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Mathematics LibreTexts

1.5.5E: Transformation of Functions

  • Page ID
    30242
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    Section 1.5 Exercises

    Describe how each function is a transformation of the original function \(f(x)\)

    1. \(f(x - 49)\)

    2. \(f(x + 43)\)

    3. \(f(x+3)\)

    4. \(f(x-4)\)

    5. \(f(x)+5\)

    6. \(f(x)+8\)

    7. \(f(x)-2\)

    8. \(f(x)-7\)

    9. \(f(x-2)+3\)

    10. \(f(x+4)-1\)

    11. Write a formula for \(f(x)=\sqrt{x}\) shifted up 1 unit and left 2 units.

    12. Write a formula for \(f(x)=|x|\) shifted down 3 units and right 1 unit.

    13. Write a formula for \(f(x)=\dfrac{1}{x}\) shifted down 4 units and right 3 units.

    14. Write a formula for \(\; f(x)=\dfrac{1}{x^{2} }\) shifted up 2 units and left 4 units.

    15. Tables of values for \(f(x)\), \(g(x)\), and \(h(x)\) are given below. Write \(g(x)\) and \(h(x)\) as transformations of \(f(x)\).

    屏幕快照 2019-06-14 下午3.07.52.png

    16. Tables of values for \(f(x)\), \(g(x)\), and \(h(x)\) are given below. Write \(g(x)\) and \(h(x)\) as transformations of \(f(x)\).

    屏幕快照 2019-06-14 下午3.08.46.png

    The graph of \(f(x)=2^{x}\) is shown. Sketch a graph of each transformation of \(f(x)\).

    17. \(g(x)=2^{x} +1\)屏幕快照 2019-06-14 下午3.10.53.png

    18. \(h(x)=2^{x} -3\)

    19. \(w(x)=2^{x-1}\)

    20. \(q(x)=2^{x+3}\)

    Sketch a graph of each function as a transformation of a toolkit function.

    21. \(f(t)=(t+1)^{2} -3\)

    22. \(h(x)=|x-1|+4\)

    23. \(k(x=(x-2)^{3} -1\)

    24. \(m(t)=3+\sqrt{t+2}\)

    Write an equation for each function graphed below.

    25. 屏幕快照 2019-06-14 下午3.13.43.png

    26. 屏幕快照 2019-06-14 下午3.14.17.png

    27. 屏幕快照 2019-06-14 下午3.14.58.png

    28.屏幕快照 2019-06-14 下午3.15.37.png

    Find a formula for each of the transformations of the square root whose graphs are given below.

    29. 屏幕快照 2019-06-14 下午3.16.39.png

    30. 屏幕快照 2019-06-14 下午3.17.08.png

    The graph of \(f(x)=2^{x}\) is shown. Sketch a graph of each transformation of \(f(x)\)

    屏幕快照 2019-06-14 下午3.18.02.png

    31. \(g(x)=-2^{x} +1\)

    32. \(h(x)=2^{-x}\)

    33. Starting with the graph of \(f(x)= 6^{x}\) write the equation of the graph that results from

    a. reflecting \(f(x)\) about the \(x\)-axis and the \(y\)-axis

    b. reflecting \(f(x)\) about the \(x\)-axis, shifting left 2 units, and down 3 units

    34. Starting with the graph of \(f(x)= 4^{x}\) write the equation of the graph that results from

    a. reflecting \(f(x)\) about the \(x\)-axis

    b. reflecting \(f(x)\) about the \(y\)-axis, shifting right 4 units, and up 2 units

    Write an equation for each function graphed below.

    35. 屏幕快照 2019-06-14 下午3.21.50.png 36. 屏幕快照 2019-06-14 下午3.22.23.png

    37. 屏幕快照 2019-06-14 下午3.23.15.png 38.屏幕快照 2019-06-14 下午3.23.48.png

    39. For each equation below, determine if the function is Odd, Even, or Neither.

    a. \(f(x)=3 x^{4}\)

    b. \(g(x)=\sqrt{x}\)

    c. \(h(x)=\dfrac{1}{x} +3 x\)

    40. For each equation below, determine if the function is Odd, Even, or Neither.

    a. \(f(x)=(x-2)^{2}\)

    b. \(g(x)=2 x^{4}\)

    c. \(h(x)=2 x-x^{3}\)

    Describe how each function is a transformation of the original function \(f(x)\).

    41. \(-f(x)\)

    42. \(f(-x)\)

    43. \(4f(x)\)

    44. \(6f(x)\)

    45. \(f(5x)\)

    46. \(f(2x)\)

    47. \(f(\dfrac{1}{3} x)\)

    48. \(f(\dfrac{1}{5} x)\)

    49. \(3f(-x)\)

    50. \(-f(3x)\)

    Write a formula for the function that results when the given toolkit function is transformed as described.

