{"id":76,"date":"2025-02-13T22:43:28","date_gmt":"2025-02-13T22:43:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/quadratic-functions\/"},"modified":"2026-01-12T20:41:16","modified_gmt":"2026-01-12T20:41:16","slug":"quadratic-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/quadratic-functions\/","title":{"raw":"Quadratic Functions: Learn It 1","rendered":"Quadratic Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify key characteristics of parabolas from the graph.<\/li>\r\n \t<li>Understand how the graph of a parabola is related to its quadratic function<\/li>\r\n \t<li>Draw the graph of a quadratic function.<\/li>\r\n \t<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\r\n<\/ul>\r\n<\/section>Quadratic functions are a fundamental concept in algebra that describe parabolic relationships. They are second-degree polynomial functions, meaning the highest power of the variable is [latex]2[\/latex]. The graph of a quadratic function is modeled by a vertical (up or down opening) parabola. These functions have numerous real-world applications, from describing the path of a projectile to modeling revenue in economics.\r\n<h2>Key Features of a Parabola's Graph<\/h2>\r\nThe graph of a quadratic function is a U-shaped curve called a\u00a0<strong>parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the\u00a0<strong>vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the\u00a0<strong>minimum value<\/strong>\u00a0of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the\u00a0<strong>maximum value<\/strong>. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the\u00a0<strong>axis of symmetry<\/strong>.\r\n\r\n<img class=\"aligncenter wp-image-4957\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222439\/4.2.1.L1-Graph-1-295x300.png\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"484\" height=\"492\" \/>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]-axis.The [latex]x[\/latex]-intercepts are the points at which the parabola crosses the [latex]x[\/latex]-axis. If they exist, the [latex]x[\/latex]-intercepts represent the <strong>zeros<\/strong>, or <strong>roots<\/strong>, of the quadratic function, the values of [latex]x[\/latex] at which [latex]y=0[\/latex].<\/section><section class=\"textbox proTip\">The places where a function's graph crosses the horizontal axis are the places where the function value equals zero. You've seen that these values are called\u00a0<em>horizontal intercepts<\/em>, [latex]x[\/latex]<em>-intercepts<\/em>,\u00a0and\u00a0<em>zeros<\/em> so far. They can also be referred to as the\u00a0<em>roots<\/em> of a function.<\/section><section class=\"textbox example\">Determine the vertex, axis of symmetry, zeros, and [latex]y[\/latex]-intercept of the parabola shown below.<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170333\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"487\" height=\"517\" \/><\/center>[reveal-answer q=\"366804\"]Show Solution[\/reveal-answer][hidden-answer a=\"366804\"]The vertex is the turning point of the graph. We can see that the vertex is at [latex](3,1)[\/latex]. The axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is [latex]x=3[\/latex]. This parabola does not cross the [latex]x[\/latex]-axis, so it has no zeros. It crosses the [latex]y[\/latex]-axis at [latex](0, 7)[\/latex] so this is the [latex]y[\/latex]-intercept.[\/hidden-answer]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318783[\/ohm_question]<\/section><section><\/section>\r\n<h2>Finding the Domain and Range of a Quadratic Function<\/h2>\r\nAny number can be the input value of a quadratic function. Therefore the <strong>domain<\/strong> of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum at the vertex, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the <strong>range<\/strong> will consist of all [latex]y[\/latex]-values greater than or equal to the [latex]y[\/latex]-coordinate of the vertex or less than or equal to the [latex]y[\/latex]-coordinate at the turning point, depending on whether the parabola opens up or down.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>domain and range of a quadratic function<\/h3>\r\nThe<strong> domain of any quadratic function<\/strong> is all real numbers.\r\n\r\n&nbsp;\r\n\r\nDetermining the <strong>range of a quadratic formula<\/strong> is different depending on which form the quadratic function is in:\r\n\r\n<strong>General Form<\/strong>\r\n<ul>\r\n \t<li>The range of a quadratic function written in general form with a positive [latex]a[\/latex] value is [latex]f\\left(x\\right)\\ge f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left[f\\left(-\\frac{b}{2a}\\right),\\infty \\right)[\/latex]<\/li>\r\n \t<li>The range of a quadratic function written in general form with a negative [latex]a[\/latex] value is [latex]f\\left(x\\right)\\le f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left(-\\infty ,f\\left(-\\frac{b}{2a}\\right)\\right][\/latex].<\/li>\r\n<\/ul>\r\n<strong>Standard Form<\/strong>\r\n<ul>\r\n \t<li>The range of a quadratic function written in standard form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex] with a positive [latex]a[\/latex] value is [latex]f\\left(x\\right)\\ge k[\/latex] or [latex][k,\\infty)[\/latex].