{"id":86,"date":"2025-02-13T22:43:36","date_gmt":"2025-02-13T22:43:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/modeling-using-variation\/"},"modified":"2026-01-14T19:31:48","modified_gmt":"2026-01-14T19:31:48","slug":"modeling-using-variation","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/modeling-using-variation\/","title":{"raw":"Variation: Learn It 1","rendered":"Variation: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve direct variation problems.<\/li>\r\n \t<li>Solve inverse variation problems.<\/li>\r\n \t<li>Solve problems involving joint variation.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Direct Variation<\/h2>\r\nDirect variation is a fundamental concept in algebra that describes a linear relationship between two variables. In direct variation, one variable is a constant multiple of another. This concept is often used in real-world situations where two quantities vary in a consistent manner.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>direct variation<\/h3>\r\nIf [latex]x[\/latex]<em>\u00a0<\/em>and [latex]y[\/latex]\u00a0are related by an equation of the form\r\n<p style=\"text-align: center;\">[latex]y=k{x}^{n}[\/latex]<\/p>\r\nthen we say that the relationship is <strong>direct variation<\/strong> and [latex]y[\/latex]\u00a0<strong>varies directly<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex].\r\n<ul>\r\n \t<li>In direct variation relationships, there is a nonzero constant ratio [latex]k=\\dfrac{y}{{x}^{n}}[\/latex], where [latex]k[\/latex]\u00a0is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers [latex]16\\%[\/latex] commission on her sales. Her earnings depend on the amount of her sales.\r\n[latex]\\\\[\/latex]\r\nNicole\u2019s earnings can be found by multiplying her sales by her commission. The formula [latex]e = 0.16s[\/latex] tells us her earnings, [latex]e[\/latex], come from the product of [latex]0.16[\/latex], her commission, and the sale price of the vehicle, [latex]s[\/latex].\r\n[latex]\\\\[\/latex]\r\nIf we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.\r\n<table style=\"width: 104%;\" summary=\"..\">\r\n<thead>\r\n<tr>\r\n<th style=\"width: 14.4781%;\">[latex]s[\/latex], sales prices<\/th>\r\n<th style=\"width: 31.6498%;\">[latex]e = 0.16s[\/latex]<\/th>\r\n<th style=\"width: 52.5253%;\">Interpretation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 14.4781%;\">[latex]$4,600[\/latex]<\/td>\r\n<td style=\"width: 31.6498%;\">[latex]e=0.16(4,600)=736[\/latex]<\/td>\r\n<td style=\"width: 52.5253%;\">A sale of a [latex]$4,600[\/latex] vehicle results in [latex]$736[\/latex] earnings.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 14.4781%;\">[latex]$9,200[\/latex]<\/td>\r\n<td style=\"width: 31.6498%;\">[latex]e=0.16(9,200)=1,472[\/latex]<\/td>\r\n<td style=\"width: 52.5253%;\">A sale of a [latex]$9,200[\/latex] vehicle results in [latex]$1472[\/latex] earnings.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 14.4781%;\">[latex]$18,400[\/latex]<\/td>\r\n<td style=\"width: 31.6498%;\">[latex]e=0.16(18,400)=2,944[\/latex]<\/td>\r\n<td style=\"width: 52.5253%;\">A sale of a [latex]$18,400[\/latex] vehicle results in [latex]$2944[\/latex] earnings.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222950\/CNX_Precalc_Figure_03_09_0012.jpg\" alt=\"Graph of y=(0.16)x where the horizontal axis is labeled,\" width=\"350\" height=\"330\" \/>\r\n\r\nNotice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from [latex]$4,600[\/latex] to [latex]$9,200[\/latex], and we double the earnings from [latex]$736[\/latex] to [latex]$1,472[\/latex].\r\n\r\nAs the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called <strong>direct variation<\/strong>. Each variable in this type of relationship <strong>varies directly <\/strong>with the other.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a description of a direct variation problem, solve for an unknown.\r\n<\/strong>\r\n<ol id=\"fs-id1165137724401\">\r\n \t<li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\r\n \t<li>Determine the constant of variation. You may need to divide [latex]y[\/latex]\u00a0by the specified power of [latex]x[\/latex]\u00a0to determine the constant of variation.<\/li>\r\n \t<li>Use the constant of variation to write an equation for the relationship.<\/li>\r\n \t<li>Substitute known values into the equation to find the unknown.