{"id":841,"date":"2025-07-15T20:52:36","date_gmt":"2025-07-15T20:52:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=841"},"modified":"2026-01-12T21:10:41","modified_gmt":"2026-01-12T21:10:41","slug":"quadratic-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/quadratic-functions-learn-it-4\/","title":{"raw":"Quadratic Functions: Learn It 4","rendered":"Quadratic Functions: Learn It 4"},"content":{"raw":"<h2>Transformations of Quadratic Functions General form<\/h2>\r\nOften times when given quadratic functions they will be in the general form, [latex]f(x)=ax^2+bx+c[\/latex] where [latex]a \\ne 0[\/latex].\r\n<h4>How Changing [latex]a[\/latex] Impacts the Graph of a Parabola<\/h4>\r\n<p class=\"whitespace-pre-wrap break-words\">Changing [latex]a[\/latex] affects the width of the parabola and whether it opens up ([latex]a&gt;0[\/latex]) or down ([latex]a&lt;0[\/latex]).<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>changing [latex]a[\/latex] of a parabola<\/h3>\r\nChanging [latex]a[\/latex] changes the width of the parabola.\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Width of the parabola:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">When [latex]|a| &gt; 1[\/latex], the parabola becomes narrower (steeper)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">When [latex]0 &lt; |a| &lt; 1[\/latex], the parabola becomes wider (flatter)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The larger the absolute value of [latex]a[\/latex], the narrower the parabola<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Direction of opening:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If a [latex]&gt; 0[\/latex], the parabola opens upward (U-shaped)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]a &lt; 0[\/latex], the parabola opens downward (inverted U-shape)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Stretching and compressing:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Multiplying '[latex]a[\/latex]' by a factor stretches the graph vertically by that factor<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Dividing '[latex]a[\/latex]' by a factor compresses the graph vertically by that factor<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section>In the following example, we show how changing the value of [latex]a[\/latex] will affect the graph of the function.\r\n\r\n<section class=\"textbox example\">Match each function with its graph.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex] \\displaystyle f(x)=3{{x}^{2}}[\/latex]<\/li>\r\n \t<li>[latex] \\displaystyle f(x)=-3{{x}^{2}}[\/latex]<\/li>\r\n \t<li>[latex] \\displaystyle f(x)=\\frac{1}{2}{{x}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li><img class=\"alignnone size-full wp-image-12894\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180349\/large-graph-a-1.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\r\n \t<li><img class=\"alignnone size-full wp-image-12892\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180238\/large-graph-b-2.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\r\n \t<li><img class=\"alignnone wp-image-12893\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180315\/large-graph-c.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\r\n<\/ol>\r\n[reveal-answer q=\"850525\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"850525\"]\r\n\r\nFunction [latex]a[\/latex]\u00a0matches graph\u00a0[latex]2[\/latex]\r\n\r\nFunction [latex]b[\/latex]\u00a0matches graph\u00a0[latex]1[\/latex]\r\n\r\nFunction [latex]c[\/latex]\u00a0matches graph\u00a0[latex]3[\/latex]\r\n\r\nFunction [latex]a[\/latex]: [latex] \\displaystyle f(x)=3{{x}^{2}}[\/latex] means that inputs are squared and then multiplied by three, so the outputs will be greater than they would have been for [latex]f(x)=x^2[\/latex]. \u00a0This results in a parabola that has been squeezed, so graph [latex]2[\/latex] is the best match for this function.\r\n\r\n<img class=\"wp-image-12892 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180238\/large-graph-b-2.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/>\r\n\r\n&nbsp;\r\n\r\nFunction [latex]b[\/latex]:\u00a0[latex] \\displaystyle f(x)=-3{{x}^{2}}[\/latex]\u00a0means that inputs are squared and then multiplied by negative three, so the outputs will be farther away from the [latex]x[\/latex]-axis than they would have been for [latex]f(x)=x^2[\/latex], but negative in value, so graph\u00a0[latex]1[\/latex] is the best match for this function.