The number of miles does not need to be a whole

number. The maximum distance Tony can travel on

15 gallons is 15 gal · 25 mi _ 1 gal

= 375 mi, so the domain

is 0 ≤ x ≤ 375. The range is 0 ≤ y ≤ 15.

26. No; each domain value is not paired with exactly one range value. This is not a linear function.

27. Yes; each domain value is paired with exactly one range value; yes; a constant change of +2 in x corresponds to a constant change of -2 in y.

28. Yes; each domain value is paired with exactly one range value; no; a constant change of +0.25 in y does not correspond to a constant change in x.

29. Yes; each domain value is paired with exactly one range value; yes; a constant change of +4 in x corresponds to a constant change of +0 in y.

30. 2x - 8y = 16 The equation can be written in standard form, so the

function is linear. A = 2; B = -8; C = 16

31. y = 4x + 2 ____ - 4x ________ -4x y - 4x = 2 -4x + y = 2 The equation can be written in standard form, so the

function is linear. A = -4; B = 1; C = 2


CS10_A1_MESK710372_C04.indd 106 3/30/11 10:57:17 PM

32. 2x = y _

3 - 4

____

- y _

3

_______ -

y _

3

2x - y _

3 = -4

2x - 1 _ 3 y = -4

The equation can be written in standard form, so the function is linear.

A = 2; B = - 1 _ 3 ; C = -4

33. This is not linear, because x appears in the denominator of a fraction.

34. x + 4 _ 2 =

y - 4 _

3

3(x + 4) = 2(y - 4) 3(x) + 3(4) = 2(y) + 2(-4) 3x + 12 = 2y - 8 _______ -2y _______ -2y 3x + 12 - 2y = -8 ___________ - 12 ____ -12 3x - 2y = -20 The equation can be written in standard form, so the

function is linear. A = 3; B = -2; C = -20

35. x = 7 x + 0y = 7 The equation can be written in standard form, so it is

linear, but it is not a function because there is more than one value of y for x.

A = 1; B = 0; C = 7

36. This is not linear, because x and y are multiplied together.

37. 3x - 5 + y = 2y - 4 _________ -2y _______ -2y 3x - 5 - y = -4 _________ + 5 ___ +5 3x - y = 1 The equation can be written in standard form, so the

function is linear. A = 3; B = -1; C = 1

38. y = -x + 2 ___ + x ______ +x y + x = 2 x + y = 2 The equation can be written in standard form, so the

function is linear. A = 1; B = 1; C = 2

39. 5x = 2y - 3 ____ - 2y _______ -2y 5x - 2y = -3 The equation can be written in standard form, so the

function is linear. A = 5; B = -2; C = -3

40. 2y = -6 0x + 2y = -6 The equation can be written in standard form, so the

function is linear. A = 0; B = 2; C = -6

41. This is not a linear equation because x appears in a radical sign.

42. x y = 3x + 7 (x, y)

-3 y = 3(-3) + 7 = -2 (-3, -2)

-2 y = 3(-2) + 7 = 1 (-2, 1)

-1 y = 3(-1) + 7 = 4 (-1, 4)

2

-2

x

y

0 -4

43. x y = x + 25 (x, y)

-2 y = -2 + 25 = 23 (-2, 23)

0 y = 0 + 25 = 25 (0, 25)

2 y = 2 + 25 = 27 (2, 27)

10

20

x

y

0 2 -2

44. x y = 8 - x (x, y)

-2 y = 8 - (-2) = 10 (-2, 10)

0 y = 8 - (0) = 8 (0, 8)

2 y = 8 - (2) = 6 (2, 6)

6

-6

x

y

0 6 -6

45. x y = 2x (x, y)

-1 y = 2(-1) = -2 (-1, -2)

0 y = 2(0) = 0 (0, 0)

1 y = 2(1) = 2 (1, 2)

2

x

y

2 -2


CS10_A1_MESK710372_C04.indd 107 3/30/11 10:57:20 PM

46. -2y = -3x + 6

-2y

_ -2

= -3x + 6 _ -2

y = 3 _ 2 x - 3

x y = 3 _ 2 x - 3 (x, y)

0 y = 3 _ 2 (0) - 3 = -3 (0, -3)

2 y = 3 _ 2 (2) - 3 = 0 (2, 0)

4 y = 3 _ 2 (4) - 3 = 3 (4, 3)

2

-2

x

y

0 2 -2

47. y - x = 4 ____ + x ___ +x y = 4 + x

x y = x + 4 (x, y)

-2 y = (-2) + 4 = 2 (-2, 2)

0 y = 0 + 4 = 4 (0, 4)

2 y = 2 + 4 = 6 (2, 6)

4

-4

x

y

0 4 -4

48. y - 2x = -3 _____ + 2x ____ +2x y = -3 + 2x

x y = -3 + 2x (x, y)

0 y = -3 + 2(0) = -3 (0, -3)

1 y = -3 + 2(1) = -1 (1, -1)

2 y = -3 + 2(2) = 1 (2, 1)

2

-2

x

y

0 2 -2

49. x = 5 + y ___ - 5 ______ -5 x - 5 = y

x y = x - 5 (x, y)

-4 y = -4 - 5 = -9 (-4, -9)

0 y = 0 - 5 = -5 (0, -5)

4 y = 4 - 5 = -1 (4, -1)

-4

-8

x y

0 4 -4

50. 2.5x - y = 0 A = 2.5; B = -1; C = 0

51a. f(x) = 8x

b. x f(x) = 8x (x, f(x))

0 f(x) = 8(0) = 0 (0, 0)

2 f(x) = 8(2) = 16 (2, 16)

4 f(x) = 8(4) = 32 (4, 32)

6 f(x) = 8(6) = 48 (6, 48)

8 f(x) = 8(8) = 64 (8, 64)

Molly’s Earnings

0 2 4 6 8

10

20

30

40

Time worked (h)

Pay

($)

The number of hours does not need to be a whole number, so the domain is x ≥ 0. The range is y ≥ 0.

52. Possible answer:

x y = 2x - 1

-2 -5

-1 -3

0 -1

1 1

2 3

2

x

y

0 2 -2

The table gives some ordered pairs (x, y) that satisfy the equation y = 2x - 1. The graph is a representation of all ordered pairs (x, y) that satisfy y = 2x - 1.

53. Possible answer: The value in cents of x dimes is y = 10x. Since you can have only a whole number of dimes, the domain and range are restricted to whole numbers.

54a. Each constant change in time (+3 minutes) corresponds with a constant change in calories (+27 calories).


CS10_A1_MESK710372_C04.indd 108 3/30/11 10:57:23 PM

b.

Juan’s Workout

0 10 20

40

80

120

160

Time (min)

Calo

ries

bur

ned

c. The graph forms a line.

55. No; each constant change in time (+1 s) is not accompanied by a constant change in height.

56. Yes; the equation can be written in standard form with A = 1, B = 0, and C = 9. No; all solutions are ordered pairs with x-value 9. The x-value 9 corresponds to more than one y-value.

teSt prep

57. C; C is not linear, because x appears in the denominator of a fraction.

58. G; Each second, sound will move 331 meters. So the distance covered is 331 times the number of seconds.

59. Possible answer: 3x + 2y = 7; it is a linear equation because it can be written in standard form with A = 3, B = 2, and C = 7.

A table of values also shows it is a linear function:

x y

-3 8

-2 6.5

-1 5

0 3.5

1 2

2 0.5

Each constant change in x (+1) is accompanied by a constant change in y (-1.5).

The graph shows a linear function.

2

4

x

y

0 2-2

Both the graph and the table show solutions to the equation.

challenge and extend

60. y = 0; x = 0; the first describes a linear function, but the second does not.

61. Perimeter of a square

Side Length Perimeter

1 4

2 8

3 12

4 16

62. area of a square

Side Length Area

1 1

2 4

3 9

4 16

linear not linear

63. Volume of a Cube

Side Length Volume

1 1

2 8

3 27

4 64

not linear

usIng Intercepts

CHECK IT OUT!

1a. The y-intercept is 3. The x-intercept is -2.

b. -3x + 5y = 30 -3x + 5(0) = 30 -3x + 0 = 30 -3x = 30

-3x _ -3

= 30 _ -3

x = -10 The x-intercept is -10.

-3x + 5y = 30 -3(0) + 5y = 30 0 + 5y = 30 5y = 30

5y

_ 5

= 30 _ 5

y = 6 The y-intercept is 6.

c. 4x + 2y = 16 4x + 2(0) = 16 4x + 0 = 16 4x = 16

4x _ 4 = 16 _

4

x = 4 The x-intercept is 4.

4x + 2y = 16 4(0) + 2y = 16 0 + 2y = 16 2y = 16

2y

_ 2

= 16 _ 2


2a. 2x + 3y = 60 ________ -2x ____ -2x 3y = 60 - 2x

3y

_ 3 = 60 - 2x _

3

y = 20 - 2 _ 3 x

x 0 6 15 24 30

y = 20 - 2 _ 3 x 20 16 10 4 0

School Store Purchases

0 10 20 30

10

20

Pens

Not

eboo

ks

The x-intercept is 30. The y-intercept is 20.


4-2

CS10_A1_MESK710372_C04.indd 109 3/30/11 10:57:26 PM

b. x-intercept: pens that can be purchased if no notebooks are purchased.

y-intercept: notebooks that can be purchased if no pens are purchased.

3a. x-intercept: -3x + 4y = -12 -3x + 4(0) = -12 -3x = -12

-3x _ -3

= -12 _ -3

x = 4

y-intercept: -3x + 4y = -12 -3(0) + 4y = -12 4y = -12

4y

_ 4 = -12 _

4

y = -3

-2

x y

0 2 4

b. y = 1 _ 3 x - 2

3(y) = 3 ( 1 _ 3 x - 2)

3y = x - 6 -x + 3y = -6 x-intercept: -x + 3y = -6 -x + 3(0) = -6 -x = -6 -1(-x) = -1(-6) x = 6

y-intercept: -x + 3y = -6 -(0) + 3y = -6 3y = -6

3y

_ 3 = -6 _

3

y = -2

2

-4

x

y

0 2 6

THInK and dIsCUss

1. (4, 0) and (0, 2)

2. 4.318; -21.5489

3.

1. Find the x-intercept by letting y equal 0 and solving for x.

Graphing Ax + By = C Using Intercepts

2. Find the y-intercept by letting x equal 0 and solving for y.

3. Graph the line by plotting the points containing the intercepts and then connecting the points with a straight line.


1. y-intercept

2. The x-intercept is -5. The y-intercept is 1.

3. The x-intercept is 2. The y-intercept is -4.

4. The x-intercept is -3. The y-intercept is -2.

5. 2x - 4y = 4 2x - 4(0) = 4 2x - 0 = 4 2x = 4

2x _ 2 = 4 _

2


2x - 4y = 4 2(0) - 4y = 4 0 - 4y = 4 -4y = 4

-4y

_ -4

= 4 _ -4

y = -1 The y-intercept is -1.

6. -2y = 3x - 6 ____ -3x _______ -3x -3x - 2y = -6

-3x - 2y = -6 -3x - 2(0) = -6 -3x - 0 = -6 -3x = -6

-3x _ -3

= -6 _ -3


-3x - 2y = -6 -3(0) - 2y = -6 0 - 2y = -6 -2y = -6

-2y

_ -2

= -6 _ -2


7. 4y + 5x = 2y - 3x + 16 ________ -2y _____________ -2y 2y + 5x = -3x + 16 _______ + 3x ________ +3x 2y + 8x = 16 8x + 2y = 16 8x + 2y = 16 8x + 2(0) = 16 8x + 0 = 16 8x = 16

8x _ 8 = 16 _

8


8x + 2y = 16 8(0) + 2y = 16 0 + 2y = 16 2y = 16

2y

_ 2 = 16 _

2


8a. x 0 1 2 3 4 5

f(x) = -25 + 5x -25 -20 -15 -10 -5 0

Refrigeration Tank Temperature

0 2 4 6

-15

-20

-25

-10

-5

Time (h)

Tem

pera

ture

(°C)

The x-intercept is 5. The y-intercept is -25.

b. x-intercept: time when temperature is 0°C y-intercept: initial temperature


CS10_A1_MESK710372_C04.indd 110 3/30/11 10:57:29 PM

9. x-intercept: 4x - 5y = 20 4x - 5(0) = 20 4x = 20

4x _ 4 = 20 _

4

x = 5

y-intercept: 4x - 5y = 20 4(0) - 5y = 20 -5y = 20

-5y

_ -5

= 20 _ -5

y = -4

-2

-4

x

y

0 2

10. y = 2x + 4 -2x + y = 4

x-intercept: -2x + y = 4 -2x + 0 = 4 -2x = 4

-2x _ -2

= 4 _ -2

x = -2

y-intercept: -2x + y = 4 -2(0) + y = 4 y = 4

4

x

y

0 2 -4

11. x-intercept:

1 _ 3 x - 1 _

4 y = 2

1 _ 3 x - 1 _

4 (0) = 2

1 _ 3 x = 2

3 ( 1 _ 3 x) = 3(2)

x = 6

y-intercept:

1 _ 3 x - 1 _

4 y = 2

1 _ 3 (0) - 1 _

4 y = 2

- 1 _ 4 y = 2

-4 (- 1 _ 4 y) = -4(2)

y = -8

-4

-8

x y

0 4 -4

12. x-intercept: -5y + 2x = -10 -5(0) + 2x = -10 2x = -10

2x _ 2 = -10 _

2

x = -5

y-intercept: -5y + 2x = -10 -5y + 2(0) = -10 -5y = -10

-5y

_ -5

= -10 _ -5

y = 2

x

y

0 -2 -4 -2



14. The x-intercept is -5. The y-intercept is -1.


16. 6x + 3y = 12 6x + 3(0) = 12 6x + 0 = 12 6x = 12

6x _ 6 = 12 _

6


6x + 3y = 12 6(0) + 3y = 12 0 + 3y = 12 3y = 12

3y

_ 3

= 12 _ 3


17. 4y - 8 = 2x ______ -2x -2x 4y - 2x - 8 = 0 __________ + 8 ___ +8 4y - 2x = 8 -2x + 4y = 8 -2x + 4y = 8 -2x + 4(0) = 8 -2x + 0 = 8 -2x = 8

-2x _ -2

= 8 _ -2


-2x + 4y = 8 -2(0) + 4y = 8 0 + 4y = 8

4y

_ 4

= 8 _ 4


18. -2y + x = 2y - 8 _______ -2y _______ -2y -4y + x = -8 x - 4y = -8 x - 4y = -8 x - 4(0) = -8 x - 0 = -8 x = -8 The x-intercept is -8.

x - 4y = -8 0 - 4y = -8 -4y = -8

-4y

_ -4

= -8 _ -4


19. 4x + y = 8 4x + 0 = 8 4x = 8

4x _ 4 = 8 _

4


4x + y = 8 4(0) + y = 8 0 + y = 8 y = 8 The y-intercept is 8.

20. y - 3x = -15 0 - 3x = -15 -3x = -15

-3x _ -3

= -15 _ -3


y - 3x = -15 y - 3(0) = -15 y - 0 = -15 y = -15 The y-intercept is -15.


CS10_A1_MESK710372_C04.indd 111 3/30/11 10:57:32 PM

21. 2x + y = 10x - 1 ________ -10x ________ -10x -8x + y = -1 -8x + y = -1 -8x + 0 = -1 -8x = -1

-8x _ -8

= -1 _ -8

x = 1 _ 8

The x-intercept is 1 _ 8 .

-8x + y = -1 -8(0) + y = -1 0 + y = -1 y = -1 The y-intercept is -1.

22a. x 0 3 6 9 12

f(x) = 300 - 25x 300 225 150 75 0

Bass Population

0 4 8 12

70

140

210

280

Time (yr)

Popu

lati

on


b. x-intercept: time when bass population is 0 y-intercept: number of bass originally in the lake

23a. x 0 5 15 20 25

f(x) = 5 - 1 _ 5 x 5 4 2 1 0

5K Race

0 10 20

1

2

3

4

Time (min)

Dist

ance

to fi

nish

line

(km

)


b. x-intercept: total time to run the race (when the distance to the finish line is 0)

y-intercept: total length of the race (when time is 0)

24. x-intercept: 4x - 6y = 12 4x - 6(0) = 12 4x = 12

4x _ 4

= 12 _ 4

x = 3

y-intercept: 4x - 6y = 12 4(0) - 6y = 12 -6y = 12

-6y

_ -6

= 12 _ -6

y = -2

2

4

-2

-4

x

y

0 4 -2

25. x-intercept: 2x + 3y = 18 2x + 3(0) = 18 2x = 18

2x _ 2 = 18 _

2

x = 9

y-intercept: 2x + 3y = 18 2(0) + 3y = 18 3y = 18

3y

_ 3 = 18 _

3

y = 6

2

4

6

x 0 4 8

y

26. x-intercept:

1 _ 2 x - 4y = 4

1 _ 2 x - 4(0) = 4

1 _ 2 x = 4

2 ( 1 _ 2 x) = 2(4)

x = 8

y-intercept:

1 _ 2 x - 4y = 4

1 _ 2 (0) - 4y = 4

-4y = 4

-4y

_ -4

= 4 _ -4

y = -1

4

-4

x

y

4 8

27. x-intercept: y - x = -1 0 - x = -1 -x = -1 -1(-x) = -1(-1) x = 1

y-intercept: y - x = -1 y - 0 = -1 y = -1

2

x

y

0 2 -2

28. x-intercept: 5x + 3y = 15 5x + 3(0) = 15 5x = 15

5x _ 5 = 15 _

5

x = 3

y-intercept: 5x + 3y = 15 5(0) + 3y = 15 3y = 15

3y

_ 3 = 15 _

3

y = 5

2

4

x

y

0 2 -2


CS10_A1_MESK710372_C04.indd 112 3/30/11 10:57:36 PM

29. x-intercept: x - 3y = -1 x - 3(0) = -1 x = -1

2

-2

x

y

0 2

y-intercept: x - 3y = -1 0 - 3y = -1 -3y = -1

-3y

_ -3

= -1 _ -3

y = 1 _ 3

30a. y = x + 4

b. y = x + 4 ___ -x ______ -x -x + y = 4

-x + y = 4 -(0) + y = 4 0 + y = 4 y = 4 The y-intercept is 4.

c. The original height of the bamboo plant.

