{"id":916,"date":"2022-02-28T01:06:14","date_gmt":"2022-02-28T01:06:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=916"},"modified":"2025-11-19T14:38:18","modified_gmt":"2025-11-19T14:38:18","slug":"2-2-2-graph-linear-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/2-2-2-graph-linear-functions\/","title":{"raw":"2.2.2: Graphing Linear Functions","rendered":"2.2.2: Graphing Linear Functions"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Graph a linear function using a table of values<\/li>\r\n \t<li>Graph a linear function using slope and [latex]y[\/latex]-intercept<\/li>\r\n \t<li>Graph vertical and horizontal lines<\/li>\r\n \t<li>Identify parallel and perpendicular lines from a graph<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphing Linear Functions Using Tables<\/h2>\r\nOne way to graph any function is to create a\u00a0<em><strong>table of values<\/strong><\/em>. This is particularly useful when we do not know the general shape of the graphed function. We have already seen that the graph of a <em><strong>linear function<\/strong><\/em> is a line, but let's make a table to see how it can be helpful.\r\n\r\nTo build a table of values we make a table consisting of two columns; one for [latex]x[\/latex] and the other for [latex]y=f(x)[\/latex]:\r\n<table class=\" aligncenter\" style=\"width: 20.054%; height: 84px;\">\r\n<thead>\r\n<tr style=\"height: 14px;\">\r\n<th style=\"text-align: center; height: 14px; width: 13.3065%;\"><strong>[latex]x[\/latex]<\/strong><\/th>\r\n<th style=\"text-align: center; height: 14px; width: 34.1398%;\"><strong>[latex]f(x)[\/latex]<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 13.3065%;\"><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 34.1398%;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; height: 14px; width: 13.3065%;\"><\/td>\r\n<td style=\"text-align: center; height: 14px; width: 34.1398%;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe then choose several values for [latex]x[\/latex]\u00a0and enter them in separate rows in the [latex]x[\/latex] column. The values we choose are totally up to us \u2013 there is no \u201cright\u201d or \u201cwrong\u201d values to pick, just go for it.\u00a0It is a good idea to include negative values, positive values, and zero and ensure that we have enough points to determine the shape of the graph.\r\n\r\nSuppose we'd like to graph the function [latex]f(x)=3x+2[\/latex]. We start by adding our chosen [latex]x[\/latex]-values to the table:\r\n<table class=\" aligncenter\" style=\"width: 22.9446%; height: 178px;\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"text-align: center; height: 12px; width: 89.2595%;\"><strong>[latex]x[\/latex]<\/strong><\/th>\r\n<th style=\"text-align: center; height: 12px; width: 148.271%;\"><strong>[latex]f(x)=3x+2[\/latex]<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 89.2595%;\">[latex]\u22122[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 148.271%;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 89.2595%;\">[latex]\u22121[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 148.271%;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 89.2595%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 148.271%;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 89.2595%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 148.271%;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 89.2595%;\">[latex]3[\/latex]<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 148.271%;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe then evaluate the function for each chosen value of [latex]x[\/latex], and write the result in the [latex]f(x)[\/latex] column in the same row as\u00a0 the [latex]x[\/latex] value we used.\r\n\r\nTo evaluate the function values, we replace [latex]x[\/latex] in the function with one of our chosen values:\r\n<table class=\" aligncenter\" style=\"width: 25.7623%; height: 251px;\">\r\n<tbody>\r\n<tr style=\"height: 28px;\">\r\n<th style=\"text-align: center; width: 81.0879%; height: 28px;\"><strong>[latex]x[\/latex]\r\n<\/strong><\/th>\r\n<th style=\"text-align: center; width: 242.681%; height: 28px;\"><strong>[latex]f(x)=3x+2[\/latex]\r\n<\/strong><\/th>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; width: 81.0879%; height: 14px;\">[latex]\u22122[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 242.681%; height: 14px;\">[latex]f(-2)=3(-2)+2=\u22124[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; width: 81.0879%; height: 14px;\">[latex]\u22121[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 242.681%; height: 14px;\">[latex]f(-1)=3(-1)+2=\u22121[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; width: 81.0879%; height: 14px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 242.681%; height: 14px;\">[latex]f(0)=3(0)+2=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"text-align: center; width: 81.0879%; height: 14px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 242.681%; height: 14px;\">[latex]f(1)=3(1)+2=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"text-align: center; width: 81.0879%; height: 34px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 242.681%; height: 34px;\">[latex]f(3)=3(3)+2=11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote that your table of values may be different from someone else\u2019s as you may each choose different values for [latex]x[\/latex].\r\n<p style=\"text-align: left;\">Now that we have a table of values, we can use it to help us draw both the shape and location of the function by plotting [latex](x, f(x))[\/latex] as coordinate points [latex](x, y)[\/latex]. Each point we plot is a point on the graph. But there are an infinite number of points that make up the graph, so when we are sure the graph is a line, we join the dots using a straightedge to represent the entire graph.<\/p>\r\n<p style=\"text-align: center;\"><img class=\"alignleft\" 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>\r\n\r\n<div class=\"wp-nocaption aligncenter\" style=\"text-align: center;\"><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\" \/><\/div>\r\n<div style=\"text-align: left;\"><\/div>\r\n<div><\/div>\r\n<div style=\"text-align: center;\"><\/div>\r\n<div class=\"textbox examples\" style=\"text-align: center;\">\r\n<h3>Example 1<\/h3>\r\nGraph [latex]f(x)=\u2212x+1[\/latex].\r\n\r\nStart with a table of values. Choose any values for [latex]x[\/latex] that make sense to you, but remember that it is helpful to include [latex]0[\/latex], some positive values, and some negative values.\r\n<table style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<th style=\"text-align: center;\"><strong>[latex]x[\/latex]\r\n<\/strong><\/th>\r\n<th style=\"text-align: center;\"><strong>[latex]f(x)=-x+1[\/latex]\r\n<\/strong><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]\u22122[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]f(-2)=-(-2)+1=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]f(-1)=-(-1)+1=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]f(0)=-(0)+1=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]f(1)=-(1)+1=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]f(2)=-(2)+1=\u22121[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the points:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Use a straightedge to draw the line:\r\n\r\n<img class=\"alignleft\" 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\r\n<img class=\"aligncenter\" 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\n&nbsp;\r\n\r\n<\/div>\r\n<p style=\"text-align: left;\">The following video shows an example of how to graph a linear function on a coordinate plane.<\/p>\r\n<p style=\"text-align: center;\"><iframe src=\"https:\/\/www.youtube.com\/embed\/sfzpdThXpA8?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n\r\n<div class=\"textbox examples\" style=\"text-align: center;\">\r\n<h3>Example 2<\/h3>\r\nBuild a table of values then graph the function [latex]g(x)=3x-4[\/latex].\r\n<h4>Solution<\/h4>\r\nChoose any [latex]x[\/latex]-values that include positives, negatives and 0, then determine the corresponding function values.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 50%; text-align: center;\">[latex]g(x)=3x-4[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">-3<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]g(-3)=3(-3)-4=-13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">-2<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]g(-2)=3(-2)-4=-10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">0<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]g(0)=3(0)=-4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">1<\/td>\r\n<td style=\"width: 50%; text-align: center;\">\u00a0[latex]g(1)=3(1)-4=-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">4<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex]g(4)=3(4)-4=8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow plot the points and join the dots:\r\n\r\n<img class=\"aligncenter wp-image-1218 size-medium\" style=\"background-color: #f3e1e3;\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/gx3x-4-268x300.png\" alt=\"Line creating by joining plotted points\" width=\"268\" height=\"300\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\" style=\"text-align: center;\">\r\n<h3>Try It 1<\/h3>\r\nBuild a table of values then graph the function [latex]h(x)=-2x+5[\/latex].