{"id":1415,"date":"2025-07-25T01:10:52","date_gmt":"2025-07-25T01:10:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1415"},"modified":"2026-03-18T05:52:24","modified_gmt":"2026-03-18T05:52:24","slug":"rational-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions-fresh-take\/","title":{"raw":"Rational Functions: Fresh Take","rendered":"Rational Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the domains of rational functions.<\/li>\r\n \t<li>Identify vertical and horizontal asymptotes.<\/li>\r\n \t<li>Identify slant asymptotes.<\/li>\r\n \t<li>Graph rational functions.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Rational Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A rational function is a quotient of two polynomial functions: [latex]f(x) = \\frac{P(x)}{Q(x)}[\/latex], where [latex]Q(x) \\neq 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Includes all real numbers except where [latex]Q(x) = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Zero in the denominator makes the function undefined<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Behavior near vertical asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Function values approach infinity or negative infinity as [latex]x[\/latex] approaches points where [latex]Q(x) = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">End behavior:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Determined by the degrees of [latex]P(x)[\/latex] and [latex]Q(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">May approach a horizontal asymptote as [latex]x[\/latex] approaches infinity<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Arrow notation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used to describe function behavior as [latex]x[\/latex] approaches specific values or infinity<\/li>\r\n \t<li>Arrow notation shortcuts:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x \\to a^-[\/latex]: x approaches a from the left<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x \\to a^+[\/latex]: x approaches a from the right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x \\to \\pm\\infty[\/latex]: x approaches positive\/negative infinity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) \\to \\pm\\infty[\/latex]: function values increase\/decrease without bound<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Local and End Behavior of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Occur when the denominator equals zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represent local behavior near undefined points<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Represent end behavior as [latex]x[\/latex] approaches infinity or negative infinity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determined by comparing degrees of numerator and denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">End Behavior:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Describes function behavior for very large positive or negative [latex]x[\/latex] values<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Arrow Notation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used to concisely describe limits and asymptotic behavior<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function.[reveal-answer q=\"785291\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"785291\"]End behavior: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to 0[\/latex]; Local behavior: as [latex]x\\to 0, f\\left(x\\right)\\to \\infty [\/latex] (there are no [latex]x[\/latex]- or [latex]y[\/latex]-intercepts)[\/hidden-answer]<\/section><section aria-label=\"Example\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dcafcaab-ORga2Ooxfvw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ORga2Ooxfvw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dcafcaab-ORga2Ooxfvw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780729&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dcafcaab-ORga2Ooxfvw&amp;vembed=0&amp;video_id=ORga2Ooxfvw&amp;video_target=tpm-plugin-dcafcaab-ORga2Ooxfvw\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Arrow+Notation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cArrow Notation\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/section>\r\n<h2>Domain and Its Effect on Vertical Asymptotes<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Domain of Rational Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Includes all real numbers except those that make the denominator zero<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Occur when the denominator equals zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represent values where the function is undefined<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Removable Discontinuities:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Occur when a factor cancels out between numerator and denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Create \"holes\" in the graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factoring:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Essential for identifying both vertical asymptotes and removable discontinuities<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphical Representation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical asymptotes: function approaches infinity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Removable discontinuities: open circles on the graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the domain of [latex]f\\left(x\\right)=\\dfrac{4x}{5\\left(x - 1\\right)\\left(x - 5\\right)}[\/latex].[reveal-answer q=\"553731\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"553731\"]The domain is all real numbers except [latex]x=1[\/latex] and [latex]x=5[\/latex].[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the vertical asymptotes of the graph of [latex]k\\left(x\\right)=\\dfrac{5+2{x}^{2}}{2-x-{x}^{2}}[\/latex].[reveal-answer q=\"787718\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"787718\"]First, factor the numerator and denominator.