{"id":947,"date":"2021-10-06T15:39:11","date_gmt":"2021-10-06T15:39:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=947"},"modified":"2022-01-20T01:16:53","modified_gmt":"2022-01-20T01:16:53","slug":"9-1-quadratic-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/9-1-quadratic-equations\/","title":{"raw":"9.1: Quadratic Equations","rendered":"9.1: Quadratic Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Describe a quadratic equation<\/li>\r\n \t<li>Look for patterns in quadratic growth<\/li>\r\n \t<li>Graph the quadratic equation [latex]y=x^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>KEYWORDS<\/h3>\r\n<ul>\r\n \t<li><strong>Quadratic<\/strong>: a degree two polynomial of the form [latex]ax^2+bx+c[\/latex]\u00a0where [latex]a, b, c[\/latex] are real numbers with [latex]a\\ne0[\/latex]<\/li>\r\n \t<li><strong>Quadratic equation in 2 variables<\/strong>: a degree two equation of the [latex]y=ax^2+bx+c=0[\/latex]\u00a0where [latex]a, b, c[\/latex] are real numbers with [latex]a\\ne0[\/latex]<\/li>\r\n \t<li><strong>Parabola<\/strong>: the shape of the graph of a quadratic equation\u00a0[latex]y=ax^2+bx+c[\/latex]<\/li>\r\n \t<li><strong>Line of symmetry<\/strong><strong>:\u00a0<\/strong>a line that separates mirror images of each side of a graph<\/li>\r\n \t<li><strong>Vertex<\/strong>: the turning point of a parabola<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Quadratic Equations and Solutions<\/h2>\r\nThe single defining feature of a <em><strong>quadratic<\/strong><\/em> is that it is a polynomial of degree two in one variable. This means it takes the form [latex]ax^2+bx+c[\/latex], where [latex]a, b, c[\/latex] are real numbers with [latex]a\\ne0[\/latex]. If we introduce a second variable [latex]y[\/latex] and set it equal to a quadratic, we get a <strong><em>quadratic equation<\/em><\/strong> of the form [latex]y=ax^2+bx+c[\/latex].\u00a0 SInce a quadratic equation has two variables, [latex]x[\/latex] and [latex]y[\/latex], the solutions will be ordered pairs [latex](x, y)[\/latex].\r\n\r\nWe can discover solutions to a quadratic equations by picking values for [latex]x[\/latex] and finding the corresponding [latex]y[\/latex]-values.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nDetermine the solutions to [latex]y=3x^2-4x+1[\/latex] when,\r\n<ol>\r\n \t<li>[latex]x=1[\/latex]<\/li>\r\n \t<li>[latex]x=0[\/latex]<\/li>\r\n \t<li>[latex]x=-2[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n1. [latex]y=3x^2-4x+1[\/latex]\u00a0 \u00a0 \u00a0Plug in [latex]x=1[\/latex]\r\n\r\n[latex]\\;\\;y=3(1)^2-4(1)+1=3-4+1=0[\/latex]\r\n\r\n[latex]\\;\\;(1, 0)[\/latex] is a solution.\r\n\r\n&nbsp;\r\n\r\n2.\u00a0[latex]y=3x^2-4x+1[\/latex]\u00a0 \u00a0 \u00a0Plug in [latex]x=0[\/latex]\r\n\r\n[latex]\\;\\;y=3(0)^2-4(0)+1=0-0+1=1[\/latex]\r\n\r\n[latex]\\;\\;(0, 1)[\/latex] is a solution.\r\n\r\n&nbsp;\r\n\r\n3.\u00a0[latex]y=3x^2-4x+1[\/latex]\u00a0 \u00a0 \u00a0Plug in [latex]x=-2[\/latex]\r\n\r\n[latex]\\;\\;y=3(-2)^2-4(-2)+1=3(4)+8+1=12+8+1=21[\/latex]\r\n\r\n[latex]\\;\\;(-2, 21)[\/latex] is a solution.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDetermine the solutions to [latex]y=-2x^2+3x-5[\/latex] when,\r\n<ol>\r\n \t<li>[latex]x=1[\/latex]<\/li>\r\n \t<li>[latex]x=3[\/latex]<\/li>\r\n \t<li>[latex]x=-2[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm572\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm572\"]\r\n<ol>\r\n \t<li>[latex](1, -4)[\/latex]<\/li>\r\n \t<li>[latex](3, 22)[\/latex]<\/li>\r\n \t<li>[latex](-2, -3)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Patterns in Quadratic Growth<\/h2>\r\nIn order to look at patterns within quadratic equations, we will begin by examining the simplest form of a quadratic equation: \u00a0[latex]y=x^2[\/latex]. By examining this particular equation, we can extrapolate information to make generalizations for all quadratic equations.