{"id":15305,"date":"2021-10-11T22:52:38","date_gmt":"2021-10-11T22:52:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/introduction-quadratic-functions\/"},"modified":"2021-12-29T19:07:55","modified_gmt":"2021-12-29T19:07:55","slug":"equations-and-graphs-of-quadratic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/equations-and-graphs-of-quadratic-functions\/","title":{"raw":"Equations and Graphs of Quadratic Functions","rendered":"Equations and Graphs of Quadratic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul class=\"ul1\">\r\n \t<li class=\"li2\"><span class=\"s1\">Recognize characteristics of parabolas.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Understand how the graph of a parabola is related to the equation of quadratic function.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nCurved antennas, such as the ones shown in the photo, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170328\/CNX_Precalc_Figure_03_02_0012.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/> An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)[\/caption]\r\n\r\nThe simplest example of a quadratic function, that you have likely come across before, is\u00a0[latex]f\\left(x\\right)={x}^{2}[\/latex]. Before we talk about more general equation of a quadratic function, we will look at its graph.\r\n<h2>Characteristics of Parabolas<\/h2>\r\nThe graph of a quadratic function is a U-shaped curve called a <strong>parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the <strong>vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the <strong>minimum value<\/strong> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <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 <strong>axis of symmetry<\/strong>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170330\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/>\r\n\r\nThe [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]\u00a0at which [latex]y=0[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying the Characteristics of a Parabola<\/h3>\r\nDetermine the vertex, axis of symmetry, zeros, and <em>y<\/em>-intercept of the parabola shown below.\r\n\r\n<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\" \/>\r\n\r\n[reveal-answer q=\"366804\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"366804\"]\r\n\r\nThe 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 (0, 7) so this is the [latex]y[\/latex]-intercept.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Equations of Quadratic Functions<\/h2>\r\nThe <strong>general form of a quadratic function<\/strong> presents the function in the form\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/p>\r\nwhere [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers and [latex]a\\ne 0[\/latex]. If [latex]a&gt;0[\/latex], the parabola opens upward. If [latex]a&lt;0[\/latex], the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.\r\n\r\nThe axis of symmetry is defined by [latex]x=-\\dfrac{b}{2a}[\/latex]. If we use the quadratic formula, [latex]x=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex], to solve [latex]a{x}^{2}+bx+c=0[\/latex] for the [latex]x[\/latex]-intercepts, or zeros, we find the value of [latex]x[\/latex]\u00a0halfway between them is always [latex]x=-\\dfrac{b}{2a}[\/latex], the equation for the axis of symmetry.\r\n\r\nThe figure below shows\u00a0the graph of the quadratic function written in general form as [latex]y={x}^{2}+4x+3[\/latex]. In this form, [latex]a=1,\\text{ }b=4[\/latex], and [latex]c=3[\/latex]. Because [latex]a&gt;0[\/latex], the parabola opens upward. The axis of symmetry is [latex]x=-\\dfrac{4}{2\\left(1\\right)}=-2[\/latex]. This also makes sense because we can see from the graph that the vertical line [latex]x=-2[\/latex] divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, [latex]\\left(-2,-1\\right)[\/latex]. The [latex]x[\/latex]-intercepts, those points where the parabola crosses the [latex]x[\/latex]-axis, occur at [latex]\\left(-3,0\\right)[\/latex] and [latex]\\left(-1,0\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170335\/CNX_Precalc_Figure_03_02_0042.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.\" width=\"487\" height=\"555\" \/>\r\n\r\nThe <strong>standard form of a quadratic function<\/strong> presents the function in the form\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\nwhere [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0052.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=-3(x+2)^2+4.