{"id":184,"date":"2023-06-21T13:22:41","date_gmt":"2023-06-21T13:22:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/the-graph-of-a-quadratic-function\/"},"modified":"2024-01-08T19:08:16","modified_gmt":"2024-01-08T19:08:16","slug":"the-graph-of-a-quadratic-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/the-graph-of-a-quadratic-function\/","title":{"raw":"R3.3   The Graph of a Quadratic Function","rendered":"R3.3   The Graph of a Quadratic Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph quadratic functions using tables and transformations<\/li>\r\n \t<li>Identify important features of\u00a0the graph of a quadratic function of the form [latex]f(x)=ax^2+bx+c[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\nQuadratic functions form parabolas when graphed in the coordinate plane, so you can be sure fairly quickly when graphing a quadratic if you have the correct shape. As with lines in the plane, creating a table of input and output values then plotting points will reveal the shape. But unlike straight lines between points, the parabola is a smooth curve.\u00a0 Later in this module, you'll learn some other good methods for sketching a quick, accurate graph of a quadratic equation.\r\n\r\nIt is helpful to have an idea about what the shape of the graph of a quadratic function should be so you can be sure that you have chosen enough points to plot as a guide when sketching the graph. Let's start with the most basic quadratic function,\u00a0[latex]f(x)=x^{2}[\/latex].\r\nGraph [latex]f(x)=x^{2}[\/latex].\r\nStart with a table of values. Then think of each row of the table as an ordered pair.\r\n<table style=\"width: 20%;\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\"><i>x<\/i><\/th>\r\n<th style=\"text-align: center;\"><i>f<\/i>(<i>x<\/i>)<\/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]4[\/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]1[\/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]0[\/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]1[\/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]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the points [latex](-2,4), (-1,1), (0,0), (1,1), (2,4)[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232437\/image013.gif\" alt=\"Graph with the point negative 2, 4; the point negative 1, 1; the point 0, 0; the point 1,1; the point 2,4.\" width=\"322\" height=\"353\" \/>\r\n\r\nSince the points are <i>not<\/i> on a line, you cannot use a straight edge. Connect the points as best as you can using a <i>smooth curve<\/i> (not a series of straight lines). You may want to find and plot additional points (such as the ones in blue here). Placing arrows on the tips of the lines implies that they continue in that direction forever.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232441\/image014.gif\" alt=\"A curved U-shaped line through the points from the previous graph.\" width=\"322\" height=\"353\" \/>\r\n\r\nNotice that the shape is similar to the letter U. This is called a parabola. One-half of the parabola is a mirror image of the other half. The lowest point on this graph is called the vertex. The vertical line that goes through the vertex is called the line of reflection. In this case, that line is the\u00a0<i>y<\/i>-axis.\r\n\r\nThe equations for quadratic functions have the form [latex]f(x)=ax^{2}+bx+c[\/latex]\u00a0where [latex] a\\ne 0[\/latex]. In the basic graph above, [latex]a=1[\/latex], [latex]b=0[\/latex], and [latex]c=0[\/latex].\r\n\r\nIn the following video, we show an example of plotting a quadratic function using a table of values.\r\n\r\nhttps:\/\/youtu.be\/wYfEzOJugS8\r\n\r\nChanging <i>a<\/i> changes the width of the parabola and whether it opens up ([latex]a&gt;0[\/latex]) or down ([latex]a&lt;0[\/latex]). If a is positive, the vertex is the lowest point, if a is negative, the vertex is the highest point. In the following example, we show how changing the value of a will affect the graph of the function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMatch each function with its graph.\r\n\r\na)\u00a0[latex] \\displaystyle f(x)=3{{x}^{2}}[\/latex]\r\n\r\nb)\u00a0[latex] \\displaystyle f(x)=-3{{x}^{2}}[\/latex]\r\n\r\nc) [latex] \\displaystyle f(x)=\\frac{1}{2}{{x}^{2}}[\/latex]\r\n\r\n1)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232443\/image016.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\n2)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232442\/image015.