{"id":2135,"date":"2022-01-28T00:20:47","date_gmt":"2022-01-28T00:20:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=2135"},"modified":"2022-02-02T02:03:51","modified_gmt":"2022-02-02T02:03:51","slug":"9-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/9-3\/","title":{"raw":"9.3: Graphing Transformations","rendered":"9.3: Graphing Transformations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine the role that the values [latex]a, b [\/latex] and [latex]c[\/latex] play in the graph of [latex]y=ax^2+bx+c[\/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>Parent equation<\/strong>: the simplest form of a general equation<\/li>\r\n \t<li><strong>Parabola<\/strong>: the shape of any quadratic equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Graphing Quadratic Equations Using Transformations<\/span>\r\n\r\nIn the previous section we learned that the shape of\u00a0[latex]y=x^2[\/latex] is called a parabola. The turning point on the graph is called the <strong>vertex<\/strong>. The vertical line that passes through the vertex and splits the parabola into two mirror images is called the <strong>line of symmetry<\/strong>. All quadratic functions have graphs in the shape of a parabola.\u00a0 However, in the general equation [latex]y=ax^2+bx+c[\/latex] the values of [latex]a[\/latex],\u00a0[latex]b[\/latex], and\u00a0[latex]c[\/latex] change the direction, shape, and position of the graph.\u00a0 Let's discover how changing these values can transform a graph.\r\n<h3>How\u00a0[latex]a[\/latex] affects the graph of\u00a0[latex]y=ax^{2}+bx+c[\/latex]<\/h3>\r\nFigure 1 shows an animation of the graph [latex]y=ax^2[\/latex] as the value of [latex]a[\/latex] moves between [latex]-10[\/latex] and [latex]10[\/latex].\u00a0 Click on the\u00a0<em>desmos<\/em> logo at the bottom right corner of the graph to manipulate the value of [latex]a[\/latex].\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/omdyntadcw?embed\" width=\"500\" height=\"500\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">Figure 1: Animation of [latex]y=ax^2[\/latex] as [latex]a[\/latex] changes.<\/span><\/p>\r\nThe value of [latex]a[\/latex] tells us whether the parabola opens upwards ([latex]a&gt;0[\/latex]) or downwards ([latex]a&lt;0[\/latex]). If [latex]a[\/latex] is positive, the vertex is the turning point and the lowest point on the graph and the graph opens upward.\u00a0 If [latex]a[\/latex] is negative, the vertex is the turning point and the highest point on the graph and the graph opens downward.\r\n\r\n<img class=\"aligncenter wp-image-2118 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27201953\/parabola-a-values-224x300.png\" alt=\"graph of a parabola with positive and negative values of a\" width=\"224\" height=\"300\" \/>\r\n<p style=\"text-align: center;\">Figure 2. Graph of [latex]y=ax^2[\/latex] with different values of [latex]a[\/latex].<\/p>\r\nThe value of [latex]a[\/latex] also tells us about the width of the graph.\u00a0 When [latex]|a|&gt;1[\/latex], as in [latex]y=2x^2[\/latex] in figure 2, the graph will appear more narrow than\u00a0[latex]y=x^2[\/latex].\u00a0 When [latex]|a|&lt;1[\/latex], as in [latex]y=-\\frac{3}{4} x^2[\/latex] in figure 2, the graph will appear wider than\u00a0\u00a0[latex]y=x^2[\/latex].\r\n<div class=\"Example\">\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<div class=\"Example\">\r\n\r\nMatch each function with its graph.\r\n\r\n1. [latex] \\displaystyle y=3{{x}^{2}}[\/latex]\r\n\r\n2. [latex] \\displaystyle y=-2{{x}^{2}}[\/latex]\r\n\r\n3. [latex] \\displaystyle y=-8{x}^{2}[\/latex]\r\n\r\n4.\u00a0[latex] \\displaystyle y=7{x}^{2}[\/latex]\r\n\r\n[caption id=\"attachment_2100\" align=\"alignleft\" width=\"117\"]<img class=\"wp-image-2100 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20175937\/y7x%5E2-117x300.png\" alt=\"y=7x^2\" width=\"117\" height=\"300\" \/> Graph A[\/caption]\r\n\r\n[caption id=\"attachment_2102\" align=\"alignleft\" width=\"129\"]<img class=\"wp-image-2102 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20175946\/y-2x%5E2-129x300.png\" alt=\"Graph C\" width=\"129\" height=\"300\" \/> Graph B[\/caption]\r\n\r\n[caption id=\"attachment_2101\" align=\"alignleft\" width=\"117\"]<img class=\"wp-image-2101 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20175941\/y-8x%5E2-117x300.png\" alt=\"y=-8x^2\" width=\"117\" height=\"300\" \/> Graph C[\/caption]\r\n\r\n[caption id=\"attachment_2103\" align=\"alignleft\" width=\"136\"]<img class=\"wp-image-2103 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20175950\/y3x%5E2-136x300.png\" alt=\"Graph D\" width=\"136\" height=\"300\" \/> Graph D[\/caption]\r\n<h4 style=\"text-align: left;\">Solution<\/h4>\r\n<p style=\"text-align: left;\">1. [latex] \\displaystyle y=3{{x}^{2}}[\/latex] opens upwards and passes through the point [latex](1, 3)[\/latex]: Graph D<\/p>\r\n&nbsp;\r\n\r\n2. [latex] \\displaystyle y=-2{{x}^{2}}[\/latex]nopens downwards and passes through the point [latex](1, -2)[\/latex]: Graph B\r\n\r\n&nbsp;\r\n\r\n3. [latex] \\displaystyle y=-8{x}^{2}[\/latex] opens downwards and passes through the point [latex](1, -8)[\/latex]: Graph C\r\n\r\n&nbsp;\r\n\r\n4.\u00a0[latex] \\displaystyle y=7{x}^{2}[\/latex] opens upwards and passes through the point [latex](1, 7)[\/latex]: Graph A\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<h3><span style=\"font-size: 1rem; font-weight: normal; orphans: 1; text-align: initial; color: #373d3f;\">We can compare the graphs to the graph of the parent equation [latex]y=x^2[\/latex]. If [latex]|a|&gt;1[\/latex], the graph of [latex]y=ax^2[\/latex] will be thinner than that of [latex]y=x^2[\/latex]. If [latex]0&lt;|a|&lt;1[\/latex], the graph of [latex]y=ax^2[\/latex] will be broader than that of [latex]y=x^2[\/latex]. And remember that when [latex]a&lt;0[\/latex] the graph opens downwards.<\/span><\/h3>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nGiven the graph of [latex]y=x^2[\/latex] shown in black, determine the relative value of [latex]a[\/latex] in the graph of [latex]y=ax^2[\/latex] shown in blue.\r\n\r\n[caption id=\"attachment_2105\" align=\"alignleft\" width=\"145\"]<img class=\"wp-image-2105 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20214525\/y9x%5E2-198x300.png\" alt=\"y=9x^2\" width=\"145\" height=\"220\" \/> Graph 1[\/caption]\r\n\r\n[caption id=\"attachment_2106\" align=\"alignleft\" width=\"147\"]<img class=\"wp-image-2106 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20214625\/y0.2x%5E2-199x300.png\" alt=\"y=0.2x^2\" width=\"147\" height=\"222\" \/> Graph 2[\/caption]\r\n\r\n[caption id=\"attachment_2107\" align=\"alignleft\" width=\"166\"]<img class=\"wp-image-2107 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20214734\/yax%5E2-with-a1-224x300.png\" alt=\"y=ax^2 with a&gt;1\" width=\"166\" height=\"222\" \/> Graph 3[\/caption]\r\n\r\nGraph 1. The blue graph is narrower than the graph of\u00a0[latex]y=x^2[\/latex] and opens upwards, which means that [latex]a&gt;1[\/latex].