{"id":6675,"date":"2021-12-21T18:36:54","date_gmt":"2021-12-21T18:36:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=6675"},"modified":"2021-12-21T19:17:30","modified_gmt":"2021-12-21T19:17:30","slug":"3d-graphing-a-line-given-a-point-and-a-slope","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/3d-graphing-a-line-given-a-point-and-a-slope\/","title":{"raw":"3D Graphing a Line Given a Point and a Slope","rendered":"3D Graphing a Line Given a Point and a Slope"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph a line given the slope and a point on the line<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.\r\n\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 \" \/>\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 \" \/>\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 unlabeled points are drawn at \" \/>\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&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>Graph a line given a point and a slope<\/h3>\r\n<ol id=\"eip-id1168468510671\" 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<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\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 \" \/>\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 \" \/>\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 \" \/>\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<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 \" \/>\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 \" \/>\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 \" \/>\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\nIn the following video we show another example of how to graph a line given a point on the line and the slope of the line.\r\n\r\nhttps:\/\/youtu.be\/ngpAgpMjozw","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph a line given the slope and a point on the line<\/li>\n<\/ul>\n<\/div>\n<p>In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.<\/p>\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 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\" \/><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 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\" \/><br \/>\nThen we connect the points with a line and draw arrows at the ends to show it continues.<\/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\/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 unlabeled points are drawn at\" \/><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<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>Graph a line given a point and a slope<\/h3>\n<ol id=\"eip-id1168468510671\" 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<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\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 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\" \/><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 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\" \/><br \/>\nConnect the points with a line.<\/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\/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\" \/><\/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<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 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\" \/><\/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 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\" \/><br \/>\nMark the second point. Connect the two points with a line.<\/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\/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\" \/><\/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>In the following video we show another example of how to graph a line given a point on the line and the slope of the line.<\/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","protected":false},"author":359705,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-6675","chapter","type-chapter","status-publish","hentry"],"part":6629,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6675","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/users\/359705"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6675\/revisions"}],"predecessor-version":[{"id":6686,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6675\/revisions\/6686"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/parts\/6629"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapters\/6675\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/media?parent=6675"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/pressbooks\/v2\/chapter-type?post=6675"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/contributor?post=6675"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/wp-json\/wp\/v2\/license?post=6675"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}