\" width=\"487\" height=\"630\" data-media-type=\"image\/jpg\" \/> <b>Figure 5<\/b>[\/caption]\r\n<p id=\"fs-id1165137894543\">As with the general form, if [latex]a&gt;0[\/latex], the parabola opens upward and the vertex is a minimum. If [latex]a&lt;0[\/latex], the parabola opens downward, and the vertex is a maximum. Figure 5\u00a0is the\u00a0graph of the quadratic function written in standard form as [latex]y=-3{\\left(x+2\\right)}^{2}+4[\/latex]. Since [latex]x-h=x+2[\/latex] in this example, [latex]h=-2[\/latex]. In this form, [latex]a=-3,\\text{ }h=-2[\/latex], and [latex]k=4[\/latex]. Because [latex]a&lt;0[\/latex], the parabola opens downward. The vertex is at [latex]\\left(-2,\\text{ 4}\\right)[\/latex].<\/p>\r\nThe standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.\r\n<div id=\"fs-id1165137749882\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"title\">A General Note: Forms of Quadratic Functions<\/h3>\r\n<p id=\"fs-id1165135333154\">A quadratic function is a function of degree two. The graph of a <strong>quadratic function<\/strong> is a parabola. The <strong>general form of a quadratic function<\/strong> is [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex] where <em>a,\u00a0b<\/em>, and <em>c<\/em>\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/p>\r\n<p id=\"fs-id1165137666538\">The <strong>standard form of a quadratic function<\/strong> is [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex].<\/p>\r\n<p id=\"fs-id1165137762385\">The vertex [latex]\\left(h,k\\right)[\/latex] is located at<\/p>\r\n\r\n<div id=\"eip-301\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]h=-\\frac{b}{2a},\\text{ }k=f\\left(h\\right)=f\\left(\\frac{-b}{2a}\\right)[\/latex].<\/div>\r\n<\/div>\r\n<h2>Given a quadratic function in general form, find the vertex of the parabola.<\/h2>\r\nOne reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, [latex]k[\/latex], and where it occurs, [latex]h[\/latex]. If we are given the general form of a quadratic function:\r\n<p style=\"text-align: center;\">[latex]f(x)=ax^2+bx+c[\/latex]<\/p>\r\nWe can define the vertex, [latex](h,k)[\/latex], by doing the following:\r\n<ul>\r\n \t<li>Identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].<\/li>\r\n \t<li>Find [latex]h[\/latex], the [latex]x[\/latex]-coordinate of the vertex, by substituting [latex]a[\/latex] and [latex]b[\/latex]\u00a0into [latex]h=-\\dfrac{b}{2a}[\/latex].<\/li>\r\n \t<li>Find [latex]k[\/latex], the [latex]y[\/latex]-coordinate of the vertex, by evaluating [latex]k=f\\left(h\\right)=f\\left(-\\dfrac{b}{2a}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Vertex of a Quadratic Function<\/h3>\r\nFind the vertex of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}-6x+7[\/latex]. Rewrite the quadratic in standard form (vertex form).\r\n\r\n[reveal-answer q=\"466886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"466886\"]\r\n\r\nThe horizontal coordinate of the vertex will be at\r\n<p style=\"text-align: center;\">[latex]\\begin{align}h&amp;=-\\dfrac{b}{2a}\\ \\\\[2mm] &amp;=-\\dfrac{-6}{2\\left(2\\right)} \\\\[2mm]&amp;=\\dfrac{6}{4} \\\\[2mm]&amp;=\\dfrac{3}{2} \\end{align}[\/latex]<\/p>\r\nThe vertical coordinate of the vertex will be at\r\n<p style=\"text-align: center;\">[latex]\\begin{align}k&amp;=f\\left(h\\right) \\\\[2mm]&amp;=f\\left(\\dfrac{3}{2}\\right) \\\\[2mm]&amp;=2{\\left(\\dfrac{3}{2}\\right)}^{2}-6\\left(\\dfrac{3}{2}\\right)+7 \\\\[2mm]&amp;=\\dfrac{5}{2}\\end{align}[\/latex]<\/p>\r\nSo the vertex is [latex]\\left(\\dfrac{3}{2},\\dfrac{5}{2}\\right)[\/latex]\r\n\r\nRewriting into standard form, the stretch factor will be the same as the [latex]a[\/latex] in the original quadratic.\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=2{\\left(x-\\frac{3}{2}\\right)}^{2}+\\frac{5}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the equation [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex], write the equation in general form and then in standard form.\r\n\r\n[reveal-answer q=\"713769\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"713769\"]\r\n\r\n[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\n[\/hidden-answer]\r\n\r\n<\/div>\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 domain 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 range 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<div class=\"textbox\">\r\n<h3>A General Note: Domain and Range of a Quadratic Function<\/h3>\r\nThe domain of any <strong>quadratic function<\/strong> is all real numbers.\r\n\r\nThe range of a quadratic function written in general form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex] 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]; the range of a quadratic function written in general form with a negative [latex]a[\/latex]\u00a0value 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].