{"id":4959,"date":"2020-11-10T20:39:20","date_gmt":"2020-11-10T20:39:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebra\/?post_type=chapter&#038;p=4959"},"modified":"2021-06-28T19:35:58","modified_gmt":"2021-06-28T19:35:58","slug":"quadratic-functions-edit-example","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/chapter\/quadratic-functions-edit-example\/","title":{"raw":"Graphs of Quadratic Functions","rendered":"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 its 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<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<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 is 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>Transformations of Quadratic Functions<\/h2>\r\n<p id=\"fs-id1165137676320\">The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\n<p style=\"text-align: left;\">where [latex]\\left(h,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>\r\nThe standard form is useful for determining how the graph is transformed from the graph of [latex]y={x}^{2}[\/latex]. The figure below\u00a0is the graph of this basic function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201255\/CNX_Precalc_Figure_03_02_0062.jpg\" alt=\"Graph of y=x^2.\" width=\"487\" height=\"480\" \/>\r\n\r\n&nbsp;\r\n<h2>Shift Up and Down by Changing the Value of [latex]k[\/latex]<\/h2>\r\n<p id=\"fs-id1165137770279\">You can represent a vertical (up, down) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, [latex]k[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=x^2 + k[\/latex]<\/p>\r\nIf [latex]k&gt;0[\/latex], the graph shifts upward, whereas if [latex]k&lt;0[\/latex], the graph shifts downward.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted up 4 units. Also, determine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units.\r\n\r\n[reveal-answer q=\"941360\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"941360\"]\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted up 4 units is\r\n\r\n[latex]f(x)=x^2+4[\/latex]\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units is\r\n\r\n[latex]f(x)=x^2-4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>\u00a0Shift left and right by changing the value of [latex]h[\/latex]<\/h2>\r\n<p id=\"fs-id1165137770279\">You can represent a horizontal (left, right) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, [latex]h[\/latex], to the variable [latex]x[\/latex], before squaring.<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=(x-h)^2 [\/latex]<\/p>\r\nIf [latex]h&gt;0[\/latex], the graph shifts toward the right and if [latex]h&lt;0[\/latex], the graph shifts to the left.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted right 2 units. Also, determine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted left 2 units.\r\n\r\n[reveal-answer q=\"978434\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"978434\"]\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted right 2 units is\r\n<p style=\"text-align: center;\">[latex]f(x)=(x-2)^2[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The equation for the graph of\u00a0[latex]f(x)=^2[\/latex] that has been shifted left 2 units is<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=(x+2)^2[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<h2>Stretch or compress by changing the value of [latex]a[\/latex].<\/h2>\r\n<p id=\"fs-id1165137770279\">You can represent a stretch or compression (narrowing, widening)\u00a0of the graph of [latex]f(x)=x^2[\/latex] by\u00a0multiplying the squared variable by a constant, [latex]a[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=ax^2 [\/latex]<\/p>\r\nThe magnitude of [latex]a[\/latex]\u00a0indicates the stretch of the graph. If [latex]|a|&gt;1[\/latex], the point associated with a particular [latex]x[\/latex]-value shifts farther from the [latex]x[\/latex]<em>-<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch. But if [latex]|a|&lt;1[\/latex], the point associated with a particular [latex]x[\/latex]-value shifts closer to the [latex]x[\/latex]<em>-<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex]. Also, determine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3.\r\n\r\n[reveal-answer q=\"391411\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"391411\"]\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex] is\r\n<p style=\"text-align: center;\">[latex]f(x)=\\frac{1}{2}x^2[\/latex]<\/p>\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3 is\r\n<p style=\"text-align: center;\">[latex]f(x)=3x^2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\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=\"eip-173\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}&amp;a{\\left(x-h\\right)}^{2}+k=a{x}^{2}+bx+c\\\\ &amp;a{x}^{2}-2ahx+\\left(a{h}^{2}+k\\right)=a{x}^{2}+bx+c \\end{align}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">For the two sides to be equal, the corresponding coefficients must be equal. In particular, the coefficients of [latex]x[\/latex] must be equal.