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}k\\left(x\\right)&amp;=\\dfrac{5+2{x}^{2}}{2-x-{x}^{2}} \\\\[1mm] &amp;=\\dfrac{5+2{x}^{2}}{\\left(2+x\\right)\\left(1-x\\right)} \\end{align}[\/latex]<\/p>\r\nTo find the vertical asymptotes, we determine where this function will be undefined by setting the denominator equal to zero:\r\n<p style=\"text-align: center;\">[latex]\\left(2+x\\right)\\left(1-x\\right)=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=-2,1[\/latex]<\/p>\r\nNeither [latex]x=-2[\/latex] nor [latex]x=1[\/latex] are zeros of the numerator, so the two values indicate two vertical asymptotes. The graph below confirms the location of the two vertical asymptotes.\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\/11\/02213927\/CNX_Precalc_Figure_03_07_0102.jpg\" alt=\"Graph of k(x)=(5+2x)^2\/(2-x-x^2) with its vertical asymptotes at x=-2 and x=1 and its horizontal asymptote at y=-2.\" width=\"487\" height=\"514\" \/> Graph of k(x) with asymptotes labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>Watch the following video to see more examples of finding the domain of a rational function.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fhbhcheg-v0IhvIzCc_I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/v0IhvIzCc_I?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fhbhcheg-v0IhvIzCc_I\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=6454925&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fhbhcheg-v0IhvIzCc_I&amp;vembed=0&amp;video_id=v0IhvIzCc_I&amp;video_target=tpm-plugin-fhbhcheg-v0IhvIzCc_I\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+The+Domain+of+Rational+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: The Domain of Rational Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Horizontal Asymptotes and Intercepts<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">End Behavior:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Describes how a function behaves as [latex]x[\/latex] approaches positive or negative infinity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For rational functions, determined by the ratio of leading terms in numerator and denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Represent the [latex]y[\/latex]-value the function approaches as [latex]x[\/latex] approaches infinity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determined by comparing degrees of numerator and denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Three Cases for Horizontal Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Case [latex]1[\/latex]: Degree(numerator) [latex]&lt;[\/latex] Degree(denominator)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Case [latex]2[\/latex]: Degree(numerator) [latex]&gt;[\/latex] Degree(denominator)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Case [latex]3[\/latex]: Degree(numerator) [latex]=[\/latex] Degree(denominator)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Slant Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Occur when degree of numerator is exactly one more than degree of denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Crossing Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Rational functions can cross horizontal asymptotes, unlike vertical asymptotes<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Intercept ([latex]y[\/latex]-intercept):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Found by evaluating the function at [latex]x = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">May not exist if the function is undefined at [latex]x = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Intercepts ([latex]x[\/latex]-intercepts):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Found by solving [latex]r(x) = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Occur when the numerator equals zero (if the denominator is not also zero)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Existence of Intercepts:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A rational function may not have vertical or horizontal intercepts<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-lg font-bold\"><strong>How to Find Intercepts<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Evaluate [latex]r(0)[\/latex] by substituting [latex]0[\/latex] for all [latex]x[\/latex] in the function<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If defined, the y-intercept is [latex](0, r(0))[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">x-intercepts:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set the numerator equal to zero and solve for [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check that these [latex]x[\/latex]-values don't make the denominator zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The [latex]x[\/latex]-intercepts are [latex](x, 0)[\/latex] for each valid solution<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the vertical and horizontal asymptotes of the function:\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{\\left(2x - 1\\right)\\left(2x+1\\right)}{\\left(x - 2\\right)\\left(x+3\\right)}[\/latex]<\/p>\r\n[reveal-answer q=\"5590\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"5590\"]\r\n\r\nVertical asymptotes at [latex]x=2[\/latex] and [latex]x=-3[\/latex]; horizontal asymptote at [latex]y=4[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>Watch this video to see more worked examples of determining which kind of horizontal asymptote a rational function will have.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-baeccgdf-A1tApZSE8nI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/A1tApZSE8nI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-baeccgdf-A1tApZSE8nI\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847034&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-baeccgdf-A1tApZSE8nI&amp;vembed=0&amp;video_id=A1tApZSE8nI&amp;video_target=tpm-plugin-baeccgdf-A1tApZSE8nI\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Determine+Horizontal+Asymptotes+of+Rational+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine Horizontal Asymptotes of Rational Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>Watch the following video to see more worked examples of finding asymptotes, intercepts and holes of rational functions.