{"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":"2023-11-08T13:18:52","modified_gmt":"2023-11-08T13:18:52","slug":"outcome-graphs-of-functions","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/chapter\/outcome-graphs-of-functions\/","title":{"raw":"3.2 - Graphs of Functions","rendered":"3.2 &#8211; Graphs of Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>(3.2.1) - Graphing functions using a table of values\r\n<ul>\r\n \t<li>Linear functions<\/li>\r\n \t<li>Quadratic functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>(3.2.2) - Finding function values from a graph<\/li>\r\n \t<li>(3.2.3) - Vertical line test<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h1>(3.2.1) - Graphing functions using a table of values<\/h1>\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<h3>Graphing linear functions<\/h3>\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<h3>Graphing quadratic functions<\/h3>\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>\r\n<h1>(3.2.2) -\u00a0Finding Function Values from a Graph<\/h1>\r\nEvaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Reading Function Values from a Graph<\/h3>\r\nGiven the graph below,\r\n<ol>\r\n \t<li>Evaluate [latex]f\\left(2\\right)[\/latex].<\/li>\r\n \t<li>Solve [latex]f\\left(x\\right)=4[\/latex].<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191001\/CNX_Precalc_Figure_01_01_0072.jpg\" alt=\"Graph of a positive parabola centered at (1, 0).\" width=\"487\" height=\"445\" \/>\r\n[reveal-answer q=\"915833\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"915833\"]\r\n<ol>\r\n \t<li>To evaluate [latex]f\\left(2\\right)[\/latex], locate the point on the curve where [latex]x=2[\/latex], then read the <em>y<\/em>-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right)[\/latex], so [latex]f\\left(2\\right)=1[\/latex].<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/><\/li>\r\n \t<li>To solve [latex]f\\left(x\\right)=4[\/latex], we find the output value [latex]4[\/latex] on the vertical axis. Moving horizontally along the line [latex]y=4[\/latex], we locate two points of the curve with output value [latex]4:[\/latex] [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(3,4\\right)[\/latex]. These points represent the two solutions to [latex]f\\left(x\\right)=4:[\/latex] [latex]x=-1[\/latex] or [latex]x=3[\/latex]. This means [latex]f\\left(-1\\right)=4[\/latex] and [latex]f\\left(3\\right)=4[\/latex], or when the input is [latex]-1[\/latex] or [latex]\\text{3,}[\/latex] the output is [latex]\\text{4}\\text{.}[\/latex]See Figure 8.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191006\/CNX_Precalc_Figure_01_01_0092.jpg\" alt=\"Graph of an upward-facing\u00a0parabola with a vertex at (0,1) and\u00a0labeled points at (-1, 4) and (3,4). A\u00a0line at y = 4 intersects the parabola at the labeled points.\" width=\"487\" height=\"445\" \/><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUsing the graph, solve [latex]f\\left(x\\right)=1[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/>\r\n[reveal-answer q=\"529772\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"529772\"]\r\n\r\n[latex]x=0[\/latex] or [latex]x=2[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom9\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2471&amp;theme=oea&amp;iframe_resize_id=mom9\" width=\"100%\" height=\"550\"><\/iframe>\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2886&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"600\"><\/iframe>\r\n\r\n<\/div>\r\n<h1>(3.2.3) - Vertical line test<\/h1>\r\nWhen both the independent quantity (input) and the dependent quantity (output) are real numbers, a function can be represented by a graph in the coordinate plane. The independent value is plotted on the <i>x<\/i>-axis and the dependent value is plotted on the <i>y<\/i>-axis. The fact that each input value has exactly one output value means graphs of functions have certain characteristics. For each input on the graph, there will be exactly one output. For a function defined as [latex]y = f(x)[\/latex], or [latex]y[\/latex] is a function of [latex]x[\/latex], we would write ordered pairs [latex](x, f(x))[\/latex] using function notation instead of [latex](x,y)[\/latex] as you may have seen previously.\r\n\r\n<img class=\"wp-image-2679 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16195741\/Screen-Shot-2016-07-16-at-12.56.58-PM-300x281.png\" alt=\"Coordinate axes with y-axis labeled f(x) and the x axis labeled x.