<\/div>\r\n<div id=\"eip-144\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]-2ah=b,\\text{ so }h=-\\dfrac{b}{2a}[\/latex].<\/div>\r\n<p id=\"fs-id1165134118295\">This is the [latex]x[\/latex] coordinate of the vertexr and [latex]x=-\\dfrac{b}{2a}[\/latex] is the\u00a0<strong>axis of symmetry<\/strong> we defined earlier. Setting the constant terms equal gives us:<\/p>\r\n\r\n<div id=\"eip-313\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}a{h}^{2}+k&amp;=c \\\\[2mm] k&amp;=c-a{h}^{2} \\\\ &amp;=c-a-{\\left(\\dfrac{b}{2a}\\right)}^{2} \\\\ &amp;=c-\\dfrac{{b}^{2}}{4a} \\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165137476446\">In practice, though, it is usually easier to remember that [latex]k[\/latex]\u00a0is the output value of the function when the input is [latex]h[\/latex], so [latex]f\\left(h\\right)=f\\left(-\\dfrac{b}{2a}\\right)=k[\/latex].<\/p>\r\n[embed]https:\/\/www.youtube.com\/watch?v=vAPPYoBV2Ow&feature=emb_title[\/embed]\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29461&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox tryit\">\r\n<h3>\u00a0Try It<\/h3>\r\nA coordinate grid has been superimposed over the quadratic path of a basketball in the picture below. Find an equation for the path of the ball. Does the shooter make the basket?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170344\/CNX_Precalc_Figure_03_02_0082.jpg\" alt=\"Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.\" width=\"487\" height=\"424\" \/> (credit: modification of work by Dan Meyer)[\/caption]\r\n\r\n[reveal-answer q=\"283194\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283194\"]\r\n\r\nThe path passes through the origin and has vertex at [latex]\\left(-4,\\text{ }7\\right)[\/latex], so [latex]\\left(h\\right)x=-\\frac{7}{16}{\\left(x+4\\right)}^{2}+7[\/latex]. To make the shot, [latex]h\\left(-7.5\\right)[\/latex] would need to be about 4 but [latex]h\\left(-7.5\\right)\\approx 1.64[\/latex]; he doesn\u2019t make it.\r\n\r\n[\/hidden-answer]\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 its 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<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<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 is 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>Transformations of Quadratic Functions<\/h2>\n<p id=\"fs-id1165137676320\">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 style=\"text-align: left;\">where [latex]\\left(h,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<p>The standard form is useful for determining how the graph is transformed from the graph of [latex]y={x}^{2}[\/latex]. The figure below\u00a0is the graph of this basic function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201255\/CNX_Precalc_Figure_03_02_0062.jpg\" alt=\"Graph of y=x^2.\" width=\"487\" height=\"480\" \/><\/p>\n<p>&nbsp;<\/p>\n<h2>Shift Up and Down by Changing the Value of [latex]k[\/latex]<\/h2>\n<p id=\"fs-id1165137770279\">You can represent a vertical (up, down) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, [latex]k[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=x^2 + k[\/latex]<\/p>\n<p>If [latex]k>0[\/latex], the graph shifts upward, whereas if [latex]k<0[\/latex], the graph shifts downward.\n\n\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted up 4 units. Also, determine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q941360\">Show Solution<\/span><\/p>\n<div id=\"q941360\" class=\"hidden-answer\" style=\"display: none\">\n<p>The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted up 4 units is<\/p>\n<p>[latex]f(x)=x^2+4[\/latex]<\/p>\n<p>The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units is<\/p>\n<p>[latex]f(x)=x^2-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Shift left and right by changing the value of [latex]h[\/latex]<\/h2>\n<p id=\"fs-id1165137770279\">You can represent a horizontal (left, right) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, [latex]h[\/latex], to the variable [latex]x[\/latex], before squaring.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=(x-h)^2[\/latex]<\/p>\n<p>If [latex]h>0[\/latex], the graph shifts toward the right and if [latex]h<0[\/latex], the graph shifts to the left.\n\n\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted right 2 units. Also, determine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted left 2 units.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q978434\">Show Solution<\/span><\/p>\n<div id=\"q978434\" class=\"hidden-answer\" style=\"display: none\">\n<p>The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted right 2 units is<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=(x-2)^2[\/latex]<\/p>\n<p style=\"text-align: left;\">The equation for the graph of\u00a0[latex]f(x)=^2[\/latex] that has been shifted left 2 units is<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=(x+2)^2[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<h2>Stretch or compress by changing the value of [latex]a[\/latex].