{"id":319,"date":"2019-07-15T22:44:10","date_gmt":"2019-07-15T22:44:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/direct-variation\/"},"modified":"2019-07-15T22:44:10","modified_gmt":"2019-07-15T22:44:10","slug":"direct-variation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/direct-variation\/","title":{"raw":"Direct Variation","rendered":"Direct Variation"},"content":{"raw":"\n<div class=\"textbox learning-objectives\"><h3>Learning Outcomes<\/h3><ul><li>Solve a direct variation problem<\/li><li>Use a constant of variation to describe the relationship between two variables<\/li><\/ul><\/div>A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section we will look at relationships, such as this one, between earnings, sales, and commission rate.\n\nIn the example above, Nicole\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 0.16, her commission, and the sale price of the vehicle, [latex]s[\/latex]. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.\n\n<table summary=\"..\"><thead><tr><th>[latex]s[\/latex], sales prices<\/th><th>[latex]e = 0.16s[\/latex]<\/th><th>Interpretation<\/th><\/tr><\/thead><tbody><tr><td>$4,600<\/td><td>[latex]e=0.16(4,600)=736[\/latex]<\/td><td>A sale of a $4,600 vehicle results in $736 earnings.<\/td><\/tr><tr><td>$9,200<\/td><td>[latex]e=0.16(9,200)=1,472[\/latex]<\/td><td>A sale of a $9,200 vehicle results in $1472 earnings.<\/td><\/tr><tr><td>$18,400<\/td><td>[latex]e=0.16(18,400)=2,944[\/latex]<\/td><td>A sale of a $18,400 vehicle results in $2944 earnings.<\/td><\/tr><\/tbody><\/table>Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. 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.\n\nThe graph below&nbsp;represents the data for Nicole\u2019s potential earnings. We say that earnings vary directly with the sales price of the car. The formula [latex]y=k{x}^{n}[\/latex] is used for direct variation. The value [latex]k[\/latex] is a nonzero constant greater than zero and is called the <strong>constant of variation<\/strong>. In this case, [latex]k=0.16[\/latex]&nbsp;and [latex]n=1[\/latex].\n\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=\"487\" height=\"459\">\n\n<div class=\"textbox\"><h3>A General Note: Direct Variation<\/h3>If [latex]x[\/latex]<em>&nbsp;<\/em>and [latex]y[\/latex]&nbsp;are related by an equation of the form\n\n<p style=\"text-align: center\">[latex]y=k{x}^{n}[\/latex]\n\nthen we say that the relationship is <strong>direct variation<\/strong> and [latex]y[\/latex]&nbsp;<strong>varies directly<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex]. In direct variation relationships, there is a nonzero constant ratio [latex]k=\\dfrac{y}{{x}^{n}}[\/latex], where [latex]k[\/latex]&nbsp;is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.\n\n<\/p><\/div><div class=\"textbox examples\"><h3>recall isolating a variable in a formula<\/h3>We've learned to solve certain formulas for one of the variables. For example, in the formula that relates distance, rate, and time, [latex]d=rt[\/latex], we can solve the equation for rate, [latex]r=\\dfrac{d}{t}[\/latex] or for time, [latex]t=\\dfrac{d}{r}[\/latex]. We say we are&nbsp;<em>isolating&nbsp;<\/em>the variable of interest in these cases.\n\nThe same idea applies when solving a direct variation problem for the constant of variation, [latex]k[\/latex]. Given a direct variation such as [latex]y=kx^n[\/latex], we can <em>isolate<\/em> the constant of variation using the properties of equality to obtain [latex]\\dfrac{y}{x^n}=k[\/latex].\n\n<\/div><div class=\"textbox\"><h3>How To: Given a description of a direct variation problem, solve for an unknown.<strong>\n<\/strong><\/h3><ol id=\"fs-id1165137724401\"><li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li><li>Determine the constant of variation. You may need to divide [latex]y[\/latex]&nbsp;by the specified power of [latex]x[\/latex]&nbsp;to determine the constant of variation.<\/li><li>Use the constant of variation to write an equation for the relationship.<\/li><li>Substitute known values into the equation to find the unknown.<\/li><\/ol><\/div><div class=\"textbox exercises\"><h3>Example: Solving a Direct Variation Problem<\/h3>The quantity [latex]y[\/latex]&nbsp;varies directly with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]&nbsp;when [latex]x=2[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 6.\n\n[reveal-answer q=\"647220\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"647220\"]\n\nThe 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]&nbsp;by the cube of [latex]x[\/latex].\n\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]\n\nNow use the constant to write an equation that represents this relationship.\n\n<\/p><p style=\"text-align: center\">[latex]y=\\dfrac{25}{8}{x}^{3}[\/latex]\n\nSubstitute [latex]x=6[\/latex]&nbsp;and solve for [latex]y[\/latex].\n\n<\/p><p style=\"text-align: center\">[latex]\\begin{align}y&amp;=\\dfrac{25}{8}{\\left(6\\right)}^{3} \\\\[1mm] &amp;=675\\hfill \\end{align}[\/latex]\n\n<\/p><h4>Analysis of the Solution<\/h4>The graph of this equation is a simple cubic, as shown below.\n\n&nbsp;\n\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=\"487\" height=\"367\">\n\n[\/hidden-answer]\n\n<\/div><div class=\"textbox\"><h3>Q &amp; A<\/h3><strong>Do the graphs of all direct variation equations look like the example above?<\/strong>\n\n<em>No. Direct variation equations are power functions\u2014they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through <\/em>[latex](0, 0)[\/latex]<em>.<\/em>\n\n<\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>The quantity [latex]y[\/latex]&nbsp;varies directly with the square of [latex]y[\/latex]. If [latex]y=24[\/latex]&nbsp;when [latex]x=3[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 4.\n\n[reveal-answer q=\"536994\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"536994\"]\n\n[latex]\\dfrac{128}{3}[\/latex]\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91391&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\n\n\n\n<\/div>Watch this video to see a quick lesson in direct variation. &nbsp;You will see more worked examples.