\r\n\r\nGiven an [latex]x[\/latex]-value, we can find the corresponding [latex]y[\/latex]-value by plugging in the [latex]x[\/latex]-value to the equation [latex]y=x^2[\/latex].\r\n\r\nThe following table represents solutions of the equation [latex]y=x^2[\/latex].\r\n<table style=\"border-collapse: collapse; width: 18.0062%; height: 154px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.7075%; height: 22px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 18.5374%; height: 22px;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.7075%; height: 22px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 18.5374%; height: 22px;\">[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.7075%; height: 22px;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 18.5374%; height: 22px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.7075%; height: 22px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 18.5374%; height: 22px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.7075%; height: 22px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 18.5374%; height: 22px;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"width: 18.7075%; height: 16px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 18.5374%; height: 16px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 18.7075%; height: 14px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 18.5374%; height: 14px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 18.7075%; height: 14px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 18.5374%; height: 14px;\">[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTable 1. Table of values of [latex]y=x^2[\/latex]\r\n\r\nIn a linear equation, like [latex]y=mx+b[\/latex], a one unit increase in [latex]x[\/latex] corresponds to a constant change in [latex]y[\/latex]. However, the table of values shows us that for a change of [latex]+1[\/latex] in the [latex]x[\/latex]-value, the change in the [latex]y[\/latex]-value is not constant.\r\n\r\nFor example if [latex]x[\/latex] increases one unit from [latex]1[\/latex] to [latex]2[\/latex], the corresponding change in the value of [latex]y[\/latex] is [latex]+3[\/latex]. However, if\u00a0[latex]x[\/latex] increases one unit from [latex]2[\/latex] to [latex]3[\/latex], the corresponding change in the value of [latex]y[\/latex] is [latex]+5[\/latex]. This demonstrates that the rate of change is not always the same, thus this change is <strong>non-linear<\/strong>. The growth demonstrated in the above table is called\u00a0<strong><em>quadratic growth<\/em>.<\/strong>\r\n<table style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">[caption id=\"attachment_1372\" align=\"alignnone\" width=\"300\"]<img class=\"wp-image-1372 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/22164850\/change-in-x-and-y-values-for-quadratic-from-1-to-2-300x174.jpg\" alt=\"Demonstrating the change in the x and y-values for the quadratic equation from x is 1 to x is 2.\" width=\"300\" height=\"174\" \/> Figure 1: The change in [latex]y[\/latex] for the quadratic equation [latex]y=x^2[\/latex] from [latex]x = 1 \\text{ to } x = 2[\/latex].[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n<div class=\"mceTemp\">[caption id=\"\" align=\"alignnone\" width=\"300\"]<img class=\"wp-image-1373 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/22164853\/change-in-x-and-y-values-for-quadratic-from-2-to-3-300x178.jpg\" alt=\"The change in x and y-values in the quadratic equation from x is 2 to x is 3.\" width=\"300\" height=\"178\" \/> Figure 2: The change in [latex]y[\/latex] for the quadratic equation [latex]y=x^2[\/latex] from [latex]x = 2 \\text{ to } x = 3[\/latex].[\/caption]<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAnother pattern in quadratic growth comes from squaring real numbers. If a real number is squared, the result is always positive. Thus if the [latex]x[\/latex]-value is positive or negative, the corresponding [latex]y[\/latex]-value is always positive. If we consider [latex]x=2[\/latex] this gives us a corresponding [latex]y[\/latex]-value of [latex]4[\/latex]. The [latex]x[\/latex]-value of [latex]-2[\/latex] also gives us a corresponding [latex]y[\/latex]-value of [latex]4[\/latex]. This property of squaring leads to a symmetry within quadratic growth. It gives us symmetrical points on the graph of any quadratic equation. Two sets of symmetrical points are demonstrated in the following table.