{"id":18227,"date":"2022-04-08T05:03:17","date_gmt":"2022-04-08T05:03:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18227"},"modified":"2022-04-28T23:14:57","modified_gmt":"2022-04-28T23:14:57","slug":"cr-5-rectangular-coordinate-system-and-graphing","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-5-rectangular-coordinate-system-and-graphing\/","title":{"raw":"CR.5: Rectangular Coordinate System and Graphing","rendered":"CR.5: Rectangular Coordinate System and Graphing"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define the components of the Cartesian coordinate system.<\/li>\r\n \t<li>Plot points on the Cartesian coordinate plane.<\/li>\r\n \t<li>Plot linear equations in two variables on the coordinate plane.<\/li>\r\n \t<li>Use intercepts to plot lines.<\/li>\r\n \t<li>Use a graphing utility to graph a linear equation on a coordinate plane.<\/li>\r\n \t<li>Find the slope of a line passing through two points.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAn old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly\u2019s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers\u2014the displacement from the horizontal axis and the displacement from the vertical axis.\r\n\r\nWhile there is evidence that ideas similar to Descartes\u2019 grid system existed centuries earlier, it was Descartes who introduced the components that comprise the <strong>Cartesian coordinate system<\/strong>, a grid system having perpendicular axes. Descartes named the horizontal axis the <strong><em>x-<\/em>axis<\/strong> and the vertical axis the <strong><em>y-<\/em>axis<\/strong>.\r\n\r\nThe Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em>x<\/em>-axis and the <em>y<\/em>-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a <strong>quadrant<\/strong>; the quadrants are numbered counterclockwise as shown in the figure below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042358\/CNX_CAT_Figure_02_01_002.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"487\" height=\"442\" \/> <b>The Cartesian coordinate system with all four quadrants labeled.<\/b>[\/caption]\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]92752[\/ohm_question]\r\n\r\n<\/div>\r\nThe center of the plane is the point at which the two axes cross. It is known as the <strong>origin\u00a0<\/strong>or point [latex]\\left(0,0\\right)[\/latex]. From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em>x-<\/em>axis and up the <em>y-<\/em>axis; decreasing, negative numbers to the left on the <em>x-<\/em>axis and down the <em>y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in the figure below.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042401\/CNX_CAT_Figure_02_01_003.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5.\" width=\"487\" height=\"442\" \/>\r\n\r\nEach point in the plane is identified by its <strong><em>x-<\/em>coordinate<\/strong>,\u00a0or horizontal displacement from the origin, and its <strong><em>y-<\/em>coordinate<\/strong>, or vertical displacement from the origin. Together we write them as an <strong>ordered pair<\/strong> indicating the combined distance from the origin in the form [latex]\\left(x,y\\right)[\/latex]. An ordered pair is also known as a coordinate pair because it consists of <em>x\u00a0<\/em>and <em>y<\/em>-coordinates. For example, we can represent the point [latex]\\left(3,-1\\right)[\/latex] in the plane by moving three units to the right of the origin in the horizontal direction and one unit down in the vertical direction.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042403\/CNX_CAT_Figure_02_01_004.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. The point (3, -1) is labeled. An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.\" width=\"487\" height=\"442\" \/> <b>An illustration of how to plot the point (3,-1).<\/b>[\/caption]\r\n\r\nWhen dividing the axes into equally spaced increments, note that the <em>x-<\/em>axis may be considered separately from the <em>y-<\/em>axis. In other words, while the <em>x-<\/em>axis may be divided and labeled according to consecutive integers, the <em>y-<\/em>axis may be divided and labeled by increments of 2 or 10 or 100. In fact, the axes may represent other units such as years against the balance in a savings account or quantity against cost. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Cartesian Coordinate System<\/h3>\r\nA two-dimensional plane where the\r\n<ul>\r\n \t<li><em>x<\/em>-axis is the horizontal axis<\/li>\r\n \t<li><em>y<\/em>-axis is the vertical axis<\/li>\r\n<\/ul>\r\nA point in the plane is defined as an ordered pair, [latex]\\left(x,y\\right)[\/latex], such that <em>x <\/em>is determined by its horizontal distance from the origin and <em>y <\/em>is determined by its vertical distance from the origin.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Plotting Points in a Rectangular Coordinate System<\/h3>\r\nPlot the points [latex]\\left(-2,4\\right)[\/latex], [latex]\\left(3,3\\right)[\/latex], and [latex]\\left(0,-3\\right)[\/latex] in the coordinate plane.