{"id":93,"date":"2019-08-20T17:02:21","date_gmt":"2019-08-20T21:02:21","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/modeling-using-variation\/"},"modified":"2022-06-01T10:39:27","modified_gmt":"2022-06-01T14:39:27","slug":"modeling-using-variation","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/modeling-using-variation\/","title":{"raw":"Modeling Using Variation","rendered":"Modeling Using Variation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Solve direct variation problems.<\/li>\n \t<li>Solve inverse variation problems.<\/li>\n \t<li>Solve problems involving joint variation.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135356540\">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\n<div id=\"fs-id1165137730306\" class=\"bc-section section\">\n<h3>Solving Direct Variation Problems<\/h3>\n<p id=\"fs-id1165137823230\">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. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. See <a class=\"autogenerated-content\" href=\"#Table_03_09_01\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_03_09_01\" summary=\"..\">\n<thead>\n<tr>\n<th>[latex]\\,s\\,[\/latex], sales price<\/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\\left(4,600\\right)=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\\left(9,200\\right)=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\\left(18,400\\right)=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 id=\"fs-id1165135188294\">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 id=\"fs-id1165137937533\"><a class=\"autogenerated-content\" href=\"#Figure_03_09_001\">(Figure)<\/a> 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]and[latex]\\,n=1.\\,[\/latex]We saw functions like this one when we discussed power functions.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/8bfb65f3c8025c36830c5246429f7688276ed66d\/CNX_Precalc_Figure_03_09_001.jpg\" alt=\"Graph of y=(0.16)x where the horizontal axis is labeled, \u201cs, Sales Price in Dollars\u201d, and the vertical axis is labeled, \u201ce, Earnings, $\u201d.\" width=\"487\" height=\"459\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<div id=\"fs-id1165137730075\" class=\"textbox key-takeaways\">\n<h3>Direct Variation<\/h3>\n<p id=\"fs-id1165137827458\">If[latex]\\,x\\,\\text{and}\\,y\\,[\/latex]are related by an equation of the form<\/p>\n\n<div id=\"fs-id1165135437156\">[latex]\\,y=k{x}^{n}\\,[\/latex]<\/div>\n<p id=\"fs-id1165133155266\">then we say that the relationship is direct variation and [latex]\\,y\\,[\/latex] varies directly with, or is proportional to, the[latex]\\,n\\text{th}\\,[\/latex]power of[latex]\\,x.\\,[\/latex]In direct variation relationships, there is a nonzero constant ratio[latex]\\,k=\\frac{y}{{x}^{n}},\\,[\/latex]where[latex]\\,k\\,[\/latex]is called the constant of variation, which help defines the relationship between the variables.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137550958\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137723932\"><strong>Given a description of a direct variation problem, solve for an unknown.\n<\/strong><\/p>\n\n<ol id=\"fs-id1165137724401\" type=\"1\">\n \t<li>Identify the input,[latex]\\,x,[\/latex]and the output,[latex]\\,y.\\,[\/latex]<\/li>\n \t<li>Determine the constant of variation. You may need to divide[latex]\\,y\\,[\/latex]by the specified power of[latex]\\,x\\,[\/latex]to determine the constant of variation.<\/li>\n \t<li>Use the constant of variation to write an equation for the relationship.<\/li>\n \t<li>Substitute known values into the equation to find the unknown.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_09_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137676066\">\n<div id=\"fs-id1165137434564\">\n<h3>Solving a Direct Variation Problem<\/h3>\n<p id=\"fs-id1165137849016\">The quantity[latex]\\,y\\,[\/latex]varies directly with the cube of[latex]\\,x.\\,[\/latex]If[latex]\\,y=25\\,[\/latex]when[latex]\\,x=2,\\,[\/latex]find[latex]\\,y\\,[\/latex]when[latex]\\,x\\,[\/latex]is 6.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137642960\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137642960\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137642960\"]\n<p id=\"fs-id1165137659713\">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]by the cube of[latex]\\,x.\\,[\/latex]<\/p>\n\n<div id=\"eip-id1165134084945\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill k&amp; =&amp; \\frac{y}{{x}^{3}}\\hfill \\\\ &amp; =&amp; \\frac{25}{{2}^{3}}\\hfill \\\\ &amp; =&amp; \\frac{25}{8}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137628102\">Now use the constant to write an equation that represents this relationship.<\/p>\n\n<div id=\"eip-id1165135440091\" class=\"unnumbered\">[latex]y=\\frac{25}{8}{x}^{3}[\/latex]<\/div>\n<p id=\"fs-id1165135432964\">Substitute[latex]\\,x=6\\,[\/latex]and solve for[latex]\\,y.[\/latex]<\/p>\n\n<div id=\"eip-id1165135207297\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill y&amp; =&amp; \\frac{25}{8}{\\left(6\\right)}^{3}\\hfill \\\\ &amp; =&amp; 675\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135533140\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134557390\">The graph of this equation is a simple cubic, as shown in <a class=\"autogenerated-content\" href=\"#Figure_03_09_002\">(Figure)<\/a>.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/b4093a150bd577af57b691e9c4b951e271614aa9\/CNX_Precalc_Figure_03_09_002.jpg\" alt=\"Graph of y=25\/8(x^3) with the labeled points (2, 25) and (6, 675).\" width=\"487\" height=\"367\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736204\" class=\"precalculus qa textbox shaded\">\n<p id=\"eip-id1165137772190\"><strong>Do the graphs of all direct variation equations look like <a class=\"autogenerated-content\" href=\"#Example_03_09_01\">(Figure)<\/a>?<\/strong><\/p>\n<p id=\"fs-id1165137596402\"><em>No. Direct variation equations are power functions\u2014they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through[latex]\\,\\left(0,0\\right).[\/latex]\n<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1165137783766\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_09_01\">\n<div id=\"fs-id1165137645447\">\n<p id=\"fs-id1165135160334\">The quantity[latex]\\,y\\,[\/latex]varies directly with the square of[latex]\\,x.\\,[\/latex]If[latex]\\,y=24\\,[\/latex]when[latex]\\,x=3,\\,[\/latex]find[latex]\\,y\\,[\/latex]when[latex]\\,x\\,[\/latex]is 4.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137659774\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137659774\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137659774\"]\n<p id=\"fs-id1165137558700\">[latex]\\frac{128}{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137758211\" class=\"bc-section section\">\n<h3>Solving Inverse Variation Problems<\/h3>\n<p id=\"fs-id1165137734583\">Water temperature in an ocean varies inversely to the water\u2019s depth. The formula[latex]\\,T=\\frac{14,000}{d}\\,[\/latex]gives us the temperature in degrees Fahrenheit at a depth in feet below Earth\u2019s surface. Consider the Atlantic Ocean, which covers 22% of Earth\u2019s surface. At a certain location, at the depth of 500 feet, the temperature may be 28\u00b0F.<\/p>\n<p id=\"fs-id1165137761800\">If we create <a class=\"autogenerated-content\" href=\"#Table_03_09_02\">(Figure)<\/a>, we observe that, as the depth increases, the water temperature decreases.<\/p>\n\n<table id=\"Table_03_09_02\" summary=\"..\">\n<thead>\n<tr>\n<th>[latex]d,\\,[\/latex]depth<\/th>\n<th>[latex]T=\\frac{\\text{14,000}}{d}[\/latex]<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>500 ft<\/td>\n<td>[latex]\\frac{14,000}{500}=28[\/latex]<\/td>\n<td>At a depth of 500 ft, the water temperature is 28\u00b0 F.<\/td>\n<\/tr>\n<tr>\n<td>1000 ft<\/td>\n<td>[latex]\\frac{14,000}{1000}=14[\/latex]<\/td>\n<td>At a depth of 1,000 ft, the water temperature is 14\u00b0 F.<\/td>\n<\/tr>\n<tr>\n<td>2000 ft<\/td>\n<td>[latex]\\frac{14,000}{2000}=7[\/latex]<\/td>\n<td>At a depth of 2,000 ft, the water temperature is 7\u00b0 F.