\r\n\r\n<img class=\"wp-image-12894 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180349\/large-graph-a-1.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/>\r\n\r\n&nbsp;\r\n\r\nFunction [latex]c[\/latex]:\u00a0[latex] \\displaystyle f(x)=\\frac{1}{2}{{x}^{2}}[\/latex] means that inputs are squared then multiplied by [latex]\\dfrac{1}{2}[\/latex], so the outputs are less than they would be for\u00a0[latex]f(x)=x^2[\/latex]. \u00a0This results in a parabola that has been opened wider than[latex]f(x)=x^2[\/latex]. Graph\u00a0[latex]3[\/latex] is the best match for this function.\r\n\r\n<img class=\"wp-image-12893 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180315\/large-graph-c.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h4>How Changing [latex]c[\/latex] Impacts the Graph of a Parabola<\/h4>\r\n<p class=\"whitespace-pre-wrap break-words\">Changing [latex]c[\/latex] affects the vertical position of the entire parabola.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>changing [latex]c[\/latex] of a parabola<\/h3>\r\nChanging [latex]c[\/latex] moves the parabola up or down so that the [latex]y[\/latex] intercept is ([latex]0, c[\/latex]).\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]c[\/latex] represents the [latex]y[\/latex]-intercept of the parabola<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Positive [latex]c[\/latex] shifts the parabola up<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Negative c shifts the parabola down<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The magnitude of c determines the amount of vertical shift<\/li>\r\n<\/ul>\r\n<\/section>In the next example, we show how changes to\u00a0[latex]c[\/latex] affect the graph of the function.\r\n\r\n<section class=\"textbox example\">Match each of the following functions with its graph.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex] \\displaystyle f(x)={{x}^{2}}+3[\/latex]<\/li>\r\n \t<li>[latex] \\displaystyle f(x)={{x}^{2}}-3[\/latex]<\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li><img class=\"alignnone wp-image-12898\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180552\/large-graph-d.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\r\n \t<li><img class=\"alignnone wp-image-12899\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180617\/large-graph-g.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\r\n<\/ol>\r\n[reveal-answer q=\"393290\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"393290\"]\r\n\r\nFunction [latex]a[\/latex]: [latex] \\displaystyle f(x)={{x}^{2}}+3[\/latex] means square the inputs then add three, so every output will be moved up [latex]3[\/latex] units. The graph that matches this function best is [latex]2[\/latex].\r\n\r\n<img class=\"wp-image-12899 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180617\/large-graph-g.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/>\r\n\r\n&nbsp;\r\n\r\nFunction\u00a0[latex]b[\/latex]:\u00a0[latex] \\displaystyle f(x)={{x}^{2}}-3[\/latex] \u00a0means square the inputs then subtract\u00a0three, so every output will be moved down\u00a0[latex]3[\/latex] units. The graph that matches this function best is\u00a0[latex]1[\/latex].\r\n\r\n<img class=\"wp-image-12898 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180552\/large-graph-d.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h4>How Changing [latex]b[\/latex] Impacts the Graph of a Parabola<\/h4>\r\nChanging [latex]b[\/latex] moves the line of reflection, which is the vertical line that passes through the vertex ( the high or low point) of the parabola. It may help to know how to calculate the vertex of a parabola to understand how changing the value of [latex]b[\/latex] in a function will change its graph.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>changing [latex]b[\/latex] of a parabola<\/h3>\r\nChanging [latex]b[\/latex] affects the horizontal position of the vertex and the axis of symmetry of the parabola.