{"id":3126,"date":"2025-08-15T22:08:19","date_gmt":"2025-08-15T22:08:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3126"},"modified":"2026-01-13T16:22:43","modified_gmt":"2026-01-13T16:22:43","slug":"graphs-of-polynominal-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-polynominal-functions-apply-it\/","title":{"raw":"Graphs of Polynominal Functions: Apply It","rendered":"Graphs of Polynominal Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph polynomial functions.<\/li>\r\n \t<li>Use the Intermediate Value Theorem.<\/li>\r\n \t<li>Write the formula for a polynomial function given its graph.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div class=\"h-8\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Audio Waveform Analysis<\/h2>\r\n<p class=\"whitespace-normal break-words\">Sound engineers and audio equipment designers use functions to model complex audio waveforms. By analyzing intercepts, graphing behavior, and using mathematical theorems, they can design better speakers, eliminate unwanted noise, and create high-quality audio equipment.<\/p>\r\n\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">Audio waveforms cross zero displacement multiple times per cycle, and zero crossings (x-intercepts) are crucial for understanding sound quality and eliminating distortion.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311017 [\/ohm_question]<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Graphing Complete Waveform Behavior<\/h3>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311018[\/ohm_question]<\/section>\r\n<p class=\"whitespace-normal break-words\"><strong>Audio Engineering Insight:<\/strong> This waveform shows the string vibrates through 4 complete zero crossings in 4 milliseconds, indicating the fundamental frequency characteristics.<\/p>\r\n\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Writing Waveform Equations from Oscilloscope Data<\/h3>\r\n<p class=\"whitespace-normal break-words\">An audio engineer observes a speaker waveform on an oscilloscope and needs to write its equation.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Use the observed data to write a polynomial model.<\/p>\r\n<p class=\"whitespace-normal break-words\">Observed data:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Zero crossings at [latex]t = -1, 2, 4[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">At [latex]t = -1[\/latex]: graph touches axis and bounces (doesn't cross)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">At [latex]t = 2[\/latex] and [latex]t = 4[\/latex]: graph crosses axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Maximum amplitude: [latex]f(0) = 6[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"627786\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"627786\"]\r\n\r\nFirst, identify the factors from zero crossings\r\n\r\n[latex]t = -1[\/latex] (touches axis) \u2192 [latex](t + 1)^2[\/latex]\r\n[latex]t = 2[\/latex] (crosses axis) \u2192 [latex](t - 2)[\/latex]\r\n[latex]t = 4[\/latex] (crosses axis) \u2192 [latex](t - 4)[\/latex]\r\n\r\nNow, we need to write the general form:\r\n[latex]f(t) = a(t + 1)^2(t - 2)(t - 4)[\/latex]\r\n\r\nFinally, we need to solve for the amplitude coefficient using one of the given points (that aren't one of the x-intercepts). We'll use [latex]f(0)=6[\/latex]\r\n\r\n[latex]\\begin{align} 6 &amp;= a(0 + 1)^2(0 - 2)(0 - 4)\\ 6 &amp;= a(1)(-2)(-4)\\ 6 &amp;= 8a\\ a &amp;= \\frac{3}{4} \\end{align}[\/latex] Answer: The speaker waveform equation is [latex]f(t) = \\frac{3}{4}(t + 1)^2(t - 2)(t - 4)[\/latex][\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph polynomial functions.<\/li>\n<li>Use the Intermediate Value Theorem.<\/li>\n<li>Write the formula for a polynomial function given its graph.<\/li>\n<\/ul>\n<\/section>\n<div class=\"h-8\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Audio Waveform Analysis<\/h2>\n<p class=\"whitespace-normal break-words\">Sound engineers and audio equipment designers use functions to model complex audio waveforms. By analyzing intercepts, graphing behavior, and using mathematical theorems, they can design better speakers, eliminate unwanted noise, and create high-quality audio equipment.<\/p>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">Audio waveforms cross zero displacement multiple times per cycle, and zero crossings (x-intercepts) are crucial for understanding sound quality and eliminating distortion.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311017\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311017&theme=lumen&iframe_resize_id=ohm311017&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Graphing Complete Waveform Behavior<\/h3>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311018\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311018&theme=lumen&iframe_resize_id=ohm311018&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"whitespace-normal break-words\"><strong>Audio Engineering Insight:<\/strong> This waveform shows the string vibrates through 4 complete zero crossings in 4 milliseconds, indicating the fundamental frequency characteristics.<\/p>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Writing Waveform Equations from Oscilloscope Data<\/h3>\n<p class=\"whitespace-normal break-words\">An audio engineer observes a speaker waveform on an oscilloscope and needs to write its equation.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Use the observed data to write a polynomial model.<\/p>\n<p class=\"whitespace-normal break-words\">Observed data:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Zero crossings at [latex]t = -1, 2, 4[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">At [latex]t = -1[\/latex]: graph touches axis and bounces (doesn&#8217;t cross)<\/li>\n<li class=\"whitespace-normal break-words\">At [latex]t = 2[\/latex] and [latex]t = 4[\/latex]: graph crosses axis<\/li>\n<li class=\"whitespace-normal break-words\">Maximum amplitude: [latex]f(0) = 6[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q627786\">Show Solution<\/button><\/p>\n<div id=\"q627786\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, identify the factors from zero crossings<\/p>\n<p>[latex]t = -1[\/latex] (touches axis) \u2192 [latex](t + 1)^2[\/latex]<br \/>\n[latex]t = 2[\/latex] (crosses axis) \u2192 [latex](t - 2)[\/latex]<br \/>\n[latex]t = 4[\/latex] (crosses axis) \u2192 [latex](t - 4)[\/latex]<\/p>\n<p>Now, we need to write the general form:<br \/>\n[latex]f(t) = a(t + 1)^2(t - 2)(t - 4)[\/latex]<\/p>\n<p>Finally, we need to solve for the amplitude coefficient using one of the given points (that aren&#8217;t one of the x-intercepts). We&#8217;ll use [latex]f(0)=6[\/latex]<\/p>\n<p>[latex]\\begin{align} 6 &= a(0 + 1)^2(0 - 2)(0 - 4)\\ 6 &= a(1)(-2)(-4)\\ 6 &= 8a\\ a &= \\frac{3}{4} \\end{align}[\/latex] Answer: The speaker waveform equation is [latex]f(t) = \\frac{3}{4}(t + 1)^2(t - 2)(t - 4)[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"module-header":"apply_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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3126"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3126\/revisions"}],"predecessor-version":[{"id":3600,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3126\/revisions\/3600"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3126\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3126"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3126"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3126"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}