Solution manual for graphical approach to college algebra 4th edition by john hornsby lial rockswold

Solution manual for Graphical Approach to College Algebra 4th Edition by
John Hornsby Lial Rockswold
Chapter 2: Analysis of Graphs of Functions
2.1: Graphs of Basic Functions and Relations; Symmetry
1.

1 q, q2

2.
3.

1 q, q2; 3 0, q2
10, 02

4.

3 0, q2; 3 0, q2

5.
6.

increases
1 q, 0 4 ; 3 0, q2

7.

x-axis

8.

even

9.

odd

10. y-axis; origin
11. The domain can be all real numbers, therefore the function is continuous for the interval: 1 q, q2.
12. The domain can be all real numbers, therefore the function is continuous for the interval: 1 q, q2.
13. The domain can only be values where x

0, therefore the function is continuous for the interval: 3 0, q2.

14. The domain can only be values where x

0, therefore The function is continuous for the interval: 1 q, 0 4 .

15. The domain can be all real numbers except 3, therefore the function is continuous for the interval: 1
q, 32; 1 3, q2.
16. The domain can be all real numbers except 1, therefore the function is continuous for the interval:
1 q, 12; 11, q2.
17. (a) The function is increasing for the interval: 3 3, q2.
(b) The function is decreasing for the interval: 1 q, 3 4 .
(c) The function is never constant, therefore: none
(d) The domain can be all real numbers, therefore the interval: 1 q, q2.
(e) The range can only be values where y

0, therefore the interval: 3 0, q2.

18. (a) The function is increasing for the interval: 3 4, q2.
(b) The function is decreasing for the interval: 1 q, 1 4 .

(c) The function is constant for the interval: 3 1, 4 4 .
(d) The domain can be all real numbers, therefore the interval: 1 q, q2.
(e) The range can only be values where y

3, therefore the interval: 3 3, q2.

19. (a) The function is increasing for the interval: 1 q, 1 4 .
(b) The function is decreasing for the interval: 3 4, q2.
(c) The function is constant for the interval: 3 1, 4 4 .
(d) The domain can be all real numbers, therefore the interval: 1 q, q2.
(e) The range can only be values where y

3, therefore the interval: 1 q, 3 4 .


58

CHAPTER 2

Analysis of Graphs of Functions

20. (a) The function never is increasing, therefore: none
(b) The function is always decreasing, therefore the interval: 1 q, q2.
(c) The function is never constant, therefore: none
(d) The domain can be all real numbers, therefore the interval: 1 q, q2.
(e) The range can be all real numbers, therefore the interval: 1 q, q2.
21. (a) The function never is increasing, therefore: none
(b) The function is decreasing for the intervals: 1 q, 2 4 ; 3 3, q2.
(c) The function is constant for the interval: 1 2, 32.
(d) The domain can be all real numbers, therefore the interval: 1 q, q2.

(e) The range can only be values where y

2, therefore the interval: 1 q, 1.5 4 ´ 3 2, q2.

1.5 or y

22. (a) The function is increasing for the interval: 13, q 2.
(b) The function is decreasing for the interval: 1 q, 32.
(c) The function is constant for the interval: 1 3, 3 4 .
(d) The domain can be all real numbers except 3, therefore the interval: 1 q, 32 ´ 1 3, q2.
(e) The range can only be values where y 7 1, therefore the interval: 11, q2.
x5, See Figure 23. As x increases for the interval: 1 q, q2, y increases, therefore increasing.

23. Graph f 1x2

24. Graph f 1x2x3, See Figure 24. As x increases for the interval: 1 q, q2, y decreases, therefore decreasing.
x4, See Figure 25. As x increases for the interval: 1 q, 0 4 , y decreases, therefore decreasing.

25. Graph f 1x2

x4, See Figure 26. As x increases for the interval: 3 0, q2, y increases, therefore increasing.

26. Graph f 1x2
10, 10 by

Xscl

3

1

4

10, 10

3

Yscl

10, 10 by

4

1

Xscl

3

1

4

10, 10

3

Yscl

10, 10 by

4

1

Xscl

3

1

4

10, 10

3

Yscl

4

10, 10 by

1

Xscl

3

1

4

10, 10

Yscl

3

4

1

Figure 23
Figure 24
Figure 25
Figure 26
27. Graph f 1x210x 0 See Figure 27. As x increases for the interval: 1 q, 0 4 , y increases, therefore increasing.
28. Graph f 1x210x 0 , See Figure 28. As x increases for the interval: 3 0, q2, y decreases, therefore decreasing.
29. Graph f 1x23 x, See Figure 29. As x increases for the interval: 1 q, q2, y decreases, therefore decreasing.
30. Graph f 1x2x, See Figure 30. As x increases for the interval: 3 0, q2, y decreases, therefore decreasing.
10, 10 by

Xscl

3

1

4

10, 10

3

Yscl

Figure 27

10, 10 by

4

1

Xscl

3

1

4

10, 10

3

Yscl

Figure 28

10, 10 by

4

1

Xscl

3

1

4

10, 10

3

Yscl

Figure 29

4

10, 10 by

1

Xscl

3

1

4

10, 10

3

Figure 30

Yscl

4

1


Graphs of Basic Functions and Relations; Symmetry

SECTION 2.1

59

3
31. Graph f 1x2 1 2 x , See Figure 31. As x increases for the interval: 1 q, q2, y decreases, therefore decreasing.
f 1x2 x
3 q2

2
32. Graph
As x increases for the interval: 1,
2x, See Figure 32.
, y increases, therefore increasing.

33. Graph f 1x2 2
34. Graph f 1x2 0
10, 10

3

Xscl

1

See Figure 33. As x increases for the interval: 1 q, 0 4 , y increases, therefore increasing.

x

x 1 0 , See Figure 34. As x increases for the interval: 1 q, 1 4 , y decreases, therefore decreasing.

by 10, 10

4

Yscl

3

10, 10 by

4

1

Xscl

3

Figure 31

1

4

10, 10

3

Yscl

4

10, 10 by

3

1

Xscl

Figure 32

35. (a) No

(b) Yes

(c) No

36. (a) Yes

(b) No

(c) No

37. (a) Yes

(b) No

(c) No

38. (a) No

(b) No

(c) Yes

39. (a) Yes

(b) Yes

(c) Yes

40. (a) Yes

(b) Yes

(c) Yes

41. (a) No

(b) No

(c) Yes

1

4

10, 10

3

Yscl

Figure 33

4

10, 10 by

1

3

Xscl

1

4

10, 10

3

Yscl

Figure 34

42. (a) No
(b) Yes
(c) No
43. If f is an even function then f 1 x2 f 1x2 or opposite domains have the same range. See Figure 43.
44. If g is an odd function then g1 x2g1x2 or opposite domains have the opposite range. See Figure 44.
x g(x)
–5 13
–3
1
–2 –5

0
0
2
5
3 –1
5 –13

x ƒ(x)
–3
21
–2 –12
–1 –25

1

–25

2 –12
3 21

Figure 43
Figure 44
45. (a) Since f 1 x2 f 1x2, this is an even function and is symmetric with respect to the y-axis. See Figure 45a.
(b) Since f 1 x2f 1x2, this is an odd function and is symmetric with respect to the origin. See Figure 45b.
y

y

2

2

0

x

x
0

2

Figure 45a

2

Figure 45b

4

1


60

CHAPTER 2

Analysis of Graphs of Functions

46. (a) Since this is an odd function the graph is symmetric with respect to the origin. See Figure 46a.
(b) Since this is an even function the graph is symmetric with respect to the y-axis. See Figure 46b.

y

y

2

2

x

0

x

0

2

Figure 46a
47. Since f 1 x2 f 1x2, it is even.

