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2.3e: Exercises - Transformations

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A: Concepts

Exercise 2.3e.A 

1) When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?
2) When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?
3) When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?
4) When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x-axis from a reflection with respect to the y-axis?
5) How can you determine whether a function is odd or even from the formula of the function?

Answers to Odd Exercises:

1. A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

3. A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

5. For a function f, substitute (−x) for (x) in f(x). Simplify. If the resulting function is the same as the original function, f(x)=f(x), then the function is even. If the resulting function is the opposite of the original function, f(x)=f(x), then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

B: Describe transformations of a function written in function notation

Exercise 2.3e.B 

 Describe how the graph of the function is a transformation of the graph of the original function f.

6) y=f(x49)

7) y=f(x+43)

8) y=f(x+3)

9) y=f(x4)

10) y=f(x)+5

11) y=f(x)+8

12) y=f(x)2

13) y=f(x)7

14) y=f(x2)+3

15) y=f(x+4)1

16) g(x)=f(x)

17) g(x)=f(x)

18) g(x)=6f(x)

19) g(x)=4f(x)

20) g(x)=f(2x)

21) g(x)=f(5x)

22) g(x)=f(15x)

23) g(x)=f(13x)

24) g(x)=f(3x)

25) g(x)=3f(x)
Answers to Odd Exercises:

7. The graph of f(x+43) is a horizontal shift to the left 43 units of the graph of f.

9. The graph of f(x4) is a horizontal shift to the right 4 units of the graph of f.

11. The graph of f(x)+8 is a vertical shift up 8 units of the graph of f.

13. The graph of f(x)7 is a vertical shift down 7 units of the graph of f.

15. The graph of f(x+4)1 is a horizontal shift left 4 units and vertical shift down 1 unit of the graph of f.

17. The graph of g is a vertical reflection (across the x-axis) of the graph of f.

19. The graph of g is a vertical stretch by a factor of 4 of the graph of f.

21. The graph of g is a horizontal compression by a factor of 15 of the graph of f.

23. The graph of g is a horizontal stretch by a factor of 3 of the graph of f.

25. The graph of g is a horizontal reflection across the y-axis and a vertical stretch by a factor of 3 of the graph of f.

C: Graph transformations of a basic function

Exercise 2.3e.C 

 Begin by graphing the basic quadratic function f(x)=x2. State the transformations needed to apply to f to graph the function below. Then use transformations to graph the function.

27. g(x)=x2+1

28. g(x)=x24

29. g(x)=(x5)2

30. g(x)=(x+1)2

31. g(x)=(x5)2+2

32. g(x)=(x+2)25

33. f(t)=(t+1)23

34. f(x)=(x+2)2

35. f(x)=x2+6 

36. g(x)=2x2

37. g(x)=4(x+1)25

38. g(x)=5(x+3)22

39. h(x)=12(x1)2

40. h(x)=13(x+2)2

41. f(x)=(12x3)2+1 

42. g(x)=(2x+3)24

 Begin by graphing the square root function f(x)=x. State the transformations needed to apply to f to graph the function below. Then use transformations to graph the function.

43. g(x)=x5

44. g(x)=x5

45. g(x)=x2+1

46. g(x)=x+2+3

47. a(x)=x+4

48. m(t)=3t+2

49. h(x)=x+2

50. g(x)=x+2

51. g(x)=12x3

52. h(x)=x2+1

53. f(x)=4x1+2

54. f(x)=5x+2

55.  k(x)=2x+51

56.1 a(x)=13x4

56.2 b(x)=3x+2

 Begin by graphing the absolute value function f(x)=|x|. State the transformations needed to apply to f to graph the function below. Then use transformations to graph the function.

57. h(x)=|x+4|

58. h(x)=|x4|

59. h(x)=|x1|3

60. h(x)=|x+2|5

61. g(x)=|x1|

62. h(x)=|x1|+4

63. f(x)=3|x|

64. f(x)=|x|3

65. h(x)=2|x4|+3

66. n(x)=13|x2|

67. h(x)=|3x+4|2

68. g(x)=|13x2|+1

 Begin by graphing the standard cubic function f(x)=x3. State the transformations needed to apply to f to graph the function below. Then use transformations to graph the function.

69. h(x)=(x2)3

70. h(x)=x3+4

71. h(x)=(x1)34

72. h(x)=(x+1)3+3

73. g(x)=(x+2)3

74. k(x)=(x2)31

75. g(x)=x3+4

76. m(x)=12x3

77. g(x)=14(x+3)31

78. q(x)=(14x)3+1

79. p(x)=(13x)33

 Begin by graphing the appropriate parent function : the basic cube root function f(x)=3x,  constant function f(x)=0,  or linear function f(x)=x. Then use transformations of this graph to graph the given function.