\r\n\r\nThe 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]\u00a0value is [latex]f\\left(x\\right)\\ge k[\/latex]; the range of a quadratic function written in standard form with a negative [latex]a[\/latex]\u00a0value is [latex]f\\left(x\\right)\\le k[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a quadratic function, find the domain and range.<\/h3>\r\n<ol>\r\n \t<li>The domain of any quadratic function as all real numbers.<\/li>\r\n \t<li>Determine whether [latex]a[\/latex] is positive or negative. If [latex]a[\/latex]\u00a0is positive, the parabola has a minimum. If [latex]a[\/latex]\u00a0is negative, the parabola has a maximum.<\/li>\r\n \t<li>Determine the maximum or minimum value of the parabola, [latex]k[\/latex].<\/li>\r\n \t<li>If 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]. If 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<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain and Range of a Quadratic Function<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=-5{x}^{2}+9x - 1[\/latex].\r\n\r\n[reveal-answer q=\"40392\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"40392\"]\r\n\r\nAs with any quadratic function, the domain is all real numbers or [latex]\\left(-\\infty,\\infty\\right)[\/latex].\r\n\r\nBecause [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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=2{\\left(x-\\dfrac{4}{7}\\right)}^{2}+\\dfrac{8}{11}[\/latex].\r\n\r\n[reveal-answer q=\"307368\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"307368\"]\r\n\r\nThe domain is all real numbers. The range is [latex]f\\left(x\\right)\\ge \\dfrac{8}{11}[\/latex], or [latex]\\left[\\dfrac{8}{11},\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=120300&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Intercepts of Quadratic Functions<\/h2>\r\nWhen graphing parabolas, it's often helpful to find intercepts of quadratic function. Recall that we find the <span class=\"s1\">[latex]y[\/latex]<\/span>-intercept of a function by evaluating the function at an input of zero, and we find the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts at locations where the output is zero. Notice\u00a0that the number of <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts can vary depending upon the location of the graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170357\/CNX_Precalc_Figure_03_02_0132.jpg\" alt=\"Three graphs where the first graph shows a parabola with no x-intercept, the second is a parabola with one \u2013intercept, and the third parabola is of two x-intercepts.\" width=\"975\" height=\"317\" \/> Number of <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts of a parabola[\/caption]Mathematicians also define <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts as roots of the quadratic function. This is where you will apply the methods of solving quadratic equations you reviewed in the\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/why-it-matters-complex-numbers\/\">Quadratic equations<\/a>\u00a0section.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a quadratic function [latex]f\\left(x\\right)[\/latex], find the <em><span class=\"s1\">Y<\/span><\/em>-\u00a0and <em><span class=\"s1\">X<\/span><\/em>-intercepts.<\/h3>\r\n<ol>\r\n \t<li>Evaluate [latex]f\\left(0\\right)[\/latex] to find the <span class=\"s1\">[latex]y[\/latex]<\/span>-intercept.<\/li>\r\n \t<li>Solve the quadratic equation [latex]f\\left(x\\right)=0[\/latex] to find the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the <em>y<\/em>- and <em>x<\/em>-Intercepts of a Parabola<\/h3>\r\nFind the <span class=\"s1\">[latex]y[\/latex]<\/span>- and <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts of the quadratic [latex]f\\left(x\\right)=3{x}^{2}+5x - 2[\/latex].\r\n\r\n[reveal-answer q=\"14680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"14680\"]\r\n\r\nWe find the <span class=\"s1\">[latex]y[\/latex]<\/span>-intercept by evaluating [latex]f\\left(0\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]f\\left(0\\right)=3{\\left(0\\right)}^{2}+5\\left(0\\right)-2=-2[\/latex]<\/p>\r\nSo the <span class=\"s1\">[latex]y[\/latex]<\/span>-intercept is at [latex]\\left(0,-2\\right)[\/latex].\r\n\r\nFor the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts, or roots, we find all solutions of [latex]f\\left(x\\right)=0[\/latex].\r\n<p style=\"text-align: center;\">[latex]3{x}^{2}+5x - 2=0[\/latex]<\/p>\r\nIn this case, the quadratic can be factored easily, providing the simplest method for solution.