\" width=\"408\" height=\"382\" \/>\r\n\r\nWe can identify whether the graph of a relation represents a function\u00a0because for each <i>x<\/i>-coordinate there will be exactly one <i>y<\/i>-coordinate.\r\n<p align=\"center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232417\/image001.jpg\" alt=\"A graph of a semicircle. Four vertical lines cross the semicircle at one point each.\" width=\"304\" height=\"307\" \/><\/p>\r\nWhen a vertical line is placed across the plot of this relation, it does not intersect the graph more than once for any values of [latex]x[\/latex].\r\n\r\nIf, on the other hand, a graph shows two or more intersections with a vertical line, then an input (<i>x<\/i>-coordinate) can have more than one output (<i>y<\/i>-coordinate), and [latex]y[\/latex] is not a function of [latex]x[\/latex]. Examining the graph of a relation to determine if a vertical line would intersect with more than one point is a quick way to determine if the relation shown by the graph is a function. This method is often called the \u201cvertical line test.\u201d\r\n\r\nYou try it.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the vertical line test to determine whether the relation plotted on this graph is a function.\r\n\r\n<img class=\"size-medium wp-image-2680 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16200323\/Screen-Shot-2016-07-16-at-1.02.50-PM-300x275.png\" alt=\"Graph with circle plotted - center at (0,0) radius = 2, more points include (2,0), (-2,0)\" width=\"300\" height=\"275\" \/>\r\n[reveal-answer q=\"28965\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"28965\"]\r\n\r\nThis relationship cannot be a function, because some of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-coordinates.\r\n<p align=\"center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232418\/image002.jpg\" alt=\"A circle with four vertical lines through it. Three of the lines cross the circle at two points, and one line crosses the edge of the circle at one point.\" width=\"340\" height=\"343\" \/><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe vertical line method can also be applied to a set of ordered pairs plotted on a coordinate plane to determine if the relation is a function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the ordered pairs\r\n\r\n[latex]\\{(\u22121,3),(\u22122,5),(\u22123,3),(\u22125,\u22123)\\}[\/latex], plotted on the graph below. Use the vertical line test to determine whether the set of ordered pairs represents a function.\r\n<p align=\"center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232420\/image003.jpg\" alt=\"The points (\u22121,3); (\u22122,5); (\u22123,3); and (\u22125,\u22123).\" width=\"329\" height=\"328\" \/><\/p>\r\n<p align=\"center\">[reveal-answer q=\"114452\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"114452\"]<\/p>\r\n<p align=\"center\">Drawing vertical lines through each point results in each line only touching one point. This means that none of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-coordinates, so this is a function.<\/p>\r\n<p align=\"center\"><img class=\"size-full wp-image-2514 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/14213709\/Edit_Page_%E2%80%B9_Intermediate_Algebra_%E2%80%94_Pressbooks.png\" alt=\"Edit_Page_\u2039_Intermediate_Algebra_\u2014_Pressbooks\" width=\"298\" height=\"294\" \/><\/p>\r\n<p align=\"center\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nIn another set of ordered pairs, [latex]\\{(3,\u22121),(5,\u22122),(3,\u22123),(\u22123,5)\\}[\/latex], one of the inputs, 3, can produce two different outputs, [latex]\u22121[\/latex] and [latex]\u22123[\/latex]. You know what that means\u2014this set of ordered pairs is not a function. A plot confirms this.\r\n<p align=\"center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232421\/image004.jpg\" alt=\"The point negative 3, 5; the point 5, negative 2. A line through the points negative 1, 3; and the point negative 3, negative 3.\" width=\"337\" height=\"336\" \/><\/p>\r\nNotice that a vertical line passes through two plotted points. One <i>x<\/i>-coordinate has multiple <i>y<\/i>-coordinates. This relation is not a function.\r\n\r\nIn the following video we show another example of determining whether a graph represents a function using the vertical line test.