\r\n\r\n[reveal-answer q=\"hjm221\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm221\"]\r\n\r\n<img class=\"alignleft wp-image-1220 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-292x300.png\" alt=\"Values of x and h(x)\" width=\"292\" height=\"300\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nYour chosen points may be different.\r\n\r\n<img class=\"aligncenter wp-image-1219 size-medium\" style=\"background-color: #f3e1e3;\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx-2x5-273x300.png\" alt=\"y=-2x+5\" width=\"273\" height=\"300\" \/>\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\" style=\"text-align: center;\">\r\n<h3>Try It 2<\/h3>\r\nBuild a table of values then graph the function [latex]h(x)=4x-5[\/latex].\r\n\r\n[reveal-answer q=\"hjm618\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm618\"]\r\n\r\n<img class=\"aligncenter wp-image-1221 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-2-242x300.png\" alt=\"Values of x and h(x)\" width=\"242\" height=\"300\" \/>\r\n\r\nYour chosen points may be different.\r\n\r\n<img class=\"aligncenter wp-image-1222 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx4x-5-256x300.png\" alt=\"A line\" width=\"256\" height=\"300\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Graphing Linear Functions Using Slope and y-Intercept<\/h2>\r\nAnother way to graph a linear function is by using its slope\u00a0and [latex]y[\/latex]-intercept. The slope and\u00a0[latex]y[\/latex]-intercept of a linear function define a unique function, consequently there is only one possible graph that represents the function. The\u00a0[latex]y[\/latex]-intercept gives us a starting point on the coordinate plane, while the slope determines the direction we go in to determine another point on the line. Once we have two points on the line, we have a unique line. However, it is always better to determine three or more points.\r\n\r\nAs an example, consider the function [latex]f\\left(x\\right)=\\dfrac{1}{2}x+1[\/latex].\r\n\r\nThe function is in <em><strong>slope-intercept form<\/strong><\/em>, [latex]f(x)=mx+b[\/latex]. so the slope is [latex]\\dfrac{1}{2}[\/latex] and the [latex]y[\/latex]-intercept is the point (0, 1). Because the slope is positive, we know the graph will slant upward from left to right. The [latex]y[\/latex]<em>-<\/em>intercept is the point on the graph where the graph crosses the [latex]y[\/latex]-axis. We can begin graphing by plotting the point\u00a0[latex](0, 1)[\/latex]. We know that the slope [latex]m=\\dfrac{\\text{rise}}{\\text{run}}[\/latex], so the slope [latex]m=\\dfrac{1}{2}[\/latex] means that the rise is\u00a0[latex]1[\/latex] and the run is\u00a0[latex]2[\/latex]. Starting from our [latex]y[\/latex]-intercept\u00a0[latex](0, 1)[\/latex], we can move vertically using the rise\u00a0[latex]1[\/latex] and then move horizontally using the run\u00a0[latex]2[\/latex], or first move horizontally using the run\u00a0[latex]2[\/latex] and then move vertically using the rise\u00a0[latex]1[\/latex] to get to the point (2, 2). We then repeat this process until we have a few points. Figure 1 shows this process and the result that all found points lie on a line. To form the complete line with an infinite number of coordinate points, we draw a line through the points (figure 1).\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"617\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201048\/CNX_Precalc_Figure_02_02_0032.jpg\" alt=\"The line y = (1\/2)x +1 showing the &quot;rise&quot;, or change in the y direction as 1 and the &quot;run&quot;, or change in x direction as 2, and the y-intercept at (0,1)\" width=\"617\" height=\"340\" \/> Figure 1. Graph of [latex]f\\left(x\\right)=\\dfrac{1}{2}x+1[\/latex][\/caption]\r\n<div class=\"textbox\">\r\n<h3>Graphical Interpretation of a Linear Function<\/h3>\r\nThe function [latex]f\\left(x\\right)=mx+b[\/latex] is interpreted graphically as a line where,\r\n<ul>\r\n \t<li>[latex]b[\/latex]\u00a0is the [latex]y[\/latex]-coordinate of the [latex]y[\/latex]-intercept of the graph and indicates the point [latex](0,b)[\/latex] at which the graph crosses the [latex]y[\/latex]-axis.<\/li>\r\n \t<li>[latex]m[\/latex] is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAll linear functions cross the [latex]y[\/latex]-axis and therefore have [latex]y[\/latex]-intercepts. The slope then locks in the direction and steepness of the line. This means that if we know the slope and\u00a0[latex]y[\/latex]-intercept, we know the unique graphical representation of the function.\r\n\r\nNote: At this point you may be thinking that a vertical line parallel to the [latex]y[\/latex]-axis does not have a [latex]y[\/latex]-intercept. However, a vertical line is not a function as it represents a one-to-many mapping.\r\n<div class=\"textbox\">\r\n<h3>graphING A linear function using the\u00a0<em>y<\/em>-intercept and slope<\/h3>\r\n<ol>\r\n \t<li>Evaluate the function at [latex]x=0[\/latex] to find the [latex]y[\/latex]<em>-<\/em>intercept.<\/li>\r\n \t<li>Identify the slope.<\/li>\r\n \t<li>Plot the point represented by the [latex]y[\/latex]<em>-<\/em>intercept.<\/li>\r\n \t<li>Identify the rise and the run from the slope and use these to determine at least one more point on the line.<\/li>\r\n \t<li>Draw the unique line that passes through all of the points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nGraph [latex]f\\left(x\\right)=-\\dfrac{2}{3}x+5[\/latex] using the\u00a0[latex]y[\/latex]<em>-<\/em>intercept and slope.\r\n<h4>Solution<\/h4>\r\nEvaluate the function at [latex]x=0[\/latex]\u00a0to find the\u00a0[latex]y[\/latex]<em>-<\/em>intercept. [latex]f(0)=-\\dfrac{2}{3}(0)+5=5[\/latex], so the graph will cross the [latex]y[\/latex]-axis at\u00a0[latex](0, 5)[\/latex].\r\n\r\nAccording to the equation for the function, the slope of the line is [latex]-\\dfrac{2}{3}[\/latex]. This tells us that the \u201crise\u201d of [latex]\u20132[\/latex] units accompanies the \u201crun\u201d of [latex]3[\/latex] units. We can now graph the function by first plotting the [latex]y[\/latex]-intercept. Then, from the initial value\u00a0[latex](0, 5)[\/latex], we move down\u00a0[latex]2[\/latex] units and to the right\u00a0[latex]3[\/latex] units. We can extend the line to the left and right by using this relationship to plot additional points and then drawing a line through the points.\r\n<div class=\"wp-nocaption aligncenter\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201050\/CNX_Precalc_Figure_02_02_0042.jpg\" alt=\"The line y = (-2\/3)x + 5 showing the change of -2 in y and change of 3 in x.\" width=\"487\" height=\"318\" \/><\/div>\r\nThe graph slants downward from left to right, which means it has a negative slope as expected.\r\n\r\n<\/div>\r\nThis should remind you of a staircase. As long as the rise and run of the staircase are constant from step to step, the stairs form a line from one floor to the next.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\n<iframe id=\"ohm79774\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=79774&amp;theme=oea&amp;iframe_resize_id=ohm79774&amp;show_question_numbers\" width=\"100%\" height=\"250\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\u00a0 \u00a0 \u00a0Note: The graph in OHM will accept only 2 points to draw a line.\r\n\r\n<\/div>\r\nThe following video shows an example of how to graph a linear function given the [latex]y-[\/latex]intercept and the slope.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/N6lEPh11gk8?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nGraph [latex]f\\left(x\\right)=-\\dfrac{3}{4}x+6[\/latex]\u00a0using the slope and [latex]y[\/latex]-intercept.\r\n<h4>Solution<\/h4>\r\nThe function is of the form [latex]f(x)=mx+b[\/latex], so the slope of this function is [latex]-\\dfrac{3}{4}[\/latex] and the [latex]y[\/latex]-intercept is [latex](0,6)[\/latex]. We can start graphing by plotting the [latex]y[\/latex]-intercept and counting down three units (rise = -3) and right\u00a0[latex]4[\/latex] units (run = 4). The first stop is the point [latex](4,3)[\/latex], and the next stop is the point [latex](8,0)[\/latex]. Then join the dots to form the line.\r\n<div class=\"wp-nocaption size-medium wp-image-2482 aligncenter\"><img class=\"aligncenter wp-image-2482 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/13204212\/Screen-Shot-2016-07-13-at-1.35.32-PM-300x229.png\" alt=\"A line\" width=\"300\" height=\"229\" \/><\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\" style=\"text-align: center;\">\r\n<h3>\u00a0Try It 4<\/h3>\r\nIdentify the slope and the [latex]y[\/latex]-intercept, then graph the function [latex]d\\left(x\\right)=\\dfrac{2}{3}x+3[\/latex]\u00a0using the slope and [latex]y[\/latex]-intercept.\r\n<p style=\"text-align: left;\">[reveal-answer q=\"hjm764\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm764\"]<\/p>\r\nSlope = [latex]\\frac{2}{3}[\/latex]\r\n\r\n[latex]y[\/latex]-intercept = (0, 3)\r\n\r\nGraph:\u00a0\u00a0<img class=\"aligncenter wp-image-1224\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30010803\/rise-run-graph-1024x592.png\" alt=\"A line using rise and run\" width=\"771\" height=\"446\" \/>\r\n\r\nNotice that a rise\/run combination of 2\/3 can also be thought of as \u20132\/\u20133. So instead of running 3 to the right and 2 up, we can run 3 to the left and 2 down. This is shown by the purple lines starting at (0, 3) and moving to (\u20133, 1).