\r\n<ul>\r\n \t<li>Axis of symmetry:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The axis of symmetry is given by [latex]x=\\dfrac{-b}{2a}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Changing [latex]b[\/latex] shifts this axis left or right<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">A positive [latex]b[\/latex] shifts the vertex left of the y-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">A negative [latex]b[\/latex] shifts the vertex right of the y-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The larger the absolute value of [latex]b[\/latex], the greater the shift<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox recall\">To find the vertex of the parabola, use the formula:<center>[latex] \\displaystyle \\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)[\/latex]<\/center>For example, if the function being considered is [latex]f(x)=2x^2-3x+4[\/latex], to find the vertex, first calculate [latex]\\Large\\frac{-b}{2a}[\/latex].\r\n[latex]\\\\[\/latex]\r\n[latex]a = 2[\/latex], and [latex]b = -3[\/latex], therefore,<center>\u00a0[latex]\\dfrac{-b}{2a}=\\dfrac{-(-3)}{2(2)}=\\dfrac{3}{4}[\/latex]<\/center>\r\nThis is the [latex]x[\/latex] value of the vertex.\r\n[latex]\\\\[\/latex]\r\nNow evaluate the function at [latex]x =\\Large\\frac{3}{4}[\/latex] to get the corresponding [latex]y[\/latex]-value for the vertex.<center>[latex]f\\left( \\dfrac{-b}{2a} \\right)=2\\left(\\dfrac{3}{4}\\right)^2-3\\left(\\dfrac{3}{4}\\right)+4=2\\left(\\dfrac{9}{16}\\right)-\\dfrac{9}{4}+4=\\dfrac{18}{16}-\\dfrac{9}{4}+4=\\dfrac{9}{8}-\\dfrac{9}{4}+4=\\dfrac{9}{8}-\\dfrac{18}{8}+\\dfrac{32}{8}=\\dfrac{23}{8}[\/latex]<\/center>The vertex is at the point [latex]\\left(\\dfrac{3}{4},\\dfrac{23}{8}\\right)[\/latex]. \u00a0This means that the vertical line of reflection passes through this point as well.<\/section><section class=\"textbox proTip\">It is not easy to tell how changing the values for [latex]b[\/latex] will change the graph of a quadratic function, but if you find the vertex, you can tell how the graph will change.<\/section>In the next example, we show how changing [latex]b[\/latex] can change the graph of the quadratic function.\r\n\r\n<section class=\"textbox example\">Match each of the following functions with its graph.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex] \\displaystyle f(x)={{x}^{2}}+2x[\/latex]<\/li>\r\n \t<li>[latex] \\displaystyle f(x)={{x}^{2}}-2x[\/latex]<\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li><img class=\"alignnone wp-image-12900\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180715\/large-graph-e.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\r\n \t<li><img class=\"alignnone wp-image-12901\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180738\/large-graph-f.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\r\n<\/ol>\r\n[reveal-answer q=\"857922\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"857922\"]\r\n\r\nFind the vertex of function [latex]a[\/latex]:\r\n\r\n<center>[latex] \\displaystyle f(x)={{x}^{2}}+2x[\/latex].<\/center><center>[latex]a = 1, b = 2[\/latex]<\/center>[latex]x[\/latex]-value:\r\n\r\n<center>[latex]\\dfrac{-b}{2a}=\\dfrac{-2}{2(1)}=-1[\/latex]<\/center>[latex]y[\/latex]-value:\r\n\r\n<center>[latex]f(\\dfrac{-b}{2a})=(-1)^2+2(-1)=1-2=-1[\/latex].<\/center>Vertex = [latex](-1,-1)[\/latex], which means the graph that best fits this function is graph 1\r\n\r\n<img class=\"wp-image-12900 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180715\/large-graph-e.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/>\r\n\r\n&nbsp;\r\n\r\nFind the vertex of function [latex]b[\/latex]:\r\n\r\n<center>[latex]f(x)={{x}^{2}}-2x[\/latex].<\/center><center>[latex]a = 1, b = -2[\/latex]<\/center>[latex]x[\/latex]-value:\r\n\r\n<center>[latex]\\dfrac{-b}{2a}=\\dfrac{2}{2(1)}=1[\/latex]<\/center>[latex]y[\/latex]-value:\r\n\r\n<center>[latex]f(\\dfrac{-b}{2a})=(1)^2-2(1)=1-2=-1[\/latex].<\/center>Vertex = [latex](1,-1)[\/latex], which means the graph that best fits this function is graph 2\r\n\r\n<img class=\"wp-image-12901 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180738\/large-graph-f.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\">Note that the vertex can change if the value for [latex]c[\/latex] changes because the [latex]y[\/latex]-value of the vertex is calculated by substituting the [latex]x[\/latex]-value into the function.<\/section><section class=\"textbox example\">Graph [latex]f(x)=\u22122x^{2}+3x\u20133[\/latex].