{"id":3793,"date":"2020-02-08T20:24:29","date_gmt":"2020-02-08T20:24:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3793"},"modified":"2021-02-05T23:54:45","modified_gmt":"2021-02-05T23:54:45","slug":"finding-the-slope-of-a-line-from-its-graph","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/finding-the-slope-of-a-line-from-its-graph\/","title":{"raw":"Finding the Slope of a Line From Its Graph","rendered":"Finding the Slope of a Line From Its Graph"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Given the graph of a line, determine the slope of the line<\/li>\r\n \t<li>Identify the slope of a horizontal line given it's equation<\/li>\r\n \t<li>Identify the slope of a vertical line given it's equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>Now we\u2019ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.<\/p>\r\nTo find the slope, we must count out the rise and the run. But where do we start?\r\n\r\nWe locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the slope of the line shown:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224557\/CNX_BMath_Figure_11_04_020.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points \" \/>\r\n\r\nSolution\r\nLocate two points on the graph, choosing points whose coordinates are integers. We will use [latex]\\left(0,-3\\right)[\/latex] and [latex]\\left(5,1\\right)[\/latex].\r\n\r\nStarting with the point on the left, [latex]\\left(0,-3\\right)[\/latex], sketch a right triangle, going from the first point to the second point, [latex]\\left(5,1\\right)[\/latex].\r\n<table id=\"eip-id1168466130951\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224558\/CNX_BMath_Figure_11_04_021.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points \" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the rise on the vertical leg of the triangle.<\/td>\r\n<td>The rise is [latex]4[\/latex] units.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run on the horizontal leg.<\/td>\r\n<td>The run is [latex]5[\/latex] units.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values of the rise and run.<\/td>\r\n<td>[latex]m={\\Large\\frac{4}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the line is [latex]{\\Large\\frac{4}{5}}[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice that the slope is positive since the line slants upward from left to right.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147014[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>Find the slope from a graph<\/h3>\r\n<ol id=\"eip-id1168469837806\" class=\"stepwise\">\r\n \t<li>Locate two points on the line whose coordinates are integers.<\/li>\r\n \t<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/li>\r\n \t<li>Count the rise and the run on the legs of the triangle.<\/li>\r\n \t<li>Take the ratio of rise to run to find the slope. [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the slope of the line shown:\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224602\/CNX_BMath_Figure_11_04_024.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points \" \/>\r\n[reveal-answer q=\"966343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"966343\"]\r\n\r\nSolution\r\nLocate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use [latex](0, 5)[\/latex] and [latex](3, 3)[\/latex]. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.\r\n<table id=\"eip-id1168465988432\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224603\/CNX_BMath_Figure_11_04_025.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points \" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the rise \u2013 it is negative.<\/td>\r\n<td>The rise is [latex]\u22122[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run.<\/td>\r\n<td>The run is [latex]3[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values of the rise and run.<\/td>\r\n<td>[latex]m={\\Large\\frac{-2}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]m=-{\\Large\\frac{2}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the line is [latex]-{\\Large\\frac{2}{3}}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that the slope is negative since the line slants downward from left to right.\r\n\r\nWhat if we had chosen different points? Let\u2019s find the slope of the line again, this time using different points. We will use the points [latex]\\left(-3,7\\right)[\/latex] and [latex]\\left(6,1\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224606\/CNX_BMath_Figure_11_04_043_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points \" \/>\r\nStarting at [latex]\\left(-3,7\\right)[\/latex], sketch a right triangle to [latex]\\left(6,1\\right)[\/latex].\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224607\/CNX_BMath_Figure_11_04_044_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points \" \/>\r\n<table id=\"eip-id1168469716067\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Count the rise.<\/td>\r\n<td>The rise is [latex]\u22126[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run.