{"id":16513,"date":"2019-10-03T17:08:39","date_gmt":"2019-10-03T17:08:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-slope-intercept-form-of-a-line\/"},"modified":"2024-04-30T23:16:15","modified_gmt":"2024-04-30T23:16:15","slug":"read-or-watch-slope-intercept-form-of-a-line","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-slope-intercept-form-of-a-line\/","title":{"raw":"Graphing a Line Using Slope and a Point","rendered":"Graphing a Line Using Slope and a Point"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Draw the graph of a line using slope and a point on the line<\/li>\r\n \t<li>Write the equation of a line using slope and y-intercept<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen graphing a line we found one method we could use is to make a table of values.\u00a0 We also learned how to graph lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.\u00a0 However, if we can identify some properties of the line, we may be able to make a graph much quicker and easier.\r\n<h2>Graphing a Line Using Slope and a Point on the Line<\/h2>\r\nAnother method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nGraph the line passing through the point [latex]\\left(1,-1\\right)[\/latex] whose slope is [latex]m=\\Large\\frac{3}{4}[\/latex].\r\n\r\nSolution\r\nPlot the given point, [latex]\\left(1,-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\/25224624\/CNX_BMath_Figure_11_04_050_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -3 to 4. A labeled point is drawn at \u201cordered pair 1, -1\u201d.\" width=\"241\" height=\"225\" \/>\r\nUse the slope formula [latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex] to identify the rise and the run.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\\\ m=\\frac{3}{4}\\hfill \\\\ \\frac{\\text{rise}}{\\text{run}}=\\frac{3}{4}\\hfill \\\\ \\\\ \\\\ \\text{rise}=3\\hfill \\\\ \\text{run}=4\\hfill \\end{array}[\/latex]<\/p>\r\nStarting at the point we plotted, count out the rise and run to mark the second point. We count [latex]3[\/latex] units up and [latex]4[\/latex] units right.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224625\/CNX_BMath_Figure_11_04_051_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two line segments are drawn. A vertical line segment connects the points \u201cordered pair 1, -1\u201d and \u201corder pair \u201c1, 2\u201d. It is labeled \u201c3\u201d. A horizontal line segment starts at the top of the vertical line segment and goes to the right, connecting the points \u201cordered pair 1, 2\u201d and \u201cordered pair 5, 2\u201d. It is labeled \u201c4\u201d.\" width=\"263\" height=\"269\" \/>\r\nThen we connect the points with a line and draw arrows at the ends to show it continues.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224627\/CNX_BMath_Figure_11_04_033.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -3 to 5. The y-axis runs from -1 to 7. Two labeled points are drawn at \u201cordered pair 1, -1\u201d and \u201cordered pair 5, 2\u201d. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects \u201cordered pair 1, -1\u201d and \u201cordered pair 1, 2 \u201d. A horizontal line segment connects \u201cordered pair 1, 2\u201d and \u201cordered pair 5, 2\u201d.\" width=\"234\" height=\"241\" \/>\r\nWe can check our line by starting at any point and counting up [latex]3[\/latex] and to the right [latex]4[\/latex]. We should get to another point on the line.\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]147024[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Graph a line given a point and a slope<\/h3>\r\n<ol class=\"stepwise\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol class=\"stepwise\">\r\n \t<li>Plot the given point.<\/li>\r\n \t<li>Use the slope formula to identify the rise and the run.<\/li>\r\n \t<li>Starting at the given point, count out the rise and run to mark the second point.<\/li>\r\n \t<li>Connect the points with a line.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nGraph the line passing through the point [latex]\\left(-1,-3\\right)[\/latex] whose slope is [latex]m=4[\/latex]\r\n[reveal-answer q=\"804097\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"804097\"]\r\n\r\nSolution\r\nPlot the given point.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224638\/CNX_BMath_Figure_11_04_054_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The point \u201cordered pair -1, -3\u201d is labeled.\" width=\"263\" height=\"270\" \/>\r\n<table id=\"eip-id1168469472029\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Identify the rise and the run.<\/td>\r\n<td>[latex]m=4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]4[\/latex] as a fraction.