\r\n[reveal-answer q=\"380739\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"380739\"]\r\n\r\nTo plot the point [latex]\\left(-2,4\\right)[\/latex], begin at the origin. The <em>x<\/em>-coordinate is \u20132, so move two units to the left. The <em>y<\/em>-coordinate is 4, so then move four units up in the positive <em>y <\/em>direction.\r\n\r\nTo plot the point [latex]\\left(3,3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 3, so move three units to the right. The <em>y<\/em>-coordinate is also 3, so move three units up in the positive <em>y <\/em>direction.\r\n\r\nTo plot the point [latex]\\left(0,-3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 0. This tells us not to move in either direction along the <em>x<\/em>-axis. The <em>y<\/em>-coordinate is \u20133, so move three units down in the negative <em>y<\/em> direction.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042406\/CNX_CAT_Figure_02_01_005.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5. The points (-2, 4); (3, 3); and (0, -3) are labeled. Arrows extend from the origin to the points.\" width=\"487\" height=\"442\" \/>\r\n<h4>Analysis of the Solution<\/h4>\r\nNote that when either coordinate is zero, the point must be on an axis. If the <em>x<\/em>-coordinate is zero, the point is on the <em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the <em>x<\/em>-axis.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]92753[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=7JMXi_FxA2o\r\n<h2>Graphing Linear Equations by Using a Table<\/h2>\r\nWe can plot a set of points to represent an equation. When such an equation contains both an <em>x <\/em>variable and a <em>y <\/em>variable, it is called an <strong>equation in two variables<\/strong>. Its graph is called a <strong>graph in two variables<\/strong>. Any graph on a two-dimensional plane is a graph in two variables.\r\n\r\nSuppose we want to graph the equation [latex]y=2x - 1[\/latex]. We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of <em>y<\/em>. Each pair of <em>x\u00a0<\/em>and <em>y-<\/em>values is an ordered pair that can be plotted. The table below\u00a0lists values of <em>x<\/em> from \u20133 to 3 and the resulting values for <em>y<\/em>.\r\n<table summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=2x - 1[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\r\n<td>[latex]\\left(-3,-7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\r\n<td>[latex]\\left(-2,-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\r\n<td>[latex]\\left(-1,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\r\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\r\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\r\n<td>[latex]\\left(3,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can plot these points from the table. The points for this particular equation form a line, so we can connect them.<strong>\u00a0<\/strong>This is not true for all equations.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042408\/CNX_CAT_Figure_02_01_006.jpg\" alt=\"This is a graph of a line on an x, y coordinate plane. The x- and y-axis range from negative 8 to 8. A line passes through the points (-3, -7); (-2, -5); (-1, -3); (0, -1); (1, 1); (2, 3); and (3, 5).\" width=\"731\" height=\"669\" \/>\r\n\r\nNote that the <em>x-<\/em>values chosen are arbitrary regardless of the type of equation we are graphing. Of course, some situations may require particular values of <em>x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.\r\n<div class=\"textbox\">\r\n<h3>How To: Given an equation, graph by plotting points<\/h3>\r\n<ol>\r\n \t<li>Make a table with one column labeled <em>x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\r\n \t<li>Enter <em>x-<\/em>values down the first column using positive and negative values. Selecting the <em>x-<\/em>values in numerical order will make graphing easier.<\/li>\r\n \t<li>Select <em>x-<\/em>values that will yield <em>y-<\/em>values with little effort, preferably ones that can be calculated mentally.<\/li>\r\n \t<li>Plot the ordered pairs.<\/li>\r\n \t<li>Connect the points if they form a line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing an Equation in Two Variables by Plotting Points<\/h3>\r\nGraph the equation [latex]y=-x+2[\/latex] by plotting points.\r\n[reveal-answer q=\"792137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792137\"]\r\n\r\nFirst, we construct a table similar to the one below. Choose <em>x<\/em> values and calculate <em>y.<\/em>\r\n<table summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=-x+2[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\r\n<td>[latex]\\left(-5,7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\r\n<td>[latex]\\left(-3,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\r\n<td>[latex]\\left(-1,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\r\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\r\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\r\n<td>[latex]\\left(3,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\r\n<td>[latex]\\left(5,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow, plot the points. Connect them if they form a line.