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137645896\">We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be <strong>inversely proportional<\/strong> and each term <strong>varies inversely<\/strong> with the other. Inversely proportional relationships are also called <strong>inverse variations<\/strong>.<\/p>\n<p id=\"fs-id1165137805474\">For our example, <a class=\"autogenerated-content\" href=\"#Figure_03_09_003\">(Figure)<\/a> depicts the <span class=\"no-emphasis\">inverse variation<\/span>. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula[latex]\\,y=\\frac{k}{x}\\,[\/latex]for inverse variation in this case uses[latex]\\,k=14,000.\\,[\/latex]<span id=\"fs-id1165135161265\"><\/span><\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/9e5691ee3d78b06ea7ff5f12e359208ead427ef2\/CNX_Precalc_Figure_03_09_003.jpg\" alt=\"Graph of y=(14000)\/x where the horizontal axis is labeled, \u201cDepth, d (ft)\u201d, and the vertical axis is labeled, \u201cTemperature, T (Degrees Fahrenheit)\u201d.\" width=\"487\" height=\"312\"> <strong>Figure 3.<\/strong>[\/caption]\n\n<div id=\"fs-id1165135397976\" class=\"textbox key-takeaways\">\n<h3>Inverse Variation<\/h3>\n<p id=\"fs-id1165137536242\">If[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]are related by an equation of the form<\/p>\n\n<div id=\"fs-id1165137571596\">[latex]y=\\frac{k}{{x}^{n}}[\/latex]<\/div>\n<p id=\"fs-id1165137843973\">where[latex]\\,k\\,[\/latex]is a nonzero constant, then we say that[latex]\\,y[\/latex] varies inversely with the[latex]\\,n\\text{th}\\,[\/latex]power of[latex]\\,x.\\,[\/latex]In inversely proportional relationships, or inverse variations, there is a constant multiple[latex]\\,k={x}^{n}y.\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"Example_03_09_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137641735\">\n<div id=\"fs-id1165137658061\">\n<h3>Writing a Formula for an Inversely Proportional Relationship<\/h3>\n<p id=\"fs-id1165131797298\">A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137654947\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137654947\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137654947\"]\n<p id=\"fs-id1165137827766\">Recall that multiplying speed by time gives distance. If we let[latex]\\,t\\,[\/latex]represent the drive time in hours, and[latex]\\,v\\,[\/latex]represent the velocity (speed or rate) at which the tourist drives, then[latex]\\,vt=\\text{distance}\\text{.}\\,[\/latex]Because the distance is fixed at 100 miles,[latex]\\,vt=100\\,[\/latex] so[latex]t=100\/v.\\,[\/latex]Because time is a function of velocity, we can write[latex]\\,t\\left(v\\right).[\/latex]<\/p>\n\n<div id=\"eip-id1165134094568\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill t\\left(v\\right)&amp; =&amp; \\frac{100}{v}\\hfill \\\\ &amp; =&amp; 100{v}^{-1}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137748472\">We can see that the constant of variation is 100 and, although we can write the relationship using the negative exponent, it is more common to see it written as a fraction. We say that time varies inversely with velocity.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135187117\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137677962\"><strong>Given a description of an indirect variation problem, solve for an unknown.\n<\/strong><\/p>\n\n<ol id=\"fs-id1165137662822\" type=\"1\">\n \t<li>Identify the input,[latex]\\,x,\\,[\/latex]and the output,[latex]\\,y.[\/latex]<\/li>\n \t<li>Determine the constant of variation. You may need to multiply[latex]\\,y\\,[\/latex]by the specified power of[latex]\\,x\\,[\/latex]to determine the constant of variation.<\/li>\n \t<li>Use the constant of variation to write an equation for the relationship.<\/li>\n \t<li>Substitute known values into the equation to find the unknown.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_09_03\" class=\"textbox examples\">\n<div id=\"fs-id1165134328944\">\n<div>\n<h3>Solving an Inverse Variation Problem<\/h3>\n<p id=\"fs-id1165135209804\">A quantity[latex]\\,y\\,[\/latex]varies inversely with the cube of[latex]\\,x.\\,[\/latex]If[latex]\\,y=25\\,[\/latex]when[latex]\\,x=2,\\,[\/latex]find[latex]\\,y\\,[\/latex]when[latex]\\,x\\,[\/latex]is 6.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137547532\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137547532\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137547532\"]\n<p id=\"fs-id1165137627457\">The general formula for inverse variation with a cube is[latex]\\,y=\\frac{k}{{x}^{3}}.\\,[\/latex]The constant can be found by multiplying[latex]\\,y\\,[\/latex]by the cube of[latex]\\,x.\\,[\/latex]<\/p>\n\n<div id=\"eip-id1165132213474\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill k&amp; =&amp; {x}^{3}y\\hfill \\\\ &amp; =&amp; {2}^{3}\\cdot 25\\hfill \\\\ &amp; =&amp; 200\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135188786\">Now we use the constant to write an equation that represents this relationship.<\/p>\n\n<div id=\"eip-id1165133333885\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill y&amp; =&amp; \\frac{k}{{x}^{3}},\\,\\,k=200\\hfill \\\\ y\\hfill &amp; =&amp; \\frac{200}{{x}^{3}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137653904\">Substitute[latex]\\,x=6\\,[\/latex]and solve for[latex]\\,y.[\/latex]<\/p>\n\n<div id=\"eip-id1165131878567\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill y&amp; =&amp; \\frac{200}{{6}^{3}}\\hfill \\\\ &amp; =&amp; \\frac{25}{27}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137573081\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137852181\">The graph of this equation is a rational function, as shown in <a class=\"autogenerated-content\" href=\"#Figure_03_09_004\">(Figure)<\/a>.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/cnx.org\/resources\/66fddde286812d980caabaca4f75faa30f86dd71\/CNX_Precalc_Figure_03_09_004.jpg\" alt=\"Graph of y=25\/(x^3) with the labeled points (2, 25) and (6, 25\/27).\" width=\"488\" height=\"292\"> <strong>Figure 4.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135178098\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_09_02\">\n<div id=\"fs-id1165137810877\">\n<p id=\"fs-id1165137810878\">A quantity[latex]\\,y\\,[\/latex]varies inversely with the square of[latex]\\,x.\\,[\/latex]If[latex]\\,y=8\\,[\/latex]when[latex]\\,x=3,\\,[\/latex]find[latex]\\,y\\,[\/latex]when[latex]\\,x\\,[\/latex]is 4.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137683037\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137683037\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137683037\"]\n<p id=\"fs-id1165137683038\">[latex]\\,\\frac{9}{2}\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134054044\" class=\"bc-section section\">\n<h3>Solving Problems Involving Joint Variation<\/h3>\n<p id=\"fs-id1165137558033\">Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable[latex]\\,c,[\/latex]cost, varies jointly with the number of students,[latex]\\,n,[\/latex]and the distance,[latex]\\,d.\\,[\/latex]<\/p>\n\n<div id=\"fs-id1165135177639\" class=\"textbox key-takeaways\">\n<h3>Joint Variation<\/h3>\n<p id=\"fs-id1165135195246\">Joint variation occurs when a variable varies directly or inversely with multiple variables.<\/p>\n<p id=\"fs-id1165137678943\">For instance, if[latex]\\,x\\,[\/latex]varies directly with both[latex]\\,y\\,[\/latex]and[latex]\\,z,\\,[\/latex] we have[latex]\\,x=kyz.\\,[\/latex]If[latex]\\,x\\,[\/latex]varies directly with[latex]\\,y\\,[\/latex]and inversely with[latex]z,[\/latex]we have[latex]\\,x=\\frac{ky}{z}.\\,[\/latex]Notice that we only use one constant in a joint variation equation.<\/p>\n\n<\/div>\n<div id=\"Example_03_09_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137673524\">\n<div id=\"fs-id1165135394333\">\n<h3>Solving Problems Involving Joint Variation<\/h3>\n<p id=\"fs-id1165137452990\">A quantity[latex]\\,x\\,[\/latex]varies directly with the square of[latex]\\,y\\,[\/latex]and inversely with the cube root of[latex]\\,z.\\,[\/latex]If[latex]\\,x=6\\,[\/latex]when[latex]\\,y=2\\,[\/latex] and[latex]\\,z=8,\\,[\/latex]find[latex]\\,x\\,[\/latex]when[latex]\\,y=1\\,[\/latex] and [latex]\\,z=27.