{"id":11042,"date":"2015-07-14T17:56:33","date_gmt":"2015-07-14T17:56:33","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=11042"},"modified":"2018-06-18T19:45:09","modified_gmt":"2018-06-18T19:45:09","slug":"solutions-11","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/solutions-11\/","title":{"raw":"Solutions Chapter 3: Quadratic Functions","rendered":"Solutions Chapter 3: Quadratic Functions"},"content":{"raw":"

Solutions to Try Its<\/h2>\r\n1.\u00a0The 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\u2019t make it.\r\n\r\n2.\u00a0[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\r\n\r\n3.\u00a0The 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].\r\n\r\n4.\u00a0y<\/em>-intercept at (0, 13), No x-<\/em>intercepts\r\n\r\n5. a.\u00a03 seconds \u00a0b.\u00a0256 feet \u00a0c.\u00a07 seconds\r\n

Solutions to Odd-Numbered Exercises<\/h2>\r\n1.\u00a0When written in that form, the vertex can be easily identified.\r\n\r\n3.\u00a0If [latex]a=0[\/latex] then the function becomes a linear function.\r\n\r\n5.\u00a0If possible, we can use factoring. Otherwise, we can use the quadratic formula.\r\n\r\n7.\u00a0[latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-2[\/latex], Vertex [latex]\\left(-1,-4\\right)[\/latex]\r\n\r\n9.\u00a0[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]\r\n\r\n11.\u00a0[latex]f\\left(x\\right)=3{\\left(x - 1\\right)}^{2}-12[\/latex], Vertex [latex]\\left(1,-12\\right)[\/latex]\r\n\r\n13.\u00a0[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]\r\n\r\n15.\u00a0Minimum is [latex]-\\frac{17}{2}[\/latex] and occurs at [latex]\\frac{5}{2}[\/latex]. Axis of symmetry is [latex]x=\\frac{5}{2}[\/latex].\r\n\r\n17.\u00a0Minimum is [latex]-\\frac{17}{16}[\/latex] and occurs at [latex]-\\frac{1}{8}[\/latex]. Axis of symmetry is [latex]x=-\\frac{1}{8}[\/latex].\r\n\r\n19.\u00a0Minimum is [latex]-\\frac{7}{2}[\/latex] and occurs at \u20133. Axis of symmetry is [latex]x=-3[\/latex].\r\n\r\n21.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[2,\\infty \\right)[\/latex].\r\n\r\n23.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-5,\\infty \\right)[\/latex].\r\n\r\n25.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-12,\\infty \\right)[\/latex].\r\n\r\n27.\u00a0[latex]\\left\\{2i\\sqrt{2},-2i\\sqrt{2}\\right\\}[\/latex]\r\n\r\n29.\u00a0[latex]\\left\\{3i\\sqrt{3},-3i\\sqrt{3}\\right\\}[\/latex]\r\n\r\n31.\u00a0[latex]\\left\\{2+i,2-i\\right\\}[\/latex]\r\n\r\n33.\u00a0[latex]\\left\\{2+3i,2 - 3i\\right\\}[\/latex]\r\n\r\n35.\u00a0[latex]\\left\\{5+i,5-i\\right\\}[\/latex]\r\n\r\n37.\u00a0[latex]\\left\\{2+2\\sqrt{6}, 2 - 2\\sqrt{6}\\right\\}[\/latex]\r\n\r\n39.\u00a0[latex]\\left\\{-\\frac{1}{2}+\\frac{3}{2}i, -\\frac{1}{2}-\\frac{3}{2}i\\right\\}[\/latex]\r\n\r\n41.\u00a0[latex]\\left\\{-\\frac{3}{5}+\\frac{1}{5}i, -\\frac{3}{5}-\\frac{1}{5}i\\right\\}[\/latex]\r\n\r\n43.\u00a0[latex]\\left\\{-\\frac{1}{2}+\\frac{1}{2}i\\sqrt{7}, -\\frac{1}{2}-\\frac{1}{2}i\\sqrt{7}\\right\\}[\/latex]\r\n\r\n45.\u00a0[latex]f\\left(x\\right)={x}^{2}-4x+4[\/latex]\r\n\r\n47.\u00a0[latex]f\\left(x\\right)={x}^{2}+1[\/latex]\r\n\r\n49.\u00a0[latex]f\\left(x\\right)=\\frac{6}{49}{x}^{2}+\\frac{60}{49}x+\\frac{297}{49}[\/latex]\r\n\r\n51.\u00a0[latex]f\\left(x\\right)=-{x}^{2}+1[\/latex]\r\n\r\n53.\u00a0Vertex [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].\r\n\r\n\"Graph\r\n\r\n55.\u00a0Vertex [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].\r\n\r\n\"Graph\r\n\r\n57.\u00a0Vertex [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].\r\n\r\n\"Graph\r\n\r\n59.\u00a0[latex]f\\left(x\\right)={x}^{2}-4x+1[\/latex]\r\n\r\n61.\u00a0[latex]f\\left(x\\right)=-2{x}^{2}+8x - 1[\/latex]\r\n\r\n63.\u00a0[latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}-3x+\\frac{7}{2}[\/latex]\r\n\r\n65.\u00a0[latex]f\\left(x\\right)={x}^{2}+1[\/latex]\r\n\r\n67.\u00a0[latex]f\\left(x\\right)=2-{x}^{2}[\/latex]\r\n\r\n69.\u00a0[latex]f\\left(x\\right)=2{x}^{2}[\/latex]\r\n\r\n71.\u00a0The graph is shifted up or down (a vertical shift).\r\n\r\n73.\u00a050 feet\r\n\r\n75.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-2,\\infty \\right)[\/latex].\r\n\r\n77.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex] Range is [latex]\\left(-\\infty ,11\\right][\/latex].\r\n\r\n79.\u00a0[latex]f\\left(x\\right)=2{x}^{2}-1[\/latex]\r\n\r\n81.\u00a0[latex]f\\left(x\\right)=3{x}^{2}-9[\/latex]\r\n\r\n83.\u00a0[latex]f\\left(x\\right)=5{x}^{2}-77[\/latex]\r\n\r\n85.\u00a050 feet by 50 feet. Maximize [latex]f\\left(x\\right)=-{x}^{2}+100x[\/latex].\r\n\r\n87.\u00a0125 feet by 62.5 feet. Maximize [latex]f\\left(x\\right)=-2{x}^{2}+250x[\/latex].\r\n\r\n89. 6 and \u20136; product is \u201336; maximize [latex]f\\left(x\\right)={x}^{2}+12x[\/latex].\r\n\r\n91.\u00a02909.56 meters\r\n\r\n93.\u00a0$10.70","rendered":"

