{"id":2950,"date":"2024-02-09T20:33:08","date_gmt":"2024-02-09T20:33:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=2950"},"modified":"2026-03-24T19:47:53","modified_gmt":"2026-03-24T19:47:53","slug":"10-2-graphs-of-quadratic-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/10-2-graphs-of-quadratic-equations\/","title":{"raw":"10.2: Graphs of Quadratic Equations","rendered":"10.2: Graphs of Quadratic Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Graph quadratic equations of the form [latex]y=ax^2+bx+c[\/latex]<\/li>\r\n \t<li>Identify important features of\u00a0the graphs of quadratic equations<\/li>\r\n \t<li>Determine the maximum or minimum value of a quadratic equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>Keywords<\/h1>\r\n<ul>\r\n \t<li><strong>Parent equation<\/strong>: the simplest form of a general equation<\/li>\r\n \t<li><strong>Parabola<\/strong>: the shape of any quadratic equation<\/li>\r\n \t<li><strong>Vertex<\/strong>: the turning point of a parabola<\/li>\r\n \t<li><strong>Line of symmetry<\/strong>: a line that cuts the graph into two mirror images<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphing Quadratic Equations Using Tables<\/h2>\r\nThe simplest form of a quadratic equation is [latex]y=ax^2[\/latex]. This is also referred to as the\u00a0<em><strong>parent equation<\/strong><\/em> of any quadratic equation. The basic shape of a quadratic equation is a <em><strong>parabola<\/strong><\/em>. It has a <em><strong>vertex<\/strong><\/em> where the parabola turns and a <em><strong>line of symmetry<\/strong><\/em> that runs through the parabola and splits the graph into two mirror images. We discovered all of this in the last section by determining solutions of the equation and plotting the solution points. We can use this technique for any quadratic equation.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nComplete a table of values for the equation [latex]y=2x^2[\/latex], then graph the equation.\r\n<h4>Solution<\/h4>\r\nTo create a table of values, we can choose any [latex]x[\/latex]-values and find the corresponding [latex]y[\/latex]-values.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 50%; text-align: center;\">[latex]y=2x^2[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=2(0)^2=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=2(1)^2=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=2(-1)^2=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=2(2)^2=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=2(-2)^2=8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo create the graph, we plot the points and join the dots.\r\n\r\n<img class=\"aligncenter wp-image-2095 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y2x^2-with-points-190x300.png\" alt=\"y=2x^2 with points\" width=\"190\" height=\"300\" \/>\r\n\r\n<\/div>\r\nNotice that it is still a parabola. All quadratic equations take the shape of a parabola.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nComplete a table of values for the equation [latex]y=-x^2+4x-3[\/latex], then graph the equation. State the vertex and axis of symmetry.\r\n<h4>Solution<\/h4>\r\nTo create a table of values, we can choose any [latex]x[\/latex]-values and find the corresponding [latex]y[\/latex]-values.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 50%; text-align: center;\">[latex]y=-x^2+4x-3[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=(0)^2+4(0)-3=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(1)^2+4(-1)-3=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(-1)^2+4(-1)-3=--8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(2)^2+4(2)-3=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(-2)^2+4(-2)-3=-15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo create the graph, we plot the points.\r\n\r\n<img class=\"aligncenter wp-image-2140 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/First-points-182x300.png\" alt=\"Plot of first points\" width=\"182\" height=\"300\" \/>\r\n\r\nUnfortunately, the points we chose do not show any symmetry or a turning point. However, the graph looks like it will turn to the right of [latex]x=2[\/latex], so let's find a few other points that lie to the right of [latex]x=2[\/latex].\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 50%; text-align: center;\">[latex]y=-x^2+4x-3[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(3)^2+4(3)-3=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(4)^2+4(4)-3=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">\u00a0[latex]y=-(5)^2+4(5)-3=--8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow we have some symmetry and can join the dots to create the graph.\r\n\r\n<img class=\"aligncenter wp-image-2141 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Last-graph-242x300.png\" alt=\"y=-x^2+4x-3\" width=\"242\" height=\"300\" \/>\r\n\r\n&nbsp;\r\n\r\nThe vertex is at [latex](2, 1)[\/latex] and the axis of symmetry is the vertical line [latex]x=2[\/latex].\r\n\r\nNotice also, that the point [latex](-2, -15)[\/latex] has a twin as a mirror image at [latex](6, -15)[\/latex].\r\n\r\n<\/div>\r\nIn the first example, the parabola opens upwards, while in the second example, the parabola opens downwards. This is determined by the value of [latex]a[\/latex] in the equation [latex]y=ax^2+bx+c[\/latex]. When [latex]a&gt;0[\/latex], the parabola opens upwards. When\u00a0[latex]a&lt;0[\/latex], the parabola opens downwards.\u00a0 Notice that [latex]a\\neq0[\/latex] because that would turn the quadratic equation [latex]y=ax^2+bx+c[\/latex] into a linear equation [latex]y=bx+c[\/latex].\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nComplete a table of values for the equation [latex]y=-x^2+7[\/latex], then graph the equation. State the vertex and axis of symmetry.\r\n\r\n[reveal-answer q=\"hjm886\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm886\"]\r\n\r\nFirst notice that [latex]a=-1[\/latex] so the parabola will open downwards.