\\,[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div id=\"eip-id1165131924522\" class=\"unnumbered\">[reveal-answer q=\"928767\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"928767\"]Begin by writing an equation to show the relationship between the variables.\n<div id=\"eip-id1165133305360\" class=\"unnumbered\">[latex]x=\\frac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/div>\nSubstitute[latex]\\,x=6,\\,[\/latex][latex]y=2,\\,[\/latex]and[latex]\\,z=8\\,[\/latex]to find the value of the constant[latex]\\,k.\\,[\/latex]\n<div id=\"eip-id1165133420152\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill 6&amp; =&amp; \\frac{k{2}^{2}}{\\sqrt[3]{8}}\\hfill \\\\ \\hfill 6&amp; =&amp; \\frac{4k}{2}\\hfill \\\\ \\hfill 3&amp; =&amp; k\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137863719\">Now we can substitute the value of the constant into the equation for the relationship.<\/p>\n\n<div id=\"eip-id1165134503062\" class=\"unnumbered\">[latex]x=\\frac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/div>\n<p id=\"fs-id1165137742401\">To find[latex]\\,x\\,[\/latex]when[latex]\\,y=1\\,[\/latex]and[latex]\\,z=27,\\,[\/latex]we will substitute values for[latex]\\,y\\,[\/latex]and[latex]\\,z\\,[\/latex]into our equation.<\/p>\n\n<div id=\"eip-id1165131924522\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill x&amp; =&amp; \\hfill \\frac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}}\\\\ &amp; =&amp; 1\\hfill \\end{array}[\/latex]<\/div>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135508625\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_09_03\">\n<div id=\"fs-id1165137588085\">\n<p id=\"fs-id1165137588086\">A quantity[latex]\\,x\\,[\/latex]varies directly with the square of[latex]\\,y\\,[\/latex]and inversely with[latex]\\,z.\\,[\/latex]If[latex]\\,x=40\\,[\/latex]when[latex]\\,y=4\\,[\/latex]and[latex]\\,z=2,\\,[\/latex]find[latex]\\,x\\,[\/latex]when[latex]\\,y=10\\,[\/latex]\nand[latex]\\,z=25.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137755838\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137755838\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137755838\"]\n<p id=\"fs-id1165137755840\">[latex]\\,x=20\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137425403\" class=\"precalculus media\">\n<p id=\"fs-id1165137844171\">Access these online resources for additional instruction and practice with direct and inverse variation.<\/p>\n\n<ul id=\"fs-id1165137854996\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/directvariation\">Direct Variation<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/inversevariatio\">Inverse Variation<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/directinverse\">Direct and Inverse Variation<\/a><\/li>\n<\/ul>\n<\/div>\n<p id=\"eip-251\">Visit <a href=\"http:\/\/openstaxcollege.org\/l\/PreCalcLPC03\">this website<\/a> for additional practice questions from Learningpod.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137898092\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165133094986\" summary=\"..\">\n<tbody>\n<tr>\n<td>Direct variation<\/td>\n<td>[latex]y=k{x}^{n},k\\text{ is a nonzero constant}.[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Inverse variation<\/td>\n<td>[latex]y=\\frac{k}{{x}^{n}},k\\text{ is a nonzero constant}.[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137419773\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137723142\">\n \t<li>A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See <a class=\"autogenerated-content\" href=\"#Example_03_09_01\">(Figure)<\/a>.<\/li>\n \t<li>Two variables that are directly proportional to one another will have a constant ratio.<\/li>\n \t<li>A relationship where one quantity is a constant divided by another quantity is called inverse variation. See <a class=\"autogenerated-content\" href=\"#Example_03_09_02\">(Figure)<\/a>.<\/li>\n \t<li>Two variables that are inversely proportional to one another will have a constant multiple. See <a class=\"autogenerated-content\" href=\"#Example_03_09_03\">(Figure)<\/a>.<\/li>\n \t<li>In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See <a class=\"autogenerated-content\" href=\"#Example_03_09_04\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165131958288\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137430832\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165134389954\">\n<div id=\"fs-id1165134389956\">\n<p id=\"fs-id1165137612339\">What is true of the appearance of graphs that reflect a direct variation between two variables?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137759992\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137759992\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137759992\"]\n<p id=\"fs-id1165137646694\">The graph will have the appearance of a power function.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137386819\">\n<div id=\"fs-id1165137386821\">\n<p id=\"fs-id1165137581165\">If two variables vary inversely, what will an equation representing their relationship look like?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137641727\">\n<div id=\"fs-id1165135522169\">\n<p id=\"fs-id1165135522171\">Is there a limit to the number of variables that can vary jointly? Explain.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134192951\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134192951\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134192951\"]\n<p id=\"fs-id1165137598495\">No. Multiple variables may jointly vary.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131959628\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137507782\">For the following exercises, write an equation describing the relationship of the given variables.<\/p>\n\n<div id=\"fs-id1165137611285\">\n<div id=\"fs-id1165137611287\">\n<p id=\"fs-id1165137535129\">[latex]\\,y\\,[\/latex]varies directly as[latex]\\,x\\,[\/latex]and when[latex]\\,x=6, y=12.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176447\">\n<div>\n<p id=\"fs-id1165135250657\">[latex]\\,y\\,[\/latex]varies directly as the square of[latex]\\,x\\,[\/latex]and when[latex]\\,x=4,\\, y=80.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134381645\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134381645\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134381645\"]\n<p id=\"fs-id1165134381647\">[latex]y=5{x}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134108533\">\n<div id=\"fs-id1697066\">\n<p id=\"fs-id1165135332727\">[latex]\\,y\\,[\/latex]varies directly as the square root of[latex]\\,x\\,[\/latex]and when[latex]\\,x=36, \\,y=24.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134192961\">\n<div id=\"fs-id1165135496491\">\n<p id=\"fs-id1165135496493\">[latex]\\,y\\,[\/latex]varies directly as the cube of[latex]\\,x\\,[\/latex]and when[latex]\\,x=36,\\,y=24.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137760761\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137760761\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137760761\"]\n<p id=\"fs-id1165137760763\">[latex]y=10{x}^{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137809885\">\n<div id=\"fs-id1165137679904\">\n<p id=\"fs-id1165137679906\">[latex]\\,y\\,[\/latex]varies directly as the cube root of[latex]\\,x\\,[\/latex]and when[latex]\\,x=27,\\, y=15.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137924366\">\n<div id=\"fs-id1165137924368\">\n<p id=\"fs-id1165137748522\">[latex]\\,y\\,[\/latex]varies directly as the fourth power of[latex]\\,x\\,[\/latex]and when[latex]\\,x=1,\\, y=6.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137667538\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137667538\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137667538\"]\n<p id=\"fs-id1165137740851\">[latex]y=6{x}^{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137465246\">\n<p id=\"fs-id1165137465248\">[latex]\\,y\\,[\/latex]varies inversely as[latex]\\,x\\,[\/latex]and when[latex]\\,x=4,\\, y=2.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447334\">\n<div id=\"fs-id1165137447336\">\n<p id=\"fs-id1165133030852\">[latex]\\,y\\,[\/latex]varies inversely as the square of[latex]\\,x\\,[\/latex]and when[latex]\\,x=3,\\, y=2.