<\/td>\r\n<td>[latex]\\Large\\frac{\\text{rise}}{\\text{run}}\\normalsize=\\Large\\frac{4}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\text{rise}=4\\text{ run}=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCount 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\/25224639\/CNX_BMath_Figure_11_04_055_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The y-axis runs from -4 to 2. A vertical line segment connects points at \u201cordered pair -1, -3\u201d and \u201cordered pair -1, 1\u201d and is labeled \u201cup 4\u201d. A horizontal line segment connects \u201cordered pair -1, 1\u201d and \u201cordered pair 0, 1\u201d and is labeled \u201cover 1\u201d.\" width=\"263\" height=\"269\" \/>\r\nMark the second point. Connect the two points with a line.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224641\/CNX_BMath_Figure_11_04_056_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at \u201cordered pair -1, -3\u201d and \u201cordered pair -1, 1\u201d. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects \u201cordered pair -1, -3\u201d and \u201cordered pair -1, 1 \u201d. It is labeled \u201cup 4\u201d A horizontal line segment connects \u201cordered pair -1, 1\u201d and \u201cordered pair 0, 1\u201d. It is labeled \u201cover 1\u201d\" width=\"264\" height=\"270\" \/>\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]147026[\/ohm_question]\r\n\r\n<\/div>\r\nYou can watch the video below for another example of how to graph a line given a point and a slope.\r\n\r\nhttps:\/\/youtu.be\/ngpAgpMjozw\r\n\r\nA special case for graphing using the point-slope method is when the given point is the y-intercept. We will give some examples here and then we will show below why this case is so important.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>example<\/h3>\r\nGraph the line with [latex]y[\/latex] -intercept [latex]\\left(0,2\\right)[\/latex] and slope [latex]m=-\\Large\\frac{2}{3}[\/latex]\r\n[reveal-answer q=\"489314\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"489314\"]\r\n\r\nSolution\r\nPlot the given point, the [latex]y[\/latex] -intercept [latex]\\left(0,2\\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\/25224631\/CNX_BMath_Figure_11_04_052_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 4. The y-axis runs from -1 to 3. The point \u201cordered pair 0, 2\u201d is labeled.\" width=\"261\" height=\"268\" \/>\r\nUse the slope formula [latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex] to identify the rise and the run.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\\\ m=-\\frac{2}{3}\\hfill \\\\ \\frac{\\text{rise}}{\\text{run}}=\\frac{-2}{3}\\hfill \\\\ \\\\ \\\\ \\text{rise}=-2\\hfill \\\\ \\text{run}=3\\hfill \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Starting at [latex]\\left(0,2\\right)[\/latex], count the rise and the run and mark the second point.<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224632\/CNX_BMath_Figure_11_04_053_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line segment connects points at \u201cordered pair 0, 2\u201d and \u201cordered pair 0, 0\u201d and is labeled \u201cdown 2\u201d. A horizontal line segment connects \u201cordered pair 0, 0\u201d and \u201cordered pair 0, 3\u201d and is labeled \u201cright 3\u201d.\" width=\"263\" height=\"268\" \/>\r\nConnect the points with a line.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224633\/CNX_BMath_Figure_11_04_036.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at \u201cordered pair 0, 2\u201d and \u201cordered pair 3, 0\u201d. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects \u201cordered pair 0, 2\u201d and \u201cordered pair 0, 0 \u201d. A horizontal line segment connects \u201cordered pair 0, 0\u201d and \u201cordered pair 3, 0\u201d.\" width=\"263\" height=\"270\" \/>\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]147025[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Slope-Intercept Form<\/h2>\r\nWe will now show you what is so special about the case in which the given point is the y-intercept.\u00a0 The slope can be represented by m and the <em>y<\/em>-intercept, where it crosses the axis and [latex]x=0[\/latex], can be represented by [latex](0,b)[\/latex] where <em>b<\/em> is the value where the graph crosses the vertical <em>y<\/em>-axis. Any other point on the line can be represented by [latex](x,y)[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Slope-Intercept Form of a Linear Equation<\/h3>\r\nIn the equation [latex]y=mx+b[\/latex],\r\n<ul id=\"eip-id1168468510671\" class=\"stepwise\">\r\n \t<li>m is the slope of the graph.<\/li>\r\n \t<li>b is the y value of the y-intercept of the graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p style=\"text-align: left;\">This formula is known as the slope-intercept equation.\u00a0If we know the slope and the <em>y<\/em>-intercept we can easily find the equation that represents the line.<\/p>\r\n\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the equation of the line that has a slope of [latex] \\displaystyle \\frac{1}{2}[\/latex] and a <i>y<\/i>-intercept of [latex]\u22125[\/latex].