2.3e: Exercises - Transformations

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A: Concepts

Exercise $\PageIndex{A}$

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 $\PageIndex{B}$

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

6) $y=f(x−49)$

7) $y=f(x+43)$

8) $y=f(x+3)$

9) $y=f(x-4)$

10) $y=f(x)+5$

11) $y=f(x)+8$

12) $y=f(x)−2$

13) $y=f(x)−7$

14) $y=f(x−2)+3$

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) $g(x)=f \left(\tfrac{1}{5}x \right)$

23) $g(x)=f \left(\tfrac{1}{3}x \right)$

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 $f(x-4)$ 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 $\frac{1}{5}$ 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 $\PageIndex{C}$

$ \bigstar$ Begin by graphing the basic quadratic function $f(x)=x^2$. State the transformations needed to apply to $f$ to graph the function below. Then use transformations to graph the function.

27. $g(x) = x^{2} + 1$

28. $g(x) = x^{2} − 4$

29. $g(x) = (x − 5)^{2}$

30. $g(x) = (x + 1)^{2}$

31. $g(x) = (x − 5)^{2} + 2$

32. $g(x) = (x + 2)^{2} − 5$

33. $f(t)=(t+1)^2−3$

34. $f ( x ) = - ( x + 2 ) ^ { 2 }$

35. $f ( x ) = - x ^ { 2 } + 6$

36. $g ( x ) = - 2 x ^ { 2 }$

37. $g(x)=4(x+1)^2−5$

38. $g(x)=5(x+3)^2−2$

39. $h ( x ) = \tfrac { 1 } { 2 } ( x - 1 ) ^ { 2 }$

40. $h ( x ) = \tfrac { 1 } { 3 } ( x + 2 ) ^ { 2 }$

41. $f ( x ) = ( -\tfrac{1}{2}x - 3 ) ^ 2 + 1$

42. $g(x)=(-2x+3)^2 -4 $

$ \bigstar$ Begin by graphing the square root function $f(x)=\sqrt{x}$. State the transformations needed to apply to $f$ to graph the function below. Then use transformations to graph the function.

43. $g(x) = \sqrt{x} − 5$

44. $g(x) = \sqrt{x − 5}$

45. $g(x) = \sqrt{x − 2} + 1$

46. $g(x) = \sqrt{x + 2} + 3$

47. $a(x)=\sqrt{−x+4}$

48. $m(t)=3-\sqrt{t+2}$

49. $h ( x ) = \sqrt { - x } + 2$

50. $g ( x ) = - \sqrt { x } + 2$

51. $g ( x ) = - \frac { 1 } { 2 } \sqrt { x - 3 }$

52. $h ( x ) = - \sqrt { x - 2 } + 1$

53. $f ( x ) = 4 \sqrt { x - 1 } + 2$

54. $f ( x ) = - 5 \sqrt { x + 2 }$

55. $k(x) = \sqrt{2x+5} - 1$

56.1 $a(x) = \sqrt{\tfrac{1}{3} x - 4} $

56.2 $b(x) = \sqrt{3-x}+2 $

$ \bigstar$ 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. $h(x) = |x − 4|$

59. $h(x) = |x − 1| − 3$

60. $h(x) = |x + 2| − 5$

61. $g ( x ) = - | x - 1 |$

62. $h(x)=|x−1|+4$

63. $f ( x ) = - 3 | x |$

64. $f ( x ) = - | x | - 3$

65. $h(x)=−2|x−4|+3$

66. $n(x)=\dfrac{1}{3}|x−2|$

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

68. $g(x) = | \tfrac{1}{3}x-2| + 1 $

$ \bigstar$ Begin by graphing the standard cubic function $f(x) = x^3 $. State the transformations needed to apply to $f$ to graph the function below. Then use transformations to graph the function.

69. $h(x) = (x − 2)^{3}$

70. $h(x) = x^{3} + 4$

71. $h(x) = (x − 1)^{3} − 4$

72. $h(x) = (x + 1)^{3} + 3$

73. $g ( x ) = - ( x + 2 ) ^ { 3 }$

74. $k(x)=(x−2)^3−1$

75. $g ( x ) = - x ^ { 3 } + 4$

76. $m(x)=\tfrac{1}{2}x^3$

77. $g ( x ) = - \frac { 1 } { 4 } ( x + 3 ) ^ { 3 } - 1$

78. $q(x)=\big(\tfrac{1}{4}x\big)^3+1$

79. $p(x)=\big(\tfrac{1}{3}x\big)^3−3$

$ \bigstar$ Begin by graphing the appropriate parent function : the basic cube root function $f(x)=\sqrt[3]{x}$, constant function $f(x)=0$, or linear function $f(x)=x$. Then use transformations of this graph to graph the given function.

