{"id":4631,"date":"2017-06-07T18:53:47","date_gmt":"2017-06-07T18:53:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-slope-from-a-graph\/"},"modified":"2017-08-16T02:59:54","modified_gmt":"2017-08-16T02:59:54","slug":"read-slope-from-a-graph","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-slope-from-a-graph\/","title":{"raw":"Find the Slope from a Graph","rendered":"Find the Slope from a Graph"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Find the Slope from a Graph\r\n<ul>\r\n \t<li>Identify rise and run from a graph<\/li>\r\n \t<li>Distinguish between graphs of lines with negative and positive slopes<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Identify slope from a graph<\/h2>\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. 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:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185338\/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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185340\/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. 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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185341\/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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185343\/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 1 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 4 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, 4, 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>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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185344\/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 4.5. 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 6 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 0.75.[\/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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185346\/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 moving 3 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 2 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].[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the example above, 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].","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Find the Slope from a Graph\n<ul>\n<li>Identify rise and run from a graph<\/li>\n<li>Distinguish between graphs of lines with negative and positive slopes<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Identify slope from a graph<\/h2>\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. 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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185338\/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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185340\/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. 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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185341\/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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185343\/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 1 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 4 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, 4, 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>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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185344\/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 4.5. 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 6 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 0.75.<\/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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185346\/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 moving 3 units in a negative direction.<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{run}=2[\/latex]<\/td>\n<td>From there, run 2 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><\/div>\n<\/div>\n<\/div>\n<p>In the example above, 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\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-4631\">\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":17533,"menu_order":5,"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":"e8f64f40-8c02-44b6-830c-24108f5fcfd2","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4631","chapter","type-chapter","status-publish","hentry"],"part":4587,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4631","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4631\/revisions"}],"predecessor-version":[{"id":5078,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4631\/revisions\/5078"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/parts\/4587"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4631\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/media?parent=4631"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapter-type?post=4631"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/contributor?post=4631"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/license?post=4631"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}