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137662476\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137662476\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137662476\"]\n<p id=\"fs-id1165137706986\">[latex]y=\\frac{18}{{x}^{2}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137766904\">\n<div id=\"fs-id1165135528413\">\n<p id=\"fs-id1165135528415\">[latex]\\,y\\,[\/latex]varies inversely as the cube of[latex]\\,x\\,[\/latex]and when[latex]\\,x=2,\\, y=5.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137805102\">\n<div id=\"fs-id1165137805103\">\n<p id=\"fs-id1165137805104\">[latex]\\,y\\,[\/latex]varies inversely as the fourth power of[latex]\\,x\\,[\/latex]and when[latex]\\,x=3,\\, y=1.\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135524553\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135524553\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135524553\"]\n<p id=\"fs-id1165135524556\">[latex]y=\\frac{81}{{x}^{4}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137898971\">\n<div id=\"fs-id1165137656213\">\n<p id=\"fs-id1165137656216\">[latex]\\,y\\,[\/latex]varies inversely as the square root of[latex]\\,x\\,[\/latex]and when[latex]\\,x=25, \\,y=3.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627360\">\n<div id=\"fs-id1165137627362\">\n<p id=\"fs-id1165137843982\">[latex]\\,y\\,[\/latex]varies inversely as the cube root of[latex]\\,x\\,[\/latex]and when[latex]\\,x=64, \\,y=5.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137664262\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137664262\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137664262\"]\n<p id=\"fs-id1165137664264\">[latex]y=\\frac{20}{\\sqrt[3]{x}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137541589\">[latex]\\,y\\,[\/latex]varies jointly with[latex]\\,x\\,[\/latex]and[latex]\\,z\\,[\/latex]and when[latex]\\,x=2\\,[\/latex]and[latex]\\,z=3,\\, y=36.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137658781\">\n<div id=\"fs-id1165137507651\">\n<p id=\"fs-id1165137507654\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x, z,\\,[\/latex]and[latex]\\,w\\,[\/latex]and when[latex]\\,x=1,\\, z=2,\\, w=5,\\,[\/latex]then[latex]\\,y=100.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137563934\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137563934\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137563934\"]\n<p id=\"fs-id1165137563936\">[latex]y=10xzw[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135686723\">\n<p id=\"fs-id1165135237103\">[latex]\\,y\\,[\/latex]varies jointly as the square of[latex]\\,x\\,[\/latex]and the square of[latex]\\,z\\,[\/latex]and when[latex]\\,x=3\\,[\/latex]and[latex]\\,z=4,\\,[\/latex]then[latex]\\,y=72.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203444\">\n<div id=\"fs-id1165135408490\">\n<p id=\"fs-id1165135408493\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x\\,[\/latex]and the square root of[latex]\\,z\\,[\/latex]and when[latex]\\,x=2\\,[\/latex]and[latex]\\,z=25,\\,[\/latex]then[latex]\\,y=100.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134323767\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134323767\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134323767\"]\n<p id=\"fs-id1165134323770\">[latex]y=10x\\sqrt{z}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137675512\">\n<div id=\"fs-id1165135445761\">\n<p id=\"fs-id1165135445763\">[latex]\\,y[\/latex]varies jointly as the square of[latex]\\,x[\/latex]the cube of[latex]\\,z[\/latex]and the square root of[latex]\\,W.\\,[\/latex]When[latex]\\,x=1,z=2,\\,[\/latex]and[latex]\\,w=36,\\,[\/latex]then[latex]\\,y=48.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134047625\">\n<div id=\"fs-id1165134047627\">\n<p id=\"fs-id1165137758007\">[latex]\\,y\\,[\/latex]\nvaries jointly as[latex]\\,x \\text{and} z\\,[\/latex]and inversely as[latex]\\,w.\\,[\/latex]When[latex]\\,x=3, \\,z=5,\\,[\/latex]and[latex]\\,w=6,\\,[\/latex]then[latex]\\,y=10.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137723403\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137723403\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137723403\"]\n[latex]y=4\\frac{xz}{w}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137661782\">\n<div id=\"fs-id1165137661783\">\n\n[latex]\\,y\\,[\/latex]varies jointly as the square of[latex]\\,x\\,[\/latex]and the square root of[latex]\\,z\\,[\/latex]and inversely as the cube of[latex]\\,w\\text{.\\hspace{0.17em}}[\/latex]When[latex]\\,x=3,z=4,\\,[\/latex]and[latex]\\,w=3,\\,[\/latex]then[latex]\\,y=6.[\/latex]\n\n<\/div>\n<\/div>\n<div id=\"eip-387\">\n<div>\n<p id=\"eip-646\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x\\,[\/latex]and[latex]\\,z\\,[\/latex]and inversely as the square root of[latex]\\,w\\,[\/latex]and the square of[latex]\\,t\\text{\\hspace{0.17em}.}[\/latex]When[latex]\\,x=3,z=1,w=25,\\,[\/latex]and[latex]\\,t=2,\\,[\/latex]then[latex]\\,y=6.[\/latex]<\/p>\n\n<div class=\"textbox shaded\">[reveal-answer q=\"767549\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"767549\"][latex]y=40\\frac{xz}{\\sqrt{w}{t}^{2}}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137741107\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165137476436\">For the following exercises, use the given information to find the unknown value.<\/p>\n\n<div id=\"fs-id1165137476439\">\n<div id=\"fs-id1165135511345\">\n<p id=\"fs-id1165135511347\">[latex]\\,y\\,[\/latex]varies directly as[latex]\\,x.\\,[\/latex]When[latex]\\,x=3,\\,[\/latex]then[latex]\\,y=12.\\,[\/latex]Find[latex]\\,y\\,[\/latex]wneh[latex]\\,x=20.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137619342\">\n<div id=\"fs-id1165137619344\">\n<p id=\"fs-id1165137677831\">[latex]\\,y\\,[\/latex]varies directly as the square of[latex]\\,x.\\,[\/latex]When[latex]\\,x=2,\\,[\/latex]then[latex]\\,y=16.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]x=8.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135194956\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135194956\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135194956\"]\n<p id=\"fs-id1165137476412\">[latex]y=256[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137638624\">\n<div id=\"fs-id1165137638626\">\n<p id=\"fs-id1165137805207\">[latex]\\,y\\,[\/latex]varies directly as the cube of[latex]\\,x.\\,[\/latex]When[latex]\\,x=3,\\,[\/latex]then[latex]\\,y=5. \\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=4.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137722717\">\n<div>\n<p id=\"fs-id1165137737434\">[latex]\\,y\\,[\/latex]varies directly as the square root of[latex]\\,x.\\,[\/latex]When[latex]\\,x=16,\\,[\/latex]then[latex]\\,y=4.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=36.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137571788\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137571788\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137571788\"]\n[latex]y=6[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137812576\">\n<p id=\"fs-id1165137693975\">[latex]\\,y\\,[\/latex]varies directly as the cube root of[latex]\\,x.\\,[\/latex]When[latex]\\,x=125,\\,[\/latex]then[latex]\\,y=15.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=1,000.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137476513\">\n<div id=\"fs-id1165137476515\">\n<p id=\"fs-id1165137476517\">[latex]\\,y\\,[\/latex]varies inversely with[latex]\\,x.\\,[\/latex]When[latex]\\,x=3,\\,[\/latex]then[latex]\\,y=2.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137445714\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137445714\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137445714\"]\n<p id=\"fs-id1165137651705\">[latex]y=6[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627408\">\n<div id=\"fs-id1165137627410\">\n<p id=\"fs-id1165134306730\">[latex]\\,y\\,[\/latex]varies inversely with the square of[latex]\\,x.\\,[\/latex]When[latex]\\,x=4,\\,[\/latex]then[latex]\\,y=3.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=2.