\r\n<table style=\"border-collapse: collapse; width: 50%; height: 42px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 12.6805%; height: 14px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 11.9159%; height: 14px;\">[latex]y=x^2[\/latex]<\/td>\r\n<td style=\"width: 11.0663%; height: 14px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 12.8505%; height: 14px;\">[latex]y=x^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 12.6805%; height: 14px;\">[latex]-4[\/latex]<\/td>\r\n<td style=\"width: 11.9159%; height: 14px;\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 11.0663%;\">[latex]-5[\/latex]<\/td>\r\n<td style=\"width: 12.8505%;\">[latex]25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 12.6805%;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 11.9159%;\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 11.0663%;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 12.8505%;\">[latex]25[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Graphing [latex]y=x^2[\/latex] from a Table of Values<\/h2>\r\nUsing the table of values for the equation [latex]y=x^2[\/latex] above, we can plot these points in order to see part of the simplest quadratic equation.\r\n\r\n[caption id=\"attachment_1367\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1367\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/22162256\/quadratic-from-values-300x274.png\" alt=\"By plotting the points from the table of values we can see the shape of the quadratic parent function\" width=\"300\" height=\"274\" \/> Figure 3: Graph of the quadratic equation [latex]y=x^2[\/latex] from Table 1.[\/caption]By connecting the points, the shape of a quadratic equation becomes clearer. This slightly U-shape is indicative of all quadratic equations, and is called a\u00a0<strong><em>parabola<\/em>.<\/strong>\r\n\r\n[caption id=\"attachment_1368\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1368\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/22162727\/quadratic-equation-graphed-300x274.png\" alt=\"This image is the graph of the quadratic equation.\" width=\"300\" height=\"274\" \/> Figure 4: The graph of the equation [latex]y=x^2[\/latex].[\/caption]From this graph it is clear that a parabola has symmetry. The line that runs vertically through the middle of the parabola and separates the two symmetrical sides of the parabola is called the\u00a0<em><strong>line of symmetry<\/strong><\/em>. For the graph of [latex]y=x^2[\/latex], the line of symmetry is the vertical line with equation [latex]x=0[\/latex].\r\n\r\n[caption id=\"attachment_1369\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1369\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/22163334\/Quadratic-Equation-with-Line-of-Symmetry-and-Symmetrical-Points-300x274.png\" alt=\"This is a graph of the line of symmetry and two symmetrical points.\" width=\"300\" height=\"274\" \/> Figure 5: Quadratic Equation with the Line of Symmetry and Symmetrical Points[\/caption]\r\n\r\nIn Figure 5, we can see the line of symmetry in red and how it splits the parabola into two mirror image halves. On both sides of the line of symmetry are corresponding points that are also symmetric to one another.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n1. What point is symmetric to [latex](-3, 9)[\/latex] on the graph of [latex]y=x^2[\/latex]?\r\n\r\n2.\u00a0What point is symmetric to [latex](5, 25)[\/latex] on the graph of [latex]y=x^2[\/latex]?\r\n\r\n3.\u00a0What point is symmetric to [latex](0, 0)[\/latex] on the graph of [latex]y=x^2[\/latex]?