{"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&#038;p=11042"},"modified":"2021-12-29T19:08:17","modified_gmt":"2021-12-29T19:08:17","slug":"solutions-quadratic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/solutions-quadratic-functions\/","title":{"raw":"Solutions: Quadratic Functions","rendered":"Solutions: Quadratic Functions"},"content":{"raw":"<h2>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.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-2,\\infty \\right)[\/latex].\r\n\r\n29.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex] Range is [latex]\\left(-\\infty ,11\\right][\/latex].\r\n\r\n31.\u00a0[latex]f\\left(x\\right)={x}^{2}+1[\/latex]\r\n\r\n33.\u00a0[latex]f\\left(x\\right)=\\frac{6}{49}{x}^{2}+\\frac{60}{49}x+\\frac{297}{49}[\/latex]\r\n\r\n35.\u00a0[latex]f\\left(x\\right)=-{x}^{2}+1[\/latex]\r\n\r\n37.\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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005209\/CNX_Precalc_Figure_03_02_201.jpg\" alt=\"Graph of f(x) = x^2-2x\" \/>\r\n\r\n39.\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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005209\/CNX_Precalc_Figure_03_02_203.jpg\" alt=\"Graph of f(x)x^2-5x-6\" \/>\r\n\r\n41.\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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005209\/CNX_Precalc_Figure_03_02_205.jpg\" alt=\"Graph of f(x)=-2x^2+5x-8\" \/>\r\n\r\n43.\u00a050 feet\r\n\r\n45.\u00a050 feet by 50 feet. Maximize [latex]f\\left(x\\right)=-{x}^{2}+100x[\/latex].\r\n\r\n47.\u00a0125 feet by 62.5 feet. Maximize [latex]f\\left(x\\right)=-2{x}^{2}+250x[\/latex].\r\n\r\n49. 6 and \u20136; product is \u201336; maximize [latex]f\\left(x\\right)={x}^{2}+12x[\/latex].\r\n\r\n51.\u00a02909.56 meters\r\n\r\n53.\u00a0$10.70","rendered":"<h2>Solutions to Odd-Numbered Exercises<\/h2>\n<p>1.\u00a0When written in that form, the vertex can be easily identified.<\/p>\n<p>3.\u00a0If [latex]a=0[\/latex] then the function becomes a linear function.<\/p>\n<p>5.\u00a0If possible, we can use factoring. Otherwise, we can use the quadratic formula.<\/p>\n<p>7.\u00a0[latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-2[\/latex], Vertex [latex]\\left(-1,-4\\right)[\/latex]<\/p>\n<p>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<p>11.\u00a0[latex]f\\left(x\\right)=3{\\left(x - 1\\right)}^{2}-12[\/latex], Vertex [latex]\\left(1,-12\\right)[\/latex]<\/p>\n<p>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<p>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<p>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<p>19.\u00a0Minimum is [latex]-\\frac{7}{2}[\/latex] and occurs at \u20133. Axis of symmetry is [latex]x=-3[\/latex].<\/p>\n<p>21.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[2,\\infty \\right)[\/latex].<\/p>\n<p>23.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-5,\\infty \\right)[\/latex].<\/p>\n<p>25.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-12,\\infty \\right)[\/latex].<\/p>\n<p>27.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. Range is [latex]\\left[-2,\\infty \\right)[\/latex].<\/p>\n<p>29.\u00a0Domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex] Range is [latex]\\left(-\\infty ,11\\right][\/latex].<\/p>\n<p>31.\u00a0[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\n<p>33.\u00a0[latex]f\\left(x\\right)=\\frac{6}{49}{x}^{2}+\\frac{60}{49}x+\\frac{297}{49}[\/latex]<\/p>\n<p>35.\u00a0[latex]f\\left(x\\right)=-{x}^{2}+1[\/latex]<\/p>\n<p>37.\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<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005209\/CNX_Precalc_Figure_03_02_201.jpg\" alt=\"Graph of f(x) = x^2-2x\" \/><\/p>\n<p>39.\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<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005209\/CNX_Precalc_Figure_03_02_203.jpg\" alt=\"Graph of f(x)x^2-5x-6\" \/><\/p>\n<p>41.\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<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005209\/CNX_Precalc_Figure_03_02_205.jpg\" alt=\"Graph of f(x)=-2x^2+5x-8\" \/><\/p>\n<p>43.\u00a050 feet<\/p>\n<p>45.\u00a050 feet by 50 feet. Maximize [latex]f\\left(x\\right)=-{x}^{2}+100x[\/latex].<\/p>\n<p>47.\u00a0125 feet by 62.5 feet. Maximize [latex]f\\left(x\\right)=-2{x}^{2}+250x[\/latex].<\/p>\n<p>49. 6 and \u20136; product is \u201336; maximize [latex]f\\left(x\\right)={x}^{2}+12x[\/latex].<\/p>\n<p>51.\u00a02909.56 meters<\/p>\n<p>53.\u00a0$10.70<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-11042\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-11042","chapter","type-chapter","status-publish","hentry"],"part":15410,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11042","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":11,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11042\/revisions"}],"predecessor-version":[{"id":16040,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11042\/revisions\/16040"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/15410"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11042\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=11042"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=11042"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=11042"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=11042"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}