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545751\">\n<div id=\"fs-id1165135545753\">\n<p id=\"fs-id1165135545756\">[latex]\\,y\\,[\/latex]varies inversely with the cube of[latex]\\,x.\\,[\/latex]When[latex]\\,x=3,\\,[\/latex]then[latex]\\,y=1.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137431582\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137431582\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137431582\"]\n<p id=\"fs-id1165135535029\">[latex]y=27[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137932431\">\n<div id=\"fs-id1165137932433\">\n<p id=\"fs-id1165135403502\">[latex]\\,y\\,[\/latex]varies inversely with the square root of[latex]\\,x.\\,[\/latex]When[latex]\\,x=64,\\,[\/latex]then[latex]\\,y=12.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=36.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696700\">\n<div id=\"fs-id1165137579619\">\n<p id=\"fs-id1165137579622\">[latex]\\,y\\,[\/latex]varies inversely with the cube root of[latex]\\,x.\\,[\/latex]When[latex]\\,x=27,\\,[\/latex]then[latex]\\,y=5.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=125.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137542372\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137542372\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137542372\"]\n<p id=\"fs-id1165137542374\">[latex]y=3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137706810\">\n<div>\n<p id=\"fs-id1165137885945\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x \\text{and} z.[\/latex]When[latex]\\,x=4\\,[\/latex]and[latex]\\,z=2,\\,[\/latex]then[latex]\\,y=16.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=3\\,[\/latex]and[latex]\\,z=3.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134401069\">\n<div id=\"fs-id1165134401071\">\n<p id=\"fs-id1165137863353\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x, z, \\text{and} w.[\/latex]When[latex]\\,x=2,\\,[\/latex][latex]z=1,\\,[\/latex]and[latex]\\,w=12,\\,[\/latex]then[latex]\\,y=72.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=1,\\,[\/latex][latex]z=2,\\,[\/latex] and[latex]\\,w=3.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137406882\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137406882\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137406882\"]\n<p id=\"fs-id1165137444724\">[latex]y=18[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135411328\">\n<div id=\"fs-id1165135411330\">\n<p id=\"fs-id1165137393586\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x\\,[\/latex]and the square of[latex]\\,\\mathrm{z.}\\,[\/latex]When[latex]\\,x=2\\,[\/latex]and[latex]\\,z=4,\\,[\/latex]then[latex]\\,y=144.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=4\\,[\/latex]and[latex]\\,z=5.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737695\">\n<div>\n<p id=\"fs-id1165137662667\">[latex]\\,y\\,[\/latex]varies jointly as the square of[latex]\\,x\\,[\/latex]and the square root of[latex]\\,z.\\,[\/latex]When[latex]\\,x=2\\,[\/latex]and[latex]\\,z=9,\\,[\/latex]then[latex]\\,y=24.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=3\\,[\/latex]and[latex]\\,z=25.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135543455\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135543455\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135543455\"]\n<p id=\"fs-id1165135543457\">[latex]y=90[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137637576\">\n<div>\n<p id=\"fs-id1165137637581\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x\\,[\/latex]and[latex]\\,z\\,[\/latex]and inversely as[latex]\\,w.\\,[\/latex]When[latex]\\,x=5,\\,[\/latex][latex]z=2, \\,[\/latex]and[latex] \\,w=20,\\,[\/latex]then[latex]y=4.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=3\\,[\/latex]and[latex]\\,z=8, \\,[\/latex]and[latex]\\, w=48.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440221\">\n<div id=\"fs-id1165135194226\">\n<p id=\"fs-id1165135194229\">[latex]\\,y\\,[\/latex]varies jointly as the square of[latex]\\,x\\,[\/latex]and the cube of[latex]\\,z\\,[\/latex]and inversely as the square root of[latex]\\,w\\.[\/latex]When[latex]\\,x=2,\\,[\/latex][latex]z=2,\\,[\/latex]and[latex]\\,w=64,\\,[\/latex]then[latex]\\,y=12.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=1,\\,[\/latex][latex]z=3,\\,[\/latex]and[latex] \\,w=4.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137470711\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137470711\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137470711\"]\n<p id=\"fs-id1165137470713\">[latex]y=\\frac{81}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451415\">\n<div id=\"fs-id1165137663674\">\n<p id=\"fs-id1165137663676\">[latex]\\,y\\,[\/latex]varies jointly as the square of[latex]\\,x\\,[\/latex]and of[latex]\\,z\\,[\/latex]and inversely as the square root of[latex]\\,w\\,[\/latex]and of[latex]\\,t\\.[\/latex]When[latex]\\,x=2,\\,[\/latex][latex]z=3,\\,[\/latex][latex]w=16,\\,[\/latex]and[latex]\\,t=3,\\,[\/latex]then[latex]\\,y=1.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=3,\\,[\/latex][latex]z=2,\\,[\/latex][latex] w=36,\\,[\/latex]and[latex]\\,t=5.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135496464\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137758637\">For the following exercises, use a calculator to graph the equation implied by the given variation.<\/p>\n\n<div>\n<div id=\"fs-id1165137560300\">\n<p id=\"fs-id1165137560302\">[latex]\\,y\\,[\/latex]varies directly with the square of[latex]\\,x\\,[\/latex]and when[latex]\\,x=2, y=3.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137832227\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137832227\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137832227\"]\n<p id=\"fs-id1165137748367\">[latex]y=\\frac{3}{4}{x}^{2}[\/latex]<\/p>\n<img src=\"https:\/\/cnx.org\/resources\/56c4b015811357d76fcb67eea800b48dbc5db5bc\/CNX_Precalc_Figure_03_09_201.jpg\" alt=\"Graph of y=3\/4(x^2).\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137785175\">\n<div id=\"fs-id1165137785177\">\n<p id=\"fs-id1165137811898\">[latex]\\,y\\,[\/latex]varies directly as the cube of[latex]\\,x\\,[\/latex]and when[latex]\\,x=2, y=4.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137481150\">\n<div id=\"fs-id1165137735571\">\n<p id=\"fs-id1165137735573\">[latex]\\,y\\,[\/latex]varies directly as the square root of[latex]\\,x\\,[\/latex]and when[latex]\\,x=36, y=2.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137936864\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137936864\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137936864\"]\n<p id=\"fs-id1165137936866\">[latex]y=\\frac{1}{3}\\sqrt{x}[\/latex]<\/p>\n<img src=\"https:\/\/cnx.org\/resources\/fc13afa0b8f2db0496a47e9bc91b52dc008cc086\/CNX_Precalc_Figure_03_09_203.jpg\" alt=\"Graph of y=1\/3sqrt(x).\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137460988\">\n<div id=\"fs-id1165137460990\">\n<p id=\"fs-id1165137460992\">[latex]\\,y\\,[\/latex]varies inversely with[latex]\\,x\\,[\/latex]and when[latex]\\,x=6, y=2.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137472903\">\n<div id=\"fs-id1165137472905\">\n<p id=\"fs-id1165137638128\">[latex]\\,y\\,[\/latex]varies inversely as the square of[latex]\\,x\\,[\/latex]and when[latex]\\,x=1, y=4.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134248054\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134248054\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134248054\"]\n<p id=\"fs-id1165134248055\">[latex]y=\\frac{4}{{x}^{2}}[\/latex]<\/p>\n<img src=\"https:\/\/cnx.org\/resources\/53473051e0a2f6be7fe18c2acd62a8912b7a4961\/CNX_Precalc_Figure_03_09_205.jpg\" alt=\"Graph of y=4\/(x^2).\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193362\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165137762704\">For the following exercises, use Kepler\u2019s Law, which states that the square of the time,[latex]\\,T,\\,[\/latex]required for a planet to orbit the Sun varies directly with the cube of the mean distance,[latex]\\,a,\\,[\/latex]that the planet is from the Sun.