{"id":1863,"date":"2016-06-28T04:25:35","date_gmt":"2016-06-28T04:25:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1863"},"modified":"2018-01-03T16:37:44","modified_gmt":"2018-01-03T16:37:44","slug":"outcome-graphs-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/outcome-graphs-of-functions\/","title":{"raw":"Graphs of Functions","rendered":"Graphs of Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Graph linear functions using a table of values<\/li>\r\n \t<li>Graph a quadratic function using a table of values<\/li>\r\n \t<li>Identify\u00a0how multiplication can change the graph of a radical function<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen both the input (independent variable) and the output (dependent variable) are real numbers, a function can be represented by a coordinate graph. The input is plotted on the horizontal\u00a0<i>x<\/i>-axis and the output is plotted on the vertical\u00a0<i>y<\/i>-axis.\r\n\r\nA helpful first step in graphing a function is to make a table of values. This is particularly useful when you don\u2019t know the general shape the function will have. You probably already know that a linear function will be a straight line, but let\u2019s make a table first to see how it can be helpful.\r\n\r\nWhen making a table, it\u2019s a good idea to include negative values, positive values, and zero to ensure that you do have a linear function.\r\n\r\nMake a table of values for [latex]f(x)=3x+2[\/latex].\r\n\r\nMake a two-column table. Label the columns <i>x<\/i> and <i>f<\/i>(<i>x<\/i>).\r\n<table style=\"width: 20%\">\r\n<thead>\r\n<tr>\r\n<th><i>x<\/i><\/th>\r\n<th><i>f<\/i>(<i>x<\/i>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nChoose several values for <i>x<\/i> and put them as separate rows in the <i>x<\/i> column. These are YOUR CHOICE - there is no \"right\" or \"wrong\" values to pick, just go for it.\r\n\r\n<i>Tip:<\/i> It\u2019s always good to include 0, positive values, and negative values, if you can.\r\n<table style=\"width: 20%\">\r\n<thead>\r\n<tr>\r\n<th><i>x<\/i><\/th>\r\n<th><i>f<\/i>(<i>x<\/i>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEvaluate the function for each value of <i>x<\/i>, and write the result in the <i>f<\/i>(<i>x<\/i>) column next to the <i>x<\/i> value you used.\r\n\r\nWhen [latex]x=0[\/latex], [latex]f(0)=3(0)+2=2[\/latex],\r\n\r\n[latex]f(1)=3(1)+2=5[\/latex],\r\n\r\n[latex]f(\u22121)=3(\u22121)+2=\u22123+2=\u22121[\/latex],\u00a0and so on.\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><i>x<\/i><\/td>\r\n<td><i>f<\/i>(<i>x<\/i>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]\u22124[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n(Note that your table of values may be different from someone else\u2019s. You may each choose different numbers for <i>x<\/i>.)\r\n\r\nNow that you have a table of values, you can use them to help you draw both the shape and location of the function. <i>Important:<\/i> The graph of the function will show all possible values of <i>x<\/i> and the corresponding values of <i>y<\/i>. This is why the graph is a line and not just the dots that make up the points in our table.\r\n\r\nGraph [latex]f(x)=3x+2[\/latex].\r\nUsing the table of values we created above you can think of <i>f<\/i>(<i>x<\/i>) as <i>y,<\/i> each row forms an ordered pair that you can plot on a coordinate grid.\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><i>x<\/i><\/td>\r\n<td><i>f<\/i>(<i>x<\/i>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]\u22124[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the points.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232424\/image005.gif\" alt=\"The points negative 2, negative 4; the point negative 1, negative 1; the point 0, 2; the point 1, 5; the point 3, 11.\" width=\"322\" height=\"353\" \/>\r\n\r\nSince the points lie on a line, use a straight edge to draw the line. Try to go through each point without moving the straight edge.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232426\/image006.gif\" alt=\"A line through the points in the previous graph.\" width=\"322\" height=\"353\" \/>\r\n\r\nLet\u2019s try another one. Before you look at the answer, try to make the table yourself and draw the graph on a piece of paper.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGraph [latex]f(x)=\u2212x+1[\/latex].\r\n\r\n[reveal-answer q=\"748367\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"748367\"]Start with a table of values. You can choose different values for <i>x<\/i>, but once again, it\u2019s helpful to include 0, some positive values, and some negative values.\r\n\r\nIf you think of <i>f<\/i>(<i>x<\/i>) as <i>y,<\/i> each row forms an ordered pair that you can plot on a coordinate grid.\r\n<p style=\"text-align: center\">[latex]f(\u22122)=\u2212(\u22122)+1=2+1=3\\\\f(\u22121)=\u2212(\u22121)+1=1+1=2\\\\f(0)=\u2212(0)+1=0+1=1\\\\f(1)=\u2212(1)+1=\u22121+1=0\\\\f(2)=\u2212(2)+1=\u22122+1=\u22121[\/latex]<\/p>\r\n\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><i>x<\/i><\/td>\r\n<td><i>f<\/i>(<i>x<\/i>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the points.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232428\/image007.gif\" alt=\"The point negative 2, 3; the point negative 1, 2; the point 0, 1; the point 1, 0; the point 2, negative 1.\" width=\"322\" height=\"353\" \/>\r\n<h4>Answer<\/h4>\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232430\/image008.gif\" alt=\"Line through the points in the last graph.