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">The quantity [latex]y[\/latex]\u00a0varies directly with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]\u00a0when [latex]x=2[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is [latex]6[\/latex].[reveal-answer q=\"647220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"647220\"]The general formula for direct variation with a cube is [latex]y=k{x}^{3}[\/latex]. The constant can be found by dividing [latex]y[\/latex]\u00a0by the cube of [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align} k&amp;=\\dfrac{y}{{x}^{3}} \\\\[1mm] &amp;=\\dfrac{25}{{2}^{3}}\\\\[1mm] &amp;=\\dfrac{25}{8}\\end{align}[\/latex]<\/p>\r\nNow use the constant to write an equation that represents this relationship.\r\n<p style=\"text-align: center;\">[latex]y=\\dfrac{25}{8}{x}^{3}[\/latex]<\/p>\r\nSubstitute [latex]x=6[\/latex]\u00a0and solve for [latex]y[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\dfrac{25}{8}{\\left(6\\right)}^{3} \\\\[1mm] &amp;=675\\hfill \\end{align}[\/latex]<\/p>\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nThe graph of this equation is a simple cubic, as shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222952\/CNX_Precalc_Figure_03_09_0022.jpg\" alt=\"Graph of y=25\/8(x^3) with the labeled points (2, 25) and (6, 675).\" width=\"399\" height=\"301\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Do the graphs of all direct variation equations look like the example above?\r\n<\/strong>\r\n\r\n<hr \/>\r\n\r\nNo. Direct variation equations are power functions\u2014they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through [latex](0, 0)[\/latex].\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]318957[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]318958[\/ohm_question]<\/section><\/div>\r\n<section id=\"fs-id1165137419773\" class=\"key-concepts\">\r\n<dl id=\"fs-id1165137432958\" class=\"definition\">\r\n \t<dd id=\"fs-id1165135439853\"><\/dd>\r\n<\/dl>\r\n<\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve direct variation problems.<\/li>\n<li>Solve inverse variation problems.<\/li>\n<li>Solve problems involving joint variation.<\/li>\n<\/ul>\n<\/section>\n<h2>Direct Variation<\/h2>\n<p>Direct variation is a fundamental concept in algebra that describes a linear relationship between two variables. In direct variation, one variable is a constant multiple of another. This concept is often used in real-world situations where two quantities vary in a consistent manner.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>direct variation<\/h3>\n<p>If [latex]x[\/latex]<em>\u00a0<\/em>and [latex]y[\/latex]\u00a0are related by an equation of the form<\/p>\n<p style=\"text-align: center;\">[latex]y=k{x}^{n}[\/latex]<\/p>\n<p>then we say that the relationship is <strong>direct variation<\/strong> and [latex]y[\/latex]\u00a0<strong>varies directly<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex].<\/p>\n<ul>\n<li>In direct variation relationships, there is a nonzero constant ratio [latex]k=\\dfrac{y}{{x}^{n}}[\/latex], where [latex]k[\/latex]\u00a0is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers [latex]16\\%[\/latex] commission on her sales. Her earnings depend on the amount of her sales.<br \/>\n[latex]\\\\[\/latex]<br \/>\nNicole\u2019s earnings can be found by multiplying her sales by her commission. The formula [latex]e = 0.16s[\/latex] tells us her earnings, [latex]e[\/latex], come from the product of [latex]0.16[\/latex], her commission, and the sale price of the vehicle, [latex]s[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\nIf we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.<\/p>\n<table style=\"width: 104%;\" summary=\"..\">\n<thead>\n<tr>\n<th style=\"width: 14.4781%;\">[latex]s[\/latex], sales prices<\/th>\n<th style=\"width: 31.6498%;\">[latex]e = 0.16s[\/latex]<\/th>\n<th style=\"width: 52.5253%;\">Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 14.4781%;\">[latex]$4,600[\/latex]<\/td>\n<td style=\"width: 31.6498%;\">[latex]e=0.16(4,600)=736[\/latex]<\/td>\n<td style=\"width: 52.5253%;\">A sale of a [latex]$4,600[\/latex] vehicle results in [latex]$736[\/latex] earnings.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.4781%;\">[latex]$9,200[\/latex]<\/td>\n<td style=\"width: 31.6498%;\">[latex]e=0.16(9,200)=1,472[\/latex]<\/td>\n<td style=\"width: 52.5253%;\">A sale of a [latex]$9,200[\/latex] vehicle results in [latex]$1472[\/latex] earnings.