31a. The y-intercept is approximately 7.5. The x-intercept is approximately 600.

b. x-intercept: the number of years after 1800 when there will be no acres of tropical forest

y-intercept: million acres of tropical forest in 1800

32a. m 0 20 40 60 80

b = 412 - 4m 412 332 252 172 92

BUA111FT-BKMADD-A732

0

300

340

380

420

642

Account Balance

Time (mo)

Bala

nce

($)

The number of months must be a whole number so the domain is (0, 1, 2, 3, ...}. The range is {$412, $408, $404, $400, ...}.

b. y = 412 - 4x ____ +4x ________ + 4x 4x + y = 412 4x + y = 412 4x + 0 = 412 4x = 412

4x _ 4 = 412 _

4


4x + y = 412 4(0) + y = 412 0 + y = 412 y = 412

The y-intercept is 412. x-intercept: number of months from that time until

the account has $0 y-intercept: balance when bank employee noticed

the account

c. After 103 months or 8 years and 7 months.

33a.

4

-4

-8

x

y

0 -4 -8 4 8

x = -6 x = 1 x = 5

x = -6: x-intercept: -6, no y-intercept

x = 1: x-intercept: 1, no y-intercept

x = 5: x-intercept: 5, no y-intercept

b.

4

-4

-8

x

y

0 -4 -8 4 8

y = -3

y = 2

y = 7 y = -3: no x-intercept,

y-intercept: -3 y = 2: no x-intercept, y-intercept: 2 y = 7: no x-intercept, y-intercept: 7

c. Horizontal: For y = c, the y-intercept is c and there is no x-intercept.

Vertical: For x = k, the x-intercept is k, and there is no y-intercept.

34. -2x - y = 4 -2x - 0 = 4 -2x = 4

-2x _ -2

= 4 _ -2


-2x - y = 4 -2(0) - y = 4 0 - y = 4 -y = 4 -1(-y) = -1(4) y = -4 The y-intercept is -4.

Graph D has an x-intercept of -2 and a y-intercept of -4.

35. y = 4 - 2x ____ +2x ______ + 2x 2x + y = 4 2x + y = 4 2x + 0 = 4 2x = 4

2x _ 2 = 4 _

2


2x + y = 4 2(0) + y = 4 0 + y = 4 y = 4 The y-intercept is 4.

Graph A has an x-intercept of 2 and a y-intercept of 4.

36. 2y + 4x = 8 2(0) + 4x = 8 0 + 4x = 8 4x = 8

4x _ 4 = 8 _

4


2y + 4x = 8 2y + 4(0) = 8 2y + 0 = 8 2y = 8

2y

_ 2

= 8 _ 2


Graph A has an x-intercept of 2 and a y-intercept of 4.

37. 4x - 2y = 8 4x - 2(0) = 8 4x - 0 = 8 4x = 8

4x _ 4 = 8 _

4


4x - 2y = 8 4(0) - 2y = 8 0 - 2y = 8 -2y = 8

-2y

_ -2

= 8 _ -2


Graph B has an x-intercept of 2 and a y-intercept of -4.

38a. The x-intercept is 20. The y-intercept is 1.75.

b. x-intercept: time remaining when Kristyn started her workout

y-intercept: total distance Kristyn covered

39. Possible answer: Jen wants to save $60. Each week she will earn $12. The function shows how much money Jen has left to save each week.


CS10_A1_MESK710372_C04.indd 113 3/30/11 10:57:39 PM

teSt prep

40. D; Notice that -2(9) = -18 = 0 - 18 = 9(0) - 18. So, (9, 0) is on -2x = 9y - 18 and therefore is the x-intercept.

41. F; The y-intercept is -200 so Jamie owed her uncle $200. The x-intercept is 40 so Jamie was paying her uncle for 40 weeks.

42. 60x + 55y = 660 60(0) + 55y = 660 0 + 55y = 660 55y = 660

55y

_ 55

= 660 _ 55



43. x-intercept:

1 _ 2 x + 1 _

5 y = 1

1 _ 2 x + 1 _

5 (0) = 1

1 _ 2

x = 1

2 ( 1 _ 2

x) = 2(1)

x = 2

y-intercept:

1 _ 2 x + 1 _

5 y = 1

1 _ 2 (0) + 1 _

5 y = 1

1 _ 5 y = 1

5 ( 1 _ 5 y) = 5(1)

y = 5

2

4

x

y

0 2 -2

44. x-intercept: 0.5x - 0.2y = 0.75 0.5x - 0.2(0) = 0.75 0.5x = 0.75

0.5x _ 0.5

= 0.75 _ 0.5

x = 1.5

y-intercept: 0.5x - 0.2y = 0.75 0.5(0) - 0.2y = 0.75 -0.2y = 0.75

-0.2y

_ -0.2

= 0.75 _ -0.2

y = -3.75

-2

-4

x y

0 2 -2

45. y = 3 _ 8 x + 6

____

- 3 _ 8 x

________ - 3 _

8 x

- 3 _ 8 x + y = 6

x-intercept:

- 3 _ 8 x + y = 6

- 3 _ 8 x + 0 = 6

- 3 _ 8 x = 6

- 8 _ 3 (- 3 _

8 x) = - 8 _

3 (6)

x = -16

y-intercept:

- 3 _ 8 x + y = 6

- 3 _ 8 (0) + y = 6

y = 6

-6

x

y

0 -6 -12

46. Ax + By = C Ax + B(0) = C Ax + 0 = C Ax = C

Ax _ A

= C _ A

x = C _ A

The x-intercept is C _ A

.

Ax + By = C A(0) +By = C 0 + By = C By = C

By

_ B

= C _ B

y = C _ B

The y-intercept is C _ B

.

47. 22x - 380y = 20,900 22x - 380(0) = 20,900 22x - 0 = 20,900 22x = 20,900

22x _ 22

= 20,900

_ 22


22x - 380y = 20,900 22(0) - 380y = 20,900 0 - 380y = 20,900 -380y = 20,900

-380y

_ -380

= 20,900

_ -280

y = -55 The y-interecpt is -55.

Possible answer: scale on the x-axis should include numbers from 0 to a number a little greater than 950; scale on y-axis should include numbers from a little less than -55 to 0.


CS10_A1_MESK710372_C04.indd 114 3/30/11 10:57:42 PM

rAte oF chAnge And sLope

CHECK IT OUT!

1. dependent: balance independent: day day 1 to day 6

change in balance

__ change in day

= 285 - 550 _ 6 - 1

= -265 _ 5 = -53

day 6 to day 16

change in balance

__ change in day

= 210 - 285 _ 16 - 6

= -75 _ 10

= -7.5

day 16 to day 22

change in balance

__ change in day

= 210 - 210 _ 22 - 16

= 0 _ 6 = 0

day 22 to day 30

change in balance

__ change in day

= 175 - 210 _ 30 - 22

= -35 _ 8 = -4.375

The balance decreases at the greatest rate from day 1 to day 6.

2. Bank Balance

0 6 12 18 24

120

240

360

480

Day

Bala

nce

($) -$53/day

-$7.50/day $0/day

-$4.38/day

3. slope = -2 _ 5 = - 2 _

5

4a. rise _ run = 8 _ 0

The slope is undefined.

b. rise _ run = 0 _ 4 = 0

The slope is 0.

5a. The slope is undefined. b. The slope is positive.

THInK and dIsCUss

1. 6 units; 5 units; 6 _ 5 2. decreased

3. Possible answer: 5 __ 2 , because it is less steep.

4.

y y

Slope

x

y

x

y

x x

Positive: N

egative: Zero:

Undefined:


1. constant

2. dependent: volume independent: time hour 0 to hour 1

change in volume

__ change in time

= 9 - 12 _ 1 - 0

= -3 _ 1

= -3

hour 1 to hour 3

change in volume

__ change in time

= 5 - 9 _ 3 - 1

= -4 _ 2

= -2

hour 3 to hour 6

change in volume

__ change in time

= 1 - 5 _ 6 - 3

= -4 _ 3

= - 4 _ 3

hour 6 to hour 7

change in volume

__ change in time

= 1 - 1 _ 7 - 6

= 0 _ 1

= 0

The volume decreased at the greatest rate from hour 0 to hour 1.

3. Heart Rate

0 2 4 6 8

70

90

110

130

Time (min)

Hea

rt r

ate

(bea

ts/m

in)

14 beats/min min

-31 beats/min min

beats/min

18 beats/min min

- 20 3

min


4-3

CS10_A1_MESK710372_C04.indd 115 3/30/11 10:57:44 PM

4. slope = 3 _ 6 = 1 _

2 5. slope = -3 _

4 = - 3 _

4

6. rise _ run = 0 _ 6

= 0

The slope is 0.

7. rise _ run = 7 _ 0


8. The slope is negative. 9. The slope is undefined.

10. The slope is zero. 11. The slope is positive.


12. dependent: length independent: age month 3 to month 9

change in length

__ change in age

= 27.5 - 23.5 _ 9 - 3

= 4 _ 6 = 0.7

month 9 to month 18

change in length

__ change in age

= 31.6 - 27.5 _ 18 - 9

= 4.1 _ 9 = 0.5

month 18 to month 26

change in length

__ change in age

= 34.5 - 31.6 _ 26 - 18

= 2.9 _ 8 = 0.4

month 26 to month 33

change in length

__ change in age

= 36.7 - 34.5 _ 33 - 26

= 2.2 _ 7 = 0.3

The baby increased in length at the greatest rate from month 3 to month 9.

13. Elevator Movement

0 10 20 30

20

40

60

Time (s)

Dis

tanc

e (m

)

3 m s

m 45 7 s

m - 35 4 s

m 8 3 s

14. slope = -7 ___ 7 = -1

15. slope = 6 _ 6 = 1

16. rise _ run = 3 _ 0


17. rise _ run = 0 _ 5 = 0

The slope is 0.

18. The slope is positive. 19. The slope is positive.

20. Let ℓ represent the length.

slope = rise _ run

0.73 = 1 _ ℓ

0.73 _ 1 = 1 _

ℓ

0.73ℓ = 1

0.73ℓ _ 0.73

= 1 _ 0.73

ℓ ≈ 1.37 The horizontal run that corresponds to a vertical

change of 1 unit is 1.37.

21. Possible answer: slope is the ratio of change in y to change in x, and, for a line, it is always constant.

22a. slope = -40 _ 40

= -1

b. This maximum heart rate decreases by 1 beat per minute every year.

23. slope = rise _ run = 8 1 _

2 ___

9 = 17 ___

18 , or ≈ 0.94444

24a.

9 ft

16 ft

b. slope = rise _ run = 16 _ 9

25. Possible answer: The slope of a horizontal line will always be 0 because the y-coordinates of any two points will be the same. Therefore, the numerator in the slope formula will always be 0. The slope of a vertical line will always be undefined because the x-coordinates of any two points will be the same. Therefore, the denominator in the slope formula will always be 0. Since you cannot divide by 0, the slope will always be undefined.

26a. Road Trip

0 1 2 3 4

40

80

120

160

Time (h)

Dis

tanc

e (m

i)

50 mi h 30 mi

h

40 mi h

40 mi h

0 mi h

b. The slope is greatest between hour 4 and hour 5. Therefore, the rate of change is greatest between hour 4 and hour 5. Therefore, the car’s average speed was the greatest during the 5th hour.

27a. Possible answer: (16, 420)

b. Possible answer: (26, 650)

c. Possible answer:

change in files

__ change in time

= 650 - 420 _ 26 - 16

= 230 _ 10

= 23

28a. Walk toward or away from the motion detector at a constant rate. A line has constant slope, and in this case slope represents distance/time, or rate. So keeping the rate constant will result in a line.

b. For a positive slope, walk away from the detector. For a negative slope, walk towards the detector.

c. Stand still-as time passes, your distance from the detector does not change. This graph is a horizontal line.

teSt prep

29. C; The slope of line D is undefined so D is incorrect. Line C is the steepest so the absolute value of its slope is the greatest.

30. D; Since line D is a vertical line, it has a run of 0.


CS10_A1_MESK710372_C04.indd 116 3/30/11 10:57:46 PM

31. G; The slope of F is zero so F is incorrect. The slope of H is negative so H is incorrect. Choosing points (0, -2) and (1, 2) on G give a rise of 4 and a run of 1. So the slope of G is 4.


32. The slope of the hill is constant. Let r represent the rise of Jade’s stride.

slope of hill = Tara’s rise _ Tara’s run

= 8 _ 32

= 1 _ 4

slope of hill = Jade’s rise _ Jade’s run

1 _ 4 = r _

36

36 = 4r

36 _ 4 = 4r _

4

9 = r Jade’s rise is 9 inches.

33a. Electricity Costs

0 400 800 1200 1600

30

60

90

120

Energy used (kWh)

Cost

($) 14¢

kWh

14¢ kWh

14¢ kWh

0¢ kWh

3.5¢ kWh

b. dependent: cost independent: energy 0 kWh to 200 kWh

change in cost

__ change in energy

= 3 - 3 _ 200 - 0

= 0 _ 200

= 0

200 kWh to 400 kWh

change in cost

__ change in energy

= 31 - 3 _ 400 - 200

= 28 _ 200

= 0.14

400 kWh to 600 kWh

change in cost

__ change in energy

= 59 - 31 _ 600 - 400

= 28 _ 200

= 0.14

600 kWh to 1000 kWh

change in cost

__ change in energy

= 115 - 59 _ 1000 - 600

= 56 _ 400

= 0.14

1000 kWh to 2000 kWh

change in cost

__ change in energy

= 150 - 115 __ 2000 - 1000

= 35 _ 1000

= 0.035

The rates of change for 200 kWh to 400 kWh, 400 kWh to 600 kWh, and 600 kWh to 1000 kWh

are equivalent.

c. The cost in dollars per kilowatt hour.

d. Up to 200 kWh costs $3.00. Between 200 and 1000 kWh costs $0.14 per kWh. Between 1000 and 2000 kWh costs $0.035 per kWh.

the sLope FormuLA

CHECK IT OUT!

1a. m = y 2 - y 1

_ x 2 - x 1

= -2 - (-2)

_ 7 - (-2)

= 0 _ 9

= 0

b. m = y 2 - y 1

_ x 2 - x 1

= -4 - (-7)

_ 6 - 5

= 3 _ 1

= 3

c. m = y 2 - y 1

_ x 2 - x 1

= 2 _ 5 - 7 _

5 _____

1 _ 4 - 3 _

4

= -1 ___ - 1 _

2

= 2

2a. m = y 2 - y 1

_ x 2 - x 1

= 6 - 4 _ 8 - 4

= 2 _ 4

= 1 __ 2

b. m = y 2 - y 1

_ x 2 - x 1

= -2 - 4 _ 0 - (-2)

= -6 _ 2

= -3

c. Let (0, 1) be ( x 1 , y 1 ) and (2, 5) be ( x 2 , y 2 ).

m = y 2 - y 1

_ x 2 - x 1

= 5 - 1 _ 2 - 0

= 4 _ 2

= 2

d. Let (0, 0) be ( x 1 , y 1 ) and (2, -3) be ( x 2 , y 2 ).

m = y 2 - y 1

_ x 2 - x 1

= -3 - 0 _ 2 - 0

= -3 _ 2

= - 3 _ 2

3. m = y 2 - y 1

_ x 2 - x 1

= 20 - 10 _ 50 - 30

= 10 _ 20

= 1 _ 2

A slope of 1 _ 2 means the height of the plant is

increasing at a rate of 1 cm every 2 days.

4. Find the x-intercept. 2x + 3y = 12 2x + 3(0) = 12 2x = 12

2x _ 2 = 12 _

2

x = 6

Find the y-intercept. 2x + 3y = 12 2(0) + 3y = 12 3y = 12

3y

_ 3

= 12 _ 3

y = 4

m = y 2 - y 1

_ x 2 - x 1 = 4 - 0 _ 0 - 6

= 4 _ -6

= - 2 _ 3


4-4

CS10_A1_MESK710372_C04.indd 117 3/30/11 10:57:48 PM

THInK and dIsCUss

1. y-values; x-values 2. vertical line

3. Finding Slope

From a graph: Begin at any point on the line. Count rise and run to another point on the line. Slope is the ratio of rise to run.

From a table: Choose any two points from the table and substitute their coordinates into the slope formula.

From an equation: Find the x- and y- intercepts. Substitute the points containing the intercepts into the slope formula.


1. m = y 2 - y 1

_ x 2 - x 1

= 9 - 6 _ 6 - 3

= 3 _ 3

= 1

2. m = y 2 - y 1

_ x 2 - x 1

= 4 - 7 _ 4 - 2

= -3 _ 2

= - 3 _ 2

3. m = y 2 - y 1

_ x 2 - x 1

= -1 - (-5)

_ -9 - (-1)

= 4 _ -8

= - 1 _ 2

4. m = y 2 - y 1

_ x 2 - x 1

= 2 - (-1)

_ 4 - (-2)

= 3 _ 6

= 1 _ 2

5. Let (0, 25) be ( x 1 , y 1 ) and (2, 45) be ( x 2 , y 2 ).

m = y 2 - y 1

_ x 2 - x 1

= 45 - 25 _ 2 - 0

= 20 _ 2

= 10

6. m = y 2 - y 1

_ x 2 - x 1

= 160 - 80 _ 12 - 4

= 80 _ 8

= 10 A slope of 10 means the money earned is increasing

at a rate of $10/h.

7. m = y 2 - y 1

_ x 2 - x 1

= 9 - 3 __ 4860 - 1620

= 6 _ 3240

= 1 _ 540

A slope of 1 _ 540

means for each jar of peanut butter,

540 peanuts are needed.