2

Figure 46b

48. Since f 1 x2f 1x2, it is odd.
49. If f 1x2 x4 7x2 6, then f 1 x2 1 x24 71 x22 6 1 f 1 x2 x4 7x2 6. Since f 1 x2 f 1x2, the function is even.
50. If f 1x2 2x6 8x2 , then f 1 x2 21 x26 81 x22 1 f 1 x2 2x6 8x2. Since f 1 x2 f 1x2, the function is even.

51. If f 1x2 x6 4x4 5, then f 1 x2 1 x26 41 x24 5 1 f 1 x2 x6 4x4 5. Since f 1 x2 f 1x2, the function is even.

52. If f 1x2

82,then f 1 x2

8. Since2 f 1 x2

f 1x2, the function2 is even.

53. If f 1x20 5x 0 , then f 1 x20 51 x 0 1 f 1 x20 5x 0 . Since f 1 x2f 1x2, the function is even.
54. If f 1x2x2 1, then f 1 x21 x221 1 f 1 x2x2 1. Since f 1 x2f 1x2, the function is even.
55. If f 1x2 3x3 x, then f 1 x2 31 x23 1 x2 1 f 1 x2
3x3
x and
f 1x213x3x2 1 f 1x23x3
56. If f 1x2

x5

f 1x21 x52x3
57. If f 1x2

3x5

f 1x213x5x3
58. If f 1x2

x3

3x, then f 1 x2

2x3

3x2 1 f 1x2

x52x3

7x, then f 1 x2

x3

7x2 1 f 1x23x5

x3

4x, then f 1 x2

1 x23

f 1x21x34x2 1 f 1x2x3

1 x25

21 x23

31 x2 1 f 1 x2

x5

2x , then f 1 x2

31 x25

1 x23

71 x2 1 f 1 x2

3x5

41 x2 1 f 1 x2

1
21 x2 1 f 1 x2

x3

1
2x and f 1x2a

f 1x2a4x

1
1

x2

xb

x3

7x and

7x. Since f 1 x2f 1x2, the function is odd.

1
4xx, then f 1

3x and

3x. Since f 1 x2f 1x2, the function is odd.

4x and

1
2x b 1f 1x2

f 1 x2 f 1x2, the function is odd.
60. If f 1x2

2x3

4x. Since f 1 x2f 1x2, the function is odd.

1
59. If f 1x2

x. Since f 1 x2f 1x2, the function is odd.

41

1 f 1x2

x2

4x

1

1

x

1
2

1 f 1 x2 4xx and

x. Since f 1 x2 f 1x2, the function is odd.

1
2x Since


61. If f 1x2

Graphs of Basic Functions and Relations; Symmetry
2x, then f 1 x2
1 x23 21 x2 1 f 1 x2 x3
2x and

x3

f 1x21 x32x2 1 f 1x2
origin. Graph f 1x2
62. If f 1x2

x3
x

3

2x. The graph supports symmetry with respect to the origin.

f 1x21x52x32 1 f 1x2x5
the origin.
f 1x2

3

5

f 12x2

4

.5 x

1 x25

21 x23 1 f 1 x2

x5

2x3 and

2x3. Since f 1 x2f 1x2, the function is symmetric with respect to

Graph
63. If

x

. The graph supports symmetry with respect to the origin.

2x x2.5 1 x2

4

f1

then

2x1,

21 x2

2

1 1 f 1 x2

.5 x

4

24x

2

2

Since

1.

f 1 x2 f 1x2, the function is symmetric with respect to the y-axis. Graph f 1x2.5 x
supports symmetry with respect to the y-axis.

2x

2
1, then f 1 x2 .75 1 x22
64. If f 1x2 .75 x 0 x 0
1 x 2 01 1 f 1 x2
f 1 x2 f 1x2, the function is symmetric with respect to the y-axis. Graph f 1x2.75 x
supports symmetry with respect to the y-axis.
0

1. The graph

.75 x2 4 0 x 0 1.

Since
0 x 0 1. The graph

3
3
x 3, then f 1 x2
x 3 and f 1x2
1x3x
1 x231 x23
65. If f 1x2 x
1 f 1 x2 x
1 f 1x2x3 x 3. Since f 1x2f 1 x2f 1x2, the function is not symmetric with respect to the y-axis

or origin. Graph f 1x2
66. If f 1x2 x4
1 f 1x2x4

5x
5x

x3

f 1x2x6

32

3. The graph supports no symmetry with respect to the y-axis or origin.
1 x24

51 x2

21f1

x2

x4

5x

2 and f 1x2

1x4

5x

22

2. Since f 1x2f 1 x2f 1x2, the function is not symmetric with respect to the y-axis
5x

4x3, then f 1 x2

x6

2. The graph supports no symmetry with respect to the y-axis or origin.

1 x26

41 x23 1 f 1 x2

x6

4x3 and f 1x2

1x6

4x32 1

4x3 Since f 1x2f 1 x2f 1x2, the function is not symmetric with respect to the y-axis or ori-

gin. Graph f 1x2
68. If f 1x2

x

2, then f 1 x2

or origin. Graph f 1x2 x4
67. If f 1x2

61

2x. Since f 1 x2f 1x2, the function is symmetric with respect to the

2x3, then f 1 x2

x5

SECTION 2.1

x6

4x3. The graph supports no symmetry with respect to the y-axis or origin.

3x, then f 1 x2

x3

1 x23

31 x2 1 f 1 x2

x3

3x and

f 1x2 1x3 3x2 1 f 1x2 x3 3x. Since f 1 x2 f 1x2, the function is symmetric with respect to the origin. Graph f
1x2 x3 3x. The graph supports symmetry with respect to the origin.
69. If f 1x2 6, then f 1 x2 6. Since f 1 x2 f 1x2, the function is symmetric with respect to the y-axis. Graph f
1x2 6. The graph supports symmetry with respect to the y-axis.
70. If f 1x2 0 x 0 0 x 0 , then f 1 x2 0 1 x2 0

0

x 0 1 f 1 x2 0 x 0 .

Since f 1 x2

f 1x2, the function is

symmetric with respect to the y-axis. Graph f 1x2 0 x 0 . The graph supports symmetry with respect to the y-axis.
1
1
1
1
1
1 12
1 f 1 x2 4x3 and f1x2a
71. If f 1x2
4x3 , then f 1 x2
4 x 3
4x3 b 1 f 1x2
4x3 . Since
f 1 x2 f 1x2, the function is symmetric with respect to the origin. Graph f 1x2
supports symmetry with respect to the origin.
72.

If f 1x2

2

x2

1 f 1x2

x, then f 1 x2

2 1 x2

2

1 f 1 x2

2

function is symmetric with respect to the y-axis. Graph f 1x22x .
respect to the y-axis.

x2 1 f 1 x2 x.

4x3 . The graph

Since f 1 x2

f 1x2, the

2

The graph supports symmetry with


62
CHAPTER 2
Analysis of Graphs of Functions
73. (a) Functions where f 1 x2 f 1x2 are even, therefore exercises: 63, 64, 69, 70, and 72 are even.
(b) Functions where f 1 x2f 1x2 are odd, therefore exercises: 61, 62, 68, and 71 are odd.
(c) Functions where f 1x2 f 1 x2 f 1x2 are neither odd or even, therefore exercises: 65, 66, and 67 are
neither odd or even.
74. Answers may vary. If a function f is even, then f 1x2 f 1 x2 for all x in the domain. Its graph is symmetric
with respect to the y-axis. If a function f is odd, then f 1 x2 f 1x2 for all x in the domain. Its graph is

symmetric with respect to the origin.

2.2: Vertical and Horizontal Shifts of Graphs
1. The equation y
2. The equation y

3

shifted 3 units upward is: y x2 3.
shifted 2 units downward is: y x3 2.

1 x shifted 4 units downward is: y

3. The equation y
4. The equation y

1x

3

0

4.

