{"id":1693,"date":"2022-04-21T17:39:07","date_gmt":"2022-04-21T17:39:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=1693"},"modified":"2026-01-22T17:35:03","modified_gmt":"2026-01-22T17:35:03","slug":"4-1-quadratic-functions-and-their-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/4-1-quadratic-functions-and-their-graphs\/","title":{"raw":"4.1: Quadratic Functions and Their Graphs","rendered":"4.1: Quadratic Functions and Their Graphs"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-402\" class=\"standard post-402 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1 style=\"text-align: center;\">Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Graph the quadratic function [latex]f(x)=x^2[\/latex]<\/li>\r\n \t<li>Identify the vertex, line of symmetry, [latex]x[\/latex]-intercept(s), [latex]y[\/latex]-intercept, and vertex of a parabola<\/li>\r\n \t<li>Determine if a parabola has a\u00a0minimum or maximum and find that value<\/li>\r\n \t<li>Identify the symmetric point given a point on the parabola and the line of symmetry<\/li>\r\n \t<li>Find the [latex]x[\/latex]-coordinate of the vertex using two symmetric points<\/li>\r\n \t<li>Identify the domain and range of a quadratic function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphing a Quadratic Function<\/h2>\r\nOne way to graph a function is to make a table of values and transfer the resulting coordinate pairs onto the coordinate plane. Then we can connect the plotted points to sketch the graph of the function.\r\n\r\nThe function [latex]f(x)=x^2[\/latex] is a polynomial function of degree 2. Such polynomial functions are also called <em><strong>quadratic function<\/strong><\/em><em><strong>s<\/strong>.\u00a0<\/em>To graph the quadratic function [latex]f(x)=x^2[\/latex], we can generate the table of values in table 1. We can choose any [latex]x[\/latex]-values we want, but it is best to cover both positive and negative values of [latex]x[\/latex] so that the graph may be comprehensive rather than only showing part of the graph.\r\n<table style=\"border-collapse: collapse; width: 0%; height: 180px;\" border=\"1\"><caption>Table of values for the function [latex]f(x)=x^2[\/latex]<\/caption>\r\n<tbody>\r\n<tr style=\"height: 18px;\">\r\n<th style=\"width: 100.703px; text-align: center; height: 18px;\"><strong>[latex]x[\/latex]<\/strong><\/th>\r\n<th style=\"width: 139.812px; text-align: center; height: 18px;\"><strong>[latex]f(x)=x^2[\/latex]<\/strong><\/th>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">-3<\/td>\r\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">-2<\/td>\r\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">-1<\/td>\r\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">0<\/td>\r\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">1<\/td>\r\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">2<\/td>\r\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">3<\/td>\r\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px;\">\r\n<td style=\"width: 265.515625px; text-align: left; height: 36px;\" colspan=\"2\">Table 1. Table of values for [latex]f(x)=x^2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow, we take each [latex](x, f(x))[\/latex] pair on the table as a coordinate pair [latex](x, y)[\/latex] and plot them on the coordinate plane. We cannot connect these points using straight lines because the function is not linear (i.e., the exponent of the independent variable [latex]x[\/latex] is not 1). That is, the slopes between any two points are not a constant. For example, we use a curve instead of a straight line to connect the two points (1,1) and (2,4) because you can find another point (1.5, 2.25) on the graph between the two points, and the slope between point (1,1) and (1.5, 2.25) is 2.5 but the slope between (1.5, 2.25) and (2, 4) is 3.5. Therefore, we cannot connect the two points (1,1) and (2,4) with a straight line because the slopes between the two points are different (i.e., [latex]2.5 \\neq 3.5[\/latex]). Since it is a polynomial, we will use a curve to connect these points on the coordinate plane (figure 1). The shape of this curve is called a\u00a0<em><strong>parabola<\/strong><\/em>.\r\n\r\n<\/div>\r\n<\/div>\r\n[caption id=\"attachment_1780\" align=\"aligncenter\" width=\"310\"]<img class=\"wp-image-1780\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/23004124\/4-1-SimpleFunction4-300x300.png\" alt=\"An upward-opening parabola with the lowest point at (0, 0).\" width=\"310\" height=\"310\" \/> Figure 1. The graph of the function [latex]f(x)=x^2[\/latex].[\/caption]\r\n<div id=\"post-402\" class=\"standard post-402 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n\r\nThe function [latex]f(x)=x^2[\/latex] is the parent function of all quadratic functions, which can be written in the form [latex]f(x)=ax^2+bx+c[\/latex], where [latex]a, b, c[\/latex] are real numbers with [latex]a\\ne0[\/latex]. The parabola will open upwards if [latex]a&gt;0[\/latex] and will open downwards if [latex]a&lt;0[\/latex] (figure 2).\r\n\r\n[caption id=\"attachment_2013\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2013\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/05190040\/desmos-graph-2022-05-05T130025.155-300x300.png\" alt=\"two parabolas, one opening downward and the other opening upward\" width=\"300\" height=\"300\" \/> Figure 2. Parabola opens up when [latex]a&gt;0[\/latex] and down when [latex]a&lt;0[\/latex] on graph of [latex]f(x)=ax^2+bx+c[\/latex][\/caption]\r\n<h2>Features of the Graph of a Quadratic Function<\/h2>\r\nOne important feature of the graph of a quadratic function (a <em><strong>parabola<\/strong><\/em>) is that it has a turning point, called the <em><strong>vertex<\/strong><\/em>. If the parabola opens up, the vertex represents the lowest point on the graph, or the <em><strong>minimum value<\/strong><\/em> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <em><strong>maximum value<\/strong><\/em>. In either case, the vertex is a turning point on the graph. The graph is also <em><strong>symmetric<\/strong><\/em> about a vertical line that passes through the vertex, called the <em><strong>line (or axis) of symmetry<\/strong><\/em> (figure 3).