\r\n\r\n[reveal-answer q=\"624715\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624715\"]Substitute the slope (<i>m<\/i>) into [latex]y=mx+b[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\displaystyle y=\\frac{1}{2}x+b[\/latex]<\/p>\r\nSubstitute the [latex]y[\/latex]-intercept (b) into the equation.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle y=\\frac{1}{2}x-5[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]y=\\frac{1}{2}x-5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can also easily find the equation by looking at a graph and finding the slope and <em>y<\/em>-intercept.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite the equation of the line in the graph by identifying the slope and <em>y<\/em>-intercept.\r\n<img class=\"aligncenter wp-image-3198 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24164757\/CNX_BMath_Figure_06_05_17_img-02-300x16.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -10 to 10. A line is drawn through the points (0, 3) and (3, 1). A triangle is drawn under these two points to illustrate the rise and run.\" width=\"300\" height=\"16\" \/>\r\n[reveal-answer q=\"96446\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96446\"]Identify the point where the graph crosses the y-axis [latex](0,3)[\/latex]. This means the [latex]y[\/latex]-intercept is [latex]3[\/latex].\r\n\r\nIdentify one other point and draw a slope triangle to find the slope.\r\n\r\nThe slope is [latex]\\frac{-2}{3}[\/latex]\r\n\r\nSubstitute the slope and y value of the intercept into the slope-intercept equation.\r\n<p style=\"text-align: center;\">[latex]y=mx+b\\\\y=\\frac{-2}{3}x+b\\\\y=\\frac{-2}{3}x+3[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]y=\\frac{-2}{3}x+3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can also move in the opposite direction.\u00a0 When we are given an equation in the slope-intercept form of [latex]y=mx+b[\/latex], we can easily identify the slope\u00a0and <em>y<\/em>-intercept and graph the equation from this information.\u00a0 \u00a0 When we have an equation in slope-intercept form we can graph it by first plotting\u00a0the y-intercept, then using the slope, find a second point and connecting the dots.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGraph [latex]y=\\frac{1}{2}x-4[\/latex] using the slope-intercept equation.\r\n\r\n[reveal-answer q=\"420487\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"420487\"]First, plot the [latex]y[\/latex]-intercept.\r\n\r\n<img class=\"aligncenter wp-image-3202 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/26042734\/SVG_Grapher2-300x294.png\" alt=\"The y-intercept plotted at negative 4 on the y axis.\" width=\"300\" height=\"294\" \/>\r\n\r\nNow use the slope to count up or down and over left or right to the next point. This slope is [latex]\\frac{1}{2}[\/latex], so you can count up one and right two\u2014both positive because both parts of the slope are positive.\r\n\r\nConnect the dots.\r\n<img class=\"aligncenter wp-image-3203 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/26043304\/SVG_Grapher3-300x289.png\" alt=\"A line crosses through negative 4 on the y-axis and has a slope of 1\/2.\" width=\"300\" height=\"289\" \/>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n(NOTE: it is\u00a0important for the equation to first be in slope intercept form. If it is not, we will\u00a0have to solve it for [latex]y[\/latex] so we can identify the slope and the [latex]y[\/latex]-intercept.)\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]79774[\/ohm_question]\r\n\r\n<\/div>\r\nYou can watch the video below for another example of how to write the equation of the line, when given a graph, by identifying the slope and y-intercept.\r\n\r\nhttps:\/\/youtu.be\/GIn7vbB5AYo\r\n","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Draw the graph of a line using slope and a point on the line<\/li>\n<li>Write the equation of a line using slope and y-intercept<\/li>\n<\/ul>\n<\/div>\n<p>When graphing a line we found one method we could use is to make a table of values.\u00a0 We also learned how to graph lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.\u00a0 However, if we can identify some properties of the line, we may be able to make a graph much quicker and easier.<\/p>\n<h2>Graphing a Line Using Slope and a Point on the Line<\/h2>\n<p>Another method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Graph the line passing through the point [latex]\\left(1,-1\\right)[\/latex] whose slope is [latex]m=\\Large\\frac{3}{4}[\/latex].<\/p>\n<p>Solution<br \/>\nPlot the given point, [latex]\\left(1,-1\\right)[\/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\/277\/2017\/04\/25224624\/CNX_BMath_Figure_11_04_050_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -3 to 4. A labeled point is drawn at \u201cordered pair 1, -1\u201d.