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 style=\"text-align: left;\">Vertical and Horizontal Lines<\/h3>\r\n<p style=\"text-align: left;\">We saw in the last section that a <em><strong>horizontal line<\/strong><\/em> has a slope of zero and therefore the function [latex]f(x)=b[\/latex] represents a horizontal line that passes through the [latex]y[\/latex]-intercept [latex](0,b)[\/latex].<\/p>\r\n<p style=\"text-align: left;\">For example, suppose we want to graph the function [latex]f(x)=-2[\/latex]. No matter what [latex]x[\/latex]-value we choose, the function value will always equal \u20132. Therefore, the points [latex]\\left(-2,-2\\right),\\left(0,-2\\right),\\left(3,-2\\right)[\/latex], and [latex]\\left(5,-2\\right)[\/latex] all lie on the graph. If we plot these points we see that they form a horizontal line:<\/p>\r\n\r\n[caption id=\"attachment_1226\" align=\"aligncenter\" width=\"434\"]<img class=\"wp-image-1226\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30014406\/fx-2-300x154.png\" alt=\"Horizontal line through (0, -2)\" width=\"434\" height=\"223\" \/> Figure 2. Graph of function [latex]f(x)=-2[\/latex].[\/caption]\r\n<p style=\"text-align: left;\">We also saw in the last section that a <em><strong>vertical line<\/strong><\/em> does NOT represent a function, but it is a relation. In order for a line to be vertical, all of the [latex]x[\/latex]-values must be the same. Therefore, the equation of a\u00a0vertical line is given as [latex]x=c[\/latex] where [latex]c[\/latex]<em>\u00a0<\/em>is a constant. The slope of a vertical line is undefined (since the run is zero, and we can't divide by 0), and regardless of the [latex]y[\/latex]<em>-<\/em>value of any point on the line, the [latex]x[\/latex]<em>-<\/em>coordinate of the point will always be\u00a0<em>c<\/em>.<\/p>\r\n<p style=\"text-align: left;\">For example, suppose we want to graph the equation [latex]x=-3[\/latex]. The graph will contain the following points: [latex]\\left(-3,-5\\right),\\left(-3,1\\right),\\left(-3,3\\right)[\/latex], and [latex]\\left(-3,5\\right)[\/latex]. If we plot these points, we see they form a vertical line:<\/p>\r\n\r\n[caption id=\"attachment_1227\" align=\"aligncenter\" width=\"298\"]<img class=\"wp-image-1227\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30015216\/x-3-216x300.png\" alt=\"Vertical line through (-3, 0)\" width=\"298\" height=\"414\" \/> Figure 3. Graph of equation [latex]x=-3[\/latex].[\/caption]\r\n<div class=\"textbox\" style=\"text-align: center;\">\r\n<h3>VERTICAL AND HORIZONTAL LINES<\/h3>\r\nA linear function whose graph has a slope of zero forms a <strong>horizontal line<\/strong>. The function is of the form [latex]f(x)=b[\/latex] where [latex]b[\/latex] is a real number and is represented graphically by the [latex]y[\/latex]-intercept [latex](0, b)[\/latex].\r\n\r\n&nbsp;\r\n\r\nA <strong>vertical line<\/strong> represents the equation [latex]x=c[\/latex] where [latex]c[\/latex] is a real number. This equation is NOT a function.\r\n\r\n<\/div>\r\n<div style=\"text-align: center;\">\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nGraph:\r\n<ol>\r\n \t<li>[latex]f(x)=4[\/latex]<\/li>\r\n \t<li>[latex]x=-1[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>The function [latex]f(x)=4[\/latex] is represented by a horizontal line that passes through (0, 4).<\/li>\r\n \t<li>The equation [latex]x=-1[\/latex] is represented by a vertical line that passes through (\u20131, 0).<\/li>\r\n<\/ol>\r\nBoth graphs are shown:\r\n\r\n<img class=\"aligncenter wp-image-1229\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30155649\/Horizontal-and-vertical-300x297.png\" alt=\"Horizontal line through (0, 4) and vertical line through (-1, 0)\" width=\"412\" height=\"408\" \/>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nGraph:\r\n<ol>\r\n \t<li>[latex]f(x)=-6[\/latex]<\/li>\r\n \t<li>[latex]x=4[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm708\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm708\"]\r\n\r\n<img class=\"aligncenter wp-image-1277\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/31222209\/x4-and-y%E2%80%936-250x300.png\" alt=\"The lines y=-6 and x = 4\" width=\"316\" height=\"379\" \/>\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<h2 style=\"text-align: left;\">Parallel and Perpendicular Lines<\/h2>\r\n<p style=\"text-align: left;\"><em><strong>Parallel lines<\/strong><\/em> have the same slope but different\u00a0<em>y-<\/em>intercepts. Lines that are\u00a0parallel to each other will never intersect. For example, figure 4 shows the graphs of three lines with the same slope, [latex]m=2[\/latex]. Notice that each line goes in exactly the same direction because they have the same slope. Notice also that each line has a different [latex]y[\/latex]-intercept. If these lines all had the same slope and the same\u00a0[latex]y[\/latex]-intercept, they would all lie on top of each other and be the same line. Such lines are referred to as <em><strong>coincidental lines<\/strong><\/em>, and we will see the significance of such lines later in this course.<\/p>\r\n\r\n<div class=\"wp-caption aligncenter\" style=\"width: 497px; text-align: center;\">\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/17232102\/CNX_CAT_Figure_02_02_004.jpg\" alt=\"Three parallel lines with same slopes but different y-intercepts\" width=\"487\" height=\"593\" data-media-type=\"image\/jpg\" \/>Figure 4. Parallel lines.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Parallel Lines<\/h3>\r\nTwo or more lines are parallel to each other if they have identical slopes but different\u00a0[latex]y[\/latex]-intercepts.\r\n\r\nIf one line has a slope of [latex]m_1[\/latex] and a\u00a0[latex]y[\/latex]-intercept at [latex](0, b_1)[\/latex] and another line has a\u00a0slope of [latex]m_2[\/latex] and a\u00a0[latex]y[\/latex]-intercept at [latex](0, b_2)[\/latex], the lines are parallel if [latex]m_1=m_2[\/latex] and [latex]b_1\\ne b_2[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nDetermine if the graph represents two parallel lines:\r\n\r\n<img class=\"aligncenter wp-image-1232\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30163041\/y23x4-and-y23x-3-298x300.png\" alt=\"Two lines that might be parallel\" width=\"421\" height=\"424\" \/>\r\n<h4>Solution<\/h4>\r\nThe blue line passes through the points (\u20136, 0) and (0, 4). [latex]m_1=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{4-0}{0-(-6)}=\\dfrac{4}{6}=\\dfrac{2}{3}[\/latex].\r\n\r\nThe green line passes through the points (0, \u20133) and (6, 1). [latex]m_1=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{1-(-3)}{6-0)}=\\dfrac{4}{6}=\\dfrac{2}{3}[\/latex].\r\n\r\nSince the slopes are equal, the lines are parallel.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nDetermine if the graph represents two parallel lines:\r\n\r\n<img class=\"aligncenter wp-image-1233\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30164219\/y-4x6-and-y-3.75x-3-222x300.png\" alt=\"Two lines that might be parallel\" width=\"368\" height=\"495\" \/>\r\n\r\n[reveal-answer q=\"hjm060\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm060\"]\r\n\r\nThe blue line passes through the points (0, 6) and (1, 2). By counting, its run is 1 and its rise is \u20134, resulting in a slope of [latex]m_1=\\dfrac{\\text{rise}}{\\text{run}}=\\dfrac{-4}{1}=-4[\/latex].\r\n\r\nThe green line passes through the points (0, \u20133) and (\u20134, 12). Therefore, its run is -4 and its rise is 15, resulting in a slope of [latex]m_1=\\dfrac{\\text{rise}}{\\text{run}}=\\dfrac{15}{-4}=-\\dfrac{15}{4}[\/latex].\r\n\r\nSince the slopes are not equal, the lines are not parallel.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p style=\"text-align: left;\">Lines that are\u00a0<em><strong>perpendicular\u00a0<\/strong><\/em>intersect to form a [latex]{90}^{\\circ }[\/latex] angle (a right angle). The product of the slopes of perpendicular lines always equals \u20131.<\/p>\r\nIf one line has a slope of [latex]m_1[\/latex] and another line has a\u00a0slope of [latex]m_2[\/latex], the lines are perpendicular if [latex]m_1 \\cdot m_2=-1[\/latex]. This means that [latex]m_1=-\\frac{1}{m_2}[\/latex] and\u00a0[latex]m_2=-\\frac{1}{m_1}[\/latex], i.e. the slopes are the negative reciprocal of each other.\r\n<p style=\"text-align: left;\">For example, in figure 5 the orange line has a slope of [latex]m_1=3[\/latex] and the blue line has a slope of [latex]m_2=-\\frac{1}{3}[\/latex]. The product of these two slopes is [latex]3\\cdot \\left ( -\\frac{1}{3}\\right )=-1[\/latex]. Notice that [latex]3=\\frac{3}{1}[\/latex] and [latex]-\\frac{1}{3}[\/latex] are negative reciprocals of one another. i.e. if we turn one slope upside down and take it's opposite, we get the other slope.<\/p>\r\n\r\n<div class=\"wp-caption aligncenter\" style=\"width: 497px; text-align: center;\">\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/17232106\/CNX_CAT_Figure_02_02_005.jpg\" alt=\"Two perpendicular lines\" width=\"487\" height=\"329\" data-media-type=\"image\/jpg\" \/>Figure 5. Perpendicular lines\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Perpendicular Lines<\/h3>\r\nTwo lines are perpendicular to each other if the product of their slopes equals \u20131.