\r\n<p style=\"text-align: center;\">[latex]\\left(3x - 1\\right)\\left(x+2\\right)=0[\/latex]\r\n[latex]\\begin{align}&amp;3x - 1=0&amp;x+2=0\\end{align}[\/latex]\r\n[latex]x=\\frac{1}{3}\\hspace{8mm}[\/latex] or [latex]\\hspace{8mm}x=-2[\/latex]<\/p>\r\nSo the <em>roots<\/em>\u00a0are at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nBy graphing the function, we can confirm that the graph crosses the <span class=\"s1\">[latex]y[\/latex]<\/span>-axis at [latex]\\left(0,-2\\right)[\/latex]. We can also confirm that the graph crosses the <span class=\"s1\">[latex]x[\/latex]<\/span>-axis at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170400\/CNX_Precalc_Figure_03_02_0142.jpg\" alt=\"Graph of a parabola which has the following intercepts (-2, 0), (1\/3, 0), and (0, -2).\" width=\"487\" height=\"480\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the above example\u00a0the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a quadratic function, find the <em>x<\/em>-intercepts by rewriting in standard form.<\/h3>\r\n<ol>\r\n \t<li>Substitute <span class=\"s1\">[latex]a[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]b[\/latex]<\/span>\u00a0into [latex]h=-\\dfrac{b}{2a}[\/latex].<\/li>\r\n \t<li>Substitute [latex]x=h[\/latex]\u00a0into the general form of the quadratic function to find <span class=\"s1\">[latex]k[\/latex]<\/span>.<\/li>\r\n \t<li>Rewrite the quadratic in standard form using <span class=\"s1\">[latex]h[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]k[\/latex]<\/span>.<\/li>\r\n \t<li>Solve for when the output of the function will be zero to find the <span class=\"s1\">[latex]x[\/latex]<\/span><em>-<\/em>intercepts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Roots\u00a0of a Parabola<\/h3>\r\nFind the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}+4x - 4[\/latex].\r\n\r\n[reveal-answer q=\"201989\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"201989\"]\r\n\r\nWe begin by solving for when the output will be zero.\r\n<p style=\"text-align: center;\">[latex]0=2{x}^{2}+4x - 4[\/latex]<\/p>\r\nBecause the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\nWe know that [latex]a=2[\/latex]. Then we solve for\u00a0<span class=\"s1\">[latex]h[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]k[\/latex].<\/span>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;h=-\\dfrac{b}{2a}=-\\dfrac{4}{2\\left(2\\right)}=-1\\\\[2mm]&amp;\\end{align}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}k&amp;=f\\left(h\\right)\\\\&amp;=f\\left(-1\\right)\\\\&amp;=2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4\\\\&amp;=-6\\end{align}[\/latex]<\/p>\r\nSo now we can rewrite in standard form.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=2{\\left(x+1\\right)}^{2}-6[\/latex]<\/p>\r\nWe can now solve for when the output will be zero.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;0=2{\\left(x+1\\right)}^{2}-6 \\\\ &amp;6=2{\\left(x+1\\right)}^{2} \\\\ &amp;3={\\left(x+1\\right)}^{2} \\\\ &amp;x+1=\\pm \\sqrt{3} \\\\ &amp;x=-1\\pm \\sqrt{3} \\end{align}[\/latex]<\/p>\r\nThe graph has [latex]x[\/latex]<em>-<\/em>intercepts at [latex]\\left(-1-\\sqrt{3},0\\right)[\/latex] and [latex]\\left(-1+\\sqrt{3},0\\right)[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170402\/CNX_Precalc_Figure_03_02_0152.jpg\" alt=\"Graph of a parabola which has the following x-intercepts (-2.732, 0) and (0.732, 0).\" width=\"487\" height=\"517\" \/>\r\n\r\nWe can check our work by graphing the given function on a graphing utility and observing the roots.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=121416&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<section id=\"fs-id1165135426424\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165134570662\">\r\n \t<li>A polynomial function of degree two is called a quadratic function.<\/li>\r\n \t<li>The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.<\/li>\r\n \t<li>The axis of symmetry is the vertical line passing through the vertex.<\/li>\r\n \t<li>Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.<\/li>\r\n \t<li>The vertex can be found from an equation representing a quadratic function.<\/li>\r\n \t<li>The domain of a quadratic function is all real numbers. The range varies with the function.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135449657\" class=\"definition\">\r\n \t<dt><strong>axis of symmetry<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135449662\">a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by [latex]x=-\\dfrac{b}{2a}[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135502777\" class=\"definition\">\r\n \t<dt><strong>general form of a quadratic function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135502783\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137931314\" class=\"definition\">\r\n \t<dt><strong>standard form of a quadratic function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137931319\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex], where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135623614\" class=\"definition\">\r\n \t<dt><\/dt>\r\n<\/dl>\r\n<\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n<li class=\"li2\"><span class=\"s1\">Recognize characteristics of parabolas.