\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive tool showing a parabola with a movable vertex to explore its line of symmetry\" src=\"https:\/\/www.desmos.com\/calculator\/jfwuga89rp?embed\" width=\"500\" height=\"400\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 3. The vertex is always on the axis of symmetry.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: left;\">The intersection points between the graph and the [latex]x[\/latex]-axis are the <strong>[latex]x[\/latex]-intercepts<\/strong>. There can be 0, 1, or 2 [latex]x[\/latex]-intercepts, depending where the graph lies on the coordinate plane. The intersection point between the graph and the [latex]y[\/latex]-axis is the <strong>[latex]y[\/latex]-intercept<\/strong> (figure 4).<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">No x-intercepts<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">One x-intercept<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Two x-intercepts<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1993\" align=\"aligncenter\" width=\"260\"]<img class=\"wp-image-1993\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/04214034\/desmos-graph-2022-05-04T153934.273-300x300.png\" alt=\"Parabola opening up with the lowest point above the x-axis\" width=\"260\" height=\"260\" \/> Parabola opening up with a minimum function value and no [latex]x[\/latex]-intercepts.[\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1995\" align=\"aligncenter\" width=\"260\"]<img class=\"wp-image-1995\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/04214632\/desmos-graph-2022-05-04T154604.177-300x300.png\" alt=\"parabola opening up with lowest point on the x-axis\" width=\"260\" height=\"260\" \/> Parabola opening up with a minimum function value and one [latex]x[\/latex]-intercepts.[\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1994\" align=\"aligncenter\" width=\"260\"]<img class=\"wp-image-1994\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/04214045\/desmos-graph-2022-05-04T153905.777-300x300.png\" alt=\"parabola opening down with the highest point above the x-axis\" width=\"260\" height=\"260\" \/> Parabola opening down with a maximum function value and two [latex]x[\/latex]-intercepts[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\" colspan=\"3\">\r\n<div class=\"mceTemp\">Figure 4. Parabolas with 0, 1, and 2 [latex]x[\/latex]-intercepts<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nThe parabola in figure 5 has [latex]x[\/latex]-intercepts of (\u20131, 0) and (3, 0). The\u00a0[latex]y[\/latex]-intercept is (0, \u20133) and the vertex is (1, \u20134). The equation for the line of symmetry is [latex]x=1[\/latex] because it is a vertical line that passes through the vertex (1, \u20134). The minimum value of the function occurs at the vertex and has a value of \u20134.\r\n<div class=\"wp-nocaption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"300\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170330\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"A parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"300\" height=\"296\" \/> Figure 5. Features of a parabola[\/caption]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nDetermine the vertex, line of symmetry, [latex]x[\/latex]-intercepts, [latex]y[\/latex]-intercept and minimum value of the parabola.\r\n<h4 class=\"wp-nocaption aligncenter\" style=\"text-align: left;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170333\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"A parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"320\" height=\"340\" \/>Solution<\/h4>\r\nThe graph crosses the\u00a0[latex]y[\/latex]-axis at 7, so the\u00a0[latex]y[\/latex]-intercept is (0, 7).\r\n\r\nThe graph never crosses the\u00a0[latex]x[\/latex]-axis, so there are no\u00a0[latex]x[\/latex]-intercepts.\r\n\r\nThe graph turns at (3, 1) so the vertex is (3, 1).\r\n\r\nThe graph has a minimum value at the vertex of 1.\r\n\r\nThe axis of symmetry is the vertical line through the vertex (3, 1), so is the line [latex]x=3[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nDetermine the vertex, line of symmetry, [latex]x[\/latex]-intercepts, [latex]x[\/latex]-intercept and maximum value of the parabola.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-1997\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/04230952\/desmos-graph-2022-05-04T170926.392-300x300.png\" alt=\"parabola opening downwards\" width=\"260\" height=\"260\" \/>[reveal-answer q=\"hjm202\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm202\"]<\/p>\r\nvertex = [latex](1,4)[\/latex]\r\n\r\nline of symmetry:\u00a0[latex]x=1[\/latex]\r\n\r\n[latex]x[\/latex]-intercepts =\u00a0[latex](-1,0)[\/latex] and\u00a0[latex](3,0)[\/latex]\r\n\r\n[latex]y[\/latex]-intercept =\u00a0[latex](0,3)[\/latex]\r\n\r\nmaximum value = 4\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Identifying Symmetric Points<\/h2>\r\nSince the graph of a quadratic function is symmetric across the line of symmetry, every point on the graph has a reflection called its <em><strong>symmetric point<\/strong><\/em>. Given any point [latex]A[\/latex] on a parabola, the symmetric point [latex]A^{\\prime}[\/latex] of the given point [latex]A[\/latex] will be on the other side of the line of symmetry, the same distance from the line of symmetry as the distance of point\u00a0[latex]A[\/latex] from the line of symmetry.