{"id":16509,"date":"2019-10-03T17:08:36","date_gmt":"2019-10-03T17:08:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-find-slope-from-a-graph\/"},"modified":"2020-10-22T09:16:07","modified_gmt":"2020-10-22T09:16:07","slug":"read-or-watch-find-slope-from-a-graph","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-or-watch-find-slope-from-a-graph\/","title":{"raw":"9.2.a - Finding the Slope of a Line from its Graph","rendered":"9.2.a &#8211; Finding the Slope of a Line from its Graph"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the slope of a line from its graph<\/li>\r\n \t<li>Find the slope of horizontal and vertical lines<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe mathematical definition of <b>slope<\/b> is very similar to our everyday one. In math, slope is used to describe the steepness and direction of lines.\u00a0\u00a0In this section, we will begin exploring the concept of slope using a geoboard and then we will learn how to measure the slopes of lines on a rectangular coordinate system.\r\n\r\nUsing rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.\u00a0We\u2019ll start by stretching a rubber band between two pegs to make a line as shown in the image below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224529\/CNX_BMath_Figure_11_04_001.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.\" \/>\r\nDoes it look like a line?\r\n\r\nNow we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in the image below. We carefully make a [latex]90^ \\circ [\/latex] angle around the third peg, so that one side is vertical and the other is horizontal.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224530\/CNX_BMath_Figure_11_04_002.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4,and column 4 row 2.\" \/>\r\nTo find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the <em>rise<\/em> and the horizontal distance is called the <em>run<\/em>, as shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224531\/CNX_BMath_Figure_11_04_003.png\" alt=\"This figure shows two arrows. The first arrow is vertical and is labeled \" \/>\r\nTo help remember the terms, it may help to think of the images shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224533\/CNX_BMath_Figure_11_04_004.png\" alt=\"...\" \/>\r\nOn our geoboard, the rise is [latex]2[\/latex] units because the rubber band goes up [latex]2[\/latex] spaces on the vertical leg. See the image below.\r\n\r\nWhat is the run? (Be sure to count the spaces between the pegs rather than the pegs themselves!) The rubber band goes across [latex]3[\/latex] spaces on the horizontal leg, so the run is [latex]3[\/latex] units.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224536\/CNX_BMath_Figure_11_04_005.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4, and column 4 row 2. The triangle has a rise of 2 units and a run of 3 units.\" \/>\r\nThe slope of a line is the ratio of the rise to the run. So the slope of our line is [latex]{\\Large\\frac{2}{3}}[\/latex]. In mathematics, the slope is always represented by the letter [latex]m[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Slope of a line<\/h3>\r\nThe slope of a line is [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex].\r\nThe rise measures the vertical change and the run measures the horizontal change.\r\n\r\n<\/div>\r\nWhat is the slope of the line on the geoboard in the image above?\r\n<p style=\"text-align: center\">[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]\r\n[latex]m={\\Large\\frac{2}{3}}[\/latex]<\/p>\r\n<p style=\"text-align: center\">The line has slope [latex]{\\Large\\frac{2}{3}}[\/latex]<\/p>\r\nWhen we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle.\r\n\r\nIf we start by going up when we connect the second peg, the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.\r\n\r\nSince the slope formula is rise over run, it may be easier to always count out the rise first and then the run.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWhat is the slope of the line on the geoboard shown?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224537\/CNX_BMath_Figure_11_04_006.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 5 row 2.\" \/>\r\n\r\nSolution\r\nUse the definition of slope.\r\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]\r\n\r\nStart at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg.\r\nCount the rise and the run as shown.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224538\/CNX_BMath_Figure_11_04_007.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 5,and column 5 row 2.\" \/>\r\n[latex]\\begin{array}{cccc}\\text{The rise is }3\\text{ units}.