{"id":42,"date":"2023-11-08T13:35:41","date_gmt":"2023-11-08T13:35:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/"},"modified":"2026-02-13T17:33:41","modified_gmt":"2026-02-13T17:33:41","slug":"3-3-graphing-quadratic-functions-in-vertex-form","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/3-3-graphing-quadratic-functions-in-vertex-form\/","title":{"raw":"3.3 Graphing Quadratic Functions in Vertex Form","rendered":"3.3 Graphing Quadratic Functions in Vertex Form"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the vertex, axis of symmetry, [latex]y[\/latex]-intercept, and\/or minimum or maximum value of a quadratic function in the vertex form [latex]f(x)=a{(x-h)}^{2}+k[\/latex].<\/li>\r\n \t<li>Graph quadratic functions in vertex form.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<figure id=\"Figure_03_02_001\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0012.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/> <b>Figure 1.<\/b> An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)[\/caption]<\/figure>\r\n<p id=\"fs-id1165134339909\">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>\r\n\r\n<h2>Quadratic Functions<\/h2>\r\nBelow are three forms of the same quadratic function.\r\n\r\nVertex Form: [latex]f(x)=-2(x-3)^2+2[\/latex]\r\n\r\nIntercept Form: [latex]f(x)=-2(x-2)(x-4)[\/latex]\r\n\r\nGeneral Form: [latex]f(x)=-2x^2+12x-16[\/latex]\r\n\r\nWhat do all of these functions have in common? What makes all of the above functions QUADRATIC functions? A <strong>Quadratic Function\u00a0<\/strong>is a polynomial function of degree [latex]2[\/latex]. The verb <em>quadrare<\/em> in Latin means \"to make square.\" The quadratic term [latex]-2x^2[\/latex] can be read \"negative two multiplied by [latex]x[\/latex] to the second power\" or more simply, \"negative two [latex]x[\/latex] squared.\" The graph of a quadratic function is a curve called a parabola.\r\n<table style=\"border-collapse: collapse; width: 100%; font-size: 110%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><span style=\"text-decoration: underline;\"><strong>Quadratic Functions<\/strong><\/span><\/td>\r\n<td style=\"width: 50%; text-align: center;\"><span style=\"text-decoration: underline;\"><strong>Non-Quadratic Functions<\/strong><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]f(x)=x^2[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=3x+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=5x-4x^2-1[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]f(x)=x^4-3x^3+5x^2-4x+1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]g(x)=4(x+1)(x-3)[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=|x^2-4|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(x+4)^2-5[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]g(x)=x^3-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Recognize characteristics of parabolas<\/h2>\r\n<p id=\"fs-id1165137727999\">The graph of a quadratic function is a U-shaped curve called a <strong>parabola<\/strong>. In this section, we will learn how to graph parabolas. One important feature of the graph is that it has an extreme point, called the <strong>vertex<\/strong>. If the parabola opens upward, the vertex represents the lowest point on the graph, or the <strong>minimum value<\/strong> of the quadratic function. If the parabola opens downward, the vertex represents the highest point on the graph, or the <strong>maximum value<\/strong>. In either case, the vertex is the turning point on the graph. The graph of a parabola is symmetric with respect to a vertical line drawn through the vertex, called the <strong>axis of symmetry<\/strong>. These features are illustrated in Figure 2.<span id=\"fs-id1165134118332\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/> <b>Figure 2<\/b>[\/caption]\r\n<p id=\"fs-id1165137549127\">The <strong>[latex]y[\/latex]-intercept<\/strong> is the point\u00a0at which the parabola crosses the [latex]y[\/latex]-axis, the value of [latex]y[\/latex] at which [latex]x = 0[\/latex]; [latex]( 0,\\text{__} )[\/latex]. <strong>The [latex]x[\/latex]-intercept(s)<\/strong>\u00a0are the point(s) at which the parabola touches or 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]; [latex]( \\text{__}, 0)[\/latex]. The intercepts are illustrated in Figure 2.<\/p>\r\n\r\n<div id=\"Example_03_02_01\" class=\"example\">\r\n<div id=\"fs-id1165131959514\" class=\"exercise\">\r\n<div id=\"fs-id1165135541748\" class=\"problem textbox shaded\">\r\n<h3 style=\"text-align: center;\">Example<\/h3>\r\nDetermine the vertex, axis of symmetry, [latex]x[\/latex]-intercepts (if any), and [latex]y[\/latex]-intercept of the parabola shown in Figure 3.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"Graph of a parabola facing up with the U shape bottoming out at (3,1). Additional points: (0, 7) and (6,7).\" width=\"487\" height=\"517\" \/> <b>Figure 3<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"300488\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"300488\"]\r\n\r\nThe <strong>vertex, [latex](h,k)[\/latex]<\/strong> is the turning point of the graph. We can see that the vertex is at [latex](3, 1)[\/latex]. Which is the lowest point on the graph and [latex]k=1[\/latex] is the minimum value of this quadratic function.\r\n\r\nThe <strong>axis of symmetry<\/strong> is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is the line [latex]x= 3[\/latex].\r\n\r\nThis parabola does not cross the [latex]x[\/latex]-axis, so it has no real\u00a0<strong>[latex]x[\/latex]-intercepts<\/strong>.\r\n\r\nIt crosses the [latex]y[\/latex]-axis at [latex](0, 7)[\/latex] so this is the <strong>[latex]y[\/latex]-intercept<\/strong>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>A parent function is the simplest form of the type of function given. Figure 4 is the graph of [latex]y=x^2[\/latex], which is the parent function of quadratic functions.<\/div>\r\n<div><\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img style=\"font-size: 1em; orphans: 1; widows: 2;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010712\/CNX_Precalc_Figure_03_02_0062.jpg\" alt=\"Parabola shape with vertex at (0,0) and labeled as y = x squared.\" width=\"487\" height=\"700\" \/> <b>Figure 4<\/b>[\/caption]\r\n\r\n&nbsp;\r\n<h2>\u00a0Understand how the graph of a parabola is related to its quadratic function<\/h2>\r\n<p id=\"fs-id1165137676320\">The <strong>vertex\u00a0form <\/strong>of a quadratic function presents 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 id=\"fs-id1303104\">where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/p>\r\nHow do the [latex]a[\/latex], [latex]h[\/latex], and [latex]k[\/latex] change the parabola?\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nClick on \"Desmos\" in the lower right corner of the graph below. That will open the graph in a new tab where you can explore the ways the [latex]a[\/latex], [latex]h[\/latex], and [latex]k[\/latex] change the graph. Use the sliders on the left side to change the look of the parabola. As you move each slider, notice how it is changing the graph from the original graph of [latex]y=x^2[\/latex].\r\n\r\n<iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/sl8vf5s2mx?