\" width=\"322\" height=\"353\" \/>\r\n\r\nSince the points lie on a line, use a straight edge to draw the line. Try to go through each point without moving the straight edge.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show another example of how to graph a linear function on a set of coordinate axes.\r\n\r\nhttps:\/\/youtu.be\/sfzpdThXpA8\r\n\r\nThese graphs are representations of a linear function. Remember that a function is a correspondence between two variables, such as <i>x<\/i> and <i>y<\/i>. These will be discussed in further detail in the next module.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Linear Function<\/h3>\r\nA <strong>linear function<\/strong> is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line\r\n\r\n[latex]f\\left(x\\right)=mx+b[\/latex]\r\n\r\nwhere [latex]b[\/latex]\u00a0is the initial or starting value of the function (when input, [latex]x=0[\/latex]), and [latex]m[\/latex]\u00a0is the constant rate of change, or <strong>slope<\/strong> of the function. The <strong><em>y<\/em>-intercept<\/strong> is at [latex]\\left(0,b\\right)[\/latex].\r\n\r\n<\/div>\r\n<h2>Graph Quadratic Functions<\/h2>\r\nQuadratic\u00a0functions can also be graphed. It\u2019s helpful to have an idea what the shape should be, so you can be sure that you\u2019ve chosen enough points to plot as a guide. Let\u2019s 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 the table as ordered pairs.\r\n<table style=\"width: 20%\">\r\n<thead>\r\n<tr>\r\n<th><i>x<\/i><\/th>\r\n<th><i>f<\/i>(<i>x<\/i>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[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 can\u2019t use a straight edge. Connect the points as best 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 like the letter U. This is called a parabola. One-half of the parabola is a mirror image of the other half. The line that goes down the middle is called the line of reflection, in this case that line is they <i>y<\/i>-axis. The lowest point on this graph is called the vertex.\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\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\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 the following functions with their 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\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\/18232443\/image016.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\/18232442\/image015.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\nc)\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 Answer[\/reveal-answer]\r\n[hidden-answer a=\"534119\"]\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 the graph b) 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 greater than they would have been for [latex]f(x)=x^2[\/latex] so graph a) \u00a0is 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]\\frac{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 c) 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<h4>Answer<\/h4>\r\nFunction a) matches graph b)\r\n\r\nFunction b) matches graph a)\r\n\r\nFunction c) matches graph c)\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 (0, <i>c<\/i>). 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 the following functions with their 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\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\/18232447\/image019.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\/18232446\/image018.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n[reveal-answer q=\"393290\"]Show Answer[\/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 3 units. the graph that matches this function best 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\/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 3 units. the graph that matches this function best 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\/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>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Graph linear functions using a table of values<\/li>\n<li>Graph a quadratic function using a table of values<\/li>\n<li>Identify\u00a0how multiplication can change the graph of a radical function<\/li>\n<\/ul>\n<\/div>\n<p>When both the input (independent variable) and the output (dependent variable) are real numbers, a function can be represented by a coordinate graph. The input is plotted on the horizontal\u00a0<i>x<\/i>-axis and the output is plotted on the vertical\u00a0<i>y<\/i>-axis.<\/p>\n<p>A helpful first step in graphing a function is to make a table of values. This is particularly useful when you don\u2019t know the general shape the function will have. You probably already know that a linear function will be a straight line, but let\u2019s make a table first to see how it can be helpful.<\/p>\n<p>When making a table, it\u2019s a good idea to include negative values, positive values, and zero to ensure that you do have a linear function.<\/p>\n<p>Make a table of values for [latex]f(x)=3x+2[\/latex].<\/p>\n<p>Make a two-column table. Label the columns <i>x<\/i> and <i>f<\/i>(<i>x<\/i>).<\/p>\n<table style=\"width: 20%\">\n<thead>\n<tr>\n<th><i>x<\/i><\/th>\n<th><i>f<\/i>(<i>x<\/i>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Choose several values for <i>x<\/i> and put them as separate rows in the <i>x<\/i> column. These are YOUR CHOICE &#8211; there is no &#8220;right&#8221; or &#8220;wrong&#8221; values to pick, just go for it.<\/p>\n<p><i>Tip:<\/i> It\u2019s always good to include 0, positive values, and negative values, if you can.