\u00a0 Since the line of symmetry is vertical, the point [latex]A[\/latex] and [latex]A^{\\prime}[\/latex] lie on the same horizontal line that is perpendicular to the line of symmetry. Therefore, the point [latex]A^{\\prime}[\/latex] has the same [latex]y[\/latex]-coordinate as the given point [latex]A[\/latex] (figure 6).\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/mhcfaphmbe?embed\" width=\"500\" height=\"500\" frameborder=\"0\" aria-hidden=\"true\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 6. Move point P to see its symmetric point.<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nFind the symmetric point of a given point (2, 10) on a parabola with vertex (5, 1).\r\n<h4>Solution<\/h4>\r\nDraw a quick sketch of a parabola with its vertex at (5, 1) and that passes through the point (2, 10).\r\n\r\nOn the graph of the quadratic function, we have the line of symmetry [latex]x=5[\/latex] passing through the vertex.\r\n\r\n<img class=\"aligncenter wp-image-1778\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/23003341\/4-1-SymmetricPoint-300x300.png\" alt=\"Parabola with vertex (5, 1) that passes through the point (2, 10)\" width=\"198\" height=\"198\" \/>\r\n\r\nThe symmetric point of (2, 10) will be on the other side of the line of symmetry where the [latex]y[\/latex]-coordinate of the symmetric point will share the same [latex]y[\/latex]-value of 10.\r\n\r\nWhat is left now is to find the [latex]x[\/latex]-coordinate of the symmetric point.\r\n\r\nThe distance between the given point (2, 10) and the line of symmetry [latex]x=5[\/latex] is 3 (i.e., 5 \u2013 2 = 3).\r\n\r\nSo starting at [latex]x=5[\/latex] and moving to the right 3 units gives us the [latex]x[\/latex]-coordinate of the symmetric point, 8 (i.e., 5 + 3 = 8).\r\n\r\nTherefore, the symmetric point of the given point (2, 10) is (8, 10).\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nFind the symmetric point of a given point (0, \u20134) on a parabola with vertex (\u20133, 5).\r\n\r\n[reveal-answer q=\"hjm947\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm947\"]\r\n\r\nSymmetric point of (0, \u20134) is (\u20136, \u20134).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Finding the [latex]x[\/latex]-coordinate of the Vertex Using Two Symmetric Points<\/span><\/h2>\r\n<p style=\"text-align: left;\">Any two symmetric points on a parabola\u00a0<span style=\"font-size: 1em;\">are the same distance away from the line of symmetry. Consequently, any two symmetric points are the same distance away from<\/span><span style=\"font-size: 1rem;\">\u00a0the [latex]x[\/latex]-coordinate of the vertex. Therefore, the [latex]x[\/latex]-coordinate of the vertex is the midpoint between any two symmetric points on a parabola. This means that we can identify the\u00a0[latex]x[\/latex]-coordinate of the vertex if we know two symmetric points on a parabola.\u00a0<\/span><span style=\"font-size: 1rem;\">\u00a0<\/span><span style=\"font-size: 1em;\">To find the halfway point between two given values, [latex]x_1[\/latex] and [latex]x_2[\/latex], we find the mean (or average) of the values: [latex]\\dfrac{x_1+x_2}{2}[\/latex].\u00a0<\/span><span style=\"font-size: 1rem;\">To find the corresponding [latex]y[\/latex]-coordinate of the vertex, we need the function equation so we can find the function value at the vertex.<\/span><\/p>\r\n<p style=\"text-align: left;\">For example, given two symmetric points (2, 10) and (8, 10) on the graph of the function [latex]f(x)=x^2-10x+26[\/latex], the [latex]x[\/latex]-coordinate of the vertex will be halfway between the two symmetric points (figure 7).<\/p>\r\n\r\n\r\n[caption id=\"attachment_2005\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2005 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T195241.168-300x300.png\" alt=\"midpoint between two points on a parabola\" width=\"300\" height=\"300\" \/> Figure 7. Axis of symmetry is halfway between two symmetric points.[\/caption]\r\n\r\nThe halfway point between [latex]x=2[\/latex] and [latex]x=8[\/latex] is:\r\n<p style=\"text-align: center;\">[latex]x=\\dfrac{2+8}{2} = \\dfrac{10}{2} = 5[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Knowing that the [latex]x[\/latex]-coordinate of the vertex is 5, the [latex]y[\/latex]-coordinate of the vertex will be [latex]f(5)[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}f(5) &amp;= (5)^2-10(5)+26 \\\\&amp;= 25 - 50 + 26 \\\\&amp;= 1\\end{aligned}[\/latex].<\/p>\r\n<p style=\"text-align: left;\">Therefore, the vertex of the function is at the point\u00a0 (5, 1).<\/p>\r\n<strong>Note.<\/strong> If we did not know the equation of the function, or were unable to find the equation, we would not have been able to determine the [latex]y[\/latex]-coordinate of the vertex. This is because there are an infinite number of parabolas that pass through the\u00a0two symmetric points (2, 10) and (8, 10) (figure 8).\r\n\r\n[caption id=\"attachment_2006\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2006 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T200123.203-300x300.png\" alt=\"3 parabolas passing through the same 2 symmetric points\" width=\"300\" height=\"300\" \/> Figure 8. Example of parabolas that pass through (2, 10) and (8, 10)[\/caption]\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nFind the vertex of the parabola representing the function [latex]f(x)=x^2-8x+13[\/latex], which passes through the symmetric points (2, 1) and (6, 1).\r\n<h4>Solution<\/h4>\r\nThe axis of symmetry is halfway between [latex]x=2[\/latex] and\u00a0[latex]x=6[\/latex]:\r\n<p style=\"text-align: center;\">[latex]x=\\dfrac{2+6}{2}=4[\/latex]<\/p>\r\nTherefore, the [latex]x[\/latex]-coordinate of the vertex is 4.