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe>\r\n<ul>\r\n \t<li>How is the [latex]a[\/latex] changing the graph? (Hint: explore what happens when [latex]a[\/latex] is negative, when [latex]a[\/latex] is more than 0 but less than 1, and when [latex]a[\/latex] is positive.)<\/li>\r\n \t<li>How is the [latex]h[\/latex] changing the graph? (Make sure to move the sliders in both positive and negative directions.)<\/li>\r\n \t<li>How is the [latex]k[\/latex] changing the graph?\u00a0(Make sure to move the sliders in both positive and negative directions.)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphing Quadratic Functions of the form [latex]f(x)={x}^{2}+k[\/latex]<\/h2>\r\nLet's begin by looking at the graph of quadratic functions of the form [latex]f(x)={x}^{2}+k[\/latex] such as the functions [latex]\\require{color}\\color{MidnightBlue}{g(x)=x^2+3}[\/latex] and [latex]\\color{ForestGreen}{h(x)=x^2-4}[\/latex]. We will compare these two functions to the parent function [latex]\\color{BrickRed}{f(x)=x^2}[\/latex].\r\n\r\n[caption id=\"attachment_1128\" align=\"aligncenter\" width=\"1024\"]<a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/desmos-graph-4\/\" rel=\"attachment wp-att-1128\"><img class=\"wp-image-1128 size-large\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-4-1024x1024.png\" alt=\"Three upward parabolas are shown: g of x = x squared plus three in blue with vertex at (0,3), f of x = x squared in red with vertex (0,0), and h of x = x squared minus 4 in green with vertex (0, negative 4).\" width=\"1024\" height=\"1024\" \/><\/a> <b>Figure 5<\/b>[\/caption]\r\n\r\nThe vertical shift of the graph depends on the value of [latex]k[\/latex] in the function [latex]f(x)=x^2+k[\/latex]. When [latex]k[\/latex] is positive, the graph is shifted up [latex]k[\/latex] units. When [latex]k[\/latex] is negative, the graph is shifted down [latex]|k|[\/latex] units. Notice in Figure 5 that the graph of [latex]\\color{MidnightBlue}{g(x)=x^2+3}[\/latex] is identical to the graph of [latex]\\color{BrickRed}{f}[\/latex] except that it is shifted vertically up 3 units and that the graph of [latex]\\color{ForestGreen}{h(x)=x^2-4}[\/latex] is identical to the graph of [latex]\\color{BrickRed}{f}[\/latex] except that it is shifted vertically down 4 units.\r\n<h2>Graphing Quadratic Functions of the form [latex]f(x)=(x-h)^2[\/latex]<\/h2>\r\nWe now will look at the graph of quadratic functions of the form [latex]f(x)=(x-h)^2[\/latex]. Our goal is to determine the effect adding or subtracting a real number, [latex]h[\/latex] from [latex]x[\/latex], has on the parent function [latex]f(x)=x^2[\/latex]. Let's look at the functions [latex]\\color{MidnightBlue}{g(x)=(x-3)^2}[\/latex] and [latex]\\color{forestGreen}{h(x)=(x+5)^2}[\/latex]. We will compare these two functions to the parent function [latex]\\color{BrickRed}{f(x)=x^2}[\/latex].\r\n\r\n[caption id=\"attachment_1132\" align=\"aligncenter\" width=\"800\"]<a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/desmos-graph-5\/\" rel=\"attachment wp-att-1132\"><img class=\"wp-image-1132 size-full\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-5.png\" alt=\"Three upward parabolas are shown: g of x = (x minus 3) squared in blue with vertex at (3,0), f of x = x squared in red with vertex (0,0), and h of x = (x +5) squared in green with vertex (negative 5,0).\" width=\"800\" height=\"800\" \/><\/a> <b>Figure 6<\/b>[\/caption]\r\n\r\nThe horizontal shift of the graph depends on the value of [latex]h[\/latex] in the function [latex]f(x)=(x-h)^2[\/latex].\u00a0When [latex]h[\/latex] is subtracted from [latex]x[\/latex], the graph is shifted right [latex]h[\/latex] units. When [latex]h[\/latex] is added to [latex]x[\/latex], the graph is shifted left [latex]h[\/latex] units. Notice in Figure 6 that the graph of [latex]\\color{MidnightBlue}{g(x)=(x-3)^2}[\/latex] is identical to the graph of [latex]\\color{BrickRed}{f}[\/latex] except that it is shifted horizontally right [latex]3[\/latex] units and that the graph of [latex]\\color{forestGreen}{h(x)=(x+5)^2}[\/latex] is identical to the graph of [latex]\\color{BrickRed}{f}[\/latex] except that it is shifted horizontally left [latex]5[\/latex] units.\r\n<h2>Combining Horizontal and Vertical Shifts<\/h2>\r\nLet's now look at combining horizontal and vertical shifts that are of the form [latex]f(x)=(x-h)^2+k[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>ExAMPLE<\/h3>\r\nGraph the function [latex]g(x)={(x+2)}^{2}-3[\/latex] using transformations and label the vertex.\r\n\r\n[reveal-answer q=\"612378\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"612378\"]\r\n\r\nLet's start with the graph of [latex]f(x)=x^2[\/latex]. Since [latex]2[\/latex] is added to [latex]x[\/latex], we will shift the graph horizontally left [latex]2[\/latex] units because we are subtracting a negative [latex]2[\/latex] (which is the same as adding [latex]2[\/latex]) for [latex]h[\/latex], Therefore [latex]h=-2[\/latex], which is why the graph shifts to the left [latex]2[\/latex] units. This can be written as the function [latex]f(x)={(x-(-2))}^{2}[\/latex] or simplified to be [latex]f(x)=(x+2)^2[\/latex].\r\n\r\nNow let's look at the vertical shift of the graph. Remember that [latex]k[\/latex] is what determines the vertical shift. Since [latex]k[\/latex] is negative, the graph is shifted down [latex]|-3|[\/latex] units. In this case, the graph will be shifted vertically down [latex]3[\/latex] units.\r\n\r\nBelow are the graphs of [latex]f(x)=x^2[\/latex] and [latex]g(x)={(x+2)}^{2}-3[\/latex]. Let's compare them so that we can visualize the shifts that are occurring. This will also help us determine the vertex of the parabola, which is an important part of the parabola because it is the turning point on the graph. The vertex is a minimum value at [latex](-2,-3)[\/latex].\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/desmos-graph-22\/\" rel=\"attachment wp-att-1192\"><img class=\"alignnone wp-image-1192 size-full\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-22-e1720565197210.png\" alt=\"Graph of f of x equals x squared, with a parabola opening upward with vertex at (0,0).\" width=\"400\" height=\"400\" \/><\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/desmos-graph-21\/\" rel=\"attachment wp-att-1191\"><img class=\"alignnone wp-image-1191 size-full\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-21-e1720565024649.png\" alt=\"Graph of g of x equals quantity (x plus 2) squared minus 3. Parabola opens upward with vertex labeled at (negative 2, negative 3).\" width=\"400\" height=\"400\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Graphing Quadratic Functions of the form [latex]f(x)=a{x}^{2}[\/latex]<\/h2>\r\nWe now will look at the graph of quadratic functions of the form [latex]f(x)=a{x}^{2}[\/latex].