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042411\/CNX_CAT_Figure_02_01_007.jpg\" alt=\"This image is a graph of a line on an x, y coordinate plane. The x-axis includes numbers that range from negative 7 to 7. The y-axis includes numbers that range from negative 5 to 8. A line passes through the points: (-5, 7); (-3, 5); (-1, 3); (0, 2); (1, 1); (3, -1); and (5, -3).\" width=\"731\" height=\"556\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConstruct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].\r\n[reveal-answer q=\"811886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811886\"]\r\n<table summary=\"The table shows 6 rows and 3 columns. The entries in the first row are: x; y = x divided by 2 plus 2, (x,y). The entries in the second row are: negative 2; y = (negative 2) divided by 2 plus 2 = 1; (-2, 1). The entries in the third row are: negative 1; y = (negative 1) divided by 2 plus 2 = 3\/2; (-1,3\/2). The entries in the fourth row are: 0; y = (0)\/2 + 2 = 2; (0,2). The entries in the fifth row are: 1; y = (1)\/2 + 2 = 5\/2; (1,5\/2). The entries in the sixth row are: 2; y = (2)\/2 + 2 = 3; (2,3).\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\r\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\r\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042413\/CNX_CAT_Figure_02_01_008.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3\/2); (0, 2); (1, 5\/2); and (2, 3).\" width=\"487\" height=\"442\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=110939&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Using Intercepts to Plot Lines in the Coordinate Plane<\/h2>\r\nThe <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong><em>x-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong><em>y-<\/em>intercept<\/strong> is the point where the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.\r\n\r\nTo determine the <em>x-<\/em>intercept, we set <em>y <\/em>equal to zero and solve for <em>x<\/em>. Similarly, to determine the <em>y-<\/em>intercept, we set <em>x <\/em>equal to zero and solve for <em>y<\/em>. For example, lets find the intercepts of the equation [latex]y=3x - 1[\/latex].\r\n\r\nTo find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}y=3x - 1\\hfill &amp; \\hfill \\\\ 0=3x - 1\\hfill &amp; \\hfill \\\\ 1=3x\\hfill &amp; \\hfill \\\\ \\frac{1}{3}=x\\hfill &amp; \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill &amp; x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nTo find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill &amp; \\hfill \\\\ y=3\\left(0\\right)-1\\hfill &amp; \\hfill \\\\ y=-1\\hfill &amp; \\hfill \\\\ \\left(0,-1\\right)\\hfill &amp; y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nWe can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an equation, find the intercepts<\/h3>\r\n<ol>\r\n \t<li>Find the <em>x<\/em>-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\r\n \t<li>Find the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Intercepts of the Given Equation<\/h3>\r\nFind the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.\r\n\r\n[reveal-answer q=\"814560\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814560\"]\r\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nSet [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nPlot both points and draw a line passing through them.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].\r\n\r\n[reveal-answer q=\"80464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"80464\"]\r\n\r\n<em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" width=\"487\" height=\"447\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92757&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Using a Graphing Utility to Plot Lines<\/h2>\r\nhttps:\/\/www.geogebra.org\/graphing\r\n\r\nhttps:\/\/www.desmos.com\/calculator\r\n\r\nhttps:\/\/www.mathway.com\/Graph\r\n\r\nhttps:\/\/www.symbolab.com\/graphing-calculator\r\n\r\n[embed]https:\/\/youtu.be\/cEIOdi2R4fE[\/embed]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it now<\/h3>\r\nThese graphing utilities have features that allow you to turn a constant (number) into a variable. Follow these steps to learn how:\r\n<ol>\r\n \t<li>Graph the line [latex]y=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].<\/li>\r\n \t<li>On the next line enter\u00a0[latex]y=-a x-\\frac{4}{3}[\/latex]. You will see a button pop up that says \"add slider: a\", click on the button. You will see the next line populated with the variable a and the interval on which a can take values.<\/li>\r\n \t<li>What part of a line does the variable a represent? The slope or the y-intercept?