<\/h2>\n<p id=\"fs-id1165137770279\">You can represent a stretch or compression (narrowing, widening)\u00a0of the graph of [latex]f(x)=x^2[\/latex] by\u00a0multiplying the squared variable by a constant, [latex]a[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ax^2[\/latex]<\/p>\n<p>The magnitude of [latex]a[\/latex]\u00a0indicates the stretch of the graph. If [latex]|a|>1[\/latex], the point associated with a particular [latex]x[\/latex]-value shifts farther from the [latex]x[\/latex]<em>&#8211;<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch. But if [latex]|a|<1[\/latex], the point associated with a particular [latex]x[\/latex]-value shifts closer to the [latex]x[\/latex]<em>&#8211;<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex]. Also, determine the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q391411\">Show Solution<\/span><\/p>\n<div id=\"q391411\" class=\"hidden-answer\" style=\"display: none\">\n<p>The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=\\frac{1}{2}x^2[\/latex]<\/p>\n<p>The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3 is<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=3x^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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=\"eip-173\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}&a{\\left(x-h\\right)}^{2}+k=a{x}^{2}+bx+c\\\\ &a{x}^{2}-2ahx+\\left(a{h}^{2}+k\\right)=a{x}^{2}+bx+c \\end{align}[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: left;\">For the two sides to be equal, the corresponding coefficients must be equal. In particular, the coefficients of [latex]x[\/latex] must be equal.<\/div>\n<div id=\"eip-144\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]-2ah=b,\\text{ so }h=-\\dfrac{b}{2a}[\/latex].<\/div>\n<p id=\"fs-id1165134118295\">This is the [latex]x[\/latex] coordinate of the vertexr and [latex]x=-\\dfrac{b}{2a}[\/latex] is the\u00a0<strong>axis of symmetry<\/strong> we defined earlier. Setting the constant terms equal gives us:<\/p>\n<div id=\"eip-313\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}a{h}^{2}+k&=c \\\\[2mm] k&=c-a{h}^{2} \\\\ &=c-a-{\\left(\\dfrac{b}{2a}\\right)}^{2} \\\\ &=c-\\dfrac{{b}^{2}}{4a} \\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165137476446\">In practice, though, it is usually easier to remember that [latex]k[\/latex]\u00a0is the output value of the function when the input is [latex]h[\/latex], so [latex]f\\left(h\\right)=f\\left(-\\dfrac{b}{2a}\\right)=k[\/latex].<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Find the Equation of a Quadratic Function from a Graph\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vAPPYoBV2Ow?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29461&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox tryit\">\n<h3>\u00a0Try It<\/h3>\n<p>A coordinate grid has been superimposed over the quadratic path of a basketball in the picture below. Find an equation for the path of the ball. Does the shooter make the basket?<\/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\/wp-content\/uploads\/sites\/896\/2016\/11\/02170344\/CNX_Precalc_Figure_03_02_0082.jpg\" alt=\"Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.\" width=\"487\" height=\"424\" \/><\/p>\n<p class=\"wp-caption-text\">(credit: modification of work by Dan Meyer)<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283194\">Show Solution<\/span><\/p>\n<div id=\"q283194\" class=\"hidden-answer\" style=\"display: none\">\n<p>The path passes through the origin and has vertex at [latex]\\left(-4,\\text{ }7\\right)[\/latex], so [latex]\\left(h\\right)x=-\\frac{7}{16}{\\left(x+4\\right)}^{2}+7[\/latex]. To make the shot, [latex]h\\left(-7.5\\right)[\/latex] would need to be about 4 but [latex]h\\left(-7.5\\right)\\approx 1.64[\/latex]; he doesn\u2019t make it.<\/p>\n<\/div>\n<\/div>\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","protected":false},"author":167848,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4959","chapter","type-chapter","status-publish","hentry"],"part":764,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4959","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4959\/revisions"}],"predecessor-version":[{"id":5264,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4959\/revisions\/5264"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/parts\/764"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4959\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/media?parent=4959"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapter-type?post=4959"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/contributor?post=4959"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/license?post=4959"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}