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Understand how the graph of a parabola is related to the equation of quadratic function.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>Curved antennas, such as the ones shown in the photo, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170328\/CNX_Precalc_Figure_03_02_0012.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\">An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)<\/p>\n<\/div>\n<p>The simplest example of a quadratic function, that you have likely come across before, is\u00a0[latex]f\\left(x\\right)={x}^{2}[\/latex]. Before we talk about more general equation of a quadratic function, we will look at its graph.<\/p>\n<h2>Characteristics of Parabolas<\/h2>\n<p>The graph of a quadratic function is a U-shaped curve called a <strong>parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the <strong>vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the <strong>minimum value<\/strong> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <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 <strong>axis of symmetry<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170330\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/><\/p>\n<p>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]\u00a0at which [latex]y=0[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the Characteristics of a Parabola<\/h3>\n<p>Determine the vertex, axis of symmetry, zeros, and <em>y<\/em>-intercept of the parabola shown below.<\/p>\n<p><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\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q366804\">Show Solution<\/span><\/p>\n<div id=\"q366804\" class=\"hidden-answer\" style=\"display: none\">\n<p>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 (0, 7) so this is the [latex]y[\/latex]-intercept.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Equations of Quadratic Functions<\/h2>\n<p>The <strong>general form of a quadratic function<\/strong> presents the function in the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/p>\n<p>where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers and [latex]a\\ne 0[\/latex]. If [latex]a>0[\/latex], the parabola opens upward. If [latex]a<0[\/latex], the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.\n\nThe axis of symmetry is defined by [latex]x=-\\dfrac{b}{2a}[\/latex]. If we use the quadratic formula, [latex]x=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex], to solve [latex]a{x}^{2}+bx+c=0[\/latex] for the [latex]x[\/latex]-intercepts, or zeros, we find the value of [latex]x[\/latex]\u00a0halfway between them is always [latex]x=-\\dfrac{b}{2a}[\/latex], the equation for the axis of symmetry.\n\nThe figure below shows\u00a0the graph of the quadratic function written in general form as [latex]y={x}^{2}+4x+3[\/latex]. In this form, [latex]a=1,\\text{ }b=4[\/latex], and [latex]c=3[\/latex]. Because [latex]a>0[\/latex], the parabola opens upward. The axis of symmetry is [latex]x=-\\dfrac{4}{2\\left(1\\right)}=-2[\/latex]. This also makes sense because we can see from the graph that the vertical line [latex]x=-2[\/latex] divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, [latex]\\left(-2,-1\\right)[\/latex]. The [latex]x[\/latex]-intercepts, those points where the parabola crosses the [latex]x[\/latex]-axis, occur at [latex]\\left(-3,0\\right)[\/latex] and [latex]\\left(-1,0\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170335\/CNX_Precalc_Figure_03_02_0042.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.\" width=\"487\" height=\"555\" \/><\/p>\n<p>The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\n<p>where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0052.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=-3(x+2)^2+4.\" width=\"487\" height=\"630\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137894543\">As with the general form, if [latex]a>0[\/latex], the parabola opens upward and the vertex is a minimum. If [latex]a<0[\/latex], the parabola opens downward, and the vertex is a maximum. Figure 5\u00a0is the\u00a0graph of the quadratic function written in standard form as [latex]y=-3{\\left(x+2\\right)}^{2}+4[\/latex]. Since [latex]x-h=x+2[\/latex] in this example, [latex]h=-2[\/latex]. In this form, [latex]a=-3,\\text{ }h=-2[\/latex], and [latex]k=4[\/latex]. Because [latex]a<0[\/latex], the parabola opens downward. The vertex is at [latex]\\left(-2,\\text{ 4}\\right)[\/latex].