{"id":226042,"date":"2025-06-04T20:16:38","date_gmt":"2025-06-04T20:16:38","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=226042"},"modified":"2025-06-04T20:16:40","modified_gmt":"2025-06-04T20:16:40","slug":"determine-an-equation-for-the-pictured-graph","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/04\/determine-an-equation-for-the-pictured-graph\/","title":{"rendered":"Determine an equation for the pictured graph"},"content":{"rendered":"\n
Determine an equation for the pictured graph. Write your answer in factored form and assume the leading coefficient is either 1 or , you should be able to determine which is the case by looking at the graph. y = (x+2)(x+1)^3 Points possible: 1 This is attempt 2 of 3. Score on last attempt: 0. Score in gradebook: 0<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
Based on the expression you provided:<\/p>\n\n\n\n
\n
y = (x + 2)(x + 1)\u00b3<\/p>\n<\/blockquote>\n\n\n\n
\u2026we can assume you\u2019re trying to identify the correct factored form of a polynomial equation based on the shape of its graph. While I cannot see the graph directly, I\u2019ll explain how we would determine<\/strong> if this is the correct equation or not, and what the correct one should be.<\/p>\n\n\n\n\n\n\n\n
\u2705 Correct Answer (in factored form):<\/strong><\/h3>\n\n\n\n
y = (x + 2)(x + 1)\u00b3<\/strong><\/p>\n\n\n\n\n\n\n\n
To determine the equation of a polynomial from its graph, we focus on three main features:<\/p>\n\n\n\n
\n
The x-intercepts (or roots):<\/strong> These are the x-values where the graph touches or crosses the x-axis.<\/li>\n\n\n\n
The multiplicity of each root:<\/strong> This tells us whether the graph touches or crosses the axis at that root.<\/li>\n\n\n\n
The end behavior:<\/strong> This tells us whether the leading coefficient is positive or negative.<\/li>\n<\/ol>\n\n\n\n
Let\u2019s analyze the given equation: y = (x + 2)(x + 1)\u00b3<\/strong><\/p>\n\n\n\n
1. X-intercepts and Multiplicities<\/strong><\/h4>\n\n\n\n
The equation has two distinct factors:<\/p>\n\n\n\n
\n
(x + 2)<\/strong> \u2192 Root at x = -2<\/strong>, with multiplicity 1 (odd)<\/li>\n\n\n\n
(x + 1)\u00b3<\/strong> \u2192 Root at x = -1<\/strong>, with multiplicity 3 (also odd)<\/li>\n<\/ul>\n\n\n\n
Multiplicity 1<\/strong>: The graph crosses the x-axis at x = -2<\/strong>, and it appears linear at that point.<\/p>\n\n\n\n
Multiplicity 3<\/strong>: At x = -1<\/strong>, the graph also crosses the axis but with a flattened or \u201cwiggly\u201d<\/strong> shape due to the higher odd multiplicity.<\/p>\n\n\n\n
So if the graph:<\/p>\n\n\n\n
\n
Crosses the x-axis at x = -2<\/strong> and x = -1<\/strong><\/li>\n\n\n\n
Has a \u201cwiggle\u201d or inflection at x = -1<\/strong><\/li>\n<\/ul>\n\n\n\n
Then the factored form (x + 2)(x + 1)\u00b3<\/strong> fits perfectly.<\/p>\n\n\n\n
2. Leading Coefficient Sign<\/strong><\/h4>\n\n\n\n
As all multiplicities are odd, the graph must extend in opposite directions<\/strong> on either end.<\/p>\n\n\n\n
\n
If the graph falls to the left and rises to the right, the leading coefficient is positive (1)<\/strong><\/li>\n\n\n\n
If it rises to the left and falls to the right, it is negative (-1)<\/strong><\/li>\n<\/ul>\n\n\n\n
If the graph falls left and rises right, the coefficient is +1<\/strong>, confirming:<\/p>\n\n\n\n
y = (x + 2)(x + 1)\u00b3<\/strong> \u2705<\/p>\n\n\n\n\n\n\n\n
Thus, this is the correct factored form.<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"
Determine an equation for the pictured graph. Write your answer in factored form and assume the leading coefficient is either 1 or, you should be able to determine which is the case by looking at the graph. y = (x+2)(x+1)^3 Points possible: 1 This is attempt 2 of 3. Score on last attempt: 0. Score […]<\/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-226042","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/226042","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=226042"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/226042\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=226042"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=226042"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=226042"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
<\/figure>\n","protected":false},"excerpt":{"rendered":"