\r\n\r\n&nbsp;\r\n\r\nGraph 2. The blue graph opens downwards so has a negative value of [latex]a[\/latex].\u00a0 It is also broader that [latex]y=x^2[\/latex] so [latex]|a|&lt;0[\/latex]. This means that [latex]-1&lt;a&lt;0[\/latex].\r\n\r\n&nbsp;\r\n\r\nGraph 3. The blue graph is narrower than\u00a0the graph of\u00a0[latex]y=x^2[\/latex] and opens upwards, which means that [latex]a&gt;1[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDetermine whether each statement is true or false. Support your answer.\r\n\r\n1. The graph of [latex]y=4x^2[\/latex] is narrower than the graph of [latex]y=x^2[\/latex].\r\n\r\n2.\u00a0The graph of [latex]y=-3x^2[\/latex] is broader than the graph of [latex]y=x^2[\/latex].\r\n\r\n3. The graph of [latex]y=-7x^2[\/latex] opens in the same direction as\u00a0[latex]y=x^2[\/latex].\r\n\r\n4.\u00a0The graph of [latex]y=-\\frac{2}{3}x^2[\/latex] opens in the same direction as\u00a0[latex]y=-x^2[\/latex] and is broader than [latex]y=x^2[\/latex].\r\n\r\n[reveal-answer q=\"hjm009\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm009\"]\r\n<ol>\r\n \t<li>True. When [latex]a&gt;1[\/latex] the graph is narrower than the graph of [latex]y=x^2[\/latex].<\/li>\r\n \t<li>True. When [latex]a&lt;-1[\/latex] the graph is narrower than the graph of [latex]y=x^2[\/latex].<\/li>\r\n \t<li>False.\u00a0The graph of [latex]y=-7x^2[\/latex] opens downwards since\u00a0[latex]a&lt;-1[\/latex]. The graph of\u00a0[latex]y=x^2[\/latex].opens upwards.<\/li>\r\n \t<li>True. When [latex]0&lt;a&lt;1[\/latex], the graph is broader than [latex]y=x^2[\/latex] and since both equations have a negative [latex]a[\/latex] value so open downwards.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Think about It<\/h3>\r\nWe said that:\r\n\r\nIf [latex]|a|&gt;1[\/latex], the graph of [latex]y=ax^2[\/latex] will be thinner than that of [latex]y=x^2[\/latex]. If [latex]0&lt;|a|&lt;1[\/latex], the graph of [latex]y=ax^2[\/latex] will be broader than that of [latex]y=x^2[\/latex]. And remember that when [latex]a&lt;0[\/latex] the graph opens downwards.\r\n\r\nWhat about if [latex]a=0[\/latex]? What happens to the graph of [latex]y=ax^2[\/latex] when [latex]a=0[\/latex]?\r\n\r\n[reveal-answer q=\"hjm218\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm218\"]\r\n\r\nWhen [latex]a=0[\/latex], the equation [latex]y=ax^2[\/latex] becomes [latex]y=0[\/latex] whose graph is a horizontal line through the origin. i.e. the [latex]x[\/latex]-axis. The equation is no longer quadratic; it is now a linear equation.\r\n\r\nIndeed, the general equation [latex]y=ax^2+bx+c[\/latex] becomes the linear equation [latex]y=bx+c[\/latex] when [latex]a=0[\/latex] and is no longer quadratic.\r\n\r\nThat is why [latex]a\\neq 0[\/latex] in the quadratic equation [latex]y=ax^2+bx+c[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>How [latex]c[\/latex] affects graphs of quadratic equations<\/h3>\r\nWhen a quadratic equation is in the form\u00a0[latex]y=ax^2+c[\/latex], [latex]b=0[\/latex] and there is no [latex]x[\/latex]-term. Figure 3 shows an animation of the graph [latex]y=x^2+c[\/latex] as the value of [latex]c[\/latex] moves between [latex]-10[\/latex] and [latex]10[\/latex].\u00a0\u00a0Click on the\u00a0<em>desmos<\/em> logo at the bottom right corner of the graph to manipulate the value of [latex]c[\/latex].\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/ypyttckxyu?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 3: Animation of\u00a0<span style=\"color: #000000;\">[latex]y=x^2+c[\/latex] as [latex]c[\/latex] changes.<\/span><\/p>\r\n<p style=\"text-align: left;\">Changing the value of [latex]c[\/latex] moves the parabola up or down the [latex]y[\/latex]-axis. If [latex]c&gt;0[\/latex], the graph of [latex]y=x^2[\/latex] moves up the axis [latex]c[\/latex] units.\u00a0If [latex]c&lt;0[\/latex], the graph of [latex]y=x^2[\/latex] moves down the axis [latex]c[\/latex] units.<\/p>\r\n<img class=\"aligncenter size-medium wp-image-2120\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27203851\/parabola-c-values1-229x300.png\" alt=\"parabola with different c values\" width=\"229\" height=\"300\" \/>\r\n<p style=\"text-align: center;\">Figure 4.\u00a0Graph of [latex]y=x^2+c[\/latex] with different values of [latex]c[\/latex].<\/p>\r\n&nbsp;\r\n<div class=\"examples\">\r\n<h3>Example<\/h3>\r\nExplain how the graph of the given equation is different from the graph of [latex]y=x^2[\/latex].\r\n\r\n1. [latex] \\displaystyle y={{x}^{2}}+3[\/latex]\r\n\r\n2. [latex] \\displaystyle y={{x}^{2}}-4[\/latex]\r\n<h4>Solution<\/h4>\r\n1. The graph of\u00a0[latex] \\displaystyle y={{x}^{2}}+3[\/latex] looks exactly like\u00a0the graph of [latex]y=x^2[\/latex] but it has been moved vertically up by [latex]3[\/latex] units.\r\n\r\n2.\u00a0The graph of [latex] \\displaystyle y={{x}^{2}}-4[\/latex] looks exactly like\u00a0the graph of [latex]y=x^2[\/latex] but it has been moved vertically down by [latex]4[\/latex] units.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nExplain how the graph of the given equation is different from the graph of [latex]y=x^2[\/latex].\r\n\r\n1. [latex]y=x^2-7[\/latex]\r\n\r\n2. [latex]y=x^2+1[\/latex]\r\n\r\n[reveal-answer q=\"hjm312\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm312\"]\r\n<ol>\r\n \t<li>The graph of [latex]y=x^2-7[\/latex] looks exactly like\u00a0the graph of [latex]y=x^2[\/latex] but it has been moved vertically down by [latex]7[\/latex] units.<\/li>\r\n \t<li>The graph of [latex]y=x^2+1[\/latex] looks exactly like\u00a0the graph of [latex]y=x^2[\/latex] but it has been moved vertically up by [latex]1[\/latex] unit.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDetermine the equation of each graph, using the graph of [latex]y=x^2[\/latex] shown in black as the parent graph.\r\n\r\n[caption id=\"attachment_2110\" align=\"alignleft\" width=\"166\"]<img class=\"wp-image-2110 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/21005421\/yx%5E2-10-166x300.png\" alt=\"y=x^2-10\" width=\"166\" height=\"300\" \/> Graph 1[\/caption]\r\n\r\n[caption id=\"attachment_2111\" align=\"aligncenter\" width=\"133\"]<img class=\"wp-image-2111 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/21005425\/yx%5E25-133x300.png\" alt=\"y=x^2+5\" width=\"133\" height=\"300\" \/> Graph 2[\/caption]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"hjm035\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm035\"]\r\n\r\nGraph 1: [latex[y=x^2-10[\/latex]\r\n\r\nGraph 2: [latex]y=x^2+5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen we are graphing any quadratic equation, it is also useful to know that [latex]c[\/latex] corresponds to the [latex]y[\/latex]-intercept of the graph of any quadratic equation.