chapter linear functions 4 solutions key functions solutions key are you ready? 1. e 2. c 3. a 4. b...
TRANSCRIPT
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 = $4.75/lb
25. 32 grams
_ 4 servings
= 8 g/serving
26. 120 pictures
__ 5 rolls
= 24 pictures/roll
IdentIFyIng LIneAr FunctIons
CHECK IT OUT!
1a. Yes; each domain value is paired with exactly one range value; yes
b. Yes; each domain value is paired with exactly one range value; yes
c. No; each domain value is not paired with exactly one range value; no, not a linear function.
2. Yes; a constant change of +2 in x corresponds to a constant change of -1 in y.
3a. y = 5x - 9 ____ -5x _______ -5x y - 5x = -9 -5x + y = -9 The equation can be written in standard form, so the
function is linear.
x y = 5x - 9 (x, y)
0 y = 5(0) - 9 = -9 (0, -9)
1 y = 5(1) - 9 = -4 (1, -4)
2 y = 5(2) - 9 = 1 (2, 1)
Plot the points and connect them with a straight line.
-4
-6
-8
-2
x y 0 2 -2
b. y = 12 0x + y = 12 The equation can be written in standard form, so the
function is linear.
x y = 12 (x, y)
-1 y = 12 (-1, 12)
0 y = 12 (0, 12)
1 y = 12 (1, 12)
Plot the points and connect them with a straight line.
8
4
x
y
0 2 -2
c. This is not linear, because x appears in an exponent.
103 Holt McDougal Algebra 1
4CHAPTER
4-1
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4. x f(x) = 3x + 10 (x, f(x))
0 f(x) = 3(0) + 10 = 10 (0, 10)
1 f(x) = 3(1) + 10 = 13 (1, 13)
2 f(x) = 3(2) + 10 = 16 (2, 16)
3 f(x) = 3(3) + 10 = 19 (3, 19)
4 f(x) = 3(4) + 10 = 22 (4, 22)
5 f(x) = 3(5) + 10 = 25 (5, 25)
6 f(x) = 3(6) + 10 = 28 (6, 28)
7 f(x) = 3(7) + 10 = 31 (7, 31)
Rental Payment
0 2 4 6
8
16
24
Manicures
Rent
al p
aym
ent
($)
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
3. Yes; each domain value is paired with exactly one range value; yes
4. 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)
Plot the points and connect them with a straight line.
-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)
Plot the points and connect them with a straight line.
2
x
y
0 2 -2
11. This is not linear, because x has an exponent other than 1.
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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)
Plot the points and connect them with a straight line.
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
16. Yes; each domain value is paired with exactly one range value; yes
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)
Plot the points and connect them with a straight line.
4
2
x
y
0 2 -2
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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)
Plot the points and connect them with a straight line.
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)
Plot the points and connect them with a straight line.
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
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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
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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).
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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
y = 8 The y-intercept is 8.
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.
109 Holt McDougal Algebra 1
4-2
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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.
ExErCIsEsguided practice
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
x = 2 The x-intercept is 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
x = 2 The x-intercept is 2.
-3x - 2y = -6 -3(0) - 2y = -6 0 - 2y = -6 -2y = -6
-2y
_ -2
= -6 _ -2
y = 3 The y-intercept is 3.
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
x = 2 The x-intercept is 2.
8x + 2y = 16 8(0) + 2y = 16 0 + 2y = 16 2y = 16
2y
_ 2 = 16 _
2
y = 8 The y-intercept is 8.
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
110 Holt McDougal Algebra 1
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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
practice and problem Solving
13. The x-intercept is -1. The y-intercept is 3.
14. The x-intercept is -5. The y-intercept is -1.
15. The x-intercept is -4. The y-intercept is 2.
16. 6x + 3y = 12 6x + 3(0) = 12 6x + 0 = 12 6x = 12
6x _ 6 = 12 _
6
x = 2 The x-intercept is 2.
6x + 3y = 12 6(0) + 3y = 12 0 + 3y = 12 3y = 12
3y
_ 3
= 12 _ 3
y = 4 The y-intercept is 4.
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
x = -4 The x-intercept is -4.
-2x + 4y = 8 -2(0) + 4y = 8 0 + 4y = 8
4y
_ 4
= 8 _ 4
y = 2 The y-intercept is 2.
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
y = 2 The y-intercept is 2.
19. 4x + y = 8 4x + 0 = 8 4x = 8
4x _ 4 = 8 _
4
x = 2 The x-intercept is 2.
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
x = 5 The x-intercept is 5.
y - 3x = -15 y - 3(0) = -15 y - 0 = -15 y = -15 The y-intercept is -15.
111 Holt McDougal Algebra 1
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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
The x-intercept is 12. The y-intercept is 300.
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
)
The x-intercept is 25. The y-intercept is 5.
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
112 Holt McDougal Algebra 1
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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
x = 103 The x-intercept is 103.