Solutions to Try Its<\/h2>\n

1.\u00a0The 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\u2019t make it.<\/p>\n

2.\u00a0[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<\/p>\n

3.\u00a0The 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].<\/p>\n

4.\u00a0y<\/em>-intercept at (0, 13), No x-<\/em>intercepts<\/p>\n

5. a.\u00a03 seconds \u00a0b.\u00a0256 feet \u00a0c.\u00a07 seconds<\/p>\n

Solutions to Odd-Numbered Exercises<\/h2>\n

1.\u00a0When written in that form, the vertex can be easily identified.<\/p>\n

3.\u00a0If [latex]a=0[\/latex] then the function becomes a linear function.<\/p>\n

5.\u00a0If possible, we can use factoring. Otherwise, we can use the quadratic formula.<\/p>\n

7.\u00a0[latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-2[\/latex], Vertex [latex]\\left(-1,-4\\right)[\/latex]<\/p>\n

9.\u00a0[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]<\/p>\n

11.\u00a0[latex]f\\left(x\\right)=3{\\left(x - 1\\right)}^{2}-12[\/latex], Vertex [latex]\\left(1,-12\\right)[\/latex]<\/p>\n

13.\u00a0[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]<\/p>\n

15.\u00a0Minimum is [latex]-\\frac{17}{2}[\/latex] and occurs at [latex]\\frac{5}{2}[\/latex]. Axis of symmetry is [latex]x=\\frac{5}{2}[\/latex].<\/p>\n

17.\u00a0Minimum is [latex]-\\frac{17}{16}[\/latex] and occurs at [latex]-\\frac{1}{8}[\/latex]. Axis of symmetry is [latex]x=-\\frac{1}{8}[\/latex].<\/p>\n

19.\u00a0Minimum is [latex]-\\frac{7}{2}[\/latex] and occurs at \u20133. Axis of symmetry is [latex]x=-3[\/latex].<\/p>\n

