{"id":991,"date":"2024-05-01T20:42:21","date_gmt":"2024-05-01T20:42:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=991"},"modified":"2025-08-21T23:10:02","modified_gmt":"2025-08-21T23:10:02","slug":"graphing-and-analyzing-linear-equations-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphing-and-analyzing-linear-equations-learn-it-3\/","title":{"raw":"Graphing and Analyzing Linear Equations: Learn It 3","rendered":"Graphing and Analyzing Linear Equations: Learn It 3"},"content":{"raw":"<h2>Using Intercepts to Plot Lines in the Coordinate Plane<\/h2>\r\nThe <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong>[latex]x[\/latex]<em>-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong>[latex]y[\/latex]<em>-<\/em>intercept<\/strong> is the point where the graph crosses the [latex]y[\/latex]<em>-<\/em>axis. At this point, the [latex]x[\/latex]<em>-<\/em>coordinate is zero.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>intercepts<\/h3>\r\n<div>\r\n\r\nThe <strong>intercepts<\/strong> of a graph are points at which the graph crosses the axes.\r\n<ul>\r\n \t<li>The <strong>[latex]x[\/latex]-intercept<\/strong> is the point at which the graph crosses the [latex]x[\/latex]-axis. At this point, the [latex]y[\/latex]-coordinate is zero.<\/li>\r\n \t<li>The <strong>[latex]y[\/latex]-intercept<\/strong> is the point at which the graph crosses the [latex]y[\/latex]-axis. At this point, the [latex]x[\/latex]-coordinate is zero.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section class=\"textbox proTip\">\r\n<ul>\r\n \t<li>Find the [latex]x[\/latex]-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\r\n \t<li>Find the [latex]y[\/latex]<em>-<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\">Find the intercepts of the equation [latex]y=3x - 1[\/latex].\r\n<ul>\r\n \t<li>To find the [latex]x[\/latex]<em>-<\/em>intercept, set [latex]y=0[\/latex].<\/li>\r\n<\/ul>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}y=3x - 1\\hfill &amp; \\hfill \\\\ 0=3x - 1\\hfill &amp; \\hfill \\\\ 1=3x\\hfill &amp; \\hfill \\\\ \\frac{1}{3}=x\\hfill &amp; \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill &amp; x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\n<ul>\r\n \t<li>To find the [latex]y[\/latex]<em>-<\/em>intercept, set [latex]x=0[\/latex].<\/li>\r\n<\/ul>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill &amp; \\hfill \\\\ y=3\\left(0\\right)-1\\hfill &amp; \\hfill \\\\ y=-1\\hfill &amp; \\hfill \\\\ \\left(0,-1\\right)\\hfill &amp; y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[reveal-answer q=\"259463\"]Graph of the linear equation.[\/reveal-answer]\r\n[hidden-answer a=\"259463\"]\r\n<div>You can now graph the equation using the intercepts. All you need to do is plot both points and draw a line passing through them.<\/div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/> Graph of y = 3x - 1[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.[reveal-answer q=\"814560\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814560\"]\r\nSet [latex]y=0[\/latex] to find the [latex]x[\/latex]<em>-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nSet [latex]x=0[\/latex] to find the [latex]y[\/latex]<em>-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nPlot both points and draw a line passing through them.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/> Graph of y = -3x - 4 with the x-intercept and y-intercept labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18921[\/ohm2_question]<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong><strong>Using a Graphing Utility to Plot Lines<\/strong><\/strong>Graphing utilities are powerful tools that allow you to visualize mathematical concepts and plot lines quickly and accurately. Whether you're checking your work, exploring different equations, or just trying to understand how changes in variables affect a graph, these online tools can help. Below are some popular graphing utilities you can use to plot lines and analyze functions. Simply enter your equation, and the utility will generate the graph for you.\r\n\r\n<a href=\"https:\/\/www.geogebra.org\/graphing\">https:\/\/www.geogebra.org\/graphing<\/a>\r\n\r\n<a href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/a>\r\n\r\n<a href=\"https:\/\/www.mathway.com\/Graph\">https:\/\/www.mathway.com\/Graph<\/a>\r\n\r\n<a href=\"https:\/\/www.symbolab.com\/graphing-calculator\">https:\/\/www.symbolab.com\/graphing-calculator<\/a>\r\n\r\n<strong>Try it now<\/strong>\r\n\r\nThese graphing utilities have features that allow you to turn a constant (number) into a variable. Follow these steps to learn how:\r\n<ol>\r\n \t<li>Graph the line [latex]y=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].<\/li>\r\n \t<li>On the next line enter\u00a0[latex]y=-a x-\\frac{4}{3}[\/latex]. You will see a button pop up that says \"add slider: a\", click on the button. You will see the next line populated with the variable a and the interval on which a can take values.<\/li>\r\n \t<li>What part of a line does the variable a represent? The slope or the y-intercept?<\/li>\r\n<\/ol>\r\n<\/section>","rendered":"<h2>Using Intercepts to Plot Lines in the Coordinate Plane<\/h2>\n<p>The <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong>[latex]x[\/latex]<em>&#8211;<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong>[latex]y[\/latex]<em>&#8211;<\/em>intercept<\/strong> is the point where the graph crosses the [latex]y[\/latex]<em>&#8211;<\/em>axis. At this point, the [latex]x[\/latex]<em>&#8211;<\/em>coordinate is zero.