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
x = -2 The x-intercept is -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
x = 2 The x-intercept is 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
x = 2 The x-intercept is 2.
2y + 4x = 8 2y + 4(0) = 8 2y + 0 = 8 2y = 8
2y
_ 2
= 8 _ 2
y = 4 The y-intercept is 4.
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
x = 2 The x-intercept is 2.
4x - 2y = 8 4(0) - 2y = 8 0 - 2y = 8 -2y = 8
-2y
_ -2
= 8 _ -2
y = -4 The y-intercept is -4.
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.
113 Holt McDougal Algebra 1
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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
y = 12 The y-intercept is 12.
challenge and extend
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
x = 950 The x-intercept is 950.
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.
114 Holt McDougal Algebra 1
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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:
ExErCIsEsguided practice
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
115 Holt McDougal Algebra 1
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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
The slope is undefined.
8. The slope is negative. 9. The slope is undefined.
10. The slope is zero. 11. The slope is positive.
practice and problem Solving
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
The slope is undefined.
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.
116 Holt McDougal Algebra 1
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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.
challenge and extend
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
117 Holt McDougal Algebra 1
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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.
ExErCIsEsguided practice
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
Find the y-intercept: 5x + 9y = 90 5(0) + 9y = 90 9y = 90
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
practice and problem Solving
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
The slope is undefined.
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
118 Holt McDougal Algebra 1
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.
18. Find the x-intercept: 7x + 13y = 91 7x + 13(0) = 91 7x = 91
7x _ 7 = 91 _
7
x = 13
Find the y-intercept: 7x + 13y = 91 7(0) + 13y = 91 13y = 91
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
Find the y-intercept: 13x + 5y = 130 13(0) + 5y = 130 5y = 130
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
Find the y-intercept: 9x + 3y = 7 9(0) + 3y = 7 3y = 7
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
119 Holt McDougal Algebra 1
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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
for each ordered pair.
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.
120 Holt McDougal Algebra 1
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).
ExErCIsEsguided practice
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
it can be written in the form y = kx. The constant of variation is -4.
4. x + y = 0 ______ -x ___ -x y = -x This equation represents a direct variation because
it can be written in the form y = kx. The constant of variation is -1.
5. No; possible answer: the value of y _ x is not the same
for each ordered pair.
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
Graph the points and connect.
Cameron’s Wages
0 2 4 6 8
20
40
60
80
Time worked (h)
Am
ount
ear
ned
($)
practice and problem Solving
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
This equation represents a direct variation because it can be written in the form y = kx. The constant of
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.
13. Yes; possible answer: the value of y _ x is the same for
each ordered pair.
14. Yes; possible answer: the value of y _ x is the same for
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)
Graph the points and connect.
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.
121 Holt McDougal Algebra 1
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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)
Graph the points and connect.
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)
Graph the points and connect.
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)
Graph the points and connect.
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)
Graph the points and connect.
4
2
x
y
0 2 -2
4
1
The value of k is 4, and the graph shows that the slope of the line is 4.
24. y = kx 21 = k(7) 3 = k The equation is y = 3x.
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)
Graph the points and connect.
4
2 3
1
-4
x
y
4 2 -4 -2
The value of k is 3, and the graph shows that the slope of the line is 3.
25. y = kx 2 = k(1) 2 = k The equation is y = 2x.
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)
Graph the points and connect.
4
2 2
1
-4
x
y
4 2 -4 -2
The value of k is 2, and the graph shows that the slope of the line is 2.
122 Holt McDougal Algebra 1
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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)
Graph the points and connect.
2
x
y
0 4 2 -4
-8
1
-2
The value of k is -8, and the graph shows that the slope of the line is -8.
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)
Graph the points and connect.
2
4
6
x
y
0 4 2 -4
7
1 -2
The value of k is 7, and the graph shows that the slope of the line is 7.
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
The equation is y = - 2 _ 9 x.
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)
Graph the points and connect.
-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
The equation is y = 3 _ 2 x.
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)
Graph the points and connect.
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
.
123 Holt McDougal Algebra 1
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31. y = kx 4 = k(3)
4 _ 3 = k
The equation is y = 4 _ 3 x.
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)
Graph the points and connect.
2
-4
4
x
y
4 -4
4
3 -2
The value of k is 4 _ 3 , and the
graph shows that the slope of
the line is 4 _ 3 .
32. y = kx 1 = k(5)
1 _ 5 = k
The equation is y = 1 _ 5 x.
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)
Graph the points and connect.
2
-2
-4
4
x
y
0 6 -2 1
5
The value of k is 1 _ 5 , and the
graph shows that the slope of
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)
Graph the points and connect.
2
-1
-2
x
y
0 2 4 -2 -4
6
The value of k is -6, and the graph shows that the slope of the line is -6.
34. y = kx
1 _ 2 = k(-1)
- 1 _ 2 = k
The equation is 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)
Graph the points and connect.