<\/td>\r\n<td>The run is [latex]9[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values of the rise and run.<\/td>\r\n<td>[latex]m={\\Large\\frac{-6}{9}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the fraction.<\/td>\r\n<td>[latex]m=-{\\Large\\frac{2}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the line is [latex]-{\\Large\\frac{2}{3}}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt does not matter which points you use\u2014the slope of the line is always the same. The slope of a line is constant!\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147015[\/ohm_question]\r\n\r\n<\/div>\r\nThe lines in the previous examples had [latex]y[\/latex] -intercepts with integer values, so it was convenient to use the <em>y<\/em>-intercept as one of the points we used to find the slope. In the next example, the [latex]y[\/latex] -intercept is a fraction. The calculations are easier if we use two points with integer coordinates.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the slope of the line shown:\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224611\/CNX_BMath_Figure_11_04_045_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. A line passes through the points \" \/>\r\n[reveal-answer q=\"439279\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"439279\"]\r\n\r\nSolution\r\n<table id=\"eip-id1170321819050\" class=\"unnumbered unstyled\" summary=\"...\">\r\n<tbody>\r\n<tr>\r\n<td>Locate two points on the graph whose coordinates are integers.<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex] and [latex]\\left(7,6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Which point is on the left?<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Starting at [latex]\\left(2,3\\right)[\/latex] , sketch a right angle to [latex]\\left(7,6\\right)[\/latex] as shown below.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467128258\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224612\/CNX_BMath_Figure_11_04_046_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. Two unlabeled points are drawn at \" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the rise.<\/td>\r\n<td>The rise is [latex]3[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run.<\/td>\r\n<td>The run is [latex]5[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values of the rise and run.<\/td>\r\n<td>[latex]m={\\Large\\frac{3}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the line is [latex]{\\Large\\frac{3}{5}}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147016[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show another example of how to find the slope of a line given a graph. This graph has a positive slope.\r\n\r\nhttps:\/\/youtu.be\/zPognXmmaEo\r\n\r\nIn the following video we show another example of how to find the slope of a line given a graph. This graph has a negative slope.\r\n\r\nhttps:\/\/youtu.be\/dmla9Lj4rqg\r\n<h2>Finding the Slope of Horizontal and Vertical Lines<\/h2>\r\n<p>Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.<\/p>\r\n\r\n<ul id=\"fs-id1705241\">\r\n \t<li>horizontal line [latex]y=b[\/latex]; all the [latex]y[\/latex] -coordinates are the same.<\/li>\r\n \t<li>vertical line [latex]x=a[\/latex]; all the [latex]x[\/latex] -coordinates are the same.<\/li>\r\n<\/ul>\r\nSo how do we find the slope of the horizontal line [latex]y=4?[\/latex] One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let\u2019s see what happens. We\u2019ll use the two points [latex]\\left(0,4\\right)[\/latex] and [latex]\\left(3,4\\right)[\/latex] to count the rise and run.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224616\/CNX_BMath_Figure_11_04_028.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 5. The y-axis runs from -1 to 7. A horizontal line passes through the labeled points \" \/>\r\n<table id=\"eip-id1168469889849\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>What is the rise?<\/td>\r\n<td>The rise is [latex]0[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What is the run?<\/td>\r\n<td>The run is [latex]3[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What is the slope?<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]m={\\Large\\frac{0}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]m=0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe slope of the horizontal line [latex]y=4[\/latex] is [latex]0[\/latex].\r\n\r\nAll horizontal lines have slope [latex]0[\/latex] . When the [latex]y[\/latex] -coordinates are the same, the rise is [latex]0[\/latex] .\r\n<div class=\"textbox shaded\">\r\n<h3>Slope of a Horizontal Line<\/h3>\r\nThe slope of a horizontal line, [latex]y=b[\/latex], is [latex]0[\/latex].\r\n\r\n<\/div>\r\nNow we\u2019ll consider a vertical line, such as the line [latex]x=3[\/latex], shown below. We\u2019ll use the two points [latex]\\left(3,0\\right)[\/latex] and [latex]\\left(3,2\\right)[\/latex] to count the rise and run.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224617\/CNX_BMath_Figure_11_04_029.