{"id":1291,"date":"2015-11-12T18:35:30","date_gmt":"2015-11-12T18:35:30","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1291"},"modified":"2015-11-12T18:35:30","modified_gmt":"2015-11-12T18:35:30","slug":"introduction-to-graphs-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/introduction-to-graphs-of-polynomial-functions\/","title":{"raw":"Introduction to Graphs of Polynomial Functions","rendered":"Introduction to Graphs of Polynomial Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\n<h3>LEARNING OBJECTIVES<\/h3>\nBy the end of this lesson, you will be able to:\n<ul><li>Recognize characteristics of graphs of polynomial functions.<\/li>\n\t<li>Use factoring to \ufb01nd zeros of polynomial functions.<\/li>\n\t<li>Identify zeros and their multiplicities.<\/li>\n\t<li>Determine end behavior.<\/li>\n\t<li>Understand the relationship between degree and turning points.<\/li>\n\t<li>Graph polynomial functions.<\/li>\n\t<li>Use the Intermediate Value Theorem.<\/li>\n<\/ul><\/div>\n<p id=\"fs-id1165135545777\">The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below<b>.<\/b><\/p>\n\n<table id=\"Table_03_04_01\" summary=\"Two rows and nine columns. The first row is labeled,\"><tbody><tr><td><strong>Year<\/strong><\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<td>2013<\/td>\n<\/tr><tr><td><strong>Revenues<\/strong><\/td>\n<td>52.4<\/td>\n<td>52.8<\/td>\n<td>51.2<\/td>\n<td>49.5<\/td>\n<td>48.6<\/td>\n<td>48.6<\/td>\n<td>48.7<\/td>\n<td>47.1<\/td>\n<\/tr><\/tbody><\/table><p id=\"fs-id1165134040487\">The revenue can be modeled by the polynomial function<\/p>\n\n<div id=\"eip-679\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]R\\left(t\\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t - 205.332\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137659450\">where <em>R<\/em>\u00a0represents the revenue in millions of dollars and <em>t<\/em>\u00a0represents the year, with <em>t<\/em> = 6\u00a0corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.<\/p>\n\n<section id=\"fs-id1165135510712\" data-depth=\"1\"\/>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>LEARNING OBJECTIVES<\/h3>\n<p>By the end of this lesson, you will be able to:<\/p>\n<ul>\n<li>Recognize characteristics of graphs of polynomial functions.<\/li>\n<li>Use factoring to \ufb01nd zeros of polynomial functions.<\/li>\n<li>Identify zeros and their multiplicities.<\/li>\n<li>Determine end behavior.<\/li>\n<li>Understand the relationship between degree and turning points.<\/li>\n<li>Graph polynomial functions.<\/li>\n<li>Use the Intermediate Value Theorem.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135545777\">The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below<b>.<\/b><\/p>\n<table id=\"Table_03_04_01\" summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<td>2013<\/td>\n<\/tr>\n<tr>\n<td><strong>Revenues<\/strong><\/td>\n<td>52.4<\/td>\n<td>52.8<\/td>\n<td>51.2<\/td>\n<td>49.5<\/td>\n<td>48.6<\/td>\n<td>48.6<\/td>\n<td>48.7<\/td>\n<td>47.1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165134040487\">The revenue can be modeled by the polynomial function<\/p>\n<div id=\"eip-679\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]R\\left(t\\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t - 205.332\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137659450\">where <em>R<\/em>\u00a0represents the revenue in millions of dollars and <em>t<\/em>\u00a0represents the year, with <em>t<\/em> = 6\u00a0corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.<\/p>\n<section id=\"fs-id1165135510712\" data-depth=\"1\"><\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1291\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1291","chapter","type-chapter","status-publish","hentry"],"part":1290,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1291","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1291\/revisions"}],"predecessor-version":[{"id":2408,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1291\/revisions\/2408"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1290"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1291\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=1291"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1291"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1291"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=1291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}