\u00a0Our goal is to determine the effect that multiplying by a non-zero real number, [latex]a[\/latex] has on the parent function [latex]f(x)=x^2[\/latex]. Let's look at the functions [latex]\\color{MidnightBlue}{g(x)=2{x}^{2}}[\/latex] and [latex]\\color{forestGreen}{h(x)=\\dfrac{1}{2}{x}^{2}}[\/latex]. We will compare these two functions to the parent function [latex]\\color{BrickRed}{f(x)=x^2}[\/latex]. Does multiplying by [latex]2[\/latex] or [latex]\\dfrac{1}{2}[\/latex] cause the vertex to move to a different location? Does multiplying by [latex]2[\/latex] or [latex]\\dfrac{1}{2}[\/latex] change the shape of the parabola?\r\n\r\n[caption id=\"attachment_1194\" align=\"aligncenter\" width=\"800\"]<img class=\"wp-image-1194 size-full\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-23.png\" alt=\"Three upward parabolas are shown: g of x = 2 x squared in blue as skinniest, f of x = x squared in red with parent function width, and h of x = one-half x squared in green that opens widest. All have vertex at (0,0).\" width=\"800\" height=\"800\" \/> <b>Figure 7<\/b>[\/caption]\r\n\r\nIn Figure 7, we can see that the vertex remained in the same place, at [latex](0,0)[\/latex], but the shape of the parabola changed. The graph of [latex]\\color{MidnightBlue}{g(x)=2{x}^{2}}[\/latex] is the same as the [latex]\\color{BrickRed}{f}[\/latex] function but stretched vertically by a factor of [latex]2[\/latex] (the blue graph appears narrower than [latex]\\color{BrickRed}{f(x)=x^2}[\/latex]). As [latex]a[\/latex] gets larger, the parabola gets narrower, or vertically stretched. What happens to the graph when [latex]a[\/latex] is between zero and one ([latex]0&lt;a&lt;1[\/latex])? The graph of [latex]\\color{forestGreen}{h(x)=\\dfrac{1}{2}{x}^{2}}[\/latex] is the same as the [latex]\\color{BrickRed}{f}[\/latex] function but compressed vertically by a factor of [latex]\\dfrac{1}{2}[\/latex] (the green graph appears wider than [latex]\\color{BrickRed}{f(x)=x^2}[\/latex]).\r\n\r\nWhat happens when [latex]a[\/latex] is a negative number? Let's look at the functions [latex]\\color{MidnightBlue}{g(x)=-5x^2}[\/latex] and [latex]\\color{ForestGreen}{h(x)= -\\dfrac{1}{8}x^2}[\/latex].\u00a0We will compare these two functions to the function [latex]\\color{BrickRed}{f(x)=-x^2}[\/latex]. What do you notice about the graphs in Figure 8?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"600\"]<img class=\"alignnone size-full wp-image-1197\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-24-e1720578899103.png\" alt=\"\" width=\"600\" height=\"600\" \/> <b>Figure 8<\/b>[\/caption]\r\n\r\nThe magnitude of [latex]a[\/latex]\u00a0indicates the stretch of the graph. If [latex]|a|&gt;1[\/latex], there is a vertical stretch and the graph appears to become narrower. If [latex]|a|&lt;1[\/latex], there is a vertical compression and the graph appears to become wider.\r\n\r\nIn Figure 8, we notice that when [latex]a&lt;0[\/latex], is reflected across the [latex]x[\/latex]-axis, the parabola opens downward The vertex at [latex](0,0)[\/latex] is the highest point on the graph and [latex]k=0[\/latex] is the maximum value of this quadratic function. The graph of [latex]\\color{MidnightBlue}{g(x)=-5x^2}[\/latex] is the same as the graph [latex]\\color{BrickRed}{f(x)=-x^2}[\/latex] with a vertical stretch by a factor of [latex]|-5|[\/latex] (the blue graph appears narrower than [latex]\\color{BrickRed}{f(x)=-x^2}[\/latex]). The graph of [latex]\\color{ForestGreen}{h(x)=-\\dfrac{1}{8}x^2}[\/latex] is the same as the graph\u00a0[latex]\\color{BrickRed}{f(x)=-x^2}[\/latex] but compressed vertically by a factor of [latex]|-\\dfrac{1}{8}|[\/latex] (the green graph appears wider than [latex]\\color{BrickRed}{f(x)=-x^2}[\/latex]).\r\n<h2>Graphing Quadratic Functions of the form [latex]f(x)=a(x-h)^2+k[\/latex]<\/h2>\r\nWhat is [latex]a[\/latex], [latex]h[\/latex], and [latex]k[\/latex] in figure 9? The quadratic function of the graph below written in vertex form is [latex]y=-3{\\left(x+2\\right)}^{2}+4[\/latex].\u00a0In this form, [latex]a=-3,\\text{ }h=-2[\/latex], and [latex]k=4[\/latex]. Since [latex]a&lt;0[\/latex], the parabola opens downward. The vertex is at [latex]\\left(-2,\\text{ 4}\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0052.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=-3(x+2)^2+4.\" width=\"487\" height=\"630\" \/> <b>Figure 9 [latex]-3(x+2)^2+4[\/latex]<\/b>[\/caption]\r\n<p id=\"fs-id1165137770279\">If [latex]k&gt;0[\/latex], the graph shifts upward, whereas if [latex]k&lt;0[\/latex], the graph shifts downward. In Figure 9, [latex]k&gt;0[\/latex], so the graph is shifted [latex]4[\/latex] units upward.<\/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. In Figure 9, [latex]h&lt;0[\/latex], so the graph is shifted [latex]2[\/latex] units to the left.\r\n\r\nIn Figure 9, [latex]|a|&gt;1[\/latex]. The magnitude of [latex]a[\/latex] indicates the stretch of the graph. When [latex]|a|&gt;1[\/latex] there is a vertical stretch causing the graph to become narrower.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>ExAMPLE<\/h3>\r\nCompare the graph of [latex]f(x)=\\dfrac{1}{2}{(x-4)}^{2}-6[\/latex] to the parent function [latex]f(x)=x^2[\/latex] by stating the transformations. Find the vertex, axis of symmetry, two other points on graph, and then graph the function. Determine if the vertex is a maximum or minimum.\r\n\r\n[reveal-answer q=\"315217\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"315217\"]\r\n\r\n&nbsp;\r\n\r\nQuadratic Equations in form of [latex]f(x)=a(x-h)^2+k[\/latex] are said to be in vertex form where the vertex is [latex](h,k)[\/latex].\r\n\r\n<span style=\"text-decoration: underline;\"><strong>Transformations:<\/strong><\/span>\r\n\r\n[latex]a=\\dfrac{1}{2}[\/latex], [latex]h=4[\/latex], and [latex]k=-6[\/latex]\r\n<ul>\r\n \t<li>[latex]a=\\dfrac{1}{2}[\/latex] means the graph has a vertical compression by a factor of [latex]\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]h=4[\/latex] means the graph is shifted right [latex]4[\/latex] units<\/li>\r\n \t<li>[latex]k=-6[\/latex] means the graph is shifted down [latex]6[\/latex] units<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\nThe vertex is [latex](h,k)[\/latex]; in this case it would be [latex](4, -6)[\/latex]. When [latex]a&gt;0[\/latex] the parabola opens upward. The vertex is the minimum. The minimum value is [latex]k=-6[\/latex].\r\n\r\nThe axis of symmetry is a vertical line which passes through the [latex]x[\/latex]-coordinate of the vertex. The line of symmetry is [latex]x=4[\/latex].\r\n\r\nLet's find the [latex]y[\/latex]-intercept. Let [latex]x=0[\/latex] and solve for [latex]y[\/latex].\r\n\r\n$$\\begin{align}\r\n\\require{color}f(\\color{Green}{0}\\color{black}{)} &amp;= \\dfrac{1}{2}(\\color{Green}{0}\\color{black}{-4)^{2}-6} \\\\&amp;= \\dfrac{1}{2}(-4)^{2}-6 \\\\ &amp;= \\dfrac{1}{2}(16)-6 \\\\ &amp;= 8-6 \\\\ &amp;= 2\r\n\r\n\\end{align}$$\r\n\r\nThe [latex]y[\/latex]-intercept is the point [latex](0,2)[\/latex].\r\n\r\nUsing [latex]x=4[\/latex] as the axis of symmetry, the point symmetric to [latex](0,2)[\/latex] is [latex](8,2)[\/latex].\r\n\r\nThe graph is given below.\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/desmos-graph-25\/\" rel=\"attachment wp-att-1220\"><img class=\"alignnone wp-image-1220 size-full\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-25-e1720981193775.png\" alt=\"A upward facing parabola graphed with vertex (4, negative 6) labeled. A vertical dashed line through the vertex is labeled x = 4 and axis of symmetry. Additional points labeled at (0,2) and (8,2) are shown.