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line passes through the labeled points \" \/>\r\n<table id=\"eip-id1168468686751\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>What is the rise?<\/td>\r\n<td>The rise is [latex]2[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What is the run?<\/td>\r\n<td>The run is [latex]0[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What is the slope?<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]m={\\Large\\frac{2}{0}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBut we can\u2019t divide by [latex]0[\/latex]. Division by [latex]0[\/latex] is undefined. So we say that the slope of the vertical line [latex]x=3[\/latex] is undefined. The slope of all vertical lines is undefined, because the run is [latex]0[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Slope of a Vertical Line<\/h3>\r\nThe slope of a vertical line, [latex]x=a[\/latex], is undefined.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the slope of each line:\r\n\r\n1. [latex]x=8[\/latex]\r\n2. [latex]y=-5[\/latex]\r\n\r\nSolution\r\n1. [latex]x=8[\/latex]\r\nThis is a vertical line, so its slope is undefined.\r\n\r\n2. [latex]y=-5[\/latex]\r\nThis is a horizontal line, so its slope is [latex]0[\/latex].\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147020[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Quick Guide to the Slopes of Lines<\/h3>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224619\/CNX_BMath_Figure_11_04_049_img.png\" alt=\"The figure shows 4 arrows. The first rises from left to right with the arrow point upwards. It is labeled \" \/>\r\n\r\n<\/div>\r\nThe following example shows you how to determine the slope of horizontal and vertical lines that are plotted on the coordinate axes.\r\n\r\nhttps:\/\/youtu.be\/dJuFWXn7zJM","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Given the graph of a line, determine the slope of the line<\/li>\n<li>Identify the slope of a horizontal line given it&#8217;s equation<\/li>\n<li>Identify the slope of a vertical line given it&#8217;s equation<\/li>\n<\/ul>\n<\/div>\n<p>Now we\u2019ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.<\/p>\n<p>To find the slope, we must count out the rise and the run. But where do we start?<\/p>\n<p>We locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the slope of the line shown:<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224557\/CNX_BMath_Figure_11_04_020.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points\" \/><\/p>\n<p>Solution<br \/>\nLocate two points on the graph, choosing points whose coordinates are integers. We will use [latex]\\left(0,-3\\right)[\/latex] and [latex]\\left(5,1\\right)[\/latex].<\/p>\n<p>Starting with the point on the left, [latex]\\left(0,-3\\right)[\/latex], sketch a right triangle, going from the first point to the second point, [latex]\\left(5,1\\right)[\/latex].<\/p>\n<table id=\"eip-id1168466130951\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224558\/CNX_BMath_Figure_11_04_021.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the rise on the vertical leg of the triangle.<\/td>\n<td>The rise is [latex]4[\/latex] units.<\/td>\n<\/tr>\n<tr>\n<td>Count the run on the horizontal leg.<\/td>\n<td>The run is [latex]5[\/latex] units.<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values of the rise and run.<\/td>\n<td>[latex]m={\\Large\\frac{4}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The slope of the line is [latex]{\\Large\\frac{4}{5}}[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Notice that the slope is positive since the line slants upward from left to right.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147014\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147014&theme=oea&iframe_resize_id=ohm147014&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>Find the slope from a graph<\/h3>\n<ol id=\"eip-id1168469837806\" class=\"stepwise\">\n<li>Locate two points on the line whose coordinates are integers.<\/li>\n<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/li>\n<li>Count the rise and the run on the legs of the triangle.<\/li>\n<li>Take the ratio of rise to run to find the slope. [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the slope of the line shown:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224602\/CNX_BMath_Figure_11_04_024.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q966343\">Show Solution<\/span><\/p>\n<div id=\"q966343\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nLocate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use [latex](0, 5)[\/latex] and [latex](3, 3)[\/latex]. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/p>\n<table id=\"eip-id1168465988432\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224603\/CNX_BMath_Figure_11_04_025.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the rise \u2013 it is negative.<\/td>\n<td>The rise is [latex]\u22122[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Count the run.<\/td>\n<td>The run is [latex]3[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values of the rise and run.