\r\n\r\nIf one line has a slope of [latex]m_1[\/latex] and another line has a\u00a0slope of [latex]m_2[\/latex], the lines are perpendicular if [latex]m_1 \\cdot m_2=-1[\/latex].\r\n\r\nThis means that [latex]m_1=-\\frac{1}{m_2}[\/latex] and\u00a0[latex]m_2=-\\frac{1}{m_1}[\/latex], i.e. the slopes are the negative reciprocal of each other.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 7<\/h3>\r\nDetermine if the graph represents two perpendicular lines:\r\n\r\n<img class=\"aligncenter wp-image-1234\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30164924\/y-52x6-and-y35x-3-300x278.png\" alt=\"Two lines intersecting that might be perpendicular\" width=\"452\" height=\"419\" \/>\r\n<h4>Solution<\/h4>\r\nThe blue line passes through the points (0, 6) and (4, \u20134).\r\n\r\nThis results in a slope of [latex]m_1=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{6-(-4)}{0-4}=\\dfrac{10}{-4}=-\\dfrac{5}{2}[\/latex].\r\n\r\nThe green line passes through the points (0, \u20133) and (5, 0), resulting in a slope of [latex]m_1=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{0-(-3)}{5-0}=\\dfrac{3}{5}[\/latex].\r\n\r\n[latex]m_1\\cdot m_2=-\\dfrac{5}{2}\\cdot\\dfrac{3}{5}=-\\dfrac{3}{2}\\ne -1[\/latex]\r\n\r\nSince the product of the slopes is not equal to \u20131, the lines are not perpendicular.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\nDetermine if the graph represents two perpendicular lines:\r\n\r\n<img class=\"aligncenter wp-image-1235\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30165819\/y-43x6-and-y34x-4-275x300.png\" alt=\"Two lines that might be perpendicular\" width=\"424\" height=\"463\" \/>\r\n\r\n[reveal-answer q=\"hjm902\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm902\"]\r\n\r\nSlope of blue line: [latex]m_1=-\\frac{4}{3}[\/latex]\r\n\r\nSlope of green line: [latex]m_2=\\frac{3}{4}[\/latex]\r\n\r\nSince [latex]m_1\\cdot m_2=-1[\/latex], the lines are perpendicular.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Example 8<\/h3>\r\n<p style=\"text-align: left;\">Graph the equations, and state whether the resulting lines are parallel, perpendicular, or neither: [latex]3y=-4x+3[\/latex] and [latex]3x - 4y=8[\/latex].<\/p>\r\n\r\n<h4 style=\"text-align: left;\">Solution<\/h4>\r\nWe can graph the lines using a table of values. To determine the corresponding [latex]y[\/latex]-values we must substitute the chosen [latex]x[\/latex]-value and solve the resulting equation in one variable.\r\n\r\nTable of values for line 1:\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"vertical-align: top; width: 25%;\"><span style=\"font-size: 12px; text-align: initial;\">[latex]x=-3[\/latex]<\/span><\/th>\r\n<th style=\"width: 25%; vertical-align: top;\"><span style=\"font-size: 12px; orphans: 1;\">[latex]x=0[\/latex]<\/span><\/th>\r\n<th style=\"width: 25%; vertical-align: top;\"><span style=\"font-size: 12px; orphans: 1;\">[latex]x=3[\/latex]<\/span><\/th>\r\n<th style=\"width: 25%; vertical-align: top;\">Resulting Table<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"vertical-align: top; width: 25%;\">[latex]\\begin{aligned}3y&amp;=-4x+3\\\\ 3y&amp;=-4(-3)+3\\\\3y&amp;=12+3\\\\ 3y&amp;=15\\\\y&amp;=5\\end{aligned}[\/latex]<\/td>\r\n<td style=\"width: 25%; vertical-align: top;\">[latex]\\begin{aligned}3y&amp;= -4x+3\\\\ 3y&amp; =-4(0)+3\\\\3y&amp;=3\\\\ y&amp;=1\\end{aligned}[\/latex]<\/td>\r\n<td style=\"width: 25%; vertical-align: top;\">[latex]\\begin{aligned}3y&amp; =-4x+3 \\\\3y&amp;=-4(-)+3\\\\ 3y&amp;=-12+3\\\\ 3y&amp;=-9\\\\y&amp;=-3\\end{aligned}[\/latex]<\/td>\r\n<td style=\"width: 25%; vertical-align: top;\">&nbsp;\r\n<table style=\"width: 25%; height: 93px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 50%; height: 12px;\"><span style=\"font-size: 12px;\">[latex]x[\/latex]<\/span><\/th>\r\n<th style=\"width: 50%; height: 12px;\"><span style=\"font-size: 12px;\">[latex]3y=-4x+3[\/latex]<\/span><\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]\u20133[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">0<\/td>\r\n<td style=\"width: 50%; height: 12px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%;\">3<\/td>\r\n<td style=\"width: 50%; height: 12px;\">\u20133<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTable of values for line 2:\r\n<table style=\"width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"vertical-align: top; width: 26.0361%;\">[latex]x=-4[\/latex]<\/th>\r\n<th style=\"vertical-align: top; width: 24.8673%;\">[latex]x=0[\/latex]<\/th>\r\n<th style=\"width: 24.5748%; vertical-align: top;\">[latex]x=4[\/latex]<\/th>\r\n<th style=\"width: 24.4244%; vertical-align: top;\">Resulting Table<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"vertical-align: top; width: 26.0361%;\"><span style=\"font-size: 12px; orphans: 2;\">[latex]\\begin{aligned}3x - 4y&amp;=8\\\\3(-4)-4y&amp;=8\\\\-12-4y&amp;=8\\\\-4y&amp;=20\\\\y&amp;=-5\\end{aligned}[\/latex]<\/span><\/td>\r\n<td style=\"vertical-align: top; width: 24.8673%;\"><span style=\"font-size: 12px; orphans: 2;\">[latex]\\begin{aligned}3x - 4y&amp;=8\\\\3(0)-4y&amp;=8\\\\-4y&amp;=8\\\\y&amp;=- 2\\end{aligned}[\/latex]<\/span><\/td>\r\n<td style=\"width: 24.5748%; vertical-align: top;\"><span style=\"font-size: 12px; orphans: 2;\">[latex]\\begin{aligned}3x - 4y&amp;=8\\\\3(4)-4y&amp;=8\\\\12-4y&amp;=8\\\\-4y&amp;=-4\\\\y&amp;=1\\end{aligned}[\/latex]<\/span><\/td>\r\n<td style=\"width: 24.4244%; vertical-align: top;\">\r\n<table style=\"width: 93.1191%; height: 48px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 43.7863%; height: 12px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"height: 12px; width: 50%;\">[latex]3x - 4y=8[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 43.7863%; height: 12px;\">\u20134<\/td>\r\n<td style=\"width: 225.585%; height: 12px;\">\u20135<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 43.7863%; height: 12px;\">0<\/td>\r\n<td style=\"width: 225.585%; height: 12px;\">\u20132<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 43.7863%; height: 12px;\">4<\/td>\r\n<td style=\"width: 225.585%; height: 12px;\">1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter wp-image-1237\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30173537\/Example-2.2-287x300.png\" alt=\"Two intersecting lines\" width=\"443\" height=\"463\" \/>Now plot the points and graph each line:\r\n<div class=\"wp-nocaption aligncenter\"><\/div>\r\nFrom the graph, we can see that the lines appear perpendicular, but we must compare the slopes.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{m}_{1}=-\\frac{4}{3}\\hfill \\\\ {m}_{2}=\\frac{3}{4}\\hfill \\\\ {m}_{1}\\cdot {m}_{2}=\\left(-\\frac{4}{3}\\right)\\left(\\frac{3}{4}\\right)=-1\\hfill \\end{array}[\/latex]<\/p>\r\nThe slopes are negative reciprocals of each other, confirming that the lines are perpendicular.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox tryit\" style=\"text-align: center;\">\r\n<h3>Try It 8<\/h3>\r\nGraph the equations, and state whether the resulting lines are parallel, perpendicular, or neither:\u00a0 [latex]2y-x=10[\/latex] and [latex]2y=x+4[\/latex].\r\n\r\n[reveal-answer q=\"hjm633\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm633\"]\r\n\r\n<img class=\"aligncenter wp-image-1241\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30175940\/2x-y10-and-2yx4-288x300.png\" alt=\"Two lines\" width=\"415\" height=\"432\" \/>\r\n\r\nSlope of blue line = [latex]\\frac{1}{2}[\/latex]\r\n\r\nSlope of green line =\u00a0[latex]\\frac{1}{2}[\/latex]\r\n\r\nSince the slopes are equal, the lines are parallel.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Graph a linear function using a table of values<\/li>\n<li>Graph a linear function using slope and [latex]y[\/latex]-intercept<\/li>\n<li>Graph vertical and horizontal lines<\/li>\n<li>Identify parallel and perpendicular lines from a graph<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing Linear Functions Using Tables<\/h2>\n<p>One way to graph any function is to create a\u00a0<em><strong>table of values<\/strong><\/em>. This is particularly useful when we do not know the general shape of the graphed function. We have already seen that the graph of a <em><strong>linear function<\/strong><\/em> is a line, but let&#8217;s make a table to see how it can be helpful.<\/p>\n<p>To build a table of values we make a table consisting of two columns; one for [latex]x[\/latex] and the other for [latex]y=f(x)[\/latex]:<\/p>\n<table class=\"aligncenter\" style=\"width: 20.054%; height: 84px;\">\n<thead>\n<tr style=\"height: 14px;\">\n<th style=\"text-align: center; height: 14px; width: 13.3065%;\"><strong>[latex]x[\/latex]<\/strong><\/th>\n<th style=\"text-align: center; height: 14px; width: 34.1398%;\"><strong>[latex]f(x)[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 13.3065%;\"><\/td>\n<td style=\"text-align: center; height: 14px; width: 34.1398%;\"><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; height: 14px; width: 13.3065%;\"><\/td>\n<td style=\"text-align: center; height: 14px; width: 34.1398%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We then choose several values for [latex]x[\/latex]\u00a0and enter them in separate rows in the [latex]x[\/latex] column. The values we choose are totally up to us \u2013 there is no \u201cright\u201d or \u201cwrong\u201d values to pick, just go for it.\u00a0It is a good idea to include negative values, positive values, and zero and ensure that we have enough points to determine the shape of the graph.