4
2
-2
-2
-4
x
y
2 4 -2 -4
1
The value of k is - 1 _ 2 , and the
graph shows that the slope of
the line is - 1 _ 2 .
35. y = kx 2 = k(7) 2 __
7 = k
The equation is y = 2 _ 7 x.
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)
Graph the points and connect.
4
-2
2
-4
y
0 2
2
6 7 x
The value of k is 2 _ 7 , and the
graph shows that the slope of
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
124 Holt McDougal Algebra 1
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
challenge and extend
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
1. No; a constant change of +1 in x corresponds to different changes in y.
2. Yes; a constant change of +1 in x corresponds to a constant change of -2 in y.
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
125 Holt McDougal Algebra 1
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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
126 Holt McDougal Algebra 1
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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
and 1 unit right and plot another point. Draw the line connecting the two points.
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
ExErCIsEsguided practice
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
and 1 unit right and plot another point. Draw the line connecting the two points.
2
-2
x
y
0 2 -2
127 Holt McDougal Algebra 1
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12. 2x + y = 4 _______ -2x ____ -2x y = -2x + 4 Plot (0, 4). Count 2 units down
and 1 unit right and plot another point. Draw the line connecting the two points.
2
4
x
y
0 2 -2
13a. An equation is y = 18x + 10.
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.
practice and problem Solving
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
and 1 unit right and plot another point. Draw the line connecting the two points.
4
8
x
y
0 4 -4
26a. An equation is y = 35x + 175.
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.
128 Holt McDougal Algebra 1
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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 = -3 The y-intercept is -3.
y = mx + b
y = 1 _ 3 x + (-3)
y = 1 _ 3 x - 3
challenge and extend
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.)
129 Holt McDougal Algebra 1
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
ExErCIsEsguided practice
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
130 Holt McDougal Algebra 1
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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.
practice and problem Solving
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)
131 Holt McDougal Algebra 1
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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
132 Holt McDougal Algebra 1
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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.
challenge and extend
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
133 Holt McDougal Algebra 1
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
ExErCIsEsguided practice
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)
practice and problem Solving
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
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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.
challenge and extend
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;
135 Holt McDougal Algebra 1
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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.
136 Holt McDougal Algebra 1
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ExErCIsEsguided practice
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
These lines are perpendicular because the product of their slopes is -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
These lines are perpendicular because the product of their slopes is -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
These lines are perpendicular because the product of their slopes is -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
practice and problem Solving
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.
137 Holt McDougal Algebra 1
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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
These lines are perpendicular because the product of their slopes is -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
These lines are perpendicular because the product of their slopes is -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
These lines are perpendicular because the product of their slopes is -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
These lines are perpendicular because the product of their slopes is -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
These lines are perpendicular because the product of their slopes is -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.
138 Holt McDougal Algebra 1
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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
139 Holt McDougal Algebra 1
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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
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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.
141 Holt McDougal Algebra 1
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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.
challenge and extend
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)
142 Holt McDougal Algebra 1
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
ExErCIsEsguided practice
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
The graph of g(x) = x + 2 is the result of translating the graph of f(x) = x, 2 units up.
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).
143 Holt McDougal Algebra 1
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
To find g(x), multiply the value of m by -1.
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.
practice and problem Solving
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.
144 Holt McDougal Algebra 1
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 graph of g(x) = 2 _ 3 x + 2 is
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
To find g(x), multiply the value of m by -1.
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
To find g(x), multiply the value of m by -1.
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.
145 Holt McDougal Algebra 1
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.
challenge and extend
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.
146 Holt McDougal Algebra 1
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
These lines are perpendicular because the product of their slopes is -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
These lines are perpendicular because the product of their slopes is -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)
147 Holt McDougal Algebra 1
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 graph of g(x) = 1 _ 2 x + 4 is
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
4. No; a constant change of +2 in x corresponds to different changes in y.
5. Yes; a constant change of +1 in x corresponds to a constant change of +2 in y.
6. 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
x = 3 The x-intercept is 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
y = 3 The y-intercept is 3.
148 Holt McDougal Algebra 1
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
21. Find the x-intercept: 4x + 3y = 24 4x + 3(0) = 24 4x = 24
4x ___ 4 = 24 ___
4
x = 6
Find the y-intercept: 4x + 3y = 24 4(0) + 3y = 24 3y = 24
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
The slope is undefined.
32. m = y 2 - y 1
_______ x 2 - x 1
= -3 - (-3)
_________ 5 - 1
= 0 __ 4
= 0
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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
This equation represents a direct variation because it can be written in the form y = kx. The constant of
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 ($
)
Graph the points and connect.
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)
150 Holt McDougal Algebra 1
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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
These lines are perpendicular because the product of their slopes is -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
These lines are perpendicular because the product of their slopes is -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)
The graph of g(x) = x + 4 is the result of translating the graph of f(x) = x, 4 units up.
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.
151 Holt McDougal Algebra 1
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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.
152 Holt McDougal Algebra 1
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