<\/li>\r\n \t<li>The range of a quadratic function written in standard form with a negative [latex]a[\/latex] value is [latex]f\\left(x\\right)\\le k[\/latex] or or [latex](-\\infty,k][\/latex].<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox questionHelp\"><strong>How to: Determine the Domain and Range from the Vertex<\/strong>\r\n<ol>\r\n \t<li>The domain of any quadratic function is all real numbers.<\/li>\r\n \t<li>Determine whether [latex]a[\/latex] is positive or negative.\r\nIf [latex]a[\/latex] is positive, the parabola has a minimum.\r\nIf [latex]a[\/latex] is negative, the parabola has a maximum.<\/li>\r\n \t<li>Determine the maximum or minimum value of the parabola, [latex]k[\/latex].\r\nIf the parabola has a minimum, the range is given by [latex]f\\left(x\\right)\\ge k[\/latex], or [latex]\\left[k,\\infty \\right)[\/latex].\r\nIf the parabola has a maximum, the range is given by [latex]f\\left(x\\right)\\le k[\/latex], or [latex]\\left(-\\infty ,k\\right][\/latex].<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Find the domain and range of [latex]f\\left(x\\right)=-5{x}^{2}+9x - 1[\/latex].[reveal-answer q=\"40392\"]Show Solution[\/reveal-answer][hidden-answer a=\"40392\"]As with any quadratic function, the domain is all real numbers or [latex]\\left(-\\infty,\\infty\\right)[\/latex].Because [latex]a[\/latex] is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the [latex]x[\/latex]-value of the vertex.\r\n<p style=\"text-align: center;\">[latex]h=-\\dfrac{b}{2a}=-\\dfrac{9}{2\\left(-5\\right)}=\\dfrac{9}{10}[\/latex]<\/p>\r\nThe maximum value is given by [latex]f\\left(h\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]f\\left(\\dfrac{9}{10}\\right)=5{\\left(\\dfrac{9}{10}\\right)}^{2}+9\\left(\\dfrac{9}{10}\\right)-1=\\dfrac{61}{20}[\/latex]<\/p>\r\nThe range is [latex]f\\left(x\\right)\\le \\dfrac{61}{20}[\/latex], or [latex]\\left(-\\infty ,\\dfrac{61}{20}\\right][\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318784[\/ohm_question]<\/section><section><section id=\"fs-id1165135426424\" class=\"key-concepts\">\r\n<dl id=\"fs-id1165135623634\" class=\"definition\">\r\n \t<dd id=\"fs-id1165135623639\"><\/dd>\r\n<\/dl>\r\n<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify key characteristics of parabolas from the graph.<\/li>\n<li>Understand how the graph of a parabola is related to its quadratic function<\/li>\n<li>Draw the graph of a quadratic function.<\/li>\n<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\n<\/ul>\n<\/section>\n<p>Quadratic functions are a fundamental concept in algebra that describe parabolic relationships. They are second-degree polynomial functions, meaning the highest power of the variable is [latex]2[\/latex]. The graph of a quadratic function is modeled by a vertical (up or down opening) parabola. These functions have numerous real-world applications, from describing the path of a projectile to modeling revenue in economics.<\/p>\n<h2>Key Features of a Parabola&#8217;s Graph<\/h2>\n<p>The graph of a quadratic function is a U-shaped curve called a\u00a0<strong>parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the\u00a0<strong>vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the\u00a0<strong>minimum value<\/strong>\u00a0of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the\u00a0<strong>maximum value<\/strong>. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the\u00a0<strong>axis of symmetry<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4957\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222439\/4.2.1.L1-Graph-1-295x300.png\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"484\" height=\"492\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222439\/4.2.1.L1-Graph-1-295x300.png 295w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222439\/4.2.1.L1-Graph-1-65x66.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222439\/4.2.1.L1-Graph-1-225x229.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222439\/4.2.1.L1-Graph-1-350x356.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02222439\/4.2.1.L1-Graph-1.png 662w\" sizes=\"(max-width: 484px) 100vw, 484px\" \/><\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]-axis.The [latex]x[\/latex]-intercepts are the points at which the parabola crosses the [latex]x[\/latex]-axis. If they exist, the [latex]x[\/latex]-intercepts represent the <strong>zeros<\/strong>, or <strong>roots<\/strong>, of the quadratic function, the values of [latex]x[\/latex] at which [latex]y=0[\/latex].<\/section>\n<section class=\"textbox proTip\">The places where a function&#8217;s graph crosses the horizontal axis are the places where the function value equals zero. You&#8217;ve seen that these values are called\u00a0<em>horizontal intercepts<\/em>, [latex]x[\/latex]<em>-intercepts<\/em>,\u00a0and\u00a0<em>zeros<\/em> so far. They can also be referred to as the\u00a0<em>roots<\/em> of a function.<\/section>\n<section class=\"textbox example\">Determine the vertex, axis of symmetry, zeros, and [latex]y[\/latex]-intercept of the parabola shown below.