\\hfill &amp; &amp; &amp; m={\\Large\\frac{3}{\\text{run}}}\\hfill \\\\ \\text{The run is}4\\text{ units}.\\hfill &amp; &amp; &amp; m={\\Large\\frac{3}{4}}\\hfill \\\\ &amp; &amp; &amp; \\text{The slope is }{\\Large\\frac{3}{4}}\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWhat is the slope of the line on the geoboard shown?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224542\/CNX_BMath_Figure_11_04_010.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.\" \/>\r\n[reveal-answer q=\"698295\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"698295\"]\r\n\r\nSolution\r\nUse the definition of slope.\r\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]\r\n\r\nStart at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224543\/CNX_BMath_Figure_11_04_011.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 4 row 4.\" \/>\r\n[latex]\\begin{array}{cccc}\\text{The rise is }-1.\\hfill &amp; &amp; &amp; m={\\Large\\frac{-1}{\\text{run}}}\\hfill \\\\ \\text{The run is}3.\\hfill &amp; &amp; &amp; m={\\Large\\frac{-1}{3}}\\hfill \\\\ &amp; &amp; &amp; m=-{\\Large\\frac{1}{3}}\\hfill \\\\ &amp; &amp; &amp; \\text{The slope is }-{\\Large\\frac{1}{3}}\\hfill \\end{array}[\/latex]\r\n\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]147013[\/ohm_question]\r\n\r\n<\/div>\r\n<h2 id=\"title1\">Finding the Slope of a Line from its Graph<\/h2>\r\nBy just looking at the graph of a line, you can learn some things about its slope, especially relative to other lines graphed on the same coordinate plane. Consider the graphs of the three lines shown below:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064257\/image024-1.jpg\" alt=\"Three different lines on a graph. Line A is tilted upward. Line B is sharply titled upward. Line C is sharply tilted downward.\" width=\"345\" height=\"342\" \/>\r\n\r\nFirst, let\u2019s look at lines A and B. If you imagined these lines to be hills, you would say that line B is steeper than line A. Line B has a greater slope than line A.\r\n\r\nNext, notice that lines A and B slant up as you move from left to right. We say these two lines have a positive slope. Line C slants down from left to right. Line C has a negative slope. Using two of the points on the line, you can find the slope of the line by finding the rise and the run. The vertical change between two points is called the <b>rise<\/b>, and the horizontal change is called the <b>run<\/b>. The slope equals the rise divided by the run: [latex] \\displaystyle \\text{Slope }=\\frac{\\text{rise}}{\\text{run}}[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064258\/image025-1.jpg\" alt=\"A line that crosses the points (2,1) and (6,3). A blue line labeled Rise goes up two units from the point (2,1). A red line labeled Run goes left from the point (6,3) so that it forms a triangle with the main line and the Rise line. A formula says slope equals rise over run.\" width=\"305\" height=\"294\" \/>\r\n\r\nYou can determine the slope of a line from its graph by looking at the rise and run.\u00a0 (Notice the similarity between the image above and the Geoboard examples we looked at earlier.) One characteristic of a line is that its slope is constant all the way along it. So, you can choose any 2 points along the graph of the line to figure out the slope. Let\u2019s look at an example.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nUse the graph to find the slope of the line.\r\n\r\n<b><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064259\/image026-1.jpg\" alt=\"A line that crosses the points (2,1) and (6,3). A blue line labeled Rise goes up two units from the point (2,1). A red line labeled Run goes left from the point (6,3) so that it forms a triangle with the main line and the Rise line. A formula says slope equals rise over run.\" width=\"305\" height=\"294\" \/><\/b>\r\n\r\n[reveal-answer q=\"606472\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"606472\"]Start from a point on the line, such as [latex](2,1)[\/latex] and move vertically until in line with another point on the line, such as [latex](6,3)[\/latex]. The rise is 2 units. It is positive as you moved up.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\text{rise}=2[\/latex]<\/td>\r\n<td>Start from a point on the line, such as [latex](2,1)[\/latex] and move vertically until in line with another point on the line, such as [latex](6,3)[\/latex]. The rise is 2 units. It is positive as you moved up.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{run}=4[\/latex]<\/td>\r\n<td>Next, move horizontally to the point [latex](6,3)[\/latex]. Count the number of units. The run is 4 units. It is positive as you moved to the right.