8. Find the x-intercept: 8x + 2y = 96 8x + 2(0) = 96 8x = 96

8x _ 8 = 96 _

8

x = 12

Find the y-intercept: 8x + 2y = 96 8(0) + 2y = 96 2y = 96

2y

_ 2 = 96 _

2

y = 48

m = y 2 - y 1

_ x 2 - x 1 = 48 - 0 _ 0 - 12

= 48 _ -12

= -4

9. 5x = 90 - 9y ____ + 9y _______ + 9y 5x + 9y = 90 Find the x-intercept: 5x + 9y = 90 5x + 9(0) = 90 5x = 90

5x _ 5 = 90 _

5

x = 18


9y

_ 9 = 90 _

9

y = 10

m = y 2 - y 1

_ x 2 - x 1 = 10 - 0 _ 0 - 18

= 10 _ -18

= - 5 _ 9

10. 5y = 160 + 9x ____ -9x ________ - 9x -9x + 5y = 160 Find the x-intercept: -9x + 5y = 160 -9x + 5(0) = 160 -9x = 160

-9x _ -9

= 160 _ -9

x = - 160 ____ 9

Find the y-intercept: -9x + 5y = 160 -9(0) + 5y = 160 5y = 160

5y

_ 5 = 160 _

5

y = 32

m = y 2 - y 1

_ x 2 - x 1 = 32 - 0 __ 0 - (- 160 _

9 )

= 32 _ 160 _

9

= 9 _ 5


11. m = y 2 - y 1

_ x 2 - x 1

= 1 - 5 _ 3 - 2

= -4 _ 1

= -4

12. m = y 2 - y 1

_ x 2 - x 1

= -5 - (-5)

_ 6 - (-9)

= 0 _ 15

= 0

13. m = y 2 - y 1

_ x 2 - x 1

= -1 - 4 _ 3 - 3

= -5 _ 0


14. Let (2, 22) be ( x 1 , y 1 ) and (4, 29) be ( x 2 , y 2 ).

m = y 2 - y 1

_ x 2 - x 1

= 29 - 22 _ 4 - 2

= 7 _ 2

15. m = y 2 - y 1

_ x 2 - x 1

= 2 - (-1)

_ 0 - 4

= 3 _ -4

= - 3 _ 4


CS10_A1_MESK710372_C04.indd 118 3/30/11 10:57:50 PM

16. m = y 2 - y 1

_ x 2 - x 1

= -15 - (-40)

__ 5 - (-40)

= 25 _ 45

= 5 _ 9

A slope of 5 _ 9 means the temperature in Celsius is

increasing at a rate of 5°C for each 9°F.

17. m = y 2 - y 1

_ x 2 - x 1

= 212.9 - 207.5 __ -500 - 2500

= 5.4 _ -3000

= - 9 _ 5000

A slope of - 9 _ 5000

means the boiling point is

decreasing at a rate of 9°F for each 5000 ft above sea level.


7x _ 7 = 91 _

7

x = 13


13y

_ 13

= 91 _ 13

y = 7

m = y 2 - y 1

_ x 2 - x 1 = 7 - 0 _ 0 - 13

= 7 _ -13

= - 7 _ 13

19. 5y = 130 - 13x _____ +13x _________ + 13x 13x + 5y = 130

Find the x-intercept: 13x + 5y = 130 13x + 5(0) = 130 13x = 130

13x _ 13

= 130 _ 13

x = 10


5y

_ 5 = 130 _

5

y = 26

m = y 2 - y 1

_ x 2 - x 1 = 26 - 0 _ 0 - 10

= 26 _ -10

= - 13 _ 5

20. 7 - 3y = 9x ______ + 3y ____ +3y 7 = 9x + 3y Find the x-intercept: 9x + 3y = 7 9x + 3(0) = 7 9x = 7

9x _ 9 = 7 _

9

x = 7 _ 9


3y

_ 3 = 7 _

3

y = 7 _ 3

m = y 2 - y 1

_ x 2 - x 1 = 7 _ 3 - 0

_____ 0 - 7 _

9 =

7 _ 3 ___

- 7 _ 9 = -3

21. Student B is incorrect. Student B did not use the same coordinate pair order in the denominator as in the numerator.

22a. The rate of change for each interval is 4 chirps/min

__ 1°F.

b. yes; 4

23a. The distance of Car 1 is increasing at a faster rate than the distance of Car 2. So Car 1 is going faster. Since Car 1 traveled 20 mi more in 1 h than Car 2, Car 1 is traveling 20 mi/h faster than Car 2.

b. The speed and the slope are both equal to the distance divided by time.

c. Since Car 1 is traveling 20 mi/h faster, the distance between the cars is changing at a rate of 20 mi/h.

24. Possible answer: Given the 2 points ( x 1 , y 1 ) and ( x 2 , y 2 ), you could substitute into the slope formula or graph the two points, connect with a line, and count the rise and the run.

25a. y = 220 - x

b. Age-Based Maximum Heart Rate

0 18 36 54 72 90

120

160

200

Age (yr) M

axim

um h

eart

rat

e (b

eats

/min

) A slope of -1 means for each additional year, the maximum heart rate decreases 1 beat/min.

teSt prep

26. D; By finding the intercepts, you obtain the points (-2, 0) and (0, -3). By substituting into the slope

formula you obtain a slope of - 3 _ 2

.

27. G; The slope of the line connecting (-6, 5) and

(-3, 4) is - 1 _ 3 so a line with slope of - 1 _

3 could pass

through these points.

28. 1 _ 2 , or 0.5

m = y 2 - y 1

_ x 2 - x 1

= 5 - 2 _ 5 - (-1)

= 3 _ 6

= 1 _ 2

29. m = y 2 - y 1

_ x 2 - x 1

= b - 0 _ 0 - a

= b _ -a

= - b _ a

30. m = y 2 - y 1

_ x 2 - x 1

= 3y - y

_ x - 2x

= 2y

_ -x

= - 2y

_ x

31. m = y 2 - y 1

_ x 2 - x 1

= 3 - y - y

_ x + 2 - x

= 3 - 2y

_ 2

= 3 _ 2

- y


CS10_A1_MESK710372_C04.indd 119 3/30/11 10:57:52 PM

32. m = y 2 - y 1

_ x 2 - x 1

-1 = 8 - 2 _ -5 - x

-1 _ 1 = 6 _

-5 - x

-1(-5 - x) = 6 5 + x = 6 ______ -5 ___ -5 x = 1

33. m = y 2 - y 1

_ x 2 - x 1

1 _ 2 = 3x - x _

6 - 4

1 _ 2 = 2x _

2

1 _ 2 = x

34. m = y 2 - y 1

_ x 2 - x 1

-1 = x - (-3)

_ 3 - 1

-1 = x + 3 _ 2

2(-1) = 2 ( x + 3 _ 2 )

-2 = x + 3 ___ -3 _____ - 3 -5 = x

35. m = y 2 - y 1

_ x 2 - x 1

1 _ 7 =

x - (-4) _

x - (-10)

1 _ 7 = x + 4 _

x + 10

x + 10 = 7(x + 4) x + 10 = 7x + 28 _______ -x _______ -x 10 = 6x + 28 ____ -28 _______ - 28 -18 = 6x

-18 _ 6 = 6x _

6

-3 = x

36. Let (x, y) represent the other point.

m = y 2 - y 1

_ x 2 - x 1

1 _ 2 =

y - 2 _

x - 1

x - 1 = 2(y - 2) x - 1 = 2y - 4 ____ + 1 ______ + 1 x = 2y - 3 Since any point will do, let y = 3. x = 2(3) - 3 = 6 - 3 = 3 So one possible point is (3, 3).

37. m = y 2 - y 1

_ x 2 - x 1

= 2 - 4 _ 0 - (-2)

= -2 _ 2

= -1

m = y 2 - y 1

_ x 2 - x 1

-1 = x - 1 - 2 _ 3 - 0

-1 = x - 3 _ 3

3(-1) = 3 ( x - 3 _ 3 )

-3 = x - 3 ___ +3 _____ + 3 0 = x

dIrect vArIAtIon

CHECK IT OUT!

1a. 3y = 4x + 1

3y

_ 3 = 4x + 1 _

3

y = 4 _ 3 x + 4 _

3

This equation does not represent a direct variation because it cannot be written in the form y = kx.

b. 3x = -4y

3x _ -4

= -4y

_ -4

- 3 _ 4 x = y

y = - 3 _ 4 x

This equation represents a direct variation because it can be written in the form y = kx. The constant of

variation is - 3 _ 4 .

c. y + 3x = 0 _____ - 3x ____ -3x y = -3x This equation represents a direct variation because

it can be written in the form y = kx. The constant of variation is -3.

2a. No; possible answer: the value of y _ x is not the same

for each ordered pair.

b. Yes; possible answer: the value of y _ x is the same for

each ordered pair.

c. No; possible answer: the value of y _ x is not the same


3. 4.5 _ 0.5

= y _

10

0.5y = 45 y = 90

4. y = 4x

x y = 4x (x, y)

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

1 y = 4(1) = 4 (1, 4)

2 y = 4(2) = 8 (2, 8)

Graph the points and connect.

Perimeter of a Square

0 1 2 3 4

2

4

6

8

Side length

Peri

met

er

THInK and dIsCUss

1. It can written in the standard form kx - y = 0 with A = k, B = -1, and C = 0.


4-5

CS10_A1_MESK710372_C04.indd 120 3/30/11 10:57:54 PM

2. Possible answer: For any value of k, (0, 0) is a solution of y = kx.

3.

y __ x

Recognizing a Direct Variation

From an Equation:The equation can bewritten in the formy = kx for some nonzero value of k.

From Ordered Pairs:An equation describingthe ordered pairs can be written in the formy = kx. Also, the ratio is constant for each ordered pair.

From a Graph:The graph is a line through (0, 0).


1. direct variation

2. This equation does not represent a direct variation because it cannot be written in the form y = kx.

3. 2y = -8x

2y

_ 2 = -8x _

2

y = -4x This equation represents a direct variation because


4. x + y = 0 ______ -x ___ -x y = -x This equation represents a direct variation because


5. No; possible answer: the value of y _ x is not the same


6. Yes; possible answer: the value of y _ x is the same for

each ordered pair.

7. -3 _ 1 =

y _

-6

y = 18

8. 6 _ 18

= y _

12

18y = 72 y = 4

9. y = 7x

x y = 7x y

0 y = 7(0) = 0 0

1 y = 7(1) = 7 7

2 y = 7(2) = 14 14


Cameron’s Wages

0 2 4 6 8

20

40

60

80

Time worked (h)

Am

ount

ear

ned

($)


10. This equation represents a direct variation because it can be written in the form y = kx. The constant of

variation is 1 _ 6 .

11. 4y = x

4y

_ 4 = x _

4

y = 1 _ 4 x


variation is 1 _ 4 .

12. x = 2y - 12 ____ + 12 _______ + 12 x + 12 = 2y

x + 12 _ 2 =

2y _

2

1 _ 2 x + 6 = y

y = 1 _ 2 x + 6

This equation does not represent a direct variation because it cannot be written in the form y = kx.


each ordered pair.


each ordered pair.

15. 8 _ -32

= y _

64

-32y = 512 y = -16

16. 1 _ 2 __

3 =

y _

1

3y = 1 _ 2

y = 1 _ 6

17. y = 2.50x

x y = 2.50x (x, y)

0 y = 2.50(0) = 0 (0, 0)

1 y = 2.50(1) = 2.50 (1, 2.50)

2 y = 2.50(2) = 5.00 (2, 5.00)


Cost of Gasoline

0 2 4 6 8

2

4

6

8

Amount (gal)

Cost

($)

18. Yes; it can be written as y = 15 _ 4 x.

19. No; it cannot be written in the form y = kx.


CS10_A1_MESK710372_C04.indd 121 3/30/11 10:57:56 PM

20. y = kx 10 = k(2) 5 = k The equation is y = 5x.

x y = 5x (x, y)

0 y = 5(0) = 0 (0, 0)

1 y = 5(1) = 5 (1, 5)

2 y = 5(2) = 10 (2, 10)


2

4

x

y

0 2 -2 1

5

The value of k is 5, and the graph shows that the slope of the line is 5.

21. y = kx 9 = k(-3) -3 = k The equation is y = -3x.

x y = -3x (x, y)

-2 y = -3(-2) = 6 (-2, 6)

-1 y = -3(-1) = 3 (-1, 3)

0 y = -3(0) = 0 (0, 0)


4

x

y

0 2 -2

3 -1

The value of k is -3, and the graph shows that the slope of the line is -3.

22. y = kx 2 = k(8)

1 _ 4 = k

The equation is y = 1 _ 4 x.

x y = 1 _ 4 x (x, y)

0 y = 1 _ 4 (0) = 0 (0, 0)

4 y = 1 _ 4 (4) = 1 (4, 1)

8 y = 1 _ 4 (8) = 2 (8, 2)


4 1

2

-2

x

y

0 4

The value of k is 1 _ 4 , and the

graph shows that the slope of the

line is 1 _ 4 .

23. y = kx 6 = k(1.5) 4 = k The equation is y = 4x.

x y = 4x (x, y)

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

1 y = 4(1) = 4 (1, 4)

2 y = 4(2) = 8 (2, 8)


4

2

x

y

0 2 -2

4

1



x y = 3x (x, y)

-1 y = 3(-1) = -3 (-1, -3)

0 y = 3(0) = 0 (0, 0)

1 y = 3(1) = 3 (1, 3)


4

2 3

1

-4

x

y

4 2 -4 -2



x y = 2x (x, y)

-1 y = 2(-1) = -2 (-1, -2)

0 y = 2(0) = 0 (0, 0)

1 y = 2(1) = 2 (1, 2)


4

2 2

1

-4

x

y

4 2 -4 -2



CS10_A1_MESK710372_C04.indd 122 3/30/11 10:57:58 PM

26. y = kx -16 = k(2) -8 = k The equation is y = -8x.

x y = -8x (x, y)

-2 y = -8(-2) = 16 (-2, 16)

-1 y = -8(-1) = 8 (-1, 8)

0 y = -8(0) = 0 (0, 0)


2

x

y

0 4 2 -4

-8

1

-2


27. y = kx

1 = k ( 1 _ 7 )

7 = k The equation is y = 7x.

x y = 7x (x, y)

0 y = 7(0) = 0 (0, 0)

1 y = 7(1) = 7 (1, 7)

2 y = 7(2) = 14 (2, 14)


2

4

6

x

y

0 4 2 -4

7

1 -2


28. y = kx 9 = k(-2)

- 9 _ 2 = k

The equation is y = - 9 _ 2 x.

x y = - 9 _ 2 x (x, y)

-2 y = - 9 _ 2 (-2) = 9 (-2, 9)

0 y = - 9 _ 2 (0) = 0 (0, 0)

2 y = - 9 _ 2 (2) = -9 (2, -9)

Graph the points and connect. The value of k is - 9 _

2 , and the

8

9

x

y

0 4 -4 -2

graph shows that the slope of

the line is - 9 __ 2 .

29. y = kx -2 = k(9)

- 2 _ 9 = k


x y = - 2 _ 9 x (x, y)

0 y = - 2 _ 9 (0) = 0 (0, 0)

9 y = - 2 _ 9 (9) = -2 (9, -2)

18 y = - 2 _ 9 (18) = -4 (18, -4)


-2 -2

-4

2

4

x

y

0 8 6

9

The value of k is - 2 _ 9

, and

the graph shows that the

slope of the line is - 2 _ 9

.

30. y = kx 6 = k(4)

3 _ 2 = k


x y = 3 _ 2 x (x, y)

-2 y = 3 _ 2 (-2) = -3 (-2, -3)

0 y = 3 _ 2 (0) = 0 (0, 0)

2 y = 3 _ 2 (2) = 3 (2, 3)


2

-4

4

x

y

4 2 2 -4

3

-2

The value of k is 3 _ 2

, and the

graph shows that the

slope of the line is 3 _ 2

.


CS10_A1_MESK710372_C04.indd 123 3/30/11 10:58:01 PM

31. y = kx 4 = k(3)

4 _ 3 = k


x y = 4 _ 3 x (x, y)

-3 y = 4 _ 3 (-3) = -4 (-3, -4)

0 y = 4 _ 3 (0) = 0 (0, 0)

3 y = 4 _ 3 (3) = 4 (3, 4)


2

-4

4

x

y

4 -4

4

3 -2



the line is 4 _ 3 .

32. y = kx 1 = k(5)

1 _ 5 = k


x y = 1 _ 5 x (x, y)

0 y = 1 _ 5 (0) = 0 (0, 0)

5 y = 1 _ 5 (5) = 1 (5, 1)

10 y = 1 _ 5 (10) = 2 (10, 2)


2

-2

-4

4

x

y

0 6 -2 1

5



the line is 1 _ 5 .

33. y = kx -6 = k(1) -6 = k The equation is y = -6x.

x y = -6x (x, y)

-1 y = -6(-1) = 6 (-1, 6)

0 y = -6(0) = 0 (0, 0)

1 y = -6(1) = -6 (1, -6)


2

-1

-2

x

y

0 2 4 -2 -4

6


34. y = kx

1 _ 2 = k(-1)

- 1 _ 2 = k


x y = - 1 _ 2 x (x, y)

-2 y = - 1 _ 2 (-2) = 1 (-2, 1)

0 y = - 1 _ 2 (0) = 0 (0, 0)

2 y = - 1 _ 2 (2) = -1 (2, -1)


4

2

-2

-2

-4

x

y

2 4 -2 -4

1

The value of k is - 1 _ 2 , and the


the line is - 1 _ 2 .

35. y = kx 2 = k(7) 2 __

7 = k


x y = 2 _ 7 x (x, y)

0 y = 2 _ 7 (0) = 0 (0, 0)

7 y = 2 _ 7 (7) = 2 (7, 2)

14 y = 2 _ 7 (14) =4 (14, 4)


4

-2

2

-4

y

0 2

2

6 7 x



the line is 2 _ 7 .