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hgagbdgc-UnVZs2EaEjI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/UnVZs2EaEjI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hgagbdgc-UnVZs2EaEjI\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847035&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hgagbdgc-UnVZs2EaEjI&amp;vembed=0&amp;video_id=UnVZs2EaEjI&amp;video_target=tpm-plugin-hgagbdgc-UnVZs2EaEjI\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+the+Intercepts%2C+Asymptotes%2C+and+Hole+of+a+Rational+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Intercepts, Asymptotes, and Hole of a Rational Function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Graphing Rational Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Numerator and Denominator Roles:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Numerator reveals [latex]x[\/latex]-intercepts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Denominator reveals vertical asymptotes<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Effect of Factor Powers:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Factors with powers [latex]&gt; 1[\/latex] affect graph shape at intercepts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Odd-degree factors in denominator: opposite behavior on either side of asymptote<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Even-degree factors in denominator: same behavior on both sides of asymptote<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Asymptote Behavior:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Odd-degree factor: graph approaches positive infinity on one side, negative infinity on the other<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Even-degree factor: graph approaches either positive or negative infinity on both sides<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Step-by-Step Graphing Process<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Simplify the Function:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Factor and reduce common terms in numerator and denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Find the Domain:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify [latex]x[\/latex]-values where denominator equals zero<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Identify Vertical Asymptotes and Holes:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set denominator to zero and solve for [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check if numerator is also zero at these [latex]x[\/latex]-values (indicates holes, not asymptotes)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Find Horizontal or Slant Asymptote:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Compare degrees of numerator ([latex]n[\/latex]) and denominator ([latex]d[\/latex]):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]n &lt; d[\/latex]: Horizontal asymptote at [latex]y = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]n = d[\/latex]: Horizontal asymptote at [latex]y[\/latex] = ratio of leading coefficients<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]n = d + 1[\/latex]: Slant asymptote (find using polynomial long division)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Find Intercepts:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts: Set numerator to zero, solve for [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept: Evaluate function at [latex]x = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Plot Key Points:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Calculate points on both sides of vertical asymptotes<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Sketch the Graph:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Draw asymptotes (dashed lines)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot intercepts and key points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Draw curve, ensuring proper asymptote behavior<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]f\\left(x\\right)=\\dfrac{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}{2{\\left(x - 1\\right)}^{2}\\left(x - 3\\right)}[\/latex], use the characteristics of polynomials and rational functions to describe its behavior and sketch the function.[reveal-answer q=\"451047\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"451047\"]Horizontal asymptote at [latex]y=\\frac{1}{2}[\/latex]. Vertical asymptotes at [latex]x=1[\/latex] and [latex]x=3[\/latex]. [latex]y[\/latex]-intercept at [latex]\\left(0,\\frac{4}{3}.\\right)[\/latex]\r\n<p id=\"fs-id1165135168380\"><em>x<\/em>-intercepts at [latex]\\left(2,0\\right) \\text{ and }\\left(-2,0\\right)[\/latex]. [latex]\\left(-2,0\\right)[\/latex] is a zero with multiplicity 2, and the graph bounces off the [latex]x[\/latex]-axis at this point. [latex]\\left(2,0\\right)[\/latex] is a single zero and the graph crosses the axis at this point.<\/p>\r\n\r\n\r\n[caption id=\"attachment_2947\" align=\"aligncenter\" width=\"731\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/02213511\/CNX_Precalc_Figure_03_07_023.jpg\"><img class=\"wp-image-2947 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/02213511\/CNX_Precalc_Figure_03_07_023.jpg\" alt=\"cnx_precalc_figure_03_07_023\" width=\"731\" height=\"477\" \/><\/a> Graph of f(x)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>Watch the following video to see another worked example of how to match different kinds of rational functions with their graphs.