<\/p>\n<p>The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.<\/p>\n<div id=\"fs-id1165137749882\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Forms of Quadratic Functions<\/h3>\n<p id=\"fs-id1165135333154\">A quadratic function is a function of degree two. The graph of a <strong>quadratic function<\/strong> is a parabola. The <strong>general form of a quadratic function<\/strong> is [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex] where <em>a,\u00a0b<\/em>, and <em>c<\/em>\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/p>\n<p id=\"fs-id1165137666538\">The <strong>standard form of a quadratic function<\/strong> is [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex].<\/p>\n<p id=\"fs-id1165137762385\">The vertex [latex]\\left(h,k\\right)[\/latex] is located at<\/p>\n<div id=\"eip-301\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]h=-\\frac{b}{2a},\\text{ }k=f\\left(h\\right)=f\\left(\\frac{-b}{2a}\\right)[\/latex].<\/div>\n<\/div>\n<h2>Given a quadratic function in general form, find the vertex of the parabola.<\/h2>\n<p>One reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, [latex]k[\/latex], and where it occurs, [latex]h[\/latex]. If we are given the general form of a quadratic function:<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ax^2+bx+c[\/latex]<\/p>\n<p>We can define the vertex, [latex](h,k)[\/latex], by doing the following:<\/p>\n<ul>\n<li>Identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].<\/li>\n<li>Find [latex]h[\/latex], the [latex]x[\/latex]-coordinate of the vertex, by substituting [latex]a[\/latex] and [latex]b[\/latex]\u00a0into [latex]h=-\\dfrac{b}{2a}[\/latex].<\/li>\n<li>Find [latex]k[\/latex], the [latex]y[\/latex]-coordinate of the vertex, by evaluating [latex]k=f\\left(h\\right)=f\\left(-\\dfrac{b}{2a}\\right)[\/latex]<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Vertex of a Quadratic Function<\/h3>\n<p>Find the vertex of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}-6x+7[\/latex]. Rewrite the quadratic in standard form (vertex form).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q466886\">Show Solution<\/span><\/p>\n<div id=\"q466886\" class=\"hidden-answer\" style=\"display: none\">\n<p>The horizontal coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}h&=-\\dfrac{b}{2a}\\ \\\\[2mm] &=-\\dfrac{-6}{2\\left(2\\right)} \\\\[2mm]&=\\dfrac{6}{4} \\\\[2mm]&=\\dfrac{3}{2} \\end{align}[\/latex]<\/p>\n<p>The vertical coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}k&=f\\left(h\\right) \\\\[2mm]&=f\\left(\\dfrac{3}{2}\\right) \\\\[2mm]&=2{\\left(\\dfrac{3}{2}\\right)}^{2}-6\\left(\\dfrac{3}{2}\\right)+7 \\\\[2mm]&=\\dfrac{5}{2}\\end{align}[\/latex]<\/p>\n<p>So the vertex is [latex]\\left(\\dfrac{3}{2},\\dfrac{5}{2}\\right)[\/latex]<\/p>\n<p>Rewriting into standard form, the stretch factor will be the same as the [latex]a[\/latex] in the original quadratic.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=2{\\left(x-\\frac{3}{2}\\right)}^{2}+\\frac{5}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the equation [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex], write the equation in general form and then in standard form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q713769\">Show Solution<\/span><\/p>\n<div id=\"q713769\" class=\"hidden-answer\" style=\"display: none\">\n<p>[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<\/div>\n<\/div>\n<\/div>\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 domain 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 range 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<div class=\"textbox\">\n<h3>A General Note: Domain and Range of a Quadratic Function<\/h3>\n<p>The domain of any <strong>quadratic function<\/strong> is all real numbers.<\/p>\n<p>The range of a quadratic function written in general form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex] 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]; the range of a quadratic function written in general form with a negative [latex]a[\/latex]\u00a0value 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].<\/p>\n<p>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]\u00a0value is [latex]f\\left(x\\right)\\ge k[\/latex]; the range of a quadratic function written in standard form with a negative [latex]a[\/latex]\u00a0value is [latex]f\\left(x\\right)\\le k[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a quadratic function, find the domain and range.<\/h3>\n<ol>\n<li>The domain of any quadratic function as all real numbers.<\/li>\n<li>Determine whether [latex]a[\/latex] is positive or negative. If [latex]a[\/latex]\u00a0is positive, the parabola has a minimum. If [latex]a[\/latex]\u00a0is negative, the parabola has a maximum.