\r\n<h4><strong>Solution<\/strong><\/h4>\r\n<img class=\"aligncenter size-medium wp-image-2091\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20010150\/yx%5E2-with-points-184x300.png\" alt=\"y=x^2 with points\" width=\"184\" height=\"300\" \/>\r\n<ol>\r\n \t<li>[latex](-3, 9)[\/latex] has a symmetric point at\u00a0[latex](3, 9)[\/latex].<\/li>\r\n \t<li>[latex](5, 25)[\/latex] has a symmetric point at\u00a0[latex](-5, 25)[\/latex].<\/li>\r\n \t<li>[latex](0, 0)[\/latex] has a symmetric point at\u00a0[latex](0, 0)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n1. What point is symmetric to [latex](-4, 16)[\/latex] on the graph of [latex]y=x^2[\/latex]?\r\n\r\n2.\u00a0What point is symmetric to [latex](2, 4)[\/latex] on the graph of [latex]y=x^2[\/latex]?\r\n\r\n3.\u00a0What point is symmetric to [latex](7, 49)[\/latex] on the graph of [latex]y=x^2[\/latex]?\r\n\r\n[reveal-answer q=\"hjm152\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm152\"]\r\n<ol>\r\n \t<li>[latex](4, 16)[\/latex]<\/li>\r\n \t<li>[latex](-2, 4)[\/latex]<\/li>\r\n \t<li>[latex](-7, 49)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice also, that the graph of [latex]y=x^2[\/latex] has a turning point at [latex](0, 0)[\/latex]. The turning point of a parabola is called the <em><strong>vertex<\/strong><\/em>.\r\n\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Describe a quadratic equation<\/li>\n<li>Look for patterns in quadratic growth<\/li>\n<li>Graph the quadratic equation [latex]y=x^2[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>KEYWORDS<\/h3>\n<ul>\n<li><strong>Quadratic<\/strong>: a degree two polynomial of the form [latex]ax^2+bx+c[\/latex]\u00a0where [latex]a, b, c[\/latex] are real numbers with [latex]a\\ne0[\/latex]<\/li>\n<li><strong>Quadratic equation in 2 variables<\/strong>: a degree two equation of the [latex]y=ax^2+bx+c=0[\/latex]\u00a0where [latex]a, b, c[\/latex] are real numbers with [latex]a\\ne0[\/latex]<\/li>\n<li><strong>Parabola<\/strong>: the shape of the graph of a quadratic equation\u00a0[latex]y=ax^2+bx+c[\/latex]<\/li>\n<li><strong>Line of symmetry<\/strong><strong>:\u00a0<\/strong>a line that separates mirror images of each side of a graph<\/li>\n<li><strong>Vertex<\/strong>: the turning point of a parabola<\/li>\n<\/ul>\n<\/div>\n<h2>Quadratic Equations and Solutions<\/h2>\n<p>The single defining feature of a <em><strong>quadratic<\/strong><\/em> is that it is a polynomial of degree two in one variable. This means it takes the form [latex]ax^2+bx+c[\/latex], where [latex]a, b, c[\/latex] are real numbers with [latex]a\\ne0[\/latex]. If we introduce a second variable [latex]y[\/latex] and set it equal to a quadratic, we get a <strong><em>quadratic equation<\/em><\/strong> of the form [latex]y=ax^2+bx+c[\/latex].\u00a0 SInce a quadratic equation has two variables, [latex]x[\/latex] and [latex]y[\/latex], the solutions will be ordered pairs [latex](x, y)[\/latex].<\/p>\n<p>We can discover solutions to a quadratic equations by picking values for [latex]x[\/latex] and finding the corresponding [latex]y[\/latex]-values.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Determine the solutions to [latex]y=3x^2-4x+1[\/latex] when,<\/p>\n<ol>\n<li>[latex]x=1[\/latex]<\/li>\n<li>[latex]x=0[\/latex]<\/li>\n<li>[latex]x=-2[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>1. [latex]y=3x^2-4x+1[\/latex]\u00a0 \u00a0 \u00a0Plug in [latex]x=1[\/latex]<\/p>\n<p>[latex]\\;\\;y=3(1)^2-4(1)+1=3-4+1=0[\/latex]<\/p>\n<p>[latex]\\;\\;(1, 0)[\/latex] is a solution.<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0[latex]y=3x^2-4x+1[\/latex]\u00a0 \u00a0 \u00a0Plug in [latex]x=0[\/latex]<\/p>\n<p>[latex]\\;\\;y=3(0)^2-4(0)+1=0-0+1=1[\/latex]<\/p>\n<p>[latex]\\;\\;(0, 1)[\/latex] is a solution.<\/p>\n<p>&nbsp;<\/p>\n<p>3.\u00a0[latex]y=3x^2-4x+1[\/latex]\u00a0 \u00a0 \u00a0Plug in [latex]x=-2[\/latex]<\/p>\n<p>[latex]\\;\\;y=3(-2)^2-4(-2)+1=3(4)+8+1=12+8+1=21[\/latex]<\/p>\n<p>[latex]\\;\\;(-2, 21)[\/latex] is a solution.