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Mathematics LibreTexts

2.3e: Exercises - Transformations

  • Page ID
    45447
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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: \([1, ∞)\)

    0e393f0d6e151259a123b1e505dec86b.png
    \(g(x) = x^{2} + 1\)

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

    8fc7f879a8ba5f12d0b98f348e5adadb.png
    \(g(x) = (x − 5)^{2}\)

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

    57a5fd7bcf0e225b10961c6534cd4545.png
    \(g(x) = (x − 5)^{2} + 2\)

    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 \( \frac{1}{2}\)

    ba1195d282dfeb1b813eeec2a0ff6e74.png
    \(h(x) = \frac { 1 } { 2 } ( x - 1 ) ^ { 2 }\)

    #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: \([−5, ∞)\)

    457665c1ea5709240bd4c6e1685a1985.png
    \(g(x) = \sqrt{x} − 5\)

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

    da6d3f21b303aeb0b29fe4975b48a64f.png
    \(g(x) = \sqrt{x − 2} + 1\)

    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 \(2\)

    73fbbd2cc539ff2ab99df22497167aec.png
    \(h ( x ) = \sqrt { - x } + 2\)

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

    576a64916f5e6c8c4b851efadf07189d.png\(g ( x ) = - \frac { 1 } { 2 } \sqrt { x - 3 }\)

     

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

    3a6abc9abd61596a77ecfe672f89976d.png
    \(f(x) = 4 \sqrt { x - 1 } + 2\)

    #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: \([0, ∞)\)

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

    .

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

    424b66df0df22a96fd88c4957413d44e.png
    \(h(x) = |x − 1| − 3\)

    .

    61. Right \(1\), Reflect over \(x\)-axis

    8e5290466d22bfaad7a33f4ffcc1c2d0.png
    \(g(x) = - | x - 1 |\)

    .

    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. \(h(x) = |-3(x-\tfrac{4}{3})| -2 \) \( \longrightarrow\) 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: \(ℝ\)

    01b74b05906d95ff14c5aa6de0ae7b4f.png
    \(h(x) = (x − 2)^{3}\)

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

    a4f584febcd95dc5ef92bbe2ef80df7c.png
    \(h(x) = (x − 1)^{3} − 4\)

     

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

    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: \(\mathbb{R}\)

    ed14f13811bfb7c397b768ab1e6d718a.png
    \(f(x) = x + 3\)

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

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

    Reciprocal Function

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

    75fa23d883d738eeb47a020057002b8f.png
    \(f(x) = \frac{1}{x−2}\)

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

    53d3a12d61be06d8913ae13668760ebb.png
    \(f(x) = \frac{1}{x} + 5\)

    \( \star \)

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

    0eac4ad67881e57bfa8e7dc46c933e8e.png
    \(f(x) = \frac{1}{x+1} − 2\)

    97

    97. Left \(2\) units, reflect over x-axis

    \(f ( x ) = - \frac { 1 } { x + 2 }\)

    #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.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) \(g(x)=f(x−2)\)

    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 \(f\)

    1. \(a(x) = f(x)+1 \)
    2. \(b(x) = f(x+1) \)
    3. \(c(x) = f(x)+2 \)
    4. \(d(x) = -f(x)\)
    5. \(e(x) = f(1-x)-2\)
    6. \(f(x) = 2f(x) \)
    7. \(g(x) = -f(x)\)
    8. \(h(x) = \tfrac{1}{2} f(x+2)+3\) 
    1. \(i(x) = f(x)-1 \)
    2. \(j(x) = f(x-1) \)
    3. \(k(x) = f(x)-2 \)
    4. \(l(x) = f(-x)\)
    5. \(m(x) = -f(x-1)+2\)
    6. \(n(x) = f(2x) \)
    7. \(o(x) = f(-x)\)
    8. \(p(x) = f( \tfrac{1}{2} x-2)-3\)
    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. \(f ( x ) = - ( x - 2 ) ^ { 2 } + 2\)

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

    \( \star \)

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

    \( \star \)

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

    \( \star \)

    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 \((0, \infty)\)

    \( \star \)

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