\" width=\"600\" height=\"600\" \/><\/a>\r\n\r\nRemember, instead of finding the [latex]y[\/latex]-intercept, you could pick an [latex]x[\/latex]-value on either side of the vertex. The [latex]x[\/latex]-value of the vertex is [latex]4[\/latex], so let's pick [latex]x[\/latex] values such as [latex]2,3,5,[\/latex] and [latex]6[\/latex]. Let's create a table of values using these [latex]x[\/latex] values. Let's start by letting [latex]x=3[\/latex] and [latex]x=5[\/latex]. You can see below that we end up with fractional values for the [latex]y[\/latex]-coordinate. These are perfectly acceptable values but sometimes hard to graph precisely on a coordinate plane.\r\n<table style=\"border-collapse: collapse; width: 25%; border: 1px solid black; height: 72px; font-size: 110%;\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">\u00a0[latex]x[\/latex]<\/th>\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]f(x)[\/latex] or [latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #ff6600;\"><strong>[latex]3[\/latex]<\/strong><\/span><\/td>\r\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px;\"><span style=\"color: #ff6600;\"><strong>[latex]\\begin{align} &amp;= \\dfrac{1}{2}(3-4)^2-6\\\\ &amp;= \\dfrac{1}{2}(-1)^2-6\\\\ &amp;= \\dfrac{1}{2}-6\\\\ &amp;=-\\dfrac{11}{2}\\end{align}[\/latex]<\/strong><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><strong><span style=\"color: #339966;\">[latex]4[\/latex]<\/span><\/strong><\/td>\r\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><strong><span style=\"color: #339966;\">[latex]-6[\/latex]<\/span><\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><strong><span style=\"color: #ff6600;\">[latex]5[\/latex]<\/span><\/strong><\/td>\r\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><strong><span style=\"color: #ff6600;\">[latex]\\begin{align} &amp;= \\dfrac{1}{2}(5-4)^2-6\\\\ &amp;= \\dfrac{1}{2}(1)^2-6\\\\ &amp;= \\dfrac{1}{2}-6\\\\ &amp;=-\\dfrac{11}{2}\\end{align}[\/latex]<\/span><\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow let's fill in the chart when [latex]x=2[\/latex] and [latex]x=6[\/latex].\r\n<table style=\"border-collapse: collapse; width: 25%; border: 1px solid black; height: 72px; font-size: 110%;\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">\u00a0[latex]x[\/latex]<\/th>\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]f(x)[\/latex] or [latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #33cccc;\"><strong>[latex]2[\/latex]<\/strong><\/span><\/td>\r\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #33cccc;\"><strong>[latex]\\begin{align} &amp;= \\dfrac{1}{2}(2-4)^2-6\\\\ &amp;= \\dfrac{1}{2}(-2)^2-6\\\\ &amp;= \\dfrac{1}{2}(4)-6\\\\ &amp;= 2-6\\\\ &amp;= -4\\end{align}[\/latex]<\/strong><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #000000;\">[latex]3[\/latex]<\/span><\/td>\r\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px;\"><span style=\"color: #000000;\">[latex]\\begin{align} &amp;= \\dfrac{1}{2}(3-4)^2-6\\\\ &amp;= \\dfrac{1}{2}(-1)^2-6\\\\ &amp;= \\dfrac{1}{2}-6\\\\ &amp;=-\\dfrac{11}{2}\\end{align}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #000000;\">[latex]4[\/latex]<\/span><\/td>\r\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #000000;\">[latex]-6[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #000000;\">[latex]5[\/latex]<\/span><\/td>\r\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #000000;\">[latex]\\begin{align} &amp;= \\dfrac{1}{2}(5-4)^2-6\\\\ &amp;= \\dfrac{1}{2}(1)^2-6\\\\ &amp;= \\dfrac{1}{2}-6\\\\ &amp;=-\\dfrac{11}{2}\\end{align}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #33cccc;\"><strong>[latex]6[\/latex]<\/strong><\/span><\/td>\r\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #33cccc;\"><strong>[latex]\\begin{align} &amp;= \\dfrac{1}{2}(6-4)^2-6\\\\ &amp;= \\dfrac{1}{2}(2)^2-6\\\\ &amp;= \\dfrac{1}{2}(4)-6\\\\ &amp;= 2-6\\\\ &amp;= -4\\end{align}[\/latex]<\/strong><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo, you could also plot the points <span style=\"color: #33cccc;\">[latex](2,-4)[\/latex]<\/span> and <span style=\"color: #33cccc;\">[latex](6,-4)[\/latex].<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/hvyH-WJtMpc?si=cNY0lRvfBGHh83HO\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]288159[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"Example_03_02_01\" class=\"example\">\r\n<div id=\"fs-id1165131959514\" class=\"exercise\"><section id=\"fs-id1165134205927\" class=\"key-equations\"><\/section><section id=\"fs-id1165135426424\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\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. The [latex]x[\/latex]-intercepts, are the points at which the parabola crosses the [latex]x[\/latex]-axis. The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y-[\/latex]axis.<\/li>\r\n \t<li>Vertex form is useful to easily identify the vertex of a parabola.<\/li>\r\n \t<li>A quadratic function\u2019s minimum or maximum value is given by the [latex]y[\/latex]-value of the vertex.<\/li>\r\n<\/ul>\r\n<\/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=-\\frac{b}{2a}[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137931314\" class=\"definition\">\r\n \t<dt>quadratic function<\/dt>\r\n \t<dd>a polynomial function of degree 2. The graph of a quadratic function is a curve called a parabola.<\/dd>\r\n \t<dt><strong>vertex 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><strong>vertex<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135623619\">the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/dd>\r\n<\/dl>\r\n<\/section><\/div>\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the vertex, axis of symmetry, [latex]y[\/latex]-intercept, and\/or minimum or maximum value of a quadratic function in the vertex form [latex]f(x)=a{(x-h)}^{2}+k[\/latex].<\/li>\n<li>Graph quadratic functions in vertex form.<\/li>\n<\/ul>\n<\/div>\n<figure id=\"Figure_03_02_001\" class=\"medium\">\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0012.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1165134339909\">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<h2>Quadratic Functions<\/h2>\n<p>Below are three forms of the same quadratic function.<\/p>\n<p>Vertex Form: [latex]f(x)=-2(x-3)^2+2[\/latex]<\/p>\n<p>Intercept Form: [latex]f(x)=-2(x-2)(x-4)[\/latex]<\/p>\n<p>General Form: [latex]f(x)=-2x^2+12x-16[\/latex]<\/p>\n<p>What do all of these functions have in common? What makes all of the above functions QUADRATIC functions? A <strong>Quadratic Function\u00a0<\/strong>is a polynomial function of degree [latex]2[\/latex]. The verb <em>quadrare<\/em> in Latin means &#8220;to make square.&#8221; The quadratic term [latex]-2x^2[\/latex] can be read &#8220;negative two multiplied by [latex]x[\/latex] to the second power&#8221; or more simply, &#8220;negative two [latex]x[\/latex] squared.&#8221; The graph of a quadratic function is a curve called a parabola.