36. Let w represent its weight on Earth.

767 _ 291

= w _ 155

291w = 118,885 w ≈ 409 The Mars rover weighed about 409 lb. on Earth.

37a. y = 15x


CS10_A1_MESK710372_C04.indd 124 3/30/11 10:58:04 PM

b. x y = 15x (x, y)

0 y = 15(0) = 0 (0, 0)

1 y = 15(1) = 15 (1, 15)

2 y = 15(2) = 30 (2, 30)

Graph the points and connect. Washing Machine Efficiency

0 2 4 6 8

20

40

60

80

Loads of laundry

Wat

er s

aved

(gal

)

(0, 0) (1, 15)

(2, 30)

(3, 45)

(4, 60)

(5, 75)

(6, 90)

No; possible answer: Mischa cannot wash a fraction of a load of laundry, so only points whose x-coord. is a whole number make sense in this situation.

c. 2 loads _ 1 week

· 15 gal

_ 1 load

· 52 weeks _ 1 year

= 1560 gal

38. Possible answer: Yes; since the ratio y _ x is the same

for all ordered pairs, 2x must correspond to 2y.

39. Possible answer: The ratio y _ x is the same for all

ordered pairs in a direct variation, so you can write a proportion using any two ordered pairs.

40a. y = 3x

b. It is written in the form y = kx, where k = 3. This value represents the speed at which Rhea is walking.

teSt prep

41. C; y = 4x + 1 cannot be written in the form y = kx.

42. F; In F, the value of y _ x is the same for each ordered

pair, so it is a direct variation.

43. B; Since 13 _ 2 = 32.50 _

5 = 6.5, B is correct.

44. 4.5 Let h represent the number of hours.

3 _ 180

= h _ 270

180h = 810 h = 4.5


45a. y = 20x 120 = 20x

120 _ 20

= 20x _ 20

6 = x

y = 60x 120 = 60x

120 _ 60

= 60x _ 60

2 = x 6 - 2 = 4 gal

b.

Gas Mileage

0 2 4 6

20

40

60

80

Gas used (gal)

Dis

tanc

e (m

i)

Hybrid

SUV

No; the lines begin at (0, 0) and then move away from each other.

c. y = 20x 15,000 = 20x

15,000

_ 20

= 20x _ 20

750 = x

y = 60x 15,000 = 60x

15,000

_ 60

= 60x _ 60

250 = x

46. ax + by = c ________ -ax ____ -ax by = -ax + c

by

_ b = -ax + c _

b

y = - a _ b x + c _

b

For the equation to be a direct variation, it must be able to be written in the form y = kx. So c = 0 if it is a direct variation.

reAdy to go on? section A Quiz



3. x-intercept: 2x - 4y = 16 2x - 4(0) = 16 2x = 16

2x _ 2 = 16 _

2

x = 8

y-intercept: 2x - 4y = 16 2(0) - 4y = 16 -4y = 16

-4y

_ -4

= 16 _ -4

y = -4

4

-4

x

y

0 8

4. x-intercept: -3y + 6x = -18 -3(0) + 6x = -18 6x = -18

6x _ 6 = -18 _

6

x = -3

y-intercept: -3y + 6x = -18 -3y + 6(0) = -18 -3y = -18

-3y

_ -3

= -18 _ -3

y = 6

2

4

6

x

y

0 -2 -4


CS10_A1_MESK710372_C04.indd 125 3/30/11 10:58:06 PM

5. y = -3x + 3 ____ +3x _______ +3x 3x + y = 3

x-intercept: 3x + y = 3 3x + 0 = 3 3x = 3

3x _ 3 = 3 _

3

x = 1

y-intercept: 3x + y = 3 3(0) + y = 3 y = 3

2

4

x

y

0 2 -2

6. Rain Gauge

0 1 2 3 4

0.2

0.4

0.6

0.8

Time (h)

Rain

(in.

)

0.2 in./h

0.2 in./h

0.3 in./h

0.1 in./h

0.2 in./h

7. m = y 2 - y 1

_ x 2 - x 1

= 17.5 - 7 _ 5 - 2

= 10.5 _ 3

= 3.5 A slope of 3.5 means peppers cost $3.50/pound.

8. m = y 2 - y 1

_ x 2 - x 1

= 21 - 13 _ 4 - 2

= 8 _ 2

= 4 A slope of 4 means the speed of the car is 4 ft/s.

9. m = y 2 - y 1

_ x 2 - x 1

= 54 - 36 _ 1 - 4

= 18 _ -3

= -6 A slope of -6 means the temperature decreases at

a rate of 6°F/mi.

10. x = -4 + 3 _______ 2 = - 1 __

2

y = 6 + 8 _____ 2 = 14 ___

2 = 7

The midpoint is at (- 1 __ 2 , 7)

11. d = √ ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2

d = √ (12 - 3) 2 + (18 - 6) 2

d = √ 9 2 + 12 2

d = √ 81 + 144

d = √ 225

d = 15

d = 15 · 10 = 150 The distance is 150 ft.

12. no 13. yes; 1 _ 2

sLope-Intercept Form

CHECK IT OUT!

1a. Plot (0, -3). Count 2 units up and 1 unit right and plot another point. Draw the line connecting the two points.

2

-2

x

y

0 2 -2

b. Plot (0, 1). Count 2 units down and 3 units right and plot another point. Draw the line connecting the two points.

2

-2

x

y

0 2 -2

2a. y = mx + b y = -12x - 1 __

2

b. y = mx + b y = x c. y = mx + b 1 = 8(-3) + b 1 = -24 + b ____ +24 _______ +24 25 = b

y = 8x - 25

3a. y = 2 _ 3 x is in the form y = mx + b.

Plot (0, 0). Count 2 units up and 3 units right and plot another point. Draw the line connecting the two points.

2

-2

x

y

0 2 -2


4-6

CS10_A1_MESK710372_C04.indd 126 3/30/11 10:58:09 PM

b. 6x + 2y = 10 ________ -6x ____ -6x 2y = -6x + 10

2y

_ 2 = -6x + 10 _

2

y = -3x + 5 Plot (0, 5). Count 3 units down

and 1 unit right and plot another point. Draw the line connecting the two points.

2

4

x

y

0 -2

c. y = -4 is in the form y = mx + b. Plot (0, -4). Count 0 units up


2

-2

x

y

0 2 -2

4a. An equation is y = 18x + 200.

b. The y-intercept is 200. This is the cost for the deposit.

The slope is 18. This is the cost per person.

c. y = 18x + 200 = 18(200) + 200 = 3800 The cost of catering an event for 200 guests is

$3800.

THInK and dIsCUss

1. (0, b) 2. (0, -23.75)

3.

1. Plot the point (0, b).

2. Find a second point on the line by using the slope m to move horizontally and vertically from (0, b).

3. Draw the line connecting the two points.

Graphing the Line Described by y = mx + b


1. Plot (0, -3). Count 1 unit up and 3 units right and plot another point. Draw the line connecting the two points.

-2

-4

x y

0 2 -2

2. Plot (0, 3.5). Count 0.5 units up and 1 unit right and plot another point. Draw the line connecting the two points.

2

4

x

y

0 2 -2

3. Plot (0, -1). Count 5 units up and 1 unit right and plot another point. Draw the line connecting the two points.

2

-2

x

y

0 2 -2

4. Plot (0, 2). Count 2 units down and 1 unit right and plot another point. Draw the line connecting the two points.

2

-2

x

y

0 2 -2

5. y-intercept = -2

m = 3 - 0 ________ 1 - (-2)

m = 3 __ 3 = 1

y = mx + b y = 1x - 2 y = x - 2

6. y = mx + b y = 8x + 2

7. y = mx + b y = 0x - 3 y = -3

8. y = mx + b 7 = 5(2) + b 7 = 10 + b ____ -10 _______ -10 -3 = b

y = 5x - 3 9. y = mx + b -3 = -2(1) + b -3 = -2 + b ___ +2 ______ +2 -1 = b

y = -2x - 1

10. y = 2 _ 5 x - 6 is in the form y = mx + b.

Plot (0, -6). Count 2 units up and 5 units right and plot another point. Draw the line connecting the two points.

-2

-4

x y 0 2 -2

11. 3x - y = 1 _______ -3x ____ -3x -y = -3x + 1 -1(-y) = -1(-3x + 1) y = 3x - 1 Plot (0, -1). Count 3 units up


2

-2

x

y

0 2 -2


CS10_A1_MESK710372_C04.indd 127 3/30/11 10:58:13 PM

12. 2x + y = 4 _______ -2x ____ -2x y = -2x + 4 Plot (0, 4). Count 2 units down


2

4

x

y

0 2 -2


b. The y-intercept is 10. This is the distance she has already biked.

The slope is 18. This is Helen’s speed.

c. y = 18x + 10 = 18(2) + 10 = 46 Helen will have biked 46 mi after 2 hours.


14. Plot (0, 7). Count 1 unit up and 4 units right and plot another point. Draw the line connecting the two points.

2

4

6

8

x

y

0 2 -2

15. Plot (0, -3). Count 6 units down and 1 unit right and plot another point. Draw the line connecting the two points.

-2

-4

x y

2 -2

16. Plot (0, -4). Count 1 unit up and 1 unit right and plot another point. Draw the line connecting the two points.

-2

-4

x y

0 2 4

17. Plot (0, 6). Count 4 units down and 5 units right and plot another point. Draw the line connecting the two points.

2

4

6

x

y

0 2 -2

18. y-intercept = 3

m = 3 - 0 ______ 0 - 3

m = 3 ___ -3

= -1

y = mx + b y = -1x + 3 y = -x + 3 19. y = mx + b y = 5x - 9

20. y = mx + b

y = - 2 _ 3 x + 2

21. y = mx + b

4 = - 1 _ 2 (6) + b

4 = -3 + b ___ +3 ________ +3 7 = b

y = - 1 _ 2 x + 7

22. y = mx + b -8 = 0(6) + b -8 = b y = -8

23. - 1 _ 2 x + y = 4

________

+ 1 _ 2 x

____ + 1 _

2 x

y = 1 _ 2 x + 4

Plot (0, 4). Count 1 unit up and 2 units right and plot another point. Draw the line connecting the two points.

2

4

x

y

0 2 -2

24. 2 _ 3 x + y = 2

________

- 2 _ 3 x

____ - 2 _

3 x

y = - 2 _ 3 x + 2

Plot (0, 2). Count 2 units down and 3 units right and plot another point. Draw the line connecting the two points.

2

4

x

y

0 2 -2

25. 2x + y = 8 _______ -2x ____ -2x y = -2x + 8 Plot (0, 8). Count 2 units down


4

8

x

y

0 4 -4


b. The y-intercept is 175. This is the cost of the enrollment fee.

The slope is 35. This is the monthly cost for the health club.

c. y = 35x + 175 = 35(12) + 175 = 595 The cost for a one year membership is $595.


CS10_A1_MESK710372_C04.indd 128 3/30/11 10:58:16 PM

27a.

0 2 4 6 8 10 12 14 16 18

x

y

42

86

10

18

121416

b. y-intercept = 10 y = mx + b

m = 18 - 14 _______ 2 - 1

m = 4 __ 1 = 4

y = 4x + 10

28. possible

x

y

0 2

29. possible

-2

x 0 2

y

30. Impossible; lines with the same slope are parallel and therefore cannot intersect.

31. Impossible; if a linear function does not have a y-intercept, then its graph does not intersect the y-axis. The y-axis is vertical so only lines that do not intersect the y-axis are also vertical. But vertical lines cannot be graphs of functions. All nonvertical lines will intersect the y-axis, so every linear function will have a y-intercept.

32. B; the y-intercept is -1 and the slope is 1 __

2 .

33. C; the y-intercept is 1 and the slope is 1 __

2 .

34. A; the y-intercept is -1 and the slope is 2.

35. Possible answer: x = -4; no; because it has an undefined slope and no y-intercept.

36a. x y

0 3

2 4

4 5

6 6

8 7

10 8

b. y = 1 _ 2 x + 3

c. The y-intercept is 3. This is the distance from Sam’s house to Ricardo’s house.

The slope is 1 _ 2 . This is the boys’ walking speed.

teSt prep

37. B; The y-intercept of y = 1 _ 2 x - 2 is -2. Since

0 + 4(-2) = -8, (0, -2) is on x + 4y = -8, so it is the y-intercept.

38. J; First subtract x from both sides to isolate -y. Then multiply both sides by -1 to get rid of the minus sign.

39. B; Since 2(0) + 3 = 3, (0, 3) is on 2x + y = 3. So 2x + y = 3 has a y-intercept of 3.

40. -6x = -2y + 5 ____ - 5 _______ - 5 -6x - 5 = -2y

-6x - 5 _ -2

= -2y

_ -2

3x + 5 _ 2 = y

y = 3x + 5 _ 2

The slope is 3.

41. Find the slope: 3x - 9y = 9 ________ -3x ____ -3x -9y = -3x + 9

-9y

_ -9

= -3x + 9 _ -9

y = 1 _ 3 x - 1

The slope is 1 _ 3 .

Find the y-intercept: 8x - 2y = 6 8(0) - 2y = 6 -2y = 6

-2y

_ -2

= 6 _ -2


y = mx + b

y = 1 _ 3 x + (-3)

y = 1 _ 3 x - 3


42. Ax + By = C _________ -Ax ____ -Ax By = -Ax + C

By

_ B

= -Ax + C _ B

y = - A _ B

x + C _ B

The slope is - A _ B

. The y-intercept is C _ B

.

43. nx + 5 = 3y

nx + 5 _ 3 =

3y _

3

n _ 3 x + 5 _

3 = y

y = n _ 3 x + 5 _

3

n _ 3 = -2

3 ( n _ 3 ) = 3(-2)

n = -6

44. 0; Any number minus 0 is the number itself; x; Addition Property of Equality (Add b to both sides.)


CS10_A1_MESK710372_C04.indd 129 3/30/11 10:58:19 PM

poInt-sLope Form

CHECK IT OUT!

1a. y - y 1 = m(x - x 1 )

y - 1 = 2 (x - 1 _ 2 )

b. y - y 1 = m(x - x 1 ) y - (-4) = 0(x - 3) y + 4 = 0(x - 3)

2a.

2

-2

x

y

0 2-2

b.

2

y

x

-2

-2

2

0

3a. m = y 2 - y 1

_______ x 2 - x 1 = -1 - 3 _______ 0 - 6

= -4 ___ -6

= 2 __ 3

y - y 1 = m(x - x 1 )

y - 3 = 2 _ 3 (x - 6)

y - 3 = 2 _ 3 x - 4

_____ + 3 ______ + 3

y = 2 _ 3 x - 1

b. m = y 2 - y 1

_______ x 2 - x 1 = 10 - (-2)

_________ 3 - 1

= 12 ___ 2 = 6

y - y 1 = m(x - x 1 ) y - 10 = 6(x - 3) y - 10 = 6x - 18 _____ + 10 _______ + 10 y = 6x - 8

4. m = y 2 - y 1

_ x 2 - x 1 = -3 -15 _______ -4 -2

= -18 ____ -6

= 3

y - 15 = 3(x - 2) y - 15 = 3x - 6 +15 +15 y = 3x + 9 x-intercept: 0 = 3x + 9 ___ -9 ______ -9

9 __ 3 = 3x ___

3

3 = x y-intercept: y = 3(0) + 9 y = 0 + 9 y = 9

5. m = y 2 - y 1

_ x 2 - x 1 = 17.25 - 12.75 __ 5 - 3

= 4.5 _ 2 = 2.25

y - y 1 = m(x - x 1 ) y - 28.50 = 2.25(x - 10) y - 28.50 = 2.25x - 22.50 ________ + 28.50 ____________ + 28.50 y = 2.25x + 6

y = 2.25x + 6 = 2.25(21) + 6 = 53.25 The cost of an ad that is 21 lines long is $53.25.

THInK and dIsCUss

1. Both are based on the slope and a point. Slope-int.: uses the point that contains the y-int.: point-slope: can use any point.

2. Point-slope: when you know the slope and a point; Slope-int.: when you know the slope and the y-int.

3.

If you know two points on the line: Use the two points in the slope formula to find the slope. Then use the slope and one of the points to write the equation in point-slope form.

If you know the slope and a point on the line: Use the slope and the point to write the equation in point-slope form.

If you know the slope and y-intercept: If the slope is m and the y-intercept is b, then the equation is y = mx + b.

Writing the Equation of a Line


1. y - y 1 = m(x - x 1 )

y - (-6) = 1 _ 5 (x - 2)

y + 6 = 1 _ 5 (x - 2)

2. y - y 1 = m(x - x 1 ) y - 5 = -4(x - 1)

3. y - y 1 = m(x - x 1 ) y - (-7) = 0(x - 3) y + 7 = 0(x - 3)

4.

2

4

x

y

0 2 4

5. 2

-2

x

y

0-2-6

6.