<\/li>\n<li>Determine the maximum or minimum value of the parabola, [latex]k[\/latex].<\/li>\n<li>If 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]. If 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<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain and Range of a Quadratic Function<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q40392\">Show Solution<\/span><\/p>\n<div id=\"q40392\" class=\"hidden-answer\" style=\"display: none\">\n<p>As with any quadratic function, the domain is all real numbers or [latex]\\left(-\\infty,\\infty\\right)[\/latex].<\/p>\n<p>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<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the domain and range of [latex]f\\left(x\\right)=2{\\left(x-\\dfrac{4}{7}\\right)}^{2}+\\dfrac{8}{11}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q307368\">Show Solution<\/span><\/p>\n<div id=\"q307368\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is all real numbers. The range is [latex]f\\left(x\\right)\\ge \\dfrac{8}{11}[\/latex], or [latex]\\left[\\dfrac{8}{11},\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=120300&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Intercepts of Quadratic Functions<\/h2>\n<p>When graphing parabolas, it&#8217;s often helpful to find intercepts of quadratic function. Recall that we find the <span class=\"s1\">[latex]y[\/latex]<\/span>-intercept of a function by evaluating the function at an input of zero, and we find the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts at locations where the output is zero. Notice\u00a0that the number of <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts can vary depending upon the location of the graph.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170357\/CNX_Precalc_Figure_03_02_0132.jpg\" alt=\"Three graphs where the first graph shows a parabola with no x-intercept, the second is a parabola with one \u2013intercept, and the third parabola is of two x-intercepts.\" width=\"975\" height=\"317\" \/><\/p>\n<p class=\"wp-caption-text\">Number of <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts of a parabola<\/p>\n<\/div>\n<p>Mathematicians also define <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts as roots of the quadratic function. This is where you will apply the methods of solving quadratic equations you reviewed in the\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/why-it-matters-complex-numbers\/\">Quadratic equations<\/a>\u00a0section.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a quadratic function [latex]f\\left(x\\right)[\/latex], find the <em><span class=\"s1\">Y<\/span><\/em>&#8211;\u00a0and <em><span class=\"s1\">X<\/span><\/em>-intercepts.<\/h3>\n<ol>\n<li>Evaluate [latex]f\\left(0\\right)[\/latex] to find the <span class=\"s1\">[latex]y[\/latex]<\/span>-intercept.<\/li>\n<li>Solve the quadratic equation [latex]f\\left(x\\right)=0[\/latex] to find the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the <em>y<\/em>&#8211; and <em>x<\/em>-Intercepts of a Parabola<\/h3>\n<p>Find the <span class=\"s1\">[latex]y[\/latex]<\/span>&#8211; and <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts of the quadratic [latex]f\\left(x\\right)=3{x}^{2}+5x - 2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q14680\">Show Solution<\/span><\/p>\n<div id=\"q14680\" class=\"hidden-answer\" style=\"display: none\">\n<p>We find the <span class=\"s1\">[latex]y[\/latex]<\/span>-intercept by evaluating [latex]f\\left(0\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(0\\right)=3{\\left(0\\right)}^{2}+5\\left(0\\right)-2=-2[\/latex]<\/p>\n<p>So the <span class=\"s1\">[latex]y[\/latex]<\/span>-intercept is at [latex]\\left(0,-2\\right)[\/latex].<\/p>\n<p>For the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts, or roots, we find all solutions of [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3{x}^{2}+5x - 2=0[\/latex]<\/p>\n<p>In this case, the quadratic can be factored easily, providing the simplest method for solution.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(3x - 1\\right)\\left(x+2\\right)=0[\/latex]<br \/>\n[latex]\\begin{align}&3x - 1=0&x+2=0\\end{align}[\/latex]<br \/>\n[latex]x=\\frac{1}{3}\\hspace{8mm}[\/latex] or [latex]\\hspace{8mm}x=-2[\/latex]<\/p>\n<p>So the <em>roots<\/em>\u00a0are at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>By graphing the function, we can confirm that the graph crosses the <span class=\"s1\">[latex]y[\/latex]<\/span>-axis at [latex]\\left(0,-2\\right)[\/latex]. We can also confirm that the graph crosses the <span class=\"s1\">[latex]x[\/latex]<\/span>-axis at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170400\/CNX_Precalc_Figure_03_02_0142.jpg\" alt=\"Graph of a parabola which has the following intercepts (-2, 0), (1\/3, 0), and (0, -2).