<\/p>\n\n<div id=\"fs-id1165137401766\">\n<div id=\"fs-id1165137805286\">\n<p id=\"fs-id1165137805288\">Using Earth\u2019s time of 1 year and mean distance of 93 million miles, find the equation relating [latex]\\,T\\,[\/latex] and [latex]\\,a.\\,[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135433006\">\n<div id=\"fs-id1165137812556\">\n<p id=\"fs-id1165137677528\">Use the result from the previous exercise to determine the time required for Mars to orbit the Sun if its mean distance is 142 million miles.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137871569\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137871569\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137871569\"]\n<p id=\"fs-id1165137871571\">1.89 years<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137417733\">\n<div id=\"fs-id1165137417735\">\n<p id=\"fs-id1165137417737\">Using Earth\u2019s distance of 150 million kilometers, find the equation relating[latex]\\,T\\,[\/latex]and[latex]\\,a.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137552664\">\n<div id=\"fs-id1165137552667\">\n<p id=\"fs-id1165137552669\">Use the result from the previous exercise to determine the time required for Venus to orbit the Sun if its mean distance is 108 million kilometers.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137668810\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137668810\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137668810\"]\n<p id=\"fs-id1165137807323\">0.61 years<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503720\">\n<div id=\"fs-id1165135500737\">\n<p id=\"fs-id1165135500739\">Using Earth\u2019s distance of 1 astronomical unit (A.U.), determine the time for Saturn to orbit the Sun if its mean distance is 9.54 A.U.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137407207\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id1165137407212\">For the following exercises, use the given information to answer the questions.<\/p>\n\n<div id=\"fs-id1165137662862\">\n<div id=\"fs-id1165137662864\">\n<p id=\"fs-id1165137843829\">The distance[latex]\\,s\\,[\/latex]that an object falls varies directly with the square of the time,[latex]\\,t,\\,[\/latex]of the fall. If an object falls 16 feet in one se<strong>c<\/strong>ond, how long for it to fall 144 feet?<\/p>\n\n<\/div>\n<div id=\"fs-id1165133101738\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133101738\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133101738\"]\n<p id=\"fs-id1165133101740\">3 seconds<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810169\">\n<div id=\"fs-id1165137810171\">\n<p id=\"fs-id1165134374051\">The velocity[latex]\\,v\\,[\/latex]of a falling object varies directly to the time,[latex]\\,t,\\,[\/latex]of the fall. If after 2 seconds, the velocity of the object is 64 feet per second, what is the velocity after 5 seconds?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731890\">\n<div id=\"fs-id1165137731892\">\n<p id=\"fs-id1165137731894\">The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137728416\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137728416\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137728416\"]\n<p id=\"fs-id1165137728418\">48 inches<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137619719\">\n<div id=\"fs-id1165137619721\">\n<p id=\"fs-id1165137658372\">The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135613164\">\n<div id=\"fs-id1165135613167\">\n<p id=\"fs-id1165135613169\">The weight of an object above the surface of Earth varies inversely with the square of the distance from the center of Earth. If a body weighs 50 pounds when it is 3960 miles from Earth\u2019s center, what would it weigh it were 3970 miles from Earth\u2019s center?<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137760890\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137760890\"]\n<p id=\"fs-id1165137760890\">49.75 pounds<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545895\">\n<div id=\"fs-id1165135545898\">\n<p id=\"fs-id1165137598943\">The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot-candles at a distance of 3 meters. Find the intensity level at 8 meters.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137836546\">\n<div id=\"fs-id1165137836548\">\n<p id=\"fs-id1165134118360\">The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135194664\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135194664\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135194664\"]\n<p id=\"fs-id1165135194666\">33.33 amperes<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137535610\">\n<div id=\"fs-id1165137535612\">\n<p id=\"fs-id1165137535614\">The force exerted by the wind on a plane surface varies jointly with the square of the velocity of the wind and with the area of the plane surface. If the area of the surface is 40 square feet surface and the wind velocity is 20 miles per hour, the resulting force is 15 pounds. Find the force on a surface of 65 square feet with a velocity of 30 miles per hour.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137676536\">\n<div id=\"fs-id1165137676537\">\n<p id=\"fs-id1165137676538\">The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137641112\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137641112\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137641112\"]\n<p id=\"fs-id1165137676540\">2.88 inches<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137605795\">\n<div id=\"fs-id1165137605797\">\n<p id=\"fs-id1165137462259\">The kinetic energy[latex]\\,K\\,[\/latex]of a moving object varies jointly with its mass[latex]\\,m\\,[\/latex]and the square of its velocity[latex]\\,v.\\,[\/latex]If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135208752\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id1165137548191\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f929bd85-ed50-4a67-a528-40008aa14d0c\">Quadratic Functions<\/a><\/h4>\n<p id=\"fs-id1165137639859\">For the following exercises, write the quadratic function in standard form. Then give the vertex and axes intercepts. Finally, graph the function.<\/p>\n\n<div id=\"fs-id1165137477136\">\n<div id=\"fs-id1165137477138\">\n<p id=\"fs-id1165137477140\">[latex]f\\left(x\\right)={x}^{2}-4x-5[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137935589\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137935589\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137935589\"]\n<p id=\"fs-id1165137811941\">[latex]f\\left(x\\right)={\\left(x-2\\right)}^{2}-9\\,\\text{vertex} \\left(2,\u20139\\right), \\text{intercepts} \\left(5,0\\right); \\left(\u20131,0\\right); \\left(0,\u20135\\right)[\/latex]<\/p>\n<img src=\"https:\/\/cnx.org\/resources\/aa64c0c5507f70d1697a35a9c5ce60ec4ba29f5f\/CNX_Precalc_Figure_03_09_206.jpg\" alt=\"Graph of f(x)=x^2-4x-5.\">\n<div>[\/hidden-answer]<span id=\"fs-id1165137460981\"><\/span><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137863438\">\n<div id=\"fs-id1165137626743\">\n<p id=\"fs-id1165137626745\">[latex]f\\left(x\\right)=-2{x}^{2}-4x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137923495\">For the following exercises, find the equation of the quadratic function using the given information.<\/p>\n\n<div id=\"fs-id1165137923498\">\n<div id=\"fs-id1165135445905\">\n<p id=\"fs-id1165135445907\">The vertex is[latex]\\left(\u20132,3\\right)[\/latex]and a point on the graph is[latex]\\,\\left(3,6\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137724885\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137724885\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137724885\"]\n<p id=\"fs-id1165137398179\">[latex]f\\left(x\\right)=\\frac{3}{25}{\\left(x+2\\right)}^{2}+3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135457900\">\n<div id=\"fs-id1165135457902\">\n<p id=\"fs-id1165135457904\">The vertex is[latex]\\,\\left(\u20133,6.5\\right)\\,[\/latex]and a point on the graph is[latex]\\,\\left(2,6\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135443862\">For the following exercises, complete the task.