Solutions 13: Quadratic Functions

Solutions to Try Its

1. The path passes through the origin and has vertex at [latex]\left(-4,\text{ }7\right)[/latex], so [latex]\left(h\right)x=-\frac{7}{16}{\left(x+4\right)}^{2}+7[/latex]. To make the shot, [latex]h\left(-7.5\right)[/latex] would need to be about 4 but [latex]h\left(-7.5\right)\approx 1.64[/latex]; he doesn’t make it.

2. [latex]g\left(x\right)={x}^{2}-6x+13[/latex] in general form; [latex]g\left(x\right)={\left(x - 3\right)}^{2}+4[/latex] in standard form

3. The domain is all real numbers. The range is [latex]f\left(x\right)\ge \frac{8}{11}[/latex], or [latex]\left[\frac{8}{11},\infty \right)[/latex].

4. y-intercept at (0, 13), No x-intercepts

5. a. 3 seconds  b. 256 feet  c. 7 seconds

Solutions to Odd-Numbered Exercises

1. When written in that form, the vertex can be easily identified.

3. If [latex]a=0[/latex] then the function becomes a linear function.

5. If possible, we can use factoring. Otherwise, we can use the quadratic formula.

7. [latex]f\left(x\right)={\left(x+1\right)}^{2}-2[/latex], Vertex [latex]\left(-1,-4\right)[/latex]

9. [latex]f\left(x\right)={\left(x+\frac{5}{2}\right)}^{2}-\frac{33}{4}[/latex], Vertex [latex]\left(-\frac{5}{2},-\frac{33}{4}\right)[/latex]

11. [latex]f\left(x\right)=3{\left(x - 1\right)}^{2}-12[/latex], Vertex [latex]\left(1,-12\right)[/latex]

13. [latex]f\left(x\right)=3{\left(x-\frac{5}{6}\right)}^{2}-\frac{37}{12}[/latex], Vertex [latex]\left(\frac{5}{6},-\frac{37}{12}\right)[/latex]

15. Minimum is [latex]-\frac{17}{2}[/latex] and occurs at [latex]\frac{5}{2}[/latex]. Axis of symmetry is [latex]x=\frac{5}{2}[/latex].

17. Minimum is [latex]-\frac{17}{16}[/latex] and occurs at [latex]-\frac{1}{8}[/latex]. Axis of symmetry is [latex]x=-\frac{1}{8}[/latex].

19. Minimum is [latex]-\frac{7}{2}[/latex] and occurs at –3. Axis of symmetry is [latex]x=-3[/latex].

21. Domain is [latex]\left(-\infty ,\infty \right)[/latex]. Range is [latex]\left[2,\infty \right)[/latex].

23. Domain is [latex]\left(-\infty ,\infty \right)[/latex]. Range is [latex]\left[-5,\infty \right)[/latex].

25. Domain is [latex]\left(-\infty ,\infty \right)[/latex]. Range is [latex]\left[-12,\infty \right)[/latex].

27. [latex]\left\{2i\sqrt{2},-2i\sqrt{2}\right\}[/latex]

29. [latex]\left\{3i\sqrt{3},-3i\sqrt{3}\right\}[/latex]

31. [latex]\left\{2+i,2-i\right\}[/latex]

33. [latex]\left\{2+3i,2 - 3i\right\}[/latex]

35. [latex]\left\{5+i,5-i\right\}[/latex]

37. [latex]\left\{2+2\sqrt{6}, 2 - 2\sqrt{6}\right\}[/latex]

39. [latex]\left\{-\frac{1}{2}+\frac{3}{2}i, -\frac{1}{2}-\frac{3}{2}i\right\}[/latex]

41. [latex]\left\{-\frac{3}{5}+\frac{1}{5}i, -\frac{3}{5}-\frac{1}{5}i\right\}[/latex]

43. [latex]\left\{-\frac{1}{2}+\frac{1}{2}i\sqrt{7}, -\frac{1}{2}-\frac{1}{2}i\sqrt{7}\right\}[/latex]

45. [latex]f\left(x\right)={x}^{2}-4x+4[/latex]

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

49. [latex]f\left(x\right)=\frac{6}{49}{x}^{2}+\frac{60}{49}x+\frac{297}{49}[/latex]

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

53. Vertex [latex]\left(1,\text{ }-1\right)[/latex], Axis of symmetry is [latex]x=1[/latex]. Intercepts are [latex]\left(0,0\right), \left(2,0\right)[/latex].

Graph of f(x) = x^2-2x

55. Vertex [latex]\left(\frac{5}{2},\frac{-49}{4}\right)[/latex], Axis of symmetry is [latex]\left(0,-6\right),\left(-1,0\right),\left(6,0\right)[/latex].

Graph of f(x)x^2-5x-6

57. Vertex [latex]\left(\frac{5}{4}, -\frac{39}{8}\right)[/latex], Axis of symmetry is [latex]x=\frac{5}{4}[/latex]. Intercepts are [latex]\left(0, -8\right)[/latex].

Graph of f(x)=-2x^2+5x-8

59. [latex]f\left(x\right)={x}^{2}-4x+1[/latex]

61. [latex]f\left(x\right)=-2{x}^{2}+8x - 1[/latex]

63. [latex]f\left(x\right)=\frac{1}{2}{x}^{2}-3x+\frac{7}{2}[/latex]

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

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

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

71. The graph is shifted up or down (a vertical shift).

73. 50 feet

75. Domain is [latex]\left(-\infty ,\infty \right)[/latex]. Range is [latex]\left[-2,\infty \right)[/latex].

77. Domain is [latex]\left(-\infty ,\infty \right)[/latex] Range is [latex]\left(-\infty ,11\right][/latex].

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

81. [latex]f\left(x\right)=3{x}^{2}-9[/latex]

83. [latex]f\left(x\right)=5{x}^{2}-77[/latex]

85. 50 feet by 50 feet. Maximize [latex]f\left(x\right)=-{x}^{2}+100x[/latex].

87. 125 feet by 62.5 feet. Maximize [latex]f\left(x\right)=-2{x}^{2}+250x[/latex].

89. 6 and –6; product is –36; maximize [latex]f\left(x\right)={x}^{2}+12x[/latex].

91. 2909.56 meters

93. $10.70