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>The Slope of a Line<\/h2>\r\nThe <strong>slope<\/strong> of a line refers to the ratio of the vertical change in <em>y<\/em> over the horizontal change in <em>x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.\r\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\nIf the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[\/latex], [latex]m=2[\/latex], and [latex]m=\\frac{1}{3}[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/11185922\/CNX_CAT_Figure_02_02_002.jpg\" alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\" width=\"487\" height=\"442\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Slope of a Line<\/h3>\r\nThe slope of a line, <em>m<\/em>, represents the change in <em>y<\/em> over the change in <em>x.<\/em> Given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the following formula determines the slope of a line containing these points:\r\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Slope of a Line Given Two Points<\/h3>\r\nFind the slope of a line that passes through the points [latex]\\left(2,-1\\right)[\/latex] and [latex]\\left(-5,3\\right)[\/latex].\r\n[reveal-answer q=\"688301\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688301\"]\r\n\r\nWe substitute the <em>y-<\/em>values and the <em>x-<\/em>values into the formula.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}m\\hfill&amp;=\\frac{3-\\left(-1\\right)}{-5 - 2}\\hfill \\\\ \\hfill&amp;=\\frac{4}{-7}\\hfill \\\\ \\hfill&amp;=-\\frac{4}{7}\\hfill \\end{array}[\/latex]<\/div>\r\nThe slope is [latex]-\\frac{4}{7}[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nIt does not matter which point is called [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] or [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]. As long as we are consistent with the order of the <em>y<\/em> terms and the order of the <em>x<\/em> terms in the numerator and denominator, the calculation will yield the same result.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the slope of the line that passes through the points [latex]\\left(-2,6\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].\r\n\r\n[reveal-answer q=\"196055\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"196055\"]\r\n\r\nslope[latex]=m=\\dfrac{-2}{3}=-\\dfrac{2}{3}[\/latex][\/hidden-answer]\r\n\r\n[ohm_question]1719[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define the components of the Cartesian coordinate system.<\/li>\n<li>Plot points on the Cartesian coordinate plane.<\/li>\n<li>Plot linear equations in two variables on the coordinate plane.<\/li>\n<li>Use intercepts to plot lines.<\/li>\n<li>Use a graphing utility to graph a linear equation on a coordinate plane.<\/li>\n<li>Find the slope of a line passing through two points.<\/li>\n<\/ul>\n<\/div>\n<p>An old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly\u2019s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers\u2014the displacement from the horizontal axis and the displacement from the vertical axis.<\/p>\n<p>While there is evidence that ideas similar to Descartes\u2019 grid system existed centuries earlier, it was Descartes who introduced the components that comprise the <strong>Cartesian coordinate system<\/strong>, a grid system having perpendicular axes. Descartes named the horizontal axis the <strong><em>x-<\/em>axis<\/strong> and the vertical axis the <strong><em>y-<\/em>axis<\/strong>.<\/p>\n<p>The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em>x<\/em>-axis and the <em>y<\/em>-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a <strong>quadrant<\/strong>; the quadrants are numbered counterclockwise as shown in the figure below.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042358\/CNX_CAT_Figure_02_01_002.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\"><b>The Cartesian coordinate system with all four quadrants labeled.<\/b><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm92752\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92752&theme=oea&iframe_resize_id=ohm92752&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The center of the plane is the point at which the two axes cross. It is known as the <strong>origin\u00a0<\/strong>or point [latex]\\left(0,0\\right)[\/latex]. From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em>x-<\/em>axis and up the <em>y-<\/em>axis; decreasing, negative numbers to the left on the <em>x-<\/em>axis and down the <em>y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in the figure below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042401\/CNX_CAT_Figure_02_01_003.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5.\" width=\"487\" height=\"442\" \/><\/p>\n<p>Each point in the plane is identified by its <strong><em>x-<\/em>coordinate<\/strong>,\u00a0or horizontal displacement from the origin, and its <strong><em>y-<\/em>coordinate<\/strong>, or vertical displacement from the origin. Together we write them as an <strong>ordered pair<\/strong> indicating the combined distance from the origin in the form [latex]\\left(x,y\\right)[\/latex]. An ordered pair is also known as a coordinate pair because it consists of <em>x\u00a0<\/em>and <em>y<\/em>-coordinates. For example, we can represent the point [latex]\\left(3,-1\\right)[\/latex] in the plane by moving three units to the right of the origin in the horizontal direction and one unit down in the vertical direction.