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.4781%;\">[latex]$18,400[\/latex]<\/td>\n<td style=\"width: 31.6498%;\">[latex]e=0.16(18,400)=2,944[\/latex]<\/td>\n<td style=\"width: 52.5253%;\">A sale of a [latex]$18,400[\/latex] vehicle results in [latex]$2944[\/latex] earnings.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222950\/CNX_Precalc_Figure_03_09_0012.jpg\" alt=\"Graph of y=(0.16)x where the horizontal axis is labeled,\" width=\"350\" height=\"330\" \/><\/p>\n<p>Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from [latex]$4,600[\/latex] to [latex]$9,200[\/latex], and we double the earnings from [latex]$736[\/latex] to [latex]$1,472[\/latex].<\/p>\n<p>As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called <strong>direct variation<\/strong>. Each variable in this type of relationship <strong>varies directly <\/strong>with the other.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a description of a direct variation problem, solve for an unknown.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1165137724401\">\n<li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\n<li>Determine the constant of variation. You may need to divide [latex]y[\/latex]\u00a0by the specified power of [latex]x[\/latex]\u00a0to determine the constant of variation.<\/li>\n<li>Use the constant of variation to write an equation for the relationship.<\/li>\n<li>Substitute known values into the equation to find the unknown.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">The quantity [latex]y[\/latex]\u00a0varies directly with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]\u00a0when [latex]x=2[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is [latex]6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q647220\">Show Solution<\/button><\/p>\n<div id=\"q647220\" class=\"hidden-answer\" style=\"display: none\">The general formula for direct variation with a cube is [latex]y=k{x}^{3}[\/latex]. The constant can be found by dividing [latex]y[\/latex]\u00a0by the cube of [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} k&=\\dfrac{y}{{x}^{3}} \\\\[1mm] &=\\dfrac{25}{{2}^{3}}\\\\[1mm] &=\\dfrac{25}{8}\\end{align}[\/latex]<\/p>\n<p>Now use the constant to write an equation that represents this relationship.<\/p>\n<p style=\"text-align: center;\">[latex]y=\\dfrac{25}{8}{x}^{3}[\/latex]<\/p>\n<p>Substitute [latex]x=6[\/latex]\u00a0and solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&=\\dfrac{25}{8}{\\left(6\\right)}^{3} \\\\[1mm] &=675\\hfill \\end{align}[\/latex]<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>The graph of this equation is a simple cubic, as shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222952\/CNX_Precalc_Figure_03_09_0022.jpg\" alt=\"Graph of y=25\/8(x^3) with the labeled points (2, 25) and (6, 675).\" width=\"399\" height=\"301\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Do the graphs of all direct variation equations look like the example above?<br \/>\n<\/strong><\/p>\n<hr \/>\n<p>No. Direct variation equations are power functions\u2014they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through [latex](0, 0)[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm318957\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318957&theme=lumen&iframe_resize_id=ohm318957&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm318958\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318958&theme=lumen&iframe_resize_id=ohm318958&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<section id=\"fs-id1165137419773\" class=\"key-concepts\">\n<dl id=\"fs-id1165137432958\" class=\"definition\">\n<dd id=\"fs-id1165135439853\"><\/dd>\n<\/dl>\n<\/section>\n","protected":false},"author":6,"menu_order":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":508,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/86"}],"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\/6"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/86\/revisions"}],"predecessor-version":[{"id":5361,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/86\/revisions\/5361"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/508"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/86\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=86"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=86"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=86"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=86"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}