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\displaystyle \\text{Slope}=\\frac{2}{4}=\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\text{Slope }=\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{1}{2}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nThis line will have a slope of [latex] \\displaystyle \\frac{1}{2}[\/latex] no matter which two points you pick on the line. Try measuring the slope from the origin, [latex](0,0)[\/latex], to the point [latex](6,3)[\/latex]. You will find that the [latex]\\text{rise}=3[\/latex] and the [latex]\\text{run}=6[\/latex]. The slope is [latex] \\displaystyle \\frac{\\text{rise}}{\\text{run}}=\\frac{3}{6}=\\frac{1}{2}[\/latex]. It is the same!\r\n\r\nLet\u2019s look at another example.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nUse the graph to find the slope of the two lines.<b>\u00a0<\/b>\r\n\r\n<b><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064300\/image027-1.jpg\" alt=\"A graph showing two lines with their rise and run. The first line is drawn through the points (-2,1) and (-1,5). The rise goes up from the point (-2,1) to join with the run line that goes right to the point (-1,5). The second line is drawn through the points (-1,-2) and (3,-1). The rise goes up from the point (-1,-2) to join with the run to go right to the point (3,-1).\" width=\"291\" height=\"281\" \/><\/b>\r\n\r\n[reveal-answer q=\"860733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"860733\"]Notice that both of these lines have positive slopes, so you expect your answers to be positive.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th style=\"text-align: center\" colspan=\"2\">Blue line<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{rise}=4[\/latex]<\/td>\r\n<td>Start with the blue line, going from point [latex](-2,1)[\/latex] to point [latex](-1,5)[\/latex]. This line has a rise of 4 units up, so it is positive.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{run}=1[\/latex]<\/td>\r\n<td>Run is [latex]1[\/latex] unit to the right, so it is positive.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\displaystyle \\text{Slope }=\\frac{4}{1}=4[\/latex]<\/td>\r\n<td>Substitute the values for the rise and run in the formula [latex] \\displaystyle \\text{Slope }\\frac{\\text{rise}}{\\text{run}}[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<th style=\"text-align: center\" colspan=\"2\">Red line<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{rise}=1[\/latex]<\/td>\r\n<td>The red line, going from point [latex](-1,-2)[\/latex] to point [latex](3,-1)[\/latex] has a rise of 1 unit.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{run}=4[\/latex]<\/td>\r\n<td>The red line has a run of 4 units.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\displaystyle \\text{Slope }=\\frac{1}{4}[\/latex]<\/td>\r\n<td>Substitute the values for the rise and run into the formula [latex] \\displaystyle \\text{Slope }\\frac{\\text{rise}}{\\text{run}}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Answer<\/h4>\r\nThe slope of the blue line is [latex]4[\/latex] and the slope of the red line is [latex]\\frac{1}{4}[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen you look at the two lines, you can see that the blue line is steeper than the red line. It makes sense the value of the slope of the blue line, [latex]4[\/latex], is greater than the value of the slope of the red line, [latex] \\displaystyle \\frac{1}{4}[\/latex]. The greater the slope, the steeper the line.\r\n<h2>Finding the Slope of a Line From a Graph<\/h2>\r\nhttps:\/\/youtu.be\/29BpBqsiE5w\r\n<h2 id=\"title2\">Distinguish between graphs of lines with negative and positive slopes<\/h2>\r\nDirection is important when it comes to determining slope. It\u2019s important to pay attention to whether you are moving up, down, left, or right; that is, if you are moving in a positive or negative direction. If you go up to get to your second point, the rise is positive. If you go down to get to your second point, the rise is negative. If you go right to get to your second point, the run is positive. If you go left to get to your second point, the run is negative.\r\n\r\nIn the following two examples, you will see a slope that is positive and one that is negative.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example (Advanced)<\/h3>\r\nFind the slope of the line graphed below.\r\n\r\n<b><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064305\/image029.jpg\" alt=\"Line drawn through the point (-3,-0.25) and (3,4.25).