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170333\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"487\" height=\"517\" \/><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q366804\">Show Solution<\/button><\/p>\n<div id=\"q366804\" class=\"hidden-answer\" style=\"display: none\">The vertex is the turning point of the graph. We can see that the vertex is at [latex](3,1)[\/latex]. The axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is [latex]x=3[\/latex]. This parabola does not cross the [latex]x[\/latex]-axis, so it has no zeros. It crosses the [latex]y[\/latex]-axis at [latex](0, 7)[\/latex] so this is the [latex]y[\/latex]-intercept.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318783\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318783&theme=lumen&iframe_resize_id=ohm318783&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section><\/section>\n<h2>Finding the Domain and Range of a Quadratic Function<\/h2>\n<p>Any number can be the input value of a quadratic function. Therefore the <strong>domain<\/strong> of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum at the vertex, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the <strong>range<\/strong> will consist of all [latex]y[\/latex]-values greater than or equal to the [latex]y[\/latex]-coordinate of the vertex or less than or equal to the [latex]y[\/latex]-coordinate at the turning point, depending on whether the parabola opens up or down.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>domain and range of a quadratic function<\/h3>\n<p>The<strong> domain of any quadratic function<\/strong> is all real numbers.<\/p>\n<p>&nbsp;<\/p>\n<p>Determining the <strong>range of a quadratic formula<\/strong> is different depending on which form the quadratic function is in:<\/p>\n<p><strong>General Form<\/strong><\/p>\n<ul>\n<li>The range of a quadratic function written in general form with a positive [latex]a[\/latex] value is [latex]f\\left(x\\right)\\ge f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left[f\\left(-\\frac{b}{2a}\\right),\\infty \\right)[\/latex]<\/li>\n<li>The range of a quadratic function written in general form with a negative [latex]a[\/latex] value is [latex]f\\left(x\\right)\\le f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left(-\\infty ,f\\left(-\\frac{b}{2a}\\right)\\right][\/latex].<\/li>\n<\/ul>\n<p><strong>Standard Form<\/strong><\/p>\n<ul>\n<li>The range of a quadratic function written in standard form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex] with a positive [latex]a[\/latex] value is [latex]f\\left(x\\right)\\ge k[\/latex] or [latex][k,\\infty)[\/latex].<\/li>\n<li>The range of a quadratic function written in standard form with a negative [latex]a[\/latex] value is [latex]f\\left(x\\right)\\le k[\/latex] or or [latex](-\\infty,k][\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox questionHelp\"><strong>How to: Determine the Domain and Range from the Vertex<\/strong><\/p>\n<ol>\n<li>The domain of any quadratic function is all real numbers.<\/li>\n<li>Determine whether [latex]a[\/latex] is positive or negative.<br \/>\nIf [latex]a[\/latex] is positive, the parabola has a minimum.<br \/>\nIf [latex]a[\/latex] is negative, the parabola has a maximum.<\/li>\n<li>Determine the maximum or minimum value of the parabola, [latex]k[\/latex].<br \/>\nIf the parabola has a minimum, the range is given by [latex]f\\left(x\\right)\\ge k[\/latex], or [latex]\\left[k,\\infty \\right)[\/latex].<br \/>\nIf the parabola has a maximum, the range is given by [latex]f\\left(x\\right)\\le k[\/latex], or [latex]\\left(-\\infty ,k\\right][\/latex].<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Find the domain and range of [latex]f\\left(x\\right)=-5{x}^{2}+9x - 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q40392\">Show Solution<\/button><\/p>\n<div id=\"q40392\" class=\"hidden-answer\" style=\"display: none\">As with any quadratic function, the domain is all real numbers or [latex]\\left(-\\infty,\\infty\\right)[\/latex].Because [latex]a[\/latex] is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the [latex]x[\/latex]-value of the vertex.<\/p>\n<p style=\"text-align: center;\">[latex]h=-\\dfrac{b}{2a}=-\\dfrac{9}{2\\left(-5\\right)}=\\dfrac{9}{10}[\/latex]<\/p>\n<p>The maximum value is given by [latex]f\\left(h\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(\\dfrac{9}{10}\\right)=5{\\left(\\dfrac{9}{10}\\right)}^{2}+9\\left(\\dfrac{9}{10}\\right)-1=\\dfrac{61}{20}[\/latex]<\/p>\n<p>The range is [latex]f\\left(x\\right)\\le \\dfrac{61}{20}[\/latex], or [latex]\\left(-\\infty ,\\dfrac{61}{20}\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318784\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318784&theme=lumen&iframe_resize_id=ohm318784&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section id=\"fs-id1165135426424\" class=\"key-concepts\">\n<dl id=\"fs-id1165135623634\" class=\"definition\">\n<dd id=\"fs-id1165135623639\"><\/dd>\n<\/dl>\n<\/section>\n<\/section>\n","protected":false},"author":6,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/76"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/76\/revisions"}],"predecessor-version":[{"id":5292,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/76\/revisions\/5292"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/76\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=76"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=76"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=76"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=76"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}