{"id":236020,"date":"2025-06-16T03:35:20","date_gmt":"2025-06-16T03:35:20","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=236020"},"modified":"2025-06-16T03:35:28","modified_gmt":"2025-06-16T03:35:28","slug":"write-an-equation-for-the-function-graphed-below-y","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/16\/write-an-equation-for-the-function-graphed-below-y\/","title":{"rendered":"Write an equation for the function graphed below y="},"content":{"rendered":"\n

Write an equation for the function graphed below y=<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The equation for the function graphed below is:<\/p>\n\n\n\n

y=6(x+3)(x\u22122)y = \\frac{6}{(x+3)(x-2)}<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

The given function is a rational function with vertical asymptotes at x=\u22123x = -3 and x=2x = 2. Vertical asymptotes appear where the denominator of a rational function is zero, indicating points where the function is undefined. These asymptotes suggest that the denominator of the function must include (x+3)(x\u22122)(x + 3)(x – 2).<\/p>\n\n\n\n

The behavior near the asymptotes follows typical rational function characteristics. As xx approaches \u22123-3 from the left, the function decreases toward negative infinity, while approaching \u22123-3 from the right causes it to increase toward positive infinity. A similar pattern occurs at x=2x = 2. These characteristics confirm that the denominator correctly represents the function\u2019s behavior.<\/p>\n\n\n\n

To determine the numerator, the function’s behavior at a known point is useful. The function crosses the yy-axis at y=\u22121y = -1 when x=0x = 0. Substituting x=0x = 0 into the rational function yields:<\/p>\n\n\n\n

\u22121=k(0+3)(0\u22122)-1 = \\frac{k}{(0+3)(0-2)}<\/p>\n\n\n\n

\u22121=k\u22126-1 = \\frac{k}{-6}<\/p>\n\n\n\n

k=6k = 6<\/p>\n\n\n\n

Thus, the equation is:<\/p>\n\n\n\n

y=6(x+3)(x\u22122)y = \\frac{6}{(x+3)(x-2)}<\/p>\n\n\n\n

This equation accurately captures the function\u2019s key features. The denominator establishes the vertical asymptotes at x=\u22123x = -3 and x=2x = 2. The function also intersects the yy-axis at y=\u22121y = -1, satisfying the graph\u2019s known behavior. The function\u2019s asymptotic behavior confirms the correctness of the rational representation.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

Write an equation for the function graphed below y= The Correct Answer and Explanation is: The equation for the function graphed below is: y=6(x+3)(x\u22122)y = \\frac{6}{(x+3)(x-2)} Explanation: The given function is a rational function with vertical asymptotes at x=\u22123x = -3 and x=2x = 2. Vertical asymptotes appear where the denominator of a rational function […]<\/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-236020","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236020","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=236020"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/236020\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=236020"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=236020"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=236020"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}