3

x shifted 6 units upward is: y1x

5. The equation y
y

6. The equation
7. The equation y

2

x
x

6

x 0 shifted 4 units to the right is: y0 x4 0 .
3

0

x

x

0

shifted 7 units to the left is: y

30

0

shifted 3 units to the left is: y

x

1x

3

.

72 .

8. The equation y
1 x shifted 9 units to the right is: y
1x
9. Shift the graph of f 4 units upward to obtain the graph of g.

9.

10. Shift the graph of f 4 units to the left to obtain the graph of g.
11. The equation y
12. The equation y

x2
1x

3 is y x2 shifted 3 units downward, therefore graph B.
322 is y x2 shifted 3 units to the right, therefore graph C.

13. The equation y

1x

322 is y

x2 shifted 3 units to the left, therefore graph A.

14. The equation y0 x 04 is y0 x 0 shifted 4 units upward, therefore graph A.
15. The equation y0 x4 03 is y0 x 0 shifted 4 units to the left and 3 units downward, therefore graph B.
16. The equation y0 x4 03 is y0 x 0 shifted 4 units to the right and 3 units downward, therefore graph C.
17. The equation y

1x

323 is y

x3 shifted 3 units to the right, therefore graph C.

18. The equation y

1x

223

4 is y

x3 shifted 2 units to the right and 4 units downward, therefore graph A.

19. The equation y

1x

223

4 is y

x3 shifted 2 units to the left and 4 units downward, therefore graph B.

20. If y 0 x h 0 k with h 6 0 and k 6 0, then the graph of y 0 x 0 is shifted to the left h units and k units
downward. This would place the vertex or lowest point of the absolute value graph in the third quadrant.
21. For the equation y x2 , the Domain is: 1 q, q2 and the Range is: 3 0, q2. Shifting this 3 units downward
Domain:
gives us:
22. For the equation y

x

gives us:

(a)

2

(b) Range:
1q,

q2

33,

q2

, the Domain is: 1 q, q2 and the Range is: 3 0, q2. Shifting this 3 units to the right

(a) Domain: 1q, q2

(b) Range: 30, q2


Vertical and Horizontal Shifts of Graphs
23. For the equation y

(a) Domain: 1 q, q2

(b) Range: 3 3, q2

x 0 , the Domain is: 1 q, q2 and the Range is: 3 0, q2. Shifting this 4 units to the right

0

and 3 units downward gives us:
3

(a) Domain: 1 q , q2

(b) Range:

Domain:
(a)

3

1

,
q q2

(b) Range:

,
1 q q2

and 4 units downward gives us:
(a) Domain: 1 q, q2
27. Using Y2 Y1 k and x 0, we get 19 15 k 1 k 4.

(b) Range: 1 q, q2

28. Using Y2 Y1 k and x 0, we get 53 k 1 k2.
29. From the graphs 16, 22 is a point on Y1 and 16, 12 a point on Y2. Using Y2
2

3, q2

, the Domain is: 1 q, q2 and the Range is: 1 q, q2. Shifting this 2 units to the right

For the equation y x

1

3

x , the Domain is: 1 q, q2 and the Range is: 1 q, q2. Shifting this 3 units to the right

25. For the equation y
26. gives us:

63

x 0 , the Domain is: 1 q, q2 and the Range is: 3 0, q2. Shifting this 4 units to the left

0

and 3 units downward gives us:
24. For the equation y

SECTION 2.2

Y1

k and x

6, we get

k 1 k3.

30. From the graphs 1 4, 32
8 3 k 1 k 5.

is a point on Y1 and 1 4, 82 a point on Y2. Using Y2

122 is the graph of the equation y

31. The graph of yx

x2

Y1

k and x4, we get

shifted 1 unit to the right. See Figure 31.

1x shifted 2 units to the left. See Figure 32.
x3 shifted 1 unit upward. See Figure 33.

32. The graph of y1x 2 is the graph of the equation y
33. The graph of y x3 1 is the graph of the equation y
y

y

y

2
1

1

x

0

x
–2

1

x

0

0

1

Figure 31
Figure 32
Figure 33
34. The graph of y0 x2 0 is the graph of the equation y0 x 0 shifted 2 units to the left. See Figure 34.
1x

35. The graph of y

123 is the graph of the equation y x3 shifted 1 unit to the right. See Figure 35.

0 x 0 3 is the graph of the equation y

36. The graph of y
36.
y

0 x 0 shifted 3 units downward.

y

See Figure

y

2
x
–2

0

0
–1

x
1

3
–3

Figure 34

Figure 35

x

0
–3

Figure 36


1

Analysis of Graphs of Functions 1

37. The graph of y 1x 2 1 is the graph of the equation y 1x shifted 2 units to the right and 1 unit downward. See
Figure 37.
38.
The graph of yx 3 4 is the graph of the equation yx shifted 3 units to the left and 4

64CHAPTER2

units

downward. See Figure 38.
39. The graph of y 1x 222 3 is the graph of the equation y x2 shifted 2 units to the left and 3 units upward. See
Figure 39.
y

y

y

3
x
0
–1

x

0

–3

3

–2

x

0

–4

Figure 37
Figure 38
Figure 39
40. The graph of y 1x 422 4 is the graph of the equation y x2 shifted 4 units to the right and 4 units downward.
See Figure 40.
41. The graph of y 0 x 4 0 2 is the graph of the equation y 0 x 0 shifted 4 units to the left and 2 units
downward. See Figure 41.
42. The graph of y 1x 323 1 is the graph of the equation y x3 shifted 3 units to the left and 1 unit downward.
See Figure 42.
y

y

y

2
x
0

2
x

0

– 6 – 4 –2
–2

2 4 6

–4

x
–2

0

Figure 40
43. Since h and k are positive, the equation y

1x

Figure 41
Figure 42

2
h2
k is y x shifted to the right and down, therefore: B.

44. Since h and k are positive, the equation y

1x

h22

k is y x2 shifted to the left and down, therefore: D.

45. Since h and k are positive, the equation y

1x

h22

k is y x2 shifted to the left and up, therefore: A.

46. Since h and k are positive, the equation y

1x

h22

k is y x2 shifted to the right and up, therefore: C.

2

47. The equation y f 1x2 2 is y f 1x2 shifted up 2 units or add 2 to the y-coordinate of each point as follows:
1 3, 22 1 1 3, 02; 1 1, 42 1 1 1, 62; 15, 02 1 15, 22. See Figure 47.


Vertical and Horizontal Shifts of Graphs
SECTION 2.2
48. The equation y f 1x2 2 is y f 1x2 shifted down 2 units or subtract 2 from the y-coordinate of each point as

65

follows: 1 3, 22 1 1 3, 4 2; 1 1, 42 1 1 1, 22; 15, 02 1 15, 22. See Figure 48.
y

y

(– 1, 6)
(–1, 2)

(5, 2)
x

(– 3, 0)

x

0

0

(5, –2)

(–3, –4)

Figure 47
Figure 48
49. The equation y f 1x 22 is y f 1x2 shifted left 2 units or subtract 2 from the x-coordinate of each point as
follows: 1 3, 22 1 1 5, 22; 1 1, 42 1 1 3, 42; 15, 02 1 13, 02. See Figure 49.
50. The equation y

f 1x 22 is y f 1x2 shifted right 2 units or add 2 to the x-coordinate of each point as

follows: 1 3, 22 1 1 1, 22; 1 1, 42 1 11, 42; 15, 02 1 17, 02. See Figure 50.
y

y

(– 3, 4)

(1, 4)
(3, 0)

(7, 0)
x

x

0

0

(– 5, – 2)

(–1, –2)

Figure 49

Figure 50

51. The graph is the basic function y
is: y

1x

422

3.

x2 translated 4 units to the left and 3 units up, therefore the new equation

The equation is now increasing for the interval: (a) 3 4, q2 and decreasing for the

interval: (b) 1 q, 4 4 .
52. The graph is the basic function y

1

x

translated 5 units to the left, therefore the new equation is: y1

The equation is now increasing for the3 interval: (a) 3 5, q2 and does not decrease, therefore: (b)

53. The graph is the basic function y x

translated 5 units down, therefore the new equation is: y

54. The graph is the basic function y

x0

0

3

5.

none.

translated 10 units to the left, therefore the new equation is: y0 x10 0 .