\r\n\r\nTo complete a table of values we can choose any [latex]x[\/latex]-values:\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 50%; text-align: center;\">[latex]y=-x^2+7[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">-3<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(-3)^2+7=-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">-2<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(-2)^2+7=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">-1<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(-1)^2+7=6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">0<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(0)^2+7=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">1<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(1)^2+7=6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">2<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(2)^2+7=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">3<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(3)^2+7=-2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter wp-image-2099 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y-x^27-with-points-261x300.png\" alt=\"y=-x^2+7 with points\" width=\"261\" height=\"300\" \/>\r\n\r\nFrom the graph the vertex is at [latex](0, 7)[\/latex] and the line of symmetry is the vertical line [latex]x=0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSometimes it can be messy to draw a graph using just solutions. Deciding which [latex]x[\/latex]-values to choose can be bothersome. It can also be impossible to determine exactly where the axis of symmetry and vertex lie if they do not contain integer values. That's why we use other features of the graph to help us.\u00a0 For example, it would be helpful to know exactly where the vertex and axis of symmetry lie. Or, where the graph crosses the axes. So, let's discover how to determine such features.\r\n<h2>Features of Parabolas<\/h2>\r\n<h3>Intercepts:\u00a0The [latex]y[\/latex]-Intercept<\/h3>\r\nThe [latex]y[\/latex]-intercept of any graph is found by setting [latex]x=0[\/latex] in the equation of the graph and solving for [latex]y[\/latex]. For a parabola with equation [latex]y=ax^2+bx+c[\/latex], setting [latex]x=0[\/latex] results in [latex]y=c[\/latex]. Consequently, the [latex]y[\/latex]-intercept of any parabola is always the point [latex](0, c)[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nDetermine the\u00a0[latex]y[\/latex]-intercept of the graph with equation:\r\n<ol>\r\n \t<li>[latex]y=4x^2-3x+6[\/latex]<\/li>\r\n \t<li>[latex]y=-2x^2-7[\/latex]<\/li>\r\n \t<li>[latex]y=x^2+5x[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>Since [latex]c=6[\/latex], the [latex]y[\/latex]-intercept is the point [latex](0, 6)[\/latex].<\/li>\r\n \t<li>Since [latex]c=-7[\/latex], the [latex]y[\/latex]-intercept is the point [latex](0, -7)[\/latex].<\/li>\r\n \t<li>Since [latex]c=0[\/latex], the [latex]y[\/latex]-intercept is the point [latex](0, 0)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDetermine the\u00a0[latex]y[\/latex]-intercept of the graph with equation:\r\n<ol>\r\n \t<li>[latex]y=x^2-4x+1[\/latex]<\/li>\r\n \t<li>[latex]y=-2x^2-3[\/latex]<\/li>\r\n \t<li>[latex]y=x^2-6x[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm169\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm169\"]\r\n<ol>\r\n \t<li>[latex](0, 1)[\/latex]<\/li>\r\n \t<li>[latex](0, -3)[\/latex]<\/li>\r\n \t<li>[latex](0, 0)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Intercepts:\u00a0The [latex]x[\/latex]-Intercepts<\/h3>\r\nThe [latex]x[\/latex]-intercepts of a parabola occur where the parabola crosses the [latex]x[\/latex]-axis (figure 1). Specifically, this is where [latex]y=0[\/latex]. So to find the\u00a0[latex]x[\/latex]-intercepts of a parabola, we must solve the equation [latex]ax^2+bx+c=0[\/latex].\r\n\r\n<img class=\"aligncenter wp-image-2127 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/yx^2-x-2-300x300.png\" alt=\"y=x^2-x-2\" width=\"300\" height=\"300\" \/>\r\n<p style=\"text-align: center;\">Figure 1. [latex]x[\/latex]-intercepts<\/p>\r\nProvided this quadratic equation factors, we can solve it using the zero-factor property.\u00a0 If the equation does not factor, there are alternative ways of solving it that will be taught in the next course.\r\n<div class=\"examples\">\r\n<h3>Example<\/h3>\r\nFind the [latex]x[\/latex]-intercepts for the graph of [latex]y={x}^{2}-x-6[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"padding-left: 30px;\">The [latex]x[\/latex]-intercepts are found by solving the equation [latex]x^2 - x - 6 = 0[\/latex].<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%;\">Solve.<\/th>\r\n<th style=\"width: 50%;\">[latex]x^2 - x - 6 = 0[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Factor.<\/td>\r\n<td style=\"width: 50%;\">[latex]\\left( x - 3 \\right) \\left( x+2 \\right) = 0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Use the zero-factor property.<\/td>\r\n<td style=\"width: 50%;\">[latex]x - 3 = 0[\/latex] and\u00a0 [latex]x +2 = 0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Solve the resulting equations.<\/td>\r\n<td style=\"width: 50%;\">[latex]x=3[\/latex] and\u00a0 [latex]x = -2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">The solutions are the [latex]x[\/latex]-coordinate of the point. The [latex]y[\/latex]-coordinate is 0.<\/td>\r\n<td style=\"width: 50%;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Answer<\/h4>\r\nThe [latex]x[\/latex]-intercepts are [latex]\\left(3, 0 \\right)[\/latex] and [latex] \\left( -2, 0 \\right) [\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the [latex]x[\/latex]-intercepts for the graph of [latex]y={x}^{2}-3x+2[\/latex].\r\n\r\n[reveal-answer q=\"hjm435\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm435\"]\r\n\r\nThe graph has two [latex]x[\/latex]-intercepts at [latex](1, 0)[\/latex] and [latex](2, 0)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the [latex]x[\/latex]-intercepts for the graph of [latex]y={x}^{2}-6x+9[\/latex].\r\n\r\n[reveal-answer q=\"hjm827\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm827\"]\r\n\r\nThe graph has one [latex]x[\/latex]-intercept at [latex](3, 0)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nParabolas whose equations factor have [latex]x[\/latex]-intercepts. But not <em>all<\/em> parabolas have [latex]x[\/latex]-intercepts. Consider the parabolas in figure 2:\r\n\r\n<img class=\"aligncenter wp-image-2128\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27225602\/Parabolas-and-x-intercepts-300x188.png\" alt=\"Parabolas and x-intercepts\" width=\"427\" height=\"267\" \/>\r\n<p style=\"text-align: center;\">Figure 2. Parabolas with 2, 1, or 0 [latex]x[\/latex]-intercepts.<\/p>\r\nParabolas can have two [latex]x[\/latex]-intercepts, one [latex]x[\/latex]-intercept, or no\u00a0[latex]x[\/latex]-intercepts.\r\n<div class=\"examples\">\r\n<h3>Example<\/h3>\r\nFind the [latex]x[\/latex]-intercepts of the graph of [latex]y = x^2 + 4[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"padding-left: 30px;\">To find the [latex]x[\/latex]-intercepts, we need to solve the equation\u00a0[latex]x^2 + 4 = 0[\/latex].<\/p>\r\n<p style=\"padding-left: 30px;\">However,\u00a0[latex]x^2 + 4[\/latex] does not factor.