<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[reveal-answer q=\"992003\"]Show Solution[\/reveal-answer][hidden-answer a=\"992003\"]Before making a table of values, look at the values of [latex]a[\/latex] and [latex]c[\/latex] to get a general idea of what the graph should look like.<\/span>\r\n\r\n<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\\\[\/latex]<\/span>\r\n\r\n<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]a=\u22122[\/latex], so the graph will open down and be thinner than [latex]f(x)=x^{2}[\/latex].<\/span>\r\n\r\n<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\\\[\/latex]\r\n[latex]c=\u22123[\/latex], so it will move to intersect the [latex]y[\/latex]-axis at\u00a0[latex](0,\u22123)[\/latex].\r\n[latex]\\\\[\/latex]\r\nFinding the vertex may make graphing the parabola easier. To find the vertex of the parabola, use the formula [latex] \\displaystyle \\left( \\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a} \\right) \\right)[\/latex].<\/span>\r\n<p style=\"text-align: center;\">[latex]\\text{Vertex }\\text{formula}=\\left( \\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a} \\right) \\right)[\/latex]<\/p>\r\n[latex]x[\/latex]-coordinate of vertex:\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\dfrac{-b}{2a}=\\dfrac{-(3)}{2(-2)}=\\dfrac{-3}{-4}=\\dfrac{3}{4}[\/latex]<\/p>\r\n[latex]y[\/latex]-coordinate of vertex:\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}f\\left( \\dfrac{-b}{2a} \\right)=f\\left( \\dfrac{3}{4} \\right)\\\\\\,\\,\\,f\\left( \\dfrac{3}{4} \\right)=-2{{\\left( \\dfrac{3}{4} \\right)}^{2}}+3\\left( \\dfrac{3}{4} \\right)-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2\\left( \\dfrac{9}{16} \\right)+\\dfrac{9}{4}-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\dfrac{-18}{16}+\\dfrac{9}{4}-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\dfrac{-9}{8}+\\dfrac{18}{8}-\\dfrac{24}{8}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-\\dfrac{15}{8}\\end{array}[\/latex]<\/p>\r\nVertex: [latex] \\displaystyle \\left( \\dfrac{3}{4},-\\dfrac{15}{8} \\right)[\/latex]\r\n\r\nUse the vertex, [latex] \\displaystyle \\left( \\dfrac{3}{4},-\\dfrac{15}{8} \\right)[\/latex], and the properties you described to get a general idea of the shape of the graph. You can create a table of values to verify your graph. Notice that in this table, the [latex]x[\/latex] values increase. The [latex]y[\/latex] values increase and then start to decrease again. This indicates a parabola.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]f(x)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]\u22122[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\u221217[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\u22128[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\u22123[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\u22122[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\u22125[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232454\/image024.gif\" alt=\"Vertex at negative three-fourths, negative 15-eighths. Other points are plotted: the point negative 2, negative 17; the point negative 1, negative 8; the point 0, negative 3; the point 1, negative 2; and the point 2, negative 5.\" width=\"405\" height=\"450\" \/><\/center>&nbsp;\r\n\r\nConnect the points as best you can using a <i>smooth curve<\/i>. Remember that the parabola is two mirror images, so if your points do not have pairs with the same value, you may want to include additional points (such as the ones in blue shown below). Plot points on either side of the vertex.\r\n\r\n[latex]x=\\Large\\frac{1}{2}[\/latex] and [latex]x=\\Large\\frac{3}{2}[\/latex] are good values to include.\r\n\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232456\/image025.gif\" alt=\"A parabola drawn through the points in the previous graph\" width=\"405\" height=\"450\" \/><\/center>[\/hidden-answer]\r\n\r\n<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">\u00a0<\/span>\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question height=\"400\" hide_question_numbers=1]318788[\/ohm_question]<\/section>","rendered":"<h2>Transformations of Quadratic Functions General form<\/h2>\n<p>Often times when given quadratic functions they will be in the general form, [latex]f(x)=ax^2+bx+c[\/latex] where [latex]a \\ne 0[\/latex].