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dfffeeea-vMVYaFptvkk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/vMVYaFptvkk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dfffeeea-vMVYaFptvkk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847036&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dfffeeea-vMVYaFptvkk&amp;vembed=0&amp;video_id=vMVYaFptvkk&amp;video_target=tpm-plugin-dfffeeea-vMVYaFptvkk\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Match+Equations+of+Rational+Functions+to+Graphs_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Match Equations of Rational Functions to Graphs\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Writing Rational Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Relationship between graph features and function components:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts correspond to factors in the numerator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical asymptotes correspond to factors in the denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form of a rational function: [latex]f(x) = a\\frac{(x-x_1)^{p_1}(x-x_2)^{p_2}\\cdots(x-x_n)^{p_n}}{(x-v_1)^{q_1}(x-v_2)^{q_2}\\cdots(x-v_m)^{q_m}}[\/latex] Where:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x_i[\/latex] are [latex]x[\/latex]-intercepts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]v_j[\/latex] are vertical asymptotes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]p_i[\/latex] and [latex]q_j[\/latex] are multiplicities<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex] is the stretch factor<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Behavior at intercepts and asymptotes determines multiplicities<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Any point on the graph satisfies the function equation<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Step-by-Step Process<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Identify [latex]x[\/latex]-intercepts:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Locate where graph crosses [latex]x[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine multiplicity based on behavior (passing through or bouncing)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Identify vertical asymptotes:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Locate where graph approaches infinity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine multiplicity based on behavior on each side<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Write the general form:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use [latex]x[\/latex]-intercepts for numerator factors<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use vertical asymptotes for denominator factors<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Determine stretch factor [latex]a[\/latex]:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use a clear point on the graph, often the [latex]y[\/latex]-intercept<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute point into the function and solve for [latex]a[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aabhhfhh-VU_csSy4BqM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/VU_csSy4BqM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aabhhfhh-VU_csSy4BqM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847037&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aabhhfhh-VU_csSy4BqM&amp;vembed=0&amp;video_id=VU_csSy4BqM&amp;video_target=tpm-plugin-aabhhfhh-VU_csSy4BqM\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Find+the+Equation+of+Rational+Function+From+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Find the Equation of Rational Function From a Graph\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the domains of rational functions.<\/li>\n<li>Identify vertical and horizontal asymptotes.<\/li>\n<li>Identify slant asymptotes.<\/li>\n<li>Graph rational functions.<\/li>\n<\/ul>\n<\/section>\n<h2>Rational Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A rational function is a quotient of two polynomial functions: [latex]f(x) = \\frac{P(x)}{Q(x)}[\/latex], where [latex]Q(x) \\neq 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Includes all real numbers except where [latex]Q(x) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Zero in the denominator makes the function undefined<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Behavior near vertical asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Function values approach infinity or negative infinity as [latex]x[\/latex] approaches points where [latex]Q(x) = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">End behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determined by the degrees of [latex]P(x)[\/latex] and [latex]Q(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">May approach a horizontal asymptote as [latex]x[\/latex] approaches infinity<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Arrow notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used to describe function behavior as [latex]x[\/latex] approaches specific values or infinity<\/li>\n<li>Arrow notation shortcuts:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x \\to a^-[\/latex]: x approaches a from the left<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x \\to a^+[\/latex]: x approaches a from the right<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x \\to \\pm\\infty[\/latex]: x approaches positive\/negative infinity<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(x) \\to \\pm\\infty[\/latex]: function values increase\/decrease without bound<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Local and End Behavior of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Vertical Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Occur when the denominator equals zero<\/li>\n<li class=\"whitespace-normal break-words\">Represent local behavior near undefined points<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Represent end behavior as [latex]x[\/latex] approaches infinity or negative infinity<\/li>\n<li class=\"whitespace-normal break-words\">Determined by comparing degrees of numerator and denominator<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">End Behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Describes function behavior for very large positive or negative [latex]x[\/latex] values<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Arrow Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used to concisely describe limits and asymptotic behavior<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q785291\">Show Solution<\/button><\/p>\n<div id=\"q785291\" class=\"hidden-answer\" style=\"display: none\">End behavior: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to 0[\/latex]; Local behavior: as [latex]x\\to 0, f\\left(x\\right)\\to \\infty[\/latex] (there are no [latex]x[\/latex]&#8211; or [latex]y[\/latex]-intercepts)<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dcafcaab-ORga2Ooxfvw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ORga2Ooxfvw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dcafcaab-ORga2Ooxfvw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780729&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dcafcaab-ORga2Ooxfvw&amp;vembed=0&amp;video_id=ORga2Ooxfvw&amp;video_target=tpm-plugin-dcafcaab-ORga2Ooxfvw\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Arrow+Notation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cArrow Notation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/section>\n<h2>Domain and Its Effect on Vertical Asymptotes<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Domain of Rational Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Includes all real numbers except those that make the denominator zero<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Occur when the denominator equals zero<\/li>\n<li class=\"whitespace-normal break-words\">Represent values where the function is undefined<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Removable Discontinuities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Occur when a factor cancels out between numerator and denominator<\/li>\n<li class=\"whitespace-normal break-words\">Create &#8220;holes&#8221; in the graph<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Factoring:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Essential for identifying both vertical asymptotes and removable discontinuities<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphical Representation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical asymptotes: function approaches infinity<\/li>\n<li class=\"whitespace-normal break-words\">Removable discontinuities: open circles on the graph<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the domain of [latex]f\\left(x\\right)=\\dfrac{4x}{5\\left(x - 1\\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=\"q553731\">Show Solution<\/button><\/p>\n<div id=\"q553731\" class=\"hidden-answer\" style=\"display: none\">The domain is all real numbers except [latex]x=1[\/latex] and [latex]x=5[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the vertical asymptotes of the graph of [latex]k\\left(x\\right)=\\dfrac{5+2{x}^{2}}{2-x-{x}^{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q787718\">Show Solution<\/button><\/p>\n<div id=\"q787718\" class=\"hidden-answer\" style=\"display: none\">First, factor the numerator and denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}k\\left(x\\right)&=\\dfrac{5+2{x}^{2}}{2-x-{x}^{2}} \\\\[1mm] &=\\dfrac{5+2{x}^{2}}{\\left(2+x\\right)\\left(1-x\\right)} \\end{align}[\/latex]<\/p>\n<p>To find the vertical asymptotes, we determine where this function will be undefined by setting the denominator equal to zero:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2+x\\right)\\left(1-x\\right)=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=-2,1[\/latex]<\/p>\n<p>Neither [latex]x=-2[\/latex] nor [latex]x=1[\/latex] are zeros of the numerator, so the two values indicate two vertical asymptotes. The graph below confirms the location of the two vertical asymptotes.<\/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\/11\/02213927\/CNX_Precalc_Figure_03_07_0102.jpg\" alt=\"Graph of k(x)=(5+2x)^2\/(2-x-x^2) with its vertical asymptotes at x=-2 and x=1 and its horizontal asymptote at y=-2.\" width=\"487\" height=\"514\" \/><figcaption class=\"wp-caption-text\">Graph of k(x) with asymptotes labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video to see more examples of finding the domain of a rational function.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fhbhcheg-v0IhvIzCc_I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/v0IhvIzCc_I?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fhbhcheg-v0IhvIzCc_I\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=6454925&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fhbhcheg-v0IhvIzCc_I&amp;vembed=0&amp;video_id=v0IhvIzCc_I&amp;video_target=tpm-plugin-fhbhcheg-v0IhvIzCc_I\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+The+Domain+of+Rational+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: The Domain of Rational Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Horizontal Asymptotes and Intercepts<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">End Behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Describes how a function behaves as [latex]x[\/latex] approaches positive or negative infinity<\/li>\n<li class=\"whitespace-normal break-words\">For rational functions, determined by the ratio of leading terms in numerator and denominator<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Represent the [latex]y[\/latex]-value the function approaches as [latex]x[\/latex] approaches infinity<\/li>\n<li class=\"whitespace-normal break-words\">Determined by comparing degrees of numerator and denominator<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Three Cases for Horizontal Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Case [latex]1[\/latex]: Degree(numerator) [latex]<[\/latex] Degree(denominator)<\/li>\n<li class=\"whitespace-normal break-words\">Case [latex]2[\/latex]: Degree(numerator) [latex]>[\/latex] Degree(denominator)<\/li>\n<li class=\"whitespace-normal break-words\">Case [latex]3[\/latex]: Degree(numerator) [latex]=[\/latex] Degree(denominator)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Slant Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Occur when degree of numerator is exactly one more than degree of denominator<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Crossing Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Rational functions can cross horizontal asymptotes, unlike vertical asymptotes<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Intercept ([latex]y[\/latex]-intercept):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Found by evaluating the function at [latex]x = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">May not exist if the function is undefined at [latex]x = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Intercepts ([latex]x[\/latex]-intercepts):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Found by solving [latex]r(x) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Occur when the numerator equals zero (if the denominator is not also zero)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Existence of Intercepts:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A rational function may not have vertical or horizontal intercepts<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-lg font-bold\"><strong>How to Find Intercepts<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Evaluate [latex]r(0)[\/latex] by substituting [latex]0[\/latex] for all [latex]x[\/latex] in the function<\/li>\n<li class=\"whitespace-normal break-words\">If defined, the y-intercept is [latex](0, r(0))[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">x-intercepts:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set the numerator equal to zero and solve for [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Check that these [latex]x[\/latex]-values don&#8217;t make the denominator zero<\/li>\n<li class=\"whitespace-normal break-words\">The [latex]x[\/latex]-intercepts are [latex](x, 0)[\/latex] for each valid solution<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the vertical and horizontal asymptotes of the function:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{\\left(2x - 1\\right)\\left(2x+1\\right)}{\\left(x - 2\\right)\\left(x+3\\right)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q5590\">Show Solution<\/button><\/p>\n<div id=\"q5590\" class=\"hidden-answer\" style=\"display: none\">\n<p>Vertical asymptotes at [latex]x=2[\/latex] and [latex]x=-3[\/latex]; horizontal asymptote at [latex]y=4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch this video to see more worked examples of determining which kind of horizontal asymptote a rational function will have.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-baeccgdf-A1tApZSE8nI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/A1tApZSE8nI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-baeccgdf-A1tApZSE8nI\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847034&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-baeccgdf-A1tApZSE8nI&amp;vembed=0&amp;video_id=A1tApZSE8nI&amp;video_target=tpm-plugin-baeccgdf-A1tApZSE8nI\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Determine+Horizontal+Asymptotes+of+Rational+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine Horizontal Asymptotes of Rational Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the following video to see more worked examples of finding asymptotes, intercepts and holes of rational functions.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hgagbdgc-UnVZs2EaEjI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/UnVZs2EaEjI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hgagbdgc-UnVZs2EaEjI\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847035&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hgagbdgc-UnVZs2EaEjI&amp;vembed=0&amp;video_id=UnVZs2EaEjI&amp;video_target=tpm-plugin-hgagbdgc-UnVZs2EaEjI\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+the+Intercepts%2C+Asymptotes%2C+and+Hole+of+a+Rational+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Intercepts, Asymptotes, and Hole of a Rational Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Graphing Rational Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Numerator and Denominator Roles:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Numerator reveals [latex]x[\/latex]-intercepts<\/li>\n<li class=\"whitespace-normal break-words\">Denominator reveals vertical asymptotes<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Effect of Factor Powers:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factors with powers [latex]> 1[\/latex] affect graph shape at intercepts<\/li>\n<li class=\"whitespace-normal break-words\">Odd-degree factors in denominator: opposite behavior on either side of asymptote<\/li>\n<li class=\"whitespace-normal break-words\">Even-degree factors in denominator: same behavior on both sides of asymptote<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Asymptote Behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Odd-degree factor: graph approaches positive infinity on one side, negative infinity on the other<\/li>\n<li class=\"whitespace-normal break-words\">Even-degree factor: graph approaches either positive or negative infinity on both sides<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Step-by-Step Graphing Process<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Simplify the Function:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factor and reduce common terms in numerator and denominator<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Find the Domain:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify [latex]x[\/latex]-values where denominator equals zero<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Identify Vertical Asymptotes and Holes:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set denominator to zero and solve for [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Check if numerator is also zero at these [latex]x[\/latex]-values (indicates holes, not asymptotes)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Find Horizontal or Slant Asymptote:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Compare degrees of numerator ([latex]n[\/latex]) and denominator ([latex]d[\/latex]):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]n < d[\/latex]: Horizontal asymptote at [latex]y = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]n = d[\/latex]: Horizontal asymptote at [latex]y[\/latex] = ratio of leading coefficients<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]n = d + 1[\/latex]: Slant asymptote (find using polynomial long division)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Find Intercepts:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts: Set numerator to zero, solve for [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept: Evaluate function at [latex]x = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Plot Key Points:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Calculate points on both sides of vertical asymptotes<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Sketch the Graph:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Draw asymptotes (dashed lines)<\/li>\n<li class=\"whitespace-normal break-words\">Plot intercepts and key points<\/li>\n<li class=\"whitespace-normal break-words\">Draw curve, ensuring proper asymptote behavior<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]f\\left(x\\right)=\\dfrac{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}{2{\\left(x - 1\\right)}^{2}\\left(x - 3\\right)}[\/latex], use the characteristics of polynomials and rational functions to describe its behavior and sketch the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q451047\">Show Solution<\/button><\/p>\n<div id=\"q451047\" class=\"hidden-answer\" style=\"display: none\">Horizontal asymptote at [latex]y=\\frac{1}{2}[\/latex]. Vertical asymptotes at [latex]x=1[\/latex] and [latex]x=3[\/latex]. [latex]y[\/latex]-intercept at [latex]\\left(0,\\frac{4}{3}.