\n\nhttps:\/\/youtu.be\/plFOq4JaEyI\n\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve a direct variation problem<\/li>\n<li>Use a constant of variation to describe the relationship between two variables<\/li>\n<\/ul>\n<\/div>\n<p>A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section we will look at relationships, such as this one, between earnings, sales, and commission rate.<\/p>\n<p>In the example above, Nicole\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 0.16, her commission, and the sale price of the vehicle, [latex]s[\/latex]. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.<\/p>\n<table summary=\"..\">\n<thead>\n<tr>\n<th>[latex]s[\/latex], sales prices<\/th>\n<th>[latex]e = 0.16s[\/latex]<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>$4,600<\/td>\n<td>[latex]e=0.16(4,600)=736[\/latex]<\/td>\n<td>A sale of a $4,600 vehicle results in $736 earnings.<\/td>\n<\/tr>\n<tr>\n<td>$9,200<\/td>\n<td>[latex]e=0.16(9,200)=1,472[\/latex]<\/td>\n<td>A sale of a $9,200 vehicle results in $1472 earnings.<\/td>\n<\/tr>\n<tr>\n<td>$18,400<\/td>\n<td>[latex]e=0.16(18,400)=2,944[\/latex]<\/td>\n<td>A sale of a $18,400 vehicle results in $2944 earnings.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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 $4,600 to $9,200, and we double the earnings from $736 to $1,472. 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<p>The graph below&nbsp;represents the data for Nicole\u2019s potential earnings. We say that earnings vary directly with the sales price of the car. The formula [latex]y=k{x}^{n}[\/latex] is used for direct variation. The value [latex]k[\/latex] is a nonzero constant greater than zero and is called the <strong>constant of variation<\/strong>. In this case, [latex]k=0.16[\/latex]&nbsp;and [latex]n=1[\/latex].<\/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\/02222950\/CNX_Precalc_Figure_03_09_0012.jpg\" alt=\"Graph of y=(0.16)x where the horizontal axis is labeled,\" width=\"487\" height=\"459\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Direct Variation<\/h3>\n<p>If [latex]x[\/latex]<em>&nbsp;<\/em>and [latex]y[\/latex]&nbsp;are 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]&nbsp;<strong>varies directly<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex]. In direct variation relationships, there is a nonzero constant ratio [latex]k=\\dfrac{y}{{x}^{n}}[\/latex], where [latex]k[\/latex]&nbsp;is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>recall isolating a variable in a formula<\/h3>\n<p>We&#8217;ve learned to solve certain formulas for one of the variables. For example, in the formula that relates distance, rate, and time, [latex]d=rt[\/latex], we can solve the equation for rate, [latex]r=\\dfrac{d}{t}[\/latex] or for time, [latex]t=\\dfrac{d}{r}[\/latex]. We say we are&nbsp;<em>isolating&nbsp;<\/em>the variable of interest in these cases.<\/p>\n<p>The same idea applies when solving a direct variation problem for the constant of variation, [latex]k[\/latex]. Given a direct variation such as [latex]y=kx^n[\/latex], we can <em>isolate<\/em> the constant of variation using the properties of equality to obtain [latex]\\dfrac{y}{x^n}=k[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a description of a direct variation problem, solve for an unknown.<strong><br \/>\n<\/strong><\/h3>\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]&nbsp;by the specified power of [latex]x[\/latex]&nbsp;to 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<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Direct Variation Problem<\/h3>\n<p>The quantity [latex]y[\/latex]&nbsp;varies directly with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]&nbsp;when [latex]x=2[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 6.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q647220\">Show Solution<\/span><\/p>\n<div id=\"q647220\" class=\"hidden-answer\" style=\"display: none\">\n<p>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]&nbsp;by 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]&nbsp;and 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<h4>Analysis of the Solution<\/h4>\n<p>The graph of this equation is a simple cubic, as shown below.<\/p>\n<p>&nbsp;<\/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=\"487\" height=\"367\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Do the graphs of all direct variation equations look like the example above?<\/strong><\/p>\n<p><em>No. Direct variation equations are power functions\u2014they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through <\/em>[latex](0, 0)[\/latex]<em>.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The quantity [latex]y[\/latex]&nbsp;varies directly with the square of [latex]y[\/latex]. If [latex]y=24[\/latex]&nbsp;when [latex]x=3[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 4.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q536994\">Show Solution<\/span><\/p>\n<div id=\"q536994\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{128}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm91391\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91391&#38;theme=oea&#38;iframe_resize_id=ohm91391&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see a quick lesson in direct variation. &nbsp;You will see more worked examples.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Direct Variation Applications\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/plFOq4JaEyI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-319\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 91391. <strong>Authored by<\/strong>: Jenck, Michael. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Direct Variation Applications . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/plFOq4JaEyI\">https:\/\/youtu.be\/plFOq4JaEyI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 91391\",\"author\":\"Jenck, Michael\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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