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\n3)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232445\/image017.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n[reveal-answer q=\"534119\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"534119\"]\r\n\r\nFunction a) matches graph\u00a0[latex]2[\/latex]\r\n\r\nFunction b) matches graph\u00a0[latex]1[\/latex]\r\n\r\nFunction c) matches graph\u00a0[latex]3[\/latex]\r\n\r\n&nbsp;\r\n\r\nFunction a)\u00a0[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\u00a0[latex]2[\/latex] is the best match for this function.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232442\/image015.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\nFunction b)\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=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232443\/image016.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\nFunction c)\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=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232445\/image017.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIf there is no <i>b<\/i> term, changing <i>c<\/i> moves the parabola up or down so that the <i>y<\/i> intercept is ([latex]0, c[\/latex]). In the next example, we show how changes to\u00a0<em>c\u00a0<\/em>affect the graph of the function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMatch each of the following functions with its graph.\r\n\r\na)\u00a0[latex] \\displaystyle f(x)={{x}^{2}}+3[\/latex]\r\n\r\nb)\u00a0[latex] \\displaystyle f(x)={{x}^{2}}-3[\/latex]\r\n\r\n&nbsp;\r\n\r\n1)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232447\/image019.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\n2)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232446\/image018.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n[reveal-answer q=\"393290\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"393290\"]\r\n\r\nFunction\u00a0a)\u00a0[latex] \\displaystyle f(x)={{x}^{2}}+3[\/latex] means square the inputs then add three, so every output will be moved up\u00a0[latex]3[\/latex] units. The graph that matches this function best is\u00a0[latex]2[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232446\/image018.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\nFunction\u00a0b)\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=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232447\/image019.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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\nTo find the vertex of the parabola, use the formula [latex] \\displaystyle \\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)[\/latex].\r\n\r\nFor 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\r\n[latex]a = 2[\/latex], and [latex]b = -3[\/latex], therefore\u00a0[latex]\\dfrac{-b}{2a}=\\dfrac{-(-3)}{2(2)}=\\dfrac{3}{4}[\/latex].\r\n\r\nThis is the [latex]x[\/latex] value of the vertex.\r\n\r\nNow evaluate the function at [latex]x =\\Large\\frac{3}{4}[\/latex] to get the corresponding y-value for the vertex.\r\n\r\n[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].\r\n\r\nThe 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. \u00a0It 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.\r\n\r\nIn the next example, we show how changing <em>b\u00a0<\/em>can change the graph of the quadratic function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMatch each of the following functions with its graph.\r\n\r\na)\u00a0[latex] \\displaystyle f(x)={{x}^{2}}+2x[\/latex]\r\n\r\nb)\u00a0[latex] \\displaystyle f(x)={{x}^{2}}-2x[\/latex]\r\n\r\na)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232450\/image021.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\nb)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232452\/image023.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n[reveal-answer q=\"320978\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"320978\"]\r\n\r\nFind the vertex of function a)\r\n\r\n[latex] \\displaystyle f(x)={{x}^{2}}+2x[\/latex].\r\n\r\n[latex]a = 1, b = 2[\/latex]\r\n\r\nx-value:\r\n\r\n[latex]\\dfrac{-b}{2a}=\\dfrac{-2}{2(1)}=-1[\/latex]\r\n\r\ny-value:\r\n\r\n[latex]f(\\dfrac{-b}{2a})=(-1)^2+2(-1)=1-2=-1[\/latex].\r\n\r\nVertex = [latex](-1,-1)[\/latex], which means the graph that best fits this function is a)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232450\/image021.