\r\n\r\nhttps:\/\/youtu.be\/5Z8DaZPJLKY","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>(3.2.1) &#8211; Graphing functions using a table of values\n<ul>\n<li>Linear functions<\/li>\n<li>Quadratic functions<\/li>\n<\/ul>\n<\/li>\n<li>(3.2.2) &#8211; Finding function values from a graph<\/li>\n<li>(3.2.3) &#8211; Vertical line test<\/li>\n<\/ul>\n<\/div>\n<h1>(3.2.1) &#8211; Graphing functions using a table of values<\/h1>\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<h3>Graphing linear functions<\/h3>\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<h3>Graphing quadratic functions<\/h3>\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<h1>(3.2.2) &#8211;\u00a0Finding Function Values from a Graph<\/h1>\n<p>Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Reading Function Values from a Graph<\/h3>\n<p>Given the graph below,<\/p>\n<ol>\n<li>Evaluate [latex]f\\left(2\\right)[\/latex].<\/li>\n<li>Solve [latex]f\\left(x\\right)=4[\/latex].<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191001\/CNX_Precalc_Figure_01_01_0072.jpg\" alt=\"Graph of a positive parabola centered at (1, 0).\" width=\"487\" height=\"445\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q915833\">Solution<\/span><\/p>\n<div id=\"q915833\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>To evaluate [latex]f\\left(2\\right)[\/latex], locate the point on the curve where [latex]x=2[\/latex], then read the <em>y<\/em>-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right)[\/latex], so [latex]f\\left(2\\right)=1[\/latex].<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/><\/li>\n<li>To solve [latex]f\\left(x\\right)=4[\/latex], we find the output value [latex]4[\/latex] on the vertical axis. Moving horizontally along the line [latex]y=4[\/latex], we locate two points of the curve with output value [latex]4:[\/latex] [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(3,4\\right)[\/latex]. These points represent the two solutions to [latex]f\\left(x\\right)=4:[\/latex] [latex]x=-1[\/latex] or [latex]x=3[\/latex]. This means [latex]f\\left(-1\\right)=4[\/latex] and [latex]f\\left(3\\right)=4[\/latex], or when the input is [latex]-1[\/latex] or [latex]\\text{3,}[\/latex] the output is [latex]\\text{4}\\text{.}[\/latex]See Figure 8.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18191006\/CNX_Precalc_Figure_01_01_0092.jpg\" alt=\"Graph of an upward-facing\u00a0parabola with a vertex at (0,1) and\u00a0labeled points at (-1, 4) and (3,4). A\u00a0line at y = 4 intersects the parabola at the labeled points.\" width=\"487\" height=\"445\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Using the graph, solve [latex]f\\left(x\\right)=1[\/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\/896\/2016\/10\/18191004\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q529772\">Solution<\/span><\/p>\n<div id=\"q529772\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=0[\/latex] or [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom9\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2471&amp;theme=oea&amp;iframe_resize_id=mom9\" width=\"100%\" height=\"550\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2886&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"600\"><\/iframe><\/p>\n<\/div>\n<h1>(3.2.3) &#8211; Vertical line test<\/h1>\n<p>When both the independent quantity (input) and the dependent quantity (output) are real numbers, a function can be represented by a graph in the coordinate plane. The independent value is plotted on the <i>x<\/i>-axis and the dependent value is plotted on the <i>y<\/i>-axis. The fact that each input value has exactly one output value means graphs of functions have certain characteristics. For each input on the graph, there will be exactly one output. For a function defined as [latex]y = f(x)[\/latex], or [latex]y[\/latex] is a function of [latex]x[\/latex], we would write ordered pairs [latex](x, f(x))[\/latex] using function notation instead of [latex](x,y)[\/latex] as you may have seen previously.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2679 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16195741\/Screen-Shot-2016-07-16-at-12.56.58-PM-300x281.png\" alt=\"Coordinate axes with y-axis labeled f(x) and the x axis labeled x.\" width=\"408\" height=\"382\" \/><\/p>\n<p>We can identify whether the graph of a relation represents a function\u00a0because for each <i>x<\/i>-coordinate there will be exactly one <i>y<\/i>-coordinate.<\/p>\n<p 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\/18232417\/image001.jpg\" alt=\"A graph of a semicircle. Four vertical lines cross the semicircle at one point each.