<\/p>\r\n<p style=\"padding-left: 30px;\">We can solve this equation by rewriting it:<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x^2 + 4=0[\/latex] can be rewritten as [latex]x^2 = -4 [\/latex].<\/p>\r\n<p style=\"padding-left: 30px;\">There are no real number values for [latex]x[\/latex] that when squared result in [latex]-4[\/latex].<\/p>\r\n<p style=\"padding-left: 30px;\">The only value that [latex]x[\/latex] can be are the imaginary numbers [latex]2i[\/latex] and [latex]-2i[\/latex] because [latex](2i)^2=4i^2=4(-1)=-4[\/latex] and\u00a0[latex](-2i)^2=4i^2=4(-1)=-4[\/latex].<\/p>\r\n<p style=\"padding-left: 30px;\">Since [latex]2i[\/latex] and [latex]-2i[\/latex] are complex numbers, they will not show up on the graph.<\/p>\r\n<p style=\"padding-left: 30px;\">This parabola will have no [latex]x[\/latex]-intercepts.<\/p>\r\n\r\n<\/div>\r\nIf a parabola does not intersect with the [latex]x[\/latex]-axis,and therefore has no\u00a0[latex]x[\/latex]-intercepts, there are complex number solutions to the equation [latex]y=ax^2+bx+c=0[\/latex]. Such solutions cannot be graphed on the real number line, so will not appear on the graph.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the [latex]x[\/latex]-intercepts of the graph of [latex]y = 9x^2 + 1[\/latex].\r\n\r\n[reveal-answer q=\"hjm176\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm176\"]\r\n\r\nThe graph has no [latex]x[\/latex]-intercepts.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Axis of Symmetry and the Vertex<\/h3>\r\nThe axis of symmetry and the vertex are very important features of a parabola so being able to find them will be extremely useful for graphing.\r\n<div class=\"textbox shaded\">\r\n<h3>Axis of SYmmetry and vertex<\/h3>\r\nFor the graph of the equation [latex]y=ax^2+bx+c[\/latex], the axis of symmetry is the vertical line [latex]x=-\\frac{b}{2a}[\/latex].\r\n\r\n&nbsp;\r\n\r\nThe vertex is the point where [latex]x=-\\frac{b}{2a}[\/latex], paired with the corresponding [latex]y[\/latex]-value.\r\n\r\n<\/div>\r\nFor example, consider the equation [latex]y=2x^2-3x+4[\/latex]. To find the axis of symmetry calculate [latex]x=-\\frac{b}{2a}=-\\frac{-3}{2\\cdot 2}=\\frac{3}{4}[\/latex]. The axis of symmetry is the vertical line with equation [latex]x=\\frac{3}{4}[\/latex].\r\n\r\nThe axis of symmetry passes through the vertex, so the [latex]x[\/latex]-coordinate of the vertex is also [latex]\\frac{3}{4}[\/latex]. To find the [latex]y[\/latex]-coordinate, we substitute [latex]x=\\frac{3}{4}[\/latex] into the original equation\u00a0[latex]y=2x^2-3x+4[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned}y&amp;=2x^2-3x+4 \\\\ y&amp;=2 {\\left (\\frac{3}{4}\\right )}^{2}-3\\cdot\\frac{3}{4}+4 \\\\ y&amp;=2\\cdot\\frac{9}{16}-\\frac{9}{4}+4\\\\y&amp;=\\frac{9}{8}-\\frac{18}{8}+\\frac{32}{8}\\\\y&amp;=\\frac{23}{8}\\end{aligned}\\end{equation}[\/latex]<\/p>\r\nThe vertex is at the point [latex]\\left(\\dfrac{3}{4},\\dfrac{23}{8}\\right)[\/latex]. This is verified by the graph in figure 3:\r\n\r\n<img class=\"aligncenter wp-image-2123\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27215439\/y2x%5E2-3x41-570x1024.png\" alt=\"y equals two x squared -3x+4 \" width=\"409\" height=\"735\" \/>\r\n<p style=\"text-align: center;\">Figure 3. Graph of\u00a0[latex]y=2x^2-3x+4[\/latex]<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFind the axis of symmetry and the vertex of the graph with equation [latex]y=-3x^2+x-2[\/latex].\r\n<h4>Solution<\/h4>\r\nAxis:\u00a0 [latex]x=-\\frac{b}{2a}=-\\frac{1}{2\\cdot (-3)}=-\\frac{1}{-6}=\\frac{1}{6}[\/latex]\r\n\r\nVertex: [latex]x=\\frac{1}{6}[\/latex], so\r\n\r\n[latex]\\begin{equation}\\begin{aligned}y&amp;=-3x^2+x-2 \\\\ y&amp;=-3{\\left (\\frac{1}{6}\\right )}^{2}+\\frac{1}{6}-2 \\\\ y&amp;=-\\frac{3}{1}\\frac{1}{36}+\\frac{1}{6}-2 \\\\ y&amp;=-\\frac{1}{12}+\\frac{1}{6}-2 \\\\ y&amp;=\\frac{-1+2-24}{12} \\\\ &amp;y=-\\frac{23}{12}\\end{aligned}\\end{equation}[\/latex]\r\n<h4>Answer<\/h4>\r\nThe axis of symmetry is the vertical line\u00a0[latex]x=\\frac{1}{6}[\/latex].\r\nThe vertex is the point [latex]\\left (\\frac{1}{6},-\\frac{23}{12}\\right )[\/latex].<img class=\"aligncenter wp-image-2124\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27221154\/y-3x%5E2x-2-198x300.png\" alt=\"y=-3x^2+x-2\" width=\"203\" height=\"308\" \/>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the axis of symmetry and the vertex of the graph of the equation [latex]y=2x^2+2x-4[\/latex].\r\n\r\n[reveal-answer q=\"hjm519\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm519\"]\r\n\r\nAxis: [latex]x=\\frac{1}{2}[\/latex]\r\n\r\nVertex: [latex]\\left (\\frac{1}{2}, -\\frac{9}{2}\\right )[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Putting it all Together<\/h2>\r\nWe have learned some important things about the graphs of quadratic equations that will make it easier for us to create a graph.\u00a0 These features of a parabola are summarized below:\r\n<div class=\"textbox shaded\">\r\n<h3>Graphs of Quadratic Functions<\/h3>\r\nFor [latex] \\displaystyle y=a{{x}^{2}}+bx+c[\/latex], where [latex]a, b[\/latex] and [latex]c[\/latex]\u00a0are real numbers, and [latex]a\\neq0[\/latex],\r\n<ul>\r\n \t<li>The parabola opens upward if [latex]a &gt; 0[\/latex] and downward if [latex]a &lt; 0[\/latex].<\/li>\r\n \t<li>The [latex]y[\/latex]-intercept of the parabola occurs at the point [latex](0,c)[\/latex].<\/li>\r\n \t<li>The\u00a0[latex]x[\/latex]-intercepts are found by solving the equation [latex]ax^2+bx+c=0[\/latex]. The [latex]y[\/latex]-coordinate is zero. There may be 0, 1, or 2\u00a0[latex]x[\/latex]-intercepts.<\/li>\r\n \t<li>The axis of symmetry is the vertical line [latex]x=\\frac{-b}{2a}[\/latex].<\/li>\r\n \t<li>The vertex has an [latex]x[\/latex]-coordinate of [latex]x=\\dfrac{-b}{2a}[\/latex]. The [latex]y[\/latex]-coordinate is found by substituting this [latex]x[\/latex]-value into the equation and solving for\u00a0[latex]y[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe can use the properties of parabolas to help us graph a quadratic equation of the form [latex]y=ax^2+bx+c[\/latex] without having to calculate an exhaustive table of values.\r\n<div class=\"bcc-box examples \">\r\n<h3>Example<\/h3>\r\nGraph [latex]y=\u22122x^{2}+3x\u20133[\/latex].\r\n<h4>Solution<\/h4>\r\nLet's start by considering the features of a parabola.\r\n\r\nOpening:\r\n\r\n[latex]a=-2[\/latex] so the parabola opens downwards.\r\n\r\nSince [latex]|a|&gt;1[\/latex] the graph will be narrower than the graph of [latex]y=x^2[\/latex].