{"id":200,"date":"2023-06-21T13:22:43","date_gmt":"2023-06-21T13:22:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-minima-and-maxima-of-quadratic-functions\/"},"modified":"2023-07-03T20:21:54","modified_gmt":"2023-07-03T20:21:54","slug":"summary-minima-and-maxima-of-quadratic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-minima-and-maxima-of-quadratic-functions\/","title":{"raw":"Summary: Analysis of Quadratic Functions","rendered":"Summary: Analysis of Quadratic Functions"},"content":{"raw":"\n\n<h2>Key Equations<\/h2>\nthe quadratic formula [latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]\n\nThe discriminant is defined as [latex]b^2-4ac[\/latex]\n\n&nbsp;\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>The zeros, or [latex]x[\/latex]-intercepts, are the points at which the parabola crosses the [latex]x[\/latex]-axis. The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]<em>-<\/em>axis.<\/li>\n \t<li>The vertex can be found from an equation representing a quadratic function.<\/li>\n \t<li>A quadratic function\u2019s minimum or maximum value is given by the [latex]y[\/latex]-value of the vertex.<\/li>\n \t<li>The minium or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.<\/li>\n \t<li>Some quadratic equations must be solved by using the quadratic formula.<\/li>\n \t<li>The vertex and the intercepts can be identified and interpreted to solve real-world problems.<\/li>\n \t<li>Some quadratic functions have complex roots.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135623614\" class=\"definition\">\n \t<dt><strong>discriminant<\/strong><\/dt>\n \t<dd>the value under the radical in the quadratic formula, [latex]b^2-4ac[\/latex], which tells whether the quadratic has real or complex roots<\/dd>\n \t<dt><strong>vertex<\/strong><\/dt>\n \t<dd id=\"fs-id1165135623619\">the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623624\" class=\"definition\">\n \t<dt><strong>vertex form of a quadratic function<\/strong><\/dt>\n \t<dd id=\"fs-id1165135623630\">another name for the standard form of a quadratic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623634\" class=\"definition\">\n \t<dt><strong>zeros<\/strong><\/dt>\n \t<dd id=\"fs-id1165135623639\">in a given function, the values of [latex]x[\/latex] at which [latex]y=0[\/latex], also called roots<\/dd>\n<\/dl>\n\n","rendered":"<h2>Key Equations<\/h2>\n<p>the quadratic formula [latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/p>\n<p>The discriminant is defined as [latex]b^2-4ac[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The zeros, or [latex]x[\/latex]-intercepts, are the points at which the parabola crosses the [latex]x[\/latex]-axis. The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]<em>&#8211;<\/em>axis.<\/li>\n<li>The vertex can be found from an equation representing a quadratic function.<\/li>\n<li>A quadratic function\u2019s minimum or maximum value is given by the [latex]y[\/latex]-value of the vertex.<\/li>\n<li>The minium or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.<\/li>\n<li>Some quadratic equations must be solved by using the quadratic formula.<\/li>\n<li>The vertex and the intercepts can be identified and interpreted to solve real-world problems.<\/li>\n<li>Some quadratic functions have complex roots.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135623614\" class=\"definition\">\n<dt><strong>discriminant<\/strong><\/dt>\n<dd>the value under the radical in the quadratic formula, [latex]b^2-4ac[\/latex], which tells whether the quadratic has real or complex roots<\/dd>\n<dt><strong>vertex<\/strong><\/dt>\n<dd id=\"fs-id1165135623619\">the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623624\" class=\"definition\">\n<dt><strong>vertex form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165135623630\">another name for the standard form of a quadratic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623634\" class=\"definition\">\n<dt><strong>zeros<\/strong><\/dt>\n<dd id=\"fs-id1165135623639\">in a given function, the values of [latex]x[\/latex] at which [latex]y=0[\/latex], also called roots<\/dd>\n<\/dl>\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-200\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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":395986,"menu_order":38,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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