\r\n\r\nTo find the\u00a0[latex]y[\/latex]-coordinate of the vertex, we need to find [latex]f(4)[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}f(4)&amp;=(4)^2-8(4)+13\\\\&amp;=16-32+13\\\\&amp;=-3\\end{aligned}[\/latex]<\/p>\r\nThe vertex is the point (4, \u20133).\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nFind the vertex of the parabola representing the function [latex]f(x)=-x^2-6x-3[\/latex], which passes through the symmetric points (\u20137, \u201310) and (1, \u201310).\r\n\r\n[reveal-answer q=\"hjm048\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm048\"]The vertex is (\u20133, 6).[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 style=\"text-align: left;\">Finding the Domain and Range of a Quadratic Function<\/h2>\r\n<p style=\"text-align: left;\">Any number can be the input value of a quadratic function. Therefore the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum at the vertex, the range is restricted. Since the [latex]y[\/latex]-coordinate of the vertex of a parabola will be either a maximum or a minimum, the range will be determined by the [latex]y[\/latex]-coordinate of the vertex and whether or not the parabola opens upwards or downwards. I<span style=\"font-size: 1em;\">f the parabola opens upwards, the range will\u00a0<\/span><span style=\"font-size: 1rem;\">consist of all [latex]y[\/latex]-values greater than or equal to the [latex]y[\/latex]-coordinate of the vertex. If the parabola opens downwards, the range will consist of all [latex]y[\/latex]-values less than or equal to the [latex]y[\/latex]-coordinate of the vertex.<\/span><\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Domain and Range of a quadratic function<\/h3>\r\nFor any quadratic function whose graph has a vertex at the point [latex](h, k)[\/latex], the domain is all real numbers and the range depends on the whether the parabola opens up or down.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%;\">Domain of Parabola Opening Down<\/th>\r\n<th style=\"width: 50%;\">Domain of Parabola Opening Up<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\"><img class=\"wp-image-1999 size-medium aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/05002637\/desmos-graph-2022-05-04T182434.812-300x300.png\" alt=\"parabola opening downward with vertex at (h, k)\" width=\"300\" height=\"300\" \/>\r\n<p style=\"text-align: center;\">Domain = [latex](-\\infty,+\\infty)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Range = [latex](-\\infty,k][\/latex]<\/p>\r\n<\/td>\r\n<td style=\"width: 50%;\"><img class=\"size-medium wp-image-2000 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/05002725\/desmos-graph-2022-05-04T182610.509-300x300.png\" alt=\"Parabola opening upwards with vertex at (h, k)\" width=\"300\" height=\"300\" \/>\r\n<p style=\"text-align: center;\">Domain = [latex](-\\infty,+\\infty)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Range = [latex][k,+\\infty)[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: left;\"><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nState the domain and the range of a quadratic function whose graph is a parabola that opens downwards with a vertex at the point (\u20132, 5).\r\n<h4>Solution<\/h4>\r\nFirst we should recognize that there are an infinite number of parabolas\u00a0that opens downwards with a vertex at the point (\u20132, 5)! However, they all have the same domain and range.\r\n\r\nThe domain is all real numbers: [latex]x\\in(-\\infty,+\\infty)[\/latex]\r\n\r\nBecause the parabola opens downwards it has a maximum value at the vertex. So the range is all real numbers less than or equal to 5: [latex]y\\in(-\\infty,5][\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nState the domain and the range of a quadratic function whose graph is a parabola that opens upwards with a vertex at the point (4, \u20133).\r\n\r\n[reveal-answer q=\"hjm768\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm768\"]\r\n\r\nDomain: [latex]x\\in(-\\infty,+\\infty)[\/latex]\r\n\r\nRange: [latex]y\\in[-3,+\\infty)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-402\" class=\"standard post-402 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h1 style=\"text-align: center;\">Learning Outcomes<\/h1>\n<ul>\n<li>Graph the quadratic function [latex]f(x)=x^2[\/latex]<\/li>\n<li>Identify the vertex, line of symmetry, [latex]x[\/latex]-intercept(s), [latex]y[\/latex]-intercept, and vertex of a parabola<\/li>\n<li>Determine if a parabola has a\u00a0minimum or maximum and find that value<\/li>\n<li>Identify the symmetric point given a point on the parabola and the line of symmetry<\/li>\n<li>Find the [latex]x[\/latex]-coordinate of the vertex using two symmetric points<\/li>\n<li>Identify the domain and range of a quadratic function<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing a Quadratic Function<\/h2>\n<p>One way to graph a function is to make a table of values and transfer the resulting coordinate pairs onto the coordinate plane. Then we can connect the plotted points to sketch the graph of the function.<\/p>\n<p>The function [latex]f(x)=x^2[\/latex] is a polynomial function of degree 2. Such polynomial functions are also called <em><strong>quadratic function<\/strong><\/em><em><strong>s<\/strong>.\u00a0<\/em>To graph the quadratic function [latex]f(x)=x^2[\/latex], we can generate the table of values in table 1. We can choose any [latex]x[\/latex]-values we want, but it is best to cover both positive and negative values of [latex]x[\/latex] so that the graph may be comprehensive rather than only showing part of the graph.