\" width=\"358\" height=\"343\" \/><\/b>\r\n\r\n[reveal-answer q=\"82644\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"82644\"]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\text{rise}=4.5[\/latex]<\/td>\r\n<td>Start at [latex](-3,-0.25)[\/latex] and rise [latex]4.5[\/latex]. This means moving 4.5 units in a positive direction.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{run}=6[\/latex]<\/td>\r\n<td>From there, run [latex]6[\/latex] units in a positive direction to [latex](3,4.25)[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\displaystyle \\text{Slope}=\\frac{4.5}{6}=0.75[\/latex]<\/td>\r\n<td>[latex]\\displaystyle \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Answer<\/h4>\r\nThe slope of the line is [latex]0.75[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nThe next example shows a line with a negative slope.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the slope of the line graphed below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064301\/image028.jpg\" alt=\"A downward-sloping line that goes through points A and B. Point A is (0,4) and point B is (2,1). The rise goes down three units, and the run goes right 2 units.\" width=\"308\" height=\"297\" \/>\r\n\r\n[reveal-answer q=\"924393\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"924393\"]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\text{rise}=-3[\/latex]<\/td>\r\n<td>Start at Point A, [latex](0,4)[\/latex] and rise [latex]\u22123[\/latex]. This means movin g[latex]3[\/latex] units in a negative direction.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{run}=2[\/latex]<\/td>\r\n<td>From there, run [latex]2[\/latex] units in a positive direction to Point B [latex](2,1)[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\displaystyle \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Answer<\/h4>\r\nThe slope of the line is [latex]-\\frac{3}{2}[\/latex].\r\n\r\n<strong>NOTE:<\/strong> you could have found the slope by starting at point B, running [latex]{-2}[\/latex], and then rising [latex]+3[\/latex] to arrive at point A. The result is still a slope of [latex]\\displaystyle\\frac{\\text{rise}}{\\text{run}}=\\frac{+3}{-2}=-\\frac{3}{2}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn all of the previous examples of finding the slope of a line, we were given two points.\u00a0 Now we will look at some examples where we are not automatically given two points on the line.\r\n\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<td><\/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<td><\/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<td><\/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\nDo you remember what was special about horizontal and vertical lines? Their equations had just one variable.\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<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]m=0[\/latex]<\/td>\r\n<td><\/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\" style=\"height: 48px\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 12px\">\r\n<td style=\"height: 12px;width: 191.5px\">What is the rise?<\/td>\r\n<td style=\"height: 12px;width: 191.5px\">The rise is [latex]2[\/latex].<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px\">\r\n<td style=\"height: 12px;width: 191.5px\">What is the run?<\/td>\r\n<td style=\"height: 12px;width: 191.5px\">The run is [latex]0[\/latex].<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px\">\r\n<td style=\"height: 12px;width: 191.5px\">What is the slope?<\/td>\r\n<td style=\"height: 12px;width: 191.5px\">[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px\">\r\n<td style=\"height: 12px;width: 191.5px\">[latex]m={\\Large\\frac{2}{0}}[\/latex]<\/td>\r\n<td style=\"height: 12px;width: 191.5px\"><\/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\n[reveal-answer q=\"439270\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"439270\"]\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[\/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]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=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the slope of a line from its graph<\/li>\n<li>Find the slope of horizontal and vertical lines<\/li>\n<\/ul>\n<\/div>\n<p>The mathematical definition of <b>slope<\/b> is very similar to our everyday one. In math, slope is used to describe the steepness and direction of lines.\u00a0\u00a0In this section, we will begin exploring the concept of slope using a geoboard and then we will learn how to measure the slopes of lines on a rectangular coordinate system.<\/p>\n<p>Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.