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\n&nbsp;\r\n\r\nFind the vertex of function b)\r\n\r\n[latex]f(x)={{x}^{2}}-2x[\/latex].\r\n\r\n[latex]a = 1, b = -2[\/latex]\r\n\r\nx-value:\r\n\r\n[latex]\\dfrac{-b}{2a}=\\dfrac{2}{2(1)}=1[\/latex]\r\n\r\ny-value:\r\n\r\n[latex]f(\\dfrac{-b}{2a})=(1)^2-2(1)=1-2=-1[\/latex].\r\n\r\nVertex = [latex](1,-1)[\/latex], which means the graph that best fits this function is b)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232452\/image023.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNote that the vertex can change if the value for c changes because the y-value of the vertex is calculated by substituting the x-value into the function. Here is a summary of how the changes to the values for a, b, and, c of a quadratic function can change it is graph.\r\n<h3>Properties of a Parabola<\/h3>\r\nFor [latex] \\displaystyle f(x)=a{{x}^{2}}+bx+c[\/latex], where <i>a<\/i>, <i>b<\/i>, and <i>c<\/i> are real numbers,\r\n<ul>\r\n \t<li>The parabola opens upward if [latex]a &gt; 0[\/latex] and downward if [latex]a &lt; 0[\/latex].<\/li>\r\n \t<li><em>a<\/em>\u00a0changes the width of the parabola. The parabola gets narrower if [latex]|a|&gt; 1[\/latex] and wider if [latex]|a|&lt;1[\/latex].<\/li>\r\n \t<li>The vertex depends on the values of a, b, and c. The vertex is [latex]\\left(\\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a}\\right)\\right)[\/latex].<\/li>\r\n<\/ul>\r\nIn the last example, we showed how you can use the properties of a parabola to help you make a graph without having to calculate an exhaustive table of values.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGraph [latex]f(x)=\u22122x^{2}+3x\u20133[\/latex].\r\n\r\n[reveal-answer q=\"992003\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"992003\"]\r\n\r\nBefore making a table of values, look at the values of <i>a <\/i>and <i>c<\/i> to get a general idea of what the graph should look like.\r\n\r\n[latex]a=\u22122[\/latex], so the graph will open down and be thinner than [latex]f(x)=x^{2}[\/latex].\r\n\r\n[latex]c=\u22123[\/latex], so it will move to intersect the <i>y<\/i>-axis at\u00a0[latex](0,\u22123)[\/latex].\r\n\r\nTo find the vertex of the parabola, use the formula [latex] \\displaystyle \\left( \\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a} \\right) \\right)[\/latex]. Finding the vertex may make graphing the parabola easier.\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<i>x<\/i>-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<i>y<\/i>-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 <i>x<\/i> values increase. The <i>y<\/i> 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;\"><i>x<\/i><\/th>\r\n<th style=\"text-align: center;\"><i>f<\/i>(<i>x<\/i>)<\/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<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=\"309\" height=\"343\" \/>\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<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=\"309\" height=\"344\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows another example of plotting a quadratic function using the vertex.\r\n\r\nhttps:\/\/youtu.be\/leYhH_-3rVo\r\n\r\n&nbsp;\r\n<h2>Summary<\/h2>\r\nCreating a graph of a function is one way to understand the relationship between the inputs and outputs of that function. Creating a graph can be done by choosing values for <em>x<\/em>, finding the corresponding <em>y<\/em> values, and plotting them. However, it helps to understand the basic shape of the function. Knowing how changes to the basic function equation affect the graph is also helpful.\r\n\r\nThe shape of a quadratic function is a parabola. Parabolas have the equation [latex]f(x)=ax^{2}+bx+c[\/latex], where <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> are real numbers and [latex]a\\ne0[\/latex]. The value of <em>a<\/em> determines the width and the direction of the parabola, while the vertex depends on the values of <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>. The vertex is [latex] \\displaystyle \\left( \\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a} \\right) \\right)[\/latex].","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph quadratic functions using tables and transformations<\/li>\n<li>Identify important features of\u00a0the graph of a quadratic function of the form [latex]f(x)=ax^2+bx+c[\/latex]<\/li>\n<\/ul>\n<\/div>\n<p>Quadratic functions form parabolas when graphed in the coordinate plane, so you can be sure fairly quickly when graphing a quadratic if you have the correct shape. As with lines in the plane, creating a table of input and output values then plotting points will reveal the shape. But unlike straight lines between points, the parabola is a smooth curve.