<\/p>\n<h4>How Changing [latex]a[\/latex] Impacts the Graph of a Parabola<\/h4>\n<p class=\"whitespace-pre-wrap break-words\">Changing [latex]a[\/latex] affects the width of the parabola and whether it opens up ([latex]a>0[\/latex]) or down ([latex]a<0[\/latex]).<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>changing [latex]a[\/latex] of a parabola<\/h3>\n<p>Changing [latex]a[\/latex] changes the width of the parabola.<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Width of the parabola:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">When [latex]|a| > 1[\/latex], the parabola becomes narrower (steeper)<\/li>\n<li class=\"whitespace-normal break-words\">When [latex]0 < |a| < 1[\/latex], the parabola becomes wider (flatter)<\/li>\n<li class=\"whitespace-normal break-words\">The larger the absolute value of [latex]a[\/latex], the narrower the parabola<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Direction of opening:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If a [latex]> 0[\/latex], the parabola opens upward (U-shaped)<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]a < 0[\/latex], the parabola opens downward (inverted U-shape)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Stretching and compressing:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Multiplying &#8216;[latex]a[\/latex]&#8216; by a factor stretches the graph vertically by that factor<\/li>\n<li class=\"whitespace-normal break-words\">Dividing &#8216;[latex]a[\/latex]&#8216; by a factor compresses the graph vertically by that factor<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/section>\n<p>In the following example, we show how changing the value of [latex]a[\/latex] will affect the graph of the function.<\/p>\n<section class=\"textbox example\">Match each function with its graph.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\displaystyle f(x)=3{{x}^{2}}[\/latex]<\/li>\n<li>[latex]\\displaystyle f(x)=-3{{x}^{2}}[\/latex]<\/li>\n<li>[latex]\\displaystyle f(x)=\\frac{1}{2}{{x}^{2}}[\/latex]<\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12894\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180349\/large-graph-a-1.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12892\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180238\/large-graph-b-2.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12893\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180315\/large-graph-c.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q850525\">Show Answer<\/button><\/p>\n<div id=\"q850525\" class=\"hidden-answer\" style=\"display: none\">\n<p>Function [latex]a[\/latex]\u00a0matches graph\u00a0[latex]2[\/latex]<\/p>\n<p>Function [latex]b[\/latex]\u00a0matches graph\u00a0[latex]1[\/latex]<\/p>\n<p>Function [latex]c[\/latex]\u00a0matches graph\u00a0[latex]3[\/latex]<\/p>\n<p>Function [latex]a[\/latex]: [latex]\\displaystyle f(x)=3{{x}^{2}}[\/latex] means that inputs are squared and then multiplied by three, so the outputs will be greater than they would have been for [latex]f(x)=x^2[\/latex]. \u00a0This results in a parabola that has been squeezed, so graph [latex]2[\/latex] is the best match for this function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12892 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180238\/large-graph-b-2.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Function [latex]b[\/latex]:\u00a0[latex]\\displaystyle f(x)=-3{{x}^{2}}[\/latex]\u00a0means that inputs are squared and then multiplied by negative three, so the outputs will be farther away from the [latex]x[\/latex]-axis than they would have been for [latex]f(x)=x^2[\/latex], but negative in value, so graph\u00a0[latex]1[\/latex] is the best match for this function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12894 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180349\/large-graph-a-1.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Function [latex]c[\/latex]:\u00a0[latex]\\displaystyle f(x)=\\frac{1}{2}{{x}^{2}}[\/latex] means that inputs are squared then multiplied by [latex]\\dfrac{1}{2}[\/latex], so the outputs are less than they would be for\u00a0[latex]f(x)=x^2[\/latex]. \u00a0This results in a parabola that has been opened wider than[latex]f(x)=x^2[\/latex]. Graph\u00a0[latex]3[\/latex] is the best match for this function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12893 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180315\/large-graph-c.