<\/p>\n<table style=\"border-collapse: collapse; width: 0%; height: 180px;\">\n<caption>Table of values for the function [latex]f(x)=x^2[\/latex]<\/caption>\n<tbody>\n<tr style=\"height: 18px;\">\n<th style=\"width: 100.703px; text-align: center; height: 18px;\"><strong>[latex]x[\/latex]<\/strong><\/th>\n<th style=\"width: 139.812px; text-align: center; height: 18px;\"><strong>[latex]f(x)=x^2[\/latex]<\/strong><\/th>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">-3<\/td>\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">9<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">-2<\/td>\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">4<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">-1<\/td>\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">0<\/td>\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">1<\/td>\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">2<\/td>\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">4<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 100.703125px; text-align: center; height: 18px;\">3<\/td>\n<td style=\"width: 139.8125px; text-align: center; height: 18px;\">9<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 265.515625px; text-align: left; height: 36px;\" colspan=\"2\">Table 1. Table of values for [latex]f(x)=x^2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now, we take each [latex](x, f(x))[\/latex] pair on the table as a coordinate pair [latex](x, y)[\/latex] and plot them on the coordinate plane. We cannot connect these points using straight lines because the function is not linear (i.e., the exponent of the independent variable [latex]x[\/latex] is not 1). That is, the slopes between any two points are not a constant. For example, we use a curve instead of a straight line to connect the two points (1,1) and (2,4) because you can find another point (1.5, 2.25) on the graph between the two points, and the slope between point (1,1) and (1.5, 2.25) is 2.5 but the slope between (1.5, 2.25) and (2, 4) is 3.5. Therefore, we cannot connect the two points (1,1) and (2,4) with a straight line because the slopes between the two points are different (i.e., [latex]2.5 \\neq 3.5[\/latex]). Since it is a polynomial, we will use a curve to connect these points on the coordinate plane (figure 1). The shape of this curve is called a\u00a0<em><strong>parabola<\/strong><\/em>.<\/p>\n<\/div>\n<\/div>\n<div id=\"attachment_1780\" style=\"width: 320px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1780\" class=\"wp-image-1780\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/23004124\/4-1-SimpleFunction4-300x300.png\" alt=\"An upward-opening parabola with the lowest point at (0, 0).\" width=\"310\" height=\"310\" \/><\/p>\n<p id=\"caption-attachment-1780\" class=\"wp-caption-text\">Figure 1. The graph of the function [latex]f(x)=x^2[\/latex].<\/p>\n<\/div>\n<div id=\"post-402\" class=\"standard post-402 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<p>The function [latex]f(x)=x^2[\/latex] is the parent function of all quadratic functions, which can be written in the form [latex]f(x)=ax^2+bx+c[\/latex], where [latex]a, b, c[\/latex] are real numbers with [latex]a\\ne0[\/latex]. The parabola will open upwards if [latex]a>0[\/latex] and will open downwards if [latex]a<0[\/latex] (figure 2).\n\n\n\n<div id=\"attachment_2013\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2013\" class=\"wp-image-2013\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/05190040\/desmos-graph-2022-05-05T130025.155-300x300.png\" alt=\"two parabolas, one opening downward and the other opening upward\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2013\" class=\"wp-caption-text\">Figure 2. Parabola opens up when [latex]a&gt;0[\/latex] and down when [latex]a&lt;0[\/latex] on graph of [latex]f(x)=ax^2+bx+c[\/latex]<\/p>\n<\/div>\n<h2>Features of the Graph of a Quadratic Function<\/h2>\n<p>One important feature of the graph of a quadratic function (a <em><strong>parabola<\/strong><\/em>) is that it has a turning point, called the <em><strong>vertex<\/strong><\/em>. If the parabola opens up, the vertex represents the lowest point on the graph, or the <em><strong>minimum value<\/strong><\/em> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <em><strong>maximum value<\/strong><\/em>. In either case, the vertex is a turning point on the graph. The graph is also <em><strong>symmetric<\/strong><\/em> about a vertical line that passes through the vertex, called the <em><strong>line (or axis) of symmetry<\/strong><\/em> (figure 3).<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive tool showing a parabola with a movable vertex to explore its line of symmetry\" src=\"https:\/\/www.desmos.com\/calculator\/jfwuga89rp?embed\" width=\"500\" height=\"400\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 3. The vertex is always on the axis of symmetry.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">The intersection points between the graph and the [latex]x[\/latex]-axis are the <strong>[latex]x[\/latex]-intercepts<\/strong>. There can be 0, 1, or 2 [latex]x[\/latex]-intercepts, depending where the graph lies on the coordinate plane. The intersection point between the graph and the [latex]y[\/latex]-axis is the <strong>[latex]y[\/latex]-intercept<\/strong> (figure 4).<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">No x-intercepts<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">One x-intercept<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Two x-intercepts<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1993\" style=\"width: 270px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1993\" class=\"wp-image-1993\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/04214034\/desmos-graph-2022-05-04T153934.273-300x300.png\" alt=\"Parabola opening up with the lowest point above the x-axis\" width=\"260\" height=\"260\" \/><\/p>\n<p id=\"caption-attachment-1993\" class=\"wp-caption-text\">Parabola opening up with a minimum function value and no [latex]x[\/latex]-intercepts.