\u00a0We\u2019ll start by stretching a rubber band between two pegs to make a line as shown in the image below.<\/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\/25224529\/CNX_BMath_Figure_11_04_001.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.\" \/><br \/>\nDoes it look like a line?<\/p>\n<p>Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in the image below. We carefully make a [latex]90^ \\circ[\/latex] angle around the third peg, so that one side is vertical and the other is horizontal.<\/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\/25224530\/CNX_BMath_Figure_11_04_002.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4,and column 4 row 2.\" \/><br \/>\nTo find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the <em>rise<\/em> and the horizontal distance is called the <em>run<\/em>, as shown below.<\/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\/25224531\/CNX_BMath_Figure_11_04_003.png\" alt=\"This figure shows two arrows. The first arrow is vertical and is labeled\" \/><br \/>\nTo help remember the terms, it may help to think of the images shown below.<\/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\/25224533\/CNX_BMath_Figure_11_04_004.png\" alt=\"...\" \/><br \/>\nOn our geoboard, the rise is [latex]2[\/latex] units because the rubber band goes up [latex]2[\/latex] spaces on the vertical leg. See the image below.<\/p>\n<p>What is the run? (Be sure to count the spaces between the pegs rather than the pegs themselves!) The rubber band goes across [latex]3[\/latex] spaces on the horizontal leg, so the run is [latex]3[\/latex] units.<\/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\/25224536\/CNX_BMath_Figure_11_04_005.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4, and column 4 row 2. The triangle has a rise of 2 units and a run of 3 units.\" \/><br \/>\nThe slope of a line is the ratio of the rise to the run. So the slope of our line is [latex]{\\Large\\frac{2}{3}}[\/latex]. In mathematics, the slope is always represented by the letter [latex]m[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Slope of a line<\/h3>\n<p>The slope of a line is [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex].<br \/>\nThe rise measures the vertical change and the run measures the horizontal change.<\/p>\n<\/div>\n<p>What is the slope of the line on the geoboard in the image above?<\/p>\n<p style=\"text-align: center\">[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<br \/>\n[latex]m={\\Large\\frac{2}{3}}[\/latex]<\/p>\n<p style=\"text-align: center\">The line has slope [latex]{\\Large\\frac{2}{3}}[\/latex]<\/p>\n<p>When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle.<\/p>\n<p>If we start by going up when we connect the second peg, the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.<\/p>\n<p>Since the slope formula is rise over run, it may be easier to always count out the rise first and then the run.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>What is the slope of the line on the geoboard shown?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224537\/CNX_BMath_Figure_11_04_006.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 5 row 2.\" \/><\/p>\n<p>Solution<br \/>\nUse the definition of slope.<br \/>\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/p>\n<p>Start at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg.<br \/>\nCount the rise and the run as shown.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224538\/CNX_BMath_Figure_11_04_007.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 5,and column 5 row 2.\" \/><br \/>\n[latex]\\begin{array}{cccc}\\text{The rise is }3\\text{ units}.\\hfill & & & m={\\Large\\frac{3}{\\text{run}}}\\hfill \\\\ \\text{The run is}4\\text{ units}.\\hfill & & & m={\\Large\\frac{3}{4}}\\hfill \\\\ & & & \\text{The slope is }{\\Large\\frac{3}{4}}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>What is the slope of the line on the geoboard shown?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224542\/CNX_BMath_Figure_11_04_010.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698295\">Show Solution<\/span><\/p>\n<div id=\"q698295\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nUse the definition of slope.<br \/>\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/p>\n<p>Start at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224543\/CNX_BMath_Figure_11_04_011.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 4 row 4.\" \/><br \/>\n[latex]\\begin{array}{cccc}\\text{The rise is }-1.\\hfill & & & m={\\Large\\frac{-1}{\\text{run}}}\\hfill \\\\ \\text{The run is}3.