{"id":1887,"date":"2024-06-18T17:48:51","date_gmt":"2024-06-18T17:48:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1887"},"modified":"2025-08-15T02:08:39","modified_gmt":"2025-08-15T02:08:39","slug":"module-9-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-9-background-youll-need-3\/","title":{"raw":"Power and Polynomial Functions: Background You\u2019ll Need 3","rendered":"Power and Polynomial Functions: Background You\u2019ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Identify the key features of quadratic and cubic functions<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Quadratic Function: [latex]y = x^2 [\/latex]<\/h2>\r\n[caption id=\"attachment_16077\" align=\"aligncenter\" width=\"567\"]<a href=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2015\/07\/Screenshot-2023-09-27-105554.png\"><img class=\"wp-image-16077 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/15184110\/Screenshot-2023-09-27-105554.png\" alt=\"Graph of a parabola.\" width=\"567\" height=\"348\" \/><\/a> Graph of f(x)[\/caption]\r\n<ul>\r\n \t<li><strong>Domain and Range<\/strong>\r\n<ul>\r\n \t<li><strong>Domain:<\/strong> All real numbers [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li><strong>Range:<\/strong>\u00a0[latex][0, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Behavior: <\/strong>\r\n<ul>\r\n \t<li>The function is increasing on [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li>The function is decreasing on [latex](-\\infty, 0)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Key Features<\/strong>: The minimum is [latex]y = 0[\/latex], which occurs at [latex]x=0[\/latex]. This minimum point of the parabola is called a <strong>vertex<\/strong>.<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]292837[\/ohm_question]<\/section>\r\n<h2>Cubic Function: [latex]y = x^3[\/latex]<span><\/span><\/h2>\r\n[caption id=\"\" align=\"aligncenter\" width=\"517\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005020\/CNX_Precalc_Figure_01_01_022n.jpg\" alt=\"Graph of f(x) = x^3.\" width=\"517\" height=\"319\" \/> Graph of f(x) = x^3[\/caption]\r\n<ul>\r\n \t<li><strong>Domain and Range<\/strong>\r\n<ul>\r\n \t<li><strong>Domain:<\/strong> All real numbers [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li><strong>Range:<\/strong> All real numbers [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Behavior:\u00a0 <\/strong>The function is increasing on [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]292840[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\">Identify the key features of quadratic and cubic functions<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Quadratic Function: [latex]y = x^2[\/latex]<\/h2>\n<figure id=\"attachment_16077\" aria-describedby=\"caption-attachment-16077\" style=\"width: 567px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.lumenlearning.com\/precalculus\/wp-content\/uploads\/sites\/3675\/2015\/07\/Screenshot-2023-09-27-105554.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-16077 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/15184110\/Screenshot-2023-09-27-105554.png\" alt=\"Graph of a parabola.\" width=\"567\" height=\"348\" \/><\/a><figcaption id=\"caption-attachment-16077\" class=\"wp-caption-text\">Graph of f(x)<\/figcaption><\/figure>\n<ul>\n<li><strong>Domain and Range<\/strong>\n<ul>\n<li><strong>Domain:<\/strong> All real numbers [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li><strong>Range:<\/strong>\u00a0[latex][0, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Behavior: <\/strong>\n<ul>\n<li>The function is increasing on [latex](0, \\infty)[\/latex]<\/li>\n<li>The function is decreasing on [latex](-\\infty, 0)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Key Features<\/strong>: The minimum is [latex]y = 0[\/latex], which occurs at [latex]x=0[\/latex]. This minimum point of the parabola is called a <strong>vertex<\/strong>.<\/li>\n<\/ul>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm292837\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=292837&theme=lumen&iframe_resize_id=ohm292837&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Cubic Function: [latex]y = x^3[\/latex]<span><\/span><\/h2>\n<figure style=\"width: 517px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005020\/CNX_Precalc_Figure_01_01_022n.jpg\" alt=\"Graph of f(x) = x^3.\" width=\"517\" height=\"319\" \/><figcaption class=\"wp-caption-text\">Graph of f(x) = x^3<\/figcaption><\/figure>\n<ul>\n<li><strong>Domain and Range<\/strong>\n<ul>\n<li><strong>Domain:<\/strong> All real numbers [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li><strong>Range:<\/strong> All real numbers [latex](-\\infty, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Behavior:\u00a0 <\/strong>The function is increasing on [latex](-\\infty, \\infty)[\/latex]<\/li>\n<\/ul>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm292840\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=292840&theme=lumen&iframe_resize_id=ohm292840&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":4,"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":"background_you_need","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\/1887"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1887\/revisions"}],"predecessor-version":[{"id":7741,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1887\/revisions\/7741"}],"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\/1887\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1887"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1887"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1887"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1887"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}