\u00a0 The [latex]y[\/latex]-intercept occurs when [latex]x=0[\/latex] and is the point at which the graph crosses the [latex]y[\/latex]-axis.\u00a0 Recall that we determine the value of [latex]y[\/latex] by substituting [latex]x=0[\/latex] into the equation:\r\n<p style=\"text-align: center;\">[latex]y=a(0)^2+b(0)+c=c[\/latex]<\/p>\r\nFor a quadratic equation, the [latex]y[\/latex]-intercept is always the point [latex](0,c)[\/latex].\r\n\r\n&nbsp;\r\n<h3>How [latex]b[\/latex] affects graphs of quadratic equations<\/h3>\r\nWhen a quadratic equation is in the form\u00a0[latex]y=ax^2+bx[\/latex], [latex]c=0[\/latex] and therefore the [latex]y[\/latex]-intercept will always be [latex](0, 0)[\/latex]. Figure 5 shows an animation of the graph [latex]y=x^2+bx[\/latex] as the value of [latex]b[\/latex] moves between [latex]-10[\/latex] and [latex]10[\/latex].\u00a0\u00a0Click on the\u00a0<em>desmos<\/em> logo at the bottom right corner of the graph to manipulate the value of [latex]b[\/latex].\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/podz4dwm6v?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 5. Animation of [latex]y=x^2+bx[\/latex] as [latex]b[\/latex] changes.<\/p>\r\nChanging [latex]b[\/latex] moves the vertex and the line of symmetry, which is the vertical line that passes through the vertex of the parabola. When [latex]b&gt;0[\/latex] the vertex moves to the left. When\u00a0[latex]b&lt;0[\/latex] the vertex moves to the right.\u00a0 However, the vertex doesn't just move horizontally, but vertically as well. Notice that the vertex moves following a parabolic curve!\u00a0 Moving the vertex also moves the line of symmetry, since the line of symmetry passes through the vertex.\u00a0 In addition, notice that the [latex]x[\/latex]- and [latex]y[\/latex]-intercepts<span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1em;\">stay the same<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0at [latex](0, 0)[\/latex], but the other [latex]x[\/latex]-intercept moves between\u00a0[latex]x=-10[\/latex] when\u00a0[latex]b=10[\/latex] to\u00a0[latex]x=10[\/latex] when\u00a0[latex]b=-10[\/latex].<\/span>\r\n\r\nTo help explain this movement, let's find the\u00a0[latex]x[\/latex]-intercepts by setting\u00a0[latex]y=0[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned} y&amp;=x^2+bx \\\\ 0&amp;=x^2+bx \\;\\;\\quad\\quad\\quad\\quad\\text{Substitute }x=0 \\\\ 0&amp;=x(x+b)\\;\\qquad\\quad\\quad\\text{Factor out the GCF }x\\\\ x&amp;=0\\text{ or }x+b=0\\quad\\quad\\text{Set each equation to zero}\\\\ x&amp;=0\\text{ or }x=-b\\quad\\quad\\quad\\text{Solve each equation}\\end{aligned}\\end{equation}[\/latex]<\/p>\r\nSo the\u00a0[latex]x[\/latex]-intercepts are [latex](0, 0)\\text{ and }(0, -b)[\/latex].\u00a0\u00a0[latex](0, 0)[\/latex] is a fixed point so does not move as the value of [latex]b[\/latex] changes. However, [latex](0, -b)[\/latex] changes as [latex]b[\/latex] changes.\u00a0 For example, when [latex]b=2[\/latex] the intercept changes to [latex](0, -2)[\/latex] and when\u00a0[latex]b=-5[\/latex] the intercept changes to [latex](0, 5)[\/latex]. Do you see the pattern?\r\n\r\n<img class=\"aligncenter size-medium wp-image-2122\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27210900\/parabola-b-values-266x300.png\" alt=\"parabola with different b values\" width=\"266\" height=\"300\" \/>\r\n<p style=\"text-align: center;\">Figure 6.\u00a0Graph of [latex]y=x^2+bx[\/latex] with [latex]x[\/latex]-intercepts at [latex](0, 0)[\/latex] and [latex](-b, 0)[\/latex].<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nState the [latex]x[\/latex]-intercepts of the graph of the equation:\r\n\r\n1. [latex]y=x^2+3x[\/latex]\r\n\r\n2. [latex]y=x^2-7x[\/latex]\r\n\r\n3. [latex]y=x^2+\\frac{3}{4}x[\/latex]\r\n\r\nSolution\r\n\r\nThe\u00a0[latex]x[\/latex]-intercepts of the graph of\u00a0[latex]y=x^2+bx[\/latex] are\u00a0[latex](0, 0)[\/latex] and [latex](\u2013b,0)[\/latex].\r\n\r\n1. Since [latex]b=3[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, \u20133)[\/latex].\r\n\r\n2.\u00a0Since [latex]b=-7[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, 7)[\/latex].\r\n\r\n3. Since [latex]b=\\frac{3}{4}[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, \u2013\\frac{3}{4})[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nState the [latex]x[\/latex]-intercepts of the graph of the equation:\r\n\r\n1. [latex]y=x^2+9x[\/latex]\r\n\r\n2. [latex]y=x^2-4x[\/latex]\r\n\r\n3. [latex]y=x^2+\\frac{4}{5}x[\/latex]\r\n\r\n[reveal-answer q=\"hjm624\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm624\"]\r\n\r\nThe\u00a0[latex]x[\/latex]-intercepts of the graph of\u00a0[latex]y=x^2+bx[\/latex] are\u00a0[latex](0, 0)[\/latex] and [latex](0, \u2013b)[\/latex].\r\n\r\n1. Since [latex]b=0[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, \u20139)[\/latex].\r\n\r\n2.\u00a0Since [latex]b=-4[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, 4)[\/latex].\r\n\r\n3. Since [latex]b=\\frac{4}{5}[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, \u2013\\frac{4}{5})[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nBecause of symmetry, the vertex is always exactly halfway between the two [latex]x[\/latex]-intercepts [latex](0, 0)[\/latex] and [latex](0, \u2013b)[\/latex].\u00a0 For example, in the graph [latex]y=x^2+4x[\/latex] the\u00a0[latex]x[\/latex]-intercepts are [latex](0, 0)[\/latex] and [latex](0, -4)[\/latex] (figure 7). The axis of symmetry is the vertical line that passes through the vertex and lies exactly halfway between [latex]x=-4[\/latex] and [latex]x=0[\/latex]. i.e. the line of symmetry is [latex]x=-2[\/latex].\r\n\r\n[caption id=\"attachment_2113\" align=\"aligncenter\" width=\"224\"]<img class=\"wp-image-2113 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/21013109\/yx%5E24b-with-los-224x300.png\" alt=\"y=x^2+4b with line of symmetry\" width=\"224\" height=\"300\" \/> Figure 7. [latex]y=x^2+4x[\/latex][\/caption]\r\n<p style=\"text-align: left;\">This also means that the vertex has an [latex]x[\/latex]-value of [latex]-2[\/latex] since the line of symmetry passes through the vertex. The [latex]y[\/latex]-coordinate of the vertex is found by substituting\u00a0[latex]x=-2[\/latex] into the equation\u00a0[latex]y=x^2+4x[\/latex]:\u00a0 [latex]y=(-2)^2+4(-2)=4-8=-4[\/latex]. So the vertex is the point [latex](-2, -4)[\/latex] as can be seen on the graph in figure 7.