\" width=\"241\" height=\"225\" \/><br \/>\nUse the slope formula [latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex] to identify the rise and the run.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\\\ m=\\frac{3}{4}\\hfill \\\\ \\frac{\\text{rise}}{\\text{run}}=\\frac{3}{4}\\hfill \\\\ \\\\ \\\\ \\text{rise}=3\\hfill \\\\ \\text{run}=4\\hfill \\end{array}[\/latex]<\/p>\n<p>Starting at the point we plotted, count out the rise and run to mark the second point. We count [latex]3[\/latex] units up and [latex]4[\/latex] units right.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224625\/CNX_BMath_Figure_11_04_051_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two line segments are drawn. A vertical line segment connects the points \u201cordered pair 1, -1\u201d and \u201corder pair \u201c1, 2\u201d. It is labeled \u201c3\u201d. A horizontal line segment starts at the top of the vertical line segment and goes to the right, connecting the points \u201cordered pair 1, 2\u201d and \u201cordered pair 5, 2\u201d. It is labeled \u201c4\u201d.\" width=\"263\" height=\"269\" \/><br \/>\nThen we connect the points with a line and draw arrows at the ends to show it continues.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224627\/CNX_BMath_Figure_11_04_033.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -3 to 5. The y-axis runs from -1 to 7. Two labeled points are drawn at \u201cordered pair 1, -1\u201d and \u201cordered pair 5, 2\u201d. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects \u201cordered pair 1, -1\u201d and \u201cordered pair 1, 2 \u201d. A horizontal line segment connects \u201cordered pair 1, 2\u201d and \u201cordered pair 5, 2\u201d.\" width=\"234\" height=\"241\" \/><br \/>\nWe can check our line by starting at any point and counting up [latex]3[\/latex] and to the right [latex]4[\/latex]. We should get to another point on the line.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147024\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147024&theme=oea&iframe_resize_id=ohm147024&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Graph a line given a point and a slope<\/h3>\n<ol class=\"stepwise\">\n<li style=\"list-style-type: none;\">\n<ol class=\"stepwise\">\n<li>Plot the given point.<\/li>\n<li>Use the slope formula to identify the rise and the run.<\/li>\n<li>Starting at the given point, count out the rise and run to mark the second point.<\/li>\n<li>Connect the points with a line.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Graph the line passing through the point [latex]\\left(-1,-3\\right)[\/latex] whose slope is [latex]m=4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q804097\">Show Solution<\/span><\/p>\n<div id=\"q804097\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nPlot the given point.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224638\/CNX_BMath_Figure_11_04_054_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The point \u201cordered pair -1, -3\u201d is labeled.\" width=\"263\" height=\"270\" \/><\/p>\n<table id=\"eip-id1168469472029\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Identify the rise and the run.<\/td>\n<td>[latex]m=4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write [latex]4[\/latex] as a fraction.<\/td>\n<td>[latex]\\Large\\frac{\\text{rise}}{\\text{run}}\\normalsize=\\Large\\frac{4}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\text{rise}=4\\text{ run}=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Count the rise and run.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224639\/CNX_BMath_Figure_11_04_055_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The y-axis runs from -4 to 2. A vertical line segment connects points at \u201cordered pair -1, -3\u201d and \u201cordered pair -1, 1\u201d and is labeled \u201cup 4\u201d. A horizontal line segment connects \u201cordered pair -1, 1\u201d and \u201cordered pair 0, 1\u201d and is labeled \u201cover 1\u201d.\" width=\"263\" height=\"269\" \/><br \/>\nMark the second point. Connect the two points with a line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224641\/CNX_BMath_Figure_11_04_056_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at \u201cordered pair -1, -3\u201d and \u201cordered pair -1, 1\u201d. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects \u201cordered pair -1, -3\u201d and \u201cordered pair -1, 1 \u201d. It is labeled \u201cup 4\u201d A horizontal line segment connects \u201cordered pair -1, 1\u201d and \u201cordered pair 0, 1\u201d. It is labeled \u201cover 1\u201d\" width=\"264\" height=\"270\" \/><\/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=\"ohm147026\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147026&theme=oea&iframe_resize_id=ohm147026&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>You can watch the video below for another example of how to graph a line given a point and a slope.