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042403\/CNX_CAT_Figure_02_01_004.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. The point (3, -1) is labeled. An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\"><b>An illustration of how to plot the point (3,-1).<\/b><\/p>\n<\/div>\n<p>When dividing the axes into equally spaced increments, note that the <em>x-<\/em>axis may be considered separately from the <em>y-<\/em>axis. In other words, while the <em>x-<\/em>axis may be divided and labeled according to consecutive integers, the <em>y-<\/em>axis may be divided and labeled by increments of 2 or 10 or 100. In fact, the axes may represent other units such as years against the balance in a savings account or quantity against cost. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Cartesian Coordinate System<\/h3>\n<p>A two-dimensional plane where the<\/p>\n<ul>\n<li><em>x<\/em>-axis is the horizontal axis<\/li>\n<li><em>y<\/em>-axis is the vertical axis<\/li>\n<\/ul>\n<p>A point in the plane is defined as an ordered pair, [latex]\\left(x,y\\right)[\/latex], such that <em>x <\/em>is determined by its horizontal distance from the origin and <em>y <\/em>is determined by its vertical distance from the origin.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Plotting Points in a Rectangular Coordinate System<\/h3>\n<p>Plot the points [latex]\\left(-2,4\\right)[\/latex], [latex]\\left(3,3\\right)[\/latex], and [latex]\\left(0,-3\\right)[\/latex] in the coordinate plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q380739\">Show Solution<\/span><\/p>\n<div id=\"q380739\" class=\"hidden-answer\" style=\"display: none\">\n<p>To plot the point [latex]\\left(-2,4\\right)[\/latex], begin at the origin. The <em>x<\/em>-coordinate is \u20132, so move two units to the left. The <em>y<\/em>-coordinate is 4, so then move four units up in the positive <em>y <\/em>direction.<\/p>\n<p>To plot the point [latex]\\left(3,3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 3, so move three units to the right. The <em>y<\/em>-coordinate is also 3, so move three units up in the positive <em>y <\/em>direction.<\/p>\n<p>To plot the point [latex]\\left(0,-3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 0. This tells us not to move in either direction along the <em>x<\/em>-axis. The <em>y<\/em>-coordinate is \u20133, so move three units down in the negative <em>y<\/em> direction.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042406\/CNX_CAT_Figure_02_01_005.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5. The points (-2, 4); (3, 3); and (0, -3) are labeled. Arrows extend from the origin to the points.\" width=\"487\" height=\"442\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Note that when either coordinate is zero, the point must be on an axis. If the <em>x<\/em>-coordinate is zero, the point is on the <em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the <em>x<\/em>-axis.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm92753\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92753&theme=oea&iframe_resize_id=ohm92753&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Plotting Points on the Coordinate Plane\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/7JMXi_FxA2o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Graphing Linear Equations by Using a Table<\/h2>\n<p>We can plot a set of points to represent an equation. When such an equation contains both an <em>x <\/em>variable and a <em>y <\/em>variable, it is called an <strong>equation in two variables<\/strong>. Its graph is called a <strong>graph in two variables<\/strong>. Any graph on a two-dimensional plane is a graph in two variables.<\/p>\n<p>Suppose we want to graph the equation [latex]y=2x - 1[\/latex]. We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of <em>y<\/em>. Each pair of <em>x\u00a0<\/em>and <em>y-<\/em>values is an ordered pair that can be plotted. The table below\u00a0lists values of <em>x<\/em> from \u20133 to 3 and the resulting values for <em>y<\/em>.<\/p>\n<table summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=2x - 1[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\n<td>[latex]\\left(-3,-7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\n<td>[latex]\\left(-2,-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\n<td>[latex]\\left(-1,-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\n<td>[latex]\\left(3,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can plot these points from the table. The points for this particular equation form a line, so we can connect them.<strong>\u00a0<\/strong>This is not true for all equations.