    51. \(f(x)=|x|\) reflected over the y axis and horizontally compressed by a factor of \(\dfrac{1}{4}\).

    52. \(f(x)=\sqrt{x}\) reflected over the x axis and horizontally stretched by a factor of 2.

    53. \(f(x)=\dfrac{1}{x^{2} }\) vertically compressed by a factor of \(\dfrac{1}{3}\), then shifted to the left 2 units and down 3 units.

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

    55. \(f(x)=x^{2}\) horizontally compressed by a factor of \(\dfrac{1}{2}\), then shifted to the right 5 units and up 1 unit.

    56. \(f(x)=x^{2}\) horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

    Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

    57. \(f\left(x\right)=4(x+1)^{2} -5\)

    58. \(g(x)=5(x+3)^{2} -2\)

    59. \(h(x)=-2|x-4|+3\)

    60. \(k(x)=-3\sqrt{x} -1\)

    61. \(m(x)=\dfrac{1}{2} x^{3}\)

    62. \(n(x)=\dfrac{1}{3} |x-2|\)

    63. \(p(x)=(\dfrac{1}{3} x)^{2} -3\)

    64. \(q(x)=(\dfrac{1}{4} x)^{3} +1\)

    65. \(a(x)=\sqrt{-x+4}\)

    66. \(b(x)=\sqrt[{3}]{-x-6}\)

    Determine the interval(s) on which the function is increasing and decreasing.

    67. \(f(x)=4(x+1)^{2} -5\)

    68. \(g(x)=5(x+3)^{2} -2\)

    69. \(a(x)=\sqrt{-x+4}\)

    70. \(k(x)=-3\sqrt{x} -1\)

    Determine the interval(s) on which the function is concave up and concave down.

    71. \(m(x)=-2(x+3)^{3} +1\)

    72. \(b(x)=\sqrt[{3}]{-x-6}\)

    73. \(p(x)=(\dfrac{1}{3} x)^{2} -3\)

    74. \(k(x)=-3\sqrt{x} -1\)

    The function \(f(x)\) is graphed here. Write an equation for each graph below as a transformation of \(f(x)\).

    75.屏幕快照 2019-06-14 下午3.38.50.png76.屏幕快照 2019-06-14 下午3.39.25.png77.屏幕快照 2019-06-14 下午3.40.01.png

    78.屏幕快照 2019-06-14 下午3.40.25.png79. 屏幕快照 2019-06-14 下午3.41.11.png80. 屏幕快照 2019-06-14 下午3.41.43.png

    81.屏幕快照 2019-06-14 下午3.42.36.png82.屏幕快照 2019-06-14 下午3.43.25.png 83. 屏幕快照 2019-06-14 下午3.43.59.png

    84. 屏幕快照 2019-06-14 下午3.44.41.png85.屏幕快照 2019-06-14 下午3.45.12.png 86.屏幕快照 2019-06-14 下午3.45.47.png

    Write an equation for each transformed toolkit function graphed below.

    87. 屏幕快照 2019-06-14 下午3.47.01.png88.屏幕快照 2019-06-14 下午3.47.33.png 89. 屏幕快照 2019-06-14 下午3.47.59.png

    90. 屏幕快照 2019-06-14 下午3.48.35.png91. 屏幕快照 2019-06-14 下午3.49.16.png92.屏幕快照 2019-06-14 下午3.49.47.png

    93. 屏幕快照 2019-06-14 下午3.53.14.png94.屏幕快照 2019-06-14 下午3.53.57.png 95. 屏幕快照 2019-06-14 下午3.54.44.png

    96. 屏幕快照 2019-06-14 下午3.55.27.png97. 屏幕快照 2019-06-14 下午3.57.57.png98.屏幕快照 2019-06-14 下午3.59.00.png

    Write a formula for the piecewise function graphed below.

    99. 屏幕快照 2019-06-14 下午3.59.40.png100. 屏幕快照 2019-06-14 下午4.00.25.png

    101. 屏幕快照 2019-06-14 下午4.01.04.png102. 屏幕快照 2019-06-14 下午4.01.26.png

    103. Suppose you have a function \(y = f(x)\) such that the domain of \(f(x)\) is \(1 \le x \le 6\) and the range of \(f(x)\) is (-3 \le y \le 5\). [UW]

    a. What is the domain of \(\; f(2(x-3))\;\)?

    b. What is the range of \(f(2(x-3))\) ?

    c. What is the domain of \(2f(x)-3\) ?

    d. What is the range of \(2f(x)-3\) ?

    e. Can you find constants \(B\) and \(C\) so that the domain of \(f(B(x-C))\) is \(8 \le x \le 9\)?

    f. Can you find constants \(A\) and \(D\) so that the range of \(Af(x) + D\) is 0 \(0 \le y \le 1\)?