-2

-4

xy

0 2-2

7. y - 8 = - 1 _ 3 (x - (-3))

y - 8 = - 1 _ 3 x - 1

_____ + 8 _______ + 8

y = - 1 _ 3 x + 7

8. y - 1 = 2(x - 1) y - 1 = 2x - 2 _____ + 1 ______ + 1 y = 2x - 1


4-7

CS10_A1_MESK710372_C04.indd 130 3/30/11 10:58:22 PM

9. m = y 2 - y 1

_ x 2 - x 1 = -2 - 2 _______ 2 -(-2)

= -4 ___ 4

= -1

y - 2 = -1 (x - (-2)) y - 2 = -x - 2 _____ + 2 ______ + 2 y = -x

10. m = y 2 - y 1

_ x 2 - x 1 = 3 - 1 ______ -5 - 1

= 2 ___ -6

= - 1 __ 3

y - 3 = - 1 __ 3 (x - 1)

y - 3 = - 1 __ 3 x + 1 __

3

_____ + 3 ________ + 3

y = - 1 __ 3 x + 10 ___

3

11. m = y 2 - y 1

_______ x 2 - x 1 = 4 - 0 _____ 0 - 8

= 4 ___ -8

= - 1 __ 2

y - 4 = - 1 __ 2 (x - 0)

y - 4 = - 1 __ 2 x

_____ + 4 ______ + 4

y = - 1 __ 2 x + 4

12. m = y 2 - y 1

_______ x 2 - x 1 = 3 - 0 _______ 0 -(-2)

= 3 __ 2

y - 3 = 3 __ 2 (x - 0)

y - 3 = 3 __ 2

x

_____ + 3 _______ + 3

y = 3 __ 2 x + 3

13. m = y 2 - y 1

_______ x 2 - x 1 = 4 - 2 _____ 7 - 5

= 2 __ 2 = 1

y - 2 = 1(x - 5) y - 2 = x- 5 _____ +2 _______ +2 y = x - 3 x-intercept: 0 = x - 3 ___ +3 _____ +3 x = 3 y-intercept: y = 0 - 3 y = -3

14. m = y 2 - y 1

_______ x 2 - x 1 = -5 - 5 _________ -3 -(-1)

= -10 ____ -2

= 5

y - 5 = 5(x -(-1)) y - 5 = 5x + 5 _____ +5 _______ +5 y = 5x + 10 x-intercept: 0 = 5x + 10 -10 -10

-10 ____ 5 = 5x ___

5

x = -2 y-intercept: y = 5(0) + 10 y = 10

15. m = y 2 - y 1

_ x 2 - x 1 = -9 - 9 ______ -4 - 2

= -18 ____ -6

= 3

y - 9 = 3(x - 2)

y - 9 = 3x - 6 _____ +9 ______ +9 y = 3x + 3 x-intercept: 0 = 3x + 3 ___ -3 ______ -3

-3 ___ 3 = 3x ___

3

x = -1 y-intercept: y = 3(0) + 3 y = 3

16. m = y 2 - y 1

_ x 2 - x 1 = 5 - 3 _ 10 - 0

= 2 _ 10

= 1 _ 5

y - y 1 = m(x - x 1 )

y - 6 = 1 _ 5 (x - 15)

y - 6 = 1 _ 5 x - 3

_____ + 6 ______ + 6

y = 1 _ 5 x + 3

y = 1 _ 5 x + 3

= 1 _ 5 (30) + 3 = 9

The depth of the oil after half an hour is 9 ft.


17. y - y 1 = m(x - x 1 )

y - 5 = 2 _ 9 (x - (-1))

y - 5 = 2 _ 9 (x + 1)

18. y - y 1 = m(x - x 1 ) y - (-2) = 0(x - 4) y + 2 = 0(x - 4)

19. y - y 1 = m(x - x 1 ) y - 8 = 8(x - 1)

20. 2

-2

x

y

0 2 4

21.

2

-2

x

y

0 2 4

22.

-2

2

x

y

0 2-2

23. y - (-3) = - 2 _ 7 (x - 14)

y + 3 = - 2 _ 7 x - 4

_____ - 3 _______ - 3

y = - 2 _ 7 x + 1

24. y - 1 = 4 _ 5

(x - (-15))

y - 1 = 4 _ 5

x + 12

_____ + 1 _______ +1

y = 4 _ 5

x + 13

25. y - 3 = -6 (x - 9) y - 3 = -6x + 54 _____ +3 ________ +3 y = -6x + 57

26. m= y 2 - y 1

_ x 2 - x 1 = 6 - 8 ______ -7 - 7

= -2 ____ -14

= 1 __ 7

y - 8 = 1 __ 7 (x - 7)


CS10_A1_MESK710372_C04.indd 131 3/30/11 10:58:25 PM

y - 8 = 1 __ 7 x - 1

_____ +8 ______ +8

y = 1 __ 7 x + 7

27. m = y 2 - y 1

_ x 2 - x 1 = - 4 - 7 _______ 4 - 2

= - 11 ___ 2

y - 7 = - 11 ___ 2 (x - 2)

y - 7 = - 11 ___ 2 x + 11

_____ +7 __________ +7

y = - 11 ___ 2 x + 18

28. m = y 2 - y 1

_ x 2 - x 1 = - 23 - 2 ________ 4 - (- 1)

= -25 ____ 5 = -5

y - 2 = -5(x -(-1)) y - 2 = -5x - 5 _____ +2 _______ +2 y = -5x - 3

29. m = y 2 - y 1

_ x 2 - x 1 = - 6 - 0 _______ 0 - 3

= 6 __ 3 = 2

y - 0 = 2(x - 3) y = 2x - 6

30. m = y 2 - y 1

_ x 2 - x 1 = - 1 - 0 _______ 0 - 4

= 1 __ 4

y - 0 = 1 __ 4 (x - 4)

y = 1 __ 4 x - 1

31. m = y 2 - y 1

_ x 2 - x 1 = 10 -(- 4)

_________ 6 -(- 1)

= 14 ___ 7 = 2

y - 10 = 2(x - 6) y - 10 = 2x - 12 ______ +10 _______ +10 y = x - 2 x-intercept: 0 = x - 2 ___ +2 _____ +2 x = 2 y-intercept: y = 0 - 2 y = -2

32. m = y 2 - y 1

_ x 2 - x 1 = 16 - 4 _______ - 6 - 3

= 12 ___ -9

= - 4 __ 3

y - 4 = - 4 __ 3 (x - 3)

y - 4 = - 4 __ 3 x + 4

_____ + 4 ________ + 4

y = - 4 __ 3 x + 8

x-intercept:

0 = - 4 __ 3 x + 8

___ -8 ________ -8

-8 · - 3 __ 4

= - 3 __ 4 · - 4 __

3 x

x = 6 y-intercept:

y = - 4 __ 3 ( 0) + 8

y = 8

33. m = y 2 - y 1

_ x 2 - x 1 = 6 - 15 _______ - 2 - 4

= -9 ___ -6

= 3 __ 2

y - 6 = 3 _ 2 (x - (-2))

y - 6 = 3 __ 2 x + 3

_____ +6 _______ +6

y = 3 __ 2 x + 9

x-intercept:

0 = 3 __ 2 x + 9

___ -9 _______ - 9

-9 · 2 __ 3 = 2 __

3 · 3 __

2 x

x = -6 y-intercept:

y = 3 __ 2 ( 0) + 9

y = 9 34. y = -45x + 3,600 y = -45(50) + 3,600 y = 1,350 gal

35. m = y 2 - y 1

_ x 2 - x 1 = 206 - 210 ___________ 3000 - 1000

= -4 _____ 2000

= - 1 _ 500

y - y 1 = m(x - x 1 )

y - 206 = - 1 _ 500

(x - 3000)

y - 206 = - 1 _ 500

x + 6

______ + 206 _________ +206

y = - 1 _ 500

x + 212

y = - 1 _ 500

x + 212

= - 1 _ 500

(6000) + 212 = 200

The boiling point of water at 6000 feet is 200°F.

36a. m = y 2 - y 1

_ x 2 - x 1 = 18.10 - 15.25 ____________ 2 - 5

= 2.85 ____ -3

= -0.95 y -15.25 = -0.95 (x - 5) y -15.25 = -0.95x + 4.75 ________ +15.25 _____________ + 15.25 y = -0.95 + 20 b. -0.95; the change in the amount in dollars

remaining on the card after each download c. 20; the initial amount in dollars on the card d. $15.25 ÷ 0.95 = 16 songs 37.

-2

2

x

y

0 42

38.

-2

2

x

y

0-2

39.

-2

2

x

y

0 2


CS10_A1_MESK710372_C04.indd 132 3/30/11 10:58:27 PM

40. Always 41. Never 42. Sometimes

43a. y - y 1 = m(x - x 1 ) y - 11 = 2.5 (x - 2) b. y - 11 = 2.5 (0 - 2) y - 11 = -5 _____ +11 ____ + 11 y = 6 inches c. From 2:15 to 6:30 is 4.25 hours. y - 11 = 2.5 (4.25 - 2) y - 11 = 5.625 ______ +11 ____________ + 11 y = 16.625 inches 44. Possible answer: y + 3 = - (x - 2) 45. Possible answer: y - 3 = 3 __

2 (x - 3)

46. Possible answer: y - 1 = 2 __ 5 (x - 3)

47. x -2 0 4 7

y -18 -8 12 27

48. x -4 1 0 6

y 14 4 6 -6

49. Student A is incorrect. Student A incorrectly wrote x - (-5) as x - 5 instead of x + 5.

50. Possible answer: When you know a point and the slope, you can immediately use point-slope form. When you know two points, first use them to find the slope. Then use the point-slope form, just like in the first case.

51. Possible answer: Linear equations that describe vertical lines cannot be written in point-slope form because they have an undefined slope. All non-vertical lines represent functions, and they can all be written in point-slope form.

52a. SAT Scores

0 10 20

920

960

1000

1040

1080

Years since 1980

Mea

n co

mbi

ned

scor

e

b. Possible answer: slope: 1.5; y-intercept: 994; y = 1.5x + 994.

c. y-intercept: mean score in 1985 slope: number of points by which the mean score

is increasing each year

53a. (0, 12) and (6, 8)

b. m = y 2 - y 1

_ x 2 - x 1 = 8 - 12 _ 6 - 0

= -4 _ 6 = - 2 _

3

y - y 1 = m(x - x 1 )

y - 8 = - 2 _ 3 (x - 6)

y - 8 = - 2 _ 3 x + 4

_____ + 8 _______ + 8

y = - 2 _ 3 x + 12

c. The total time to reach Sharon’s house occurs when the number of blocks to Sharon’s house is 0. So substitute 0 for y.

y = - 2 _ 3 x + 12

0 = - 2 _ 3 x + 12

____ -12 ________ - 12

-12 = - 2 _ 3 x

- 3 _ 2 (-12) = - 3 _

2 (- 2 _

3 x)

18 = x Stephen takes 18 minutes to reach Sharon’s

house.

teSt prep

54. D; Substituting the slope and point into the slope-point formula and simplifying gives D.

55. H; The slope between the two points is -2 so the answer must be F or H. By using the slope-point formula and rearranging into the slope-intercept form, you get y = -2x + 12, so the y-intercept is 12.


56. x + 4y = 8 _______ -x ___ -x 4y = -x + 8

4y

_ 4 = -x + 8 _

4

y = - 1 _ 4 x + 2

The y-intercept is 2.

m = y 2 - y 1

_ x 2 - x 1 = 7 - 2 _ 2 - 0

= 5 _ 2

The slope is 5 _ 2 .

57. y + 3x = 6 ______ - 3x ____ -3x y = -3x + 6 The slope is -3.

y - y 1 = m(x - x 1 )

y - 1 _ 2 = -3 (x - 3 _

4 )

y - 1 _ 2 = -3x + 9 _

4

_____

+ 1 _ 2

________ + 1 _

2

y = -3x + 11 _ 4


CS10_A1_MESK710372_C04.indd 133 3/30/11 10:58:29 PM

58. m = y 2 - y 1

_ x 2 - x 1 = 1 - (- 1 _

3 ) _

1 1 _ 2 - (- 1 _

2 ) =

4 _ 3 _

2 = 2 _

3

y - y 1 = m(x - x 1 )

y - 1 = 2 _ 3 (x - 1 1 _

2 )

y - 1 = 2 _ 3 (x - 3 _

2 )

y - 1 = 2 _ 3 x - 1

_____ + 1 ______ + 1

y = 2 _ 3 x

LIne oF Best FIt

CHECK IT OUT!

1.

2

2

4

8

4 6 80

y = -x + 8

y

x

12y = - x + 6

y = - 1 __ 2 x + 6: (-2)2 + (2)2 + (-2)2 + (2)2 = 16;

y = -x + 8: (-3)2 + (2)2 + (-1)2 + (4)2 = 30;

y = - 1 __ 2 x + 6 is better.

2a.

y ≈ 0.04x + 6.38

2b. Slope: cost is $0.04/yd; y-int.: $6.38 is added to the cost of every ball of yarn.

2c. x = 1000; y = 0.04(1000) + 6.38 = $46.38

3.

y ≈ -2.74x + 84.32; very well (r ≈ -0.88)

4. strong positive correlation; likely cause- and-effect (more education often contributes to higher earnings)

THInK and dIsCUss

1. 0

2. Possible answer:

r-value -0.9 -0.4 0 0.4 0.9

scatter Plot

description of Correlation

strong negative

weak negative

none weak positive

strong positive


1. residual

2. correlation coefficient

3. y = x + 1: (-1)2 + (1)2 + (1)2 + (-4)2 = 19;

y = x - 1: (1)2 + (3)2 + (3)2 + (-2)2 = 23; y = x + 1 is better.

4a.

y ≈ 1.72x + 73.35

b. Slope: for each book read, student’s average will increase 1.72 points; y-int.: a student who reads 0 books will have an average of 73.35.

c. y ≈ 1.72(15) + 73.35 ≈ 99.15, or 99

5.

y ≈ -0.53x + 8.8; very well (r ≈ -0.91)

6. strong negative correlation; likely cause-and-effect (more time playing video games often contributes to lower test averages)


7. y = -x + 8: (-1)2 + (2)2 + (-1)2 + (1)2 = 7;

y = - 1 __ 2 x + 6: (0)2 + (2)2 + (-2)2 + (-1)2 = 9;

y = -x + 8 is better.

A B C D EA B C D EA B C D EA B C D EA B C D E


4-8

CS10_A1_MESK710372_C04.indd 134 3/30/11 10:58:39 PM

8a.

y ≈ -2.23x + 181

b. Slope: the family will use 2.23 fewer gal/mo for each 1 °F increase in mean temp.; y-int.: the family will use 181 gal in a month when the mean temp is 0 °F.

c. y ≈ -2.23(20) + 181 ≈ 136 gal

9.

y ≈ 0.2x + 2; very well (r ≈ 0.94)

10. moderately strong positive correlation; unlikely cause-and-effect (time spent on one activity in week 1 probably does not affect time spent on the other activity in week 2).

11. -0.78; -0.78 is closer to -1 than 0.65 is to 1.

12. Every data point lies on the least-squares line; residuals can be positive or negative, so their sum could be 0 even when data points are not on the least-squares line.

13a.

y ≈ 0.48x + 12.03

b. Slope: a player will score 0.48 run for every hit.

c. y-int.: a player will score 12.03 runs if he has 0 hits

d. strong positive correlation; r ≈ 0.84, which is near 1.

e. y ≈ 0.48(100) + 12.03 ≈ 60 runs

14.

y ≈ 30.43x - 1875; y ≈ 30.43(89) - 1875 ≈ 834; 9 cases

15a.

y ≈ 115.36x + 1065; r ≈ 0.96

b. Slope: each year there will be 115.36 more visitors than the previous year; y-int: there were 1065 visitors in 0.

c. Yes; r ≈ 0.96, which is very close to 1.

d. No; the passage of time likely does not cause changes in the number of visitors.

16a.

y ≈ 11.28x - 2239; r ≈ 0.98

b. Slope: there will be $11.28 in sales for each visitor; y-int.: there will be -$2239 in sales if there are no visitors. (This could not actually happen.)

c. Yes; r ≈ 0.98, which is very close to 1. However, predictions for small numbers of visitors might not be useful because of the neg. y-int.

d. Yes; more visitors is likely to mean more money spent in the gift shop.

StandardiZed teSt prep

17. Since the correlation is negative and the points do not form a straight line, choice B is correct.

18.

2

20

40

80

4 6 80

y

x

100

(0)2 + (0)2 + (10)2 + (-10)2 + (0)2 = 200; The correct choice is J.


19a. (10)2 + (-5)2 + (-5)2 + (10)2 + (-10)2 + (20)2 +

(5)2 + (-15)2 = 1000

b. 10 + 5 + 5 + 10 + 10 + 20 + 5 + 15 ______________________________ 8 = 80 ___

8 = 10;


CS10_A1_MESK710372_C04.indd 135 3/30/11 10:58:50 PM

possible answer: The mean absolute deviation tells you the average vertical distance of a data point from the line of fit.

20.

2

2

4

8

6

4 6 80

y

x

sum of the residuals: 2 + x + y + 1 = 0 x + y = -3 sum of the squares of the residuals = 22 + x2 + y2 + 12 = 14 x2 + y2 = 9 So, coordinates could be at (3, 3) and (4, 2) or (3, 0) and (4, 5). Fill in the table with 3 and 2 or 0 and 5.

sLopes oF pArALLeL And perpendIcuLAr LInes

CHECK IT OUT!

1a. The lines described by y = 2x + 2 and y = 2x + 1 both have slope 2. These lines are parallel.

b. -3x + 4y = 32 ________ +3x ____ +3x 4y = 3x + 32

4y

_ 4 = 3x + 32 _

4

y = 3 _ 4 x + 8

y - 1 = 3(x + 2) y - 1 = 3x + 6 ____ + 1 ______ + 1 y = 3x + 7

The lines described by y = 3 _ 4 x + 8 and

-3x + 4y = 32 have the same slope, but they are not parallel lines. They are the same line.

The lines described by y = 3x and y - 1 = 3(x + 2) represent parallel lines. They each have a slope of 3.

2. slope of

AB = 2 - 2 _ 4 - 0

= 0 _ 4 = 0

slope of

BC = -3 - 2 _ 1 - 4

= -5 _ -3

= 5 _ 3

slope of

CD = -3 - (-3)

_ -3 - 1

= 0 _ -4

= 0

slope of

AD = -3 - 2 _ -3 - 0

= -5 _ -3

= 5 _ 3

AB is parallel to

CD because they have the same slope.

AD is parallel to

BC because they

have the same slope. Since opposite sides are parallel, ABCD is a parallelogram.

3. The graph described by y = -4 is a horizontal line, and the graph described by x = 3 is a vertical line. These lines are perpendicular.

The slope of the line described by y - 6 = 5(x + 4) is 5. The slope of the line described by

y = - 1 _ 5 x + 2 is - 1 _

5 .