<\/p>\n<p>Suppose we&#8217;d like to graph the function [latex]f(x)=3x+2[\/latex]. We start by adding our chosen [latex]x[\/latex]-values to the table:<\/p>\n<table class=\"aligncenter\" style=\"width: 22.9446%; height: 178px;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"text-align: center; height: 12px; width: 89.2595%;\"><strong>[latex]x[\/latex]<\/strong><\/th>\n<th style=\"text-align: center; height: 12px; width: 148.271%;\"><strong>[latex]f(x)=3x+2[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 89.2595%;\">[latex]\u22122[\/latex]<\/td>\n<td style=\"text-align: center; height: 12px; width: 148.271%;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 89.2595%;\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center; height: 12px; width: 148.271%;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 89.2595%;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center; height: 12px; width: 148.271%;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 89.2595%;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center; height: 12px; width: 148.271%;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 89.2595%;\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center; height: 12px; width: 148.271%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We then evaluate the function for each chosen value of [latex]x[\/latex], and write the result in the [latex]f(x)[\/latex] column in the same row as\u00a0 the [latex]x[\/latex] value we used.<\/p>\n<p>To evaluate the function values, we replace [latex]x[\/latex] in the function with one of our chosen values:<\/p>\n<table class=\"aligncenter\" style=\"width: 25.7623%; height: 251px;\">\n<tbody>\n<tr style=\"height: 28px;\">\n<th style=\"text-align: center; width: 81.0879%; height: 28px;\"><strong>[latex]x[\/latex]<br \/>\n<\/strong><\/th>\n<th style=\"text-align: center; width: 242.681%; height: 28px;\"><strong>[latex]f(x)=3x+2[\/latex]<br \/>\n<\/strong><\/th>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; width: 81.0879%; height: 14px;\">[latex]\u22122[\/latex]<\/td>\n<td style=\"text-align: center; width: 242.681%; height: 14px;\">[latex]f(-2)=3(-2)+2=\u22124[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; width: 81.0879%; height: 14px;\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center; width: 242.681%; height: 14px;\">[latex]f(-1)=3(-1)+2=\u22121[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; width: 81.0879%; height: 14px;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center; width: 242.681%; height: 14px;\">[latex]f(0)=3(0)+2=2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"text-align: center; width: 81.0879%; height: 14px;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center; width: 242.681%; height: 14px;\">[latex]f(1)=3(1)+2=5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"text-align: center; width: 81.0879%; height: 34px;\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center; width: 242.681%; height: 34px;\">[latex]f(3)=3(3)+2=11[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note that your table of values may be different from someone else\u2019s as you may each choose different values for [latex]x[\/latex].<\/p>\n<p style=\"text-align: left;\">Now that we have a table of values, we can use it to help us draw both the shape and location of the function by plotting [latex](x, f(x))[\/latex] as coordinate points [latex](x, y)[\/latex]. Each point we plot is a point on the graph. But there are an infinite number of points that make up the graph, so when we are sure the graph is a line, we join the dots using a straightedge to represent the entire graph.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft\" 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<div class=\"wp-nocaption aligncenter\" 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\/18232426\/image006.gif\" alt=\"A line through the points in the previous graph.\" width=\"322\" height=\"353\" \/><\/div>\n<div style=\"text-align: left;\"><\/div>\n<div><\/div>\n<div style=\"text-align: center;\"><\/div>\n<div class=\"textbox examples\" style=\"text-align: center;\">\n<h3>Example 1<\/h3>\n<p>Graph [latex]f(x)=\u2212x+1[\/latex].<\/p>\n<p>Start with a table of values. Choose any values for [latex]x[\/latex] that make sense to you, but remember that it is helpful to include [latex]0[\/latex], some positive values, and some negative values.<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<th style=\"text-align: center;\"><strong>[latex]x[\/latex]<br \/>\n<\/strong><\/th>\n<th style=\"text-align: center;\"><strong>[latex]f(x)=-x+1[\/latex]<br \/>\n<\/strong><\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22122[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]f(-2)=-(-2)+1=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\u22121[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]f(-1)=-(-1)+1=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]f(0)=-(0)+1=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]f(1)=-(1)+1=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]f(2)=-(2)+1=\u22121[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the points:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Use a straightedge to draw the line:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft\" 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<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\/18232430\/image008.gif\" alt=\"Line through the points in the last graph.\" width=\"322\" height=\"353\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p style=\"text-align: left;\">The following video shows an example of how to graph a linear function on a coordinate plane.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/sfzpdThXpA8?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox examples\" style=\"text-align: center;\">\n<h3>Example 2<\/h3>\n<p>Build a table of values then graph the function [latex]g(x)=3x-4[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Choose any [latex]x[\/latex]-values that include positives, negatives and 0, then determine the corresponding function values.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 50%; text-align: center;\">[latex]g(x)=3x-4[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">-3<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]g(-3)=3(-3)-4=-13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">-2<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]g(-2)=3(-2)-4=-10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">0<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]g(0)=3(0)=-4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">1<\/td>\n<td style=\"width: 50%; text-align: center;\">\u00a0[latex]g(1)=3(1)-4=-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">4<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]g(4)=3(4)-4=8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now plot the points and join the dots:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1218 size-medium\" style=\"background-color: #f3e1e3;\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/gx3x-4-268x300.png\" alt=\"Line creating by joining plotted points\" width=\"268\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/gx3x-4-268x300.png 268w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/gx3x-4-768x860.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/gx3x-4-915x1024.png 915w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/gx3x-4-65x73.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/gx3x-4-225x252.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/gx3x-4-350x392.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/gx3x-4.png 1440w\" sizes=\"auto, (max-width: 268px) 100vw, 268px\" \/><\/p>\n<\/div>\n<div class=\"textbox tryit\" style=\"text-align: center;\">\n<h3>Try It 1<\/h3>\n<p>Build a table of values then graph the function [latex]h(x)=-2x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm221\">Show Answer<\/span><\/p>\n<div id=\"qhjm221\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-1220 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-292x300.png\" alt=\"Values of x and h(x)\" width=\"292\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-292x300.png 292w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-65x67.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-225x231.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-350x360.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values.png 488w\" sizes=\"auto, (max-width: 292px) 100vw, 292px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Your chosen points may be different.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1219 size-medium\" style=\"background-color: #f3e1e3;\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx-2x5-273x300.png\" alt=\"y=-2x+5\" width=\"273\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx-2x5-273x300.png 273w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx-2x5-768x843.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx-2x5-933x1024.png 933w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx-2x5-65x71.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx-2x5-225x247.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx-2x5-350x384.