{"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":"2016-09-16T21:33:32","modified_gmt":"2016-09-16T21:33:32","slug":"outcome-graphs-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tallahassee-intermediatealgebra\/chapter\/outcome-graphs-of-functions\/","title":{"raw":"Graphs of Functions","rendered":"Graphs of Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h2>Learning Objectives<\/h2>\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 important features of\u00a0the graphs of a quadratic functions of the form [latex]f(x)=ax^2+bx+c[\/latex]<\/li>\r\n \t<li>Graph a radical 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 \t<li>Identify how addition and subtraction can change the graph of a radical function<\/li>\r\n \t<li>Define one-to-one function<\/li>\r\n \t<li>Use the horizontal\u00a0line test to determine whether a function is one-to-one<\/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>.\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>\r\nChanging <i>b<\/i> moves the line of reflection, which is the vertical line that passes through the vertex ( the high or low point) of the parabola. It may help to know how calculate the vertex of a parabola to understand how changing the value of b in a function will change it's graph.\r\n\r\nTo find the vertex of the parabola, use the formula [latex] \\displaystyle \\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)[\/latex].\r\n\r\nFor example, if the function in consideration is [latex]f(x)=2x^2-3x+4[\/latex], to find the vertex, first calculate [latex]\\frac{-b}{2a}[\/latex]\r\n\r\na = 2, and b = -3, therefore\u00a0[latex]\\frac{-b}{2a}=\\frac{-(-3)}{2(2)}=\\frac{3}{4}[\/latex].\r\n\r\nThis is the x value of the vertex.\r\n\r\nNow evaluate the function at [latex]x =\\frac{3}{4}[\/latex] to get the corresponding y-value for the vertex.\r\n\r\n[latex]f\\left( \\frac{-b}{2a} \\right)=2\\left(\\frac{3}{4}\\right)^2-3\\left(\\frac{3}{4}\\right)+4=2\\left(\\frac{9}{16}\\right)-4+4=\\frac{9}{8}[\/latex].\r\n\r\nThe vertex is at the point [latex]\\left(\\frac{3}{4},\\frac{9}{8}\\right)[\/latex]. \u00a0This means that the vertical line of reflection passes through this point as well. \u00a0It is not easy to tell how changing the values for b will change the graph of a quadratic function, but if you find the vertex, you can tell how the graph will change.\r\n\r\nIn the next example we show how changing <em>b\u00a0<\/em>can change the graph of the quadratic function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMatch the following functions with their graph.\r\n\r\na)\u00a0[latex] \\displaystyle f(x)={{x}^{2}}+2x[\/latex]\r\n\r\nb)\u00a0[latex] \\displaystyle f(x)={{x}^{2}}-2x[\/latex]\r\n\r\na)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232450\/image021.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\nb)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232452\/image023.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n[reveal-answer q=\"320978\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"320978\"]\r\n\r\nFind the vertex of function a)[latex] \\displaystyle f(x)={{x}^{2}}+2x[\/latex].\r\n\r\na = 1, b = 2\r\n\r\nx-value:\r\n\r\n[latex]\\frac{-b}{2a}=\\frac{-2}{2(1)}=-1[\/latex]\r\n\r\ny-value:\r\n\r\n[latex]f\\left(\\frac{-b}{2a}\\right)=\\left(-1)^2+2\\left(-1)=1-2=-1[\/latex].\r\n\r\nVertex = [latex]\\left(-1,-1)[\/latex], which means the graph that best fits this function is a)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232450\/image021.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\nFind the vertex of function b)[latex] \\displaystyle f(x)={{x}^{2}}-2x[\/latex].\r\n\r\na = 1, b = -2\r\n\r\nx-value:\r\n\r\n[latex]\\frac{-b}{2a}=\\frac{2}{2(1)}=1[\/latex]\r\n\r\ny-value:\r\n\r\n[latex]f\\left(\\frac{-b}{2a}\\right)=\\left(1)^2-2\\left(1)=1-2=-1[\/latex].\r\n\r\nVertex = [latex]\\left(1,-1)[\/latex], which means the graph that best fits this function is b)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232450\/image021.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNote that the vertex can change if the value for c changes because the y-value of the vertex is calculated by substituting the x-value into the function. Here is a summary of how the changes to the values for a, b, and, c of a quadratic function can change it's graph.\r\n<h3>Properties of a Parabola<\/h3>\r\nFor [latex] \\displaystyle f(x)=a{{x}^{2}}+bx+c[\/latex], where <i>a<\/i>, <i>b<\/i>, and <i>c<\/i> are real numbers.\r\n<ul>\r\n \t<li>The parabola opens upward if <i>a<\/i> &gt; 0 and downward if <i>a<\/i> &lt; 0.<\/li>\r\n \t<li><i>a <\/i>changes the width of the parabola. The parabola gets narrower if |<i>a<\/i>| is &gt; 1 and wider if |<i>a<\/i>|&lt;1.<\/li>\r\n \t<li>The vertex depends on the values of <i>a<\/i>, <i>b<\/i>, and <i>c<\/i>. The vertex is [latex]\\left(\\frac{-b}{2a},f\\left( \\frac{-b}{2a}\\right)\\right)[\/latex].<\/li>\r\n<\/ul>\r\nIn the last example we show how you can use the properties of a parabola to help you make a graph without having to calculate an exhaustive table of values.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGraph [latex]f(x)=\u22122x^{2}+3x\u20133[\/latex].\r\n\r\n[reveal-answer q=\"992003\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"992003\"]Before making a table of values, look at the values of <i>a <\/i>and <i>c<\/i> to get a general idea of what the graph should look like.\r\n\r\n[latex]a=\u22122[\/latex], so the graph will open down and be thinner than [latex]f(x)=x^{2}[\/latex].\r\n\r\n[latex]c=\u22123[\/latex], so it will move to intercept the <i>y<\/i>-axis at\u00a0[latex](0,\u22123)[\/latex].\r\n\r\nTo find the vertex of the parabola, use the formula [latex] \\displaystyle \\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)[\/latex]. Finding the vertex may make graphing the parabola easier.\r\n<p style=\"text-align: center;\">[latex]\\text{Vertex }\\text{formula}=\\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)[\/latex]<\/p>\r\n<i>x<\/i>-coordinate of vertex:\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{-b}{2a}=\\frac{-(3)}{2(-2)}=\\frac{-3}{-4}=\\frac{3}{4}[\/latex]<\/p>\r\n<i>y<\/i>-coordinate of vertex:\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}f\\left( \\frac{-b}{2a} \\right)=f\\left( \\frac{3}{4} \\right)\\\\\\,\\,\\,f\\left( \\frac{3}{4} \\right)=-2{{\\left( \\frac{3}{4} \\right)}^{2}}+3\\left( \\frac{3}{4} \\right)-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2\\left( \\frac{9}{16} \\right)+\\frac{9}{4}-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{-18}{16}+\\frac{9}{4}-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{-9}{8}+\\frac{18}{8}-\\frac{24}{8}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-\\frac{15}{8}\\end{array}\\\\[\/latex]<\/p>\r\nVertex: [latex] \\displaystyle \\left( \\frac{3}{4},-\\frac{15}{8} \\right)\\\\[\/latex]\r\n\r\nUse the vertex, [latex] \\displaystyle \\left( \\frac{3}{4},-\\frac{15}{8} \\right)\\\\[\/latex], and the properties you described to get a general idea of the shape of the graph. You can create a table of values to verify your graph. Notice that in this table, the <i>x<\/i> values increase. The <i>y<\/i> values increase and then start to decrease again. That indicates a parabola.