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<h4>How Changing [latex]c[\/latex] Impacts the Graph of a Parabola<\/h4>\n<p class=\"whitespace-pre-wrap break-words\">Changing [latex]c[\/latex] affects the vertical position of the entire parabola.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>changing [latex]c[\/latex] of a parabola<\/h3>\n<p>Changing [latex]c[\/latex] moves the parabola up or down so that the [latex]y[\/latex] intercept is ([latex]0, c[\/latex]).<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]c[\/latex] represents the [latex]y[\/latex]-intercept of the parabola<\/li>\n<li class=\"whitespace-normal break-words\">Positive [latex]c[\/latex] shifts the parabola up<\/li>\n<li class=\"whitespace-normal break-words\">Negative c shifts the parabola down<\/li>\n<li class=\"whitespace-normal break-words\">The magnitude of c determines the amount of vertical shift<\/li>\n<\/ul>\n<\/section>\n<p>In the next example, we show how changes to\u00a0[latex]c[\/latex] affect the graph of the function.<\/p>\n<section class=\"textbox example\">Match each of the following functions with its graph.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\displaystyle f(x)={{x}^{2}}+3[\/latex]<\/li>\n<li>[latex]\\displaystyle f(x)={{x}^{2}}-3[\/latex]<\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12898\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180552\/large-graph-d.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12899\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180617\/large-graph-g.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q393290\">Show Solution<\/button><\/p>\n<div id=\"q393290\" class=\"hidden-answer\" style=\"display: none\">\n<p>Function [latex]a[\/latex]: [latex]\\displaystyle f(x)={{x}^{2}}+3[\/latex] means square the inputs then add three, so every output will be moved up [latex]3[\/latex] units. The graph that matches this function best is [latex]2[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12899 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180617\/large-graph-g.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Function\u00a0[latex]b[\/latex]:\u00a0[latex]\\displaystyle f(x)={{x}^{2}}-3[\/latex] \u00a0means square the inputs then subtract\u00a0three, so every output will be moved down\u00a0[latex]3[\/latex] units. The graph that matches this function best is\u00a0[latex]1[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12898 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180552\/large-graph-d.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<h4>How Changing [latex]b[\/latex] Impacts the Graph of a Parabola<\/h4>\n<p>Changing [latex]b[\/latex] moves the line of reflection, which is the vertical line that passes through the vertex ( the high or low point) of the parabola. It may help to know how to calculate the vertex of a parabola to understand how changing the value of [latex]b[\/latex] in a function will change its graph.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>changing [latex]b[\/latex] of a parabola<\/h3>\n<p>Changing [latex]b[\/latex] affects the horizontal position of the vertex and the axis of symmetry of the parabola.<\/p>\n<ul>\n<li>Axis of symmetry:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The axis of symmetry is given by [latex]x=\\dfrac{-b}{2a}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Changing [latex]b[\/latex] shifts this axis left or right<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">A positive [latex]b[\/latex] shifts the vertex left of the y-axis<\/li>\n<li class=\"whitespace-normal break-words\">A negative [latex]b[\/latex] shifts the vertex right of the y-axis<\/li>\n<li class=\"whitespace-normal break-words\">The larger the absolute value of [latex]b[\/latex], the greater the shift<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\">To find the vertex of the parabola, use the formula:<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)[\/latex]<\/div>\n<p>For example, if the function being considered is [latex]f(x)=2x^2-3x+4[\/latex], to find the vertex, first calculate [latex]\\Large\\frac{-b}{2a}[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\n[latex]a = 2[\/latex], and [latex]b = -3[\/latex], therefore,<\/p>\n<div style=\"text-align: center;\">\u00a0[latex]\\dfrac{-b}{2a}=\\dfrac{-(-3)}{2(2)}=\\dfrac{3}{4}[\/latex]<\/div>\n<p>This is the [latex]x[\/latex] value of the vertex.