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx-2x5.png 1374w\" sizes=\"auto, (max-width: 273px) 100vw, 273px\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\" style=\"text-align: center;\">\n<h3>Try It 2<\/h3>\n<p>Build a table of values then graph the function [latex]h(x)=4x-5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm618\">Show Answer<\/span><\/p>\n<div id=\"qhjm618\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1221 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-2-242x300.png\" alt=\"Values of x and h(x)\" width=\"242\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-2-242x300.png 242w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-2-65x81.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-2-225x279.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-2-350x434.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/Table-of-values-2.png 408w\" sizes=\"auto, (max-width: 242px) 100vw, 242px\" \/><\/p>\n<p>Your chosen points may be different.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1222 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx4x-5-256x300.png\" alt=\"A line\" width=\"256\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx4x-5-256x300.png 256w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx4x-5-768x900.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx4x-5-874x1024.png 874w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx4x-5-65x76.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx4x-5-225x264.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx4x-5-350x410.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/hx4x-5.png 1294w\" sizes=\"auto, (max-width: 256px) 100vw, 256px\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Graphing Linear Functions Using Slope and y-Intercept<\/h2>\n<p>Another way to graph a linear function is by using its slope\u00a0and [latex]y[\/latex]-intercept. The slope and\u00a0[latex]y[\/latex]-intercept of a linear function define a unique function, consequently there is only one possible graph that represents the function. The\u00a0[latex]y[\/latex]-intercept gives us a starting point on the coordinate plane, while the slope determines the direction we go in to determine another point on the line. Once we have two points on the line, we have a unique line. However, it is always better to determine three or more points.<\/p>\n<p>As an example, consider the function [latex]f\\left(x\\right)=\\dfrac{1}{2}x+1[\/latex].<\/p>\n<p>The function is in <em><strong>slope-intercept form<\/strong><\/em>, [latex]f(x)=mx+b[\/latex]. so the slope is [latex]\\dfrac{1}{2}[\/latex] and the [latex]y[\/latex]-intercept is the point (0, 1). Because the slope is positive, we know the graph will slant upward from left to right. The [latex]y[\/latex]<em>&#8211;<\/em>intercept is the point on the graph where the graph crosses the [latex]y[\/latex]-axis. We can begin graphing by plotting the point\u00a0[latex](0, 1)[\/latex]. We know that the slope [latex]m=\\dfrac{\\text{rise}}{\\text{run}}[\/latex], so the slope [latex]m=\\dfrac{1}{2}[\/latex] means that the rise is\u00a0[latex]1[\/latex] and the run is\u00a0[latex]2[\/latex]. Starting from our [latex]y[\/latex]-intercept\u00a0[latex](0, 1)[\/latex], we can move vertically using the rise\u00a0[latex]1[\/latex] and then move horizontally using the run\u00a0[latex]2[\/latex], or first move horizontally using the run\u00a0[latex]2[\/latex] and then move vertically using the rise\u00a0[latex]1[\/latex] to get to the point (2, 2). We then repeat this process until we have a few points. Figure 1 shows this process and the result that all found points lie on a line. To form the complete line with an infinite number of coordinate points, we draw a line through the points (figure 1).<\/p>\n<div style=\"width: 627px\" 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\/25201048\/CNX_Precalc_Figure_02_02_0032.jpg\" alt=\"The line y = (1\/2)x +1 showing the &quot;rise&quot;, or change in the y direction as 1 and the &quot;run&quot;, or change in x direction as 2, and the y-intercept at (0,1)\" width=\"617\" height=\"340\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. Graph of [latex]f\\left(x\\right)=\\dfrac{1}{2}x+1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Graphical Interpretation of a Linear Function<\/h3>\n<p>The function [latex]f\\left(x\\right)=mx+b[\/latex] is interpreted graphically as a line where,<\/p>\n<ul>\n<li>[latex]b[\/latex]\u00a0is the [latex]y[\/latex]-coordinate of the [latex]y[\/latex]-intercept of the graph and indicates the point [latex](0,b)[\/latex] at which the graph crosses the [latex]y[\/latex]-axis.<\/li>\n<li>[latex]m[\/latex] is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points.<\/li>\n<\/ul>\n<\/div>\n<p>All linear functions cross the [latex]y[\/latex]-axis and therefore have [latex]y[\/latex]-intercepts. The slope then locks in the direction and steepness of the line. This means that if we know the slope and\u00a0[latex]y[\/latex]-intercept, we know the unique graphical representation of the function.<\/p>\n<p>Note: At this point you may be thinking that a vertical line parallel to the [latex]y[\/latex]-axis does not have a [latex]y[\/latex]-intercept. However, a vertical line is not a function as it represents a one-to-many mapping.<\/p>\n<div class=\"textbox\">\n<h3>graphING A linear function using the\u00a0<em>y<\/em>-intercept and slope<\/h3>\n<ol>\n<li>Evaluate the function at [latex]x=0[\/latex] to find the [latex]y[\/latex]<em>&#8211;<\/em>intercept.<\/li>\n<li>Identify the slope.<\/li>\n<li>Plot the point represented by the [latex]y[\/latex]<em>&#8211;<\/em>intercept.<\/li>\n<li>Identify the rise and the run from the slope and use these to determine at least one more point on the line.<\/li>\n<li>Draw the unique line that passes through all of the points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Graph [latex]f\\left(x\\right)=-\\dfrac{2}{3}x+5[\/latex] using the\u00a0[latex]y[\/latex]<em>&#8211;<\/em>intercept and slope.<\/p>\n<h4>Solution<\/h4>\n<p>Evaluate the function at [latex]x=0[\/latex]\u00a0to find the\u00a0[latex]y[\/latex]<em>&#8211;<\/em>intercept. [latex]f(0)=-\\dfrac{2}{3}(0)+5=5[\/latex], so the graph will cross the [latex]y[\/latex]-axis at\u00a0[latex](0, 5)[\/latex].<\/p>\n<p>According to the equation for the function, the slope of the line is [latex]-\\dfrac{2}{3}[\/latex]. This tells us that the \u201crise\u201d of [latex]\u20132[\/latex] units accompanies the \u201crun\u201d of [latex]3[\/latex] units. We can now graph the function by first plotting the [latex]y[\/latex]-intercept. Then, from the initial value\u00a0[latex](0, 5)[\/latex], we move down\u00a0[latex]2[\/latex] units and to the right\u00a0[latex]3[\/latex] units. We can extend the line to the left and right by using this relationship to plot additional points and then drawing a line through the points.<\/p>\n<div class=\"wp-nocaption aligncenter\"><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\/25201050\/CNX_Precalc_Figure_02_02_0042.jpg\" alt=\"The line y = (-2\/3)x + 5 showing the change of -2 in y and change of 3 in x.\" width=\"487\" height=\"318\" \/><\/div>\n<p>The graph slants downward from left to right, which means it has a negative slope as expected.<\/p>\n<\/div>\n<p>This should remind you of a staircase. As long as the rise and run of the staircase are constant from step to step, the stairs form a line from one floor to the next.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm79774\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=79774&amp;theme=oea&amp;iframe_resize_id=ohm79774&amp;show_question_numbers\" width=\"100%\" height=\"250\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\u00a0 \u00a0 \u00a0Note: The graph in OHM will accept only 2 points to draw a line.<\/p>\n<\/div>\n<p>The following video shows an example of how to graph a linear function given the [latex]y-[\/latex]intercept and the slope.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/N6lEPh11gk8?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Graph [latex]f\\left(x\\right)=-\\dfrac{3}{4}x+6[\/latex]\u00a0using the slope and [latex]y[\/latex]-intercept.<\/p>\n<h4>Solution<\/h4>\n<p>The function is of the form [latex]f(x)=mx+b[\/latex], so the slope of this function is [latex]-\\dfrac{3}{4}[\/latex] and the [latex]y[\/latex]-intercept is [latex](0,6)[\/latex]. We can start graphing by plotting the [latex]y[\/latex]-intercept and counting down three units (rise = -3) and right\u00a0[latex]4[\/latex] units (run = 4). The first stop is the point [latex](4,3)[\/latex], and the next stop is the point [latex](8,0)[\/latex]. Then join the dots to form the line.<\/p>\n<div class=\"wp-nocaption size-medium wp-image-2482 aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2482 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/13204212\/Screen-Shot-2016-07-13-at-1.35.32-PM-300x229.png\" alt=\"A line\" width=\"300\" height=\"229\" \/><\/div>\n<\/div>\n<div class=\"textbox tryit\" style=\"text-align: center;\">\n<h3>\u00a0Try It 4<\/h3>\n<p>Identify the slope and the [latex]y[\/latex]-intercept, then graph the function [latex]d\\left(x\\right)=\\dfrac{2}{3}x+3[\/latex]\u00a0using the slope and [latex]y[\/latex]-intercept.