\r\n<table>\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]\u221217[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]\u22128[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\u22123[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]\u22125[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232454\/image024.gif\" alt=\"Vertex at negative three-fourths, negative 15-eighths. Other points are plotted: the point negative 2, negative 17; the point negative 1, negative 8; the point 0, negative 3; the point 1, negative 2; and the point 2, negative 5.\" width=\"309\" height=\"343\" \/>\r\n<h4>Answer<\/h4>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232456\/image025.gif\" alt=\"A parabola drawn through the points in the previous graph\" width=\"309\" height=\"344\" \/>\r\n\r\nConnect the points as best you can, using a <i>smooth curve<\/i>. Remember that the parabola is two mirror images, so if your points don\u2019t have pairs with the same value, you may want to include additional points (such as the ones in blue here). Plot points on either side of the vertex.\r\n\r\n[latex]x=\\frac{1}{2}[\/latex] and [latex]x=\\frac{3}{2}[\/latex] are good values to include.[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows another example of plotting a quadratic function using the vertex.\r\n\r\nhttps:\/\/youtu.be\/leYhH_-3rVo\r\n\r\nCreating a graph of a function is one way to understand the relationship between the inputs and outputs of that function. Creating a graph can be done by choosing values for <em>x<\/em>, finding the corresponding <em>y<\/em> values, and plotting them. However, it helps to understand the basic shape of the function. Knowing the effect of changes to the basic function equation is also helpful.\r\n\r\nOne common shape you will see is a parabola. Parabolas have the equation [latex]f(x)=ax^{2}+bx+c[\/latex], where <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> are real numbers and [latex]a\\ne0[\/latex]. The value of a determines the width and the direction of the parabola, while the vertex depends on the values of <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>. The vertex is [latex] \\displaystyle \\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)\\\\[\/latex].\r\n<h2>Graph Radical Functions<\/h2>\r\nYou can also graph radical functions (such as square root functions) by choosing values for <i>x<\/i> and finding points that will be on the graph. Again, it\u2019s helpful to have some idea about what the graph will look like.\r\n\r\nThink about the basic square root function, [latex]f(x)=\\sqrt{x}[\/latex]. Let\u2019s take a look at a table of values for <i>x<\/i> and <i>y<\/i> and then graph the function. (Notice that all the values for <i>x<\/i> in the table are perfect squares. Since you are taking the square root of <i>x<\/i>, using perfect squares makes more sense than just finding the square roots of 0, 1, 2, 3, 4, etc.)\r\n<div align=\"center\">\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>\r\n<p align=\"center\">0<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">0<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">1<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">1<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">4<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">2<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">9<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">3<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">16<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">4<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nRecall that <i>x<\/i> can never be negative because when you square a real number, the result is always positive. For example, [latex]\\sqrt{49}[\/latex], this\u00a0means \"find the number whose square is 49.\" \u00a0Since there is no real number that we can square and get a negative, the function [latex]f(x)=\\sqrt{x}[\/latex] will be defined for [latex]x&gt;0[\/latex].\r\n\r\nTake a look at the graph.\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\/18232457\/image026.gif\" alt=\"Curved line branching up and right from the point 0,0.\" width=\"258\" height=\"288\" \/><\/p>\r\nAs with parabolas, multiplying and adding numbers makes some changes, but the basic shape is still the same. Here are some examples.<i>\u00a0<\/i>\r\n\r\nMultiplying [latex]\\sqrt{x}[\/latex] by a positive value changes the width of the half-parabola. Multiplying [latex]\\sqrt{x}[\/latex] by a negative number gives you the other half of a horizontal parabola.\r\n\r\nIn the following example we will show how multiplying a radical function by a constant can change the shape of the graph.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMatch each of the following functions to the graph that it represents.\r\n\r\na)\u00a0[latex]f(x)=-\\sqrt{x}[\/latex]\r\n\r\nb)[latex]f(x)=2\\sqrt{x}[\/latex]\r\n\r\nc)\u00a0[latex]f(x)=\\frac{1}{2}\\sqrt{x}[\/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\/18232459\/image027.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line going right and further up than the black line.\" width=\"175\" height=\"195\" \/>\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\/18232500\/image028.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line that is a mirror of the black line over the x axis so that the red line is going to the right and down.\" width=\"175\" height=\"195\" \/>\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\/18232501\/image029.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A red line that goes to the right and not as far up as the black line.\" width=\"175\" height=\"195\" \/>\r\n[reveal-answer q=\"651190\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"651190\"]\r\n\r\nFunction\u00a0a)\u00a0[latex]f(x)=-\\sqrt{x}[\/latex] means that all the outputs will be negative - the function is the negative of the square roots of the input. \u00a0This will give the other half of the parabola on it's side. \u00a0Therefore the graph\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232500\/image028.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line that is a mirror of the black line over the x axis so that the red line is going to the right and down.\" width=\"175\" height=\"195\" \/>\r\n\r\ngoes with the function\u00a0[latex]f(x)=-\\sqrt{x}[\/latex]\r\n\r\nFunction b)[latex]f(x)=2\\sqrt{x}[\/latex] means take the square root of all the inputs, then multiply by two, so the outputs will be larger than the outputs for [latex]\\sqrt{x}[\/latex]. \u00a0The graph\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232459\/image027.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line going right and further up than the black line.\" width=\"175\" height=\"195\" \/>\r\n\r\ngoes with the function\u00a0[latex]f(x)=2\\sqrt{x}[\/latex]\r\n\r\nFunction\u00a0c)\u00a0[latex]f(x)=\\frac{1}{2}\\sqrt{x}[\/latex] means take the squat=re root of the inputs then multiply by [latex]\\frac{1}{2}[\/latex]. The outputs will be smaller than the outputs for\u00a0[latex]\\sqrt{x}[\/latex]. The graph\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232501\/image029.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A red line that goes to the right and not as far up as the black line.\" width=\"175\" height=\"195\" \/>\r\n\r\ngoes with the function\u00a0[latex]f(x)=\\frac{1}{2}\\sqrt{x}[\/latex].