The equation is now increasing for the interval: (a) 3 10, q2 and decreasing for the interval: (b) 1 q, 10 4 .
55. The graph is the basic function y

1 x translated 2 units to the right and 1 unit up, therefore the new equation

1. The equation is now increasing for the interval: (a) 3 2, q2 and does not decrease,
is: y1
x 2
therefore: (b) none.
2
56. The graph is the basic function y x translated 2 units to the right and 3 units down, therefore the new

equation is: y

1x

222

the interval: (b) 1 q, 2 4 .

5.

none.

x

The equation is now increasing for the interval: (a) 1 q, q2 and does not decrease, therefore: (b)

x

3. The equation is now increasing for the interval: (a) 3 2, q2 and decreasing for


66
CHAPTER 2
57. (a) f 1x2 0: 53, 46

Analysis of Graphs of Functions

(b) f 1x2 7 0: for1 the intervals 1 q, 32 ´ 14, q2.
(c) f 1x2 6 0: for the interval 13,142.
58. (a) f 1x2

0: 5 261

(b) f 1x2 7 0: for the interval 1

2, q2.

(c) f 1x2 6 0: for the interval 1 q, 22.
59. (a) f 1x2 0: 5 4, 56
(b) f 1x20: for the intervals 1 q, 4 4 ´ 3 5, q2.
(c) f 1x20: for the interval 3 4, 5 4 .
60. (a) f 1x2

0: never, therefore: .

(b) f 1x20: for the interval 3 1, q2.
(c) f 1x20: never, therefore: .
61. The translation is 3 units to the left and 1 unit up, therefore the new equation is: y0 x3 01. The form
y
x h0 k
2
0
x
3
1
when: h3 and k 1.
will equal y
0

0

62. The equation y

has a Domain: 1 q, q2 and a Range: 3 0, q2. After the translation the Domain is still:

x

1 q, q2, but now the Range is: 3 38, q2, a positive or upward shift of 38 units. Therefore, the horizontal shift
can be any number of units, but the vertical shift is up 38. This makes h any real number and k 38.
63. (a) Since 0 corresponds to 1998, our equation using exact years would be: y
895.512006 19982 14,709 1 y

(b) y

14,709 1 y

7164

895.51x

19982

299.81x

19982

14,709.

$21,873

64. (a) Since 0 corresponds to 1998, our equation using exact years would be: y

5249.1.

(b) y
299.812007 19982 5249.1
1 y 2698.2 5249.1 1 y $ 7947.3 billion.
65. (a) Enter the year in L1 and enter tuition and fees in L 2 . The year 1991 corresponds to x 0 and so on. The
regression equation is: y 190x 2071.3.
(b) Since x 0 corresponds to 1991, the equation when the exact year is entered is:
y

1901x

19912

2071.3

(c) y
19012008 19912
2071.3 1 y 3230 2071.3 1 y $5300
66. (a) Enter the year in L1 and enter the percent of women in the workforce in L 2 . The year 1965 corresponds to
x 0 and so on.
The regression equation is: y .6162x 40.6167.
(b) Since x 0 corresponds to 1965, the equation when the exact year is entered is:
y

.61621x

19652

40.6167

(c) y
.616212010 19652 40.6167 1 y
67. See Figure 67.
1 22 1 m 4 2
3 1
2
69. Using slope-intercept form yields: y1 2
68. m

70. 11,

40.6167 1 y

68.3%

2

42

8

2
71. m

27.729

3

1

2 11

4

6 and

3, 2

1 1m

2

2

21x

32 1 y1

1

1, 4 and 3, 8

6
2

2

2

2x

6 1 y1

2x

4


Stretching, Shrinking, and Reflecting Graphs
72. Using slope-intercept form yields: y2
73. Graph y1

2x 4 and y2

4

12 1 y2

21x

4

2x

2 1 y2

SECTION 2.3

2x

67

2

2x 2. See Figure 73. The graph y2 can be obtained by shifting the graph of y1

upward 6 units. The constant, 6, comes from the 6 we added to each y-value in Exercise 70.
y
3 10, 10 4 by 3 10, 10 4
Xscl

1

Yscl

1

(3, 2)
x

0

(1, – 2)

Figure 67

Figure 73

74. c; c; the same as; upward (or positive vertical)

2.3: Stretching, Shrinking, and Reflecting Graphs
1. The function y
2.

The function y

x 2 vertically stretched by a factor of 2 is: y 2x2
1

1

1

.

1

x3 vertically shrunk by a factor of is: y 1 x3.

22
3.

The function yx reflected across the y-axis is: yx .

4.

The function y3 x reflected across the x-axis is: y3 x .

5. The function y

x 0 vertically stretched by a factor of 3 and reflected across the x-axis is: y3 0
1

0

x0.

1

x 0 vertically shrunk by a factor of 3 and reflected across the y-axis is: y
6. The function y
7. The function y x3 vertically shrunk by a factor of .25 and reflected across the y-axis is:

30 x 0 .

0

y

3
.25 1 x23 or y.25 x .

1x vertically shrunk by a factor of .2 and reflected across the x-axis is: y.2 1
x 3 (y1 shifted up 3 units), and y3 x 3 (y1 shifted down 3 units). See Figure 9.

8. The function y
Graph y1 x, y2

9.

10. Graph y1
11. Graph y1
x

x3, y2
0x0,

x3 4 (y1 shifted up 4 units), and y3
y2 0 x

x3

x.

4 (y1 shifted down 4 units). See Figure 10.
3 0 (y1 shifted right 3 units), and y3
0

3 0 (y1 shifted left 3 units). See Figure 11.
y

y

y

2

y2

y

y1
3
0

y
1

y3

y3
x

y y y
3

4
0
–4

x

Figure 10

2

x
–3

0

3

–3

Figure 9

1

Figure 11


68

CHAPTER 2

12. Graph y1

x0

0

13. Graph y11

,

Analysis of Graphs of Functions
x 03 (y1 shifted down 3 units), and y3

y2

0

x 03 (y1 shifted up 3 units). See Figure 12.

0

x , y2

1x

(y1 shifted left 6 units), and y3 1 x

6

6 (y1 shifted right 6 units). See Figure 13.

2 0 x 0 (y1 stretched vertically by a factor of 2), and y3
14. Graph y1
x 0 , y2
a factor of 2.5). See Figure 14.

2.5 0 x 0 (y1 stretched vertically by

0

y

y

y

y

y

3 y2

3

yy

2 1

y1

3

y

y2
x

0
–3

–6

y1

3

x

0

Figure 12

Figure 13

3

x

0

6

Figure 14

3

3

15. Graph y11x,
y21x
(y1 reflected across the x-axis), and y32 1x
(y1 reflected across the x-axis

and stretched vertically by a factor of 2). See Figure 15.
16. Graph y1 x2, y2 1x 222 1 (y1 shifted right 2 units and up 1 unit), and y3 1x 222 (y1 shifted left 2 units and
reflected across the x-axis). See Figure 16.
0x0,

17. Graph y1
2,

20x 10

y2

1 (y1 reflected across the x-axis, stretched vertically by a factor of

1

shifted right 1 unit, and shifted up 1 unit,), and y3
factor of

0x0

4 (y1 reflected across the x-axis, shrunk by 2

1
, and shifted down 4 units). See Figure 17.

