{"id":1643,"date":"2016-11-02T17:12:27","date_gmt":"2016-11-02T17:12:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1643"},"modified":"2017-07-07T16:50:46","modified_gmt":"2017-07-07T16:50:46","slug":"characteristics-of-parabolas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/characteristics-of-parabolas\/","title":{"raw":"Characteristics of Parabolas","rendered":"Characteristics of Parabolas"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Identify the vertex, axis of symmetry, y-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\" data-media-type=\"image\/jpg\" \/>\r\n\r\nThe <em>y<\/em>-intercept is the point at which the parabola crosses the <em>y<\/em>-axis. The <em>x<\/em>-intercepts are the points at which the parabola crosses the <em>x<\/em>-axis. If they exist, the <em>x<\/em>-intercepts represent the <strong>zeros<\/strong>, or <strong>roots<\/strong>, of the quadratic function, the values of <em>x<\/em>\u00a0at which <em>y\u00a0<\/em>= 0.\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\" data-media-type=\"image\/jpg\" \/>\r\n\r\n[reveal-answer q=\"366804\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"366804\"]\r\nThe vertex is the turning point of the graph. We can see that the vertex is at (3, 1). Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is <em>x\u00a0<\/em>= 3. This parabola does not cross the <em>x<\/em>-axis, so it has no zeros. It crosses the <em>y<\/em>-axis at (0, 7) so this is the <em>y<\/em>-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\n<iframe id=\"mom20\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=120303&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Equations of Quadratic Functions<\/h3>\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 <em>a<\/em>,\u00a0<em>b<\/em>, and <em>c<\/em>\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=-\\frac{b}{2a}[\/latex]. If we use the quadratic formula, [latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex], to solve [latex]a{x}^{2}+bx+c=0[\/latex] for the <em>x<\/em>-intercepts, or zeros, we find the value of\u00a0<em>x<\/em>\u00a0halfway between them is always [latex]x=-\\frac{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=-\\frac{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 <i>x<\/i>-intercepts, those points where the parabola crosses the <i>x<\/i>-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\" data-media-type=\"image\/jpg\" \/>\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\nUse the sliders for h and k to examine how changing the location of the vertex (h,k) of a parabola also changes the axis of symmetry, labeled as x = h. Notice that when you move k independently of h, you are only moving the vertical location of the vertex.\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/q3e3ymnpnn\r\n\r\nThe vertex of a parabola is either the maximum or minimum value of the parabola.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.\r\n\r\nUse the textbox below to determine whether the vertex of the parabola in the interactive above is a minimum or a maximum.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\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, (<em>k<\/em>), and where it occurs, (<em>x<\/em>). 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 <em>a<\/em>,\u00a0<em>b<\/em>, and\u00a0<em>c<\/em>.<\/li>\r\n \t<li>Find <em>h<\/em>, the <em>x<\/em>-coordinate of the vertex, by substituting <em>a<\/em> and <em>b<\/em>\u00a0into [latex]h=-\\frac{b}{2a}[\/latex].<\/li>\r\n \t<li>Find <em>k<\/em>, the <em>y<\/em>-coordinate of the vertex, by evaluating [latex]k=f\\left(h\\right)=f\\left(-\\frac{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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"466886\"]\r\nThe horizontal coordinate of the vertex will be at\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}h=-\\frac{b}{2a}\\hfill \\\\ \\text{ }=-\\frac{-6}{2\\left(2\\right)}\\hfill \\\\ \\text{ }=\\frac{6}{4}\\hfill \\\\ \\text{ }=\\frac{3}{2}\\hfill \\end{array}[\/latex]<\/p>\r\nThe vertical coordinate of the vertex will be at\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}k=f\\left(h\\right)\\hfill \\\\ \\text{ }=f\\left(\\frac{3}{2}\\right)\\hfill \\\\ \\text{ }=2{\\left(\\frac{3}{2}\\right)}^{2}-6\\left(\\frac{3}{2}\\right)+7\\hfill \\\\ \\text{ }=\\frac{5}{2}\\hfill \\end{array}[\/latex]<\/p>\r\nRewriting into standard form, the stretch factor will be the same as the [latex]a[\/latex] in the original quadratic.