\r\n<h4>Answer<\/h4>\r\nFunction a) goes with graph b)\r\n\r\nFunction b) goes with graph a)\r\n\r\nFunction c) goes with graph c)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAdding a value <i>outside <\/i>the radical moves the graph up or down. Think about it as adding the value to the basic <i>y<\/i> value of [latex] \\sqrt{x}[\/latex], so a positive value added moves the graph up.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMatch each of the following functions to the graph that it represents.\r\n\r\na)\u00a0[latex] f(x)=\\sqrt{x}+3[\/latex]\r\n\r\nb)[latex] f(x)=\\sqrt{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\/18232502\/image030.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line but starting at 0, negative 2.\" width=\"175\" height=\"195\" \/>\r\n\r\nb)\r\n\r\n<img class=\"wp-image-2062 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01215107\/DM_U17_Final1stEd-11-12-12-270x300.jpg\" alt=\"DM_U17_Final1stEd-11-12-12\" width=\"175\" height=\"194\" \/>\r\n\r\n[reveal-answer q=\"928501\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"928501\"]\r\n\r\nFunction a)\u00a0[latex] f(x)=\\sqrt{x}+3[\/latex] means take the square root of all the inputs and add three, so the out puts will be greater than those for [latex]\\sqrt{x}[\/latex], therefore the graph that goes with this function is\r\n\r\n<img class=\"wp-image-2062 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01215107\/DM_U17_Final1stEd-11-12-12-270x300.jpg\" alt=\"DM_U17_Final1stEd-11-12-12\" width=\"175\" height=\"194\" \/>\r\n\r\nFunction\u00a0b) [latex]f(x)=\\sqrt{x}-2[\/latex] means take the square root of the input then subtract two. The outputs will be less than those for\u00a0[latex]\\sqrt{x}[\/latex], therefore the graph that goes with this function is\r\n<img class=\"wp-image-2062 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01215107\/DM_U17_Final1stEd-11-12-12-270x300.jpg\" alt=\"DM_U17_Final1stEd-11-12-12\" width=\"175\" height=\"194\" \/>\r\n<h4>Answer<\/h4>\r\nFunction a) goes with graph b)\r\n\r\nFunction b) goes with graph a)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div align=\"center\"><\/div>\r\nAdding a value <i>inside <\/i>the radical moves the graph left or right. Think about it as adding a value to <i>x<\/i> before you take the square root\u2014so the <i>y<\/i> value gets moved to a different <i>x<\/i> value. For example, for [latex]f(x)=\\sqrt{x}[\/latex], the square root is 3 if [latex]x=9[\/latex]. For[latex]f(x)=\\sqrt{x+1}[\/latex], the square root is 3 when [latex]x+1[\/latex] is 9, so <i>x<\/i> is 8. Changing <i>x<\/i> to [latex]x+1[\/latex] shifts the graph to the left by 1 unit (from 9 to 8). Changing <i>x<\/i> to [latex]x\u22122[\/latex] shifts the graph to the right by 2 units.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMatch each of the following functions to the graph that it represents.\r\n\r\na)\u00a0[latex]f(x)=\\sqrt{x+1}[\/latex]\r\n\r\nb)[latex]f(x)=\\sqrt{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\/18232504\/image032.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0\" width=\"175\" height=\"195\" \/>\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\/18232503\/image031.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0\" width=\"175\" height=\"195\" \/>\r\n[reveal-answer q=\"602483\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"602483\"]\r\n\r\nFunction a)\u00a0[latex]f(x)=\\sqrt{x+1}[\/latex] adds one to the inputs before the square root is taken. \u00a0The outputs will be greater, so it ends up looking like a shift to the left. The graph that matches this function is\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232503\/image031.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0\" width=\"175\" height=\"195\" \/>\r\n\r\nFunction b)\u00a0[latex]f(x)=\\sqrt{x-2}[\/latex] means subtract before the square root is taken. \u00a0This makes the outputs less than they would be for the standard [latex]\\sqrt{x}[\/latex], and looks like a shift to the right. the graph that matches this function is\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232504\/image032.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0\" width=\"175\" height=\"195\" \/>\r\n<h4>Answer<\/h4>\r\nFunction a) matches graph b)\r\n\r\nFunction b) matches graph a)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div align=\"center\"><\/div>\r\nNotice that as <i>x<\/i> gets greater, adding or subtracting a number inside the square root has less of an effect on the value of <i>y<\/i>!\r\n\r\nIn the next example we will combine some of the changes that we have seen into one function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGraph [latex] f(x)=-2+\\sqrt{x-1}[\/latex].\r\n\r\n[reveal-answer q=\"493141\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"493141\"]Before making a table of values, look at the function equation to get a general idea what the graph should look like.\r\n\r\nInside the square root, you\u2019re subtracting 1, so the graph will move to the right 1 from the basic [latex] f(x)=\\sqrt{x}[\/latex] graph.\r\n\r\nYou\u2019re also adding [latex]\u22122[\/latex] outside the square root, so the graph will move down two from the basic [latex] f(x)=\\sqrt{x}[\/latex] graph.\r\n\r\nCreate a table of values. Choose values that will make your calculations easy. You want [latex]x\u20131[\/latex] to be a perfect square (0, 1, 4, 9, and so on) so you can take the square root.\r\n<table style=\"width: 20%;\">\r\n<thead>\r\n<tr>\r\n<th>x<\/th>\r\n<th>f(x)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]\u22122[\/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<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince values of <em>x<\/em> less than 1 makes the value inside the square root negative, there will be no points on the coordinate graph to the left of [latex]x=1[\/latex]. There is no need to choose x values less than 1 for your table!\r\n\r\nUse the table pairs to plot points.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232503\/image031.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0\" width=\"175\" height=\"195\" \/>\r\n\r\nConnect the points as best you can, using a smooth curve.\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\/18232504\/image032.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0\" width=\"175\" height=\"195\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Identify a One-to-One Function<\/h2>\r\nRemember that in a function, the input value must have one and only one value for the output.\u00a0There is a name for the set of input values and another name for the set of output values for a function. The set of input values is called the <b>domain of the function<\/b>. And the set of output values is called the <b>range of the function<\/b>.\r\n\r\nIn the first example we remind you how to define domain and range using a table of values.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the domain and range for the function.\r\n<table style=\"width: 20%;\">\r\n<thead>\r\n<tr>\r\n<th>\r\n<p align=\"center\"><i>x<\/i><\/p>\r\n<\/th>\r\n<th>\r\n<p align=\"center\"><i>y<\/i><\/p>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p align=\"center\">\u22125<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">\u22126<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">\u22122<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">\u22121<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">\u22121<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">0<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">0<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">3<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">5<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">15<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"130987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"130987\"]\r\nThe domain is the set of inputs or <i>x<\/i>-coordinates.