2
y

y

y

y

2

1

–2

y1
0

0

2

y1

4
2
2

x

x

y

6

y

x

–6 –3

2

y3

y3

0
–2

4 6

–4

y2

y

–6
Figure 15

18. Graph y11
x , y2 1

See Figure 18.
x2

19. Graph y1

1
of 2 ), and y3
20. Graph y1 3

0

Figure 16

x (y1 reflected across the x-axis), and y3

1 (which is y
12x22

x2 shifted down 1 unit),

3

Figure 17

y2

1
a 2 xb

1 x (y1 reflected across the y-axis).

2

1 (y1 shrunk vertically by a factor

1 (y1 stretched vertically by a factor of 22 or 4). See Figure 19.

x 0 (which is y0 x 0 reflected across the x-axis and shifted up 3 units), y230 3x 0
1

(y1 stretched vertically by a factor of 3), and y3

3

0

1

3 x 0 (y1 shrunk vertically by a factor of 3 ). See Figure 20.


Stretching, Shrinking, and Reflecting Graphs
y

SECTION 2.3

y

y

y3

y

y

69

y

1

1

3

4

y2
x

0

x

0

–2

x

0

–8

2

y3
8

–1

y2

y

y

1

2

Figure 18
3

Figure 19
3

Figure 20
3

21. Graph y11x , y2

1 x (y1 reflected across the y-axis), and y3
y-axis and shifted right 1 unit). See Figure 21.

1 1x

1 2 (y1 reflected across the

y

y3

y1

1

x
0

1

y2

Figure 21
22. Since y f 1x2 is symmetric with respect to the y-axis, for every 1x, y2 on the graph, 1 x, y2 is also on the graph.
Reflection across the y-axis reflect onto itself and will not change the graph. It will be the same.
23. 4; x
24. 6; x
1
25. 2; left;

4

; x ; 3; downward (or negative)

2
26. y ; ; x ; 6 ; upward (or positive) 5
27. 3; right; 6
28. 2; left; .5
29. The function y

x2 is vertically shrunk by a factor of

1

and shifted 7 units down, therefore: y
2

1

x2

7.

2

30. The function y x13 is vertically stretched by a factor of 3, reflected across the x-axis, and shifted 8 units
upward, therefore:1y 3x3 8.
31. The function y 1x is shifted 3 units right, vertically stretched by a factor of 4.5, and shifted 6 units down,
therefore: y 4.5 1x 3 6 .
32. The function y 3 x is shifted 2 units left, vertically stretched by a factor of 1.5, and shifted 8 units upward,

therefore: y 1.5 3 x 2 8.


70

CHAPTER 2

Analysis of Graphs of Functions
2

33. The graph y f 1x2 x has been reflected across the x-axis, shifted 5 units to the right, and shifted 2 units
downward, therefore the equation of g1x2 is: g1x2 1x 522 2.
34. The graph y f 1x2 1x3 has been shifted 4 units1 to the right and shifted 3 units upward, therefore the equation of
g1x2 is: g1x2 1x 423 3.
35. The function f 1x21x

3 1 21is f 1x2

x1shifted 3 units right and 2 units upward1. See Figure 35.

36. The function f 1x20 x2 03 is f 1x20 x 0 shifted 2 units left and 3 units downward. See Figure 36.
37. The function f 1x22x2 x is f 1x2x stretched vertically by a factor of 2. See Figure 37.
y

y

y
8

–5

2

–1

x

0
1

0

x

0

x

8

3

Figure 35

Figure 36

Figure 37

1
38. The function f 1x2

See Figure 38.

1

2 1x 22

2

is f 1x2

x

2

39. The function f 1x2 0 2x 0 2 0 x 0 is f 1x2 0

shifted 2 units left and shrunk vertically by a factor of 2 .
x 0 stretched vertically by a factor of 2.

1
40. The function f 1x2

See Figure 39.

1

2 0x 0

is f 1x2 0 x 0 shrunk vertically by a factor of 2 . See Figure 40.

y

y

y

3
3
2
x
–2

x

0

0

Figure 38
41. The function f 1x2
42. The function f 1x2 2 21x

Figure 39
11x is f 1x2
22

3

x
0

3

Figure 40

1x reflected across the x-axis and shifted 1 unit upward. See Figure 41.

1 is f 1x2

1x shifted 2 units right, stretched vertically by a factor of 2,

and shifted 1 unit downward. See Figure 42.

43. The function f 1x2
11 x 2 1x 12
and shifted 1 unit right. See Figure 43.

is f 1x21x reflected across both the x-axis and the y-axis


Stretching, Shrinking, and Reflecting Graphs
y

SECTION 2.3

y

71

y

8
1
x

0

0

1
x

0

Figure 42
1 is f 1x2

1x

–2

8

Figure 41
44. The function f 1x2 1 x

x
1

Figure 43

reflected across the y-axis and shifted 1 unit downward.

See Figure 44.
45. The function f 1x2 1 1x

12 is f 1x2

46. The function f 1x2 21 1x 3 2 is f 1x2
shifted 2 units upward. See Figure 46.

1x

reflected across the y-axis and shifted 1 unit left. See Figure 45.

1x reflected across the y-axis, shifted 3 units right, and

y

y

x
–1

–1

0

–1

y

x

0

2
x
0

3

Figure 44

Figure 45
3

Figure 46

3

47. The function f 1x21x12 is f 1x2

x shifted 1 unit right. See Figure 47.

48. The function f 1x21x223 is f 1x2
49. The function f 1x2
x3 is f 1x2

x3 shifted 2 units left. See Figure 48.

x3 reflected across the x-axis. See Figure 49.

y

0
–1

y

x
1

y

x
–2

0

x
0

2

–2

Figure 47

Figure 48

Figure 49


72

CHAPTER 2

Analysis of Graphs of Functions
3

50. The function f 1x2 1 x2

1 is f 1x2

x3 reflected across the y-axis and shifted 1 unit upward. See Figure 50.

y

3
x
0

2

Figure 50
51. (a) The equation yf 1x2 is y f 1x2
reflected across the x-axis. See Figure 51a.
(b) The equation y

f 1 x2 is y f 1x2

(c) The equation y 2 f 1x2 is y

(d) From the graph f 102

reflected across the y-axis. See Figure 51b.

f 1x2 stretched vertically by a factor of 2. See Figure 51c.

1.
y

y

y

(6, 6)
(–6, 3)
(–4, 1)

0

(–2, –1)
(–3, –2)

(3, 2)
(2, 1)

x

(4, 2)

(–2, 2)

x

0

(4, –1)

(–3, 4)
x

0

(6, –3)

Figure 51a
52. (a) The equation yf 1x2 is y f 1x2

Figure 51b

Figure 51c

reflected across the x-axis. See Figure 52a.
(b) The equation y

f 1 x2 is y f 1x2

(c) The equation y 3 f 1x2 is y

(d) From the graph f 142

reflected across the y-axis. See Figure 52b.

f 1x2 stretched vertically by a factor of 3. See Figure 52c.

1.
y

y

y

(– 2, 9)

(–4, 0)
(2, 3)

x

0

(2, –1) (6, –1)
(–2, –3)

(–6, 1)

x

0

x
0

(4, 0)

Figure 52a

(2, 3) (6, 3)

(–2, 1)

Figure 52b

(– 4, 0)

Figure 52c


Stretching, Shrinking, and Reflecting Graphs
53. (a) The equation yf 1x2 is y f 1x2 reflected across the x-axis. See Figure 53a.

SECTION 2.3

73

f 1 x2 is y f 1x2 reflected across the y-axis. See Figure 53b.

