{"id":2602,"date":"2025-08-13T18:10:33","date_gmt":"2025-08-13T18:10:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2602"},"modified":"2026-01-14T16:31:09","modified_gmt":"2026-01-14T16:31:09","slug":"rational-functions-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions-background-youll-need-2\/","title":{"raw":"Rational Functions: Background You'll Need 2","rendered":"Rational Functions: Background You&#8217;ll Need 2"},"content":{"raw":"<section><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Find the x-intercept and y-intercept of graphs.<\/span><\/li>\r\n<\/ul>\r\n<\/section><\/section>\r\n<h2>Finding Intercepts<\/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 a linear function with intercepts labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318936[\/ohm_question]<\/section>","rendered":"<section>\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Find the x-intercept and y-intercept of graphs.<\/span><\/li>\n<\/ul>\n<\/section>\n<\/section>\n<h2>Finding Intercepts<\/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 a linear function with intercepts labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318936\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318936&theme=lumen&iframe_resize_id=ohm318936&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":508,"module-header":"background_you_need","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2602"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2602\/revisions"}],"predecessor-version":[{"id":5345,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2602\/revisions\/5345"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/508"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2602\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2602"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2602"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2602"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2602"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}