\r\n<p align=\"center\">[latex]\\{\u22125,\u22122,\u22121,0,5\\}[\/latex]<\/p>\r\nThe range is the set of outputs of <i>y<\/i>-coordinates.\r\n<p align=\"center\">[latex]\\{\u22126,\u22121,0,3,15\\}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\begin{array}{l}\\text{Domain}:\\{\u22125,\u22122,\u22121,0,5\\}\\\\\\text{Range}:\\{\u22126,\u22121,0,3,15\\}\\end{array}\\\\[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show another example of finding domain and range from tabular data.\r\n\r\nhttps:\/\/youtu.be\/GPBq18fCEv4\r\n\r\nSome functions have a given output value that corresponds to two or more input values. For example, in the following stock chart the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200506\/CNX_Precalc_Figure_01_00_001n2.jpg\" alt=\"Figure of a bull and a graph of market prices.\" width=\"975\" height=\"307\" data-media-type=\"image\/jpg\" \/>\r\n<p id=\"fs-id1165135678633\">However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in.<\/p>\r\n\r\n<table style=\"width: 20%;\" summary=\"Two columns and five rows. The first column is labeled,\"><colgroup> <col data-align=\"center\" \/> <col data-align=\"center\" \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th>Letter grade<\/th>\r\n<th>Grade point average<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>A<\/td>\r\n<td>4.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B<\/td>\r\n<td>3.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C<\/td>\r\n<td>2.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<\/td>\r\n<td>1.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137561844\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\r\nTo visualize this concept, let\u2019s look again at the two simple functions sketched in (a)and (b) of Figure 10.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200453\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" data-media-type=\"image\/jpg\" \/> <b>Figure 10<\/b>[\/caption]\r\n\r\nThe function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.\r\n<div class=\"textbox\">\r\n<h3>A General Note: One-to-One Function<\/h3>\r\nA one-to-one function is a function in which each output value corresponds to exactly one input value.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhich table represents a one-to-one function?\r\n\r\na)\r\n<table style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<td>input<\/td>\r\n<td>output<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>12<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>-1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-5<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nb)\r\n<table style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<td>input<\/td>\r\n<td>output<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>8<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>16<\/td>\r\n<td>32<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>32<\/td>\r\n<td>64<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>64<\/td>\r\n<td>128<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"945171\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"945171\"]\r\n\r\nTable a) maps the output value 2 to two different input values, therefore\u00a0this is NOT a one-to-one function.\r\n\r\nTable b) maps each output to one unique input, therefore this IS a one-to-one function.\r\n<h4>Answer<\/h4>\r\nTable b) is one-to-one\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show an example of using tables of values to determine whether a function is one-to-one.\r\n\r\nhttps:\/\/youtu.be\/QFOJmevha_Y\r\n<h2 style=\"text-align: left;\">Using the Horizontal Line Test<\/h2>\r\n<p id=\"fs-id1165137871503\">An easy way to determine whether a function\u00a0is a one-to-one function is to use the <strong>horizontal line test <\/strong>on the graph of the function. \u00a0To do this, draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.<\/p>\r\n\r\n<div class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\r\n<h3 style=\"text-align: center;\"><strong>How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.<\/strong><\/h3>\r\n<ol id=\"fs-id1165137611853\" data-number-style=\"arabic\">\r\n \t<li>Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\r\n \t<li>If there is any such line, determine that the function is not one-to-one.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\n<span data-type=\"media\" data-alt=\"Graph of a polynomial.\"><span data-type=\"media\" data-alt=\"Graph of a polynomial.\">For\u00a0the following graphs, determine which represent one-to-one functions.\r\n<\/span><\/span>\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200511\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" data-media-type=\"image\/jpg\" \/>\r\n[reveal-answer q=\"783411\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"783411\"]\r\n<p id=\"fs-id1165135185190\">The function in (a) is\u00a0not one-to-one. Using the horizontal line test, as shown below, it intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)<\/p>\r\n\r\n<figure id=\"Figure_01_01_010\" class=\"small\"><span data-type=\"media\" data-alt=\"\"><span data-type=\"media\" data-alt=\"\">\u00a0<\/span><\/span><\/figure><figure id=\"Figure_01_01_010\" class=\"small\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200515\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"\" width=\"487\" height=\"445\" data-media-type=\"image\/jpg\" \/><\/figure>The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.\r\n\r\n<img class=\"size-medium wp-image-2697 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212200\/Screen-Shot-2016-07-16-at-2.21.48-PM-287x300.png\" alt=\"Graph of a line with three dashed horizontal lines passing through it.\" width=\"287\" height=\"300\" \/>\r\n\r\nThe function (c) is not one-to-one, and is in fact not a function.\r\n\r\n<img class=\" wp-image-2698 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212527\/Screen-Shot-2016-07-16-at-2.25.36-PM-237x300.png\" alt=\"Graph of a circle with two dashed lines passing through horizontally\" width=\"279\" height=\"353\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video provides another example of using the horizontal line test to determine whether a graph represents a one-to-one function.\r\nhttps:\/\/youtu.be\/tbSGdcSN8RE\r\n<h2>Summary<\/h2>\r\nIn real life and in algebra, different variables are often linked. When a change in value of one variable causes a change in the value of another variable, their interaction is called a relation. A relation has an input value which corresponds to an output value. When each input value has one and only one output value, that relation is a function. Functions can be written as ordered pairs, tables, or graphs. The set of input values is called the domain, and the set of output values is called the range.