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1995\" style=\"width: 270px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1995\" class=\"wp-image-1995\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/04214632\/desmos-graph-2022-05-04T154604.177-300x300.png\" alt=\"parabola opening up with lowest point on the x-axis\" width=\"260\" height=\"260\" \/><\/p>\n<p id=\"caption-attachment-1995\" class=\"wp-caption-text\">Parabola opening up with a minimum function value and one [latex]x[\/latex]-intercepts.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1994\" style=\"width: 270px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1994\" class=\"wp-image-1994\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/04214045\/desmos-graph-2022-05-04T153905.777-300x300.png\" alt=\"parabola opening down with the highest point above the x-axis\" width=\"260\" height=\"260\" \/><\/p>\n<p id=\"caption-attachment-1994\" class=\"wp-caption-text\">Parabola opening down with a maximum function value and two [latex]x[\/latex]-intercepts<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\" colspan=\"3\">\n<div class=\"mceTemp\">Figure 4. Parabolas with 0, 1, and 2 [latex]x[\/latex]-intercepts<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>The parabola in figure 5 has [latex]x[\/latex]-intercepts of (\u20131, 0) and (3, 0). The\u00a0[latex]y[\/latex]-intercept is (0, \u20133) and the vertex is (1, \u20134). The equation for the line of symmetry is [latex]x=1[\/latex] because it is a vertical line that passes through the vertex (1, \u20134). The minimum value of the function occurs at the vertex and has a value of \u20134.<\/p>\n<div class=\"wp-nocaption aligncenter\">\n<div style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170330\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"A parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"300\" height=\"296\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5. Features of a parabola<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Determine the vertex, line of symmetry, [latex]x[\/latex]-intercepts, [latex]y[\/latex]-intercept and minimum value of the parabola.<\/p>\n<h4 class=\"wp-nocaption aligncenter\" style=\"text-align: left;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170333\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"A parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"320\" height=\"340\" \/>Solution<\/h4>\n<p>The graph crosses the\u00a0[latex]y[\/latex]-axis at 7, so the\u00a0[latex]y[\/latex]-intercept is (0, 7).<\/p>\n<p>The graph never crosses the\u00a0[latex]x[\/latex]-axis, so there are no\u00a0[latex]x[\/latex]-intercepts.<\/p>\n<p>The graph turns at (3, 1) so the vertex is (3, 1).<\/p>\n<p>The graph has a minimum value at the vertex of 1.<\/p>\n<p>The axis of symmetry is the vertical line through the vertex (3, 1), so is the line [latex]x=3[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Determine the vertex, line of symmetry, [latex]x[\/latex]-intercepts, [latex]x[\/latex]-intercept and maximum value of the parabola.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1997\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/04230952\/desmos-graph-2022-05-04T170926.392-300x300.png\" alt=\"parabola opening downwards\" width=\"260\" height=\"260\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm202\">Show Answer<\/span><\/p>\n<div id=\"qhjm202\" class=\"hidden-answer\" style=\"display: none\">\n<p>vertex = [latex](1,4)[\/latex]<\/p>\n<p>line of symmetry:\u00a0[latex]x=1[\/latex]<\/p>\n<p>[latex]x[\/latex]-intercepts =\u00a0[latex](-1,0)[\/latex] and\u00a0[latex](3,0)[\/latex]<\/p>\n<p>[latex]y[\/latex]-intercept =\u00a0[latex](0,3)[\/latex]<\/p>\n<p>maximum value = 4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Identifying Symmetric Points<\/h2>\n<p>Since the graph of a quadratic function is symmetric across the line of symmetry, every point on the graph has a reflection called its <em><strong>symmetric point<\/strong><\/em>. Given any point [latex]A[\/latex] on a parabola, the symmetric point [latex]A^{\\prime}[\/latex] of the given point [latex]A[\/latex] will be on the other side of the line of symmetry, the same distance from the line of symmetry as the distance of point\u00a0[latex]A[\/latex] from the line of symmetry.\u00a0 Since the line of symmetry is vertical, the point [latex]A[\/latex] and [latex]A^{\\prime}[\/latex] lie on the same horizontal line that is perpendicular to the line of symmetry. Therefore, the point [latex]A^{\\prime}[\/latex] has the same [latex]y[\/latex]-coordinate as the given point [latex]A[\/latex] (figure 6).<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/mhcfaphmbe?embed\" width=\"500\" height=\"500\" frameborder=\"0\" aria-hidden=\"true\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 6. Move point P to see its symmetric point.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Find the symmetric point of a given point (2, 10) on a parabola with vertex (5, 1).<\/p>\n<h4>Solution<\/h4>\n<p>Draw a quick sketch of a parabola with its vertex at (5, 1) and that passes through the point (2, 10).<\/p>\n<p>On the graph of the quadratic function, we have the line of symmetry [latex]x=5[\/latex] passing through the vertex.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1778\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/23003341\/4-1-SymmetricPoint-300x300.png\" alt=\"Parabola with vertex (5, 1) that passes through the point (2, 10)\" width=\"198\" height=\"198\" \/><\/p>\n<p>The symmetric point of (2, 10) will be on the other side of the line of symmetry where the [latex]y[\/latex]-coordinate of the symmetric point will share the same [latex]y[\/latex]-value of 10.