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Study Guides > MTH 163, Precalculus

Solutions

Solutions to Try Its

1. The path passes through the origin and has vertex at (4, 7)\left(-4,\text{ }7\right)\\, so (h)x=716(x+4)2+7\left(h\right)x=-\frac{7}{16}{\left(x+4\right)}^{2}+7\\. To make the shot, h(7.5)h\left(-7.5\right)\\ would need to be about 4 but h(7.5)1.64h\left(-7.5\right)\approx 1.64\\; he doesn’t make it. 2. g(x)=x26x+13g\left(x\right)={x}^{2}-6x+13\\ in general form; g(x)=(x3)2+4g\left(x\right)={\left(x - 3\right)}^{2}+4\\ in standard form 3. The domain is all real numbers. The range is f(x)811f\left(x\right)\ge \frac{8}{11}\\, or [811,)\left[\frac{8}{11},\infty \right)\\. 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 a=0a=0\\ then the function becomes a linear function. 5. If possible, we can use factoring. Otherwise, we can use the quadratic formula. 7. f(x)=(x+1)22f\left(x\right)={\left(x+1\right)}^{2}-2\\, Vertex (1,4)\left(-1,-4\right)\\ 9. f(x)=(x+52)2334f\left(x\right)={\left(x+\frac{5}{2}\right)}^{2}-\frac{33}{4}\\, Vertex (52,334)\left(-\frac{5}{2},-\frac{33}{4}\right)\\ 11. f(x)=3(x1)212f\left(x\right)=3{\left(x - 1\right)}^{2}-12\\, Vertex (1,12)\left(1,-12\right)\\ 13. f(x)=3(x56)23712f\left(x\right)=3{\left(x-\frac{5}{6}\right)}^{2}-\frac{37}{12}\\, Vertex (56,3712)\left(\frac{5}{6},-\frac{37}{12}\right)\\ 15. Minimum is 172-\frac{17}{2}\\ and occurs at 52\frac{5}{2}\\. Axis of symmetry is x=52x=\frac{5}{2}\\. 17. Minimum is 1716-\frac{17}{16}\\ and occurs at 18-\frac{1}{8}\\. Axis of symmetry is x=18x=-\frac{1}{8}\\. 19. Minimum is 72-\frac{7}{2}\\ and occurs at –3. Axis of symmetry is x=3x=-3\\. 21. Domain is (,)\left(-\infty ,\infty \right)\\. Range is [2,)\left[2,\infty \right)\\. 23. Domain is (,)\left(-\infty ,\infty \right)\\. Range is [5,)\left[-5,\infty \right)\\. 25. Domain is (,)\left(-\infty ,\infty \right)\\. Range is [12,)\left[-12,\infty \right)\\. 27. {2i2,2i2}\left\{2i\sqrt{2},-2i\sqrt{2}\right\}\\ 29. {3i3,3i3}\left\{3i\sqrt{3},-3i\sqrt{3}\right\}\\ 31. {2+i,2i}\left\{2+i,2-i\right\}\\ 33. {2+3i,23i}\left\{2+3i,2 - 3i\right\}\\ 35. {5+i,5i}\left\{5+i,5-i\right\}\\ 37. {2+26,226}\left\{2+2\sqrt{6}, 2 - 2\sqrt{6}\right\}\\ 39. {12+32i,1232i}\left\{-\frac{1}{2}+\frac{3}{2}i, -\frac{1}{2}-\frac{3}{2}i\right\}\\ 41. {35+15i,3515i}\left\{-\frac{3}{5}+\frac{1}{5}i, -\frac{3}{5}-\frac{1}{5}i\right\}\\ 43. {12+12i7,1212i7}\left\{-\frac{1}{2}+\frac{1}{2}i\sqrt{7}, -\frac{1}{2}-\frac{1}{2}i\sqrt{7}\right\}\\ 45. f(x)=x24x+4f\left(x\right)={x}^{2}-4x+4\\ 47. f(x)=x2+1f\left(x\right)={x}^{2}+1\\ 49. f(x)=649x2+6049x+29749f\left(x\right)=\frac{6}{49}{x}^{2}+\frac{60}{49}x+\frac{297}{49}\\ 51. f(x)=x2+1f\left(x\right)=-{x}^{2}+1\\ 53. Vertex (1, 1)\left(1,\text{ }-1\right)\\, Axis of symmetry is x=1x=1\\. Intercepts are (0,0),(2,0)\left(0,0\right), \left(2,0\right)\\. Graph of f(x) = x^2-2x 55. Vertex (52,494)\left(\frac{5}{2},\frac{-49}{4}\right)\\, Axis of symmetry is (0,6),(1,0),(6,0)\left(0,-6\right),\left(-1,0\right),\left(6,0\right)\\. Graph of f(x)x^2-5x-6 57. Vertex (54,398)\left(\frac{5}{4}, -\frac{39}{8}\right)\\, Axis of symmetry is x=54x=\frac{5}{4}\\. Intercepts are (0,8)\left(0, -8\right)\\. Graph of f(x)=-2x^2+5x-8 59. f(x)=x24x+1f\left(x\right)={x}^{2}-4x+1\\ 61. f(x)=2x2+8x1f\left(x\right)=-2{x}^{2}+8x - 1\\ 63. f(x)=12x23x+72f\left(x\right)=\frac{1}{2}{x}^{2}-3x+\frac{7}{2}\\ 65. f(x)=x2+1f\left(x\right)={x}^{2}+1\\ 67. f(x)=2x2f\left(x\right)=2-{x}^{2}\\ 69. f(x)=2x2f\left(x\right)=2{x}^{2}\\ 71. The graph is shifted up or down (a vertical shift). 73. 50 feet 75. Domain is (,)\left(-\infty ,\infty \right)\\. Range is [2,)\left[-2,\infty \right)\\. 77. Domain is (,)\left(-\infty ,\infty \right)\\ Range is (,11]\left(-\infty ,11\right]\\. 79. f(x)=2x21f\left(x\right)=2{x}^{2}-1\\ 81. f(x)=3x29f\left(x\right)=3{x}^{2}-9\\ 83. f(x)=5x277f\left(x\right)=5{x}^{2}-77\\ 85. 50 feet by 50 feet. Maximize f(x)=x2+100xf\left(x\right)=-{x}^{2}+100x\\. 87. 125 feet by 62.5 feet. Maximize f(x)=2x2+250xf\left(x\right)=-2{x}^{2}+250x\\. 89. 6 and –6; product is –36; maximize f(x)=x2+12xf\left(x\right)={x}^{2}+12x\\. 91. 2909.56 meters 93. $10.70

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