\r\n\r\n[latex]y[\/latex]-intercept<span style=\"font-size: 1rem; text-align: initial;\">:<\/span>\r\n\r\nSince [latex]c=-3[\/latex], the [latex]y[\/latex]-intercept will be [latex](0, -3)[\/latex].\r\n\r\nAxis of symmetry:\r\n\r\n[latex]x=\\frac{-b}{2a}=\\frac{-3}{2\\cdot (-2)}=\\frac{3}{4}[\/latex].\r\n\r\nVertex:\r\n\r\n[latex]x=\\frac{3}{4}[\/latex], so [latex]y=\u22122x^{2}+3x\u20133=-2{\\left (\\frac{3}{4}\\right )}^{2}+3\\cdot \\frac{3}{4}-3=-2\\cdot \\frac{9}{16}+\\frac{9}{4}-3=\\frac{-9}{8}+\\frac{9}{4}-3=\\frac{-9+18-24}{8}=\\frac{-15}{8}[\/latex].\u00a0 The vertex is the point [latex]\\left (\\frac{3}{4}, -\\frac{15}{8}\\right )[\/latex].\r\n\r\n[latex]x[\/latex]-intercepts<span style=\"font-size: 1rem; text-align: initial;\">:\u00a0<\/span>\r\n\r\n[latex]\u22122x^{2}+3x\u20133[\/latex] does not factor, so we cannot find the\u00a0[latex]x[\/latex]-intercepts<span style=\"font-size: 1rem; text-align: initial;\">.<\/span>\r\n\r\nLet's graph everything we have:\r\n\r\n<img class=\"alignleft wp-image-2130 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/1st-draft-300x231.png\" alt=\"1st draft of graph\" width=\"300\" height=\"231\" \/>Notice that the point [latex](0, -3)[\/latex] has a mirror image on the other side of the axis of symmetry.\r\n\r\nThis twin has the same [latex]y[\/latex]-value of [latex]-3[\/latex]. To find the [latex]x[\/latex]-value, move the same number of units past the axis of symmetry to the right.\r\n\r\nThe point\u00a0[latex](0, -3)[\/latex] is [latex]\\frac{3}{4}[\/latex] of a unit to the left of the axis, so its twin is\u00a0[latex]\\frac{3}{4}[\/latex] of a unit to the right of the axis at [latex]\\left (\\frac{3}{2}, -3\\right )[\/latex].\r\n\r\nFrom the information we have found so far, we have the turning point and direction of the graph. Now let's find some solutions and use their twins to plot more points.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<table style=\"height: 92px; width: 355px;\" width=\"100\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"text-align: center; width: 294.719px; height: 12px;\"><strong>[latex]x[\/latex]<\/strong><\/th>\r\n<th style=\"text-align: center; width: 248.297px; height: 12px;\">[latex]y=\u22122x^{2}+3x\u20133[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 20px;\">\r\n<td style=\"text-align: center; width: 294.719px; height: 20px;\">[latex]\u22121[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 248.297px; height: 20px;\">[latex]\u22128[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 20px;\">\r\n<td style=\"text-align: center; width: 294.719px; height: 20px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 248.297px; height: 20px;\">[latex]\u22123[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 20px;\">\r\n<td style=\"text-align: center; width: 294.719px; height: 20px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 248.297px; height: 20px;\">[latex]\u22122[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 20px;\">\r\n<td style=\"text-align: center; width: 294.719px; height: 20px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 248.297px; height: 20px;\">[latex]\u22125[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"alignleft wp-image-2131\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/28000312\/2nd-draft-195x300.png\" alt=\"2nd draft\" width=\"273\" height=\"420\" \/>\r\n\r\n<img class=\"alignright wp-image-2132\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/28000446\/Final-graph-203x300.png\" alt=\"Final graph\" width=\"284\" height=\"420\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nFinally connect the points as best you can using a <i>smooth curve.<\/i>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nDetermine the maximum value of [latex]y[\/latex]: [latex]y=-4x^2-4x+3[\/latex]\r\n<h4>Solution<\/h4>\r\nSince the graph is a parabola that opens downwards, the maximum value will occur at the vertex.\r\n\r\nFind the vertex:\u00a0 [latex]x=\\frac{-b}{2a}=\\frac{4}{2\\cdot (-4)}=\\frac{-1}{2}[\/latex]\r\n\r\nThen find the [latex]y[\/latex]-value:\r\n\r\n[latex]y=-4x^2-4x+3\\\\y=-4{\\left (\\frac{-1}{2}\\right)}^{2}-4\\left (\\frac{-1}{2}\\right )+3\\\\y=\\frac{-4}{1}\\cdot\\frac{1}{4}+2+3\\\\y=-1+2+3\\\\y=4[\/latex]\r\n<h4>Answer<\/h4>\r\nThe maximum value is [latex]y=4[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDetermine the minimum value of [latex]y[\/latex]:\u00a0 [latex]y=x^2-8x-4[\/latex]\r\n\r\n[reveal-answer q=\"hjm057\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm057\"]\r\n\r\nThe minimum value is [latex]-20[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Graph quadratic equations of the form [latex]y=ax^2+bx+c[\/latex]<\/li>\n<li>Identify important features of\u00a0the graphs of quadratic equations<\/li>\n<li>Determine the maximum or minimum value of a quadratic equation<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>Keywords<\/h1>\n<ul>\n<li><strong>Parent equation<\/strong>: the simplest form of a general equation<\/li>\n<li><strong>Parabola<\/strong>: the shape of any quadratic equation<\/li>\n<li><strong>Vertex<\/strong>: the turning point of a parabola<\/li>\n<li><strong>Line of symmetry<\/strong>: a line that cuts the graph into two mirror images<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing Quadratic Equations Using Tables<\/h2>\n<p>The simplest form of a quadratic equation is [latex]y=ax^2[\/latex]. This is also referred to as the\u00a0<em><strong>parent equation<\/strong><\/em> of any quadratic equation. The basic shape of a quadratic equation is a <em><strong>parabola<\/strong><\/em>. It has a <em><strong>vertex<\/strong><\/em> where the parabola turns and a <em><strong>line of symmetry<\/strong><\/em> that runs through the parabola and splits the graph into two mirror images. We discovered all of this in the last section by determining solutions of the equation and plotting the solution points. We can use this technique for any quadratic equation.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Complete a table of values for the equation [latex]y=2x^2[\/latex], then graph the equation.<\/p>\n<h4>Solution<\/h4>\n<p>To create a table of values, we can choose any [latex]x[\/latex]-values and find the corresponding [latex]y[\/latex]-values.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 50%; text-align: center;\">[latex]y=2x^2[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=2(0)^2=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=2(1)^2=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=2(-1)^2=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=2(2)^2=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=2(-2)^2=8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To create the graph, we plot the points and join the dots.