{"id":1015,"date":"2025-07-21T19:16:06","date_gmt":"2025-07-21T19:16:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1015"},"modified":"2026-01-14T18:49:57","modified_gmt":"2026-01-14T18:49:57","slug":"rational-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions-learn-it-4\/","title":{"raw":"Rational Functions: Learn It 5","rendered":"Rational Functions: Learn It 5"},"content":{"raw":"<h2>Finding Intercepts of Rational Functions<\/h2>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">We can always find any horizontal intercepts of the graph of a function by setting the output equal to zero. We can always find any vertical intercept by setting the input equal to zero.<\/section>In a rational function of the form [latex]r(x)=\\dfrac{P(x)}{Q(x)}[\/latex],\r\n<ul>\r\n \t<li>Find the vertical intercept (the [latex]y[\/latex]-intercept) by evaluating [latex]r(0)[\/latex]. That is, replace all the input variables with [latex]0[\/latex] and calculate the result.<\/li>\r\n \t<li>Find the horizontal intercept(s) (the [latex]x[\/latex]-intercepts) by solving [latex]r(x)=0[\/latex]. Since the function is undefined where the denominator equals zero], set the numerator equal to zero to find the horizontal intercepts of the function.<\/li>\r\n \t<li>Note that the graph of a rational function may not possess a vertical- or horizontal-intercept.<\/li>\r\n<\/ul>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>intercepts of rational functions<\/h3>\r\nA <strong>rational function<\/strong> will have a [latex]y[\/latex]-intercept when the input is zero, if the function is defined at zero. A rational function will not have a [latex]y[\/latex]-intercept if the function is not defined at zero.\r\n[latex]\\\\[\/latex]\r\n\r\nLikewise, a rational function will have [latex]x[\/latex]-intercepts at the inputs that cause the output to be zero. Since a fraction is only equal to zero when the numerator is zero, [latex]x[\/latex]-intercepts can only occur when the numerator of the rational function is equal to zero.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the intercepts of [latex]f\\left(x\\right)=\\dfrac{\\left(x - 2\\right)\\left(x+3\\right)}{\\left(x - 1\\right)\\left(x+2\\right)\\left(x - 5\\right)}[\/latex].[reveal-answer q=\"418596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"418596\"]We can find the [latex]y[\/latex]-intercept by evaluating the function at zero\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(0\\right)&amp;=\\dfrac{\\left(0 - 2\\right)\\left(0+3\\right)}{\\left(0 - 1\\right)\\left(0+2\\right)\\left(0 - 5\\right)} \\\\[1mm] &amp;=\\frac{-6}{10} \\\\[1mm] &amp;=-\\frac{3}{5} \\\\[1mm] &amp;=-0.6 \\end{align}[\/latex]<\/p>\r\nThe [latex]x[\/latex]-intercepts will occur when the function is equal to zero. A rational function is equal to 0 when the numerator is 0, as long as the denominator is not also 0.\r\n<p style=\"text-align: center;\">[latex]\\begin{align} 0&amp;=\\frac{\\left(x - 2\\right)\\left(x+3\\right)}{\\left(x - 1\\right)\\left(x+2\\right)\\left(x - 5\\right)} \\\\[1mm] 0&amp;=\\left(x - 2\\right)\\left(x+3\\right)\\end{align}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=2, -3[\/latex]<\/p>\r\nThe [latex]y[\/latex]-intercept is [latex]\\left(0,-0.6\\right)[\/latex], the [latex]x[\/latex]-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-3,0\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213944\/CNX_Precalc_Figure_03_07_0172.jpg\" alt=\"Graph of f(x)=(x-2)(x+3)\/(x-1)(x+2)(x-5) with its vertical asymptotes at x=-2, x=1, and x=5, its horizontal asymptote at y=0, and its intercepts at (-3, 0), (0, -0.6), and (2, 0).\" width=\"731\" height=\"514\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]318949[\/ohm_question]<\/section>","rendered":"<h2>Finding Intercepts of Rational Functions<\/h2>\n<section class=\"textbox recall\" aria-label=\"Recall\">We can always find any horizontal intercepts of the graph of a function by setting the output equal to zero. We can always find any vertical intercept by setting the input equal to zero.