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; font-size: 110%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><span style=\"text-decoration: underline;\"><strong>Quadratic Functions<\/strong><\/span><\/td>\n<td style=\"width: 50%; text-align: center;\"><span style=\"text-decoration: underline;\"><strong>Non-Quadratic Functions<\/strong><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]f(x)=x^2[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=3x+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]y=5x-4x^2-1[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]f(x)=x^4-3x^3+5x^2-4x+1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]g(x)=4(x+1)(x-3)[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=|x^2-4|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(x+4)^2-5[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]g(x)=x^3-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Recognize characteristics of parabolas<\/h2>\n<p id=\"fs-id1165137727999\">The graph of a quadratic function is a U-shaped curve called a <strong>parabola<\/strong>. In this section, we will learn how to graph parabolas. One important feature of the graph is that it has an extreme point, called the <strong>vertex<\/strong>. If the parabola opens upward, the vertex represents the lowest point on the graph, or the <strong>minimum value<\/strong> of the quadratic function. If the parabola opens downward, the vertex represents the highest point on the graph, or the <strong>maximum value<\/strong>. In either case, the vertex is the turning point on the graph. The graph of a parabola is symmetric with respect to a vertical line drawn through the vertex, called the <strong>axis of symmetry<\/strong>. These features are illustrated in Figure 2.<span id=\"fs-id1165134118332\"><br \/>\n<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_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 class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137549127\">The <strong>[latex]y[\/latex]-intercept<\/strong> is the point\u00a0at which the parabola crosses the [latex]y[\/latex]-axis, the value of [latex]y[\/latex] at which [latex]x = 0[\/latex]; [latex]( 0,\\text{__} )[\/latex]. <strong>The [latex]x[\/latex]-intercept(s)<\/strong>\u00a0are the point(s) at which the parabola touches or 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]; [latex]( \\text{__}, 0)[\/latex]. The intercepts are illustrated in Figure 2.<\/p>\n<div id=\"Example_03_02_01\" class=\"example\">\n<div id=\"fs-id1165131959514\" class=\"exercise\">\n<div id=\"fs-id1165135541748\" class=\"problem textbox shaded\">\n<h3 style=\"text-align: center;\">Example<\/h3>\n<p>Determine the vertex, axis of symmetry, [latex]x[\/latex]-intercepts (if any), and [latex]y[\/latex]-intercept of the parabola shown in Figure 3.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"Graph of a parabola facing up with the U shape bottoming out at (3,1). Additional points: (0, 7) and (6,7).\" width=\"487\" height=\"517\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q300488\">Show Solution<\/span><\/p>\n<div id=\"q300488\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <strong>vertex, [latex](h,k)[\/latex]<\/strong> is the turning point of the graph. We can see that the vertex is at [latex](3, 1)[\/latex]. Which is the lowest point on the graph and [latex]k=1[\/latex] is the minimum value of this quadratic function.<\/p>\n<p>The <strong>axis of symmetry<\/strong> is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is the line [latex]x= 3[\/latex].<\/p>\n<p>This parabola does not cross the [latex]x[\/latex]-axis, so it has no real\u00a0<strong>[latex]x[\/latex]-intercepts<\/strong>.<\/p>\n<p>It crosses the [latex]y[\/latex]-axis at [latex](0, 7)[\/latex] so this is the <strong>[latex]y[\/latex]-intercept<\/strong>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>A parent function is the simplest form of the type of function given. Figure 4 is the graph of [latex]y=x^2[\/latex], which is the parent function of quadratic functions.<\/div>\n<div><\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" style=\"font-size: 1em; orphans: 1; widows: 2;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010712\/CNX_Precalc_Figure_03_02_0062.jpg\" alt=\"Parabola shape with vertex at (0,0) and labeled as y = x squared.\" width=\"487\" height=\"700\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>\u00a0Understand how the graph of a parabola is related to its quadratic function<\/h2>\n<p id=\"fs-id1165137676320\">The <strong>vertex\u00a0form <\/strong>of a quadratic function presents 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 id=\"fs-id1303104\">where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/p>\n<p>How do the [latex]a[\/latex], [latex]h[\/latex], and [latex]k[\/latex] change the parabola?<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Click on &#8220;Desmos&#8221; in the lower right corner of the graph below. That will open the graph in a new tab where you can explore the ways the [latex]a[\/latex], [latex]h[\/latex], and [latex]k[\/latex] change the graph. Use the sliders on the left side to change the look of the parabola. As you move each slider, notice how it is changing the graph from the original graph of [latex]y=x^2[\/latex].<\/p>\n<p><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/sl8vf5s2mx?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\n<ul>\n<li>How is the [latex]a[\/latex] changing the graph? (Hint: explore what happens when [latex]a[\/latex] is negative, when [latex]a[\/latex] is more than 0 but less than 1, and when [latex]a[\/latex] is positive.)<\/li>\n<li>How is the [latex]h[\/latex] changing the graph? (Make sure to move the sliders in both positive and negative directions.)<\/li>\n<li>How is the [latex]k[\/latex] changing the graph?\u00a0(Make sure to move the sliders in both positive and negative directions.)<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing Quadratic Functions of the form [latex]f(x)={x}^{2}+k[\/latex]<\/h2>\n<p>Let&#8217;s begin by looking at the graph of quadratic functions of the form [latex]f(x)={x}^{2}+k[\/latex] such as the functions [latex]\\require{color}\\color{MidnightBlue}{g(x)=x^2+3}[\/latex] and [latex]\\color{ForestGreen}{h(x)=x^2-4}[\/latex]. We will compare these two functions to the parent function [latex]\\color{BrickRed}{f(x)=x^2}[\/latex].<\/p>\n<div id=\"attachment_1128\" style=\"width: 1034px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/desmos-graph-4\/\" rel=\"attachment wp-att-1128\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1128\" class=\"wp-image-1128 size-large\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-4-1024x1024.png\" alt=\"Three upward parabolas are shown: g of x = x squared plus three in blue with vertex at (0,3), f of x = x squared in red with vertex (0,0), and h of x = x squared minus 4 in green with vertex (0, negative 4).