{"id":233659,"date":"2025-06-13T09:49:08","date_gmt":"2025-06-13T09:49:08","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233659"},"modified":"2025-06-13T09:49:11","modified_gmt":"2025-06-13T09:49:11","slug":"which-of-the-following-rational-functions-is-graphed-below-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/13\/which-of-the-following-rational-functions-is-graphed-below-2\/","title":{"rendered":"&#8216;Which of the following rational functions is graphed below"},"content":{"rendered":"\n<p>&#8216;Which of the following rational functions is graphed below? (x+1) 0 ^ Fx) = (x-1(+2) 2x 0 B Fx) = xx-2) (x1) 0 c Fx) = x(x-2) (x+1) 0 D. Fx) (x+ 2(* &#8211; 2) PREVIOUS&#8217;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/image-434.png\" alt=\"\" class=\"wp-image-233660\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To determine which rational function is represented by the graph, we will analyze the key features of the graph: vertical asymptotes, horizontal asymptotes, and intercepts.<\/p>\n\n\n\n<p><strong>1. Identify the Vertical Asymptotes:<\/strong><br>Vertical asymptotes occur where the function is undefined, which corresponds to the values of&nbsp;<em>x<\/em>&nbsp;that make the denominator of the rational function equal to zero.<br>Looking at the graph, we can see two vertical dashed lines, which represent the vertical asymptotes.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>One asymptote is at\u00a0<strong>x = -2<\/strong>.<\/li>\n\n\n\n<li>The other asymptote is at\u00a0<strong>x = 2<\/strong>.<br>These asymptotes tell us that the denominator of the function must have the factors (x &#8211; (-2)) and (x &#8211; 2), which are\u00a0<strong>(x + 2)<\/strong>\u00a0and\u00a0<strong>(x &#8211; 2)<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p><strong>2. Identify the x-intercept(s):<\/strong><br>An x-intercept is a point where the graph crosses the x-axis. At this point, the value of the function F(x) is 0. This happens when the numerator of the rational function is zero (and the denominator is not).<br>The graph crosses the x-axis at&nbsp;<strong>x = -1<\/strong>.<br>This means the numerator of the function must have a factor of (x &#8211; (-1)), which is&nbsp;<strong>(x + 1)<\/strong>.<\/p>\n\n\n\n<p><strong>3. Identify the Horizontal Asymptote:<\/strong><br>A horizontal asymptote describes the behavior of the function as&nbsp;<em>x<\/em>&nbsp;approaches positive or negative infinity.<br>The graph shows that as&nbsp;<em>x<\/em>&nbsp;gets very large in both the positive and negative directions, the curve approaches the x-axis.<br>Therefore, the horizontal asymptote is the line&nbsp;<strong>y = 0<\/strong>.<br>For a rational function, the horizontal asymptote is y = 0 when the degree of the numerator is less than the degree of the denominator.<\/p>\n\n\n\n<p><strong>4. Assemble the Function and Verify:<\/strong><br>Based on our analysis, the function should have the form:<br>F(x) = (x + 1) \/ [(x + 2)(x &#8211; 2)]<\/p>\n\n\n\n<p>Let&#8217;s check if this fits all our observations:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Vertical Asymptotes:<\/strong>\u00a0The denominator (x+2)(x-2) is zero at x = -2 and x = 2. This matches.<\/li>\n\n\n\n<li><strong>x-intercept:<\/strong>\u00a0The numerator (x+1) is zero at x = -1. This matches.<\/li>\n\n\n\n<li><strong>Horizontal Asymptote:<\/strong>\u00a0The degree of the numerator is 1, and the degree of the denominator is 2. Since 1 &lt; 2, the horizontal asymptote is y = 0. This matches.<\/li>\n\n\n\n<li><strong>y-intercept:<\/strong>\u00a0We can also check the y-intercept. Let x = 0:<br>F(0) = (0 + 1) \/ [(0 + 2)(0 &#8211; 2)] = 1 \/ (2 * -2) = -1\/4.<br>The graph appears to cross the y-axis at (0, -1\/4), which further confirms our function is correct.<\/li>\n<\/ul>\n\n\n\n<p><strong>Conclusion:<\/strong><br>The function F(x) = (x + 1) \/ [(x + 2)(x &#8211; 2)] perfectly matches the graph. Comparing this with the given choices, we find that it corresponds to option D.<\/p>\n\n\n\n<p>The correct answer is&nbsp;<strong>D. F(x) = (x + 1) \/ [(x + 2)(x &#8211; 2)]<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-296.jpeg\" alt=\"\" class=\"wp-image-233661\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>&#8216;Which of the following rational functions is graphed below? (x+1) 0 ^ Fx) = (x-1(+2) 2x 0 B Fx) = xx-2) (x1) 0 c Fx) = x(x-2) (x+1) 0 D. Fx) (x+ 2(* &#8211; 2) PREVIOUS&#8217; The Correct Answer and Explanation is: To determine which rational function is represented by the graph, we will analyze [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-233659","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233659","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=233659"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233659\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233659"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233659"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233659"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}