<\/p>\n<p>What is left now is to find the [latex]x[\/latex]-coordinate of the symmetric point.<\/p>\n<p>The distance between the given point (2, 10) and the line of symmetry [latex]x=5[\/latex] is 3 (i.e., 5 \u2013 2 = 3).<\/p>\n<p>So starting at [latex]x=5[\/latex] and moving to the right 3 units gives us the [latex]x[\/latex]-coordinate of the symmetric point, 8 (i.e., 5 + 3 = 8).<\/p>\n<p>Therefore, the symmetric point of the given point (2, 10) is (8, 10).<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Find the symmetric point of a given point (0, \u20134) on a parabola with vertex (\u20133, 5).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm947\">Show Answer<\/span><\/p>\n<div id=\"qhjm947\" class=\"hidden-answer\" style=\"display: none\">\n<p>Symmetric point of (0, \u20134) is (\u20136, \u20134).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Finding the [latex]x[\/latex]-coordinate of the Vertex Using Two Symmetric Points<\/span><\/h2>\n<p style=\"text-align: left;\">Any two symmetric points on a parabola\u00a0<span style=\"font-size: 1em;\">are the same distance away from the line of symmetry. Consequently, any two symmetric points are the same distance away from<\/span><span style=\"font-size: 1rem;\">\u00a0the [latex]x[\/latex]-coordinate of the vertex. Therefore, the [latex]x[\/latex]-coordinate of the vertex is the midpoint between any two symmetric points on a parabola. This means that we can identify the\u00a0[latex]x[\/latex]-coordinate of the vertex if we know two symmetric points on a parabola.\u00a0<\/span><span style=\"font-size: 1rem;\">\u00a0<\/span><span style=\"font-size: 1em;\">To find the halfway point between two given values, [latex]x_1[\/latex] and [latex]x_2[\/latex], we find the mean (or average) of the values: [latex]\\dfrac{x_1+x_2}{2}[\/latex].\u00a0<\/span><span style=\"font-size: 1rem;\">To find the corresponding [latex]y[\/latex]-coordinate of the vertex, we need the function equation so we can find the function value at the vertex.<\/span><\/p>\n<p style=\"text-align: left;\">For example, given two symmetric points (2, 10) and (8, 10) on the graph of the function [latex]f(x)=x^2-10x+26[\/latex], the [latex]x[\/latex]-coordinate of the vertex will be halfway between the two symmetric points (figure 7).<\/p>\n<div id=\"attachment_2005\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2005\" class=\"wp-image-2005 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T195241.168-300x300.png\" alt=\"midpoint between two points on a parabola\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T195241.168-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T195241.168-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T195241.168-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T195241.168-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T195241.168-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T195241.168-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T195241.168.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-2005\" class=\"wp-caption-text\">Figure 7. Axis of symmetry is halfway between two symmetric points.<\/p>\n<\/div>\n<p>The halfway point between [latex]x=2[\/latex] and [latex]x=8[\/latex] is:<\/p>\n<p style=\"text-align: center;\">[latex]x=\\dfrac{2+8}{2} = \\dfrac{10}{2} = 5[\/latex]<\/p>\n<p style=\"text-align: left;\">Knowing that the [latex]x[\/latex]-coordinate of the vertex is 5, the [latex]y[\/latex]-coordinate of the vertex will be [latex]f(5)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}f(5) &= (5)^2-10(5)+26 \\\\&= 25 - 50 + 26 \\\\&= 1\\end{aligned}[\/latex].<\/p>\n<p style=\"text-align: left;\">Therefore, the vertex of the function is at the point\u00a0 (5, 1).<\/p>\n<p><strong>Note.<\/strong> If we did not know the equation of the function, or were unable to find the equation, we would not have been able to determine the [latex]y[\/latex]-coordinate of the vertex. This is because there are an infinite number of parabolas that pass through the\u00a0two symmetric points (2, 10) and (8, 10) (figure 8).<\/p>\n<div id=\"attachment_2006\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2006\" class=\"wp-image-2006 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T200123.203-300x300.png\" alt=\"3 parabolas passing through the same 2 symmetric points\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T200123.203-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T200123.203-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T200123.203-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T200123.203-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T200123.203-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T200123.203-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-05-04T200123.203.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-2006\" class=\"wp-caption-text\">Figure 8. Example of parabolas that pass through (2, 10) and (8, 10)<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Find the vertex of the parabola representing the function [latex]f(x)=x^2-8x+13[\/latex], which passes through the symmetric points (2, 1) and (6, 1).<\/p>\n<h4>Solution<\/h4>\n<p>The axis of symmetry is halfway between [latex]x=2[\/latex] and\u00a0[latex]x=6[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]x=\\dfrac{2+6}{2}=4[\/latex]<\/p>\n<p>Therefore, the [latex]x[\/latex]-coordinate of the vertex is 4.<\/p>\n<p>To find the\u00a0[latex]y[\/latex]-coordinate of the vertex, we need to find [latex]f(4)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}f(4)&=(4)^2-8(4)+13\\\\&=16-32+13\\\\&=-3\\end{aligned}[\/latex]<\/p>\n<p>The vertex is the point (4, \u20133).