<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm764\">Show Answer<\/span><\/p>\n<div id=\"qhjm764\" class=\"hidden-answer\" style=\"display: none\">\n<p>Slope = [latex]\\frac{2}{3}[\/latex]<\/p>\n<p>[latex]y[\/latex]-intercept = (0, 3)<\/p>\n<p>Graph:\u00a0\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1224\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30010803\/rise-run-graph-1024x592.png\" alt=\"A line using rise and run\" width=\"771\" height=\"446\" \/><\/p>\n<p>Notice that a rise\/run combination of 2\/3 can also be thought of as \u20132\/\u20133. So instead of running 3 to the right and 2 up, we can run 3 to the left and 2 down. This is shown by the purple lines starting at (0, 3) and moving to (\u20133, 1).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3 style=\"text-align: left;\">Vertical and Horizontal Lines<\/h3>\n<p style=\"text-align: left;\">We saw in the last section that a <em><strong>horizontal line<\/strong><\/em> has a slope of zero and therefore the function [latex]f(x)=b[\/latex] represents a horizontal line that passes through the [latex]y[\/latex]-intercept [latex](0,b)[\/latex].<\/p>\n<p style=\"text-align: left;\">For example, suppose we want to graph the function [latex]f(x)=-2[\/latex]. No matter what [latex]x[\/latex]-value we choose, the function value will always equal \u20132. Therefore, the points [latex]\\left(-2,-2\\right),\\left(0,-2\\right),\\left(3,-2\\right)[\/latex], and [latex]\\left(5,-2\\right)[\/latex] all lie on the graph. If we plot these points we see that they form a horizontal line:<\/p>\n<div id=\"attachment_1226\" style=\"width: 444px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1226\" class=\"wp-image-1226\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30014406\/fx-2-300x154.png\" alt=\"Horizontal line through (0, -2)\" width=\"434\" height=\"223\" \/><\/p>\n<p id=\"caption-attachment-1226\" class=\"wp-caption-text\">Figure 2. Graph of function [latex]f(x)=-2[\/latex].<\/p>\n<\/div>\n<p style=\"text-align: left;\">We also saw in the last section that a <em><strong>vertical line<\/strong><\/em> does NOT represent a function, but it is a relation. In order for a line to be vertical, all of the [latex]x[\/latex]-values must be the same. Therefore, the equation of a\u00a0vertical line is given as [latex]x=c[\/latex] where [latex]c[\/latex]<em>\u00a0<\/em>is a constant. The slope of a vertical line is undefined (since the run is zero, and we can&#8217;t divide by 0), and regardless of the [latex]y[\/latex]<em>&#8211;<\/em>value of any point on the line, the [latex]x[\/latex]<em>&#8211;<\/em>coordinate of the point will always be\u00a0<em>c<\/em>.<\/p>\n<p style=\"text-align: left;\">For example, suppose we want to graph the equation [latex]x=-3[\/latex]. The graph will contain the following points: [latex]\\left(-3,-5\\right),\\left(-3,1\\right),\\left(-3,3\\right)[\/latex], and [latex]\\left(-3,5\\right)[\/latex]. If we plot these points, we see they form a vertical line:<\/p>\n<div id=\"attachment_1227\" style=\"width: 308px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1227\" class=\"wp-image-1227\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30015216\/x-3-216x300.png\" alt=\"Vertical line through (-3, 0)\" width=\"298\" height=\"414\" \/><\/p>\n<p id=\"caption-attachment-1227\" class=\"wp-caption-text\">Figure 3. Graph of equation [latex]x=-3[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\" style=\"text-align: center;\">\n<h3>VERTICAL AND HORIZONTAL LINES<\/h3>\n<p>A linear function whose graph has a slope of zero forms a <strong>horizontal line<\/strong>. The function is of the form [latex]f(x)=b[\/latex] where [latex]b[\/latex] is a real number and is represented graphically by the [latex]y[\/latex]-intercept [latex](0, b)[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>A <strong>vertical line<\/strong> represents the equation [latex]x=c[\/latex] where [latex]c[\/latex] is a real number. This equation is NOT a function.<\/p>\n<\/div>\n<div style=\"text-align: center;\">\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>Graph:<\/p>\n<ol>\n<li>[latex]f(x)=4[\/latex]<\/li>\n<li>[latex]x=-1[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>The function [latex]f(x)=4[\/latex] is represented by a horizontal line that passes through (0, 4).<\/li>\n<li>The equation [latex]x=-1[\/latex] is represented by a vertical line that passes through (\u20131, 0).<\/li>\n<\/ol>\n<p>Both graphs are shown:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1229\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30155649\/Horizontal-and-vertical-300x297.png\" alt=\"Horizontal line through (0, 4) and vertical line through (-1, 0)\" width=\"412\" height=\"408\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Graph:<\/p>\n<ol>\n<li>[latex]f(x)=-6[\/latex]<\/li>\n<li>[latex]x=4[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm708\">Show Answer<\/span><\/p>\n<div id=\"qhjm708\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1277\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/31222209\/x4-and-y%E2%80%936-250x300.png\" alt=\"The lines y=-6 and x = 4\" width=\"316\" height=\"379\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2 style=\"text-align: left;\">Parallel and Perpendicular Lines<\/h2>\n<p style=\"text-align: left;\"><em><strong>Parallel lines<\/strong><\/em> have the same slope but different\u00a0<em>y-<\/em>intercepts. Lines that are\u00a0parallel to each other will never intersect. For example, figure 4 shows the graphs of three lines with the same slope, [latex]m=2[\/latex]. Notice that each line goes in exactly the same direction because they have the same slope. Notice also that each line has a different [latex]y[\/latex]-intercept. If these lines all had the same slope and the same\u00a0[latex]y[\/latex]-intercept, they would all lie on top of each other and be the same line. Such lines are referred to as <em><strong>coincidental lines<\/strong><\/em>, and we will see the significance of such lines later in this course.<\/p>\n<div class=\"wp-caption aligncenter\" style=\"width: 497px; text-align: center;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/17232102\/CNX_CAT_Figure_02_02_004.jpg\" alt=\"Three parallel lines with same slopes but different y-intercepts\" width=\"487\" height=\"593\" data-media-type=\"image\/jpg\" \/>Figure 4. Parallel lines.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Parallel Lines<\/h3>\n<p>Two or more lines are parallel to each other if they have identical slopes but different\u00a0[latex]y[\/latex]-intercepts.<\/p>\n<p>If one line has a slope of [latex]m_1[\/latex] and a\u00a0[latex]y[\/latex]-intercept at [latex](0, b_1)[\/latex] and another line has a\u00a0slope of [latex]m_2[\/latex] and a\u00a0[latex]y[\/latex]-intercept at [latex](0, b_2)[\/latex], the lines are parallel if [latex]m_1=m_2[\/latex] and [latex]b_1\\ne b_2[\/latex].<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>Determine if the graph represents two parallel lines:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1232\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30163041\/y23x4-and-y23x-3-298x300.png\" alt=\"Two lines that might be parallel\" width=\"421\" height=\"424\" \/><\/p>\n<h4>Solution<\/h4>\n<p>The blue line passes through the points (\u20136, 0) and (0, 4). [latex]m_1=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{4-0}{0-(-6)}=\\dfrac{4}{6}=\\dfrac{2}{3}[\/latex].<\/p>\n<p>The green line passes through the points (0, \u20133) and (6, 1). [latex]m_1=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{1-(-3)}{6-0)}=\\dfrac{4}{6}=\\dfrac{2}{3}[\/latex].<\/p>\n<p>Since the slopes are equal, the lines are parallel.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Determine if the graph represents two parallel lines:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1233\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30164219\/y-4x6-and-y-3.75x-3-222x300.png\" alt=\"Two lines that might be parallel\" width=\"368\" height=\"495\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm060\">Show Answer<\/span><\/p>\n<div id=\"qhjm060\" class=\"hidden-answer\" style=\"display: none\">\n<p>The blue line passes through the points (0, 6) and (1, 2). By counting, its run is 1 and its rise is \u20134, resulting in a slope of [latex]m_1=\\dfrac{\\text{rise}}{\\text{run}}=\\dfrac{-4}{1}=-4[\/latex].<\/p>\n<p>The green line passes through the points (0, \u20133) and (\u20134, 12). Therefore, its run is -4 and its rise is 15, resulting in a slope of [latex]m_1=\\dfrac{\\text{rise}}{\\text{run}}=\\dfrac{15}{-4}=-\\dfrac{15}{4}[\/latex].<\/p>\n<p>Since the slopes are not equal, the lines are not parallel.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"text-align: left;\">Lines that are\u00a0<em><strong>perpendicular\u00a0<\/strong><\/em>intersect to form a [latex]{90}^{\\circ }[\/latex] angle (a right angle). The product of the slopes of perpendicular lines always equals \u20131.<\/p>\n<p>If one line has a slope of [latex]m_1[\/latex] and another line has a\u00a0slope of [latex]m_2[\/latex], the lines are perpendicular if [latex]m_1 \\cdot m_2=-1[\/latex]. This means that [latex]m_1=-\\frac{1}{m_2}[\/latex] and\u00a0[latex]m_2=-\\frac{1}{m_1}[\/latex], i.e. the slopes are the negative reciprocal of each other.<\/p>\n<p style=\"text-align: left;\">For example, in figure 5 the orange line has a slope of [latex]m_1=3[\/latex] and the blue line has a slope of [latex]m_2=-\\frac{1}{3}[\/latex]. The product of these two slopes is [latex]3\\cdot \\left ( -\\frac{1}{3}\\right )=-1[\/latex]. Notice that [latex]3=\\frac{3}{1}[\/latex] and [latex]-\\frac{1}{3}[\/latex] are negative reciprocals of one another. i.e. if we turn one slope upside down and take it&#8217;s opposite, we get the other slope.