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=a{x}^{2}+bx+c\\hfill \\\\ f\\left(x\\right)=2{x}^{2}-6x+7\\hfill \\end{array}[\/latex]<\/p>\r\nUsing the vertex to determine the shifts,\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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"713769\"][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[\/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 point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all <em>y<\/em>-values greater than or equal to the <em>y<\/em>-coordinate at the turning point or less than or equal to the <em>y<\/em>-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 <em>a<\/em>\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 <em>a<\/em>\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 <em>a<\/em>\u00a0value is [latex]f\\left(x\\right)\\ge k[\/latex]; the range of a quadratic function written in standard form with a negative <em>a<\/em>\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>Identify the domain of any quadratic function as all real numbers.<\/li>\r\n \t<li>Determine whether <em>a<\/em>\u00a0is positive or negative. If <em>a<\/em>\u00a0is positive, the parabola has a minimum. If <em>a<\/em>\u00a0is negative, the parabola has a maximum.<\/li>\r\n \t<li>Determine the maximum or minimum value of the parabola, <em>k<\/em>.<\/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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"40392\"]\r\nAs with any quadratic function, the domain is all real numbers.\r\n\r\nBecause <em>a<\/em>\u00a0is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the <em>x<\/em>-value of the vertex.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}h=-\\frac{b}{2a}\\hfill \\\\ \\text{ }=-\\frac{9}{2\\left(-5\\right)}\\hfill \\\\ \\text{ }=\\frac{9}{10}\\hfill \\end{array}[\/latex]<\/p>\r\nThe maximum value is given by [latex]f\\left(h\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(\\frac{9}{10}\\right)=5{\\left(\\frac{9}{10}\\right)}^{2}+9\\left(\\frac{9}{10}\\right)-1\\hfill \\\\ \\text{ }=\\frac{61}{20}\\hfill \\end{array}[\/latex]<\/p>\r\nThe range is [latex]f\\left(x\\right)\\le \\frac{61}{20}[\/latex], or [latex]\\left(-\\infty ,\\frac{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-\\frac{4}{7}\\right)}^{2}+\\frac{8}{11}[\/latex].\r\n\r\n[reveal-answer q=\"307368\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"307368\"]The domain is all real numbers. The range is [latex]f\\left(x\\right)\\ge \\frac{8}{11}[\/latex], or [latex]\\left[\\frac{8}{11},\\infty \\right)[\/latex].[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=120300&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Identify the vertex, axis of symmetry, y-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\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>The <em>y<\/em>-intercept is the point at which the parabola crosses the <em>y<\/em>-axis. The <em>x<\/em>-intercepts are the points at which the parabola crosses the <em>x<\/em>-axis. If they exist, the <em>x<\/em>-intercepts represent the <strong>zeros<\/strong>, or <strong>roots<\/strong>, of the quadratic function, the values of <em>x<\/em>\u00a0at which <em>y\u00a0<\/em>= 0.<\/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\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q366804\">Solution<\/span><\/p>\n<div id=\"q366804\" class=\"hidden-answer\" style=\"display: none\">\nThe vertex is the turning point of the graph. We can see that the vertex is at (3, 1). Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is <em>x\u00a0<\/em>= 3. This parabola does not cross the <em>x<\/em>-axis, so it has no zeros. It crosses the <em>y<\/em>-axis at (0, 7) so this is the <em>y<\/em>-intercept.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom20\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=120303&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Equations of Quadratic Functions<\/h3>\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 <em>a<\/em>,\u00a0<em>b<\/em>, and <em>c<\/em>\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=-\\frac{b}{2a}[\/latex]. If we use the quadratic formula, [latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex], to solve [latex]a{x}^{2}+bx+c=0[\/latex] for the <em>x<\/em>-intercepts, or zeros, we find the value of\u00a0<em>x<\/em>\u00a0halfway between them is always [latex]x=-\\frac{b}{2a}[\/latex], the equation for the axis of symmetry.<\/p>\n<p>The 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=-\\frac{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 <i>x<\/i>-intercepts, those points where the parabola crosses the <i>x<\/i>-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\" data-media-type=\"image\/jpg\" \/><\/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>Use the sliders for h and k to examine how changing the location of the vertex (h,k) of a parabola also changes the axis of symmetry, labeled as x = h. Notice that when you move k independently of h, you are only moving the vertical location of the vertex.