5 (- 1 _ 5 ) = -1

These lines are perpendicular because the product of their slopes is -1.

4. slope of

PQ = 6 - 4 _ 2 - 1

= 2 _ 1 = 2

slope of

QR = 1 - 6 _ 7 - 2

= -5 _ 5 = -1

slope of

PR = 1 - 4 _ 7 - 1

= -3 _ 6 = - 1 _

2

PQ is perpendicular to

PR because the product of their slopes is -1. Since PQR contains a right angle, PQR is a right triangle.

5a. The parallel line also has a slope of 4 _ 5 .

y - y 1 = m(x - x 1 )

y - 7 = 4 _ 5 (x - 5)

y - 7 = 4 _ 5 x - 4

_____ + 7 ______ + 7

y = 4 _ 5 x + 3

b. The perpendicular line has a slope of - 1 _ 5 , because

5 (- 1 _ 5 ) = -1.

y - y 1 = m(x - x 1 )

y - 3 = - 1 _ 5 (x - (-5))

y - 3 = - 1 _ 5 (x + 5)

y - 3 = - 1 _ 5 x - 1

_____ + 3 _______ + 3

y = - 1 _ 5 x + 2

THInK and dIsCUss

1. No; the product of their slopes is 1, not -1.

2. The slopes are the same, and the y-intercepts are different.

3.

2

-2

x

y

0

2

x

y

0 2

Parallel lines: same slopes

Perpendicular lines: Product of slopes is -1.


4-9

CS10_A1_MESK710372_C04.indd 136 3/30/11 10:58:53 PM


1. Parallel

2. The lines described by y = 6 and y = -8 both have slope 0. These lines are parallel.

The lines described by y = 6x + 5 and y = 6x - 7 both have slope 6. These lines are parallel.

3. y - 3 = 3 _ 4 (x - 5)

y - 3 = 3 _ 4 x - 15 _

4

____ + 3 _______ +3

y = 3 _ 4 x - 3 _

4

y - 4 = -2(x + 2) y - 4 = -2x - 4 ____ + 4 _______ + 4 y = -2x

The lines described by y = 3 _ 4 x - 1 and

y - 3 = 3 _ 4 (x - 5) represent parallel lines. They

each have slope 3 _ 4 .

The lines described by y = -2x and y -4 = -2(x + 2) have the same slope, but they are not parallel lines. They are the same line.

4. slope of

AB = 5 - 5 _ 4 - (-2)

= 0 _ 6 = 0

slope of

BC = 2 - 5 _ 6 - 4

= -3 _ 2 = - 3 _

2

slope of

CD = -1 - 2 _ 2 - 6

= -3 _ -4

= 3 _ 4

slope of

AD = -1 - 5 _ 2 - (-2)

= -6 _ 4 = - 3 _

2

AD and

BC are parallel because they have the same slope. Therefore, ABCD is a trapezoid.

5. The slope of the line described by y = 2 _ 3 x - 4 is

2 _ 3 . The slope of the line described by y = - 3 _

2 x + 2

is - 3 _ 2 .

2 _ 3 (- 3 _

2 ) = -1


The graph described by y = -1 is a horizontal line, and the graph described by x = 3 is a vertical line. These lines are perpendicular.

6. The slope of the line described by y = - 3 _ 7 x - 4 is

- 3 _ 7 . The slope of the line described by

y - 7 = 7 _ 3 (x - 3) is 7 _

3 .

- 3 _ 7 ( 7 _

3 ) = -1


The slope of the line described by y - 4 = -7(x + 2) is -7. The slope of the line

described by y - 1 = 1 _ 7 (x - 4) is 1 _

7 .

-7 ( 1 _ 7 ) = -1


7. slope of

PQ = 6 - 4 _ 2 - 1

= 2 _ 1 = 2

slope of

QR = 3 - 6 _ 8 - 2

= -3 _ 6 = - 1 _

2

slope of

RS = 1 - 3 _ 7 - 8

= -2 _ -1

= 2

slope of

PS = 1 - 4 _ 7 - 1

= -3 _ 6 = - 1 _

2

PQ is perpendicular to

QR because the product of their slopes is -1.

QR is perpendicular to

RS

because the product of their slopes is -1.

RS isperpendicular to

PS because the product of their

slopes is -1.

PS is perpendicular to

PQ because the product of their slopes is -1. Therefore, all the angles are right angles, and PQRS is a rectangle.

8. The perpendicular line has a slope of 2 _ 5

, because

- 5 _ 2 ( 2 _

5 ) = -1.

y - y 1 = m(x - x 1 )

y - 0 = 2 _ 5 (x - 5)

y = 2 _ 5 (x - 5)

y = 2 _ 5 x - 2


9. The lines described by x = 7 and x = -9 are both vertical. These lines are parallel.

The lines described by y = - 5 _ 6

x + 8 and

y = - 5 _ 6 x - 4 both have slope - 5 _

6 . These lines are

parallel.

10. y - 3 = -1(x + 9) y - 3 = -x - 9 ____ + 3 ______ + 3 y = -x - 6

y + 1 = 1 _ 2 x

____ - 1 ___ -1

y = 1 _ 2 x - 1

y - 6 = 1 _ 2

(x - 14)

y - 6 = 1 _ 2

x - 7

____ + 6 ______ + 6

y = 1 _ 2

x - 1

The lines described by y = -x and y - 3 = -1(x + 9) represent parallel lines. They each have slope -1. The lines described by y - 6 = 1 _

2 (x - 14) and

y + 1 = 1 _ 2 x have the same slope are the same line.


CS10_A1_MESK710372_C04.indd 137 3/30/11 10:58:54 PM

11. -x + 2y = 17 _______ +x ___ +x 2y = x + 17

2y

_ 2 = x + 17 _

2

y = 1 _ 2 x + 17 _

2

3x + y = 27 _______ -3x ____ -3x y = -3x + 27

The lines described by y = -3x + 2 and 3x + y = 27 represent parallel lines. They each

have slope -3.

The lines described by y = 1 _ 2 x - 1 and

-x + 2y =17 represent parallel lines. They each

have slope 1 _ 2 .

12. slope of

LM = 4 - 0 _ 0 - (-3)

= 4 _ 3

slope of

MN = 4 - 0 _ 0 - 3

= 4 _ -3

= - 4 _ 3

slope of

NP = -4 -0 _ 0 - 3

= -4 _ -3

= 4 _ 3

slope of

LP = -4 - 0 _ 0 - (-3)

= -4 _ 3 = - 4 _

3

LM is parallel to

NP because they have the same slope.

MN is parallel to

LP because they have the

same slope. Since opposite sides are parallel, LMNP is a parallelogram.

13. The slope of the line described by y = 6x is 6. The

slope of the line described by y = - 1 _ 6 x is - 1 _

6 .

6 (- 1 _ 6 ) = -1


The slope of the line described by y = 1 _ 6

x is 1 _ 6 . The

slope of the line described by y = -6x is -6.

1 _ 6 (-6) = -1


14. The slope of the line described by y - 9 = 3(x + 1)

is 3. The slope of the line described by y = - 1 _ 3 x + 5

is - 1 _ 3 .

3 (- 1 _ 3 ) = -1


The graph described by y = 0 is a horizontal line, and the graph described by x = 6 is a vertical line. These lines are perpendicular.

15. x - 6y = 15 _______ -x ___ -x -6y = -x + 15

-6y

_ -6

= -x + 15 _ -6

y = 1 _ 6 x - 5 _

2

3y = -x - 11

3y

_ 3 = -x - 11 _

3

y = - 1 _ 3 x - 11 _

3

The slope of the line described by x - 6y = 15 is

1 _ 6 . The slope of the line described by y = -6x - 8 is

-6.

1 _ 6 (-6) = -1


The slope of the line described by y = 3x - 2 is 3. The slope of the line described by 3y = -x - 11 is

- 1 _ 3 .

3 (- 1 _ 3 ) = -1


16. slope of

AB = -3 - (-2)

_ -3 - (-7)

= -1 _ 4 = - 1 _

4

slope of

BC = -7 - (-3)

_ (-4 - (-3)

= -4 _ -1

= 4

AB is perpendicualr to

BC because the product of their slopes is -1. Since ABC contains a right

angle, ABC is a right triangle.

17. The parallel line also has a slope of - 6 _ 7 .

y - y 1 = m(x - x 1 )

y - 0 = - 6 _ 7 (x - 0)

y = - 6 _ 7 x

18. The graph described by x = 2 is a vertical line and the graph described by y = -5 is a horizontal line. These lines are perpendicular.

19. y - 28 = 7(x - 4) y - 28 = 7x - 28 _____ + 28 _______ + 28 y = 7x The lines described by y = 7x and y - 28 = 7(x - 4) have the same slope, but are not

parallel. They are the same line, so they are neither parallel nor perpendicular.

20. The slope of the line described by y = 2x - 1 is 2.

The slope of the line described by y = 1 _ 2 x + 2 is 1 _

2 .

Since 2 ≠ 1 _ 2 and 2 ( 1 _

2 ) ≠ -1, these lines are neither

parallel nor perpendicular.


CS10_A1_MESK710372_C04.indd 138 3/30/11 10:58:56 PM

21. y - 3 = 1 _ 4 (x - 3)

y - 3 = 1 _ 4 x - 3 _

4

____ + 3 _______ + 3

y = 1 _ 4 x + 9 _

4

y + 13 = 1 _ 4 (x + 1)

y + 13 = 1 _ 4 x + 1 _

4

_____ - 13 ______ -13

y = 1 _ 4 x - 51 _

4

The lines described by y - 3 = 1 _ 4 (x - 3) and

y + 13 = 1 _ 4 (x + 1) represent parallel lines. They

both have slope 1 _ 4 .

22. The parallel line also has a slope of 3. y - y 1 = m(x - x 1 ) y - 4 = 3(x - 0) y - 4 = 3x _____ + 4 ___ +4 y = 3x + 4

23. The parallel line also has a slope of 1 _ 2 .

y - y 1 = m(x - x 1 )

y - (-3) = 1 _ 2 (x - 4)

y + 3 = 1 _ 2 (x - 4)

y + 3 = 1 _ 2 x - 2

_____ - 3 ______ - 3

y = 1 _ 2 x - 5

24. 4y = x

4y

_ 4 = x _

4

y = 1 _ 4 x

The parallel line also has a slope of 1 _ 4 .

y - y 1 = m(x - x 1 )

y - 0 = 1 _ 4 (x - 4)

y = 1 _ 4 (x - 4)

y = 1 _ 4 x - 1

25. The parallel line also has a slope of 2. y - y 1 = m(x - x 1 ) y - 7 = 2(x - 1) y - 7 = 2x - 2 _____ + 7 ______ + 7 y = 2x + 5

26. 5x - 2y = 10 ________ -5x ____ -5x -2y = -5x + 10

-2y

_ -2

= -5x + 10 _ -2

y = 5 _ 2 x - 5

The parallel line also has a slope of 5 _ 2

.

y - y 1 = m(x - x 1 )

y - (-5) = 5 _ 2 (x - 3)

y + 5 = 5 _ 2 x - 15 _

2

_____ - 5 ________ -5

y = 5 _ 2 x - 25 _

2

27. The parallel line also has a slope of 3. y - y 1 = m(x - x 1 ) y - 7 = 3 (x - (-2)) y - 7 = 3(x + 2) y - 7 = 3x + 6 _____ + 7 ______ + 7 y = 3x + 13

28. The parallel line also has a slope of 0. y - y 1 = m(x - x 1 ) y - 4 = 0(x - 2) y - 4 = 0 _____ + 4 ___ +4 y = 4

29. x + y = 1 ______ -x ___ -x y = -x + 1

The parallel line also has a slope of -1. y - y 1 = m(x - x 1 ) y - 3 = -1(x - 2) y - 3 = -x + 2 _____ + 3 ______ + 3 y = -x + 5

30. 2x + 3y = 7 ________ -2x ____ -2x 3y = -2x + 7

3y

_ 3 = -2x + 7 _

3

y = - 2 _ 3 x + 7 _

3

The parallel line also has a slope of - 2 _ 3

.

y - y 1 = m(x - x 1 )

y - 5 = - 2 _ 3 (x - 4)

y - 5 = - 2 _ 3 x + 8 _

3

_____ + 5 ________ + 5

y = - 2 _ 3 x + 23 _

3

31. The parallel line also has a slope of 4. y - y 1 = m(x - x 1 ) y - (-3) = 4(x - 5) y + 3 = 4x - 20 _____ - 3 _______ -3 y = 4x - 23


CS10_A1_MESK710372_C04.indd 139 3/30/11 10:58:57 PM

32. The parallel line also has a slope of 1 _ 2 .

y - y 1 = m(x - x 1 )

y - (-4) = 1 _ 2 (x - 0)

y + 4 = 1 _ 2 x

_____ - 4 ___ -4

y = 1 _ 2 x - 4

33. 3x + 4y = 8 ________ -3x ____ -3x 4y = -3x + 8

4y

_ 4 = -3x + 8 _

4

y = - 3 _ 4 x + 2

The parallel line also has a slope of - 3 _ 4 .

y - y 1 = m(x - x 1 )

y - (-3) = - 3 _ 4 (x - 4)

y + 3 = - 3 _ 4 (x - 4)

y + 3 = - 3 _ 4 x + 3

_____ - 3 _______ - 3

y = - 3 _ 4 x

34. The perpendicular line has a slope of 1 _ 3 , because

-3 ( 1 _ 3 ) = -1.

y - y 1 = m(x - x 1 )

y - (-2) = 1 _ 3 (x - 6)

y + 2 = 1 _ 3 (x - 6)

y + 2 = 1 _ 3 x - 2

_____ - 2 ______ - 2

y = 1 _ 3 x - 4

35. The perpendicular line has a slope of -1, because 1(-1) = -1.

y - y 1 = m(x - x 1 ) y - 2 = -1 (x - (-1)) y - 2 = -1(x + 1) y - 2 = -x - 1 _____ + 2 ______ + 2 y = -x + 1

36. 3x - 4y = 8 ________ -3x ____ -3x -4y = -3x + 8

-4y

_ -4

= -3x + 8 _ -4

y = 3 _ 4 x - 2

The perpendicular line has a slope of - 4 _ 3 ,

because 3 _ 4 (- 4 _

3 ) = -1.

y - y 1 = m(x - x 1 )

y - 5 = - 4 _ 3 (x - (-6))

y - 5 = - 4 _ 3 (x + 6)

y - 5 = - 4 _ 3 x - 8

_____ + 5 ________ + 5

y = - 4 _ 3 x - 3

37. 5x + 2y = 10 ________ -5x ____ -5x 2y = -5x + 10

2y

_ 2 = -5x + 10 _

2

y = - 5 _ 2 x + 5

The perpendicular line has a slope of 2 _ 5 , because

- 5 _ 2 ( 2 _

5 ) = -1.

y - y 1 = m(x - x 1 )

y - (-5) = 2 _ 5 (x - 3)

y + 5 = 2 _ 5 (x - 3)

y + 5 = 2 _ 5 x - 6 _

5

_____ - 5 ______ -5

y = 2 _ 5 x - 31 _

5

38. The perpendicular line has a slope of 1 _ 3 , because

-3 ( 1 _ 3 ) = -1.

y - y 1 = m(x - x 1 )

y - (-4) = 1 _ 3 (x - 2)

y + 4 = 1 _ 3 (x - 2)

y + 4 = 1 _ 3 x - 2 _

3

_____ - 4 _______ -4

y = 1 _ 3 x - 14 _

3


CS10_A1_MESK710372_C04.indd 140 3/30/11 10:58:59 PM

39. -10x + 2y = 8 _________ +10x _____ +10x 2y = 10x + 8

2y

_ 2 = 10x + 8 _

2

y = 5x + 4

The perpendicular line has a slope of - 1 _ 5 , because

5 (- 1 _ 5 ) = -1.

y - y 1 = m(x - x 1 )

y - (-3) = - 1 _ 5 (x - 4)

y + 3 = - 1 _ 5 (x - 4)

y + 3 = - 1 _ 5 x + 4 _

5

_____ - 3 ________ -3

y = - 1 _ 5 x - 11 _

5

40. 2x + 3y = 7 ________ -2x ____ -2x 3y = -2x + 7

3y

_ 3 = -2x + 7 _

3

y = - 2 _ 3 x + 7 _

3

The perpendicular line has a slope of 3 _ 2 , because

- 2 _ 3 ( 3 _

2 ) = -1.

y - y 1 = m(x - x 1 )

y - 5 = 3 _ 2 (x - 4)

y - 5 = 3 _ 2 x - 6

_____ + 5 ______ + 5

y = 3 _ 2 x - 1

41. 4x - 2y = -6 ________ -4x ____ -4x -2y = -4x - 6

-2y

_ -2

= -4x - 6 _ -2

y = 2x + 3

The perpendicular line has a slope of - 1 _ 2 , because

2 (- 1 _ 2 ) = -1.

y - y 1 = m(x - x 1 )

y - (-2) = - 1 _ 2 (x - 3)

y + 2 = - 1 _ 2 (x - 3)

y + 2 = - 1 _ 2 x + 3 _

2

_____ - 2 ________ -2

y = - 1 _ 2 x - 1 _

2

42. -2x - 8y = 16 ________ +2x ____ +2x -8y = 2x + 16

-8y

_ -8

= 2x + 16 _ -8

y = - 1 _ 4 x - 2

The perpendicular line has a slope of 4, because

- 1 _ 4 (4) = -1.

y - y 1 = m(x - x 1 ) y - 5 = 4(x - 4) y - 5 = 4x - 16 _____ + 5 _______ +5 y = 4x - 11

43. The perpendicular line has a slope of 1 _ 2

, because

-2 ( 1 _ 2 ) = -1.

y - y 1 = m(x - x 1 )

y - 5 = 1 _ 2 (x - (-2))

y - 5 = 1 _ 2 (x + 2)

y - 5 = 1 _ 2 x + 1

_____ + 5 ______ + 5

y = 1 _ 2 x + 6

44. The perpendicular line has a slope of -1, because 1(-1) = -1.

y - y 1 = m(x - x 1 ) y - 5 = -1(x - 0) y - 5 = -x _____ + 5 ___ +5 y = -x + 5

45. x + y = 2 ______ -x ___ -x y = -x + 2 The perpendicular line has a slope of 1, because -1(1) = 0.

y - y 1 = m(x - x 1 ) y - 5 = 1(x - 8) y - 5 = x - 8 _____ + 5 _____ + 5 y = x - 3

46. Since the line is parallel to the y-axis, the line is vertical. Since the line is 6 units right of the y-axis, the line is x = 6.