\" width=\"304\" height=\"307\" \/><\/p>\n<p>When a vertical line is placed across the plot of this relation, it does not intersect the graph more than once for any values of [latex]x[\/latex].<\/p>\n<p>If, on the other hand, a graph shows two or more intersections with a vertical line, then an input (<i>x<\/i>-coordinate) can have more than one output (<i>y<\/i>-coordinate), and [latex]y[\/latex] is not a function of [latex]x[\/latex]. Examining the graph of a relation to determine if a vertical line would intersect with more than one point is a quick way to determine if the relation shown by the graph is a function. This method is often called the \u201cvertical line test.\u201d<\/p>\n<p>You try it.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the vertical line test to determine whether the relation plotted on this graph is a function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2680 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16200323\/Screen-Shot-2016-07-16-at-1.02.50-PM-300x275.png\" alt=\"Graph with circle plotted - center at (0,0) radius = 2, more points include (2,0), (-2,0)\" width=\"300\" height=\"275\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q28965\">Show Answer<\/span><\/p>\n<div id=\"q28965\" class=\"hidden-answer\" style=\"display: none\">\n<p>This relationship cannot be a function, because some of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-coordinates.<\/p>\n<p 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\/18232418\/image002.jpg\" alt=\"A circle with four vertical lines through it. Three of the lines cross the circle at two points, and one line crosses the edge of the circle at one point.\" width=\"340\" height=\"343\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The vertical line method can also be applied to a set of ordered pairs plotted on a coordinate plane to determine if the relation is a function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the ordered pairs<\/p>\n<p>[latex]\\{(\u22121,3),(\u22122,5),(\u22123,3),(\u22125,\u22123)\\}[\/latex], plotted on the graph below. Use the vertical line test to determine whether the set of ordered pairs represents a function.<\/p>\n<p 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\/18232420\/image003.jpg\" alt=\"The points (\u22121,3); (\u22122,5); (\u22123,3); and (\u22125,\u22123).\" width=\"329\" height=\"328\" \/><\/p>\n<p style=\"text-align: center;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q114452\">Show Answer<\/span><\/p>\n<div id=\"q114452\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">Drawing vertical lines through each point results in each line only touching one point. This means that none of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-coordinates, so this is a function.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2514 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/14213709\/Edit_Page_%E2%80%B9_Intermediate_Algebra_%E2%80%94_Pressbooks.png\" alt=\"Edit_Page_\u2039_Intermediate_Algebra_\u2014_Pressbooks\" width=\"298\" height=\"294\" \/><\/p>\n<p style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<p>In another set of ordered pairs, [latex]\\{(3,\u22121),(5,\u22122),(3,\u22123),(\u22123,5)\\}[\/latex], one of the inputs, 3, can produce two different outputs, [latex]\u22121[\/latex] and [latex]\u22123[\/latex]. You know what that means\u2014this set of ordered pairs is not a function. A plot confirms this.<\/p>\n<p 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\/18232421\/image004.jpg\" alt=\"The point negative 3, 5; the point 5, negative 2. A line through the points negative 1, 3; and the point negative 3, negative 3.\" width=\"337\" height=\"336\" \/><\/p>\n<p>Notice that a vertical line passes through two plotted points. One <i>x<\/i>-coordinate has multiple <i>y<\/i>-coordinates. This relation is not a function.<\/p>\n<p>In the following video we show another example of determining whether a graph represents a function using the vertical line test.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 1: Use the Vertical Line Test to Determine if a Graph Represents a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5Z8DaZPJLKY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-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><li>Question ID  2471. <strong>Authored by<\/strong>: Langkamp,Greg. <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":3,"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 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