<br \/>\n[latex]\\\\[\/latex]<br \/>\nNow evaluate the function at [latex]x =\\Large\\frac{3}{4}[\/latex] to get the corresponding [latex]y[\/latex]-value for the vertex.<\/p>\n<div style=\"text-align: center;\">[latex]f\\left( \\dfrac{-b}{2a} \\right)=2\\left(\\dfrac{3}{4}\\right)^2-3\\left(\\dfrac{3}{4}\\right)+4=2\\left(\\dfrac{9}{16}\\right)-\\dfrac{9}{4}+4=\\dfrac{18}{16}-\\dfrac{9}{4}+4=\\dfrac{9}{8}-\\dfrac{9}{4}+4=\\dfrac{9}{8}-\\dfrac{18}{8}+\\dfrac{32}{8}=\\dfrac{23}{8}[\/latex]<\/div>\n<p>The vertex is at the point [latex]\\left(\\dfrac{3}{4},\\dfrac{23}{8}\\right)[\/latex]. \u00a0This means that the vertical line of reflection passes through this point as well.<\/section>\n<section class=\"textbox proTip\">It is not easy to tell how changing the values for [latex]b[\/latex] will change the graph of a quadratic function, but if you find the vertex, you can tell how the graph will change.<\/section>\n<p>In the next example, we show how changing [latex]b[\/latex] can change the graph of the quadratic function.<\/p>\n<section class=\"textbox example\">Match each of the following functions with its graph.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\displaystyle f(x)={{x}^{2}}+2x[\/latex]<\/li>\n<li>[latex]\\displaystyle f(x)={{x}^{2}}-2x[\/latex]<\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12900\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180715\/large-graph-e.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12901\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180738\/large-graph-f.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q857922\">Show Answer<\/button><\/p>\n<div id=\"q857922\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the vertex of function [latex]a[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle f(x)={{x}^{2}}+2x[\/latex].<\/div>\n<div style=\"text-align: center;\">[latex]a = 1, b = 2[\/latex]<\/div>\n<p>[latex]x[\/latex]-value:<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{-b}{2a}=\\dfrac{-2}{2(1)}=-1[\/latex]<\/div>\n<p>[latex]y[\/latex]-value:<\/p>\n<div style=\"text-align: center;\">[latex]f(\\dfrac{-b}{2a})=(-1)^2+2(-1)=1-2=-1[\/latex].<\/div>\n<p>Vertex = [latex](-1,-1)[\/latex], which means the graph that best fits this function is graph 1<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12900 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180715\/large-graph-e.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Find the vertex of function [latex]b[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]f(x)={{x}^{2}}-2x[\/latex].<\/div>\n<div style=\"text-align: center;\">[latex]a = 1, b = -2[\/latex]<\/div>\n<p>[latex]x[\/latex]-value:<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{-b}{2a}=\\dfrac{2}{2(1)}=1[\/latex]<\/div>\n<p>[latex]y[\/latex]-value:<\/p>\n<div style=\"text-align: center;\">[latex]f(\\dfrac{-b}{2a})=(1)^2-2(1)=1-2=-1[\/latex].<\/div>\n<p>Vertex = [latex](1,-1)[\/latex], which means the graph that best fits this function is graph 2<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12901 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/10\/12180738\/large-graph-f.png\" alt=\"compared to g(x)=x squared\" width=\"300\" height=\"300\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">Note that the vertex can change if the value for [latex]c[\/latex] changes because the [latex]y[\/latex]-value of the vertex is calculated by substituting the [latex]x[\/latex]-value into the function.<\/section>\n<section class=\"textbox example\">Graph [latex]f(x)=\u22122x^{2}+3x\u20133[\/latex].<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q992003\">Show Solution<\/button><\/p>\n<div id=\"q992003\" class=\"hidden-answer\" style=\"display: none\">Before making a table of values, look at the values of [latex]a[\/latex] and [latex]c[\/latex] to get a general idea of what the graph should look like.<\/span><\/p>\n<p><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\\\[\/latex]<\/span><\/p>\n<p><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]a=\u22122[\/latex], so the graph will open down and be thinner than [latex]f(x)=x^{2}[\/latex].