<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Find the vertex of the parabola representing the function [latex]f(x)=-x^2-6x-3[\/latex], which passes through the symmetric points (\u20137, \u201310) and (1, \u201310).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm048\">Show Answer<\/span><\/p>\n<div id=\"qhjm048\" class=\"hidden-answer\" style=\"display: none\">The vertex is (\u20133, 6).<\/div>\n<\/div>\n<\/div>\n<h2 style=\"text-align: left;\">Finding the Domain and Range of a Quadratic Function<\/h2>\n<p style=\"text-align: left;\">Any number can be the input value of a quadratic function. Therefore the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum at the vertex, the range is restricted. Since the [latex]y[\/latex]-coordinate of the vertex of a parabola will be either a maximum or a minimum, the range will be determined by the [latex]y[\/latex]-coordinate of the vertex and whether or not the parabola opens upwards or downwards. I<span style=\"font-size: 1em;\">f the parabola opens upwards, the range will\u00a0<\/span><span style=\"font-size: 1rem;\">consist of all [latex]y[\/latex]-values greater than or equal to the [latex]y[\/latex]-coordinate of the vertex. If the parabola opens downwards, the range will consist of all [latex]y[\/latex]-values less than or equal to the [latex]y[\/latex]-coordinate of the vertex.<\/span><\/p>\n<div class=\"textbox shaded\">\n<h3>Domain and Range of a quadratic function<\/h3>\n<p>For any quadratic function whose graph has a vertex at the point [latex](h, k)[\/latex], the domain is all real numbers and the range depends on the whether the parabola opens up or down.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%;\">Domain of Parabola Opening Down<\/th>\n<th style=\"width: 50%;\">Domain of Parabola Opening Up<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1999 size-medium aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/05002637\/desmos-graph-2022-05-04T182434.812-300x300.png\" alt=\"parabola opening downward with vertex at (h, k)\" width=\"300\" height=\"300\" \/><\/p>\n<p style=\"text-align: center;\">Domain = [latex](-\\infty,+\\infty)[\/latex]<\/p>\n<p style=\"text-align: center;\">Range = [latex](-\\infty,k][\/latex]<\/p>\n<\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2000 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/05002725\/desmos-graph-2022-05-04T182610.509-300x300.png\" alt=\"Parabola opening upwards with vertex at (h, k)\" width=\"300\" height=\"300\" \/><\/p>\n<p style=\"text-align: center;\">Domain = [latex](-\\infty,+\\infty)[\/latex]<\/p>\n<p style=\"text-align: center;\">Range = [latex][k,+\\infty)[\/latex]<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: left;\">\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>State the domain and the range of a quadratic function whose graph is a parabola that opens downwards with a vertex at the point (\u20132, 5).<\/p>\n<h4>Solution<\/h4>\n<p>First we should recognize that there are an infinite number of parabolas\u00a0that opens downwards with a vertex at the point (\u20132, 5)! However, they all have the same domain and range.<\/p>\n<p>The domain is all real numbers: [latex]x\\in(-\\infty,+\\infty)[\/latex]<\/p>\n<p>Because the parabola opens downwards it has a maximum value at the vertex. So the range is all real numbers less than or equal to 5: [latex]y\\in(-\\infty,5][\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>State the domain and the range of a quadratic function whose graph is a parabola that opens upwards with a vertex at the point (4, \u20133).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm768\">Show Answer<\/span><\/p>\n<div id=\"qhjm768\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain: [latex]x\\in(-\\infty,+\\infty)[\/latex]<\/p>\n<p>Range: [latex]y\\in[-3,+\\infty)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n\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-1693\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Graph the Quadratic Function f(x)=x^2. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Identify a Symmetric Point. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Find the x-coordinate of the Vertex Using Two Symmetric Points. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Figure 1. Vertex and Axis of Symmetry Interactive. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/xwvxml2h7z\">https:\/\/www.desmos.com\/calculator\/xwvxml2h7z<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Figure 6. Symmetric points interactive. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/szdua4agki\">https:\/\/www.desmos.com\/calculator\/szdua4agki<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Figures 2, 3, 4, 6, 7, 8. All Examples. All Try Its: hjm768, hjm048, hjm947, hjm 202. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Graph the Quadratic Function f(x)=x^2\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Identify a Symmetric Point\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Find the x-coordinate of the Vertex Using Two Symmetric Points\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"original\",\"description\":\"Figure 1. Vertex and Axis of Symmetry Interactive\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"https:\/\/www.desmos.com\/calculator\/xwvxml2h7z\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Figure 6. Symmetric points interactive\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"https:\/\/www.desmos.com\/calculator\/szdua4agki\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Figures 2, 3, 4, 6, 7, 8. All Examples. 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