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2095 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y2x^2-with-points-190x300.png\" alt=\"y=2x^2 with points\" width=\"190\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y2x^2-with-points-190x300.png 190w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y2x^2-with-points-648x1024.png 648w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y2x^2-with-points-65x103.png 65w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y2x^2-with-points-225x356.png 225w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y2x^2-with-points-350x553.png 350w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y2x^2-with-points.png 740w\" sizes=\"auto, (max-width: 190px) 100vw, 190px\" \/><\/p>\n<\/div>\n<p>Notice that it is still a parabola. All quadratic equations take the shape of a parabola.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Complete a table of values for the equation [latex]y=-x^2+4x-3[\/latex], then graph the equation. State the vertex and axis of symmetry.<\/p>\n<h4>Solution<\/h4>\n<p>To create a table of values, we can choose any [latex]x[\/latex]-values and find the corresponding [latex]y[\/latex]-values.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 50%; text-align: center;\">[latex]y=-x^2+4x-3[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=(0)^2+4(0)-3=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(1)^2+4(-1)-3=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(-1)^2+4(-1)-3=--8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(2)^2+4(2)-3=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(-2)^2+4(-2)-3=-15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To create the graph, we plot the points.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2140 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/First-points-182x300.png\" alt=\"Plot of first points\" width=\"182\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/First-points-182x300.png 182w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/First-points-768x1268.png 768w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/First-points-620x1024.png 620w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/First-points-65x107.png 65w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/First-points-225x371.png 225w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/First-points-350x578.png 350w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/First-points.png 934w\" sizes=\"auto, (max-width: 182px) 100vw, 182px\" \/><\/p>\n<p>Unfortunately, the points we chose do not show any symmetry or a turning point. However, the graph looks like it will turn to the right of [latex]x=2[\/latex], so let&#8217;s find a few other points that lie to the right of [latex]x=2[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 50%; text-align: center;\">[latex]y=-x^2+4x-3[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(3)^2+4(3)-3=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(4)^2+4(4)-3=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">\u00a0[latex]y=-(5)^2+4(5)-3=--8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now we have some symmetry and can join the dots to create the graph.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2141 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Last-graph-242x300.png\" alt=\"y=-x^2+4x-3\" width=\"242\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Last-graph-242x300.png 242w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Last-graph-768x952.png 768w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Last-graph-826x1024.png 826w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Last-graph-65x81.png 65w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Last-graph-225x279.png 225w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Last-graph-350x434.png 350w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/Last-graph.png 1226w\" sizes=\"auto, (max-width: 242px) 100vw, 242px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>The vertex is at [latex](2, 1)[\/latex] and the axis of symmetry is the vertical line [latex]x=2[\/latex].<\/p>\n<p>Notice also, that the point [latex](-2, -15)[\/latex] has a twin as a mirror image at [latex](6, -15)[\/latex].<\/p>\n<\/div>\n<p>In the first example, the parabola opens upwards, while in the second example, the parabola opens downwards. This is determined by the value of [latex]a[\/latex] in the equation [latex]y=ax^2+bx+c[\/latex]. When [latex]a>0[\/latex], the parabola opens upwards. When\u00a0[latex]a<0[\/latex], the parabola opens downwards.\u00a0 Notice that [latex]a\\neq0[\/latex] because that would turn the quadratic equation [latex]y=ax^2+bx+c[\/latex] into a linear equation [latex]y=bx+c[\/latex].\n\n\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Complete a table of values for the equation [latex]y=-x^2+7[\/latex], then graph the equation. State the vertex and axis of symmetry.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm886\">Show Answer<\/span><\/p>\n<div id=\"qhjm886\" class=\"hidden-answer\" style=\"display: none\">\n<p>First notice that [latex]a=-1[\/latex] so the parabola will open downwards.<\/p>\n<p>To complete a table of values we can choose any [latex]x[\/latex]-values:<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 50%; text-align: center;\">[latex]y=-x^2+7[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">-3<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(-3)^2+7=-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">-2<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(-2)^2+7=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">-1<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(-1)^2+7=6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">0<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(0)^2+7=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">1<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(1)^2+7=6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">2<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(2)^2+7=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">3<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]y=-(3)^2+7=-2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2099 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y-x^27-with-points-261x300.png\" alt=\"y=-x^2+7 with points\" width=\"261\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y-x^27-with-points-261x300.png 261w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y-x^27-with-points-768x881.png 