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h2>Learning Objectives<\/h2>\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 important features of\u00a0the graphs of a quadratic functions of the form [latex]f(x)=ax^2+bx+c[\/latex]<\/li>\n<li>Graph a radical function using a table of values<\/li>\n<li>Identify\u00a0how multiplication can change the graph of a radical function<\/li>\n<li>Identify how addition and subtraction can change the graph of a radical function<\/li>\n<li>Define one-to-one function<\/li>\n<li>Use the horizontal\u00a0line test to determine whether a function is one-to-one<\/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>.<\/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<p>Changing <i>b<\/i> moves the line of reflection, which is the vertical line that passes through the vertex ( the high or low point) of the parabola. It may help to know how calculate the vertex of a parabola to understand how changing the value of b in a function will change it&#8217;s graph.<\/p>\n<p>To find the vertex of the parabola, use the formula [latex]\\displaystyle \\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)[\/latex].<\/p>\n<p>For example, if the function in consideration is [latex]f(x)=2x^2-3x+4[\/latex], to find the vertex, first calculate [latex]\\frac{-b}{2a}[\/latex]<\/p>\n<p>a = 2, and b = -3, therefore\u00a0[latex]\\frac{-b}{2a}=\\frac{-(-3)}{2(2)}=\\frac{3}{4}[\/latex].<\/p>\n<p>This is the x value of the vertex.<\/p>\n<p>Now evaluate the function at [latex]x =\\frac{3}{4}[\/latex] to get the corresponding y-value for the vertex.<\/p>\n<p>[latex]f\\left( \\frac{-b}{2a} \\right)=2\\left(\\frac{3}{4}\\right)^2-3\\left(\\frac{3}{4}\\right)+4=2\\left(\\frac{9}{16}\\right)-4+4=\\frac{9}{8}[\/latex].<\/p>\n<p>The vertex is at the point [latex]\\left(\\frac{3}{4},\\frac{9}{8}\\right)[\/latex]. \u00a0This means that the vertical line of reflection passes through this point as well. \u00a0It is not easy to tell how changing the values for b will change the graph of a quadratic function, but if you find the vertex, you can tell how the graph will change.<\/p>\n<p>In the next example we show how changing <em>b\u00a0<\/em>can change the graph of the quadratic function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Match the following functions with their graph.<\/p>\n<p>a)\u00a0[latex]\\displaystyle f(x)={{x}^{2}}+2x[\/latex]<\/p>\n<p>b)\u00a0[latex]\\displaystyle f(x)={{x}^{2}}-2x[\/latex]<\/p>\n<p>a)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232450\/image021.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232452\/image023.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q320978\">Show Answer<\/span><\/p>\n<div id=\"q320978\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the vertex of function a)[latex]\\displaystyle f(x)={{x}^{2}}+2x[\/latex].<\/p>\n<p>a = 1, b = 2<\/p>\n<p>x-value:<\/p>\n<p>[latex]\\frac{-b}{2a}=\\frac{-2}{2(1)}=-1[\/latex]<\/p>\n<p>y-value:<\/p>\n<p>[latex]f\\left(\\frac{-b}{2a}\\right)=\\left(-1)^2+2\\left(-1)=1-2=-1[\/latex].<\/p>\n<p>Vertex = [latex]\\left(-1,-1)[\/latex], which means the graph that best fits this function is a)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232450\/image021.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<p>Find the vertex of function b)[latex]\\displaystyle f(x)={{x}^{2}}-2x[\/latex].<\/p>\n<p>a = 1, b = -2<\/p>\n<p>x-value:<\/p>\n<p>[latex]\\frac{-b}{2a}=\\frac{2}{2(1)}=1[\/latex]<\/p>\n<p>y-value:<\/p>\n<p>[latex]f\\left(\\frac{-b}{2a}\\right)=\\left(1)^2-2\\left(1)=1-2=-1[\/latex].<\/p>\n<p>Vertex = [latex]\\left(1,-1)[\/latex], which means the graph that best fits this function is b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232450\/image021.gif\" alt=\"compared to g(x)=x squared\" width=\"182\" height=\"197\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Note that the vertex can change if the value for c changes because the y-value of the vertex is calculated by substituting the x-value into the function. Here is a summary of how the changes to the values for a, b, and, c of a quadratic function can change it&#8217;s graph.<\/p>\n<h3>Properties of a Parabola<\/h3>\n<p>For [latex]\\displaystyle f(x)=a{{x}^{2}}+bx+c[\/latex], where <i>a<\/i>, <i>b<\/i>, and <i>c<\/i> are real numbers.<\/p>\n<ul>\n<li>The parabola opens upward if <i>a<\/i> &gt; 0 and downward if <i>a<\/i> &lt; 0.<\/li>\n<li><i>a <\/i>changes the width of the parabola. The parabola gets narrower if |<i>a<\/i>| is &gt; 1 and wider if |<i>a<\/i>|&lt;1.<\/li>\n<li>The vertex depends on the values of <i>a<\/i>, <i>b<\/i>, and <i>c<\/i>. The vertex is [latex]\\left(\\frac{-b}{2a},f\\left( \\frac{-b}{2a}\\right)\\right)[\/latex].<\/li>\n<\/ul>\n<p>In the last example we show how you can use the properties of a parabola to help you make a graph without having to calculate an exhaustive table of values.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph [latex]f(x)=\u22122x^{2}+3x\u20133[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q992003\">Show Solution<\/span><\/p>\n<div id=\"q992003\" class=\"hidden-answer\" style=\"display: none\">Before making a table of values, look at the values of <i>a <\/i>and <i>c<\/i> to get a general idea of what the graph should look like.<\/p>\n<p>[latex]a=\u22122[\/latex], so the graph will open down and be thinner than [latex]f(x)=x^{2}[\/latex].<\/p>\n<p>[latex]c=\u22123[\/latex], so it will move to intercept the <i>y<\/i>-axis at\u00a0[latex](0,\u22123)[\/latex].<\/p>\n<p>To find the vertex of the parabola, use the formula [latex]\\displaystyle \\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)[\/latex]. Finding the vertex may make graphing the parabola easier.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Vertex }\\text{formula}=\\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)[\/latex]<\/p>\n<p><i>x<\/i>-coordinate of vertex:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{-b}{2a}=\\frac{-(3)}{2(-2)}=\\frac{-3}{-4}=\\frac{3}{4}[\/latex]<\/p>\n<p><i>y<\/i>-coordinate of vertex:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}f\\left( \\frac{-b}{2a} \\right)=f\\left( \\frac{3}{4} \\right)\\\\\\,\\,\\,f\\left( \\frac{3}{4} \\right)=-2{{\\left( \\frac{3}{4} \\right)}^{2}}+3\\left( \\frac{3}{4} \\right)-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2\\left( \\frac{9}{16} \\right)+\\frac{9}{4}-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{-18}{16}+\\frac{9}{4}-3\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{-9}{8}+\\frac{18}{8}-\\frac{24}{8}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=-\\frac{15}{8}\\end{array}\\\\[\/latex]<\/p>\n<p>Vertex: [latex]\\displaystyle \\left( \\frac{3}{4},-\\frac{15}{8} \\right)\\\\[\/latex]<\/p>\n<p>Use the vertex, [latex]\\displaystyle \\left( \\frac{3}{4},-\\frac{15}{8} \\right)\\\\[\/latex], and the properties you described to get a general idea of the shape of the graph. You can create a table of values to verify your graph. Notice that in this table, the <i>x<\/i> values increase. The <i>y<\/i> values increase and then start to decrease again. That indicates a parabola.<\/p>\n<table>\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]\u221217[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u22121[\/latex]<\/td>\n<td>[latex]\u22128[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]\u22122[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\u22125[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232454\/image024.gif\" alt=\"Vertex at negative three-fourths, negative 15-eighths. Other points are plotted: the point negative 2, negative 17; the point negative 1, negative 8; the point 0, negative 3; the point 1, negative 2; and the point 2, negative 5.