(b) The equation y

(c) The equation yf 1x 12 is y f 1x2 shifted 1 unit to the left. See Figure 53c.
(d) From the graph, there are two x-intercepts: 1 and 4.
y

(–1, 3)

y

(0, 2.5)

(–2, 0)

y

(–3.5, 1.5)

(2.5, 1.5)

(3, 0)
x

0

0

(–3, 0)
(0, –2.5)

(3.5, –1.5)

x

(2, 0)
(1, –3)

(c)

The equation y

1

(2, 0)
(– 1, – 2.5)

(– 2, – 3)

Figure 53a
Figure 53b
54. (a) The equation yf 1x2 is y f 1x2 reflected across the x-axis. See Figure 54a.
(b) The equation y

x

0

(– 3, 0)

Figure 53c

f 1 x2 is y f 1x2 reflected across the y-axis. See Figure 54b.

f 1x2 is y f 1x2 shrunk vertically by a factor of 1. See Figure 54c.

2

2

(d) From the graph f 1x2 6 0 for the interval: 1 q, 02.
y

y

y

(–2, 32)

(–2, 32)
(2, 16)
x

0

0

x

0

x

(–2, –16)
(2, –32)

(2, –32)

Figure 54a
Figure 54b
55. (a) The equation yf 1x2 is y f 1x2 reflected across the x-axis. See Figure 55a.

(b) The equation y f a

(c) The equation y

Figure 54c

1

xb is y f 1x2 stretched horizontally by a factor of 3. See Figure 55b. 3
.5 f 1x2 is y f 1x2 shrunk vertically by a factor of .5. See Figure 55c.

(d) From the graph, symmetry with respect to the origin.
y

y

(–1, 0)

y

(–3, 0)

(–2, 0)

(3, 0)

x

0

(1, 0)

Figure 55a

(2, 0)

x

0

–1

1
(–1, 0)

1

(2, 0)
x

(–6, 0)

0

(6, 0)

Figure 55b

(–2, 0)
–1

(1, 0)

Figure 55c


74

CHAPTER 2

Analysis of Graphs of Functions

1

56. (a) The equation y f 12x2 is y f 1x2 stretched horizontally by a factor of . See Figure 56a. 2
(b) The equation y

f 1 x2 is y f 1x2 reflected across the y-axis. See Figure 56b.

(c) The equation y

3 f 1x2 is y f 1x2 stretched vertically by a factor of 3. See Figure 56c.

(d) From the graph, symmetry with respect to the y-axis.
y

y

y

(– 12, 0) ( 12 , 0)
(–1, 1)

(1, 1)

1

(– 4, 0) ( 14 , 0)
1

(– 12, 0) 1

, 0)

3
(4

x

0

3

( 1 , 0)

(–2, 1)

2

(– 4 , 0 )

1 is y

(b) The equation yf 1x21 is y
(c) The equation y 2 f a

1

(2, 1)
x

2

(– 32 , 0) –1

Figure 56a
57. (a) The equation y f 1x2

(2, 3)

x

0

1

(–2, 3)

3

3

(– 2 , 0 )

( 32 , 0)

Figure 56b
f 1x2 shifted 1 unit upward. See Figure 57a.

( 2 , 0)

Figure 56c

f 1x2 reflected across the x-axis and shifted 1 unit down. See Figure 57b.

xb is y f 1x2 stretched vertically by a factor of 2 and horizontally by a factor of 2. 2

See Figure 57c.
y

y

y
(–2, 4)

(–1, 3)
(1, 1)

(0, 1) (2, 1)

(–2, 1)

(4, 0)

(0, 0)

x

x

0

x

0

(–2, –1)

(1, –1)

0

(0, –1) (2, –1)

(– 4, 0)

(–1, –3)

Figure 57a
58. (a) The equation y f 1x2

2 is y

(2, – 4)

Figure 57b
f 1x2 shifted 2 units downward. See Figure 58a.

Figure 57c

(b) The equation y

f 1x

(c) The equation y

2 f 1 x2 is y f 1x2 stretched vertically by a factor of 2. See Figure 58c.

12

2 is y

f 1x2 shifted 1 unit right and 2 units upward. See Figure 58b.

y

y

y
(0, 4)

(–1, 4)
(3, 2)

(4, 3)
(–1, 0)

(–1, 1)
x

0

x

x

0

0

(3, –1)
(–2, –2)

(–2, –3)

Figure 58a

Figure 58b

Figure 58c


Stretching, Shrinking, and Reflecting Graphs
59. (a) The equation y f 12x2 1 is y f 1x2 shrunk horizontally by a factor of

1

SECTION 2.3

75

and shifted 1 unit upward. 2

See Figure 59a.
(b) The equation y 2 f a

1

xb 1 is y f 1x2 stretched vertically by a factor of 2, stretched horizontally by a 2

factor of 2, and shifted 1 unit upward. See Figure 59b.
1

1

(c) The equation y 2 f 1 x
See Figure 59c.

2 2 is yf

1x2 shrunk vertically by a factor of 2 and shifted 2 units to the right.

y

y

y

(– 4, 5)
(–1, 3)
(1, 2)

(4, 3)

(0, 1)

(4, .5)

x

x

0 (0, 0)

0

x

0

(2, –.5)

(0, –1)

Figure 59a

Figure 59b

Figure 59c

1

60. (a) The equation y f 12x2 is y f 1x2 shrunk horizontally by a factor of . See Figure 60a. 2
(b)

The equation yf a
2

1

xb 1 is y f 1x2 stretched horizontally by a factor of 2, and shifted 1 unit

downward. See Figure 60b.
(c) The equation y 2 f 1 x2 1 is y f 1x2 stretched vertically by a factor of 2 and shifted 1 unit downward. See
Figure 60c.
y

y

y
(–2, 3)

(–1, 2)

(0, 3)
(0, 2)

(– 4, 1)

(0, 1)

x

x

0

(–.5, –1)

x

0

(1, –2)

0

(–2, –2)
(4, –3)

Figure 60a
61. (a) If r is the x-intercept of y

Figure 60b

(–1, –3)

(2, –5)

Figure 60c

f 1x2 and yf 1x2 is y f 1x2 reflected across the x-axis, then r is also the

x-intercept of yf 1x2.
(b) If r is the x-intercept of y f 1x2 and y f 1 x2 is y f 1x2 reflected across the y-axis, then r is the xintercept of y f 1 x2.
(c) If r is the x-intercept of y f 1x2 and y f 1 x2 is y f 1x2 reflected across both the x-axis and y-axis, then r is the
x-intercept of y f 1 x2.


76

CHAPTER 2
Analysis of Graphs of Functions
62. (a) If b is the y-intercept of y f 1x2 and yf 1x2 is y f 1x2 reflected across the x-axis, then b is the yintercept of y f 1x2.

(b) If b is the y-intercept of y f 1x2 and y f 1 x2 is y f 1x2 reflected across the y-axis, then b is also the yintercept of y f 1 x2.
(c) If b is the y-intercept of y f 1x2 and y 5 f 1x2 is y f 1x2 stretched vertically by a factor of 5, then 5b is
the y-intercept of y 5 f 1x2.
(d) If b is the y-intercept of y f 1x2 and y 3 f 1x2 is y f 1x2 reflected across the x-axis and stretched
vertically by a factor of 3, then 3 b is the y-intercept of of y 3 f 1x2.
63. Since f 1x 22 is f 1x2 shifted 2 units to the right, the domain of f 1x 22 is: 3 1

2, 2

2 4 or 3 1, 4 4 ; and the

range is the same: 3 0, 3 4 .
64. Since 5 f 1x 12 is f 1x2 shifted 1 unit to the left, the domain of 5 f 1x 12 is: 3 1 1, 2 1 4 or 3 2, 1 4 ; and

stretched vertically by a factor of 5, the range is: 3 5102, 5132 4 or 3 0, 15 4 .
65. Since f 1x2 is f 1x2 reflected across the x-axis, the domain of f 1x 22 is the same: 3 1, 2 4 ; and the range is:
3 102, 132 4 or 3 3, 0 4 .
66. Since f 1x
3 13, 2

32

1 is f 1x2 shifted 3 units to the right, the domain of f 1x

32

1 is:

34

or 3 2, 5 4 ; and shift 1 unit upward, the range is: 3 0 1, 3 1 4or 3 1, 4 4 .
1
1
1
1
67. Since f 12 x2 is f 1x2 shrunk horizontally by a factor of 2 , the domain of f 12 x2 is: c 2 1 12, 2 122 d or c 2, 1 d ;
and the range is the same: 3 0, 3 4 .
68. Since 2 f 1x 12 is f 1x2 shifted 1 unit to the right, the domain of 2 f 1x

12 is: 3 1

1, 2

1 4 or 3 0, 3 4 ; and

stretched vertically by a factor of 2, the range is: 3 2102, 2132 4 or 3 0, 6 4 .
1
1
69. Since 3 f a 4 xb is f 1x2 stretched horizontally by a factor of 4, the domain of 3 f a 4 xb is:
3 4 1 12, 4 122 4 or 3 4, 8 4 ; and stretched vertically by a factor of 3, the range is: 3 3102, 3132 4 or 3 0, 9 4 .