\\hfill & & & m={\\Large\\frac{-1}{3}}\\hfill \\\\ & & & m=-{\\Large\\frac{1}{3}}\\hfill \\\\ & & & \\text{The slope is }-{\\Large\\frac{1}{3}}\\hfill \\end{array}[\/latex]<\/p>\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=\"ohm147013\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147013&theme=oea&iframe_resize_id=ohm147013&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2 id=\"title1\">Finding the Slope of a Line from its Graph<\/h2>\n<p>By just looking at the graph of a line, you can learn some things about its slope, especially relative to other lines graphed on the same coordinate plane. Consider the graphs of the three lines shown below:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064257\/image024-1.jpg\" alt=\"Three different lines on a graph. Line A is tilted upward. Line B is sharply titled upward. Line C is sharply tilted downward.\" width=\"345\" height=\"342\" \/><\/p>\n<p>First, let\u2019s look at lines A and B. If you imagined these lines to be hills, you would say that line B is steeper than line A. Line B has a greater slope than line A.<\/p>\n<p>Next, notice that lines A and B slant up as you move from left to right. We say these two lines have a positive slope. Line C slants down from left to right. Line C has a negative slope. Using two of the points on the line, you can find the slope of the line by finding the rise and the run. The vertical change between two points is called the <b>rise<\/b>, and the horizontal change is called the <b>run<\/b>. The slope equals the rise divided by the run: [latex]\\displaystyle \\text{Slope }=\\frac{\\text{rise}}{\\text{run}}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064258\/image025-1.jpg\" alt=\"A line that crosses the points (2,1) and (6,3). A blue line labeled Rise goes up two units from the point (2,1). A red line labeled Run goes left from the point (6,3) so that it forms a triangle with the main line and the Rise line. A formula says slope equals rise over run.\" width=\"305\" height=\"294\" \/><\/p>\n<p>You can determine the slope of a line from its graph by looking at the rise and run.\u00a0 (Notice the similarity between the image above and the Geoboard examples we looked at earlier.) One characteristic of a line is that its slope is constant all the way along it. So, you can choose any 2 points along the graph of the line to figure out the slope. Let\u2019s look at an example.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use the graph to find the slope of the line.<\/p>\n<p><b><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064259\/image026-1.jpg\" alt=\"A line that crosses the points (2,1) and (6,3). A blue line labeled Rise goes up two units from the point (2,1). A red line labeled Run goes left from the point (6,3) so that it forms a triangle with the main line and the Rise line. A formula says slope equals rise over run.\" width=\"305\" height=\"294\" \/><\/b><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q606472\">Show Solution<\/span><\/p>\n<div id=\"q606472\" class=\"hidden-answer\" style=\"display: none\">Start from a point on the line, such as [latex](2,1)[\/latex] and move vertically until in line with another point on the line, such as [latex](6,3)[\/latex]. The rise is 2 units. It is positive as you moved up.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\text{rise}=2[\/latex]<\/td>\n<td>Start from a point on the line, such as [latex](2,1)[\/latex] and move vertically until in line with another point on the line, such as [latex](6,3)[\/latex]. The rise is 2 units. It is positive as you moved up.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{run}=4[\/latex]<\/td>\n<td>Next, move horizontally to the point [latex](6,3)[\/latex]. Count the number of units. The run is 4 units. It is positive as you moved to the right.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle \\text{Slope}=\\frac{2}{4}=\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\displaystyle \\text{Slope }=\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{1}{2}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>This line will have a slope of [latex]\\displaystyle \\frac{1}{2}[\/latex] no matter which two points you pick on the line. Try measuring the slope from the origin, [latex](0,0)[\/latex], to the point [latex](6,3)[\/latex]. You will find that the [latex]\\text{rise}=3[\/latex] and the [latex]\\text{run}=6[\/latex]. The slope is [latex]\\displaystyle \\frac{\\text{rise}}{\\text{run}}=\\frac{3}{6}=\\frac{1}{2}[\/latex]. It is the same!<\/p>\n<p>Let\u2019s look at another example.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use the graph to find the slope of the two lines.