<\/p>\n\n<div id=\"fs-id1165135443865\">\n<div id=\"fs-id1165135443867\">\n<p id=\"fs-id1165137767424\">A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137725075\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137725075\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137725075\"]\n<p id=\"fs-id1165137725077\">300 meters by 150 meters, the longer side parallel to river.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137422734\">\n<div id=\"fs-id1165137422736\">\n<p id=\"fs-id1165135706774\">An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height,[latex]\\,h,\\,[\/latex]in terms of horizontal distance traveled,[latex]\\,x,\\,[\/latex]given by[latex]\\,h\\left(x\\right)=\\frac{-32}{{\\left(120\\right)}^{2}}{x}^{2}+x.\\,[\/latex]Find the maximum height the object attains.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133233027\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/e017cb02-d494-4d95-8d09-966d0968a606\">Power Functions and Polynomial Functions<\/a><\/h4>\n<p id=\"fs-id1165137596106\">For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.<\/p>\n\n<div id=\"fs-id1165137596111\">\n<div id=\"fs-id1165137927725\">\n<p id=\"fs-id1165137927727\">[latex]f\\left(x\\right)=4{x}^{5}-3{x}^{3}+2x-1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137451016\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137451016\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137451016\"]\n<p id=\"fs-id1165137430842\">Yes, degree = 5, leading coefficient = 4<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137786551\">\n<div id=\"fs-id1165137786554\">\n<p id=\"fs-id1165137595265\">[latex]f\\left(x\\right)={5}^{x+1}-{x}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135407052\">\n<div id=\"fs-id1165135407054\">\n<p id=\"fs-id1165135407057\">[latex]f\\left(x\\right)={x}^{2}\\left(3-6x+{x}^{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"125615\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"125615\"]Yes, degree = 4, leading coefficient = 1[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135181695\">For the following exercises, determine end behavior of the polynomial function.<\/p>\n\n<div id=\"fs-id1165135181698\">\n<div id=\"fs-id1165137735839\">\n<p id=\"fs-id1165137735841\">[latex]f\\left(x\\right)=2{x}^{4}+3{x}^{3}-5{x}^{2}+7[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137643403\">\n<div id=\"fs-id1165137643405\">\n<p id=\"fs-id1165137643407\">[latex]f\\left(x\\right)=4{x}^{3}-6{x}^{2}+2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134155145\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134155145\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134155145\"]\n<p id=\"fs-id1165137386933\">[latex]\\text{As}\\,x\\to -\\infty ,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137601687\">\n<div id=\"fs-id1165137601688\">\n<p id=\"fs-id1165137601689\">[latex]f\\left(x\\right)=2{x}^{2}\\left(1+3x-{x}^{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135169092\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/644f6a93-7429-4768-8752-eccf5fb8e0bb\">Graphs of Polynomial Functions<\/a><\/h4>\n<p id=\"fs-id1165137451676\">For the following exercises, find all zeros of the polynomial function, noting multiplicities.<\/p>\n\n<div id=\"fs-id1165137451679\">\n<div id=\"fs-id1165137557555\">\n<p id=\"fs-id1165137557556\">[latex]f\\left(x\\right)={\\left(x+3\\right)}^{2}\\left(2x-1\\right){\\left(x+1\\right)}^{3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137432758\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137432758\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137432758\"]\n<p id=\"fs-id1165137806236\">\u20133 with multiplicity 2,[latex]-\\frac{1}{2}[\/latex]with multiplicity 1, \u20131 with multiplicity 3<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135683516\">\n<div id=\"fs-id1165135683518\">\n<p id=\"fs-id1165137402103\">[latex]f\\left(x\\right)={x}^{5}+4{x}^{4}+4{x}^{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137645405\">\n<div id=\"fs-id1165137404519\">\n<p id=\"fs-id1165137404521\">[latex]f\\left(x\\right)={x}^{3}-4{x}^{2}+x-4[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137749873\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137749873\"]\n<p id=\"fs-id1165137749873\">4 with multiplicity 1<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137846483\">For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.<\/p>\n\n<div id=\"fs-id1165137846487\">\n<div id=\"fs-id1165137697060\"><span id=\"fs-id1165135368421\"><img src=\"https:\/\/cnx.org\/resources\/2bfb5f803ebe608bc71ee1abd0f111c8a6d38ded\/CNX_Precalc_Figure_03_09_208.jpg\" alt=\"Graph of an odd-degree polynomial with two turning points.\"><\/span><\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137742630\"><span id=\"fs-id1165134313923\"><img src=\"https:\/\/cnx.org\/resources\/dc5c225da94ba479e35a0d1bee5759184b6f78a5\/CNX_Precalc_Figure_03_09_209.jpg\" alt=\"Graph of an even-degree polynomial with two turning points.\"><\/span><\/div>\n<div id=\"fs-id1165137539727\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137539727\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137539727\"]\n<p id=\"fs-id1165137648165\">[latex]\\frac{1}{2}\\,[\/latex]with multiplicity 1, 3 with multiplicity 3<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137557643\">\n<div id=\"fs-id1165135694414\">\n<p id=\"fs-id1165135694416\">Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the function[latex]\\,f\\left(x\\right)={x}^{3}-5x+1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137920803\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/11f2bd05-8eff-4fb8-9e06-02a657fbb031\">Dividing Polynomials<\/a><\/h4>\n<p id=\"fs-id1165137749390\">For the following exercises, use long division to find the quotient and remainder.<\/p>\n\n<div id=\"fs-id1165137696726\">\n<div id=\"fs-id1165137696728\">\n<p id=\"fs-id1165137696730\">[latex]\\frac{{x}^{3}-2{x}^{2}+4x+4}{x-2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137423736\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137423736\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137423736\"]\n<p id=\"fs-id1165137423738\">[latex]\\,{x}^{2}+4\\,[\/latex] with remainder 12<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135446689\">\n<div id=\"fs-id1165135446691\">\n<p id=\"fs-id1165135446693\">[latex]\\frac{3{x}^{4}-4{x}^{2}+4x+8}{x+1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165132944971\">For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.<\/p>\n\n<div id=\"fs-id1165134378622\">\n<div id=\"fs-id1165134378624\">\n<p id=\"fs-id1165134378626\">[latex]\\frac{{x}^{3}-2{x}^{2}+5x-1}{x+3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135365932\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135365932\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135365932\"]\n<p id=\"fs-id1165135365934\">[latex]{x}^{2}-5x+20-\\frac{61}{x+3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135560695\">\n<div id=\"fs-id1165135560697\">\n<p id=\"fs-id1165137762397\">[latex]\\frac{{x}^{3}+4x+10}{x-3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137935190\">\n<div id=\"fs-id1165137935192\">\n<p id=\"fs-id1165135176562\">[latex]\\frac{2{x}^{3}+6{x}^{2}-11x-12}{x+4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137772480\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137772480\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137772480\"]\n<p id=\"fs-id1165135558432\">[latex]2{x}^{2}-2x-3[\/latex], so factored form is [latex]\\left(x+4\\right)\\left(2{x}^{2}-2x-3\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135414382\">\n<div id=\"fs-id1165135414383\">\n<p id=\"fs-id1165137735052\">[latex]\\frac{3{x}^{4}+3{x}^{3}+2x+2}{x+1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135194274\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/1bb57d2c-789f-4252-bfe4-edc0873f1e29\">Zeros of Polynomial Functions<\/a><\/h4>\n<p id=\"fs-id1165137653216\">For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.