47. Since the line is perpendicular to the y-axis, the line is horizontal. Since the line is 4 units below the

x-axis, the line is y = -4.

48. Possible answer: No, because parallel lines have no points in common. If they had the same y-intercept, they would both intersect the y-axis at the same place, and they could not be parallel.


CS10_A1_MESK710372_C04.indd 141 3/30/11 10:59:00 PM

49. Round (2.07, 8.95) to (2, 9) and (-1.9, 25.07) to (-2, 25).

m = y 2 - y 1

_ x 2 - x 1 = 25 - 9 _ -2 - 2

= 16 _ -4

= -4

The perpendicular line has a slope of about 1 _ 4 ,

because -4 ( 1 _ 4 ) = -1.

50. Possible answer: First find the slope of y - 3 = -6(x - 3). Since it is in point-slope form,

you can immediately tell that the slope is -6. Then find the y-intercept of y - 3 = -6(x - 3) by solving for y: y = -6x + 21. So the y-intercept is 21. Choose any other y-intercept b and write

y = -6x + b. This line will be parallel to y - 3 = -6(x - 3).

51a. Let x represent the number of minutes and let y represent the distance from home.

y = 50x

b. y = 50x + 30

c. Walk to Bus Stop

0 1 2 3 4

20

40

60

80

Time (min)

Step

s fr

om F

lora

’s h

ouse

Flora

Dan

No; the lines are parallel and never intersect; this means Flora and Dan will never be at the same place at the same time during their walk. Because Dan is walking at the same pace as Flora, Flora will not be able to catch up.

teSt prep

52. A; Since y - 3x + 2 and y = -3x both have a slope of -3; they are parallel.

53. H; The perpendicular line has slope - 5 _ 3 , because

3 _ 5 (- 5 _

3 ) = -1. So G and J are incorrect since the

slopes of those equations are positive. Both F and H go through (3, 3). Using the rise and run method, F

has slope - 3 _ 5 and H has slope - 5 _

3 , so H is correct.

54. 2x + y = 5 _______ -2x ____ -2x y = -2x + 5

The parallel line also has a slope of -2. y - y 1 = m(x - x 1 ) y - (-2) = -2(x - 6) y + 2 = -2(x - 6) y + 2 = -2x + 12 _____ - 2 ________ -2 y = -2x + 10 The y-intercept of the line is 10.


55. If the line containing A and B has the same slope

as the line containing B and C, then either AB is

parallel to BC or they are the same line. They cannot be parallel because they both contain B. Therefore, they must be the same line.

56. Since the lines are parallel, the slopes are equal. a + 12 = 4a _______ -a ___ -a 12 = 3a

12 _ 3 = 3a _

3

4 = a

57. Since the lines are perpendicular, the product of their slopes is -1.

- 1 _ 2 (5a + 3) = -1

- 5 _ 2 a - 3 _

2 = -1

________

+ 3 _ 2

___ + 3 _

2

- 5 _ 2 a = 1 _

2

- 2 _ 5 (- 5 _

2 a) = - 2 _

5 ( 1 _

2 )

a = - 1 _ 5

58. The slope of one diagonal is a - 0 _ a - 0

= a _ a = 1. The

slope of the other diagonal is 0 - a _ a - 0

= -a _ a = -1.

The product of the slopes is 1(-1) = -1, so the diagonals are perpendicular.

trAnsFormIng LIneAr FunctIons

CHECK IT OUT!

1. The graph of g(x) = x - 2 is the result of translating the graph of f(x) = x + 4, 6 units down.

2

-2

x 0 2 -2

y

g(x)

f (x)

2. The graph of g(x) = 1 _ 2 x - 1 is the result of rotating

the graph of f(x) = 3x - 1 about (0, -1). The graph of g(x) is less steep than the graph of f(x).

2

-2

x

y

0 2 4

g(x)

f (x)


4-10

CS10_A1_MESK710372_C04.indd 142 3/30/11 10:59:03 PM

3. To find g(x), multiply the value of m by -1.

In f(x) = 2 _ 3 x + 2, m = 2 _

3 .

2 _ 3 (-1) = - 2 _

3

g(x) = - 2 _ 3 x + 2

2

4

x

y

0 2

f(x)

g(x)

4. Multiply f(x) by -1 to get h(x) = -x. This reflects the graph across the y-axis.

Then add 2 to h(x) to get g(x) = -x + 2. This translates the graph 2 units up.

2

4

-2

-4

x

y

2 4 -2 -4

g(x) f (x)

5. If the charge per letter is lowered to $0.15, the new function is g(x) = 0.15x + 175. The original graph will be rotated about (0, 175) and become less steep.

If the trophy’s cost is raised to $180, the new function is h(x) = 0.20x + 180. The original graph will be translated 5 units up.

THInK and dIsCUss

1. translation of f(x) = x 3.45 units up

2. No; there can only be a whole number of letters, so points whose x-coordinates are not whole numbers have no meaning in this situation.

3. Transformations off(x) = x

4

-2

x

y

0 2 -2

2

x

y

2 -2

2

-2

x

y

0 2 -2

Translation:g(x ) = x + 3

Rotation:g(x ) = 2x

Reflection:g(x ) = -x


1. translation 2. rotation

3.

2

-4

-2

4

x

y

4 2 -4 -2

f (x)

g(x)

The graph of g(x) = x - 4 is the result of translating the graph of f(x) = x, 4 units down.

4.

2

-4

-2

4 y

4 2 -4

f (x)

g(x)

x

The graph of g(x) = x + 1 is the result of translating the graph of f(x) = x, 1 units up.

5.

2

-4

-2

4

4 2 -4

f (x)

g(x)

x


6. 2

-4

-6

-2

y

4 2 -4 -2

f (x)

g(x)

x

The graph of g(x) = x - 6.5 is the result of translating the graph of f(x) = x, 6.5 units down.

7.

2

-4

-2

4 y

4 2

f (x)

g(x) x

The graph of g(x) = 1 _ 4

x is the

result of rotating the graph of f(x) = x about (0, 0). The graph of g(x) is less steep than the graph of f(x).

8.

-4

-2

4 y

0 4 2 -2

f (x)

g(x) x

The graph of g(x) = x + 3 is the result of rotating the

graph of f(x) = 1 _ 5

x + 3 about

(0, 3). The graph of g(x) is steeper than the graph of f(x).

9.

-2

4

2

y

0 4 2 -2 -4

f (x) g(x)

x

The graph of g(x) = 4x - 2 is the result of rotating the graph of f(x) = 2x - 2 about (0, -2). The graph of g(x) is steeper than the graph of f(x).


CS10_A1_MESK710372_C04.indd 143 3/30/11 10:59:08 PM

10.

-2

-4

4

2

y

0 4 2

f (x)

g(x)

x

The graph of g(x) = 1 _ 2 x + 1 is

the result of rotating the graph of f(x) = x + 1 about (0, 1). The graph of g(x) is less steep than the graph of f(x).

11.

-2

-4

4

2

y

0 2 -2 f (x)

g(x)

To find g(x), multiply the value of m by -1.

In f(x) = - 1 _ 5 x, m = - 1 _

5 .

- 1 _ 5 (-1) = 1 _

5

g(x) = 1 _ 5 x

12.

-2

4

0 2 4 -4

f (x) g(x)

x

y To find g(x), multiply the value of m by -1.

In f(x) = 2x + 4, m = 2. 2(-1) = -2 g(x) = -2x + 4

13.

-2

-4

y

0 2 4 -4 -2

f (x) g(x)

x To find g(x), multiply the value of m by -1.

In f(x) = 1 _ 3 x - 6, m = 1 _

3 .

1 _ 3 (-1) = - 1 _

3

g(x) = - 1 _ 3 x - 6

14.

-2

4 y

0 2 4 -4 -2

f (x) g(x)

x


In f(x) = 5x - 1, m = 5. 5(-1) = -5 g(x) = -5x - 1

15.

-2

4

2

y

2 4 -4 -2

f (x) g(x)

x

Multiply f(x) by 2 to get h(x) = 2x. This makes it steeper. Then subtract 2 from h(x) to get g(x) = 2x - 2. This translates the graph 2 units down.

16.

-2

-4

4

2

y

2 4 -4 -2

f (x)

g(x) x

Multiply f(x) by 1 _ 3 to get

h(x) = 1 __ 3 x. This rotates the

graph and makes it less steep. Then add 1 to h(x) to

get g(x) = 1 __ 3 x + 1. This

translates the graph 1 units up.

17.

-2

-4

4 y

2 4 -4 -2

f (x)

g(x)

x

Add 1 to f(x) to get h(x) = -x. This translates the graph 1 unit up. Then multiply h(x) by 4 to get g(x) = -4x. This reflects the graph in the x-axis and makes it steeper.

18.

-2

-4

4 y

0 2 4 -2

f (x)

g(x)

x

Multiply f(x) by 1 _ 2 to get

h(x) = - 1 __ 2 x. This rotates the

graph about (0, 0) and makes it less steep. Then subtract 3 from h(x) to get g(x) = - 1 _

2 x - 3. This

translates the graph 3 units down.

19. If the reservation fee is raised to $50, the new function is g(x) = 15x + 50. The original graph will be translated 25 units up.

If the charge per person is lowered to $12, the new function is h(x) = 12x + 25. The original graph is rotated about (0, 25) and becomes less steep.


20.

-2

2

1

y

2 4 -2 -4

f (x)

g(x)

x

The graph of g(x) = x + 1 _ 2

is

the result of translating the

graph of f(x) = x, 1 _ 2 unit up.

21.

-2

-4

4

2

y

2 4 -2 -4

f (x)

g(x) x

The graph of g(x) = x - 4 is the result of translating the graph of f(x) = x, 4 units down.


CS10_A1_MESK710372_C04.indd 144 3/30/11 10:59:13 PM

22.

-2

2

1

y

0 -2 -4 f (x) g(x)

x

The graph of g(x) = 1 _ 10

x - 1

is the result of rotating the

graph of f(x) = 1 _ 5 x - 1 about

(0, -1). The graph of g(x) is less steep than the graph of f(x).

23.

-2

-4

4

2

y

0 -2

f (x) g(x)

x

2 4


the result of rotating the graph of f(x) = x + 2 about (0, 2). The graph of g(x) is less steep than the graph of f(x).

24.

-2

2

0 -2 -4

f (x) g(x)

x

2 4


In f(x) = 6x, m = 6. 6(-1) = -6 g(x) = -6x

25.

-2

2

4 y

-2 -4

f (x) g(x)

x

2 4


In f(x) = -3x - 2, m = -3 -3(-1) = 3 g(x) = 3x - 2

26.

2

4 y

-2 -4

f (x)

g(x)

x

2 4

Multiply f(x) by 2 to get h(x) = 4x. This makes the graph steeper. Then subtract 1 from h(x) to get g(x) = 4x - 1. This translates the graph 1 unit down.

27.

4

-4

-8

y

0 -2 -4

f (x) g(x)

x

2 4

Subtract 5 from f(x) to get h(x) = -7x. This translates the graph 5 units down. Then multiply h(x) by 2 to get g(x) = -14x. This makes the graph steeper.

28. If the number of parents is reduced to 0, the new

function is g(x) = 1 _ 4 x. The original graph will be

translated 2 units down. If the number of teachers is raised to 1 for every

3 students, the new function is h(x) = 1 _ 3 x + 2.

The original graph will be rotated about (0, 2) and become steeper.

29. The graph of g(x) = -x is the result of reflecting the graph of f(x) = x across the y-axis.

2

-2

x

y

2 -2

f (x)

g (x)

The graphs have oppposite slopes-same steepness but opposite in directions. The graphs have the same y-intercept.

30. The graph of g(x) = x + 8 is the result of translating the graph of f(x) = x 8 units up.

2

4

6

x

y

2 -2

f (x)

g (x) The graphs have the same slope but different y-intercepts.

31. The graph of g(x) = 3x is the result of rotating the graph of f(x) = x about (0, 0) The graph of g(x) is steeper than the graph of f(x).

2

x

y

2 -2

f (x)

g (x)

The graphs have different slopes but the same y-intercept.

32. The graph of g(x) = - 2 _ 7 x is the result of rotating the

graph of f(x) = x about (0, 0). The graph of g(x) is less steep than the graph of f(x).

2

-2

x

y

-2

f (x)

g ( x )

The graphs have different slopes but the same y-intercept.

33.

2

-2

x

y

2 -2

f (x)

g (x)

Multiply f(x) by 6 to get h(x) = 6x. This rotates the graph about (0, 0) and makes it steeper. Then subtract 3 from h(x) to get g(x) = 6x - 3. This translates the graph 3 units down. The graphs have different slopes and different intercepts, but both graphs are increasing.

34.

-2

x

2 -2

f (x)

g (x)

y Multiply f(x) by -2 to get h(x) = -2x. This rotates the graph about (0, 0) and makes it steeper. Then add 1 to h(x) to get g(x) = -2x + 1. This translates the graph 1 unit up. The graphs have different slopes and different intercepts.


CS10_A1_MESK710372_C04.indd 145 3/30/11 10:59:19 PM

35. Possible answer:

2

-2

x

y

0 2 -2

g(x) = x + 2 Answers may vary

depending on the point of rotation used.

36. g(x) = -x - 5

4

x

y

0 4

37. g(x) = 1 _ 6

x - 4

-2

x y

0 2 -2

38a. y = 12x + 20

Book Club Costs

0 2 4 6 8

20

40

60

80

Books purchased

Cost

s ($

)

(0, 20)

(1, 32) (2, 44)

(3, 56) (4, 68)

(5, 80) (6, 92)

b. y = 12x + 30

Book Club Costs

0 2 4 6 8

20

40

60

80

Books purchased

Cost

s ($

)

(0, 30)

(1, 42) (2, 54)

(3, 66) (4, 78)

(5, 90)

c. The graph in part a is translated 10 units up to get the graph in part b.

39. trans. 9 units down 40. rot. about (0, 0); ref. across y-axis

41. rot. about (0, 0) (steeper)

42. rot. about (0, 0) (less steep), and trans. 1 unit up

43. rot. about (0, 0) (steeper)

44. rot. about (0, 0) (less steep)

45a. $300 b. 0.20 = 20%

c. Commisson changes to 25%; base pay changes to $400.

46. Possible answer: reflect across the x-axis

47. Possible answer: A reflection across the y-axis multiplies the x-coordinate of each ordered pair by -1. For example, when f(x) = x is reflected across the y-axis, the x coordinate changes to -x. (i.e., (1, 0) → (-1, 0))

48a. y = 3x

x

y

b. Possible answer: Jen is walking from the stadium to the softball field, and the stadium is 100 ft closer to the field than the school is.

c. The distance from the school when the walking begins.

teSt prep

49. D; If the slope changes to 10, the new function is g(x) = 10x - 5. The x-intercept can be found by substituting 0 for g(x) which gives 0 = 10x - 5 or

x = 1 _ 2 .

50. J; Since the slope will not change by increasing the y-intercept, the new line is not steeper than the original.


51.

4

x

y

0 -2

The graph of g(x) = x + 3 is the result of translating the graph of f(x) = x, 3 units to the left.

52. The graph of g(x) = x + c is the result of translating the graph of f(x) = x, c units to the left.

The graph of g(x) = x - c is the result of translating the graph of f(x) = x, c units to the right.

reAdy to go on? section B Quiz

1. 2x + y = 5 _______ -2x ____ -2x y = -2x + 5

2

4

x

y

0 2 -2

Plot (0, 5). Count 2 units down and 1 unit right and plot another point. Draw the line connecting the two.


CS10_A1_MESK710372_C04.indd 146 3/30/11 10:59:23 PM

2. 2x - 6y = 6 ________ -2x ____ -2x -6y = -2x + 6

-6y

_ -6

= -2x + 6 _ -6

y = 1 _ 3 x - 1

2

-2

x

y

0 2 4

Plot (0, -1). Count 1 unit up and 3 units right and plot another point. Draw the line connecting the two points.

3. 3x + y = 3x - 4 _______ -3x _______ -3x y = -4

2

-2

x

y

0 2 -2

Plot (0, -4). Count 0 units up and 1 unit right and plot another point. Draw the line connecting the two points.

4a. y = 0.5x + 3

b. The y-intercept is 3. This is the entrance fee. The slope is 0.5. This is the cost per bowl of chili.

5.

2

4

x

y

0 2 -2

Plot (0, 3). Count 3 units down and 1 unit right and plot another point. Draw the line connecting the two points.

6.

2

4

x

y

0 2 -2

Plot (-3, 5). Count 2 units down and 3 units right and plot another point. Draw the line connecting the two points.

7.

2

-2

x

y

0 2 -2 -4

Plot (-3, -1). Count 2 units up and 1 unit right and plot another point. Draw the line connecting the two points.