81. g(x)=3x1

82. g(x)=3x1

83. g(x)=3x2+6

84. g(x)=3x+84

84.1 g(x)=3x+32

84.1 g(x)=3x1+2

85. g(x)=23x+3+4

86. g(x)=32x51

87. f(x)=x+3

88. h(x)=2x+1

89. g(x)=4

 Begin by graphing the basic reciprocal function f(x)=1x. State the transformations needed to apply to f to graph the function below. Then use transformations to graph the function.

91. f(x)=1x2

92. f(x)=1x+3

93. f(x)=1x+5

94. f(x)=1x3

95. f(x)=1x+12

96. f(x)=1x3+3

97. f(x)=1x+2

98. f(x)=1x

99.p(x)=1x+1+2

100.1 a(x)=2x35

100.2 b(x)=12x+6+4

Answers to Odd Numbered Exercises for the Squaring Function:
Squaring Function

27. y=x2; Shift up 1 unit; domain: ; range: [1,)

0e393f0d6e151259a123b1e505dec86b.png
g(x)=x2+1

29. y=x2; Shift right 5 units; domain: ; range: [0,)

8fc7f879a8ba5f12d0b98f348e5adadb.png
g(x)=(x5)2

31. y=x2; Shift right 5 units and up 2 units; domain: ; range: [2,)

57a5fd7bcf0e225b10961c6534cd4545.png
g(x)=(x5)2+2

33.  Shift left 1 unit and down 3 units;

Graph of \(f(t)\).
 f(t)=(t+1)23

#35 Reflect over x-axis, up 6 units.

2.3E graph #35.png

37 f(x)=x2 is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

2.3E graph #37.png

39. Shift right 1 unit, and vertically shrink by a factor of 12

ba1195d282dfeb1b813eeec2a0ff6e74.png
h(x)=12(x1)2

#41 Shift right 3, reflect over y-axis, horizontally stretch by a factor of 2, up 1 units.

2.3E graph #41.png

for # 41, if f(x)=(12x3)2+1 is rewritten as 

f(x)=(12(x+6))2+1 ,

then the transformations would be

horizontal stretch by a factor of 2, reflect in y-axis (no change), left 6, up 1.

Answers to Odd Numbered Exercises for the Square Root Function

Square Root Function

43. y=x; Shift down 5 units; domain: [0,); range: [5,)

457665c1ea5709240bd4c6e1685a1985.png
g(x)=x5

45. y=x; Shift right 2 units and up 1 unit; domain: [2,); range: [1,)

da6d3f21b303aeb0b29fe4975b48a64f.png
g(x)=x2+1

47 The graph of f(x)=x is shifted left 4 units and then reflected across the y-axis.

2.3E graph #47.png

49. Reflect over y-axis, up 2

73fbbd2cc539ff2ab99df22497167aec.png
h(x)=x+2

51. Right 3, Reflect over x axis, Vertically compressed by a factor of 1/2.

576a64916f5e6c8c4b851efadf07189d.pngg(x)=12x3

 

53. Right 1, Vertically stretched by a factor of 4, up 2

3a6abc9abd61596a77ecfe672f89976d.png
f(x)=4x1+2

#55 Horizontal compression by 1/2, shift left 2.5, down 1 unit.

2.3E graph #55.png

Answers to Odd Numbered Exercises for the Absolute Value Function:

Absolute Value Function

57. y=|x|; Shift left 4 units; domain: ; range: [0,)

86a1d10b4aad0ab79bc2c8dd55bf4f38.png
h(x)=|x+4|

.

59. y=|x|; Shift right 1 unit and down 3 units; domain: ; range: [3,)

424b66df0df22a96fd88c4957413d44e.png
h(x)=|x1|3

.

61. Right 1, Reflect over x-axis

8e5290466d22bfaad7a33f4ffcc1c2d0.png
g(x)=|x1|

.

63. Reflect over x-axis, vertically stretch by a factor of 3

b03918836eb8805c137b8a53dc8d07ff.png
f(x)=3|x|

 

65 The graph of f(x)=|x| is shifted horizontally 4 units to the right, stretched vertically by a factor of 2, reflected across the horizontal axis, then shifted up 3 units.