\" width=\"309\" height=\"343\" \/><\/p>\n<h4>Answer<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232456\/image025.gif\" alt=\"A parabola drawn through the points in the previous graph\" width=\"309\" height=\"344\" \/><\/p>\n<p>Connect the points as best you can, using a <i>smooth curve<\/i>. Remember that the parabola is two mirror images, so if your points don\u2019t have pairs with the same value, you may want to include additional points (such as the ones in blue here). Plot points on either side of the vertex.<\/p>\n<p>[latex]x=\\frac{1}{2}[\/latex] and [latex]x=\\frac{3}{2}[\/latex] are good values to include.<\/p><\/div>\n<\/div>\n<\/div>\n<p>The following video shows another example of plotting a quadratic function using the vertex.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Graph a Quadratic Function Using a Table of Value and the Vertex\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/leYhH_-3rVo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Creating a graph of a function is one way to understand the relationship between the inputs and outputs of that function. Creating a graph can be done by choosing values for <em>x<\/em>, finding the corresponding <em>y<\/em> values, and plotting them. However, it helps to understand the basic shape of the function. Knowing the effect of changes to the basic function equation is also helpful.<\/p>\n<p>One common shape you will see is a parabola. Parabolas have the equation [latex]f(x)=ax^{2}+bx+c[\/latex], where <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> are real numbers and [latex]a\\ne0[\/latex]. The value of a determines the width and the direction of the parabola, while the vertex depends on the values of <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>. The vertex is [latex]\\displaystyle \\left( \\frac{-b}{2a},f\\left( \\frac{-b}{2a} \\right) \\right)\\\\[\/latex].<\/p>\n<h2>Graph Radical Functions<\/h2>\n<p>You can also graph radical functions (such as square root functions) by choosing values for <i>x<\/i> and finding points that will be on the graph. Again, it\u2019s helpful to have some idea about what the graph will look like.<\/p>\n<p>Think about the basic square root function, [latex]f(x)=\\sqrt{x}[\/latex]. Let\u2019s take a look at a table of values for <i>x<\/i> and <i>y<\/i> and then graph the function. (Notice that all the values for <i>x<\/i> in the table are perfect squares. Since you are taking the square root of <i>x<\/i>, using perfect squares makes more sense than just finding the square roots of 0, 1, 2, 3, 4, etc.)<\/p>\n<div style=\"margin: auto;\">\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>\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">9<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">16<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Recall that <i>x<\/i> can never be negative because when you square a real number, the result is always positive. For example, [latex]\\sqrt{49}[\/latex], this\u00a0means &#8220;find the number whose square is 49.&#8221; \u00a0Since there is no real number that we can square and get a negative, the function [latex]f(x)=\\sqrt{x}[\/latex] will be defined for [latex]x>0[\/latex].<\/p>\n<p>Take a look at the graph.<\/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\/18232457\/image026.gif\" alt=\"Curved line branching up and right from the point 0,0.\" width=\"258\" height=\"288\" \/><\/p>\n<p>As with parabolas, multiplying and adding numbers makes some changes, but the basic shape is still the same. Here are some examples.<i>\u00a0<\/i><\/p>\n<p>Multiplying [latex]\\sqrt{x}[\/latex] by a positive value changes the width of the half-parabola. Multiplying [latex]\\sqrt{x}[\/latex] by a negative number gives you the other half of a horizontal parabola.<\/p>\n<p>In the following example we will show how multiplying a radical function by a constant can change the shape of the graph.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Match each of the following functions to the graph that it represents.<\/p>\n<p>a)\u00a0[latex]f(x)=-\\sqrt{x}[\/latex]<\/p>\n<p>b)[latex]f(x)=2\\sqrt{x}[\/latex]<\/p>\n<p>c)\u00a0[latex]f(x)=\\frac{1}{2}\\sqrt{x}[\/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\/18232459\/image027.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line going right and further up than the black line.\" width=\"175\" height=\"195\" \/><\/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\/18232500\/image028.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line that is a mirror of the black line over the x axis so that the red line is going to the right and down.\" width=\"175\" height=\"195\" \/><\/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\/18232501\/image029.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A red line that goes to the right and not as far up as the black line.\" width=\"175\" height=\"195\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q651190\">Show Answer<\/span><\/p>\n<div id=\"q651190\" class=\"hidden-answer\" style=\"display: none\">\n<p>Function\u00a0a)\u00a0[latex]f(x)=-\\sqrt{x}[\/latex] means that all the outputs will be negative &#8211; the function is the negative of the square roots of the input. \u00a0This will give the other half of the parabola on it&#8217;s side. \u00a0Therefore the graph<\/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\/18232500\/image028.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line that is a mirror of the black line over the x axis so that the red line is going to the right and down.\" width=\"175\" height=\"195\" \/><\/p>\n<p>goes with the function\u00a0[latex]f(x)=-\\sqrt{x}[\/latex]<\/p>\n<p>Function b)[latex]f(x)=2\\sqrt{x}[\/latex] means take the square root of all the inputs, then multiply by two, so the outputs will be larger than the outputs for [latex]\\sqrt{x}[\/latex]. \u00a0The graph<\/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\/18232459\/image027.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line going right and further up than the black line.\" width=\"175\" height=\"195\" \/><\/p>\n<p>goes with the function\u00a0[latex]f(x)=2\\sqrt{x}[\/latex]<\/p>\n<p>Function\u00a0c)\u00a0[latex]f(x)=\\frac{1}{2}\\sqrt{x}[\/latex] means take the squat=re root of the inputs then multiply by [latex]\\frac{1}{2}[\/latex]. The outputs will be smaller than the outputs for\u00a0[latex]\\sqrt{x}[\/latex]. The graph<\/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\/18232501\/image029.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A red line that goes to the right and not as far up as the black line.\" width=\"175\" height=\"195\" \/><\/p>\n<p>goes with the function\u00a0[latex]f(x)=\\frac{1}{2}\\sqrt{x}[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>Function a) goes with graph b)<\/p>\n<p>Function b) goes with graph a)<\/p>\n<p>Function c) goes with graph c)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Adding a value <i>outside <\/i>the radical moves the graph up or down. Think about it as adding the value to the basic <i>y<\/i> value of [latex]\\sqrt{x}[\/latex], so a positive value added moves the graph up.