<\/span><\/p>\n<p><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\\\[\/latex]<br \/>\n[latex]c=\u22123[\/latex], so it will move to intersect the [latex]y[\/latex]-axis at\u00a0[latex](0,\u22123)[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\nFinding the vertex may make graphing the parabola easier. To find the vertex of the parabola, use the formula [latex]\\displaystyle \\left( \\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a} \\right) \\right)[\/latex].<\/span><\/p>\n<p style=\"text-align: center;\">[latex]\\text{Vertex }\\text{formula}=\\left( \\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a} \\right) \\right)[\/latex]<\/p>\n<p>[latex]x[\/latex]-coordinate of vertex:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\dfrac{-b}{2a}=\\dfrac{-(3)}{2(-2)}=\\dfrac{-3}{-4}=\\dfrac{3}{4}[\/latex]<\/p>\n<p>[latex]y[\/latex]-coordinate of vertex:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}f\\left( \\dfrac{-b}{2a} \\right)=f\\left( \\dfrac{3}{4} \\right)\\\\\\,\\,\\,f\\left( \\dfrac{3}{4} \\right)=-2{{\\left( \\dfrac{3}{4} \\right)}^{2}}+3\\left( \\dfrac{3}{4} \\right)-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2\\left( \\dfrac{9}{16} \\right)+\\dfrac{9}{4}-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\dfrac{-18}{16}+\\dfrac{9}{4}-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\dfrac{-9}{8}+\\dfrac{18}{8}-\\dfrac{24}{8}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-\\dfrac{15}{8}\\end{array}[\/latex]<\/p>\n<p>Vertex: [latex]\\displaystyle \\left( \\dfrac{3}{4},-\\dfrac{15}{8} \\right)[\/latex]<\/p>\n<p>Use the vertex, [latex]\\displaystyle \\left( \\dfrac{3}{4},-\\dfrac{15}{8} \\right)[\/latex], and the properties you described to get a general idea of the shape of the graph. You can create a table of values to verify your graph. Notice that in this table, the [latex]x[\/latex] values increase. The [latex]y[\/latex] values increase and then start to decrease again. This indicates a parabola.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center;\">[latex]x[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]f(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22122[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u221217[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u22128[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u22122[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\u22125[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232454\/image024.gif\" alt=\"Vertex at negative three-fourths, negative 15-eighths. Other points are plotted: the point negative 2, negative 17; the point negative 1, negative 8; the point 0, negative 3; the point 1, negative 2; and the point 2, negative 5.\" width=\"405\" height=\"450\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>Connect the points as best you can using a <i>smooth curve<\/i>. Remember that the parabola is two mirror images, so if your points do not have pairs with the same value, you may want to include additional points (such as the ones in blue shown below). Plot points on either side of the vertex.<\/p>\n<p>[latex]x=\\Large\\frac{1}{2}[\/latex] and [latex]x=\\Large\\frac{3}{2}[\/latex] are good values to include.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232456\/image025.gif\" alt=\"A parabola drawn through the points in the previous graph\" width=\"405\" height=\"450\" \/><\/div>\n<\/div>\n<\/div>\n<p><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">\u00a0<\/span><\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318788\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318788&theme=lumen&iframe_resize_id=ohm318788&source=tnh\" width=\"100%\" height=\"400\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/841"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/841\/revisions"}],"predecessor-version":[{"id":5302,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/841\/revisions\/5302"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/841\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=841"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=841"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=841"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=841"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}