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042408\/CNX_CAT_Figure_02_01_006.jpg\" alt=\"This is a graph of a line on an x, y coordinate plane. The x- and y-axis range from negative 8 to 8. A line passes through the points (-3, -7); (-2, -5); (-1, -3); (0, -1); (1, 1); (2, 3); and (3, 5).\" width=\"731\" height=\"669\" \/><\/p>\n<p>Note that the <em>x-<\/em>values chosen are arbitrary regardless of the type of equation we are graphing. Of course, some situations may require particular values of <em>x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given an equation, graph by plotting points<\/h3>\n<ol>\n<li>Make a table with one column labeled <em>x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\n<li>Enter <em>x-<\/em>values down the first column using positive and negative values. Selecting the <em>x-<\/em>values in numerical order will make graphing easier.<\/li>\n<li>Select <em>x-<\/em>values that will yield <em>y-<\/em>values with little effort, preferably ones that can be calculated mentally.<\/li>\n<li>Plot the ordered pairs.<\/li>\n<li>Connect the points if they form a line.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing an Equation in Two Variables by Plotting Points<\/h3>\n<p>Graph the equation [latex]y=-x+2[\/latex] by plotting points.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q792137\">Show Solution<\/span><\/p>\n<div id=\"q792137\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we construct a table similar to the one below. Choose <em>x<\/em> values and calculate <em>y.<\/em><\/p>\n<table summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=-x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\n<td>[latex]\\left(-5,7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\n<td>[latex]\\left(-3,5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(-1,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\n<td>[latex]\\left(3,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\n<td>[latex]\\left(5,-3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now, plot the points. Connect them if they form a line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042411\/CNX_CAT_Figure_02_01_007.jpg\" alt=\"This image is a graph of a line on an x, y coordinate plane. The x-axis includes numbers that range from negative 7 to 7. The y-axis includes numbers that range from negative 5 to 8. A line passes through the points: (-5, 7); (-3, 5); (-1, 3); (0, 2); (1, 1); (3, -1); and (5, -3).\" width=\"731\" height=\"556\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Construct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q811886\">Show Solution<\/span><\/p>\n<div id=\"q811886\" class=\"hidden-answer\" style=\"display: none\">\n<table summary=\"The table shows 6 rows and 3 columns. The entries in the first row are: x; y = x divided by 2 plus 2, (x,y). The entries in the second row are: negative 2; y = (negative 2) divided by 2 plus 2 = 1; (-2, 1). The entries in the third row are: negative 1; y = (negative 1) divided by 2 plus 2 = 3\/2; (-1,3\/2). The entries in the fourth row are: 0; y = (0)\/2 + 2 = 2; (0,2). The entries in the fifth row are: 1; y = (1)\/2 + 2 = 5\/2; (1,5\/2). The entries in the sixth row are: 2; y = (2)\/2 + 2 = 3; (2,3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042413\/CNX_CAT_Figure_02_01_008.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3\/2); (0, 2); (1, 5\/2); and (2, 3).\" width=\"487\" height=\"442\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=110939&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Using Intercepts to Plot Lines in the Coordinate Plane<\/h2>\n<p>The <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong><em>x-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong><em>y-<\/em>intercept<\/strong> is the point where the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.<\/p>\n<p>To determine the <em>x-<\/em>intercept, we set <em>y <\/em>equal to zero and solve for <em>x<\/em>. Similarly, to determine the <em>y-<\/em>intercept, we set <em>x <\/em>equal to zero and solve for <em>y<\/em>. For example, lets find the intercepts of the equation [latex]y=3x - 1[\/latex].<\/p>\n<p>To find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}y=3x - 1\\hfill & \\hfill \\\\ 0=3x - 1\\hfill & \\hfill \\\\ 1=3x\\hfill & \\hfill \\\\ \\frac{1}{3}=x\\hfill & \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill & x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>To find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill & \\hfill \\\\ y=3\\left(0\\right)-1\\hfill & \\hfill \\\\ y=-1\\hfill & \\hfill \\\\ \\left(0,-1\\right)\\hfill & y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>We can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/><\/p>\n<div class=\"textbox\">\n<h3>How To: Given an equation, find the intercepts<\/h3>\n<ol>\n<li>Find the <em>x<\/em>-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\n<li>Find the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Intercepts of the Given Equation<\/h3>\n<p>Find the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814560\">Show Solution<\/span><\/p>\n<div id=\"q814560\" class=\"hidden-answer\" style=\"display: none\">\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Set [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Plot both points and draw a line passing through them.