<\/p>\n<table style=\"width: 20%\">\n<thead>\n<tr>\n<th><i>x<\/i><\/th>\n<th><i>f<\/i>(<i>x<\/i>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\u22122[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22121[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Evaluate the function for each value of <i>x<\/i>, and write the result in the <i>f<\/i>(<i>x<\/i>) column next to the <i>x<\/i> value you used.<\/p>\n<p>When [latex]x=0[\/latex], [latex]f(0)=3(0)+2=2[\/latex],<\/p>\n<p>[latex]f(1)=3(1)+2=5[\/latex],<\/p>\n<p>[latex]f(\u22121)=3(\u22121)+2=\u22123+2=\u22121[\/latex],\u00a0and so on.<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td><i>x<\/i><\/td>\n<td><i>f<\/i>(<i>x<\/i>)<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22122[\/latex]<\/td>\n<td>[latex]\u22124[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22121[\/latex]<\/td>\n<td>[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]11[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>(Note that your table of values may be different from someone else\u2019s. You may each choose different numbers for <i>x<\/i>.)<\/p>\n<p>Now that you have a table of values, you can use them to help you draw both the shape and location of the function. <i>Important:<\/i> The graph of the function will show all possible values of <i>x<\/i> and the corresponding values of <i>y<\/i>. This is why the graph is a line and not just the dots that make up the points in our table.<\/p>\n<p>Graph [latex]f(x)=3x+2[\/latex].<br \/>\nUsing the table of values we created above you can think of <i>f<\/i>(<i>x<\/i>) as <i>y,<\/i> each row forms an ordered pair that you can plot on a coordinate grid.<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td><i>x<\/i><\/td>\n<td><i>f<\/i>(<i>x<\/i>)<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22122[\/latex]<\/td>\n<td>[latex]\u22124[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22121[\/latex]<\/td>\n<td>[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]11[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the points.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232424\/image005.gif\" alt=\"The points negative 2, negative 4; the point negative 1, negative 1; the point 0, 2; the point 1, 5; the point 3, 11.\" width=\"322\" height=\"353\" \/><\/p>\n<p>Since the points lie on a line, use a straight edge to draw the line. Try to go through each point without moving the straight edge.<\/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\/18232426\/image006.gif\" alt=\"A line through the points in the previous graph.\" width=\"322\" height=\"353\" \/><\/p>\n<p>Let\u2019s try another one. Before you look at the answer, try to make the table yourself and draw the graph on a piece of paper.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph [latex]f(x)=\u2212x+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q748367\">Show Solution<\/span><\/p>\n<div id=\"q748367\" class=\"hidden-answer\" style=\"display: none\">Start with a table of values. You can choose different values for <i>x<\/i>, but once again, it\u2019s helpful to include 0, some positive values, and some negative values.<\/p>\n<p>If you think of <i>f<\/i>(<i>x<\/i>) as <i>y,<\/i> each row forms an ordered pair that you can plot on a coordinate grid.<\/p>\n<p style=\"text-align: center\">[latex]f(\u22122)=\u2212(\u22122)+1=2+1=3\\\\f(\u22121)=\u2212(\u22121)+1=1+1=2\\\\f(0)=\u2212(0)+1=0+1=1\\\\f(1)=\u2212(1)+1=\u22121+1=0\\\\f(2)=\u2212(2)+1=\u22122+1=\u22121[\/latex]<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td><i>x<\/i><\/td>\n<td><i>f<\/i>(<i>x<\/i>)<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22122[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22121[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the points.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232428\/image007.gif\" alt=\"The point negative 2, 3; the point negative 1, 2; the point 0, 1; the point 1, 0; the point 2, negative 1.\" width=\"322\" height=\"353\" \/><\/p>\n<h4>Answer<\/h4>\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\/18232430\/image008.gif\" alt=\"Line through the points in the last graph.\" width=\"322\" height=\"353\" \/><\/p>\n<p>Since the points lie on a line, use a straight edge to draw the line. Try to go through each point without moving the straight edge.\n<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the following video we show another example of how to graph a linear function on a set of coordinate axes.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Graph a Linear Function Using a Table of Values (Function Notation)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/sfzpdThXpA8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>These graphs are representations of a linear function. Remember that a function is a correspondence between two variables, such as <i>x<\/i> and <i>y<\/i>. These will be discussed in further detail in the next module.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Linear Function<\/h3>\n<p>A <strong>linear function<\/strong> is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line<\/p>\n<p>[latex]f\\left(x\\right)=mx+b[\/latex]<\/p>\n<p>where [latex]b[\/latex]\u00a0is the initial or starting value of the function (when input, [latex]x=0[\/latex]), and [latex]m[\/latex]\u00a0is the constant rate of change, or <strong>slope<\/strong> of the function. The <strong><em>y<\/em>-intercept<\/strong> is at [latex]\\left(0,b\\right)[\/latex].