768w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y-x^27-with-points-893x1024.png 893w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y-x^27-with-points-65x75.png 65w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y-x^27-with-points-225x258.png 225w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y-x^27-with-points-350x402.png 350w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/y-x^27-with-points.png 1032w\" sizes=\"auto, (max-width: 261px) 100vw, 261px\" \/><\/p>\n<p>From the graph the vertex is at [latex](0, 7)[\/latex] and the line of symmetry is the vertical line [latex]x=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Sometimes it can be messy to draw a graph using just solutions. Deciding which [latex]x[\/latex]-values to choose can be bothersome. It can also be impossible to determine exactly where the axis of symmetry and vertex lie if they do not contain integer values. That&#8217;s why we use other features of the graph to help us.\u00a0 For example, it would be helpful to know exactly where the vertex and axis of symmetry lie. Or, where the graph crosses the axes. So, let&#8217;s discover how to determine such features.<\/p>\n<h2>Features of Parabolas<\/h2>\n<h3>Intercepts:\u00a0The [latex]y[\/latex]-Intercept<\/h3>\n<p>The [latex]y[\/latex]-intercept of any graph is found by setting [latex]x=0[\/latex] in the equation of the graph and solving for [latex]y[\/latex]. For a parabola with equation [latex]y=ax^2+bx+c[\/latex], setting [latex]x=0[\/latex] results in [latex]y=c[\/latex]. Consequently, the [latex]y[\/latex]-intercept of any parabola is always the point [latex](0, c)[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Determine the\u00a0[latex]y[\/latex]-intercept of the graph with equation:<\/p>\n<ol>\n<li>[latex]y=4x^2-3x+6[\/latex]<\/li>\n<li>[latex]y=-2x^2-7[\/latex]<\/li>\n<li>[latex]y=x^2+5x[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>Since [latex]c=6[\/latex], the [latex]y[\/latex]-intercept is the point [latex](0, 6)[\/latex].<\/li>\n<li>Since [latex]c=-7[\/latex], the [latex]y[\/latex]-intercept is the point [latex](0, -7)[\/latex].<\/li>\n<li>Since [latex]c=0[\/latex], the [latex]y[\/latex]-intercept is the point [latex](0, 0)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Determine the\u00a0[latex]y[\/latex]-intercept of the graph with equation:<\/p>\n<ol>\n<li>[latex]y=x^2-4x+1[\/latex]<\/li>\n<li>[latex]y=-2x^2-3[\/latex]<\/li>\n<li>[latex]y=x^2-6x[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm169\">Show Answer<\/span><\/p>\n<div id=\"qhjm169\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex](0, 1)[\/latex]<\/li>\n<li>[latex](0, -3)[\/latex]<\/li>\n<li>[latex](0, 0)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Intercepts:\u00a0The [latex]x[\/latex]-Intercepts<\/h3>\n<p>The [latex]x[\/latex]-intercepts of a parabola occur where the parabola crosses the [latex]x[\/latex]-axis (figure 1). Specifically, this is where [latex]y=0[\/latex]. So to find the\u00a0[latex]x[\/latex]-intercepts of a parabola, we must solve the equation [latex]ax^2+bx+c=0[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2127 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/yx^2-x-2-300x300.png\" alt=\"y=x^2-x-2\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/yx^2-x-2-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/yx^2-x-2-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/yx^2-x-2-768x769.png 768w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/yx^2-x-2-1022x1024.png 1022w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/yx^2-x-2-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/yx^2-x-2-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/yx^2-x-2-350x351.png 350w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/yx^2-x-2.png 1096w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center;\">Figure 1. [latex]x[\/latex]-intercepts<\/p>\n<p>Provided this quadratic equation factors, we can solve it using the zero-factor property.\u00a0 If the equation does not factor, there are alternative ways of solving it that will be taught in the next course.<\/p>\n<div class=\"examples\">\n<h3>Example<\/h3>\n<p>Find the [latex]x[\/latex]-intercepts for the graph of [latex]y={x}^{2}-x-6[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"padding-left: 30px;\">The [latex]x[\/latex]-intercepts are found by solving the equation [latex]x^2 - x - 6 = 0[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%;\">Solve.<\/th>\n<th style=\"width: 50%;\">[latex]x^2 - x - 6 = 0[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Factor.<\/td>\n<td style=\"width: 50%;\">[latex]\\left( x - 3 \\right) \\left( x+2 \\right) = 0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Use the zero-factor property.<\/td>\n<td style=\"width: 50%;\">[latex]x - 3 = 0[\/latex] and\u00a0 [latex]x +2 = 0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Solve the resulting equations.<\/td>\n<td style=\"width: 50%;\">[latex]x=3[\/latex] and\u00a0 [latex]x = -2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">The solutions are the [latex]x[\/latex]-coordinate of the point. The [latex]y[\/latex]-coordinate is 0.<\/td>\n<td style=\"width: 50%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Answer<\/h4>\n<p>The [latex]x[\/latex]-intercepts are [latex]\\left(3, 0 \\right)[\/latex] and [latex]\\left( -2, 0 \\right)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the [latex]x[\/latex]-intercepts for the graph of [latex]y={x}^{2}-3x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm435\">Show Answer<\/span><\/p>\n<div id=\"qhjm435\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph has two [latex]x[\/latex]-intercepts at [latex](1, 0)[\/latex] and [latex](2, 0)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the [latex]x[\/latex]-intercepts for the graph of [latex]y={x}^{2}-6x+9[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm827\">Show Answer<\/span><\/p>\n<div id=\"qhjm827\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph has one [latex]x[\/latex]-intercept at [latex](3, 0)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Parabolas whose equations factor have [latex]x[\/latex]-intercepts. But not <em>all<\/em> parabolas have [latex]x[\/latex]-intercepts. Consider the parabolas in figure 2:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2128\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27225602\/Parabolas-and-x-intercepts-300x188.png\" alt=\"Parabolas and x-intercepts\" width=\"427\" height=\"267\" \/><\/p>\n<p style=\"text-align: center;\">Figure 2. Parabolas with 2, 1, or 0 [latex]x[\/latex]-intercepts.