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Determine the solutions to [latex]y=-2x^2+3x-5[\/latex] when,<\/p>\n<ol>\n<li>[latex]x=1[\/latex]<\/li>\n<li>[latex]x=3[\/latex]<\/li>\n<li>[latex]x=-2[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm572\">Show Answer<\/span><\/p>\n<div id=\"qhjm572\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex](1, -4)[\/latex]<\/li>\n<li>[latex](3, 22)[\/latex]<\/li>\n<li>[latex](-2, -3)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Patterns in Quadratic Growth<\/h2>\n<p>In order to look at patterns within quadratic equations, we will begin by examining the simplest form of a quadratic equation: \u00a0[latex]y=x^2[\/latex]. By examining this particular equation, we can extrapolate information to make generalizations for all quadratic equations.<\/p>\n<p>Given an [latex]x[\/latex]-value, we can find the corresponding [latex]y[\/latex]-value by plugging in the [latex]x[\/latex]-value to the equation [latex]y=x^2[\/latex].<\/p>\n<p>The following table represents solutions of the equation [latex]y=x^2[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 18.0062%; height: 154px;\">\n<tbody>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.7075%; height: 22px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 18.5374%; height: 22px;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.7075%; height: 22px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 18.5374%; height: 22px;\">[latex]9[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.7075%; height: 22px;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 18.5374%; height: 22px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.7075%; height: 22px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 18.5374%; height: 22px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.7075%; height: 22px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 18.5374%; height: 22px;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 18.7075%; height: 16px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 18.5374%; height: 16px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 18.7075%; height: 14px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 18.5374%; height: 14px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 18.7075%; height: 14px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 18.5374%; height: 14px;\">[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Table 1. Table of values of [latex]y=x^2[\/latex]<\/p>\n<p>In a linear equation, like [latex]y=mx+b[\/latex], a one unit increase in [latex]x[\/latex] corresponds to a constant change in [latex]y[\/latex]. However, the table of values shows us that for a change of [latex]+1[\/latex] in the [latex]x[\/latex]-value, the change in the [latex]y[\/latex]-value is not constant.<\/p>\n<p>For example if [latex]x[\/latex] increases one unit from [latex]1[\/latex] to [latex]2[\/latex], the corresponding change in the value of [latex]y[\/latex] is [latex]+3[\/latex]. However, if\u00a0[latex]x[\/latex] increases one unit from [latex]2[\/latex] to [latex]3[\/latex], the corresponding change in the value of [latex]y[\/latex] is [latex]+5[\/latex]. This demonstrates that the rate of change is not always the same, thus this change is <strong>non-linear<\/strong>. The growth demonstrated in the above table is called\u00a0<strong><em>quadratic growth<\/em>.<\/strong><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1372\" style=\"width: 310px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1372\" class=\"wp-image-1372 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/22164850\/change-in-x-and-y-values-for-quadratic-from-1-to-2-300x174.jpg\" alt=\"Demonstrating the change in the x and y-values for the quadratic equation from x is 1 to x is 2.\" width=\"300\" height=\"174\" \/><\/p>\n<p id=\"caption-attachment-1372\" class=\"wp-caption-text\">Figure 1: The change in [latex]y[\/latex] for the quadratic equation [latex]y=x^2[\/latex] from [latex]x = 1 \\text{ to } x = 2[\/latex].