<\/p>\r\nIn the more general case of [latex]y=x^2+bx[\/latex],\u00a0the line of symmetry is exactly halfway between the two [latex]x[\/latex]-intercepts [latex](0, 0)[\/latex] and [latex](0, -b)[\/latex]. i.e. the line of symmetry is [latex]x=\\frac{-b}{2}[\/latex]. Of course, this assumes that [latex]a=1[\/latex] and [latex]c=0[\/latex]. If [latex]a[\/latex] and [latex]c[\/latex] take on different values, determining the exact movement of the graph becomes a little more difficult.\r\n\r\nThe following animation shows what happens to the graph of [latex]y=ax^2+bx+c[\/latex] when the values of [latex]a, b[\/latex] and [latex]c[\/latex] are changed. Click on the\u00a0<em>desmos<\/em> logo at the bottom right corner of the graph to manipulate the values.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/fy4zu6tppg?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 8. Animation of [latex]y=ax^2+bc+c[\/latex] as [a, b[\/latex] and [latex]c[\/latex] change.<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nExplain how the graph of the given equation is different from the graph of [latex]y=x^2[\/latex].\r\n\r\n1. [latex]y=x^2-7[\/latex]\r\n\r\n2. [latex]y=6x^2[\/latex]\r\n\r\n3. [latex]y=-x^2+5[\/latex]\r\n\r\n4. [latex]y=-\\frac{1}{4}x^2+3[\/latex]\r\n\r\n5. [latex]y=4x^2-8[\/latex]\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>Since [latex]a=1[\/latex] the graph looks the same as\u00a0[latex]y=x^2[\/latex] but since [latex]c=-7[\/latex] it has been moved vertically down by 7 units.<\/li>\r\n \t<li>SInce [latex]a=6[\/latex] the graph is thinner than\u00a0[latex]y=x^2[\/latex]. The\u00a0[latex]y[\/latex]-value is 6 times larger than\u00a0[latex]y=x^2[\/latex].<\/li>\r\n \t<li>Since [latex]a=-1[\/latex] is negative the graph looks like an upside down version of\u00a0[latex]y=x^2[\/latex] and since [latex]c=5[\/latex] it has also been moved vertically up by 5 units.<\/li>\r\n \t<li>Since [latex]a=-\\frac{1}{4}[\/latex] the graph is upside down compared to\u00a0[latex]y=x^2[\/latex], is broader than\u00a0[latex]y=x^2[\/latex], and since [latex]c=3[\/latex] it has been moved vertically up by 3 units.<\/li>\r\n \t<li>Since [latex]a=4[\/latex] the graph is thinner than\u00a0[latex]y=x^2[\/latex] and since [latex]c=-8[\/latex]has been moved vertically down by 8 units.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nExplain how the graph of the given equation is different from the graph of [latex]y=x^2[\/latex].\r\n\r\n1. [latex]y=x^2-2[\/latex]\r\n\r\n2. [latex]y=3x^2[\/latex]\r\n\r\n3. [latex]y=-x^2-8[\/latex]\r\n\r\n4. [latex]y=\\frac{2}{3}x^2+5[\/latex]\r\n\r\n5. [latex]y=2x^2+7[\/latex]\r\n\r\n[reveal-answer q=\"hjm279\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm279\"]\r\n<ol>\r\n \t<li>The graph looks the same as [latex]y=x^2[\/latex] but has been moved vertically down by 2 units.<\/li>\r\n \t<li>The graph looks thinner than [latex]y=x^2[\/latex].<\/li>\r\n \t<li>The graph looks like an upside down [latex]y=x^2[\/latex] and has been moved vertically down by 8 units.<\/li>\r\n \t<li>The graph is broader than [latex]y=x^2[\/latex] and has been moved vertically up by 5 units.<\/li>\r\n \t<li>The graph is thinner than [latex]y=x^2[\/latex] but has been moved vertically up by 7 units.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nGiven the [latex]y[\/latex]-intercepts of a parabola, determine the axis of symmetry.\r\n\r\n1. (-4, 0) and (0, 0)\r\n\r\n2. (2, 0) and (8, 0)\r\n\r\n3. (-3, 0) and (7, 0)\r\n\r\nSolution\r\n\r\nThe axis of symmetry runs exactly halfway between the two intercepts.\r\n<ol>\r\n \t<li>There are 4 units between -4 and 0, so the axis is 2 units away from both points: [latex]x=-2[\/latex]<\/li>\r\n \t<li>There are 6 units between 2 and 8, so the axis is 3 units away from both points: [latex]x=5[\/latex]<\/li>\r\n \t<li>There are 10 units between -3 and 7, so the axis is 5 units away from both points: [latex]x=2[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nGiven the [latex]y[\/latex]-intercepts of a parabola, determine the axis of symmetry.\r\n\r\n1. (0, 0) and (4, 0)\r\n\r\n2. (-2, 0) and (4, 0)\r\n\r\n3. (-5, 0) and (1, 0)\r\n\r\n[reveal-answer q=\"hjm748\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm748\"]\r\n\r\nThe axis of symmetry runs exactly halfway between the two intercepts.\r\n<ol>\r\n \t<li>[latex]x=2[\/latex]<\/li>\r\n \t<li>[latex]x=1[\/latex]<\/li>\r\n \t<li>[latex]x=-2[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nGiven two points on a parabola, determine the axis of symmetry.\r\n\r\n1. (3, 4) and (9, 4)\r\n\r\n2. (-2, 1) and (6, 1)\r\n\r\n3. (-2, -5) and (3, -5)\r\n<h4>Solution<\/h4>\r\nSince the [latex]y[\/latex]-values are the same in both points, the points are twins and the axis of symmetry runs exactly halfway between them.\r\n<ol>\r\n \t<li>There are 6 units between 3 and 9, so the axis is 3 units away from both points: [latex]x=6[\/latex]<\/li>\r\n \t<li>There are 8 units between -2 and 6, so the axis is 4 units away from both points: [latex]x=2[\/latex]<\/li>\r\n \t<li>There are 5 units between -2 and 3, so the axis is 2.5 units away from both points: [latex]x=1.5[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nGiven two points on a parabola, determine the axis of symmetry.\r\n\r\n1. (-2, 7) and (11, 7)\r\n\r\n2. (-8, 1) and (-4, 1)\r\n\r\n3. (-7, -8) and (1, -8)\r\n\r\n[reveal-answer q=\"hjm164\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm164\"]\r\n<ol>\r\n \t<li>[latex]x=4.5[\/latex]<\/li>\r\n \t<li>[latex]x=-6[\/latex]<\/li>\r\n \t<li>[latex]x=-3[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine the role that the values [latex]a, b[\/latex] and [latex]c[\/latex] play in the graph of [latex]y=ax^2+bx+c[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Keywords<\/h3>\n<ul>\n<li><strong>Parent equation<\/strong>: the simplest form of a general equation<\/li>\n<li><strong>Parabola<\/strong>: the shape of any quadratic equation<\/li>\n<\/ul>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Graphing Quadratic Equations Using Transformations<\/span><\/p>\n<p>In the previous section we learned that the shape of\u00a0[latex]y=x^2[\/latex] is called a parabola. The turning point on the graph is called the <strong>vertex<\/strong>. The vertical line that passes through the vertex and splits the parabola into two mirror images is called the <strong>line of symmetry<\/strong>. All quadratic functions have graphs in the shape of a parabola.\u00a0 However, in the general equation [latex]y=ax^2+bx+c[\/latex] the values of [latex]a[\/latex],\u00a0[latex]b[\/latex], and\u00a0[latex]c[\/latex] change the direction, shape, and position of the graph.\u00a0 Let&#8217;s discover how changing these values can transform a graph.