2.3E graph #65.png

67. h(x)=|3(x43)|2 Horizontally compress by a factor of 13, right 43, down 2

2.3E graph #67.png

Answers to Odd Numbered Exercises for the Cubing Function

Cubing Function

69. y=x3 ; Shift right 2 units; domain: ; range:

01b74b05906d95ff14c5aa6de0ae7b4f.png
h(x)=(x2)3

71. y=x3; Shift right 1 unit and down 4 units; domain: ; range:

a4f584febcd95dc5ef92bbe2ef80df7c.png
h(x)=(x1)34

 

73. Left 2 units, reflect over x-axis

87909d16e900cb252e880491550fd960.png
g(x)=(x+2)3

75. Reflect over x-axis, up 4 units

92bf8584935a01fd897e3af4c08fa4fd.png
g(x)=x3+4

77. Left 3 units, reflect over x-axis, vertically shrink by a factor of 14, down 1 unit

g(x)=14(x+3)31

79. Stretch horizontally by a factor of 3 and shift vertically downward by 3 units.

2.3E graph #79.png

Answers to Odd Numbered Exercises for the Cube Root, Linear and Constant Functions

Cube Root, Linear, Constant Functions

81. y=3x; Shift down 1

2.3E graph #81.png

83. y=3x; Shift up 6 units and right 2 units; domain: ; range:

2.3E graph #83.png

85. y=3x; Left 3, reflect over x-axis, vertically stretch by a factor of 2, up 4.    

2.3E graph #85.png

87. y=x; Shift up 3 units; domain: R; range: R

ed14f13811bfb7c397b768ab1e6d718a.png
f(x)=x+3

89. Basic graph y=4; domain: ; range: {4}

dec428893d68980da985eabaf7f7fb11.png
g(x)=4
Answers to Odd Numbered Exercises for the Reciprocal Function:

Reciprocal Function

91. y=1x; Shift right 2 units; domain: (,2)(2,); range: (,0)(0,)

75fa23d883d738eeb47a020057002b8f.png
f(x)=1x2

93. y=1x; Shift up 5 units;
domain: (,0)(0,);
range: (,1)(1,)

53d3a12d61be06d8913ae13668760ebb.png
f(x)=1x+5

95. y=1x; Shift left 1 unit and down 2 units; domain: (,1)(1,); range: (,2)(2,)

0eac4ad67881e57bfa8e7dc46c933e8e.png
f(x)=1x+12

97

97. Left 2 units, reflect over x-axis

f(x)=1x+2

#99  Left 1 unit, reflect over x-axis, up 2 units.

2.3E graph #99.png

D: Graph Transformations of a Graph

Exercise 2.3e.D 

 Use the graph of f(x) shown in the Figure below to sketch a graph of each transformation of f(x).

 Graph of \(f(x)\).Given the graph of  f(x) on the right, sketch the graph for the following transformations of f

 

101. h(x)=2x3

102. a) g(x)=2x+1

       b) w(x)=2x1

2.3e #103.pngGiven the graph of  f(x) on the right, sketch the graph for the following transformations of f

 

103. a) g(x)=f(x)

       b) g(x)=f(x2)

104. a) g(x)=f(x)2

       b) g(x)=f(x+1)

 106. Given the graph of  f(x) below, sketch the graph for the following transformations of f

  1. a(x)=f(x)+1
  2. b(x)=f(x+1)
  3. c(x)=f(x)+2
  4. d(x)=f(x)
  5. e(x)=f(1x)2
  6. f(x)=2f(x)
  7. g(x)=f(x)
  8. h(x)=12f(x+2)+3 
  1. i(x)=f(x)1
  2. j(x)=f(x1)
  3. k(x)=f(x)2
  4. l(x)=f(x)
  5. m(x)=f(x1)+2
  6. n(x)=f(2x)
  7. o(x)=f(x)
  8. p(x)=f(12x2)3
2.3E graph #104.png
Answers to Odd Exercises:

101

       2.3e #101.png

103. a

CNX_Precalc_Figure_01_05_235.jpg

103. b

CNX_Precalc_Figure_01_05_237.jpg

E: Match transformations of functions with graphs

Exercise 2.3e.E 

 Match the graph to the function definition. 

107. f(x)=x+4

108. f(x)=|x2|2

109. f(x)=x+11

110. f(x)=|x2|+1

111. f(x)=x+4+1

112. f(x)=|x+2|2

caf84d0d27db512ef90d11b59b6c37dc.png

47088a9efd6814511cb0fc8d233b539f.png

 

2fe54b1c80ea84f0f721462f90455c0b.png

d44d62205d34ed371aad179b77c54a81.png

 

3622a0d2256166544a122ecd7156de36.png

 

Match the graph to the given function definition.