\" width=\"487\" height=\"480\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the above example\u00a0the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a quadratic function, find the <em>x<\/em>-intercepts by rewriting in standard form.<\/h3>\n<ol>\n<li>Substitute <span class=\"s1\">[latex]a[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]b[\/latex]<\/span>\u00a0into [latex]h=-\\dfrac{b}{2a}[\/latex].<\/li>\n<li>Substitute [latex]x=h[\/latex]\u00a0into the general form of the quadratic function to find <span class=\"s1\">[latex]k[\/latex]<\/span>.<\/li>\n<li>Rewrite the quadratic in standard form using <span class=\"s1\">[latex]h[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]k[\/latex]<\/span>.<\/li>\n<li>Solve for when the output of the function will be zero to find the <span class=\"s1\">[latex]x[\/latex]<\/span><em>&#8211;<\/em>intercepts.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Roots\u00a0of a Parabola<\/h3>\n<p>Find the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}+4x - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q201989\">Show Solution<\/span><\/p>\n<div id=\"q201989\" class=\"hidden-answer\" style=\"display: none\">\n<p>We begin by solving for when the output will be zero.<\/p>\n<p style=\"text-align: center;\">[latex]0=2{x}^{2}+4x - 4[\/latex]<\/p>\n<p>Because the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\n<p>We know that [latex]a=2[\/latex]. Then we solve for\u00a0<span class=\"s1\">[latex]h[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]k[\/latex].<\/span><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&h=-\\dfrac{b}{2a}=-\\dfrac{4}{2\\left(2\\right)}=-1\\\\[2mm]&\\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}k&=f\\left(h\\right)\\\\&=f\\left(-1\\right)\\\\&=2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4\\\\&=-6\\end{align}[\/latex]<\/p>\n<p>So now we can rewrite in standard form.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=2{\\left(x+1\\right)}^{2}-6[\/latex]<\/p>\n<p>We can now solve for when the output will be zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&0=2{\\left(x+1\\right)}^{2}-6 \\\\ &6=2{\\left(x+1\\right)}^{2} \\\\ &3={\\left(x+1\\right)}^{2} \\\\ &x+1=\\pm \\sqrt{3} \\\\ &x=-1\\pm \\sqrt{3} \\end{align}[\/latex]<\/p>\n<p>The graph has [latex]x[\/latex]<em>&#8211;<\/em>intercepts at [latex]\\left(-1-\\sqrt{3},0\\right)[\/latex] and [latex]\\left(-1+\\sqrt{3},0\\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170402\/CNX_Precalc_Figure_03_02_0152.jpg\" alt=\"Graph of a parabola which has the following x-intercepts (-2.732, 0) and (0.732, 0).\" width=\"487\" height=\"517\" \/><\/p>\n<p>We can check our work by graphing the given function on a graphing utility and observing the roots.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=121416&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<section id=\"fs-id1165135426424\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165134570662\">\n<li>A polynomial function of degree two is called a quadratic function.<\/li>\n<li>The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.<\/li>\n<li>The axis of symmetry is the vertical line passing through the vertex.<\/li>\n<li>Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.<\/li>\n<li>The vertex can be found from an equation representing a quadratic function.<\/li>\n<li>The domain of a quadratic function is all real numbers. The range varies with the function.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135449657\" class=\"definition\">\n<dt><strong>axis of symmetry<\/strong><\/dt>\n<dd id=\"fs-id1165135449662\">a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by [latex]x=-\\dfrac{b}{2a}[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135502777\" class=\"definition\">\n<dt><strong>general form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165135502783\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137931314\" class=\"definition\">\n<dt><strong>standard form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165137931319\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex], where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623614\" class=\"definition\">\n<dt><\/dt>\n<\/dl>\n<\/section>\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-15305\">\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><li>Question ID 120303, 120300. <strong>Authored 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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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