<b>\u00a0<\/b><\/p>\n<p><b><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064300\/image027-1.jpg\" alt=\"A graph showing two lines with their rise and run. The first line is drawn through the points (-2,1) and (-1,5). The rise goes up from the point (-2,1) to join with the run line that goes right to the point (-1,5). The second line is drawn through the points (-1,-2) and (3,-1). The rise goes up from the point (-1,-2) to join with the run to go right to the point (3,-1).\" width=\"291\" height=\"281\" \/><\/b><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q860733\">Show Solution<\/span><\/p>\n<div id=\"q860733\" class=\"hidden-answer\" style=\"display: none\">Notice that both of these lines have positive slopes, so you expect your answers to be positive.<\/p>\n<table>\n<tbody>\n<tr>\n<th style=\"text-align: center\" colspan=\"2\">Blue line<\/th>\n<\/tr>\n<tr>\n<td>[latex]\\text{rise}=4[\/latex]<\/td>\n<td>Start with the blue line, going from point [latex](-2,1)[\/latex] to point [latex](-1,5)[\/latex]. This line has a rise of 4 units up, so it is positive.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{run}=1[\/latex]<\/td>\n<td>Run is [latex]1[\/latex] unit to the right, so it is positive.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle \\text{Slope }=\\frac{4}{1}=4[\/latex]<\/td>\n<td>Substitute the values for the rise and run in the formula [latex]\\displaystyle \\text{Slope }\\frac{\\text{rise}}{\\text{run}}[\/latex].<\/td>\n<\/tr>\n<tr>\n<th style=\"text-align: center\" colspan=\"2\">Red line<\/th>\n<\/tr>\n<tr>\n<td>[latex]\\text{rise}=1[\/latex]<\/td>\n<td>The red line, going from point [latex](-1,-2)[\/latex] to point [latex](3,-1)[\/latex] has a rise of 1 unit.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{run}=4[\/latex]<\/td>\n<td>The red line has a run of 4 units.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle \\text{Slope }=\\frac{1}{4}[\/latex]<\/td>\n<td>Substitute the values for the rise and run into the formula [latex]\\displaystyle \\text{Slope }\\frac{\\text{rise}}{\\text{run}}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Answer<\/h4>\n<p>The slope of the blue line is [latex]4[\/latex] and the slope of the red line is [latex]\\frac{1}{4}[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<p>When you look at the two lines, you can see that the blue line is steeper than the red line. It makes sense the value of the slope of the blue line, [latex]4[\/latex], is greater than the value of the slope of the red line, [latex]\\displaystyle \\frac{1}{4}[\/latex]. The greater the slope, the steeper the line.<\/p>\n<h2>Finding the Slope of a Line From a Graph<\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine the Slope of a Line From a Graph (No Formula)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/29BpBqsiE5w?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title2\">Distinguish between graphs of lines with negative and positive slopes<\/h2>\n<p>Direction is important when it comes to determining slope. It\u2019s important to pay attention to whether you are moving up, down, left, or right; that is, if you are moving in a positive or negative direction. If you go up to get to your second point, the rise is positive. If you go down to get to your second point, the rise is negative. If you go right to get to your second point, the run is positive. If you go left to get to your second point, the run is negative.<\/p>\n<p>In the following two examples, you will see a slope that is positive and one that is negative.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example (Advanced)<\/h3>\n<p>Find the slope of the line graphed below.<\/p>\n<p><b><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064305\/image029.jpg\" alt=\"Line drawn through the point (-3,-0.25) and (3,4.25).\" width=\"358\" height=\"343\" \/><\/b><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q82644\">Show Solution<\/span><\/p>\n<div id=\"q82644\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>[latex]\\text{rise}=4.5[\/latex]<\/td>\n<td>Start at [latex](-3,-0.25)[\/latex] and rise [latex]4.5[\/latex]. This means moving 4.5 units in a positive direction.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{run}=6[\/latex]<\/td>\n<td>From there, run [latex]6[\/latex] units in a positive direction to [latex](3,4.25)[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle \\text{Slope}=\\frac{4.5}{6}=0.75[\/latex]<\/td>\n<td>[latex]\\displaystyle \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Answer<\/h4>\n<p>The slope of the line is [latex]0.75[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<p>The next example shows a line with a negative slope.