113. f(x)=3|x|

114. f(x)=(x+3)21

115. f(x)=|x+1|+2

116. f(x)=x2+1

117. f(x)=13|x|

118. f(x)=(x2)2+2

19f3c208cfdeeffde7e76281b4b28f46.png
Figure (a)
7ddabfc77a72214e9f6bea00e3b2cca0.png
Figure (b)

039e6f4a86d07a578660882bccf7ea40.png
Figure (c)
16b19343fd01aecf51c1cdea8af3ee21.png
Figure (d)

26cdff42b4eb188a4512c934fd59f9e5.png
Figure (e)
75295519ff6aaa13dced0dc6ed6e2ef7.png
Figure (f)

Answers to Odd Exercises:

part 1 answers 107e, 109d, 111f,    part 2 Answers: 113.b, 115.d, 117.f

F: Construct equations from graphs of transformed basic functions

Exercise 2.3e.F 

 Write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

119.
Graph of an absolute function. 
120.
Graph of a parabola. 
121.
Graph of a square root function. 
122.
Graph of an absolute function.
123.
Graph of a parabola 
124.
CNX_Precalc_Figure_01_05_215.jpg 
125.
Graph of an absolute function. 
126.
 2.3E graph #126.png 
127.
 Graph of a square root function. 
128.
2.3E graph #128.png 
129.
Graph of a parabola. 
 

130. (a)
Graph of a cubic function. 

130. (b)
Graph of a square root function.

130. (c)

Graph of an absolute function. 

 
Answers to Odd Exercises:
119. f(x)=|x3|2 
121. f(x)=x+31
123. f(x)=(x2)2
125.  f(x)=|x+3|2 
127. f(x)=x
129. f(x)=(x+1)2+2

 Write an equation that represents the function whose graph is given.

131.

2.3E graph #131.png

132.

2.3E graph #132.png

133.

2.3E graph #133.png 

134.

2.3E graph #134.png

135.

2.3E graph #135.png

136.

2.3E graph #136.png 

137.

2.3E graph #137.png

138.

2.3E graph #138.png

139.

2.3E graph #139.png 

Answers to Odd Exercises:
131. f(x)=12x+3 133. f(x)=(2x5)
135. f(x)=2|x2|3
137. f(x)=12(x+2)3+4 139. f(x)=1x+6+4

G: Construct a formula from a description

Exercise 2.3e.G 

 Write a formula for the function with the following transformations

141. Write a formula for the function obtained when the graph of f(x)=|x| is shifted down 3 units and right 1 unit.

142. Write a formula for the function obtained when the graph of f(x)=1x is shifted down 4 units and right 3 units.

143. Write a formula for the function obtained when the graph of f(x)=1x2 is shifted up 2 units and left 4 units.

144. Write a formula for the function obtained when the graph of f(x)=x is shifted up 1 unit and left 2 units.

145. The graph of f(x)=|x| is reflected over the y-axis and horizontally compressed by a factor of 14.

146. The graph of f(x)=x is reflected over the x-axis and horizontally stretched by a factor of 2.

147. The graph of f(x)=1x2 is vertically compressed by a factor of 13, then shifted left 2 units and down 3 units.

148. The graph of f(x)=1x is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

149. The graph of f(x)=x2 is vertically compressed by a factor of 12, then shifted to the right 5 units and up 1 unit.

150. The graph of f(x)=x2 is horizontally stretched by a factor of 3, then shifted left 4 units and down 3 units.

Answers to Odd Exercises:

141. g(x)=|x1|3

143. g(x)=1(x+4)2+2

145. g(x)=|4x|

147. g(x)=13(x+2)23

149. g(x)=12(x5)2+1

 

H: Construct equations from transformations of tabular values

Exercise 2.3e.H 

 Given tabular representations for the functions f, g, and h, write g(x) and h(x) as transformations of f(x).

155. Tabular representations for the functions f, g, and h are given below. Write g(x) and h(x) as transformations of f(x).

x:

-2 -1 0 1 2
f(x): -2 -1 -3 1 2
x: -1 0 1 2 3
g(x): -2 -1 -3 1 2
x: -2 -1 0 1 2
h(x): -1 0 -2 2 3

156. Tabular representations for the functions f, g, and h are given below. Write g(x) and h(x) as transformations of f(x).

x:

-2 -1 0 1

2

f(x): -1 -3 4 2 1
x: -3 -2 -1 0 1
g(x): -1 -3 4 2 1

x:

-2 -1 0 1

2

h(x): -2 -4 3 1 0
Answers to Odd Exercises:

155. g(x)=f(x1), h(x)=f(x)+1.

I: Identify Increasing/decreasing Intervals

Exercise 2.3e.I 

 Use transformations to determine the interval(s) on which the function is increasing and decreasing.

151. g(x)=5(x+3)22 152. f(x)=4(x+1)25 153. k(x)=3x1 154. a(x)=x+4
Answers to Odd Exercises:
151. decreasing on (,3) and increasing on (3,) 153. decreasing on (0,)


2.3e: Exercises - Transformations is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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