21.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[2,\\infty \\right)[\/latex].<\/p>\n

23.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-5,\\infty \\right)[\/latex].<\/p>\n

25.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-12,\\infty \\right)[\/latex].<\/p>\n

27.\u00a0[latex]\\left\\{2i\\sqrt{2},-2i\\sqrt{2}\\right\\}[\/latex]<\/p>\n

29.\u00a0[latex]\\left\\{3i\\sqrt{3},-3i\\sqrt{3}\\right\\}[\/latex]<\/p>\n

31.\u00a0[latex]\\left\\{2+i,2-i\\right\\}[\/latex]<\/p>\n

33.\u00a0[latex]\\left\\{2+3i,2 - 3i\\right\\}[\/latex]<\/p>\n

35.\u00a0[latex]\\left\\{5+i,5-i\\right\\}[\/latex]<\/p>\n

37.\u00a0[latex]\\left\\{2+2\\sqrt{6}, 2 - 2\\sqrt{6}\\right\\}[\/latex]<\/p>\n

39.\u00a0[latex]\\left\\{-\\frac{1}{2}+\\frac{3}{2}i, -\\frac{1}{2}-\\frac{3}{2}i\\right\\}[\/latex]<\/p>\n

41.\u00a0[latex]\\left\\{-\\frac{3}{5}+\\frac{1}{5}i, -\\frac{3}{5}-\\frac{1}{5}i\\right\\}[\/latex]<\/p>\n

43.\u00a0[latex]\\left\\{-\\frac{1}{2}+\\frac{1}{2}i\\sqrt{7}, -\\frac{1}{2}-\\frac{1}{2}i\\sqrt{7}\\right\\}[\/latex]<\/p>\n

45.\u00a0[latex]f\\left(x\\right)={x}^{2}-4x+4[\/latex]<\/p>\n

47.\u00a0[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\n

49.\u00a0[latex]f\\left(x\\right)=\\frac{6}{49}{x}^{2}+\\frac{60}{49}x+\\frac{297}{49}[\/latex]<\/p>\n

51.\u00a0[latex]f\\left(x\\right)=-{x}^{2}+1[\/latex]<\/p>\n

53.\u00a0Vertex [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].<\/p>\n

\"Graph<\/p>\n

55.\u00a0Vertex [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].<\/p>\n

\"Graph<\/p>\n

57.\u00a0Vertex [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].<\/p>\n

\"Graph<\/p>\n

59.\u00a0[latex]f\\left(x\\right)={x}^{2}-4x+1[\/latex]<\/p>\n

61.\u00a0[latex]f\\left(x\\right)=-2{x}^{2}+8x - 1[\/latex]<\/p>\n

63.\u00a0[latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}-3x+\\frac{7}{2}[\/latex]<\/p>\n

65.\u00a0[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\n

67.\u00a0[latex]f\\left(x\\right)=2-{x}^{2}[\/latex]<\/p>\n

69.\u00a0[latex]f\\left(x\\right)=2{x}^{2}[\/latex]<\/p>\n

71.\u00a0The graph is shifted up or down (a vertical shift).<\/p>\n

73.\u00a050 feet<\/p>\n

75.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-2,\\infty \\right)[\/latex].<\/p>\n

77.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex] Range is [latex]\\left(-\\infty ,11\\right][\/latex].<\/p>\n

79.\u00a0[latex]f\\left(x\\right)=2{x}^{2}-1[\/latex]<\/p>\n

81.\u00a0[latex]f\\left(x\\right)=3{x}^{2}-9[\/latex]<\/p>\n

83.\u00a0[latex]f\\left(x\\right)=5{x}^{2}-77[\/latex]<\/p>\n

85.\u00a050 feet by 50 feet. Maximize [latex]f\\left(x\\right)=-{x}^{2}+100x[\/latex].<\/p>\n

87.\u00a0125 feet by 62.5 feet. Maximize [latex]f\\left(x\\right)=-2{x}^{2}+250x[\/latex].<\/p>\n

89. 6 and \u20136; product is \u201336; maximize [latex]f\\left(x\\right)={x}^{2}+12x[\/latex].<\/p>\n

91.\u00a02909.56 meters<\/p>\n

93.\u00a0$10.70<\/p>\n\n\t\t\t

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