\\right)[\/latex]<\/p>\n<p id=\"fs-id1165135168380\"><em>x<\/em>-intercepts at [latex]\\left(2,0\\right) \\text{ and }\\left(-2,0\\right)[\/latex]. [latex]\\left(-2,0\\right)[\/latex] is a zero with multiplicity 2, and the graph bounces off the [latex]x[\/latex]-axis at this point. [latex]\\left(2,0\\right)[\/latex] is a single zero and the graph crosses the axis at this point.<\/p>\n<figure id=\"attachment_2947\" aria-describedby=\"caption-attachment-2947\" style=\"width: 731px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/02213511\/CNX_Precalc_Figure_03_07_023.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2947 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/12\/02213511\/CNX_Precalc_Figure_03_07_023.jpg\" alt=\"cnx_precalc_figure_03_07_023\" width=\"731\" height=\"477\" \/><\/a><figcaption id=\"caption-attachment-2947\" class=\"wp-caption-text\">Graph of f(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video to see another worked example of how to match different kinds of rational functions with their graphs.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dfffeeea-vMVYaFptvkk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/vMVYaFptvkk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dfffeeea-vMVYaFptvkk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847036&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dfffeeea-vMVYaFptvkk&amp;vembed=0&amp;video_id=vMVYaFptvkk&amp;video_target=tpm-plugin-dfffeeea-vMVYaFptvkk\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Match+Equations+of+Rational+Functions+to+Graphs_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Match Equations of Rational Functions to Graphs\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Writing Rational Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Relationship between graph features and function components:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts correspond to factors in the numerator<\/li>\n<li class=\"whitespace-normal break-words\">Vertical asymptotes correspond to factors in the denominator<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General form of a rational function: [latex]f(x) = a\\frac{(x-x_1)^{p_1}(x-x_2)^{p_2}\\cdots(x-x_n)^{p_n}}{(x-v_1)^{q_1}(x-v_2)^{q_2}\\cdots(x-v_m)^{q_m}}[\/latex] Where:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x_i[\/latex] are [latex]x[\/latex]-intercepts<\/li>\n<li class=\"whitespace-normal break-words\">[latex]v_j[\/latex] are vertical asymptotes<\/li>\n<li class=\"whitespace-normal break-words\">[latex]p_i[\/latex] and [latex]q_j[\/latex] are multiplicities<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex] is the stretch factor<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Behavior at intercepts and asymptotes determines multiplicities<\/li>\n<li class=\"whitespace-normal break-words\">Any point on the graph satisfies the function equation<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Step-by-Step Process<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Identify [latex]x[\/latex]-intercepts:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Locate where graph crosses [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Determine multiplicity based on behavior (passing through or bouncing)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Identify vertical asymptotes:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Locate where graph approaches infinity<\/li>\n<li class=\"whitespace-normal break-words\">Determine multiplicity based on behavior on each side<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Write the general form:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use [latex]x[\/latex]-intercepts for numerator factors<\/li>\n<li class=\"whitespace-normal break-words\">Use vertical asymptotes for denominator factors<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Determine stretch factor [latex]a[\/latex]:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use a clear point on the graph, often the [latex]y[\/latex]-intercept<\/li>\n<li class=\"whitespace-normal break-words\">Substitute point into the function and solve for [latex]a[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aabhhfhh-VU_csSy4BqM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/VU_csSy4BqM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aabhhfhh-VU_csSy4BqM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12847037&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aabhhfhh-VU_csSy4BqM&amp;vembed=0&amp;video_id=VU_csSy4BqM&amp;video_target=tpm-plugin-aabhhfhh-VU_csSy4BqM\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Find+the+Equation+of+Rational+Function+From+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Find the Equation of Rational Function From a Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: The Domain of Rational Functions\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/v0IhvIzCc_I\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Determine Horizontal Asymptotes of Rational Functions\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/A1tApZSE8nI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Find the Intercepts, Asymptotes, and Hole of a Rational Function\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/UnVZs2EaEjI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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