81. $g( x ) = \sqrt [ 3 ] {x} -1$

82. $g( x ) = \sqrt [ 3 ] { x - 1 } $

83. $g( x ) = \sqrt [ 3 ] { x - 2 } + 6$

84. $g( x ) = \sqrt [ 3 ] { x + 8 } - 4$

84.1 $g( x ) = \sqrt [ 3 ] { -x + 3 } - 2$

84.1 $g( x ) = - \sqrt [ 3 ] { x - 1 } + 2$

85. $g ( x ) = -2 \sqrt [ 3 ] { x + 3 } + 4$

86. $g ( x ) = \sqrt [ 3 ] {-2 x - 5 } - 1$

87. $f(x) = x + 3$

88. $h ( x ) = - 2 x + 1$

89. $g(x) = −4$

$ \bigstar$ Begin by graphing the basic reciprocal function $f(x)=\frac{1}{x}$. State the transformations needed to apply to $f$ to graph the function below. Then use transformations to graph the function.

91. $f(x) = \dfrac{1}{x−2}$

92. $f(x) = \dfrac{1}{x+3}$

93. $f(x) = \dfrac{1}{x} + 5$

94. $f(x) = \dfrac{1}{x} − 3$

95. $f(x) = \dfrac{1}{x+1} − 2$

96. $f(x) = \dfrac{1}{x−3} + 3$

97. $f ( x ) = - \dfrac { 1 } { x + 2 }$

98. $f ( x ) = - \dfrac { 1 } { x }$

99.$p( x ) = - \dfrac { 1 } { x + 1 } + 2$

100.1 $a(x) = \dfrac{2}{x-3} -5 $

100.2 $b(x) = \dfrac{1}{2x+6} +4 $

Answers to Odd Numbered Exercises for the Squaring Function:

Squaring Function

27. $y = x^{2}$; Shift up $1$ unit; domain: $ℝ$; range: $[1, ∞)$

29. $y = x^{2}$; Shift right $5$ units; domain: $ℝ$; range: $[0, ∞)$

31. $y = x^{2}$; Shift right $5$ units and up $2$ units; domain: $ℝ$; range: $[2, ∞)$

33. Shift left $1$ unit and down $3$ units;

$Graph of $f(t)$.$
$f(t)=(t+1)^2−3$

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

$2.3E graph #35.png$

37 $f(x)=x^2$ 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 $ \frac{1}{2}$

$h(x) = \frac { 1 } { 2 } ( x - 1 ) ^ { 2 }$

#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 $f ( x ) = ( -\tfrac{1}{2}x - 3 ) ^ 2 + 1$ is rewritten as

$f ( x ) = ( -\tfrac{1}{2}(x + 6) ) ^ 2 + 1$ ,

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. $y = \sqrt{x}$; Shift down $5$ units; domain: $[0, ∞)$; range: $[−5, ∞)$

45. $y = \sqrt{x}$; Shift right $2$ units and up $1$ unit; domain: $[2, ∞)$; range: $[1, ∞)$

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

$2.3E graph #47.png$

49. Reflect over $y$-axis, up $2$

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

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

#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: $ℝ$; range: $[0, ∞)$

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

61. Right $1$, Reflect over $x$-axis

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

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(x-\tfrac{4}{3})| -2 $ $ \longrightarrow$ Horizontally compress by a factor of $\tfrac{1}{3}$, right $ \tfrac{4}{3}$, down $2$

$2.3E graph #67.png$

Answers to Odd Numbered Exercises for the Cubing Function

Cubing Function

69. $y = x^{3}$ ; Shift right $2$ units; domain: $ℝ$; range: $ℝ$

71. $y = x^{3}$; Shift right $1$ unit and down $4$ units; domain: $ℝ$; range: $ℝ$

73. Left $2$ units, reflect over x-axis

75. Reflect over x-axis, up $4$ units

77. Left $3$ units, reflect over x-axis, vertically shrink by a factor of $\frac {1}{4}$, down $1$ unit

$g ( x ) = - \frac {1}{4} (x + 3) ^ {3} - 1$

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. $y = \sqrt [ 3 ] { x }$; Shift down $1$

$2.3E graph #81.png$

83. $y = \sqrt [ 3 ] { x }$; Shift up $6$ units and right $2$ units; domain: $ℝ$; range: $ℝ$

$2.3E graph #83.png$

85. $y = \sqrt [ 3 ] { x }$; 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: $\mathbb{R}$; range: $\mathbb{R}$

89. Basic graph $y = −4$; domain: $ℝ$; range: $\{−4\}$

Answers to Odd Numbered Exercises for the Reciprocal Function:

Reciprocal Function

91. $y = \frac{1}{x}$; Shift right $2$ units; domain: $(−∞, 2) ∪ (2, ∞)$; range: $(−∞, 0) ∪ (0, ∞)$

93. $y = \frac{1}{x}$; Shift up $5$ units;
domain: $(−∞, 0) ∪ (0, ∞)$;
range: $(−∞, 1) ∪ (1, ∞)$