<\/p>\n<h3>How\u00a0[latex]a[\/latex] affects the graph of\u00a0[latex]y=ax^{2}+bx+c[\/latex]<\/h3>\n<p>Figure 1 shows an animation of the graph [latex]y=ax^2[\/latex] as the value of [latex]a[\/latex] moves between [latex]-10[\/latex] and [latex]10[\/latex].\u00a0 Click on the\u00a0<em>desmos<\/em> logo at the bottom right corner of the graph to manipulate the value of [latex]a[\/latex].<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/omdyntadcw?embed\" width=\"500\" height=\"500\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">Figure 1: Animation of [latex]y=ax^2[\/latex] as [latex]a[\/latex] changes.<\/span><\/p>\n<p>The value of [latex]a[\/latex] tells us whether the parabola opens upwards ([latex]a>0[\/latex]) or downwards ([latex]a<0[\/latex]). If [latex]a[\/latex] is positive, the vertex is the turning point and the lowest point on the graph and the graph opens upward.\u00a0 If [latex]a[\/latex] is negative, the vertex is the turning point and the highest point on the graph and the graph opens downward.\n\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2118 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27201953\/parabola-a-values-224x300.png\" alt=\"graph of a parabola with positive and negative values of a\" width=\"224\" height=\"300\" \/><\/p>\n<p style=\"text-align: center;\">Figure 2. Graph of [latex]y=ax^2[\/latex] with different values of [latex]a[\/latex].<\/p>\n<p>The value of [latex]a[\/latex] also tells us about the width of the graph.\u00a0 When [latex]|a|>1[\/latex], as in [latex]y=2x^2[\/latex] in figure 2, the graph will appear more narrow than\u00a0[latex]y=x^2[\/latex].\u00a0 When [latex]|a|<1[\/latex], as in [latex]y=-\\frac{3}{4} x^2[\/latex] in figure 2, the graph will appear wider than\u00a0\u00a0[latex]y=x^2[\/latex].\n\n\n<div class=\"Example\">\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<div class=\"Example\">\n<p>Match each function with its graph.<\/p>\n<p>1. [latex]\\displaystyle y=3{{x}^{2}}[\/latex]<\/p>\n<p>2. [latex]\\displaystyle y=-2{{x}^{2}}[\/latex]<\/p>\n<p>3. [latex]\\displaystyle y=-8{x}^{2}[\/latex]<\/p>\n<p>4.\u00a0[latex]\\displaystyle y=7{x}^{2}[\/latex]<\/p>\n<div id=\"attachment_2100\" style=\"width: 127px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2100\" class=\"wp-image-2100 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20175937\/y7x%5E2-117x300.png\" alt=\"y=7x^2\" width=\"117\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2100\" class=\"wp-caption-text\">Graph A<\/p>\n<\/div>\n<div id=\"attachment_2102\" style=\"width: 139px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2102\" class=\"wp-image-2102 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20175946\/y-2x%5E2-129x300.png\" alt=\"Graph C\" width=\"129\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2102\" class=\"wp-caption-text\">Graph B<\/p>\n<\/div>\n<div id=\"attachment_2101\" style=\"width: 127px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2101\" class=\"wp-image-2101 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20175941\/y-8x%5E2-117x300.png\" alt=\"y=-8x^2\" width=\"117\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2101\" class=\"wp-caption-text\">Graph C<\/p>\n<\/div>\n<div id=\"attachment_2103\" style=\"width: 146px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2103\" class=\"wp-image-2103 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20175950\/y3x%5E2-136x300.png\" alt=\"Graph D\" width=\"136\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2103\" class=\"wp-caption-text\">Graph D<\/p>\n<\/div>\n<h4 style=\"text-align: left;\">Solution<\/h4>\n<p style=\"text-align: left;\">1. [latex]\\displaystyle y=3{{x}^{2}}[\/latex] opens upwards and passes through the point [latex](1, 3)[\/latex]: Graph D<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\displaystyle y=-2{{x}^{2}}[\/latex]nopens downwards and passes through the point [latex](1, -2)[\/latex]: Graph B<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]\\displaystyle y=-8{x}^{2}[\/latex] opens downwards and passes through the point [latex](1, -8)[\/latex]: Graph C<\/p>\n<p>&nbsp;<\/p>\n<p>4.\u00a0[latex]\\displaystyle y=7{x}^{2}[\/latex] opens upwards and passes through the point [latex](1, 7)[\/latex]: Graph A<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<h3><span style=\"font-size: 1rem; font-weight: normal; orphans: 1; text-align: initial; color: #373d3f;\">We can compare the graphs to the graph of the parent equation [latex]y=x^2[\/latex]. If [latex]|a|>1[\/latex], the graph of [latex]y=ax^2[\/latex] will be thinner than that of [latex]y=x^2[\/latex]. If [latex]0<|a|<1[\/latex], the graph of [latex]y=ax^2[\/latex] will be broader than that of [latex]y=x^2[\/latex]. And remember that when [latex]a<0[\/latex] the graph opens downwards.<\/span><\/h3>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Given the graph of [latex]y=x^2[\/latex] shown in black, determine the relative value of [latex]a[\/latex] in the graph of [latex]y=ax^2[\/latex] shown in blue.<\/p>\n<div id=\"attachment_2105\" style=\"width: 155px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2105\" class=\"wp-image-2105\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20214525\/y9x%5E2-198x300.png\" alt=\"y=9x^2\" width=\"145\" height=\"220\" \/><\/p>\n<p id=\"caption-attachment-2105\" class=\"wp-caption-text\">Graph 1<\/p>\n<\/div>\n<div id=\"attachment_2106\" style=\"width: 157px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2106\" class=\"wp-image-2106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20214625\/y0.2x%5E2-199x300.png\" alt=\"y=0.2x^2\" width=\"147\" height=\"222\" \/><\/p>\n<p id=\"caption-attachment-2106\" class=\"wp-caption-text\">Graph 2<\/p>\n<\/div>\n<div id=\"attachment_2107\" style=\"width: 176px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2107\" class=\"wp-image-2107\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/20214734\/yax%5E2-with-a1-224x300.png\" alt=\"y=ax^2 with a&gt;1\" width=\"166\" height=\"222\" \/><\/p>\n<p id=\"caption-attachment-2107\" class=\"wp-caption-text\">Graph 3<\/p>\n<\/div>\n<p>Graph 1. The blue graph is narrower than the graph of\u00a0[latex]y=x^2[\/latex] and opens upwards, which means that [latex]a>1[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Graph 2. The blue graph opens downwards so has a negative value of [latex]a[\/latex].\u00a0 It is also broader that [latex]y=x^2[\/latex] so [latex]|a|<0[\/latex]. This means that [latex]-1<a<0[\/latex].\n\n&nbsp;\n\nGraph 3. The blue graph is narrower than\u00a0the graph of\u00a0[latex]y=x^2[\/latex] and opens upwards, which means that [latex]a>1[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Determine whether each statement is true or false. Support your answer.