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Graph a Line Given a Point and the Slope\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ngpAgpMjozw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>A special case for graphing using the point-slope method is when the given point is the y-intercept. We will give some examples here and then we will show below why this case is so important.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>example<\/h3>\n<p>Graph the line with [latex]y[\/latex] -intercept [latex]\\left(0,2\\right)[\/latex] and slope [latex]m=-\\Large\\frac{2}{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q489314\">Show Solution<\/span><\/p>\n<div id=\"q489314\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nPlot the given point, the [latex]y[\/latex] -intercept [latex]\\left(0,2\\right)[\/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\/277\/2017\/04\/25224631\/CNX_BMath_Figure_11_04_052_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 4. The y-axis runs from -1 to 3. The point \u201cordered pair 0, 2\u201d is labeled.\" width=\"261\" height=\"268\" \/><br \/>\nUse the slope formula [latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex] to identify the rise and the run.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\\\ m=-\\frac{2}{3}\\hfill \\\\ \\frac{\\text{rise}}{\\text{run}}=\\frac{-2}{3}\\hfill \\\\ \\\\ \\\\ \\text{rise}=-2\\hfill \\\\ \\text{run}=3\\hfill \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Starting at [latex]\\left(0,2\\right)[\/latex], count the rise and the run and mark the second point.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224632\/CNX_BMath_Figure_11_04_053_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line segment connects points at \u201cordered pair 0, 2\u201d and \u201cordered pair 0, 0\u201d and is labeled \u201cdown 2\u201d. A horizontal line segment connects \u201cordered pair 0, 0\u201d and \u201cordered pair 0, 3\u201d and is labeled \u201cright 3\u201d.\" width=\"263\" height=\"268\" \/><br \/>\nConnect the points with a line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224633\/CNX_BMath_Figure_11_04_036.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at \u201cordered pair 0, 2\u201d and \u201cordered pair 3, 0\u201d. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects \u201cordered pair 0, 2\u201d and \u201cordered pair 0, 0 \u201d. A horizontal line segment connects \u201cordered pair 0, 0\u201d and \u201cordered pair 3, 0\u201d.\" width=\"263\" height=\"270\" \/><\/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=\"ohm147025\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147025&theme=oea&iframe_resize_id=ohm147025&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Slope-Intercept Form<\/h2>\n<p>We will now show you what is so special about the case in which the given point is the y-intercept.\u00a0 The slope can be represented by m and the <em>y<\/em>-intercept, where it crosses the axis and [latex]x=0[\/latex], can be represented by [latex](0,b)[\/latex] where <em>b<\/em> is the value where the graph crosses the vertical <em>y<\/em>-axis. Any other point on the line can be represented by [latex](x,y)[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Slope-Intercept Form of a Linear Equation<\/h3>\n<p>In the equation [latex]y=mx+b[\/latex],<\/p>\n<ul id=\"eip-id1168468510671\" class=\"stepwise\">\n<li>m is the slope of the graph.<\/li>\n<li>b is the y value of the y-intercept of the graph.<\/li>\n<\/ul>\n<\/div>\n<p style=\"text-align: left;\">This formula is known as the slope-intercept equation.\u00a0If we know the slope and the <em>y<\/em>-intercept we can easily find the equation that represents the line.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the equation of the line that has a slope of [latex]\\displaystyle \\frac{1}{2}[\/latex] and a <i>y<\/i>-intercept of [latex]\u22125[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q624715\">Show Solution<\/span><\/p>\n<div id=\"q624715\" class=\"hidden-answer\" style=\"display: none\">Substitute the slope (<i>m<\/i>) into [latex]y=mx+b[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle y=\\frac{1}{2}x+b[\/latex]<\/p>\n<p>Substitute the [latex]y[\/latex]-intercept (b) into the equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle y=\\frac{1}{2}x-5[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=\\frac{1}{2}x-5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can also easily find the equation by looking at a graph and finding the slope and <em>y<\/em>-intercept.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write the equation of the line in the graph by identifying the slope and <em>y<\/em>-intercept.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3198 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24164757\/CNX_BMath_Figure_06_05_17_img-02-300x16.