<\/p>\n<\/div>\n<h2>Graph Quadratic Functions<\/h2>\n<p>Quadratic\u00a0functions can also be graphed. It\u2019s helpful to have an idea what the shape should be, so you can be sure that you\u2019ve chosen enough points to plot as a guide. Let\u2019s 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 the table as ordered pairs.<\/p>\n<table style=\"width: 20%\">\n<thead>\n<tr>\n<th><i>x<\/i><\/th>\n<th><i>f<\/i>(<i>x<\/i>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\u22122[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22121[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[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 can\u2019t use a straight edge. Connect the points as best 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 like the letter U. This is called a parabola. One-half of the parabola is a mirror image of the other half. The line that goes down the middle is called the line of reflection, in this case that line is they <i>y<\/i>-axis. The lowest point on this graph is called the vertex.<\/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-2\" 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>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>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 the following functions with their 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>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\/18232443\/image016.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\/18232442\/image015.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>c)<\/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 Answer<\/span><\/p>\n<div id=\"q534119\" class=\"hidden-answer\" style=\"display: none\">\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 the graph b) 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 greater than they would have been for [latex]f(x)=x^2[\/latex] so graph a) \u00a0is 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]\\frac{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 c) 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<h4>Answer<\/h4>\n<p>Function a) matches graph b)<\/p>\n<p>Function b) matches graph a)<\/p>\n<p>Function c) matches graph c)<\/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 (0, <i>c<\/i>). 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 the following functions with their 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>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\/18232447\/image019.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\/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 Answer<\/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 3 units. the graph that matches this function best 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\/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 3 units. the graph that matches this function best 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\/18232447\/image019.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\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-1863\">\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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Graph a Linear Function Using a Table of Values (Function Notation). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/sfzpdThXpA8\">https:\/\/youtu.be\/sfzpdThXpA8<\/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>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><li>Ex: Graph a Quadratic Function Using a Table of Values. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <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><li>Determine if a Relation Given as a Table is a One-to-One Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QFOJmevha_Y\">https:\/\/youtu.be\/QFOJmevha_Y<\/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>Ex 1: Use the Vertical Line Test to Determine if a Graph Represents a Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/5Z8DaZPJLKY\">https:\/\/youtu.be\/5Z8DaZPJLKY<\/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":21,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Graph a Linear Function Using a Table of Values (Function Notation)\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/sfzpdThXpA8\",\"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\":\"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\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Graph a Quadratic Function Using a Table of Values\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/wYfEzOJugS8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Determine if a Relation Given as a Table is a One-to-One Function\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/QFOJmevha_Y\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Use the Vertical Line Test to Determine if a Graph Represents a Function\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/5Z8DaZPJLKY\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"1b65329a-c729-4ec7-9ceb-c14babb1c0af","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1863","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1863","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":25,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1863\/revisions"}],"predecessor-version":[{"id":5163,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1863\/revisions\/5163"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1863\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/media?parent=1863"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1863"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1863"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/license?post=1863"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}