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أدلة الدراسة > College Algebra

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)=x2−6x+13g\left(x\right)={x}^{2}-6x+13 in general form; g(x)=(x−3)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)2−2f\left(x\right)={\left(x+1\right)}^{2}-2, Vertex (−1,−4)\left(-1,-4\right) 9. f(x)=(x+52)2−334f\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(x−1)2−12f\left(x\right)=3{\left(x - 1\right)}^{2}-12, Vertex (1,−12)\left(1,-12\right) 13. f(x)=3(x−56)2−3712f\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,2−i}\left\{2+i,2-i\right\} 33. {2+3i,2−3i}\left\{2+3i,2 - 3i\right\} 35. {5+i,5−i}\left\{5+i,5-i\right\} 37. {2+26,2−26}\left\{2+2\sqrt{6}, 2 - 2\sqrt{6}\right\} 39. {−12+32i,−12−32i}\left\{-\frac{1}{2}+\frac{3}{2}i, -\frac{1}{2}-\frac{3}{2}i\right\} 41. {−35+15i,−35−15i}\left\{-\frac{3}{5}+\frac{1}{5}i, -\frac{3}{5}-\frac{1}{5}i\right\} 43. {−12+12i7,−12−12i7}\left\{-\frac{1}{2}+\frac{1}{2}i\sqrt{7}, -\frac{1}{2}-\frac{1}{2}i\sqrt{7}\right\} 45. f(x)=x2−4x+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)=x2−4x+1f\left(x\right)={x}^{2}-4x+1 61. f(x)=−2x2+8x−1f\left(x\right)=-2{x}^{2}+8x - 1 63. f(x)=12x2−3x+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)=2−x2f\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)=2x2−1f\left(x\right)=2{x}^{2}-1 81. f(x)=3x2−9f\left(x\right)=3{x}^{2}-9 83. f(x)=5x2−77f\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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