<\/p>\n<p>1. The graph of [latex]y=4x^2[\/latex] is narrower than the graph of [latex]y=x^2[\/latex].<\/p>\n<p>2.\u00a0The graph of [latex]y=-3x^2[\/latex] is broader than the graph of [latex]y=x^2[\/latex].<\/p>\n<p>3. The graph of [latex]y=-7x^2[\/latex] opens in the same direction as\u00a0[latex]y=x^2[\/latex].<\/p>\n<p>4.\u00a0The graph of [latex]y=-\\frac{2}{3}x^2[\/latex] opens in the same direction as\u00a0[latex]y=-x^2[\/latex] and is broader than [latex]y=x^2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm009\">Show Answer<\/span><\/p>\n<div id=\"qhjm009\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>True. When [latex]a>1[\/latex] the graph is narrower than the graph of [latex]y=x^2[\/latex].<\/li>\n<li>True. When [latex]a<-1[\/latex] the graph is narrower than the graph of [latex]y=x^2[\/latex].<\/li>\n<li>False.\u00a0The graph of [latex]y=-7x^2[\/latex] opens downwards since\u00a0[latex]a<-1[\/latex]. The graph of\u00a0[latex]y=x^2[\/latex].opens upwards.<\/li>\n<li>True. When [latex]0<a<1[\/latex], the graph is broader than [latex]y=x^2[\/latex] and since both equations have a negative [latex]a[\/latex] value so open downwards.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Think about It<\/h3>\n<p>We said that:<\/p>\n<p>If [latex]|a|>1[\/latex], the graph of [latex]y=ax^2[\/latex] will be thinner than that of [latex]y=x^2[\/latex]. If [latex]0<|a|<1[\/latex], the graph of [latex]y=ax^2[\/latex] will be broader than that of [latex]y=x^2[\/latex]. And remember that when [latex]a<0[\/latex] the graph opens downwards.\n\nWhat about if [latex]a=0[\/latex]? What happens to the graph of [latex]y=ax^2[\/latex] when [latex]a=0[\/latex]?\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm218\">Show Answer<\/span><\/p>\n<div id=\"qhjm218\" class=\"hidden-answer\" style=\"display: none\">\n<p>When [latex]a=0[\/latex], the equation [latex]y=ax^2[\/latex] becomes [latex]y=0[\/latex] whose graph is a horizontal line through the origin. i.e. the [latex]x[\/latex]-axis. The equation is no longer quadratic; it is now a linear equation.<\/p>\n<p>Indeed, the general equation [latex]y=ax^2+bx+c[\/latex] becomes the linear equation [latex]y=bx+c[\/latex] when [latex]a=0[\/latex] and is no longer quadratic.<\/p>\n<p>That is why [latex]a\\neq 0[\/latex] in the quadratic equation [latex]y=ax^2+bx+c[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>How [latex]c[\/latex] affects graphs of quadratic equations<\/h3>\n<p>When a quadratic equation is in the form\u00a0[latex]y=ax^2+c[\/latex], [latex]b=0[\/latex] and there is no [latex]x[\/latex]-term. Figure 3 shows an animation of the graph [latex]y=x^2+c[\/latex] as the value of [latex]c[\/latex] moves between [latex]-10[\/latex] and [latex]10[\/latex].\u00a0\u00a0Click on the\u00a0<em>desmos<\/em> logo at the bottom right corner of the graph to manipulate the value of [latex]c[\/latex].<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/ypyttckxyu?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 3: Animation of\u00a0<span style=\"color: #000000;\">[latex]y=x^2+c[\/latex] as [latex]c[\/latex] changes.<\/span><\/p>\n<p style=\"text-align: left;\">Changing the value of [latex]c[\/latex] moves the parabola up or down the [latex]y[\/latex]-axis. If [latex]c>0[\/latex], the graph of [latex]y=x^2[\/latex] moves up the axis [latex]c[\/latex] units.\u00a0If [latex]c<0[\/latex], the graph of [latex]y=x^2[\/latex] moves down the axis [latex]c[\/latex] units.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-2120\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27203851\/parabola-c-values1-229x300.png\" alt=\"parabola with different c values\" width=\"229\" height=\"300\" \/><\/p>\n<p style=\"text-align: center;\">Figure 4.\u00a0Graph of [latex]y=x^2+c[\/latex] with different values of [latex]c[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div class=\"examples\">\n<h3>Example<\/h3>\n<p>Explain how the graph of the given equation is different from the graph of [latex]y=x^2[\/latex].<\/p>\n<p>1. [latex]\\displaystyle y={{x}^{2}}+3[\/latex]<\/p>\n<p>2. [latex]\\displaystyle y={{x}^{2}}-4[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>1. The graph of\u00a0[latex]\\displaystyle y={{x}^{2}}+3[\/latex] looks exactly like\u00a0the graph of [latex]y=x^2[\/latex] but it has been moved vertically up by [latex]3[\/latex] units.<\/p>\n<p>2.\u00a0The graph of [latex]\\displaystyle y={{x}^{2}}-4[\/latex] looks exactly like\u00a0the graph of [latex]y=x^2[\/latex] but it has been moved vertically down by [latex]4[\/latex] units.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Explain how the graph of the given equation is different from the graph of [latex]y=x^2[\/latex].<\/p>\n<p>1. [latex]y=x^2-7[\/latex]<\/p>\n<p>2. [latex]y=x^2+1[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm312\">Show Answer<\/span><\/p>\n<div id=\"qhjm312\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The graph of [latex]y=x^2-7[\/latex] looks exactly like\u00a0the graph of [latex]y=x^2[\/latex] but it has been moved vertically down by [latex]7[\/latex] units.<\/li>\n<li>The graph of [latex]y=x^2+1[\/latex] looks exactly like\u00a0the graph of [latex]y=x^2[\/latex] but it has been moved vertically up by [latex]1[\/latex] unit.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Determine the equation of each graph, using the graph of [latex]y=x^2[\/latex] shown in black as the parent graph.<\/p>\n<div id=\"attachment_2110\" style=\"width: 176px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2110\" class=\"wp-image-2110 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/21005421\/yx%5E2-10-166x300.png\" alt=\"y=x^2-10\" width=\"166\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2110\" class=\"wp-caption-text\">Graph 1<\/p>\n<\/div>\n<div id=\"attachment_2111\" style=\"width: 143px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2111\" class=\"wp-image-2111 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/21005425\/yx%5E25-133x300.png\" alt=\"y=x^2+5\" width=\"133\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2111\" class=\"wp-caption-text\">Graph 2<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm035\">Show Answer<\/span><\/p>\n<div id=\"qhjm035\" class=\"hidden-answer\" style=\"display: none\">\n<p>Graph 1: [latex]Graph 2: [latex]y=x^2+5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When we are graphing any quadratic equation, it is also useful to know that [latex]c[\/latex] corresponds to the [latex]y[\/latex]-intercept of the graph of any quadratic equation.\u00a0 The [latex]y[\/latex]-intercept occurs when [latex]x=0[\/latex] and is the point at which the graph crosses the [latex]y[\/latex]-axis.