\" width=\"1024\" height=\"1024\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-4-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-4-150x150.png 150w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-4-300x300.png 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-4-768x768.png 768w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-4-65x65.png 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-4-225x225.png 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-4-350x350.png 350w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-4.png 2000w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/p>\n<p id=\"caption-attachment-1128\" class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<p>The vertical shift of the graph depends on the value of [latex]k[\/latex] in the function [latex]f(x)=x^2+k[\/latex]. When [latex]k[\/latex] is positive, the graph is shifted up [latex]k[\/latex] units. When [latex]k[\/latex] is negative, the graph is shifted down [latex]|k|[\/latex] units. Notice in Figure 5 that the graph of [latex]\\color{MidnightBlue}{g(x)=x^2+3}[\/latex] is identical to the graph of [latex]\\color{BrickRed}{f}[\/latex] except that it is shifted vertically up 3 units and that the graph of [latex]\\color{ForestGreen}{h(x)=x^2-4}[\/latex] is identical to the graph of [latex]\\color{BrickRed}{f}[\/latex] except that it is shifted vertically down 4 units.<\/p>\n<h2>Graphing Quadratic Functions of the form [latex]f(x)=(x-h)^2[\/latex]<\/h2>\n<p>We now will look at the graph of quadratic functions of the form [latex]f(x)=(x-h)^2[\/latex]. Our goal is to determine the effect adding or subtracting a real number, [latex]h[\/latex] from [latex]x[\/latex], has on the parent function [latex]f(x)=x^2[\/latex]. Let&#8217;s look at the functions [latex]\\color{MidnightBlue}{g(x)=(x-3)^2}[\/latex] and [latex]\\color{forestGreen}{h(x)=(x+5)^2}[\/latex]. We will compare these two functions to the parent function [latex]\\color{BrickRed}{f(x)=x^2}[\/latex].<\/p>\n<div id=\"attachment_1132\" style=\"width: 810px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/desmos-graph-5\/\" rel=\"attachment wp-att-1132\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1132\" class=\"wp-image-1132 size-full\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-5.png\" alt=\"Three upward parabolas are shown: g of x = (x minus 3) squared in blue with vertex at (3,0), f of x = x squared in red with vertex (0,0), and h of x = (x +5) squared in green with vertex (negative 5,0).\" width=\"800\" height=\"800\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-5.png 800w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-5-150x150.png 150w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-5-300x300.png 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-5-768x768.png 768w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-5-65x65.png 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-5-225x225.png 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-5-350x350.png 350w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/a><\/p>\n<p id=\"caption-attachment-1132\" class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<p>The horizontal shift of the graph depends on the value of [latex]h[\/latex] in the function [latex]f(x)=(x-h)^2[\/latex].\u00a0When [latex]h[\/latex] is subtracted from [latex]x[\/latex], the graph is shifted right [latex]h[\/latex] units. When [latex]h[\/latex] is added to [latex]x[\/latex], the graph is shifted left [latex]h[\/latex] units. Notice in Figure 6 that the graph of [latex]\\color{MidnightBlue}{g(x)=(x-3)^2}[\/latex] is identical to the graph of [latex]\\color{BrickRed}{f}[\/latex] except that it is shifted horizontally right [latex]3[\/latex] units and that the graph of [latex]\\color{forestGreen}{h(x)=(x+5)^2}[\/latex] is identical to the graph of [latex]\\color{BrickRed}{f}[\/latex] except that it is shifted horizontally left [latex]5[\/latex] units.<\/p>\n<h2>Combining Horizontal and Vertical Shifts<\/h2>\n<p>Let&#8217;s now look at combining horizontal and vertical shifts that are of the form [latex]f(x)=(x-h)^2+k[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>ExAMPLE<\/h3>\n<p>Graph the function [latex]g(x)={(x+2)}^{2}-3[\/latex] using transformations and label the vertex.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q612378\">Show Solution<\/span><\/p>\n<div id=\"q612378\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let&#8217;s start with the graph of [latex]f(x)=x^2[\/latex]. Since [latex]2[\/latex] is added to [latex]x[\/latex], we will shift the graph horizontally left [latex]2[\/latex] units because we are subtracting a negative [latex]2[\/latex] (which is the same as adding [latex]2[\/latex]) for [latex]h[\/latex], Therefore [latex]h=-2[\/latex], which is why the graph shifts to the left [latex]2[\/latex] units. This can be written as the function [latex]f(x)={(x-(-2))}^{2}[\/latex] or simplified to be [latex]f(x)=(x+2)^2[\/latex].<\/p>\n<p>Now let&#8217;s look at the vertical shift of the graph. Remember that [latex]k[\/latex] is what determines the vertical shift. Since [latex]k[\/latex] is negative, the graph is shifted down [latex]|-3|[\/latex] units. In this case, the graph will be shifted vertically down [latex]3[\/latex] units.<\/p>\n<p>Below are the graphs of [latex]f(x)=x^2[\/latex] and [latex]g(x)={(x+2)}^{2}-3[\/latex]. Let&#8217;s compare them so that we can visualize the shifts that are occurring. This will also help us determine the vertex of the parabola, which is an important part of the parabola because it is the turning point on the graph. The vertex is a minimum value at [latex](-2,-3)[\/latex].<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/desmos-graph-22\/\" rel=\"attachment wp-att-1192\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1192 size-full\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-22-e1720565197210.png\" alt=\"Graph of f of x equals x squared, with a parabola opening upward with vertex at (0,0).\" width=\"400\" height=\"400\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/desmos-graph-21\/\" rel=\"attachment wp-att-1191\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1191 size-full\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-21-e1720565024649.png\" alt=\"Graph of g of x equals quantity (x plus 2) squared minus 3. Parabola opens upward with vertex labeled at (negative 2, negative 3).\" width=\"400\" height=\"400\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Graphing Quadratic Functions of the form [latex]f(x)=a{x}^{2}[\/latex]<\/h2>\n<p>We now will look at the graph of quadratic functions of the form [latex]f(x)=a{x}^{2}[\/latex].\u00a0Our goal is to determine the effect that multiplying by a non-zero real number, [latex]a[\/latex] has on the parent function [latex]f(x)=x^2[\/latex]. Let&#8217;s look at the functions [latex]\\color{MidnightBlue}{g(x)=2{x}^{2}}[\/latex] and [latex]\\color{forestGreen}{h(x)=\\dfrac{1}{2}{x}^{2}}[\/latex]. We will compare these two functions to the parent function [latex]\\color{BrickRed}{f(x)=x^2}[\/latex]. Does multiplying by [latex]2[\/latex] or [latex]\\dfrac{1}{2}[\/latex] cause the vertex to move to a different location? Does multiplying by [latex]2[\/latex] or [latex]\\dfrac{1}{2}[\/latex] change the shape of the parabola?