<\/section>\n<p>In a rational function of the form [latex]r(x)=\\dfrac{P(x)}{Q(x)}[\/latex],<\/p>\n<ul>\n<li>Find the vertical intercept (the [latex]y[\/latex]-intercept) by evaluating [latex]r(0)[\/latex]. That is, replace all the input variables with [latex]0[\/latex] and calculate the result.<\/li>\n<li>Find the horizontal intercept(s) (the [latex]x[\/latex]-intercepts) by solving [latex]r(x)=0[\/latex]. Since the function is undefined where the denominator equals zero], set the numerator equal to zero to find the horizontal intercepts of the function.<\/li>\n<li>Note that the graph of a rational function may not possess a vertical- or horizontal-intercept.<\/li>\n<\/ul>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>intercepts of rational functions<\/h3>\n<p>A <strong>rational function<\/strong> will have a [latex]y[\/latex]-intercept when the input is zero, if the function is defined at zero. A rational function will not have a [latex]y[\/latex]-intercept if the function is not defined at zero.<br \/>\n[latex]\\\\[\/latex]<\/p>\n<p>Likewise, a rational function will have [latex]x[\/latex]-intercepts at the inputs that cause the output to be zero. Since a fraction is only equal to zero when the numerator is zero, [latex]x[\/latex]-intercepts can only occur when the numerator of the rational function is equal to zero.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the intercepts of [latex]f\\left(x\\right)=\\dfrac{\\left(x - 2\\right)\\left(x+3\\right)}{\\left(x - 1\\right)\\left(x+2\\right)\\left(x - 5\\right)}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q418596\">Show Solution<\/button><\/p>\n<div id=\"q418596\" class=\"hidden-answer\" style=\"display: none\">We can find the [latex]y[\/latex]-intercept by evaluating the function at zero<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(0\\right)&=\\dfrac{\\left(0 - 2\\right)\\left(0+3\\right)}{\\left(0 - 1\\right)\\left(0+2\\right)\\left(0 - 5\\right)} \\\\[1mm] &=\\frac{-6}{10} \\\\[1mm] &=-\\frac{3}{5} \\\\[1mm] &=-0.6 \\end{align}[\/latex]<\/p>\n<p>The [latex]x[\/latex]-intercepts will occur when the function is equal to zero. A rational function is equal to 0 when the numerator is 0, as long as the denominator is not also 0.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} 0&=\\frac{\\left(x - 2\\right)\\left(x+3\\right)}{\\left(x - 1\\right)\\left(x+2\\right)\\left(x - 5\\right)} \\\\[1mm] 0&=\\left(x - 2\\right)\\left(x+3\\right)\\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=2, -3[\/latex]<\/p>\n<p>The [latex]y[\/latex]-intercept is [latex]\\left(0,-0.6\\right)[\/latex], the [latex]x[\/latex]-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-3,0\\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\/896\/2016\/11\/02213944\/CNX_Precalc_Figure_03_07_0172.jpg\" alt=\"Graph of f(x)=(x-2)(x+3)\/(x-1)(x+2)(x-5) with its vertical asymptotes at x=-2, x=1, and x=5, its horizontal asymptote at y=0, and its intercepts at (-3, 0), (0, -0.6), and (2, 0).\" width=\"731\" height=\"514\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm318949\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318949&theme=lumen&iframe_resize_id=ohm318949&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":10,"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":"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1015"}],"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\/13"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1015\/revisions"}],"predecessor-version":[{"id":5353,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1015\/revisions\/5353"}],"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\/1015\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1015"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1015"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1015"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1015"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}