$ \star $

95. $y = \frac{1}{x}$; Shift left $1$ unit and down $2$ units; domain: $(−∞, −1) ∪ (−1, ∞)$; range: $(−∞, −2) ∪ (−2, ∞)$

97. Left $2$ units, reflect over x-axis

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

$2.3E graph #99.png$

D: Graph Transformations of a Graph

Exercise $\PageIndex{D}$

$ \bigstar$ 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. $h(x)=2^x-3$

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

b) $w(x)=2^x−1$

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

103. a) $g(x)=−f(x)$

b) $g(x)=f(x−2)$

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 $f$

$a(x) = f(x)+1 $
$b(x) = f(x+1) $
$c(x) = f(x)+2 $
$d(x) = -f(x)$
$e(x) = f(1-x)-2$
$f(x) = 2f(x) $
$g(x) = -f(x)$
$h(x) = \tfrac{1}{2} f(x+2)+3$

$i(x) = f(x)-1 $
$j(x) = f(x-1) $
$k(x) = f(x)-2 $
$l(x) = f(-x)$
$m(x) = -f(x-1)+2$
$n(x) = f(2x) $
$o(x) = f(-x)$
$p(x) = f( \tfrac{1}{2} x-2)-3$

$2.3E graph #104.png$

Answers to Odd Exercises:

101

$2.3e #101.png$

103. a

103. b

E: Match transformations of functions with graphs

Exercise $\PageIndex{E}$

$ \bigstar$ Match the graph to the function definition.

107. $f(x) = \sqrt{x + 4}$

108. $f(x) = |x − 2| − 2$

109. $f(x) = \sqrt{x + 1} -1$

110. $f(x) = |x − 2| + 1$

111. $f(x) = \sqrt{x + 4} + 1$

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. $f ( x ) = - ( x + 3 ) ^ { 2 } - 1$

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

116. $f ( x ) = - x ^ { 2 } + 1$

117. $f ( x ) = - \frac { 1 } { 3 } | x |$

118. $f ( x ) = - ( x - 2 ) ^ { 2 } + 2$

$ \star $

$ \star $

$ \star $

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 $\PageIndex{F}$

$ \bigstar$ 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.	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. $f(x)=\sqrt{x+3}−1$
121. $f(x)=(x-2)^2$

123. $f(x)=|x+3|−2$
125. $f(x)=−\sqrt{x}$

127. $f(x)=−(x+1)^2+2$
129. $f(x)=\sqrt{−x}+1$

$ \bigstar$ 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. $f ( x ) = \tfrac{1}{2} \sqrt { x +3 }$

133. $f ( x ) = \sqrt(2x-5) $
135. $f ( x ) = 2 | x-2 | - 3$

137. $f ( x ) = -\tfrac{1}{2} ( x + 2 )^3 +4$

139. $ f(x) = \dfrac{1}{x+6} + 4$

G: Construct a formula from a description

Exercise $\PageIndex{G}$

$ \bigstar$ 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 $f(x)=\dfrac{1}{x}$ is shifted down $4$ units and right $3$ units.

143. Write a formula for the function obtained when the graph of $f(x)=\dfrac{1}{x^2}$ is shifted up $2$ units and left $4$ units.

144. Write a formula for the function obtained when the graph of $f(x)=\sqrt{x}$ 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 $\dfrac{1}{4}$.

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

147. The graph of $f(x)=\dfrac{1}{x^2}$ is vertically compressed by a factor of $\dfrac{1}{3}$, then shifted left $2$ units and down $3$ units.

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

149. The graph of $f(x)=x^2$ is vertically compressed by a factor of $\dfrac{1}{2}$, then shifted to the right $5$ units and up $1$ unit.

150. The graph of $f(x)=x^2$ is horizontally stretched by a factor of $3$, then shifted left $4$ units and down $3$ units.

Answers to Odd Exercises:

141. $g(x)=|x-1|−3$

143. $g(x)=\dfrac{1}{(x+4)^2}+2$

145. $g(x)=|−4x|$

147. $g(x)=\dfrac{1}{3(x+2)^2}−3$

149. $g(x)=\dfrac{1}{2}(x-5)^2+1$

H: Construct equations from transformations of tabular values

Exercise $\PageIndex{H}$

$ \bigstar$ 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. $g(x)=f(x-1)$, $h(x)=f(x)+1$.

I: Identify Increasing/decreasing Intervals

Exercise $\PageIndex{I}$

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

151. $g(x)=5(x+3)^2−2$

152. $f(x)=4(x+1)^2−5$

153. $k(x)=−3\sqrt{x}−1$

154. $a(x)=\sqrt{−x+4}$

Answers to Odd Exercises:

151. decreasing on $(−\infty,−3)$ and increasing on $(−3,\infty)$

153. decreasing on $(0, \infty)$

$ \star $