<\/p>\n<p>Parabolas can have two [latex]x[\/latex]-intercepts, one [latex]x[\/latex]-intercept, or no\u00a0[latex]x[\/latex]-intercepts.<\/p>\n<div class=\"examples\">\n<h3>Example<\/h3>\n<p>Find the [latex]x[\/latex]-intercepts of the graph of [latex]y = x^2 + 4[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"padding-left: 30px;\">To find the [latex]x[\/latex]-intercepts, we need to solve the equation\u00a0[latex]x^2 + 4 = 0[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">However,\u00a0[latex]x^2 + 4[\/latex] does not factor.<\/p>\n<p style=\"padding-left: 30px;\">We can solve this equation by rewriting it:<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^2 + 4=0[\/latex] can be rewritten as [latex]x^2 = -4[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">There are no real number values for [latex]x[\/latex] that when squared result in [latex]-4[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">The only value that [latex]x[\/latex] can be are the imaginary numbers [latex]2i[\/latex] and [latex]-2i[\/latex] because [latex](2i)^2=4i^2=4(-1)=-4[\/latex] and\u00a0[latex](-2i)^2=4i^2=4(-1)=-4[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">Since [latex]2i[\/latex] and [latex]-2i[\/latex] are complex numbers, they will not show up on the graph.<\/p>\n<p style=\"padding-left: 30px;\">This parabola will have no [latex]x[\/latex]-intercepts.<\/p>\n<\/div>\n<p>If a parabola does not intersect with the [latex]x[\/latex]-axis,and therefore has no\u00a0[latex]x[\/latex]-intercepts, there are complex number solutions to the equation [latex]y=ax^2+bx+c=0[\/latex]. Such solutions cannot be graphed on the real number line, so will not appear on the graph.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the [latex]x[\/latex]-intercepts of the graph of [latex]y = 9x^2 + 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm176\">Show Answer<\/span><\/p>\n<div id=\"qhjm176\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph has no [latex]x[\/latex]-intercepts.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Axis of Symmetry and the Vertex<\/h3>\n<p>The axis of symmetry and the vertex are very important features of a parabola so being able to find them will be extremely useful for graphing.<\/p>\n<div class=\"textbox shaded\">\n<h3>Axis of SYmmetry and vertex<\/h3>\n<p>For the graph of the equation [latex]y=ax^2+bx+c[\/latex], the axis of symmetry is the vertical line [latex]x=-\\frac{b}{2a}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>The vertex is the point where [latex]x=-\\frac{b}{2a}[\/latex], paired with the corresponding [latex]y[\/latex]-value.<\/p>\n<\/div>\n<p>For example, consider the equation [latex]y=2x^2-3x+4[\/latex]. To find the axis of symmetry calculate [latex]x=-\\frac{b}{2a}=-\\frac{-3}{2\\cdot 2}=\\frac{3}{4}[\/latex]. The axis of symmetry is the vertical line with equation [latex]x=\\frac{3}{4}[\/latex].<\/p>\n<p>The axis of symmetry passes through the vertex, so the [latex]x[\/latex]-coordinate of the vertex is also [latex]\\frac{3}{4}[\/latex]. To find the [latex]y[\/latex]-coordinate, we substitute [latex]x=\\frac{3}{4}[\/latex] into the original equation\u00a0[latex]y=2x^2-3x+4[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned}y&=2x^2-3x+4 \\\\ y&=2 {\\left (\\frac{3}{4}\\right )}^{2}-3\\cdot\\frac{3}{4}+4 \\\\ y&=2\\cdot\\frac{9}{16}-\\frac{9}{4}+4\\\\y&=\\frac{9}{8}-\\frac{18}{8}+\\frac{32}{8}\\\\y&=\\frac{23}{8}\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>The vertex is at the point [latex]\\left(\\dfrac{3}{4},\\dfrac{23}{8}\\right)[\/latex]. This is verified by the graph in figure 3:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2123\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27215439\/y2x%5E2-3x41-570x1024.png\" alt=\"y equals two x squared -3x+4\" width=\"409\" height=\"735\" \/><\/p>\n<p style=\"text-align: center;\">Figure 3. Graph of\u00a0[latex]y=2x^2-3x+4[\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Find the axis of symmetry and the vertex of the graph with equation [latex]y=-3x^2+x-2[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Axis:\u00a0 [latex]x=-\\frac{b}{2a}=-\\frac{1}{2\\cdot (-3)}=-\\frac{1}{-6}=\\frac{1}{6}[\/latex]<\/p>\n<p>Vertex: [latex]x=\\frac{1}{6}[\/latex], so<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}y&=-3x^2+x-2 \\\\ y&=-3{\\left (\\frac{1}{6}\\right )}^{2}+\\frac{1}{6}-2 \\\\ y&=-\\frac{3}{1}\\frac{1}{36}+\\frac{1}{6}-2 \\\\ y&=-\\frac{1}{12}+\\frac{1}{6}-2 \\\\ y&=\\frac{-1+2-24}{12} \\\\ &y=-\\frac{23}{12}\\end{aligned}\\end{equation}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The axis of symmetry is the vertical line\u00a0[latex]x=\\frac{1}{6}[\/latex].<br \/>\nThe vertex is the point [latex]\\left (\\frac{1}{6},-\\frac{23}{12}\\right )[\/latex].<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2124\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27221154\/y-3x%5E2x-2-198x300.png\" alt=\"y=-3x^2+x-2\" width=\"203\" height=\"308\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the axis of symmetry and the vertex of the graph of the equation [latex]y=2x^2+2x-4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm519\">Show Answer<\/span><\/p>\n<div id=\"qhjm519\" class=\"hidden-answer\" style=\"display: none\">\n<p>Axis: [latex]x=\\frac{1}{2}[\/latex]<\/p>\n<p>Vertex: [latex]\\left (\\frac{1}{2}, -\\frac{9}{2}\\right )[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Putting it all Together<\/h2>\n<p>We have learned some important things about the graphs of quadratic equations that will make it easier for us to create a graph.\u00a0 These features of a parabola are summarized below:<\/p>\n<div class=\"textbox shaded\">\n<h3>Graphs of Quadratic Functions<\/h3>\n<p>For [latex]\\displaystyle y=a{{x}^{2}}+bx+c[\/latex], where [latex]a, b[\/latex] and [latex]c[\/latex]\u00a0are real numbers, and [latex]a\\neq0[\/latex],<\/p>\n<ul>\n<li>The parabola opens upward if [latex]a > 0[\/latex] and downward if [latex]a < 0[\/latex].<\/li>\n<li>The [latex]y[\/latex]-intercept of the parabola occurs at the point [latex](0,c)[\/latex].<\/li>\n<li>The\u00a0[latex]x[\/latex]-intercepts are found by solving the equation [latex]ax^2+bx+c=0[\/latex]. The [latex]y[\/latex]-coordinate is zero. There may be 0, 1, or 2\u00a0[latex]x[\/latex]-intercepts.<\/li>\n<li>The axis of symmetry is the vertical line [latex]x=\\frac{-b}{2a}[\/latex].<\/li>\n<li>The vertex has an [latex]x[\/latex]-coordinate of [latex]x=\\dfrac{-b}{2a}[\/latex]. The [latex]y[\/latex]-coordinate is found by substituting this [latex]x[\/latex]-value into the equation and solving for\u00a0[latex]y[\/latex].