<\/td>\n<td>[latex]m={\\Large\\frac{-2}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]m=-{\\Large\\frac{2}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The slope of the line is [latex]-{\\Large\\frac{2}{3}}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that the slope is negative since the line slants downward from left to right.<\/p>\n<p>What if we had chosen different points? Let\u2019s find the slope of the line again, this time using different points. We will use the points [latex]\\left(-3,7\\right)[\/latex] and [latex]\\left(6,1\\right)[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224606\/CNX_BMath_Figure_11_04_043_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points\" \/><br \/>\nStarting at [latex]\\left(-3,7\\right)[\/latex], sketch a right triangle to [latex]\\left(6,1\\right)[\/latex].<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224607\/CNX_BMath_Figure_11_04_044_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points\" \/><\/p>\n<table id=\"eip-id1168469716067\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Count the rise.<\/td>\n<td>The rise is [latex]\u22126[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Count the run.<\/td>\n<td>The run is [latex]9[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values of the rise and run.<\/td>\n<td>[latex]m={\\Large\\frac{-6}{9}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the fraction.<\/td>\n<td>[latex]m=-{\\Large\\frac{2}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The slope of the line is [latex]-{\\Large\\frac{2}{3}}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It does not matter which points you use\u2014the slope of the line is always the same. The slope of a line is constant!<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147015\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147015&theme=oea&iframe_resize_id=ohm147015&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The lines in the previous examples had [latex]y[\/latex] -intercepts with integer values, so it was convenient to use the <em>y<\/em>-intercept as one of the points we used to find the slope. In the next example, the [latex]y[\/latex] -intercept is a fraction. The calculations are easier if we use two points with integer coordinates.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the slope of the line shown:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224611\/CNX_BMath_Figure_11_04_045_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. A line passes through the points\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q439279\">Show Solution<\/span><\/p>\n<div id=\"q439279\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1170321819050\" class=\"unnumbered unstyled\" summary=\"...\">\n<tbody>\n<tr>\n<td>Locate two points on the graph whose coordinates are integers.<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex] and [latex]\\left(7,6\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Which point is on the left?<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Starting at [latex]\\left(2,3\\right)[\/latex] , sketch a right angle to [latex]\\left(7,6\\right)[\/latex] as shown below.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467128258\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224612\/CNX_BMath_Figure_11_04_046_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. Two unlabeled points are drawn at\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the rise.<\/td>\n<td>The rise is [latex]3[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Count the run.<\/td>\n<td>The run is [latex]5[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values of the rise and run.<\/td>\n<td>[latex]m={\\Large\\frac{3}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The slope of the line is [latex]{\\Large\\frac{3}{5}}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147016\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147016&theme=oea&iframe_resize_id=ohm147016&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of how to find the slope of a line given a graph. This graph has a positive slope.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1: Determine the Slope Given the Graph of a Line (positive slope)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/zPognXmmaEo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the following video we show another example of how to find the slope of a line given a graph. This graph has a negative slope.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 2: Determine the Slope Given the Graph of a Line (negative slope)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/dmla9Lj4rqg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Finding the Slope of Horizontal and Vertical Lines<\/h2>\n<p>Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.<\/p>\n<ul id=\"fs-id1705241\">\n<li>horizontal line [latex]y=b[\/latex]; all the [latex]y[\/latex] -coordinates are the same.