<\/p>\n<div id=\"attachment_1194\" style=\"width: 810px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1194\" class=\"wp-image-1194 size-full\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-23.png\" alt=\"Three upward parabolas are shown: g of x = 2 x squared in blue as skinniest, f of x = x squared in red with parent function width, and h of x = one-half x squared in green that opens widest. All have vertex at (0,0).\" width=\"800\" height=\"800\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-23.png 800w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-23-150x150.png 150w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-23-300x300.png 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-23-768x768.png 768w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-23-65x65.png 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-23-225x225.png 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-23-350x350.png 350w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/p>\n<p id=\"caption-attachment-1194\" class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<p>In Figure 7, we can see that the vertex remained in the same place, at [latex](0,0)[\/latex], but the shape of the parabola changed. The graph of [latex]\\color{MidnightBlue}{g(x)=2{x}^{2}}[\/latex] is the same as the [latex]\\color{BrickRed}{f}[\/latex] function but stretched vertically by a factor of [latex]2[\/latex] (the blue graph appears narrower than [latex]\\color{BrickRed}{f(x)=x^2}[\/latex]). As [latex]a[\/latex] gets larger, the parabola gets narrower, or vertically stretched. What happens to the graph when [latex]a[\/latex] is between zero and one ([latex]0<a<1[\/latex])? The graph of [latex]\\color{forestGreen}{h(x)=\\dfrac{1}{2}{x}^{2}}[\/latex] is the same as the [latex]\\color{BrickRed}{f}[\/latex] function but compressed vertically by a factor of [latex]\\dfrac{1}{2}[\/latex] (the green graph appears wider than [latex]\\color{BrickRed}{f(x)=x^2}[\/latex]).\n\nWhat happens when [latex]a[\/latex] is a negative number? Let&#8217;s look at the functions [latex]\\color{MidnightBlue}{g(x)=-5x^2}[\/latex] and [latex]\\color{ForestGreen}{h(x)= -\\dfrac{1}{8}x^2}[\/latex].\u00a0We will compare these two functions to the function [latex]\\color{BrickRed}{f(x)=-x^2}[\/latex]. What do you notice about the graphs in Figure 8?\n\n\n\n<div style=\"width: 610px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1197\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-24-e1720578899103.png\" alt=\"\" width=\"600\" height=\"600\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<p>The magnitude of [latex]a[\/latex]\u00a0indicates the stretch of the graph. If [latex]|a|>1[\/latex], there is a vertical stretch and the graph appears to become narrower. If [latex]|a|<1[\/latex], there is a vertical compression and the graph appears to become wider.\n\nIn Figure 8, we notice that when [latex]a<0[\/latex], is reflected across the [latex]x[\/latex]-axis, the parabola opens downward The vertex at [latex](0,0)[\/latex] is the highest point on the graph and [latex]k=0[\/latex] is the maximum value of this quadratic function. The graph of [latex]\\color{MidnightBlue}{g(x)=-5x^2}[\/latex] is the same as the graph [latex]\\color{BrickRed}{f(x)=-x^2}[\/latex] with a vertical stretch by a factor of [latex]|-5|[\/latex] (the blue graph appears narrower than [latex]\\color{BrickRed}{f(x)=-x^2}[\/latex]). The graph of [latex]\\color{ForestGreen}{h(x)=-\\dfrac{1}{8}x^2}[\/latex] is the same as the graph\u00a0[latex]\\color{BrickRed}{f(x)=-x^2}[\/latex] but compressed vertically by a factor of [latex]|-\\dfrac{1}{8}|[\/latex] (the green graph appears wider than [latex]\\color{BrickRed}{f(x)=-x^2}[\/latex]).\n\n\n<h2>Graphing Quadratic Functions of the form [latex]f(x)=a(x-h)^2+k[\/latex]<\/h2>\n<p>What is [latex]a[\/latex], [latex]h[\/latex], and [latex]k[\/latex] in figure 9? The quadratic function of the graph below written in vertex form is [latex]y=-3{\\left(x+2\\right)}^{2}+4[\/latex].\u00a0In this form, [latex]a=-3,\\text{ }h=-2[\/latex], and [latex]k=4[\/latex]. Since [latex]a<0[\/latex], the parabola opens downward. The vertex is at [latex]\\left(-2,\\text{ 4}\\right)[\/latex].\n\n\n\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0052.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=-3(x+2)^2+4.\" width=\"487\" height=\"630\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9 [latex]-3(x+2)^2+4[\/latex]<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137770279\">If [latex]k>0[\/latex], the graph shifts upward, whereas if [latex]k<0[\/latex], the graph shifts downward. In Figure 9, [latex]k>0[\/latex], so the graph is shifted [latex]4[\/latex] units upward.<\/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. In Figure 9, [latex]h<0[\/latex], so the graph is shifted [latex]2[\/latex] units to the left.\n\nIn Figure 9, [latex]|a|>1[\/latex]. The magnitude of [latex]a[\/latex] indicates the stretch of the graph. When [latex]|a|>1[\/latex] there is a vertical stretch causing the graph to become narrower.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>ExAMPLE<\/h3>\n<p>Compare the graph of [latex]f(x)=\\dfrac{1}{2}{(x-4)}^{2}-6[\/latex] to the parent function [latex]f(x)=x^2[\/latex] by stating the transformations. Find the vertex, axis of symmetry, two other points on graph, and then graph the function. Determine if the vertex is a maximum or minimum.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q315217\">Show Solution<\/span><\/p>\n<div id=\"q315217\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p>Quadratic Equations in form of [latex]f(x)=a(x-h)^2+k[\/latex] are said to be in vertex form where the vertex is [latex](h,k)[\/latex].<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Transformations:<\/strong><\/span><\/p>\n<p>[latex]a=\\dfrac{1}{2}[\/latex], [latex]h=4[\/latex], and [latex]k=-6[\/latex]<\/p>\n<ul>\n<li>[latex]a=\\dfrac{1}{2}[\/latex] means the graph has a vertical compression by a factor of [latex]\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]h=4[\/latex] means the graph is shifted right [latex]4[\/latex] units<\/li>\n<li>[latex]k=-6[\/latex] means the graph is shifted down [latex]6[\/latex] units<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>The vertex is [latex](h,k)[\/latex]; in this case it would be [latex](4, -6)[\/latex]. When [latex]a>0[\/latex] the parabola opens upward. The vertex is the minimum. The minimum value is [latex]k=-6[\/latex].<\/p>\n<p>The axis of symmetry is a vertical line which passes through the [latex]x[\/latex]-coordinate of the vertex. The line of symmetry is [latex]x=4[\/latex].<\/p>\n<p>Let&#8217;s find the [latex]y[\/latex]-intercept. Let [latex]x=0[\/latex] and solve for [latex]y[\/latex].<\/p>\n<p>$$\\begin{align}<br \/>\n\\require{color}f(\\color{Green}{0}\\color{black}{)} &amp;= \\dfrac{1}{2}(\\color{Green}{0}\\color{black}{-4)^{2}-6} \\\\&amp;= \\dfrac{1}{2}(-4)^{2}-6 \\\\ &amp;= \\dfrac{1}{2}(16)-6 \\\\ &amp;= 8-6 \\\\ &amp;= 2<\/p>\n<p>\\end{align}$$<\/p>\n<p>The [latex]y[\/latex]-intercept is the point [latex](0,2)[\/latex].<\/p>\n<p>Using [latex]x=4[\/latex] as the axis of symmetry, the point symmetric to [latex](0,2)[\/latex] is [latex](8,2)[\/latex].<\/p>\n<p>The graph is given below.