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Match each of the following functions to the graph that it represents.<\/p>\n<p>a)\u00a0[latex]f(x)=\\sqrt{x}+3[\/latex]<\/p>\n<p>b)[latex]f(x)=\\sqrt{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\/18232502\/image030.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line but starting at 0, negative 2.\" width=\"175\" height=\"195\" \/><\/p>\n<p>b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2062 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01215107\/DM_U17_Final1stEd-11-12-12-270x300.jpg\" alt=\"DM_U17_Final1stEd-11-12-12\" width=\"175\" height=\"194\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q928501\">Show Answer<\/span><\/p>\n<div id=\"q928501\" class=\"hidden-answer\" style=\"display: none\">\n<p>Function a)\u00a0[latex]f(x)=\\sqrt{x}+3[\/latex] means take the square root of all the inputs and add three, so the out puts will be greater than those for [latex]\\sqrt{x}[\/latex], therefore the graph that goes with this function is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2062 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01215107\/DM_U17_Final1stEd-11-12-12-270x300.jpg\" alt=\"DM_U17_Final1stEd-11-12-12\" width=\"175\" height=\"194\" \/><\/p>\n<p>Function\u00a0b) [latex]f(x)=\\sqrt{x}-2[\/latex] means take the square root of the input then subtract two. The outputs will be less than those for\u00a0[latex]\\sqrt{x}[\/latex], therefore the graph that goes with this function is<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2062 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01215107\/DM_U17_Final1stEd-11-12-12-270x300.jpg\" alt=\"DM_U17_Final1stEd-11-12-12\" width=\"175\" height=\"194\" \/><\/p>\n<h4>Answer<\/h4>\n<p>Function a) goes with graph b)<\/p>\n<p>Function b) goes with graph a)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"margin: auto;\"><\/div>\n<p>Adding a value <i>inside <\/i>the radical moves the graph left or right. Think about it as adding a value to <i>x<\/i> before you take the square root\u2014so the <i>y<\/i> value gets moved to a different <i>x<\/i> value. For example, for [latex]f(x)=\\sqrt{x}[\/latex], the square root is 3 if [latex]x=9[\/latex]. For[latex]f(x)=\\sqrt{x+1}[\/latex], the square root is 3 when [latex]x+1[\/latex] is 9, so <i>x<\/i> is 8. Changing <i>x<\/i> to [latex]x+1[\/latex] shifts the graph to the left by 1 unit (from 9 to 8). Changing <i>x<\/i> to [latex]x\u22122[\/latex] shifts the graph to the right by 2 units.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Match each of the following functions to the graph that it represents.<\/p>\n<p>a)\u00a0[latex]f(x)=\\sqrt{x+1}[\/latex]<\/p>\n<p>b)[latex]f(x)=\\sqrt{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\/18232504\/image032.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0\" width=\"175\" height=\"195\" \/><\/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\/18232503\/image031.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0\" width=\"175\" height=\"195\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q602483\">Show Answer<\/span><\/p>\n<div id=\"q602483\" class=\"hidden-answer\" style=\"display: none\">\n<p>Function a)\u00a0[latex]f(x)=\\sqrt{x+1}[\/latex] adds one to the inputs before the square root is taken. \u00a0The outputs will be greater, so it ends up looking like a shift to the left. The graph that matches this function is<\/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\/18232503\/image031.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0\" width=\"175\" height=\"195\" \/><\/p>\n<p>Function b)\u00a0[latex]f(x)=\\sqrt{x-2}[\/latex] means subtract before the square root is taken. \u00a0This makes the outputs less than they would be for the standard [latex]\\sqrt{x}[\/latex], and looks like a shift to the right. the graph that matches this function is<\/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\/18232504\/image032.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0\" width=\"175\" height=\"195\" \/><\/p>\n<h4>Answer<\/h4>\n<p>Function a) matches graph b)<\/p>\n<p>Function b) matches graph a)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"margin: auto;\"><\/div>\n<p>Notice that as <i>x<\/i> gets greater, adding or subtracting a number inside the square root has less of an effect on the value of <i>y<\/i>!<\/p>\n<p>In the next example we will combine some of the changes that we have seen into one function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Graph [latex]f(x)=-2+\\sqrt{x-1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q493141\">Show Solution<\/span><\/p>\n<div id=\"q493141\" class=\"hidden-answer\" style=\"display: none\">Before making a table of values, look at the function equation to get a general idea what the graph should look like.<\/p>\n<p>Inside the square root, you\u2019re subtracting 1, so the graph will move to the right 1 from the basic [latex]f(x)=\\sqrt{x}[\/latex] graph.<\/p>\n<p>You\u2019re also adding [latex]\u22122[\/latex] outside the square root, so the graph will move down two from the basic [latex]f(x)=\\sqrt{x}[\/latex] graph.<\/p>\n<p>Create a table of values. Choose values that will make your calculations easy. You want [latex]x\u20131[\/latex] to be a perfect square (0, 1, 4, 9, and so on) so you can take the square root.<\/p>\n<table style=\"width: 20%;\">\n<thead>\n<tr>\n<th>x<\/th>\n<th>f(x)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]\u22122[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since values of <em>x<\/em> less than 1 makes the value inside the square root negative, there will be no points on the coordinate graph to the left of [latex]x=1[\/latex]. There is no need to choose x values less than 1 for your table!<\/p>\n<p>Use the table pairs to plot points.<\/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\/18232503\/image031.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from negative 1, 0\" width=\"175\" height=\"195\" \/><\/p>\n<p>Connect the points as best you can, using a smooth curve.<\/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\/18232504\/image032.gif\" alt=\"A black curving line going up and right labeled g(x) equals the square root of x. A curved red line much like the black line starting from 2,0\" width=\"175\" height=\"195\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Identify a One-to-One Function<\/h2>\n<p>Remember that in a function, the input value must have one and only one value for the output.\u00a0There is a name for the set of input values and another name for the set of output values for a function. The set of input values is called the <b>domain of the function<\/b>. And the set of output values is called the <b>range of the function<\/b>.<\/p>\n<p>In the first example we remind you how to define domain and range using a table of values.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the domain and range for the function.<\/p>\n<table style=\"width: 20%;\">\n<thead>\n<tr>\n<th>\n<p style=\"text-align: center;\"><i>x<\/i><\/p>\n<\/th>\n<th>\n<p style=\"text-align: center;\"><i>y<\/i><\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\n<p style=\"text-align: center;\">\u22125<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">\u22126<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">\u22122<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">\u22121<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">\u22121<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">15<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q130987\">Show Solution<\/span><\/p>\n<div id=\"q130987\" class=\"hidden-answer\" style=\"display: none\">\nThe domain is the set of inputs or <i>x<\/i>-coordinates.