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the slope of the line graphed below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064301\/image028.jpg\" alt=\"A downward-sloping line that goes through points A and B. Point A is (0,4) and point B is (2,1). The rise goes down three units, and the run goes right 2 units.\" width=\"308\" height=\"297\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q924393\">Show Solution<\/span><\/p>\n<div id=\"q924393\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>[latex]\\text{rise}=-3[\/latex]<\/td>\n<td>Start at Point A, [latex](0,4)[\/latex] and rise [latex]\u22123[\/latex]. This means movin g[latex]3[\/latex] units in a negative direction.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{run}=2[\/latex]<\/td>\n<td>From there, run [latex]2[\/latex] units in a positive direction to Point B [latex](2,1)[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<td>[latex]\\displaystyle \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Answer<\/h4>\n<p>The slope of the line is [latex]-\\frac{3}{2}[\/latex].<\/p>\n<p><strong>NOTE:<\/strong> you could have found the slope by starting at point B, running [latex]{-2}[\/latex], and then rising [latex]+3[\/latex] to arrive at point A. The result is still a slope of [latex]\\displaystyle\\frac{\\text{rise}}{\\text{run}}=\\frac{+3}{-2}=-\\frac{3}{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In all of the previous examples of finding the slope of a line, we were given two points.\u00a0 Now we will look at some examples where we are not automatically given two points on the line.<\/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<td><\/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<td><\/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<td><\/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-2\" 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-3\" 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<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]m=0[\/latex]<\/td>\n<td><\/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\" style=\"height: 48px\" summary=\".\">\n<tbody>\n<tr style=\"height: 12px\">\n<td style=\"height: 12px;width: 191.5px\">What is the rise?<\/td>\n<td style=\"height: 12px;width: 191.5px\">The rise is [latex]2[\/latex].<\/td>\n<\/tr>\n<tr style=\"height: 12px\">\n<td style=\"height: 12px;width: 191.5px\">What is the run?<\/td>\n<td style=\"height: 12px;width: 191.5px\">The run is [latex]0[\/latex].<\/td>\n<\/tr>\n<tr style=\"height: 12px\">\n<td style=\"height: 12px;width: 191.5px\">What is the slope?<\/td>\n<td style=\"height: 12px;width: 191.5px\">[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px\">\n<td style=\"height: 12px;width: 191.5px\">[latex]m={\\Large\\frac{2}{0}}[\/latex]<\/td>\n<td style=\"height: 12px;width: 191.5px\"><\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q439270\">Show Solution<\/span><\/p>\n<div id=\"q439270\" class=\"hidden-answer\" style=\"display: none\">\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<\/div>\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-4\" 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-16509\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Determine the Slope of a Line From a Graph (No Formula). <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/29BpBqsiE5w\">https:\/\/youtu.be\/29BpBqsiE5w<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Determine the Slope of a Line From a Graph (No Formula)\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/29BpBqsiE5w\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"012082040f0848959b64162572e7cb1a, 538d3853a1d94e6ab24bee558d88960f, f4501dc69ac34ca2a70716f9ff5f0cbe","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16509","chapter","type-chapter","status-publish","hentry"],"part":8524,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16509","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16509\/revisions"}],"predecessor-version":[{"id":20334,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16509\/revisions\/20334"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/8524"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16509\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/media?parent=16509"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=16509"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=16509"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/license?post=16509"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}