<\/li>\n<\/ul>\n<\/div>\n<p>We can use the properties of parabolas to help us graph a quadratic equation of the form [latex]y=ax^2+bx+c[\/latex] without having to calculate an exhaustive table of values.<\/p>\n<div class=\"bcc-box examples\">\n<h3>Example<\/h3>\n<p>Graph [latex]y=\u22122x^{2}+3x\u20133[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Let&#8217;s start by considering the features of a parabola.<\/p>\n<p>Opening:<\/p>\n<p>[latex]a=-2[\/latex] so the parabola opens downwards.<\/p>\n<p>Since [latex]|a|>1[\/latex] the graph will be narrower than the graph of [latex]y=x^2[\/latex].<\/p>\n<p>[latex]y[\/latex]-intercept<span style=\"font-size: 1rem; text-align: initial;\">:<\/span><\/p>\n<p>Since [latex]c=-3[\/latex], the [latex]y[\/latex]-intercept will be [latex](0, -3)[\/latex].<\/p>\n<p>Axis of symmetry:<\/p>\n<p>[latex]x=\\frac{-b}{2a}=\\frac{-3}{2\\cdot (-2)}=\\frac{3}{4}[\/latex].<\/p>\n<p>Vertex:<\/p>\n<p>[latex]x=\\frac{3}{4}[\/latex], so [latex]y=\u22122x^{2}+3x\u20133=-2{\\left (\\frac{3}{4}\\right )}^{2}+3\\cdot \\frac{3}{4}-3=-2\\cdot \\frac{9}{16}+\\frac{9}{4}-3=\\frac{-9}{8}+\\frac{9}{4}-3=\\frac{-9+18-24}{8}=\\frac{-15}{8}[\/latex].\u00a0 The vertex is the point [latex]\\left (\\frac{3}{4}, -\\frac{15}{8}\\right )[\/latex].<\/p>\n<p>[latex]x[\/latex]-intercepts<span style=\"font-size: 1rem; text-align: initial;\">:\u00a0<\/span><\/p>\n<p>[latex]\u22122x^{2}+3x\u20133[\/latex] does not factor, so we cannot find the\u00a0[latex]x[\/latex]-intercepts<span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\n<p>Let&#8217;s graph everything we have:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-2130 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/1st-draft-300x231.png\" alt=\"1st draft of graph\" width=\"300\" height=\"231\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/1st-draft-300x231.png 300w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/1st-draft-768x593.png 768w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/1st-draft-65x50.png 65w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/1st-draft-225x174.png 225w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/1st-draft-350x270.png 350w, https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2021\/10\/1st-draft.png 884w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/>Notice that the point [latex](0, -3)[\/latex] has a mirror image on the other side of the axis of symmetry.<\/p>\n<p>This twin has the same [latex]y[\/latex]-value of [latex]-3[\/latex]. To find the [latex]x[\/latex]-value, move the same number of units past the axis of symmetry to the right.<\/p>\n<p>The point\u00a0[latex](0, -3)[\/latex] is [latex]\\frac{3}{4}[\/latex] of a unit to the left of the axis, so its twin is\u00a0[latex]\\frac{3}{4}[\/latex] of a unit to the right of the axis at [latex]\\left (\\frac{3}{2}, -3\\right )[\/latex].<\/p>\n<p>From the information we have found so far, we have the turning point and direction of the graph. Now let&#8217;s find some solutions and use their twins to plot more points.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<table style=\"height: 92px; width: 355px; width: 100px;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"text-align: center; width: 294.719px; height: 12px;\"><strong>[latex]x[\/latex]<\/strong><\/th>\n<th style=\"text-align: center; width: 248.297px; height: 12px;\">[latex]y=\u22122x^{2}+3x\u20133[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 20px;\">\n<td style=\"text-align: center; width: 294.719px; height: 20px;\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center; width: 248.297px; height: 20px;\">[latex]\u22128[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"text-align: center; width: 294.719px; height: 20px;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center; width: 248.297px; height: 20px;\">[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"text-align: center; width: 294.719px; height: 20px;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center; width: 248.297px; height: 20px;\">[latex]\u22122[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"text-align: center; width: 294.719px; height: 20px;\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center; width: 248.297px; height: 20px;\">[latex]\u22125[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-2131\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/28000312\/2nd-draft-195x300.png\" alt=\"2nd draft\" width=\"273\" height=\"420\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-2132\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/28000446\/Final-graph-203x300.png\" alt=\"Final graph\" width=\"284\" height=\"420\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Finally connect the points as best you can using a <i>smooth curve.<\/i><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Determine the maximum value of [latex]y[\/latex]: [latex]y=-4x^2-4x+3[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Since the graph is a parabola that opens downwards, the maximum value will occur at the vertex.<\/p>\n<p>Find the vertex:\u00a0 [latex]x=\\frac{-b}{2a}=\\frac{4}{2\\cdot (-4)}=\\frac{-1}{2}[\/latex]<\/p>\n<p>Then find the [latex]y[\/latex]-value:<\/p>\n<p>[latex]y=-4x^2-4x+3\\\\y=-4{\\left (\\frac{-1}{2}\\right)}^{2}-4\\left (\\frac{-1}{2}\\right )+3\\\\y=\\frac{-4}{1}\\cdot\\frac{1}{4}+2+3\\\\y=-1+2+3\\\\y=4[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The maximum value is [latex]y=4[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Determine the minimum value of [latex]y[\/latex]:\u00a0 [latex]y=x^2-8x-4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm057\">Show Answer<\/span><\/p>\n<div id=\"qhjm057\" class=\"hidden-answer\" style=\"display: none\">\n<p>The minimum value is [latex]-20[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":370291,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2950","chapter","type-chapter","status-publish","hentry"],"part":2946,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2950","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2950\/revisions"}],"predecessor-version":[{"id":3202,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2950\/revisions\/3202"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/2946"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2950\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=2950"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2950"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=2950"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=2950"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}