<\/li>\n<li>vertical line [latex]x=a[\/latex]; all the [latex]x[\/latex] -coordinates are the same.<\/li>\n<\/ul>\n<p>So how do we find the slope of the horizontal line [latex]y=4?[\/latex] One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let\u2019s see what happens. We\u2019ll use the two points [latex]\\left(0,4\\right)[\/latex] and [latex]\\left(3,4\\right)[\/latex] to count the rise and run.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224616\/CNX_BMath_Figure_11_04_028.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 5. The y-axis runs from -1 to 7. A horizontal line passes through the labeled points\" \/><\/p>\n<table id=\"eip-id1168469889849\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>What is the rise?<\/td>\n<td>The rise is [latex]0[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>What is the run?<\/td>\n<td>The run is [latex]3[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>What is the slope?<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]m={\\Large\\frac{0}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]m=0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The slope of the horizontal line [latex]y=4[\/latex] is [latex]0[\/latex].<\/p>\n<p>All horizontal lines have slope [latex]0[\/latex] . When the [latex]y[\/latex] -coordinates are the same, the rise is [latex]0[\/latex] .<\/p>\n<div class=\"textbox shaded\">\n<h3>Slope of a Horizontal Line<\/h3>\n<p>The slope of a horizontal line, [latex]y=b[\/latex], is [latex]0[\/latex].<\/p>\n<\/div>\n<p>Now we\u2019ll consider a vertical line, such as the line [latex]x=3[\/latex], shown below. We\u2019ll use the two points [latex]\\left(3,0\\right)[\/latex] and [latex]\\left(3,2\\right)[\/latex] to count the rise and run.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224617\/CNX_BMath_Figure_11_04_029.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line passes through the labeled points\" \/><\/p>\n<table id=\"eip-id1168468686751\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>What is the rise?<\/td>\n<td>The rise is [latex]2[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>What is the run?<\/td>\n<td>The run is [latex]0[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>What is the slope?<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]m={\\Large\\frac{2}{0}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>But we can\u2019t divide by [latex]0[\/latex]. Division by [latex]0[\/latex] is undefined. So we say that the slope of the vertical line [latex]x=3[\/latex] is undefined. The slope of all vertical lines is undefined, because the run is [latex]0[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Slope of a Vertical Line<\/h3>\n<p>The slope of a vertical line, [latex]x=a[\/latex], is undefined.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the slope of each line:<\/p>\n<p>1. [latex]x=8[\/latex]<br \/>\n2. [latex]y=-5[\/latex]<\/p>\n<p>Solution<br \/>\n1. [latex]x=8[\/latex]<br \/>\nThis is a vertical line, so its slope is undefined.<\/p>\n<p>2. [latex]y=-5[\/latex]<br \/>\nThis is a horizontal line, so its slope is [latex]0[\/latex].<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147020\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147020&theme=oea&iframe_resize_id=ohm147020&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Quick Guide to the Slopes of Lines<\/h3>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224619\/CNX_BMath_Figure_11_04_049_img.png\" alt=\"The figure shows 4 arrows. The first rises from left to right with the arrow point upwards. It is labeled\" \/><\/p>\n<\/div>\n<p>The following example shows you how to determine the slope of horizontal and vertical lines that are plotted on the coordinate axes.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Determine the Slope Given the Graph of a Horizontal and Vertical Line\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/dJuFWXn7zJM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3793\">\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>Question ID 147016, 147015, 147014,147013, 147020. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 2: Determine the Slope Given the Graph of a Line (negative slope). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/dmla9Lj4rqg\">https:\/\/youtu.be\/dmla9Lj4rqg<\/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: Determine the Slope Given the Graph of a Horizontal and Vertical Line. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/dJuFWXn7zJM\">https:\/\/youtu.be\/dJuFWXn7zJM<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":25777,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 147016, 147015, 147014,147013, 147020\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2: Determine the Slope Given the Graph of a Line (negative slope)\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/dmla9Lj4rqg\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Determine the Slope Given the Graph of a Horizontal and Vertical Line\",\"author\":\"James Sousa 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