1

70. Since 2 f 14 x2 is f 1x2 shrunk horizontally by a factor of , the domain of 2 f 14 x2 is: 4

1

1
1 1
1
12,
or
c
122
d
c4
4
4 , 2 d ; and reflected across the x-axis while being stretched vertically by a factor of 2, the
range is: 3 2102, 2132 4 3 0, 6 4 or 3 6, 0 4 .
71. Since f 1 x2 is f 1x2 reflected across the y-axis, the domain of f 1 x2 is: 3 1 12, 122 4 3 1, 2 4 or 3 2, 1 4 ; and
the range is the same: 3 0, 3 4 .
72. Since 2 f 1 x2 is f 1x2 reflected across the y-axis, the domain of 2 f 1 x2 is:
3 1 12, 122 4 3 1, 2 4 or 3 2, 1 4 ; and reflected across the x-axis while being stretched vertically by a factor
of 2, the range is: 3 2102, 2132 4 3 0, 6 4 or 3 6, 0 4 .

1

73. Since f 1 3x2 is f 1x2 reflected across the y-axis and shrunk horizontally by a factor of 3 , the domain of f 1 3 x2
1
1
1 2
2 1

is: c 3 1 12, 3 122 d

c3 ,

3 d or c 3 , 3 d ; and the range is the same: 3 0, 3 4 .


Stretching, Shrinking, and Reflecting Graphs
1

SECTION 2.3

1

74. Since 3 f 1x

32 is f 1x2 shifted 3 units to the right, the domain of 3 f 1x 32 is: 3 1
1
1
1

shrunk vertically by a factor of

c

, the range is:

3

3

102,

3

1x has an endpoint (0, 0), and the graph of y

75. Since y

3, 2

3 4 or 3 2, 5 4 ; and

132 d or 3 0, 1 4 .

10 1 x 20

5 is the graph of y1

x shifted 20

units right, stretched vertically by a factor of 10, and shifted 5 units upward, the endpoint of y 10 1 x 20
is: 10

77

5

52 or 120, 52. Therefore, the domain is: 3 20, q2; and the range is: 3 5, q2.

20, 10102

76. Since y
18 is the graph of y

1x has an endpoint (0, 0), and the graph of y2 1
1 x shifted
x 15
15 units left, reflected across the x-axis, stretched vertically by a factor of 2, and shifted 18 units downward, the
endpoint of y2 1

x

15

18 is: 10

15, 2102 182 or 15, 182. Therefore, the domain is:

3 15, q2; and the range, because of the reflection across the x-axis, is: 1 q, 18 4 .
77. Since y
5 is the graph of y
1x has an endpoint (0, 0), and the graph of y.5 1
1 x shifted
x 10
10 units left, reflected across the x-axis, shrunk vertically by a factor of .5, and shifted 5 units upward, the endpoint of y.5 1

x

10

5 is: 10

10, .5102

52

or 1 10, 52. Therefore, the domain is: 3 10, q2; and

the range, because of the reflection across the x-axis, is: 1 q, 5 4 .
78. Using ex. 75, the domain is: 3 h, q2; and the range is: 3 k, q2.
79. The graph of y f 1x2 is y f 1x2 reflected across the x-axis, therefore y f 1x2 is decreasing for the interval:
3 a, b 4 .
80. The graph of y f 1 x2 is y f 1x2 reflected across the y-axis, therefore y f 1 x2 is decreasing for the
interval: 3 b, a 4 .
81. The graph of y f 1 x2 is y f 1x2 reflected across both the x-axis and y-axis, therefore y f 1 x2 is
increasing for the interval: 3 b, a 4 .
82. The graph of yc f 1x2 is y
interval: 3 a, b 4 .
83. From the graph,

f 1x2 reflected across the x-axis, therefore yc f 1x2 is decreasing for the

(a) the function is increasing for the interval: 3

1, 2 4 .

(b) the function is decreasing for the interval: 1 q, 1 4 .

84. From the graph,

(c) the function is constant for the interval: 3 2, q2.
(a) the function is increasing for the interval: 1 q, 1 4 .
(b) the function is decreasing for the interval: 3 1, 2 4 .
(c) the function is constant for the interval: 3 2, q2.

85. From the graph,

(a) the function is increasing for the interval: 3 1, q2.

(b) the function is decreasing for the interval: 3 2, 1 4 .

86. From the graph,

(c) the function is constant for the interval: 1 q, 2 4 .
(a) the function is increasing for the interval: 1 q, 3 4 .
(b) the function is decreasing for the interval: 3 3, q2.
(c) the function is constant for no interval: none.


78

CHAPTER 2

Analysis of Graphs of Functions

87. From the graph, the point on y2 is approximately: 18, 102.
88. From the graph, the point on y2 is approximately: 1 27, 152.
two points on the graph to find the slope, two points are: 2, 1
and 1, 1 , therefore the slope is:
1
89. Use 1
2
1
1

2
2
1
2
1
m
2.
m 1
The
stretch
factor
is
2
and
the
graph
has
been
shifted
2 units to the left and 1
1 22
1
unit down, therefore the equation is: y 2 0 x 2 01.
two points on the graph to find the slope, two points are: 1, 2 and 5, 0 , therefore the slope is:
2
1
1
90. Use 0 2
1
1

2
2
1
m
m 5 1
.
The
shrinking
factor
is
,
the
graph
has
been
reflected across the x-axis,
4
2
2
1
shifted 1 unit to the right, and shifted 2 units upward, therefore the equation is: y
2 0 x1 02.
two points on the graph to find the slope, two points are: 0, 2 and 1, 1 , therefore the slope is:
91. Use 1 2
3
1
1
2
2
3.

m
The stretch factor is 3, the graph has been reflected across the x-axis, and
1 0
1 1m
shifted 2 units upward, therefore the equation is: y3 0 x 0

2.

two points on the graph to find the slope, two points are: 1, 2
2
92. Use 1
1
3
m

1

2

0 1 12

1

1m

and 0, 1 , therefore the slope is:
2

1

2

3. The stretch factor is 3 and the graph has been shifted 1 unit to the left and 2

units down, therefore the equation is: y

30x

1 02.

Reviewing Basic Concepts (Sections 2.1—2.3)
1.

(a) If y

f 1x2 is symmetric with respect to the origin, then another function value is: f 1 326.

(b) If y

f 1x2 is symmetric with respect to the y-axis, then another function value is: f 1 32

6.

(c) If f 1 x2 f 1x2, y f 1x2 is symmetric with respect to both the x-axis and y-axis, then another function
value is: f 1 32 6.
(d) If f 1 x2f 1x2, y f 1x2 is symmetric with respect to the y-axis, then another function value is: f 1 326.
2.

(a) The equation y 1x 722 is y x2 shifted 7 units to the right: B.
(b) The equation y x2 7 is y x2 shifted 7 units downward: D.