<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div class=\"mceTemp\">\n<div style=\"width: 310px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1373 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/22164853\/change-in-x-and-y-values-for-quadratic-from-2-to-3-300x178.jpg\" alt=\"The change in x and y-values in the quadratic equation from x is 2 to x is 3.\" width=\"300\" height=\"178\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2: The change in [latex]y[\/latex] for the quadratic equation [latex]y=x^2[\/latex] from [latex]x = 2 \\text{ to } x = 3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Another pattern in quadratic growth comes from squaring real numbers. If a real number is squared, the result is always positive. Thus if the [latex]x[\/latex]-value is positive or negative, the corresponding [latex]y[\/latex]-value is always positive. If we consider [latex]x=2[\/latex] this gives us a corresponding [latex]y[\/latex]-value of [latex]4[\/latex]. The [latex]x[\/latex]-value of [latex]-2[\/latex] also gives us a corresponding [latex]y[\/latex]-value of [latex]4[\/latex]. This property of squaring leads to a symmetry within quadratic growth. It gives us symmetrical points on the graph of any quadratic equation. Two sets of symmetrical points are demonstrated in the following table.<\/p>\n<table style=\"border-collapse: collapse; width: 50%; height: 42px;\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 12.6805%; height: 14px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 11.9159%; height: 14px;\">[latex]y=x^2[\/latex]<\/td>\n<td style=\"width: 11.0663%; height: 14px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 12.8505%; height: 14px;\">[latex]y=x^2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 12.6805%; height: 14px;\">[latex]-4[\/latex]<\/td>\n<td style=\"width: 11.9159%; height: 14px;\">[latex]16[\/latex]<\/td>\n<td style=\"width: 11.0663%;\">[latex]-5[\/latex]<\/td>\n<td style=\"width: 12.8505%;\">[latex]25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 12.6805%;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 11.9159%;\">[latex]16[\/latex]<\/td>\n<td style=\"width: 11.0663%;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 12.8505%;\">[latex]25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Graphing [latex]y=x^2[\/latex] from a Table of Values<\/h2>\n<p>Using the table of values for the equation [latex]y=x^2[\/latex] above, we can plot these points in order to see part of the simplest quadratic equation.<\/p>\n<div id=\"attachment_1367\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1367\" class=\"size-medium wp-image-1367\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/22162256\/quadratic-from-values-300x274.png\" alt=\"By plotting the points from the table of values we can see the shape of the quadratic parent function\" width=\"300\" height=\"274\" \/><\/p>\n<p id=\"caption-attachment-1367\" class=\"wp-caption-text\">Figure 3: Graph of the quadratic equation [latex]y=x^2[\/latex] from Table 1.<\/p>\n<\/div>\n<p>By connecting the points, the shape of a quadratic equation becomes clearer. This slightly U-shape is indicative of all quadratic equations, and is called a\u00a0<strong><em>parabola<\/em>.<\/strong><\/p>\n<div id=\"attachment_1368\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1368\" class=\"size-medium wp-image-1368\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/22162727\/quadratic-equation-graphed-300x274.png\" alt=\"This image is the graph of the quadratic equation.\" width=\"300\" height=\"274\" \/><\/p>\n<p id=\"caption-attachment-1368\" class=\"wp-caption-text\">Figure 4: The graph of the equation [latex]y=x^2[\/latex].<\/p>\n<\/div>\n<p>From this graph it is clear that a parabola has symmetry. The line that runs vertically through the middle of the parabola and separates the two symmetrical sides of the parabola is called the\u00a0<em><strong>line of symmetry<\/strong><\/em>. For the graph of [latex]y=x^2[\/latex], the line of symmetry is the vertical line with equation [latex]x=0[\/latex].