<\/p>\n<p style=\"text-align: center;\">[latex]\\{\u22125,\u22122,\u22121,0,5\\}[\/latex]<\/p>\n<p>The range is the set of outputs of <i>y<\/i>-coordinates.<\/p>\n<p style=\"text-align: center;\">[latex]\\{\u22126,\u22121,0,3,15\\}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\begin{array}{l}\\text{Domain}:\\{\u22125,\u22122,\u22121,0,5\\}\\\\\\text{Range}:\\{\u22126,\u22121,0,3,15\\}\\end{array}\\\\[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show another example of finding domain and range from tabular data.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex: Give the Domain and Range Given the Points in a Table\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GPBq18fCEv4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Some functions have a given output value that corresponds to two or more input values. For example, in the following stock chart the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200506\/CNX_Precalc_Figure_01_00_001n2.jpg\" alt=\"Figure of a bull and a graph of market prices.\" width=\"975\" height=\"307\" data-media-type=\"image\/jpg\" \/><\/p>\n<p id=\"fs-id1165135678633\">However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in.<\/p>\n<table style=\"width: 20%;\" summary=\"Two columns and five rows. The first column is labeled,\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/><\/colgroup>\n<thead>\n<tr>\n<th>Letter grade<\/th>\n<th>Grade point average<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>A<\/td>\n<td>4.0<\/td>\n<\/tr>\n<tr>\n<td>B<\/td>\n<td>3.0<\/td>\n<\/tr>\n<tr>\n<td>C<\/td>\n<td>2.0<\/td>\n<\/tr>\n<tr>\n<td>D<\/td>\n<td>1.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137561844\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\n<p>To visualize this concept, let\u2019s look again at the two simple functions sketched in (a)and (b) of Figure 10.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200453\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<p>The function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: One-to-One Function<\/h3>\n<p>A one-to-one function is a function in which each output value corresponds to exactly one input value.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Which table represents a one-to-one function?<\/p>\n<p>a)<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td>input<\/td>\n<td>output<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>-1<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>-5<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>b)<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td>input<\/td>\n<td>output<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td>16<\/td>\n<td>32<\/td>\n<\/tr>\n<tr>\n<td>32<\/td>\n<td>64<\/td>\n<\/tr>\n<tr>\n<td>64<\/td>\n<td>128<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q945171\">Show Solution<\/span><\/p>\n<div id=\"q945171\" class=\"hidden-answer\" style=\"display: none\">\n<p>Table a) maps the output value 2 to two different input values, therefore\u00a0this is NOT a one-to-one function.<\/p>\n<p>Table b) maps each output to one unique input, therefore this IS a one-to-one function.<\/p>\n<h4>Answer<\/h4>\n<p>Table b) is one-to-one<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show an example of using tables of values to determine whether a function is one-to-one.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Determine if a Relation Given as a Table is a One-to-One Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QFOJmevha_Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 style=\"text-align: left;\">Using the Horizontal Line Test<\/h2>\n<p id=\"fs-id1165137871503\">An easy way to determine whether a function\u00a0is a one-to-one function is to use the <strong>horizontal line test <\/strong>on the graph of the function. \u00a0To do this, draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.<\/p>\n<div class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 style=\"text-align: center;\"><strong>How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.<\/strong><\/h3>\n<ol id=\"fs-id1165137611853\" data-number-style=\"arabic\">\n<li>Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\n<li>If there is any such line, determine that the function is not one-to-one.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p><span data-type=\"media\" data-alt=\"Graph of a polynomial.\"><span data-type=\"media\" data-alt=\"Graph of a polynomial.\">For\u00a0the following graphs, determine which represent one-to-one functions.<br \/>\n<\/span><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200511\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783411\">Show Answer<\/span><\/p>\n<div id=\"q783411\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135185190\">The function in (a) is\u00a0not one-to-one. Using the horizontal line test, as shown below, it intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)<\/p>\n<figure id=\"Figure_01_01_010\" class=\"small\"><span data-type=\"media\" data-alt=\"\"><span data-type=\"media\" data-alt=\"\">\u00a0<\/span><\/span><\/figure>\n<figure id=\"Figure_01_01_010\" class=\"small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200515\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"\" width=\"487\" height=\"445\" data-media-type=\"image\/jpg\" \/><\/figure>\n<p>The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2697 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212200\/Screen-Shot-2016-07-16-at-2.21.48-PM-287x300.png\" alt=\"Graph of a line with three dashed horizontal lines passing through it.\" width=\"287\" height=\"300\" \/><\/p>\n<p>The function (c) is not one-to-one, and is in fact not a function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2698 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212527\/Screen-Shot-2016-07-16-at-2.25.36-PM-237x300.png\" alt=\"Graph of a circle with two dashed lines passing through horizontally\" width=\"279\" height=\"353\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of using the horizontal line test to determine whether a graph represents a one-to-one function.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex 1:  Determine if the Graph of a Relation is a One-to-One Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tbSGdcSN8RE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>In real life and in algebra, different variables are often linked. When a change in value of one variable causes a change in the value of another variable, their interaction is called a relation. A relation has an input value which corresponds to an output value. When each input value has one and only one output value, that relation is a function. Functions can be written as ordered pairs, tables, or graphs. The set of input values is called the domain, and the set of output values is called the range.<\/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><\/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 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