    Answer

    1. Horizontal shift right 49 units

    3. Horizontal shift left 3 units

    5. Vertical shift up 5 units

    7. Vertical shift down 2 units

    9. Horizontal shift right 2 units, Vertical shift up 3 units

    11. \(f(x + 2) + 1 = \sqrt{x + 2} + 1\)

    13. \(f(x - 3) - 4 = \dfrac{1}{x - 3} - 4\)

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

    17. Screen Shot 2019-10-01 at 8.56.34 AM.png19. Screen Shot 2019-10-01 at 8.56.55 AM.png

    21. Screen Shot 2019-10-01 at 8.57.26 AM.png23. Screen Shot 2019-10-01 at 8.57.54 AM.png

    25. \(y = |x - 3| - 2\)

    27. \(y = \sqrt{x + 3} - 1\)

    29. \(y = -\sqrt{x}\)

    31. Screen Shot 2019-10-01 at 8.58.26 AM.png

    33. a. \(-f(-x) = -6^{-x}\)
    b. \(-f(x + 2) - 3 = -6^{x + 2} - 3\)

    35. \(y = -(x + 1)^2 + 2\)

    37. \(y = \sqrt{-x} + 1\)

    39. a. Even
    b. Neither
    c. Odd

    41. Reflect \(f(x)\) about the \(x\)-axis

    43. Vertically stretch \(y\) values by 4

    45. Horizontally compress \(x\) values by 1/5

    47. Horizontally stretch \(x\) values by 3

    49. Reflect \(f(x)\) about the \(y\)-axis and vertically stretch \(y\) values by 3

    51. \(f(-4x) = |-4x|\)

    53. \(\dfrac{1}{3} f(x + 2) - 3 = \dfrac{1}{3(x + 2)^2} - 3\)

    55. \(f(2(x - 5)) + 1 = (2 (x - 5))^2 + 1\)

    57. Horizontal shift left 1 unit, vertical stretch \(y\) values by 4, vertical shift down 5 units

    Screen Shot 2019-10-01 at 9.00.14 AM.png becomes Screen Shot 2019-10-01 at 9.00.39 AM.png

    59. Horizontal shift right 4 units, vertical stretch \(y\) values by 2, reflect over \(x\) axis, vertically shift up 3 units.

    Screen Shot 2019-10-01 at 9.01.08 AM.png becomes Screen Shot 2019-10-01 at 9.01.37 AM.png

    61. Vertically compress \(y\) values by 1/2

    Screen Shot 2019-10-01 at 9.06.17 AM.png becomes Screen Shot 2019-10-01 at 9.06.43 AM.png

    63. Horizontally stretch \(x\) values by 3, vertical shift down 3 units

    Screen Shot 2019-10-01 at 9.07.21 AM.png becomes Screen Shot 2019-10-01 at 9.07.43 AM.png

    65. Reflected over the \(y\) axis, horizontally shift right 4 units \(a(x) = \sqrt{-(x - 4)}\)

    Screen Shot 2019-10-01 at 9.08.13 AM.png becomes Screen Shot 2019-10-01 at 9.08.42 AM.png

    67. This function is increasing on \((-1, \infty)\) and decreasing on \((-\infty, -1)\)

    69. This function is decreasing on \((-\infty, 4)\)

    71. This function is concave down on \((-3, \infty)\) and concave up on \((-\infty, -3)\)

    73. This function is concave up everywhere

    75. \(f(-x)\)

    77. \(3f(x)\)

    79. 2\(f(-x)\)

    81. \(2f(\dfrac{1}{2}x)\)

    83. \(2f(x) - 2\)

    85. \(-f(x + 1) + 3\)

    87. \(y = -2(x + 2)^2 + 3\)

    89. \(y = (\dfrac{1}{2} (x - 1))^3 + 2\)

    91. \(y = \sqrt{2(x + 2)} + 1\)

    93. \(y = \dfrac{-1}{(x - 2)^2} + 3\)

    95. \(y = -2|x + 1| + 3\)

    97. \(y = \sqrt[3]{-\dfrac{1}{2}(x - 2)} + 1\)

    99. \(f(x) = \begin{cases} (x+3)^2 + 1 & if & x \le -2 \\ \dfrac{1}{2}|x - 2| + 3 & if & x > -2 \end{cases}\)

    101. \(f(x) = \begin{cases} 1 & if & x < -2 \\ -2(x + 1)^2 + 4 & if & -2 \le x \le 1 \\ \sqrt[3]{x - 2} + 1 & if & x > 1 \end{cases}\)

    103a. Domain: \(3.5 \le x \le 6\)
    d. Range: \(-9 \le y \le 7\)