\u00a0 Recall that we determine the value of [latex]y[\/latex] by substituting [latex]x=0[\/latex] into the equation:<\/p>\n<p style=\"text-align: center;\">[latex]y=a(0)^2+b(0)+c=c[\/latex]<\/p>\n<p>For a quadratic equation, the [latex]y[\/latex]-intercept is always the point [latex](0,c)[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<h3>How [latex]b[\/latex] affects graphs of quadratic equations<\/h3>\n<p>When a quadratic equation is in the form\u00a0[latex]y=ax^2+bx[\/latex], [latex]c=0[\/latex] and therefore the [latex]y[\/latex]-intercept will always be [latex](0, 0)[\/latex]. Figure 5 shows an animation of the graph [latex]y=x^2+bx[\/latex] as the value of [latex]b[\/latex] moves between [latex]-10[\/latex] and [latex]10[\/latex].\u00a0\u00a0Click on the\u00a0<em>desmos<\/em> logo at the bottom right corner of the graph to manipulate the value of [latex]b[\/latex].<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/podz4dwm6v?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 5. Animation of [latex]y=x^2+bx[\/latex] as [latex]b[\/latex] changes.<\/p>\n<p>Changing [latex]b[\/latex] moves the vertex and the line of symmetry, which is the vertical line that passes through the vertex of the parabola. When [latex]b>0[\/latex] the vertex moves to the left. When\u00a0[latex]b<0[\/latex] the vertex moves to the right.\u00a0 However, the vertex doesn&#8217;t just move horizontally, but vertically as well. Notice that the vertex moves following a parabolic curve!\u00a0 Moving the vertex also moves the line of symmetry, since the line of symmetry passes through the vertex.\u00a0 In addition, notice that the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-intercepts<span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1em;\">stay the same<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0at [latex](0, 0)[\/latex], but the other [latex]x[\/latex]-intercept moves between\u00a0[latex]x=-10[\/latex] when\u00a0[latex]b=10[\/latex] to\u00a0[latex]x=10[\/latex] when\u00a0[latex]b=-10[\/latex].<\/span><\/p>\n<p>To help explain this movement, let&#8217;s find the\u00a0[latex]x[\/latex]-intercepts by setting\u00a0[latex]y=0[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned} y&=x^2+bx \\\\ 0&=x^2+bx \\;\\;\\quad\\quad\\quad\\quad\\text{Substitute }x=0 \\\\ 0&=x(x+b)\\;\\qquad\\quad\\quad\\text{Factor out the GCF }x\\\\ x&=0\\text{ or }x+b=0\\quad\\quad\\text{Set each equation to zero}\\\\ x&=0\\text{ or }x=-b\\quad\\quad\\quad\\text{Solve each equation}\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>So the\u00a0[latex]x[\/latex]-intercepts are [latex](0, 0)\\text{ and }(0, -b)[\/latex].\u00a0\u00a0[latex](0, 0)[\/latex] is a fixed point so does not move as the value of [latex]b[\/latex] changes. However, [latex](0, -b)[\/latex] changes as [latex]b[\/latex] changes.\u00a0 For example, when [latex]b=2[\/latex] the intercept changes to [latex](0, -2)[\/latex] and when\u00a0[latex]b=-5[\/latex] the intercept changes to [latex](0, 5)[\/latex]. Do you see the pattern?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-2122\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/27210900\/parabola-b-values-266x300.png\" alt=\"parabola with different b values\" width=\"266\" height=\"300\" \/><\/p>\n<p style=\"text-align: center;\">Figure 6.\u00a0Graph of [latex]y=x^2+bx[\/latex] with [latex]x[\/latex]-intercepts at [latex](0, 0)[\/latex] and [latex](-b, 0)[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>State the [latex]x[\/latex]-intercepts of the graph of the equation:<\/p>\n<p>1. [latex]y=x^2+3x[\/latex]<\/p>\n<p>2. [latex]y=x^2-7x[\/latex]<\/p>\n<p>3. [latex]y=x^2+\\frac{3}{4}x[\/latex]<\/p>\n<p>Solution<\/p>\n<p>The\u00a0[latex]x[\/latex]-intercepts of the graph of\u00a0[latex]y=x^2+bx[\/latex] are\u00a0[latex](0, 0)[\/latex] and [latex](\u2013b,0)[\/latex].<\/p>\n<p>1. Since [latex]b=3[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, \u20133)[\/latex].<\/p>\n<p>2.\u00a0Since [latex]b=-7[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, 7)[\/latex].<\/p>\n<p>3. Since [latex]b=\\frac{3}{4}[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, \u2013\\frac{3}{4})[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>State the [latex]x[\/latex]-intercepts of the graph of the equation:<\/p>\n<p>1. [latex]y=x^2+9x[\/latex]<\/p>\n<p>2. [latex]y=x^2-4x[\/latex]<\/p>\n<p>3. [latex]y=x^2+\\frac{4}{5}x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm624\">Show Answer<\/span><\/p>\n<div id=\"qhjm624\" class=\"hidden-answer\" style=\"display: none\">\n<p>The\u00a0[latex]x[\/latex]-intercepts of the graph of\u00a0[latex]y=x^2+bx[\/latex] are\u00a0[latex](0, 0)[\/latex] and [latex](0, \u2013b)[\/latex].<\/p>\n<p>1. Since [latex]b=0[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, \u20139)[\/latex].<\/p>\n<p>2.\u00a0Since [latex]b=-4[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, 4)[\/latex].<\/p>\n<p>3. Since [latex]b=\\frac{4}{5}[\/latex], the intercepts are\u00a0[latex](0, 0)[\/latex] and [latex](0, \u2013\\frac{4}{5})[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Because of symmetry, the vertex is always exactly halfway between the two [latex]x[\/latex]-intercepts [latex](0, 0)[\/latex] and [latex](0, \u2013b)[\/latex].\u00a0 For example, in the graph [latex]y=x^2+4x[\/latex] the\u00a0[latex]x[\/latex]-intercepts are [latex](0, 0)[\/latex] and [latex](0, -4)[\/latex] (figure 7). The axis of symmetry is the vertical line that passes through the vertex and lies exactly halfway between [latex]x=-4[\/latex] and [latex]x=0[\/latex]. i.e. the line of symmetry is [latex]x=-2[\/latex].<\/p>\n<div id=\"attachment_2113\" style=\"width: 234px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2113\" class=\"wp-image-2113 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/10\/21013109\/yx%5E24b-with-los-224x300.png\" alt=\"y=x^2+4b with line of symmetry\" width=\"224\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2113\" class=\"wp-caption-text\">Figure 7. [latex]y=x^2+4x[\/latex]<\/p>\n<\/div>\n<p style=\"text-align: left;\">This also means that the vertex has an [latex]x[\/latex]-value of [latex]-2[\/latex] since the line of symmetry passes through the vertex. The [latex]y[\/latex]-coordinate of the vertex is found by substituting\u00a0[latex]x=-2[\/latex] into the equation\u00a0[latex]y=x^2+4x[\/latex]:\u00a0 [latex]y=(-2)^2+4(-2)=4-8=-4[\/latex]. So the vertex is the point [latex](-2, -4)[\/latex] as can be seen on the graph in figure 7.<\/p>\n<p>In the more general case of [latex]y=x^2+bx[\/latex],\u00a0the line of symmetry is exactly halfway between the two [latex]x[\/latex]-intercepts [latex](0, 0)[\/latex] and [latex](0, -b)[\/latex]. i.e. the line of symmetry is [latex]x=\\frac{-b}{2}[\/latex]. Of course, this assumes that [latex]a=1[\/latex] and [latex]c=0[\/latex]. If [latex]a[\/latex] and [latex]c[\/latex] take on different values, determining the exact movement of the graph becomes a little more difficult.