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -10 to 10. A line is drawn through the points (0, 3) and (3, 1). A triangle is drawn under these two points to illustrate the rise and run.\" width=\"300\" height=\"16\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q96446\">Show Solution<\/span><\/p>\n<div id=\"q96446\" class=\"hidden-answer\" style=\"display: none\">Identify the point where the graph crosses the y-axis [latex](0,3)[\/latex]. This means the [latex]y[\/latex]-intercept is [latex]3[\/latex].<\/p>\n<p>Identify one other point and draw a slope triangle to find the slope.<\/p>\n<p>The slope is [latex]\\frac{-2}{3}[\/latex]<\/p>\n<p>Substitute the slope and y value of the intercept into the slope-intercept equation.<\/p>\n<p style=\"text-align: center;\">[latex]y=mx+b\\\\y=\\frac{-2}{3}x+b\\\\y=\\frac{-2}{3}x+3[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]y=\\frac{-2}{3}x+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can also move in the opposite direction.\u00a0 When we are given an equation in the slope-intercept form of [latex]y=mx+b[\/latex], we can easily identify the slope\u00a0and <em>y<\/em>-intercept and graph the equation from this information.\u00a0 \u00a0 When we have an equation in slope-intercept form we can graph it by first plotting\u00a0the y-intercept, then using the slope, find a second point and connecting the dots.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph [latex]y=\\frac{1}{2}x-4[\/latex] using the slope-intercept equation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q420487\">Show Solution<\/span><\/p>\n<div id=\"q420487\" class=\"hidden-answer\" style=\"display: none\">First, plot the [latex]y[\/latex]-intercept.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3202 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/26042734\/SVG_Grapher2-300x294.png\" alt=\"The y-intercept plotted at negative 4 on the y axis.\" width=\"300\" height=\"294\" \/><\/p>\n<p>Now use the slope to count up or down and over left or right to the next point. This slope is [latex]\\frac{1}{2}[\/latex], so you can count up one and right two\u2014both positive because both parts of the slope are positive.<\/p>\n<p>Connect the dots.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3203 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/26043304\/SVG_Grapher3-300x289.png\" alt=\"A line crosses through negative 4 on the y-axis and has a slope of 1\/2.\" width=\"300\" height=\"289\" \/>\n<\/div>\n<\/div>\n<\/div>\n<p>(NOTE: it is\u00a0important for the equation to first be in slope intercept form. If it is not, we will\u00a0have to solve it for [latex]y[\/latex] so we can identify the slope and the [latex]y[\/latex]-intercept.)<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm79774\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=79774&theme=oea&iframe_resize_id=ohm79774&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>You can watch the video below for another example of how to write the equation of the line, when given a graph, by identifying the slope and y-intercept.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Find the Equation of a Line in Slope-Intercept Form of a Line (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GIn7vbB5AYo?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-16513\">\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>Slope-Intercept Form of a Line. <strong>Authored by<\/strong>: Mathispower4u. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Beginning and Intermediate Algebra. <strong>Authored by<\/strong>: Tyler Wallace. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"\"><\/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":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Beginning and Intermediate Algebra\",\"author\":\"Tyler Wallace\",\"organization\":\"\",\"url\":\": http:\/\/wallace.ccfaculty.org\/book\/book.html\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Slope-Intercept Form of a Line\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"33769246025745d88f2fea56ee77e766, 7b3ef4f09f9341f295bfe9ae181918d3","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16513","chapter","type-chapter","status-publish","hentry"],"part":8524,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16513","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16513\/revisions"}],"predecessor-version":[{"id":20473,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16513\/revisions\/20473"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/8524"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16513\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=16513"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=16513"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=16513"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=16513"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}