3.9 Practice Question Solutions

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 [latex]\text{}f\text{}[/latex], substitute [latex]\text{}\left(-x\right)\text{}[/latex] for [latex]\text{}\left(x\right)\text{}[/latex] in [latex]\text{}f\left(x\right)\text{}[/latex]. Simplify. If the resulting function is the same as the original function, [latex]\text{}f\left(-x\right)=f\left(x\right)\text{}[/latex] then the function is even. If the resulting function is the opposite of the original function, [latex]\text{}f\left(-x\right)=-f\left(x\right)\text{}[/latex], then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

7. [latex]g\left(x\right)=|x-1|-3[/latex]

9. [latex]g\left(x\right)=\frac{1}{{\left(x+4\right)}^{2}}+2[/latex]

11. The graph of [latex]\text{}f\left(x+43\right)\text{}[/latex] is a horizontal shift to the left 43 units of the graph of [latex]\text{}f[/latex].

13. The graph of [latex]\text{}f\left(x-4\right)\text{}[/latex] is a horizontal shift to the right 4 units of the graph of [latex]\text{}f[/latex].

15. The graph of [latex]\text{}f\left(x\right)+8\text{}[/latex] is a vertical shift up 8 units of the graph of [latex]f.[/latex]

17. The graph of [latex]\text{}f\left(x\right)-7\text{}[/latex] is a vertical shift down 7 units of the graph of [latex]\text{}f.[/latex]

19. The graph of [latex]f\left(x+4\right)-1[/latex] is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of [latex]f[/latex].

21. decreasing on [latex]\text{}\left(-\infty ,-3\right)\text{}[/latex] and increasing on [latex]\text{}\left(-3,\infty \right)[/latex]

23. decreasing on [latex]\left(0,\text{}\infty \right)[/latex]

25.

27.

29.

31. [latex]g\left(x\right)=f\left(x-1\right),\text{}h\left(x\right)=f\left(x\right)+1[/latex]

33. [latex]f\left(x\right)=|x-3|-2[/latex]

35. [latex]f\left(x\right)=\sqrt{x+3}-1[/latex]

37. [latex]f\left(x\right)={\left(x-2\right)}^{2}[/latex]

39. [latex]f\left(x\right)=|x+3|-2[/latex]

41. [latex]f\left(x\right)=-\sqrt{x}[/latex]

43. [latex]f\left(x\right)=-{\left(x+1\right)}^{2}+2[/latex]

45. [latex]f\left(x\right)=\sqrt{-x}+1[/latex]

47. even

49. odd

51. even

53. The graph of [latex]\text{}g\text{}[/latex] is a vertical reflection (across the [latex]\text{}x[/latex] -axis) of the graph of [latex]\text{}f[/latex].

55. The graph of [latex]\text{}g\text{}[/latex] is a vertical stretch by a factor of 4 of the graph of [latex]\text{}f.[/latex]

57. The graph of [latex]\text{}g\text{}[/latex] is a horizontal compression by a factor of [latex]\text{}\frac{1}{5}\text{}[/latex] of the graph of [latex]\text{}f[/latex].

59. The graph of [latex]\text{}g\text{}[/latex] is a horizontal stretch by a factor of 3 of the graph of [latex]\text{}f[/latex].

61. The graph of [latex]\text{}g\text{}[/latex] is a horizontal reflection across the [latex]\text{}y[/latex] -axis and a vertical stretch by a factor of 3 of the graph of [latex]\text{}f[/latex].

63. [latex]g\left(x\right)=|-4x|[/latex]

65. [latex]g\left(x\right)=\frac{1}{3{\left(x+2\right)}^{2}}-3[/latex]

67. [latex]g\left(x\right)=\frac{1}{2}{\left(x-5\right)}^{2}+1[/latex]

69. The graph of the function [latex]\text{}f\left(x\right)={x}^{2}\text{}[/latex] is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

71. The graph of [latex]\text{}f\left(x\right)=|x|\text{}[/latex] is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

73. The graph of the function [latex]\text{}f\left(x\right)={x}^{3}\text{}[/latex] is compressed vertically by a factor of [latex]\text{}\frac{1}{2}[/latex].

75. The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

77. The graph of [latex]\text{}f\left(x\right)=\sqrt{x}\text{}[/latex] is shifted right 4 units and then reflected across the vertical line [latex]\text{}x=4[/latex].

79.

81.