\u00a0 Later in this module, you&#8217;ll learn some other good methods for sketching a quick, accurate graph of a quadratic equation.<\/p>\n<p>It is helpful to have an idea about what the shape of the graph of a quadratic function should be so you can be sure that you have chosen enough points to plot as a guide when sketching the graph. Let&#8217;s start with the most basic quadratic function,\u00a0[latex]f(x)=x^{2}[\/latex].<br \/>\nGraph [latex]f(x)=x^{2}[\/latex].<br \/>\nStart with a table of values. Then think of each row of the table as an ordered pair.<\/p>\n<table style=\"width: 20%;\">\n<thead>\n<tr>\n<th style=\"text-align: center;\"><i>x<\/i><\/th>\n<th style=\"text-align: center;\"><i>f<\/i>(<i>x<\/i>)<\/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]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the points [latex](-2,4), (-1,1), (0,0), (1,1), (2,4)[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232437\/image013.gif\" alt=\"Graph with the point negative 2, 4; the point negative 1, 1; the point 0, 0; the point 1,1; the point 2,4.\" width=\"322\" height=\"353\" \/><\/p>\n<p>Since the points are <i>not<\/i> on a line, you cannot use a straight edge. Connect the points as best as you can using a <i>smooth curve<\/i> (not a series of straight lines). You may want to find and plot additional points (such as the ones in blue here). Placing arrows on the tips of the lines implies that they continue in that direction forever.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232441\/image014.gif\" alt=\"A curved U-shaped line through the points from the previous graph.\" width=\"322\" height=\"353\" \/><\/p>\n<p>Notice that the shape is similar to the letter U. This is called a parabola. One-half of the parabola is a mirror image of the other half. The lowest point on this graph is called the vertex. The vertical line that goes through the vertex is called the line of reflection. In this case, that line is the\u00a0<i>y<\/i>-axis.<\/p>\n<p>The equations for quadratic functions have the form [latex]f(x)=ax^{2}+bx+c[\/latex]\u00a0where [latex]a\\ne 0[\/latex]. In the basic graph above, [latex]a=1[\/latex], [latex]b=0[\/latex], and [latex]c=0[\/latex].<\/p>\n<p>In the following video, we show an example of plotting a quadratic function using a table of values.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Graph a Quadratic Function Using a Table of Values\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/wYfEzOJugS8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Changing <i>a<\/i> changes the width of the parabola and whether it opens up ([latex]a>0[\/latex]) or down ([latex]a<0[\/latex]). If a is positive, the vertex is the lowest point, if a is negative, the vertex is the highest point. In the following example, we show how changing the value of a will affect the graph of the function.\n\n\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Match each function with its graph.<\/p>\n<p>a)\u00a0[latex]\\displaystyle f(x)=3{{x}^{2}}[\/latex]<\/p>\n<p>b)\u00a0[latex]\\displaystyle f(x)=-3{{x}^{2}}[\/latex]<\/p>\n<p>c) [latex]\\displaystyle f(x)=\\frac{1}{2}{{x}^{2}}[\/latex]<\/p>\n<p>1)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232443\/image016.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>2)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232442\/image015.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>3)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232445\/image017.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q534119\">Show Solution<\/span><\/p>\n<div id=\"q534119\" class=\"hidden-answer\" style=\"display: none\">\n<p>Function a) matches graph\u00a0[latex]2[\/latex]<\/p>\n<p>Function b) matches graph\u00a0[latex]1[\/latex]<\/p>\n<p>Function c) matches graph\u00a0[latex]3[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Function a)\u00a0[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\u00a0[latex]2[\/latex] is the best match for this function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232442\/image015.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>Function b)\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=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232443\/image016.