{"id":236,"date":"2024-10-18T21:13:18","date_gmt":"2024-10-18T21:13:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/polynomial-functions-apply-it-1\/"},"modified":"2024-10-18T21:13:18","modified_gmt":"2024-10-18T21:13:18","slug":"polynomial-functions-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/polynomial-functions-apply-it-1\/","title":{"raw":"Polynomial Functions: Apply It 1","rendered":"Polynomial Functions: Apply It 1"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Recognize polynomial functions, noting their degree and leading coefficient<\/li>\n\t<li>Apply various methods to divide polynomials and locate the zeros of polynomial equations<\/li>\n\t<li>Generate graphs and formulate equations for polynomial functions<\/li>\n\t<li>Explain the end behavior of polynomial functions<\/li>\n<\/ul>\n<\/section>\n<h2>Ride the Polynomial: Roller Coaster Curves and Calculations<\/h2>\n<p>Jordan, a civil engineer, is designing a new roller coaster. The track's shape, characterized by its exhilarating drops and climbs, can be modeled using polynomial functions. The design must prioritize safety while maximizing excitement and adhering to material constraints.<\/p>\n<center><img class=\"aligncenter wp-image-10561\" src=\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/09\/pexels-sandy-torchon-3973555_50-scaled.webp#fixme\" alt=\"A rollercoaster\" width=\"500\" height=\"333\"><\/center><center><\/center>\n<p>&nbsp;<\/p>\n<p>As Jordan begins the design, the initial drop is the first consideration, setting the tone for the entire ride. He sketches the initial drop modeled by the polynomial function [latex]f(x)=\u22120.5x^4+3x^3+x^2[\/latex]. The function not only dictates the steepness and curvature of this descent but also encapsulates the delicate balance between thrill and safety. This leads us to our first task: dissecting this function to understand how its degree and leading coefficient will shape the riders' experience.<\/p>\n<section class=\"textbox tryIt\">\n<p>[ohm2_question hide_question_numbers=1]13889[\/ohm2_question]<\/p>\n<\/section>\n<p>Having explored the initial drop's polynomial and its impact on the thrill factor, we now ascend to the first hill of our roller coaster. This is where we experience the anticipation build-up, a crucial element of the ride's excitement. As we crest the hill, let's shift our focus to the structural integrity and the importance of zeros in the polynomial function that models this ascent.<\/p>\n<p>The first hill following the drop can be modeled by the polynomial function [latex]g(x)=x^3\u22124x^2+4x[\/latex].<\/p>\n<section class=\"textbox tryIt\">\n<p>[ohm2_question hide_question_numbers=1]13892[\/ohm2_question]<\/p>\n<\/section>\n<p>With the zeros of the first hill pinpointed, ensuring a safe and exhilarating peak, we now dip into the smaller bumps that add variety to our ride. These minor undulations are essential for maintaining rider engagement before the next major feature. Let's calculate the vertex of the polynomial function representing this smaller bump, which will tell us the maximum height riders will experience, ensuring it provides excitement while keeping the ride's momentum.<\/p>\n<p>A smaller bump on the track is represented by the polynomial function [latex]h(x)=2x^2\u22128x+6[\/latex].<\/p>\n<section class=\"textbox tryIt\">\n<p>[ohm2_question hide_question_numbers=1]13893[\/ohm2_question]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize polynomial functions, noting their degree and leading coefficient<\/li>\n<li>Apply various methods to divide polynomials and locate the zeros of polynomial equations<\/li>\n<li>Generate graphs and formulate equations for polynomial functions<\/li>\n<li>Explain the end behavior of polynomial functions<\/li>\n<\/ul>\n<\/section>\n<h2>Ride the Polynomial: Roller Coaster Curves and Calculations<\/h2>\n<p>Jordan, a civil engineer, is designing a new roller coaster. The track&#8217;s shape, characterized by its exhilarating drops and climbs, can be modeled using polynomial functions. The design must prioritize safety while maximizing excitement and adhering to material constraints.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10561\" src=\"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-content\/uploads\/sites\/18\/2023\/09\/pexels-sandy-torchon-3973555_50-scaled.webp#fixme\" alt=\"A rollercoaster\" width=\"500\" height=\"333\" \/><\/div>\n<div style=\"text-align: center;\"><\/div>\n<p>&nbsp;<\/p>\n<p>As Jordan begins the design, the initial drop is the first consideration, setting the tone for the entire ride. He sketches the initial drop modeled by the polynomial function [latex]f(x)=\u22120.5x^4+3x^3+x^2[\/latex]. The function not only dictates the steepness and curvature of this descent but also encapsulates the delicate balance between thrill and safety. This leads us to our first task: dissecting this function to understand how its degree and leading coefficient will shape the riders&#8217; experience.<\/p>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13889\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13889&theme=lumen&iframe_resize_id=ohm13889&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>Having explored the initial drop&#8217;s polynomial and its impact on the thrill factor, we now ascend to the first hill of our roller coaster. This is where we experience the anticipation build-up, a crucial element of the ride&#8217;s excitement. As we crest the hill, let&#8217;s shift our focus to the structural integrity and the importance of zeros in the polynomial function that models this ascent.<\/p>\n<p>The first hill following the drop can be modeled by the polynomial function [latex]g(x)=x^3\u22124x^2+4x[\/latex].<\/p>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13892\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13892&theme=lumen&iframe_resize_id=ohm13892&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>With the zeros of the first hill pinpointed, ensuring a safe and exhilarating peak, we now dip into the smaller bumps that add variety to our ride. These minor undulations are essential for maintaining rider engagement before the next major feature. Let&#8217;s calculate the vertex of the polynomial function representing this smaller bump, which will tell us the maximum height riders will experience, ensuring it provides excitement while keeping the ride&#8217;s momentum.<\/p>\n<p>A smaller bump on the track is represented by the polynomial function [latex]h(x)=2x^2\u22128x+6[\/latex].<\/p>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13893\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13893&theme=lumen&iframe_resize_id=ohm13893&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"menu_order":45,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"People Riding The Roller Coaster\",\"author\":\"Sandy Torchon\",\"organization\":\"Pexels\",\"url\":\"https:\/\/www.pexels.com\/photo\/people-riding-the-roller-coaster-3973555\/\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"","content_attributions":[{"type":"cc-attribution","description":"People Riding The Roller Coaster","author":"Sandy Torchon","organization":"Pexels","url":"https:\/\/www.pexels.com\/photo\/people-riding-the-roller-coaster-3973555\/","project":"","license":"cc-by-sa","license_terms":""}],"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\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/236"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/236\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/236\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/media?parent=236"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapter-type?post=236"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/contributor?post=236"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/license?post=236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}