<\/p>\n<p>The following animation shows what happens to the graph of [latex]y=ax^2+bx+c[\/latex] when the values of [latex]a, b[\/latex] and [latex]c[\/latex] are changed. Click on the\u00a0<em>desmos<\/em> logo at the bottom right corner of the graph to manipulate the values.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/fy4zu6tppg?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 8. Animation of [latex]y=ax^2+bc+c[\/latex] as [a, b[\/latex] and [latex]c[\/latex] change.<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Explain how the graph of the given equation is different from the graph of [latex]y=x^2[\/latex].<\/p>\n<p>1. [latex]y=x^2-7[\/latex]<\/p>\n<p>2. [latex]y=6x^2[\/latex]<\/p>\n<p>3. [latex]y=-x^2+5[\/latex]<\/p>\n<p>4. [latex]y=-\\frac{1}{4}x^2+3[\/latex]<\/p>\n<p>5. [latex]y=4x^2-8[\/latex]<\/p>\n<h4>Solution<\/h4>\n<ol>\n<li>Since [latex]a=1[\/latex] the graph looks the same as\u00a0[latex]y=x^2[\/latex] but since [latex]c=-7[\/latex] it has been moved vertically down by 7 units.<\/li>\n<li>SInce [latex]a=6[\/latex] the graph is thinner than\u00a0[latex]y=x^2[\/latex]. The\u00a0[latex]y[\/latex]-value is 6 times larger than\u00a0[latex]y=x^2[\/latex].<\/li>\n<li>Since [latex]a=-1[\/latex] is negative the graph looks like an upside down version of\u00a0[latex]y=x^2[\/latex] and since [latex]c=5[\/latex] it has also been moved vertically up by 5 units.<\/li>\n<li>Since [latex]a=-\\frac{1}{4}[\/latex] the graph is upside down compared to\u00a0[latex]y=x^2[\/latex], is broader than\u00a0[latex]y=x^2[\/latex], and since [latex]c=3[\/latex] it has been moved vertically up by 3 units.<\/li>\n<li>Since [latex]a=4[\/latex] the graph is thinner than\u00a0[latex]y=x^2[\/latex] and since [latex]c=-8[\/latex]has been moved vertically down by 8 units.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Explain how the graph of the given equation is different from the graph of [latex]y=x^2[\/latex].<\/p>\n<p>1. [latex]y=x^2-2[\/latex]<\/p>\n<p>2. [latex]y=3x^2[\/latex]<\/p>\n<p>3. [latex]y=-x^2-8[\/latex]<\/p>\n<p>4. [latex]y=\\frac{2}{3}x^2+5[\/latex]<\/p>\n<p>5. [latex]y=2x^2+7[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm279\">Show Answer<\/span><\/p>\n<div id=\"qhjm279\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The graph looks the same as [latex]y=x^2[\/latex] but has been moved vertically down by 2 units.<\/li>\n<li>The graph looks thinner than [latex]y=x^2[\/latex].<\/li>\n<li>The graph looks like an upside down [latex]y=x^2[\/latex] and has been moved vertically down by 8 units.<\/li>\n<li>The graph is broader than [latex]y=x^2[\/latex] and has been moved vertically up by 5 units.<\/li>\n<li>The graph is thinner than [latex]y=x^2[\/latex] but has been moved vertically up by 7 units.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Given the [latex]y[\/latex]-intercepts of a parabola, determine the axis of symmetry.<\/p>\n<p>1. (-4, 0) and (0, 0)<\/p>\n<p>2. (2, 0) and (8, 0)<\/p>\n<p>3. (-3, 0) and (7, 0)<\/p>\n<p>Solution<\/p>\n<p>The axis of symmetry runs exactly halfway between the two intercepts.<\/p>\n<ol>\n<li>There are 4 units between -4 and 0, so the axis is 2 units away from both points: [latex]x=-2[\/latex]<\/li>\n<li>There are 6 units between 2 and 8, so the axis is 3 units away from both points: [latex]x=5[\/latex]<\/li>\n<li>There are 10 units between -3 and 7, so the axis is 5 units away from both points: [latex]x=2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Given the [latex]y[\/latex]-intercepts of a parabola, determine the axis of symmetry.<\/p>\n<p>1. (0, 0) and (4, 0)<\/p>\n<p>2. (-2, 0) and (4, 0)<\/p>\n<p>3. (-5, 0) and (1, 0)<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm748\">Show Answer<\/span><\/p>\n<div id=\"qhjm748\" class=\"hidden-answer\" style=\"display: none\">\n<p>The axis of symmetry runs exactly halfway between the two intercepts.<\/p>\n<ol>\n<li>[latex]x=2[\/latex]<\/li>\n<li>[latex]x=1[\/latex]<\/li>\n<li>[latex]x=-2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Given two points on a parabola, determine the axis of symmetry.<\/p>\n<p>1. (3, 4) and (9, 4)<\/p>\n<p>2. (-2, 1) and (6, 1)<\/p>\n<p>3. (-2, -5) and (3, -5)<\/p>\n<h4>Solution<\/h4>\n<p>Since the [latex]y[\/latex]-values are the same in both points, the points are twins and the axis of symmetry runs exactly halfway between them.<\/p>\n<ol>\n<li>There are 6 units between 3 and 9, so the axis is 3 units away from both points: [latex]x=6[\/latex]<\/li>\n<li>There are 8 units between -2 and 6, so the axis is 4 units away from both points: [latex]x=2[\/latex]<\/li>\n<li>There are 5 units between -2 and 3, so the axis is 2.5 units away from both points: [latex]x=1.5[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Given two points on a parabola, determine the axis of symmetry.<\/p>\n<p>1. (-2, 7) and (11, 7)<\/p>\n<p>2. (-8, 1) and (-4, 1)<\/p>\n<p>3. (-7, -8) and (1, -8)<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm164\">Show Answer<\/span><\/p>\n<div id=\"qhjm164\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]x=4.5[\/latex]<\/li>\n<li>[latex]x=-6[\/latex]<\/li>\n<li>[latex]x=-3[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\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-2135\">\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>Graphing Quadratic Equations Using Transformations. <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 graph created using desmos graphing calculator. <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><\/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":370291,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Graphing Quadratic Equations Using Transformations\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graph created using desmos graphing calculator\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2135","chapter","type-chapter","status-publish","hentry"],"part":665,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2135","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2135\/revisions"}],"predecessor-version":[{"id":2156,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2135\/revisions\/2156"}],"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\/2135\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=2135"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2135"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=2135"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=2135"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}