<\/p>\n<div class=\"wp-caption aligncenter\" style=\"width: 497px; text-align: center;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/17232106\/CNX_CAT_Figure_02_02_005.jpg\" alt=\"Two perpendicular lines\" width=\"487\" height=\"329\" data-media-type=\"image\/jpg\" \/>Figure 5. Perpendicular lines<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Perpendicular Lines<\/h3>\n<p>Two lines are perpendicular to each other if the product of their slopes equals \u20131.<\/p>\n<p>If one line has a slope of [latex]m_1[\/latex] and another line has a\u00a0slope of [latex]m_2[\/latex], the lines are perpendicular if [latex]m_1 \\cdot m_2=-1[\/latex].<\/p>\n<p>This means that [latex]m_1=-\\frac{1}{m_2}[\/latex] and\u00a0[latex]m_2=-\\frac{1}{m_1}[\/latex], i.e. the slopes are the negative reciprocal of each other.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 7<\/h3>\n<p>Determine if the graph represents two perpendicular lines:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1234\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30164924\/y-52x6-and-y35x-3-300x278.png\" alt=\"Two lines intersecting that might be perpendicular\" width=\"452\" height=\"419\" \/><\/p>\n<h4>Solution<\/h4>\n<p>The blue line passes through the points (0, 6) and (4, \u20134).<\/p>\n<p>This results in a slope of [latex]m_1=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{6-(-4)}{0-4}=\\dfrac{10}{-4}=-\\dfrac{5}{2}[\/latex].<\/p>\n<p>The green line passes through the points (0, \u20133) and (5, 0), resulting in a slope of [latex]m_1=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{0-(-3)}{5-0}=\\dfrac{3}{5}[\/latex].<\/p>\n<p>[latex]m_1\\cdot m_2=-\\dfrac{5}{2}\\cdot\\dfrac{3}{5}=-\\dfrac{3}{2}\\ne -1[\/latex]<\/p>\n<p>Since the product of the slopes is not equal to \u20131, the lines are not perpendicular.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<p>Determine if the graph represents two perpendicular lines:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1235\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30165819\/y-43x6-and-y34x-4-275x300.png\" alt=\"Two lines that might be perpendicular\" width=\"424\" height=\"463\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm902\">Show Answer<\/span><\/p>\n<div id=\"qhjm902\" class=\"hidden-answer\" style=\"display: none\">\n<p>Slope of blue line: [latex]m_1=-\\frac{4}{3}[\/latex]<\/p>\n<p>Slope of green line: [latex]m_2=\\frac{3}{4}[\/latex]<\/p>\n<p>Since [latex]m_1\\cdot m_2=-1[\/latex], the lines are perpendicular.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Example 8<\/h3>\n<p style=\"text-align: left;\">Graph the equations, and state whether the resulting lines are parallel, perpendicular, or neither: [latex]3y=-4x+3[\/latex] and [latex]3x - 4y=8[\/latex].<\/p>\n<h4 style=\"text-align: left;\">Solution<\/h4>\n<p>We can graph the lines using a table of values. To determine the corresponding [latex]y[\/latex]-values we must substitute the chosen [latex]x[\/latex]-value and solve the resulting equation in one variable.<\/p>\n<p>Table of values for line 1:<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"vertical-align: top; width: 25%;\"><span style=\"font-size: 12px; text-align: initial;\">[latex]x=-3[\/latex]<\/span><\/th>\n<th style=\"width: 25%; vertical-align: top;\"><span style=\"font-size: 12px; orphans: 1;\">[latex]x=0[\/latex]<\/span><\/th>\n<th style=\"width: 25%; vertical-align: top;\"><span style=\"font-size: 12px; orphans: 1;\">[latex]x=3[\/latex]<\/span><\/th>\n<th style=\"width: 25%; vertical-align: top;\">Resulting Table<\/th>\n<\/tr>\n<tr>\n<td style=\"vertical-align: top; width: 25%;\">[latex]\\begin{aligned}3y&=-4x+3\\\\ 3y&=-4(-3)+3\\\\3y&=12+3\\\\ 3y&=15\\\\y&=5\\end{aligned}[\/latex]<\/td>\n<td style=\"width: 25%; vertical-align: top;\">[latex]\\begin{aligned}3y&= -4x+3\\\\ 3y& =-4(0)+3\\\\3y&=3\\\\ y&=1\\end{aligned}[\/latex]<\/td>\n<td style=\"width: 25%; vertical-align: top;\">[latex]\\begin{aligned}3y& =-4x+3 \\\\3y&=-4(-)+3\\\\ 3y&=-12+3\\\\ 3y&=-9\\\\y&=-3\\end{aligned}[\/latex]<\/td>\n<td style=\"width: 25%; vertical-align: top;\">&nbsp;<\/p>\n<table style=\"width: 25%; height: 93px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 50%; height: 12px;\"><span style=\"font-size: 12px;\">[latex]x[\/latex]<\/span><\/th>\n<th style=\"width: 50%; height: 12px;\"><span style=\"font-size: 12px;\">[latex]3y=-4x+3[\/latex]<\/span><\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]\u20133[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">5<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">0<\/td>\n<td style=\"width: 50%; height: 12px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%;\">3<\/td>\n<td style=\"width: 50%; height: 12px;\">\u20133<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Table of values for line 2:<\/p>\n<table style=\"width: 100%;\">\n<tbody>\n<tr>\n<th style=\"vertical-align: top; width: 26.0361%;\">[latex]x=-4[\/latex]<\/th>\n<th style=\"vertical-align: top; width: 24.8673%;\">[latex]x=0[\/latex]<\/th>\n<th style=\"width: 24.5748%; vertical-align: top;\">[latex]x=4[\/latex]<\/th>\n<th style=\"width: 24.4244%; vertical-align: top;\">Resulting Table<\/th>\n<\/tr>\n<tr>\n<td style=\"vertical-align: top; width: 26.0361%;\"><span style=\"font-size: 12px; orphans: 2;\">[latex]\\begin{aligned}3x - 4y&=8\\\\3(-4)-4y&=8\\\\-12-4y&=8\\\\-4y&=20\\\\y&=-5\\end{aligned}[\/latex]<\/span><\/td>\n<td style=\"vertical-align: top; width: 24.8673%;\"><span style=\"font-size: 12px; orphans: 2;\">[latex]\\begin{aligned}3x - 4y&=8\\\\3(0)-4y&=8\\\\-4y&=8\\\\y&=- 2\\end{aligned}[\/latex]<\/span><\/td>\n<td style=\"width: 24.5748%; vertical-align: top;\"><span style=\"font-size: 12px; orphans: 2;\">[latex]\\begin{aligned}3x - 4y&=8\\\\3(4)-4y&=8\\\\12-4y&=8\\\\-4y&=-4\\\\y&=1\\end{aligned}[\/latex]<\/span><\/td>\n<td style=\"width: 24.4244%; vertical-align: top;\">\n<table style=\"width: 93.1191%; height: 48px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 43.7863%; height: 12px;\">[latex]x[\/latex]<\/th>\n<th style=\"height: 12px; width: 50%;\">[latex]3x - 4y=8[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 43.7863%; height: 12px;\">\u20134<\/td>\n<td style=\"width: 225.585%; height: 12px;\">\u20135<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 43.7863%; height: 12px;\">0<\/td>\n<td style=\"width: 225.585%; height: 12px;\">\u20132<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 43.7863%; height: 12px;\">4<\/td>\n<td style=\"width: 225.585%; height: 12px;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1237\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30173537\/Example-2.2-287x300.png\" alt=\"Two intersecting lines\" width=\"443\" height=\"463\" \/>Now plot the points and graph each line:<\/p>\n<div class=\"wp-nocaption aligncenter\"><\/div>\n<p>From the graph, we can see that the lines appear perpendicular, but we must compare the slopes.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{m}_{1}=-\\frac{4}{3}\\hfill \\\\ {m}_{2}=\\frac{3}{4}\\hfill \\\\ {m}_{1}\\cdot {m}_{2}=\\left(-\\frac{4}{3}\\right)\\left(\\frac{3}{4}\\right)=-1\\hfill \\end{array}[\/latex]<\/p>\n<p>The slopes are negative reciprocals of each other, confirming that the lines are perpendicular.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox tryit\" style=\"text-align: center;\">\n<h3>Try It 8<\/h3>\n<p>Graph the equations, and state whether the resulting lines are parallel, perpendicular, or neither:\u00a0 [latex]2y-x=10[\/latex] and [latex]2y=x+4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm633\">Show Answer<\/span><\/p>\n<div id=\"qhjm633\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1241\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/30175940\/2x-y10-and-2yx4-288x300.png\" alt=\"Two lines\" width=\"415\" height=\"432\" \/><\/p>\n<p>Slope of blue line = [latex]\\frac{1}{2}[\/latex]<\/p>\n<p>Slope of green line =\u00a0[latex]\\frac{1}{2}[\/latex]<\/p>\n<p>Since the slopes are equal, the lines are parallel.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-916\">\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>Revision and Adaptation. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All examples, Try Its . <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using Desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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>Ex: Graph a Line and ID the Slope and Intercepts (Fraction Slope). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/N6lEPh11gk8\">https:\/\/youtu.be\/N6lEPh11gk8<\/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>QID 79774: Graph linear eq. in slope-intercept form, give slope and intercept.. <strong>Authored by<\/strong>: Day, Alyson. <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, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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":422608,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Graph a Linear Function Using a Table of Values (Function Notation)\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/sfzpdThXpA8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et 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