(c) The equation y
(d) The equation y
(e)

The equation y

7 x2 is y x2 stretched vertically by a factor of 7: E.
1x 722 is y x2 shifted 7 units to the left: A.

a 1 xb2 is y

x2 stretched horizontally by a factor of 3: C.

3
3.

(a) The equation y

x2

2 is y

x2 shifted 2 units upward: B.

(b) The equation y

x2

2 is y x2 shifted 2 units downward: A.

(c) The equation y

1x

222 is y

(d) The equation y
(e) The equation y

1x 222 is y x2 shifted 2 units to the right: C.
2 x2 is y x2 stretched vertically by a factor of 2: F.

x2 shifted 2 units to the left: G.

(f) The equation yx2 is y x2 reflected across the x-axis: D.
(g) The equation y 1x 222 1 is y x2 shifted 2 units to the right and 1 unit upward: H.
(h) The equation y

1x

222

1 is y

x2 shifted 2 units to the left and 1 unit upward: E.


Stretching, Shrinking, and Reflecting Graphs
4.

SECTION 2.3

79

(a) The equation y0 x 04 is y0 x 0 shifted 4 units upward. See Figure 4a.
(b) The equation y0 x4 0 is y0 x 0 shifted 4 units to the left. See Figure 4b.
(c) The equation y0 x4 0 is y0 x 0 shifted 4 units to the right. See Figure 4c.
(d) The equation y0 x2 04 is y0 x 0 shifted 2 units to the left and 4 units downward. See Figure 4d.
(e) The equation y 0 x 2 0 4 is y 0 x 0 reflected across the x-axis, shifted 2 units to the right, and 4 units
upward. See Figure 4e.
y

y

4

y

4

4

x
–4

0

x

Figure 4a

x
0

0

–4

4

Figure 4b

y

4

Figure 4c

y

4
x
–6

5.

0
–2

2

x
–2

0
6

Figure 4d
Figure 4e
(a) The graph is the function f 1x210x 0 reflected across the x-axis, shifted 1 unit left and 3 units upward.
Therefore the equation is: y 10x 1 0 3.
(b) The graph is the function g1x2 1x reflected across the x-axis, shifted 4 units left and 2 units upward
Therefore the equation is: y x 4 21.
(c) The graph is the function g1x2 x stretched vertically by a factor of 2, shifted 4 units left and 4 units
downward. Therefore the equation is: y 2 x 4 4.

6.

(d) The graph is the function f 1x20 x 0 shrunk vertically by a factor of
1
downward. Therefore the equation is: y
2 0 x2 01.
(a) The graph of g1x2 is the graph f 1x2 shifted 2 units upward. Therefore c

1
2, shifted 2 units right and 1 unit

2.

(b) The graph of g1x2 is the graph f 1x2 shifted 4 units to the left. Therefore c4.

7.

The graph of y F1x h2 is a horizontal translation of he graph of y F1x2. The graph of y F1x2 h is not the same
as the graph of y F1x h2 because the graph of y F1x2 h is a vertical translation of the graph of y F1x2 and y
F1x h2 is a horizontal translation of the graph y F1x2.


80
8.

CHAPTER 2

Analysis of Graphs of Functions

The effect is either a stretch or a shrink, and perhaps a reflection across the x-axis. If c 7 0, there is a stretch or
shrink by a factor of c. If c 6 0, there is a stretch or shrink by a factor of 0 c 0 , and a reflection across the xaxis. If 0 c 0 7 1, a stretch occurs; when 0 c 0 6 1, a shrink occurs.

9.

(a) If f is even, then f 1x2

f 1 x2. See Figure 9a.

(b) If f is odd, then f 1 x2f 1x2. See Figure 9b.
10. (a) Since x

1 corresponds to 1992, the equation using actual year is: g1x2.2791x

19922

5.532.

(b) g120062 .27912006 19922 5.532 1 g120062 .2791142 5.532 1 g120062 1.626
ppm.
x
3
2
1
1
2
3

ƒ(x)
4
6
5
5
6
4

x ƒ(x)
3
4
2
6
1
5
1
5
2

6
3
4

Figure 9a

Figure 9b

2.4: Absolute Value Functions: Graphs, Equations, Inequalities, and Applications
1. If f 1a25,

2

then

0 f 1a2 00 5 05.

2. Since f 1x 2 2 x is an even function, f 1x2
3. If f 1x2x , then y0 f 1x2 0 1 y

2

2

x

2

and f 1x2 2 0 x 0 are the same graph.

x0 1y

0

x . Therefore the range of y

4. If the range of y f 1x2 is 3 2, q2, the range of y0 f 1x2 0
across the x-axis.
5. If the range of y

0 f 1x2 0 is: 3 0, q2.

is 3 0, q2 since all negative values of y are reflected

f 1x2 is 1 q, 2 4 , the range of y0 f 1x2

0

is 3 2, q2 since all negative values of y are

reflected across the x-axis.

6.

0 f 1x2 0 is greater than or equal to 0 for any value of x. Since 1 is less than 0, 1 cannot be in the range of f.

7.

We reflect the graph of y f 1x2 across the x-axis for all points for which y 6 0. Where y 0, the graph
remains unchanged. See Figure 7.

We reflect the graph of y f 1x2 across the x-axis for all points for which y 6 0. Where y 0, the graph

8.

remains unchanged. See Figure 8.
9.

We reflect the graph of y f 1x2 across the x-axis for all points for which y 6 0. Where y 0, the graph
remains unchanged. See Figure 9.
y

y

y

(1, 2)

(–2, 8)

(–2, 1)
0

x

(2, 0)

(–4, 0)

0

(2, 8)

x
x
0

Figure 7

Figure 8

Figure 9


Absolute Value Functions: Graphs, Equations, Inequalities, and Applications
SECTION 2.4
10. We reflect the graph of y f 1x2 across the x-axis for all points for which y 6 0. Where y 0, the graph

81

remains unchanged. See Figure 10.
0, the graph remains unchanged. That is, y0 f 1x2 0 has the same graph as yf 1x2.

11. Since for all y, y

12. We reflect the graph of y f 1x2 across the x-axis for all points for which y 6 0. Where y 0, the graph
remains unchanged. See Figure 12.
y

y

(4, 2)
(–8, 2)

(8, 2)

x

0
x

0

Figure 10
Figure 12
13. We reflect the graph of y f 1x2 across the x-axis for all points for which y 6 0. Where y 0, the graph
remains unchanged. See Figure 13.
14. We reflect the graph of y f 1x2 across the x-axis for all points for which y 6 0. Where y 0, the graph
remains unchanged. See Figure 14.
15. We reflect the graph of y f 1x2 across the x-axis for all points for which y 6 0. Where y 0, the graph
remains unchanged. See Figure 15.
y

y

(–2, 2)

y

(2, 2)

(0, 1)
x

0

(0, 1)

(2, 1)

x

(–2, 0) 0

(2, 0)

x
0

(3, 0)

Figure 13
Figure 14
Figure 15
16. We reflect the graph of y f 1x2 across the x-axis for all points for which y 6 0. Where y 0, the graph
remains unchanged.
2
17. From the graph of y1x12

From the graph of y

0

18. From the graph of y
From the graph of y

0

1x

1 12

2

2 x the domain of f 1x2 is: 1 q, q2; and the range is: 3 q, q2.
1
2 x 0 the domain of 0 f 1x2 0 is: 1 q, q2; and the range is: 3 0, q2.

2

0

2 0 the domain of 0 f 1x2 0 is: 1 q, q2; and the range is: 3 0, q2.

1x222

19. From the graph of y1

From the graph of y

2 the domain of f 1x2 is: 1 q, q2; and the range is: 3 2, q2.

2

1

1x

22

2the domain of f 1x2 is: 1 q, q2; and the range is: 1 q, 1 4 .

0 the domain of 0 f 1x2 0 is: 1 q, q2; and the range is: 3 1, q2.