8. m = y 2 - y 1

_ x 2 - x 1 = 3 - 1 _ 4 - 3

= 2 _ 1

= 2

y - y 1 = m(x - x 1 ) y - 1 = 2(x - 3) y - 1 = 2x - 6 _____ + 1 ______ + 1 y = 2x - 5

9. m = y 2 - y 1

_ x 2 - x 1 = 7 - (-1)

_ 1 - (-1)

= 8 _ 2

= 4

y - y 1 = m(x - x 1 ) y - 7 = 4(x - 1) y - 7 = 4x - 4 _____ + 7 ______ + 7 y = 4x + 3

10. m = y 2 - y 1

_ x 2 - x 1 = 5 - (-4)

_ -2 - 1

= 9 _ -3

= -3

y - y 1 = m(x - x 1 ) y - (-4) = -3(x - 1) y + 4 = -3(x - 1) y + 4 = -3x + 3 _____ - 4 _______ - 4 y = -3x - 1

11. y ≈ 0.47x - 27; extremely well, (r ≈ 0.99)

12. y = 2(x + 5) y = 2x + 10 The lines described by y = 2x + 1, y = 2x, and

y = 2(x + 5) are parallel lines. They each have slope 2.

13. -3y = x

-3y

_ -3

= x _ -3

y = - 1 _ 3 x

y + 2 = x + 4 ____ - 2 _____ - 2 y = x + 2

The lines described by -3y = x and y = - 1 _ 3

x + 1

are parallel. They each have slope - 1 _ 3

.

14. The slope of the line described by y = -4x - 1 is

-4. The slope of the line described by y = 1 _ 4

x is 1 _ 4

.

-4 ( 1 _ 4 ) = -1


15. The slope of the line described by y = - 3 _ 4

x is - 3 _ 4

.

The slope of the line described by y = 4 _ 3

x is 4 _ 3

.

- 3 _ 4 ( 4 _

3 ) = -1


The line described by y = 4 is a horizontal line and the line described by x = 3 is a vertical line. These graphs are perpendicular.

16.

-2

2

x

y

0 2 -2

y = -5x

y = 5x The graph of g(x) = -5x is the result of reflecting the graph of f(x) = 5x across the y-axis; rotation about (0, 0)


CS10_A1_MESK710372_C04.indd 147 3/30/11 10:59:26 PM

17.

-2

4

2

0 2 -2 -4

y = x + 4

y = x - 1

1 _ 2

1 _ 2


the result of translating the

graph of f(x) = 1 _ 2 x - 1, 5 units

up.

study guIde: revIew

1. translation; rotation; reflection

2. y-intercept 3. slope; y-intercept

IdEnTIfyIng lInEar fUnCTIOns


5. Yes; a constant change of +1 in x corresponds to a constant change of +2 in y.


7. No; a constant change of -1 in y corresponds to different changes in x.

8. y = -5x + 1 ____ +5x _______ +5x 5x + y = 1 A = 5; B = 1; C = 1

9. x + 2 _ 2 = -3y

2 ( x + 2 _ 2 ) = 2(-3y)

x + 2 = -6y _____ - 2 ____ -2 x = -6y - 2 _____ + 6y _______ +6y x + 6y = -2 A = 1; B = 6; C = -2

10. 4y = 7x ____ -4y ____ -4y 0 = 7x - 4y 7x - 4y = 0 A = 7; B = -4; C = 0

11. 9 = y y = 9 0x + y = 9 A = 0; B = 1; C = 9

12. x f(x) = 0.5x (x, f(x))

0 f(x) = 0.5(0) = 0 (0, 0)

1 f(x) = 0.5(1) = 0.5 (1, 0.5)

2 f(x) = 0.5(2) = 1.0 (2, 1.0)

3 f(x) = 0.5(3) = 1.5 (3, 1.5)

4 f(x) = 0.5(4) = 2.0 (4, 2.0)

5 f(x) = 0.5(5) = 2.5 (5, 2.5)

6 f(x) = 0.5(6) = 3.0 (6, 3.0)

7 f(x) = 0.5(7) = 3.5 (7, 3.5)

8 f(x) = 0.5(8) = 4.0 (8, 4.0)

9 f(x) = 0.5(9) = 4.5 (9, 4.5)

Cupcake Sales

0 2 4 6 8

1

2

3

4

Cupcakes sold A

mou

nt e

arne

d ($

)

The number of cupcakes must be a whole number, so the domain is whole numbers. The range is nonnegative multiples of 0.5.

UsIng InTErCEPTs

13. The x-intercept is 2. The y-intercept is -4.

14. The x-intercept is 5. The y-intercept is 6.

15. 3x - y = 9 3x - 0 = 9 3x = 9

3x ___ 3 = 9 __

3


3x - y = 9 3(0) - y = 9 0 - y = 9 -y = 9 -1(-y) = -1(9) y = -9 The y-intercept is -9.

16. -2x + y = 1 -2x + 0 = 1 -2x = 1

-2x ____ -2

= 1 ___ -2

x = - 1 __ 2

The x-intercept is - 1 __ 2 .

-2x + y = 1 -2(0) + y = 1 0 + y = 1 y = 1 The y-intercept is 1.

17. -x + 6y = 18 -x + 6(0) = 18 -x + 0 = 18 -x = 18 -1(-x) = -1(18) x = -18 The x-intercept is -18.

-x + 6y = 18 -(0) + 6y = 18 0 + 6y = 18 6y = 18

6y

___ 6 = 18 ___

6



4-1

4-2

CS10_A1_MESK710372_C04.indd 148 3/30/11 10:59:28 PM

18. 3x - 4y = 1 3x - 4(0) = 1 3x - 0 = 1 3x = 1

3x ___ 3 = 1 __

3

x = 1 __ 3

The x-intercept is 1 __ 3 .

3x - 4y = 1 3(0) - 4y = 1 0 - 4y = 1 -4y = 1

-4y

____ -4

= 1 ___ -4

y = - 1 __ 4

The y-intercept is - 1 __ 4 .

raTE Of CHangE and slOPE

19.

0 1 2 3 4

60

120

180

240

Time (s)

Dis

tanc

e (f

t)

Rate

112 fts

80 fts

48 fts

16 fts

20. slope = 10 ___ 2 = 5

THE slOPE fOrmUla


4x ___ 4 = 24 ___

4

x = 6


3y

___ 3 = 24 ___

3

y = 8

m = y 2 - y 1

_______ x 2 - x 1 = 8 - 0 _____ 0 - 6

= 8 ___ -6

= - 4 __ 3

22. y = -3x + 6 ____ +3x _______ +3x 3x + y = 6

Find the x-intercept: 3x + y = 6 3x + 0 = 6 3x = 6

3x ___ 3 = 6 __

3

x = 2

Find the y-intercept: 3x+ y = 6 3(0) + y = 6 y = 6

m = y 2 - y 1

_______ x 2 - x 1 = 6 - 0 _____ 0 - 2

= 6 ___ -2

= -3

23. Find the x-intercept: x + 2y = 10 x + 2(0) = 10 x = 10

Find the y-intercept: x + 2y = 10 0 + 2y = 10 2y = 10

2y

___ 2 = 10 ___

2

y = 5

m = y 2 - y 1

_______ x 2 - x 1 = 5 - 0 ______ 0 - 10

= 5 ____ -10

= - 1 __ 2

24. 3x = y + 3 ___ - y ______ -y 3x - y = 3

Find the x-intercept: 3x - y = 3 3x - 0 = 3 3x = 3

3x ___ 3 = 3 __

3

x = 1

Find the y-intercept: 3x - y = 3 3(0) - y = 3 -y = 3 -1(-y) = -1(3) y = -3

m = y 2 - y 1

_______ x 2 - x 1 = -3 - 0 _______ 0 - 1

= -3 ___ -1

= 3

25. y + 2 = 7x ______ -y ___ -y 2 = 7x - y Find the x-intercept: 7x - y = 2 7x - 0 = 2 7x = 2

7x ___ 7 = 2 __

7

x = 2 __ 7

Find the y-intercept: 7x - y = 2 7(0) - y = 2 -y = 2 -1(-y) = -1(2) y = -2

m = y 2 - y 1

_______ x 2 - x 1 = -2 - 0 _______ 0 - 2 _

7 = -2 ___

- 2 _ 7 = 7

26. 16x = 4y + 1 ____ - 4y _______ -4y 16x - 4y = 1 Find the x-intercept: 16x - 4y = 1 16x - 4(0) = 1 16x = 1

16x ____ 16

= 1 ___ 16

x = 1 ___ 16

Find the y-intercept: 16x - 4y = 1 16(0) - 4y = 1 -4y = 1

-4y

____ -4

= 1 ___ -4

y = - 1 __ 4

m = y 2 - y 1

_______ x 2 - x 1 = - 1 _

4 - 0 _______

0 - 1 __ 16

=

- 1 _ 4 ____

- 1 __ 16

= 4

27. m = y 2 - y 1

_______ x 2 - x 1

= -3 - 2 _______ 2 - 1

= -5 ___ 1

= -5

28. m = y 2 - y 1

_______ x 2 - x 1

= 7 - (-2)

________ -5 - 4

= 9 ___ -9

= -1

29. m = y 2 - y 1

_______ x 2 - x 1

= 1 - (-6)

________ 4 - (-3)

= 7 __ 7

= 1

30. m = y 2 - y 1

_______ x 2 - x 1

= 5 _ 2 - 2

_____ 3 _ 4 - 1 _

2

= 1 _ 2 __

1 _ 4

= 2

31. m = y 2 - y 1

_______ x 2 - x 1

= 7 - 2 _____ 2 - 2

= 5 __ 0


32. m = y 2 - y 1

_______ x 2 - x 1

= -3 - (-3)

_________ 5 - 1

= 0 __ 4

= 0


4-3

4-4

CS10_A1_MESK710372_C04.indd 149 3/30/11 10:59:31 PM

dIrECT VarIaTIOn

33. This equation represents a direct variation because it can be written in the form y = kx. The constant of variation is -6.

34. x - y = 0 ____ + y ___ +y x = y y = x This equation represents a direct variation because

it can be written in the form y = kx. The constant of variation is 1.

35. y + 4x = 3 _____ - 4x ____ -4x y = -4x + 3 This equation does not represent a direct variation

because it cannot be written in the form y = kx.

36. 2x = -4y

2x ___ -4

= -4y

____ -4

- 1 __ 2 x = y

y = - 1 __ 2 x


variation is - 1 __ 2 .

37. -8 ___ 2 =

y __

3

2y = -24 y = -12

38. x y = 8x (x, y)

0 y = 8(0) = 0 (0, 0)

2 y = 8(2) = 16 (2, 16)

4 y = 8(4) = 32 (4, 32)

Maleka’s Baby-sitting Earnings

0 2 4 6 8

10

20

30

40

Time (h)

Mon

ey e

arne

d ($

)


slOPE-InTErCEPT fOrm

39.

2

4

6

x

y

0 2 -2

Plot (0, 4). Count 1 unit down and 2 units right and plot another point. Draw the line connecting the two points.

40.

-2

-4

-6

x y 0 -2

Plot (0, -7). Count 3 units up and 1 unit right and plot another point. Draw the line connecting the two points.

41. y = mx + b

y = 1 __ 3 x + 5

42. y = mx + b -5 = 4(1) + b -5 = 4 + b ___ -4 ______ -4 -9 = b

y = mx + b y = 4x + (-9) y = 4x - 9

POInT-slOPE fOrm

43.

-2

-4

-6

xy0 2-2

44.

2

-2

x

y

0 2-2

45. y - y 1 = m(x - x 1 ) y - 3 = 2(x - 1) y - 3 = 2x - 2 _____ + 3 ______ + 3 y = 2x + 1

46. y - 4 = -5 (x - (-6)) y - 4 = -5(x + 6) y - 4 = -5x - 30 _____ + 4 ________ + 4 y = -5x - 26

47. m = y 2 - y 1

_______ x 2 - x 1 = 8 - 4 _____ 3 - 1

= 4 __ 2 = 2

y - 4 = 2(x - 1) y - 4 = 2x - 2 _____ + 4 ______ + 4 y = 2x + 2

48. m = y 2 - y 1

_______ x 2 - x 1 = 6 - 4 _________ -1 - (-2)

= 2 __ 1 = 2

y - 4 = 2 (x - (-2)) y - 4 = 2x + 4 _____ + 4 ______ + 4 y = 2x + 8

49. y =0.5x

( 3 __ 2 )

2

+ (1) 2 + (1) 2 + ( 1 __ 2 )

2

=4.5

y = x - 1

(2) 2 + (1) 2 + (2) 2 + (1) 2 =10

50. y ≈ 2.3x - 16; extremely well (r ≈ 0.99)


4-5

4-6

4-7

CS10_A1_MESK710372_C04.indd 150 3/30/11 10:59:34 PM

slOPEs Of ParallEl and PErPEndICUlar lInEs

51. The lines described by y = - 1 __ 3 x and y = - 1 __

3 x - 6

represent parallel lines. They each have slope - 1 __ 3 .

52. y - 2 = -4(x - 1) y - 2 = -4x + 4 ____ + 2 _______ + 2 y = -4x + 6 The lines described by y - 2 = -4(x - 1) and

y = -4x - 2 represent parallel lines. They each have slope -4.

53. The slope of the line described by y - 1 = -5(x - 6) is -5. The slope of the line

described by y = 1 __ 5 x + 2 is 1 __

5 .

-5 ( 1 __ 5 ) = -1


54. The slope of the line described by y - 2 = 3(x + 1)

is 3. The slope of the line described by y = - 1 __ 3 x is

- 1 __ 3 .

3 (- 1 __ 3 ) = -1


55. The parallel line also has a slope of 2. y - y 1 = m(x - x 1 ) y - (-1) = 2(x - 1) y + 1 = 2(x - 1) y + 1 = 2x - 2 _____ - 1 ______ - 1 y = 2x - 3

TransfOrmIng lInEar fUnCTIOns

56.

2

4

x

y

-2 -4

g(x)

f (x)


57.

2

-2

x

y

0 2 -2

g(x) f (x) The graph of g(x) = -4x is the result of reflecting the graph of f(x) = 4x across the y-axis; rotation about (0, 0)

58. 4

-4

-8

x

y

g(x)

f (x)

The graph of g(x) = - 1 __ 3 x - 2 is

the result of reflecting the graph

of f(x) = 1 __ 3 x - 2 across the

y-axis; rotation about (0, -2)

59. If the entrance fee is increased to $5, the new function is g(x) = x + 5. The original graph will be translated 2 units up.

If the cost per ride increases to $2, the new function is h(x) = 2x + 3. The original graph will be rotated about (0, 3) and will be steeper.

chApter test

1. yes

2

-2

x

y

0 2

2. no

3. x 0 3 6 9 12 15

f(x) = 45 - 3x 45 36 27 18 9 0

Lily’s Volunteer Hours

0 2 4 6 8

10

20

30

40

Week

Hou

rs r

emai

ning

The x-intercept is 15. The y-intercept is 45. x-intercept: the number of weeks that will have

passed when Lily has no volunteer hours remaining. y-intercept: original number of volunteer hours.

4.

0

10

20

30

6 842

Guppy Population

Time (mo)

Gup

pies

2 fish/mo

5 fish/mo

6 fish/mo

0 fish/mo

5. m = y 2 - y 1

_______ x 2 - x 1

= 68 - 25.5 _________ 8 - 3

= 42.5 ____ 5

= 8.5 A slope of 8.5 means the cost is $8.50 per ticket.


4-9

4-10

CS10_A1_MESK710372_C04.indd 151 3/30/11 10:59:37 PM

6. m = y 2 - y 1

_______ x 2 - x 1

= 76 - 40 _______ 2 - 5

= 36 ____ -3

= -12 A slope of -12 means the height decreases by

12 ft/s.

7. m = y 2 - y 1

_______ x 2 - x 1

= 4 - (-1)

________ 5.5 - .5

= 5 __ 5

= 1 A slope of 1 means the temperature increases by

1°F/h.

8. y = 5x

0

10

20

30

642

Space Shuttle

Dis

tanc

e (m

i)

Time (s)

9. 2x - 2y = 4 -2x -2x -2y = -2x + 4

-2y

____ -2

= -2x + 4 _______ -2

y = x - 2

2

-2

x

y

0 2 4

10. y - y 1 = m(x - x 1 ) y - 3 = 4(x - (-3)) y - 3 = 4x + 12 _____ + 3 _______ + 3 y = 4x + 15

11. m = y 2 - y 1

_______ x 2 - x 1 = -3 - 0 ______ 0 - 3

= -3 ___ -3

= 1

y - 0 = 1(x - 3) y = x - 3 12. y - y 1 = m(x - x 1 ) y - 3 = -(x - 1) 13. y - y 1 = m(x - x 1 ) y - 2 = 5(x - (-3)) y - 2 = 5(x + 3)

14. y ≈ 0.7x + 14.5; moderately well (r ≈ 0.75)

15. The parallel line also has a slope of 2. y - y 1 = m(x - x 1 ) y - 6 = 2(x - 0) y - 6 = 2x _____ + 6 ___ +6 y = 2x + 6

16.

-2

2

x

y

0 2 -2

y = 4x

y = 8x

The graph of g(x) = 4x is the result of rotating the graph of f(x) = 8x about (0, 0). The graph of g(x) is less steep than the graph of f(x).

17.

2

x 0 2 -2

y = -x - 1

y = -x + 2 The graph of g(x) = -x - 1 is the result of translating the graph of f(x) = -x + 2, 3 units down.

18.

2

x 0 2 -2

y = 6x - 1

y = 3x Multiply f(x) by 2 to get h(x) = 6x. This rotates the graph about (0, 0) and makes it steeper. Then subtract 1 from h(x) to get g(x) = 6x - 1. This translates the graph 1 unit down.


CS10_A1_MESK710372_C04.indd 152 3/30/11 10:59:40 PM

Documents

CHAPTER Linear Functions 4 Solutions Key Functions Solutions Key Are you reAdy? 1. E 2. C 3. A 4. B 5 -12. 8 4 -8 x y -8 -4 0 4 8 F A G C E B H D 13. 2x + y = 8 _____-2x ____-2x y

CHAPTER Linear Functions 4 Solutions Key Functions Solutions Key Are you reAdy? 1. E 2. C 3. A 4. B 5 -12. 8 4 -8 x y -8 -4 0 4 8 F A G C E B H D 13. 2x + y = 8 _-2x -2x y