{"id":193,"date":"2023-06-21T13:22:42","date_gmt":"2023-06-21T13:22:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/characteristics-of-parabolas\/"},"modified":"2023-09-29T02:54:57","modified_gmt":"2023-09-29T02:54:57","slug":"characteristics-of-parabolas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/characteristics-of-parabolas\/","title":{"raw":"\u25aa   Characteristics of Parabolas","rendered":"\u25aa   Characteristics of Parabolas"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the vertex, axis of symmetry, [latex]y[\/latex]-intercept, and minimum or maximum value of a parabola from it's graph.<\/li>\r\n \t<li>Identify a quadratic function written in general and vertex form.<\/li>\r\n \t<li>Given a quadratic function in general form, find the vertex.<\/li>\r\n \t<li>Define the domain and range of a quadratic function by identifying the vertex as a maximum or minimum.<\/li>\r\n<\/ul>\r\n<\/div>\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 examples\">\r\n<h3>tip for success<\/h3>\r\nThe places where a function's graph crosses the horizontal axis are the places where the function value equals zero. You've seen that these values are called\u00a0<em>horizontal intercepts<\/em>,\u00a0<em>x-intercepts<\/em>,\u00a0and\u00a0<em>zeros<\/em> so far. They can also be referred to as the\u00a0<em>roots<\/em> of a function.\r\n\r\n<\/div>\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<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]<em>x<\/em>=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<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse an online graphing calculator to help you solve the question below:\r\n[ohm_question height=\"330\"]155642[\/ohm_question]\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]\u00a0are 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<div class=\"textbox key-takeaways\" style=\"text-align: left;\">\r\n<h3>Try It<\/h3>\r\nUsing an online graphing calculator, plot the function [latex]f\\left(x\\right)=2\\left(x-h\\right)^2+k[\/latex].\r\n\r\nChange the values of [latex]h[\/latex] and [latex]k[\/latex] to examine how changing the location of the vertex [latex](h,k)[\/latex] of a parabola also changes the axis of symmetry. Notice that when you move [latex]k[\/latex] independently of [latex]h[\/latex], you are only moving the vertical location of the vertex. Experiment with values between [latex]-10[\/latex] and [latex]10[\/latex].\r\n\r\n&nbsp;\r\n\r\nThe vertex of a parabola is the location of either the maximum or minimum value of the parabola. If [latex]a&gt;0[\/latex], the parabola opens upward and the parabola has a minimum value of [latex]k[\/latex] at [latex]x=h[\/latex]. If [latex]a&lt;0[\/latex], the parabola opens downward, and the parabola has a maximum value of [latex]k[\/latex] at [latex]x=h[\/latex]. In this case, the vertex is the location of the minimum value of the function because [latex]a=2[\/latex].\r\n\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]\u00a0value 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[ohm_question height=\"300\"]120300[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the vertex, axis of symmetry, [latex]y[\/latex]-intercept, and minimum or maximum value of a parabola from it&#8217;s graph.<\/li>\n<li>Identify a quadratic function written in general and vertex form.<\/li>\n<li>Given a quadratic function in general form, find the vertex.<\/li>\n<li>Define the domain and range of a quadratic function by identifying the vertex as a maximum or minimum.<\/li>\n<\/ul>\n<\/div>\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 examples\">\n<h3>tip for success<\/h3>\n<p>The places where a function&#8217;s graph crosses the horizontal axis are the places where the function value equals zero. You&#8217;ve seen that these values are called\u00a0<em>horizontal intercepts<\/em>,\u00a0<em>x-intercepts<\/em>,\u00a0and\u00a0<em>zeros<\/em> so far. They can also be referred to as the\u00a0<em>roots<\/em> of a function.<\/p>\n<\/div>\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.<br \/>\n<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]<em>x<\/em>=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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use an online graphing calculator to help you solve the question below:<br \/>\n<iframe loading=\"lazy\" id=\"ohm155642\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=155642&theme=oea&iframe_resize_id=ohm155642&show_question_numbers\" width=\"100%\" height=\"330\"><\/iframe><\/p>\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]\u00a0are 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 class=\"textbox key-takeaways\" style=\"text-align: left;\">\n<h3>Try It<\/h3>\n<p>Using an online graphing calculator, plot the function [latex]f\\left(x\\right)=2\\left(x-h\\right)^2+k[\/latex].<\/p>\n<p>Change the values of [latex]h[\/latex] and [latex]k[\/latex] to examine how changing the location of the vertex [latex](h,k)[\/latex] of a parabola also changes the axis of symmetry. Notice that when you move [latex]k[\/latex] independently of [latex]h[\/latex], you are only moving the vertical location of the vertex. Experiment with values between [latex]-10[\/latex] and [latex]10[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>The vertex of a parabola is the location of either the maximum or minimum value of the parabola. If [latex]a>0[\/latex], the parabola opens upward and the parabola has a minimum value of [latex]k[\/latex] at [latex]x=h[\/latex]. If [latex]a<0[\/latex], the parabola opens downward, and the parabola has a maximum value of [latex]k[\/latex] at [latex]x=h[\/latex]. In this case, the vertex is the location of the minimum value of the function because [latex]a=2[\/latex].\n\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]\u00a0value 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=\"ohm120300\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=120300&theme=oea&iframe_resize_id=ohm120300&show_question_numbers\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\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-193\">\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>Question ID 120303. <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><li>Question ID 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><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><li>Vertex and Axis of Symmetry Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/q3e3ymnpnn\">https:\/\/www.desmos.com\/calculator\/q3e3ymnpnn<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":30,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Question ID 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