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>Function c)\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=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232445\/image017.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>If there is no <i>b<\/i> term, changing <i>c<\/i> moves the parabola up or down so that the <i>y<\/i> intercept is ([latex]0, c[\/latex]). In the next example, we show how changes to\u00a0<em>c\u00a0<\/em>affect the graph of the function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Match each of the following functions with its graph.<\/p>\n<p>a)\u00a0[latex]\\displaystyle f(x)={{x}^{2}}+3[\/latex]<\/p>\n<p>b)\u00a0[latex]\\displaystyle f(x)={{x}^{2}}-3[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>1)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232447\/image019.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>2)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232446\/image018.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q393290\">Show Solution<\/span><\/p>\n<div id=\"q393290\" class=\"hidden-answer\" style=\"display: none\">\n<p>Function\u00a0a)\u00a0[latex]\\displaystyle f(x)={{x}^{2}}+3[\/latex] means square the inputs then add three, so every output will be moved up\u00a0[latex]3[\/latex] units. The graph that matches this function best is\u00a0[latex]2[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232446\/image018.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>Function\u00a0b)\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=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232447\/image019.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\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<p>To find the vertex of the parabola, use the formula [latex]\\displaystyle \\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)[\/latex].<\/p>\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]<\/p>\n<p>[latex]a = 2[\/latex], and [latex]b = -3[\/latex], therefore\u00a0[latex]\\dfrac{-b}{2a}=\\dfrac{-(-3)}{2(2)}=\\dfrac{3}{4}[\/latex].<\/p>\n<p>This is the [latex]x[\/latex] value of the vertex.<\/p>\n<p>Now evaluate the function at [latex]x =\\Large\\frac{3}{4}[\/latex] to get the corresponding y-value for the vertex.<\/p>\n<p>[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].<\/p>\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. \u00a0It 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.<\/p>\n<p>In the next example, we show how changing <em>b\u00a0<\/em>can change the graph of the quadratic function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Match each of the following functions with its graph.<\/p>\n<p>a)\u00a0[latex]\\displaystyle f(x)={{x}^{2}}+2x[\/latex]<\/p>\n<p>b)\u00a0[latex]\\displaystyle f(x)={{x}^{2}}-2x[\/latex]<\/p>\n<p>a)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232450\/image021.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232452\/image023.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q320978\">Show Solution<\/span><\/p>\n<div id=\"q320978\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the vertex of function a)<\/p>\n<p>[latex]\\displaystyle f(x)={{x}^{2}}+2x[\/latex].<\/p>\n<p>[latex]a = 1, b = 2[\/latex]<\/p>\n<p>x-value:<\/p>\n<p>[latex]\\dfrac{-b}{2a}=\\dfrac{-2}{2(1)}=-1[\/latex]<\/p>\n<p>y-value:<\/p>\n<p>[latex]f(\\dfrac{-b}{2a})=(-1)^2+2(-1)=1-2=-1[\/latex].<\/p>\n<p>Vertex = [latex](-1,-1)[\/latex], which means the graph that best fits this function is a)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232450\/image021.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Find the vertex of function b)<\/p>\n<p>[latex]f(x)={{x}^{2}}-2x[\/latex].<\/p>\n<p>[latex]a = 1, b = -2[\/latex]<\/p>\n<p>x-value:<\/p>\n<p>[latex]\\dfrac{-b}{2a}=\\dfrac{2}{2(1)}=1[\/latex]<\/p>\n<p>y-value:<\/p>\n<p>[latex]f(\\dfrac{-b}{2a})=(1)^2-2(1)=1-2=-1[\/latex].<\/p>\n<p>Vertex = [latex](1,-1)[\/latex], which means the graph that best fits this function is b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232452\/image023.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Note that the vertex can change if the value for c changes because the y-value of the vertex is calculated by substituting the x-value into the function. Here is a summary of how the changes to the values for a, b, and, c of a quadratic function can change it is graph.