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q80464\">Show Solution<\/span><\/p>\n<div id=\"q80464\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" width=\"487\" height=\"447\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92757&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<h2>Using a Graphing Utility to Plot Lines<\/h2>\n<p><iframe loading=\"lazy\" class=\"resizable\" src=\"https:\/\/www.geogebra.org\/graphing\" frameborder=\"0\" width=\"500\" height=\"750\"><\/iframe><\/p>\n<p>https:\/\/www.desmos.com\/calculator<\/p>\n<p>https:\/\/www.mathway.com\/Graph<\/p>\n<p>https:\/\/www.symbolab.com\/graphing-calculator<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Learn Desmos: Lines\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/cEIOdi2R4fE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it now<\/h3>\n<p>These graphing utilities have features that allow you to turn a constant (number) into a variable. Follow these steps to learn how:<\/p>\n<ol>\n<li>Graph the line [latex]y=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].<\/li>\n<li>On the next line enter\u00a0[latex]y=-a x-\\frac{4}{3}[\/latex]. You will see a button pop up that says &#8220;add slider: a&#8221;, click on the button. You will see the next line populated with the variable a and the interval on which a can take values.<\/li>\n<li>What part of a line does the variable a represent? The slope or the y-intercept?<\/li>\n<\/ol>\n<\/div>\n<h2>The Slope of a Line<\/h2>\n<p>The <strong>slope<\/strong> of a line refers to the ratio of the vertical change in <em>y<\/em> over the horizontal change in <em>x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.<\/p>\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<p>If the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[\/latex], [latex]m=2[\/latex], and [latex]m=\\frac{1}{3}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/11185922\/CNX_CAT_Figure_02_02_002.jpg\" alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\" width=\"487\" height=\"442\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Slope of a Line<\/h3>\n<p>The slope of a line, <em>m<\/em>, represents the change in <em>y<\/em> over the change in <em>x.<\/em> Given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the following formula determines the slope of a line containing these points:<\/p>\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Slope of a Line Given Two Points<\/h3>\n<p>Find the slope of a line that passes through the points [latex]\\left(2,-1\\right)[\/latex] and [latex]\\left(-5,3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q688301\">Show Solution<\/span><\/p>\n<div id=\"q688301\" class=\"hidden-answer\" style=\"display: none\">\n<p>We substitute the <em>y-<\/em>values and the <em>x-<\/em>values into the formula.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}m\\hfill&=\\frac{3-\\left(-1\\right)}{-5 - 2}\\hfill \\\\ \\hfill&=\\frac{4}{-7}\\hfill \\\\ \\hfill&=-\\frac{4}{7}\\hfill \\end{array}[\/latex]<\/div>\n<p>The slope is [latex]-\\frac{4}{7}[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>It does not matter which point is called [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] or [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]. As long as we are consistent with the order of the <em>y<\/em> terms and the order of the <em>x<\/em> terms in the numerator and denominator, the calculation will yield the same result.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the slope of the line that passes through the points [latex]\\left(-2,6\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q196055\">Show Solution<\/span><\/p>\n<div id=\"q196055\" class=\"hidden-answer\" style=\"display: none\">\n<p>slope[latex]=m=\\dfrac{-2}{3}=-\\dfrac{2}{3}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm1719\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1719&theme=oea&iframe_resize_id=ohm1719&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n","protected":false},"author":264444,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18227","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18227","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18227\/revisions"}],"predecessor-version":[{"id":18700,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18227\/revisions\/18700"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18227\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18227"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18227"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18227"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}