<\/p>\n<div id=\"attachment_1369\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1369\" class=\"size-medium wp-image-1369\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/22163334\/Quadratic-Equation-with-Line-of-Symmetry-and-Symmetrical-Points-300x274.png\" alt=\"This is a graph of the line of symmetry and two symmetrical points.\" width=\"300\" height=\"274\" \/><\/p>\n<p id=\"caption-attachment-1369\" class=\"wp-caption-text\">Figure 5: Quadratic Equation with the Line of Symmetry and Symmetrical Points<\/p>\n<\/div>\n<p>In Figure 5, we can see the line of symmetry in red and how it splits the parabola into two mirror image halves. On both sides of the line of symmetry are corresponding points that are also symmetric to one another.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>1. What point is symmetric to [latex](-3, 9)[\/latex] on the graph of [latex]y=x^2[\/latex]?<\/p>\n<p>2.\u00a0What point is symmetric to [latex](5, 25)[\/latex] on the graph of [latex]y=x^2[\/latex]?<\/p>\n<p>3.\u00a0What point is symmetric to [latex](0, 0)[\/latex] on the graph of [latex]y=x^2[\/latex]?<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-2091\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20010150\/yx%5E2-with-points-184x300.png\" alt=\"y=x^2 with points\" width=\"184\" height=\"300\" \/><\/p>\n<ol>\n<li>[latex](-3, 9)[\/latex] has a symmetric point at\u00a0[latex](3, 9)[\/latex].<\/li>\n<li>[latex](5, 25)[\/latex] has a symmetric point at\u00a0[latex](-5, 25)[\/latex].<\/li>\n<li>[latex](0, 0)[\/latex] has a symmetric point at\u00a0[latex](0, 0)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>1. What point is symmetric to [latex](-4, 16)[\/latex] on the graph of [latex]y=x^2[\/latex]?<\/p>\n<p>2.\u00a0What point is symmetric to [latex](2, 4)[\/latex] on the graph of [latex]y=x^2[\/latex]?<\/p>\n<p>3.\u00a0What point is symmetric to [latex](7, 49)[\/latex] on the graph of [latex]y=x^2[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm152\">Show Answer<\/span><\/p>\n<div id=\"qhjm152\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex](4, 16)[\/latex]<\/li>\n<li>[latex](-2, 4)[\/latex]<\/li>\n<li>[latex](-7, 49)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice also, that the graph of [latex]y=x^2[\/latex] has a turning point at [latex](0, 0)[\/latex]. The turning point of a parabola is called the <em><strong>vertex<\/strong><\/em>.<\/p>\n<p>&nbsp;<\/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-947\">\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>Revision and Adaptation. <strong>Authored by<\/strong>: Roxanne Brinkerhoff 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>All Examples ; Try It hjm572; hjm152. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":422605,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"Roxanne Brinkerhoff and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"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\":\"\"},{\"type\":\"original\",\"description\":\"All Examples ; Try It hjm572; hjm152\",\"author\":\"\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-947","chapter","type-chapter","status-publish","hentry"],"part":665,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/947","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\/422605"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/947\/revisions"}],"predecessor-version":[{"id":2093,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/947\/revisions\/2093"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/665"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/947\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=947"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=947"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=947"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=947"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}