<\/p>\n<h3>Properties of a Parabola<\/h3>\n<p>For [latex]\\displaystyle f(x)=a{{x}^{2}}+bx+c[\/latex], where <i>a<\/i>, <i>b<\/i>, and <i>c<\/i> are real numbers,<\/p>\n<ul>\n<li>The parabola opens upward if [latex]a > 0[\/latex] and downward if [latex]a < 0[\/latex].<\/li>\n<li><em>a<\/em>\u00a0changes the width of the parabola. The parabola gets narrower if [latex]|a|> 1[\/latex] and wider if [latex]|a|<1[\/latex].<\/li>\n<li>The vertex depends on the values of a, b, and c. The vertex is [latex]\\left(\\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a}\\right)\\right)[\/latex].<\/li>\n<\/ul>\n<p>In the last example, we showed how you can use the properties of a parabola to help you make a graph without having to calculate an exhaustive table of values.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph [latex]f(x)=\u22122x^{2}+3x\u20133[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q992003\">Show Solution<\/span><\/p>\n<div id=\"q992003\" class=\"hidden-answer\" style=\"display: none\">\n<p>Before making a table of values, look at the values of <i>a <\/i>and <i>c<\/i> to get a general idea of what the graph should look like.<\/p>\n<p>[latex]a=\u22122[\/latex], so the graph will open down and be thinner than [latex]f(x)=x^{2}[\/latex].<\/p>\n<p>[latex]c=\u22123[\/latex], so it will move to intersect the <i>y<\/i>-axis at\u00a0[latex](0,\u22123)[\/latex].<\/p>\n<p>To find the vertex of the parabola, use the formula [latex]\\displaystyle \\left( \\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a} \\right) \\right)[\/latex]. Finding the vertex may make graphing the parabola easier.<\/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><i>x<\/i>-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><i>y<\/i>-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 <i>x<\/i> values increase. The <i>y<\/i> 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;\"><i>x<\/i><\/th>\n<th style=\"text-align: center;\"><i>f<\/i>(<i>x<\/i>)<\/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><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=\"309\" height=\"343\" \/><\/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<p><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=\"309\" height=\"344\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows another example of plotting a quadratic function using the vertex.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Graph a Quadratic Function Using a Table of Value and the Vertex\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/leYhH_-3rVo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2>Summary<\/h2>\n<p>Creating a graph of a function is one way to understand the relationship between the inputs and outputs of that function. Creating a graph can be done by choosing values for <em>x<\/em>, finding the corresponding <em>y<\/em> values, and plotting them. However, it helps to understand the basic shape of the function. Knowing how changes to the basic function equation affect the graph is also helpful.<\/p>\n<p>The shape of a quadratic function is a parabola. Parabolas have the equation [latex]f(x)=ax^{2}+bx+c[\/latex], where <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> are real numbers and [latex]a\\ne0[\/latex]. The value of <em>a<\/em> determines the width and the direction of the parabola, while the vertex depends on the values of <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>. The vertex is [latex]\\displaystyle \\left( \\dfrac{-b}{2a},f\\left( \\dfrac{-b}{2a} \\right) \\right)[\/latex].<\/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-184\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Graph a Quadratic Function Using a Table of Value and the Vertex. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/leYhH_-3rVo\">https:\/\/youtu.be\/leYhH_-3rVo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Graph a Quadratic Function Using a Table of Values. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/wYfEzOJugS8\">https:\/\/youtu.be\/wYfEzOJugS8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 17: Functions, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Graph a Quadratic Function Using a Table of Value and the Vertex\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/leYhH_-3rVo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 17: Functions, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and 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