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>intercepts<\/h3>\n<div>\n<p>The <strong>intercepts<\/strong> of a graph are points at which the graph crosses the axes.<\/p>\n<ul>\n<li>The <strong>[latex]x[\/latex]-intercept<\/strong> is the point at which the graph crosses the [latex]x[\/latex]-axis. At this point, the [latex]y[\/latex]-coordinate is zero.<\/li>\n<li>The <strong>[latex]y[\/latex]-intercept<\/strong> is the point at which the graph crosses the [latex]y[\/latex]-axis. At this point, the [latex]x[\/latex]-coordinate is zero.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<ul>\n<li>Find the [latex]x[\/latex]-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\n<li>Find the [latex]y[\/latex]<em>&#8211;<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">Find the intercepts of the equation [latex]y=3x - 1[\/latex].<\/p>\n<ul>\n<li>To find the [latex]x[\/latex]<em>&#8211;<\/em>intercept, set [latex]y=0[\/latex].<\/li>\n<\/ul>\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}y=3x - 1\\hfill & \\hfill \\\\ 0=3x - 1\\hfill & \\hfill \\\\ 1=3x\\hfill & \\hfill \\\\ \\frac{1}{3}=x\\hfill & \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill & x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<ul>\n<li>To find the [latex]y[\/latex]<em>&#8211;<\/em>intercept, set [latex]x=0[\/latex].<\/li>\n<\/ul>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill & \\hfill \\\\ y=3\\left(0\\right)-1\\hfill & \\hfill \\\\ y=-1\\hfill & \\hfill \\\\ \\left(0,-1\\right)\\hfill & y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q259463\">Graph of the linear equation.<\/button><\/p>\n<div id=\"q259463\" class=\"hidden-answer\" style=\"display: none\">\n<div>You can now graph the equation using the intercepts. All you need to do is plot both points and draw a line passing through them.<\/div>\n<div>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/><figcaption class=\"wp-caption-text\">Graph of y = 3x &#8211; 1<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q814560\">Show Solution<\/button><\/p>\n<div id=\"q814560\" class=\"hidden-answer\" style=\"display: none\">\nSet [latex]y=0[\/latex] to find the [latex]x[\/latex]<em>&#8211;<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Set [latex]x=0[\/latex] to find the [latex]y[\/latex]<em>&#8211;<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Plot both points and draw a line passing through them.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/><figcaption class=\"wp-caption-text\">Graph of y = -3x &#8211; 4 with the x-intercept and y-intercept labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18921\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18921&theme=lumen&iframe_resize_id=ohm18921&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong><strong>Using a Graphing Utility to Plot Lines<\/strong><\/strong>Graphing utilities are powerful tools that allow you to visualize mathematical concepts and plot lines quickly and accurately. Whether you&#8217;re checking your work, exploring different equations, or just trying to understand how changes in variables affect a graph, these online tools can help. Below are some popular graphing utilities you can use to plot lines and analyze functions. Simply enter your equation, and the utility will generate the graph for you.<\/p>\n<p><a href=\"https:\/\/www.geogebra.org\/graphing\">https:\/\/www.geogebra.org\/graphing<\/a><\/p>\n<p><a href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/a><\/p>\n<p><a href=\"https:\/\/www.mathway.com\/Graph\">https:\/\/www.mathway.com\/Graph<\/a><\/p>\n<p><a href=\"https:\/\/www.symbolab.com\/graphing-calculator\">https:\/\/www.symbolab.com\/graphing-calculator<\/a><\/p>\n<p><strong>Try it now<\/strong><\/p>\n<p>These graphing utilities have features that allow you to turn a constant (number) into a variable. Follow these steps to learn how:<\/p>\n<ol>\n<li>Graph the line [latex]y=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].<\/li>\n<li>On the next line enter\u00a0[latex]y=-a x-\\frac{4}{3}[\/latex]. You will see a button pop up that says &#8220;add slider: a&#8221;, click on the button. You will see the next line populated with the variable a and the interval on which a can take values.<\/li>\n<li>What part of a line does the variable a represent? The slope or the y-intercept?<\/li>\n<\/ol>\n<\/section>\n","protected":false},"author":12,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":75,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/991"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":21,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/991\/revisions"}],"predecessor-version":[{"id":7969,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/991\/revisions\/7969"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/75"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/991\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=991"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=991"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=991"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=991"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}