{"id":1917,"date":"2024-06-24T21:22:32","date_gmt":"2024-06-24T21:22:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1917"},"modified":"2025-08-15T02:30:56","modified_gmt":"2025-08-15T02:30:56","slug":"graphs-of-polynomial-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphs-of-polynomial-functions-learn-it-1\/","title":{"raw":"Graphs of Polynomial Functions: Learn It 1","rendered":"Graphs of Polynomial Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Identify key features like zeros, turning points, and end behavior in graphs of polynomial functions<\/li>\r\n \t<li>Find where polynomial functions equal zero using different methods, and understand what these zeros mean<\/li>\r\n \t<li>Create and explain graphs of polynomial functions, connecting how the function is written to what its graph looks like<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Recognizing Characteristics of Graphs of Polynomial Functions<\/h2>\r\nPolynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. The figure below shows a graph that represents a\u00a0<span id=\"term-00002\" class=\"no-emphasis\" data-type=\"term\">polynomial function<\/span>\u00a0and a graph that represents a function that is not a polynomial.\r\n\r\n[caption id=\"attachment_4279\" align=\"aligncenter\" width=\"761\"]<img class=\"wp-image-4279 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140655\/f9fba078e008719dba07a824f3eae23ec230ecf0.jpg\" alt=\"Graph of f(x)=x^3-0.01x.\" width=\"761\" height=\"372\" \/> Graph demonstrating characteristics of a polynomial[\/caption]\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Which of the graphs below represents a polynomial function?\r\n\r\n[caption id=\"attachment_4280\" align=\"aligncenter\" width=\"767\"]<img class=\"wp-image-4280 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140747\/52f0f965ff312bd15d54200bb4a8921ede64e108.jpg\" alt=\"Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.\" width=\"767\" height=\"770\" \/> Graphs showing four functions[\/caption]\r\n\r\n[reveal-answer q=\"888503\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"888503\"]The graphs of [latex]f [\/latex] and [latex]h[\/latex] are graphs of polynomial functions. They are smooth and continuous.The graphs of [latex]g[\/latex] and [latex]k[\/latex] are graphs of functions that are not polynomials. The graph of function [latex]g[\/latex] has a sharp corner. The graph of function [latex]k[\/latex] is not continuous.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p id=\"fs-id1165135496631\"><strong>Do all polynomial functions have as their domain all real numbers?<\/strong><\/p>\r\n\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165134342693\"><em data-effect=\"italics\">Yes. Any real number is a valid input for a polynomial function.<\/em><\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]290936[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify key features like zeros, turning points, and end behavior in graphs of polynomial functions<\/li>\n<li>Find where polynomial functions equal zero using different methods, and understand what these zeros mean<\/li>\n<li>Create and explain graphs of polynomial functions, connecting how the function is written to what its graph looks like<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Recognizing Characteristics of Graphs of Polynomial Functions<\/h2>\n<p>Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. The figure below shows a graph that represents a\u00a0<span id=\"term-00002\" class=\"no-emphasis\" data-type=\"term\">polynomial function<\/span>\u00a0and a graph that represents a function that is not a polynomial.<\/p>\n<figure id=\"attachment_4279\" aria-describedby=\"caption-attachment-4279\" style=\"width: 761px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4279 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140655\/f9fba078e008719dba07a824f3eae23ec230ecf0.jpg\" alt=\"Graph of f(x)=x^3-0.01x.\" width=\"761\" height=\"372\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140655\/f9fba078e008719dba07a824f3eae23ec230ecf0.jpg 761w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140655\/f9fba078e008719dba07a824f3eae23ec230ecf0-300x147.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140655\/f9fba078e008719dba07a824f3eae23ec230ecf0-65x32.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140655\/f9fba078e008719dba07a824f3eae23ec230ecf0-225x110.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140655\/f9fba078e008719dba07a824f3eae23ec230ecf0-350x171.jpg 350w\" sizes=\"(max-width: 761px) 100vw, 761px\" \/><figcaption id=\"caption-attachment-4279\" class=\"wp-caption-text\">Graph demonstrating characteristics of a polynomial<\/figcaption><\/figure>\n<section class=\"textbox example\" aria-label=\"Example\">Which of the graphs below represents a polynomial function?<\/p>\n<figure id=\"attachment_4280\" aria-describedby=\"caption-attachment-4280\" style=\"width: 767px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4280 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140747\/52f0f965ff312bd15d54200bb4a8921ede64e108.jpg\" alt=\"Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.\" width=\"767\" height=\"770\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140747\/52f0f965ff312bd15d54200bb4a8921ede64e108.jpg 767w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140747\/52f0f965ff312bd15d54200bb4a8921ede64e108-300x300.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140747\/52f0f965ff312bd15d54200bb4a8921ede64e108-150x150.jpg 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140747\/52f0f965ff312bd15d54200bb4a8921ede64e108-65x65.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140747\/52f0f965ff312bd15d54200bb4a8921ede64e108-225x226.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30140747\/52f0f965ff312bd15d54200bb4a8921ede64e108-350x351.jpg 350w\" sizes=\"(max-width: 767px) 100vw, 767px\" \/><figcaption id=\"caption-attachment-4280\" class=\"wp-caption-text\">Graphs showing four functions<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q888503\">Show Answer<\/button><\/p>\n<div id=\"q888503\" class=\"hidden-answer\" style=\"display: none\">The graphs of [latex]f[\/latex] and [latex]h[\/latex] are graphs of polynomial functions. They are smooth and continuous.The graphs of [latex]g[\/latex] and [latex]k[\/latex] are graphs of functions that are not polynomials. The graph of function [latex]g[\/latex] has a sharp corner. The graph of function [latex]k[\/latex] is not continuous.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p id=\"fs-id1165135496631\"><strong>Do all polynomial functions have as their domain all real numbers?<\/strong><\/p>\n<hr \/>\n<p id=\"fs-id1165134342693\"><em data-effect=\"italics\">Yes. Any real number is a valid input for a polynomial function.<\/em><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm290936\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=290936&theme=lumen&iframe_resize_id=ohm290936&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":13,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":206,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1917"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1917\/revisions"}],"predecessor-version":[{"id":7752,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1917\/revisions\/7752"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/206"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1917\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1917"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1917"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1917"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1917"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}