<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/quadratic-functions-2\/desmos-graph-25\/\" rel=\"attachment wp-att-1220\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1220 size-full\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/desmos-graph-25-e1720981193775.png\" alt=\"A upward facing parabola graphed with vertex (4, negative 6) labeled. A vertical dashed line through the vertex is labeled x = 4 and axis of symmetry. Additional points labeled at (0,2) and (8,2) are shown.\" width=\"600\" height=\"600\" \/><\/a><\/p>\n<p>Remember, instead of finding the [latex]y[\/latex]-intercept, you could pick an [latex]x[\/latex]-value on either side of the vertex. The [latex]x[\/latex]-value of the vertex is [latex]4[\/latex], so let&#8217;s pick [latex]x[\/latex] values such as [latex]2,3,5,[\/latex] and [latex]6[\/latex]. Let&#8217;s create a table of values using these [latex]x[\/latex] values. Let&#8217;s start by letting [latex]x=3[\/latex] and [latex]x=5[\/latex]. You can see below that we end up with fractional values for the [latex]y[\/latex]-coordinate. These are perfectly acceptable values but sometimes hard to graph precisely on a coordinate plane.<\/p>\n<table style=\"border-collapse: collapse; width: 25%; border: 1px solid black; height: 72px; font-size: 110%;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">\u00a0[latex]x[\/latex]<\/th>\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]f(x)[\/latex] or [latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #ff6600;\"><strong>[latex]3[\/latex]<\/strong><\/span><\/td>\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px;\"><span style=\"color: #ff6600;\"><strong>[latex]\\begin{align} &= \\dfrac{1}{2}(3-4)^2-6\\\\ &= \\dfrac{1}{2}(-1)^2-6\\\\ &= \\dfrac{1}{2}-6\\\\ &=-\\dfrac{11}{2}\\end{align}[\/latex]<\/strong><\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><strong><span style=\"color: #339966;\">[latex]4[\/latex]<\/span><\/strong><\/td>\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><strong><span style=\"color: #339966;\">[latex]-6[\/latex]<\/span><\/strong><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><strong><span style=\"color: #ff6600;\">[latex]5[\/latex]<\/span><\/strong><\/td>\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><strong><span style=\"color: #ff6600;\">[latex]\\begin{align} &= \\dfrac{1}{2}(5-4)^2-6\\\\ &= \\dfrac{1}{2}(1)^2-6\\\\ &= \\dfrac{1}{2}-6\\\\ &=-\\dfrac{11}{2}\\end{align}[\/latex]<\/span><\/strong><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]6[\/latex]<\/td>\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now let&#8217;s fill in the chart when [latex]x=2[\/latex] and [latex]x=6[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 25%; border: 1px solid black; height: 72px; font-size: 110%;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">\u00a0[latex]x[\/latex]<\/th>\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]f(x)[\/latex] or [latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #33cccc;\"><strong>[latex]2[\/latex]<\/strong><\/span><\/td>\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #33cccc;\"><strong>[latex]\\begin{align} &= \\dfrac{1}{2}(2-4)^2-6\\\\ &= \\dfrac{1}{2}(-2)^2-6\\\\ &= \\dfrac{1}{2}(4)-6\\\\ &= 2-6\\\\ &= -4\\end{align}[\/latex]<\/strong><\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #000000;\">[latex]3[\/latex]<\/span><\/td>\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px;\"><span style=\"color: #000000;\">[latex]\\begin{align} &= \\dfrac{1}{2}(3-4)^2-6\\\\ &= \\dfrac{1}{2}(-1)^2-6\\\\ &= \\dfrac{1}{2}-6\\\\ &=-\\dfrac{11}{2}\\end{align}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #000000;\">[latex]4[\/latex]<\/span><\/td>\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #000000;\">[latex]-6[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #000000;\">[latex]5[\/latex]<\/span><\/td>\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #000000;\">[latex]\\begin{align} &= \\dfrac{1}{2}(5-4)^2-6\\\\ &= \\dfrac{1}{2}(1)^2-6\\\\ &= \\dfrac{1}{2}-6\\\\ &=-\\dfrac{11}{2}\\end{align}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #33cccc;\"><strong>[latex]6[\/latex]<\/strong><\/span><\/td>\n<td style=\"width: 70%; border: 1px solid #999999; height: 12px; text-align: center;\"><span style=\"color: #33cccc;\"><strong>[latex]\\begin{align} &= \\dfrac{1}{2}(6-4)^2-6\\\\ &= \\dfrac{1}{2}(2)^2-6\\\\ &= \\dfrac{1}{2}(4)-6\\\\ &= 2-6\\\\ &= -4\\end{align}[\/latex]<\/strong><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So, you could also plot the points <span style=\"color: #33cccc;\">[latex](2,-4)[\/latex]<\/span> and <span style=\"color: #33cccc;\">[latex](6,-4)[\/latex].<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Quadratic Function Transformations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hvyH-WJtMpc?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=\"ohm288159\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288159&theme=oea&iframe_resize_id=ohm288159&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"Example_03_02_01\" class=\"example\">\n<div id=\"fs-id1165131959514\" class=\"exercise\">\n<section id=\"fs-id1165134205927\" class=\"key-equations\"><\/section>\n<section id=\"fs-id1165135426424\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\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. The [latex]x[\/latex]-intercepts, are the points at which the parabola crosses the [latex]x[\/latex]-axis. The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y-[\/latex]axis.<\/li>\n<li>Vertex form is useful to easily identify the vertex of a parabola.<\/li>\n<li>A quadratic function\u2019s minimum or maximum value is given by the [latex]y[\/latex]-value of the vertex.<\/li>\n<\/ul>\n<\/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=-\\frac{b}{2a}[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137931314\" class=\"definition\">\n<dt>quadratic function<\/dt>\n<dd>a polynomial function of degree 2. The graph of a quadratic function is a curve called a parabola.<\/dd>\n<dt><strong>vertex 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><strong>vertex<\/strong><\/dt>\n<dd id=\"fs-id1165135623619\">the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/dd>\n<\/dl>\n<\/section>\n<\/div>\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-42\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\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":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-42","chapter","type-chapter","status-publish","hentry"],"part":143,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/42","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":61,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/42\/revisions"}],"predecessor-version":[{"id":2139,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/42\/revisions\/2139"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/parts\/143"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/42\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/media?parent=42"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=42"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/contributor?post=42"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/license?post=42"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}