<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/q3e3ymnpnn<\/p>\n<p>The vertex of a parabola is either the maximum or minimum value of the parabola.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.\n\nUse the textbox below to determine whether the vertex of the parabola in the interactive above is a minimum or a maximum.\n\n<textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\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, (<em>k<\/em>), and where it occurs, (<em>x<\/em>). 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 <em>a<\/em>,\u00a0<em>b<\/em>, and\u00a0<em>c<\/em>.<\/li>\n<li>Find <em>h<\/em>, the <em>x<\/em>-coordinate of the vertex, by substituting <em>a<\/em> and <em>b<\/em>\u00a0into [latex]h=-\\frac{b}{2a}[\/latex].<\/li>\n<li>Find <em>k<\/em>, the <em>y<\/em>-coordinate of the vertex, by evaluating [latex]k=f\\left(h\\right)=f\\left(-\\frac{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\">Solution<\/span><\/p>\n<div id=\"q466886\" class=\"hidden-answer\" style=\"display: none\">\nThe horizontal coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}h=-\\frac{b}{2a}\\hfill \\\\ \\text{ }=-\\frac{-6}{2\\left(2\\right)}\\hfill \\\\ \\text{ }=\\frac{6}{4}\\hfill \\\\ \\text{ }=\\frac{3}{2}\\hfill \\end{array}[\/latex]<\/p>\n<p>The vertical coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}k=f\\left(h\\right)\\hfill \\\\ \\text{ }=f\\left(\\frac{3}{2}\\right)\\hfill \\\\ \\text{ }=2{\\left(\\frac{3}{2}\\right)}^{2}-6\\left(\\frac{3}{2}\\right)+7\\hfill \\\\ \\text{ }=\\frac{5}{2}\\hfill \\end{array}[\/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 style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=a{x}^{2}+bx+c\\hfill \\\\ f\\left(x\\right)=2{x}^{2}-6x+7\\hfill \\end{array}[\/latex]<\/p>\n<p>Using the vertex to determine the shifts,<\/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\">Solution<\/span><\/p>\n<div id=\"q713769\" class=\"hidden-answer\" style=\"display: none\">[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<\/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 point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all <em>y<\/em>-values greater than or equal to the <em>y<\/em>-coordinate at the turning point or less than or equal to the <em>y<\/em>-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 <em>a<\/em>\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 <em>a<\/em>\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 <em>a<\/em>\u00a0value is [latex]f\\left(x\\right)\\ge k[\/latex]; the range of a quadratic function written in standard form with a negative <em>a<\/em>\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>Identify the domain of any quadratic function as all real numbers.<\/li>\n<li>Determine whether <em>a<\/em>\u00a0is positive or negative. If <em>a<\/em>\u00a0is positive, the parabola has a minimum. If <em>a<\/em>\u00a0is negative, the parabola has a maximum.<\/li>\n<li>Determine the maximum or minimum value of the parabola, <em>k<\/em>.<\/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\">Solution<\/span><\/p>\n<div id=\"q40392\" class=\"hidden-answer\" style=\"display: none\">\nAs with any quadratic function, the domain is all real numbers.<\/p>\n<p>Because <em>a<\/em>\u00a0is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the <em>x<\/em>-value of the vertex.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}h=-\\frac{b}{2a}\\hfill \\\\ \\text{ }=-\\frac{9}{2\\left(-5\\right)}\\hfill \\\\ \\text{ }=\\frac{9}{10}\\hfill \\end{array}[\/latex]<\/p>\n<p>The maximum value is given by [latex]f\\left(h\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(\\frac{9}{10}\\right)=5{\\left(\\frac{9}{10}\\right)}^{2}+9\\left(\\frac{9}{10}\\right)-1\\hfill \\\\ \\text{ }=\\frac{61}{20}\\hfill \\end{array}[\/latex]<\/p>\n<p>The range is [latex]f\\left(x\\right)\\le \\frac{61}{20}[\/latex], or [latex]\\left(-\\infty ,\\frac{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-\\frac{4}{7}\\right)}^{2}+\\frac{8}{11}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q307368\">Solution<\/span><\/p>\n<div id=\"q307368\" class=\"hidden-answer\" style=\"display: none\">The domain is all real numbers. The range is [latex]f\\left(x\\right)\\ge \\frac{8}{11}[\/latex], or [latex]\\left[\\frac{8}{11},\\infty \\right)[\/latex].<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=120300&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/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-1643\">\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":21,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Question ID 120303\",\"author\":\"Lumen 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