<\/p>\n\n<div id=\"fs-id1165134248795\">\n<div id=\"fs-id1165134248797\">\n<p id=\"fs-id1165134248799\">[latex]2{x}^{3}-3{x}^{2}-18x-8=0[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137780754\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137780754\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137780754\"]\n<p id=\"fs-id1165137780756\">[latex]\\left\\{-2, 4, -\\frac{1}{2}\\right\\}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137901245\">\n<div id=\"fs-id1165137901248\">\n<p id=\"fs-id1165137901250\">[latex]3{x}^{3}+11{x}^{2}+8x-4=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137452024\">\n<div id=\"fs-id1165135640601\">\n<p id=\"fs-id1165135640603\">[latex]2{x}^{4}-17{x}^{3}+46{x}^{2}-43x+12=0[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137654771\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137654771\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137654771\"]\n[latex]\\left\\{1, 3, 4, \\frac{1}{2}\\right\\}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135531610\">\n<div id=\"fs-id1165135531612\">\n<p id=\"fs-id1165137770047\">[latex]4{x}^{4}+8{x}^{3}+19{x}^{2}+32x+12=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137407778\">For the following exercises, use Descartes\u2019 Rule of Signs to find the possible number of positive and negative solutions.<\/p>\n\n<div id=\"fs-id1165134573782\">\n<div id=\"fs-id1165134573784\">\n<p id=\"fs-id1165134573786\">[latex]{x}^{3}-3{x}^{2}-2x+4=0[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134108392\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134108392\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134108392\"]\n<p id=\"fs-id1165134108394\">0 or 2 positive, 1 negative<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135613231\">\n<div id=\"fs-id1165135613233\">\n<p id=\"fs-id1165134224004\">[latex]2{x}^{4}-{x}^{3}+4{x}^{2}-5x+1=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134224073\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/28d4cfda-bec3-48cf-9695-2cc71939e22a\">Rational Functions<\/a><\/h4>\n<p id=\"fs-id1165135176618\">For the following exercises, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph of the function.<\/p>\n\n<div id=\"fs-id1165135176623\">\n<div id=\"fs-id1165135516926\">\n<p id=\"fs-id1165135516928\">[latex]f\\left(x\\right)=\\frac{x+2}{x-5}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137740757\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137740757\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137740757\"]\n<p id=\"fs-id1165137740759\">Intercepts [latex]\\left(\u20132,0\\right)\\text{and}\\left(0,-\\frac{2}{5}\\right)[\/latex], Asymptotes[latex]\\,x=5\\,[\/latex]\nand[latex]\\,y=1.[\/latex]<\/p>\n\n<div><\/div>\n<img src=\"https:\/\/cnx.org\/resources\/37cb1aa7a7578e54135f0defd76697c6ec0236cc\/CNX_Precalc_Figure_03_09_210.jpg\" alt=\"Graph of f(x)=(x+1)\/(x-5).\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135205950\">\n<div id=\"fs-id1165135529121\">\n<p id=\"fs-id1165135529123\">[latex]f\\left(x\\right)=\\frac{{x}^{2}+1}{{x}^{2}-4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137863121\">\n<div id=\"fs-id1165137863124\">\n<p id=\"fs-id1165137863126\">[latex]f\\left(x\\right)=\\frac{3{x}^{2}-27}{{x}^{2}+x-2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137452244\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137452244\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137452244\"]\n<p id=\"fs-id1165137452246\">Intercepts (3, 0), (-3, 0), and[latex]\\,\\left(0,\\frac{27}{2}\\right)\\,[\/latex], Asymptotes[latex]\\,x=1, x=\u20132, y=3.[\/latex]<\/p>\n\n<div><\/div>\n<img src=\"https:\/\/cnx.org\/resources\/1cb79c1497ffbdf7e97bf632533f2d5253e8671c\/CNX_Precalc_Figure_03_09_212.jpg\" alt=\"Graph of f(x)=(3x^2-27)\/(x^2+x-2).\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134226797\">\n<div>[latex]f\\left(x\\right)=\\frac{x+2}{{x}^{2}-9}[\/latex]<\/div>\n<\/div>\nFor the following exercises, find the slant asymptote.\n<div id=\"fs-id1165137842426\">\n<div id=\"fs-id1165137842428\">[latex]f\\left(x\\right)=\\frac{{x}^{2}-1}{x+2}[\/latex]<\/div>\n<div id=\"fs-id1165137642577\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137642577\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137642577\"]\n<p id=\"fs-id1165135252057\">[latex]y=\\text{ }x-2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438219\">\n<div id=\"fs-id1165135508577\">\n<p id=\"fs-id1165135508579\">[latex]f\\left(x\\right)=\\frac{2{x}^{3}-{x}^{2}+4}{{x}^{2}+1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/ce233d35-8468-4ded-97dc-b34c3475a26a\">Inverses and Radical Functions<\/a><\/h4>\nFor the following exercises, find the inverse of the function with the domain given.\n<div>\n<div id=\"fs-id1165135458657\">\n<p id=\"fs-id1165135458659\">[latex]f\\left(x\\right)={\\left(x-2\\right)}^{2},\\,x\\ge 2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135640618\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135640618\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135640618\"]\n<p id=\"fs-id1165135593176\">[latex]{f}^{-1}\\left(x\\right)=\\sqrt{x}+2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133103950\">\n<div id=\"fs-id1165133103952\">\n<p id=\"fs-id1165133103954\">[latex]f\\left(x\\right)={\\left(x+4\\right)}^{2}-3,\\,x\\ge -4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137770318\">\n<div id=\"fs-id1165137770321\">\n<p id=\"fs-id1165137770323\">[latex]f\\left(x\\right)={x}^{2}+6x-2,\\,x\\ge -3[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135210050\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135210050\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135210050\"]\n<p id=\"fs-id1165135210052\">[latex]{f}^{-1}\\left(x\\right)=\\sqrt{x+11}-3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135301653\">\n<div id=\"fs-id1165135301655\">\n<p id=\"fs-id1165137619912\">[latex]f\\left(x\\right)=2{x}^{3}-3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137749717\">\n<div id=\"fs-id1165137749719\">\n<p id=\"fs-id1165137692176\">[latex]f\\left(x\\right)=\\sqrt{4x+5}-3[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134224016\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134224016\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134224016\"]\n<p id=\"fs-id1165134224018\">[latex]{f}^{-1}\\left(x\\right)=\\frac{{\\left(x+3\\right)}^{2}-5}{4},\\,x\\ge -3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135591017\">\n<div id=\"fs-id1165135342998\">\n<p id=\"fs-id1165135343000\">[latex]f\\left(x\\right)=\\frac{x-3}{2x+1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135245788\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/c941f485-13ab-481a-a029-2160ef470707\">Modeling Using Variation<\/a><\/h4>\n<p id=\"fs-id1165137462872\">For the following exercises, find the unknown value.<\/p>\n\n<div id=\"fs-id1165137462875\">\n<div id=\"fs-id1165137462877\">\n<p id=\"fs-id1165137939566\">[latex]\\,y\\,[\/latex] varies directly as the square of[latex]\\,x.\\,[\/latex] If when[latex]\\,x=3,\\text{ }y=36,\\,[\/latex]find[latex]\\,y\\,[\/latex]if[latex]\\,x=4.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135319534\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135319534\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135319534\"]\n<p id=\"fs-id1165135203742\">[latex]y=64[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137872027\">\n<div id=\"fs-id1165137872029\">\n<p id=\"fs-id1165137696624\">[latex]\\,y\\,[\/latex] varies inversely as the square root of[latex]\\,x\\,[\/latex] If when[latex]\\,x=25,\\text{ }y=2,\\,[\/latex]find[latex]\\,y\\,[\/latex]if[latex]\\,x=4.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135209414\">\n<div>\n\n[latex]\\,y\\,[\/latex]varies jointly as the cube of[latex]\\,x\\,[\/latex]and as[latex]\\,z.\\,[\/latex]If when[latex]\\,x=1\\,[\/latex]and[latex]\\,z=2,\\,[\/latex][latex]y=6,\\,[\/latex]find[latex]\\,y\\,[\/latex]if[latex]\\,x=2\\,[\/latex]and[latex]\\,z=3.[\/latex]\n\n<\/div>\n<div id=\"fs-id1165137934401\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137934401\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137934401\"]\n<p id=\"fs-id1165137452780\">[latex]y\\text{ }=\\text{ }72[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137456499\">\n<div id=\"fs-id1165137456502\">\n\n[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x\\,[\/latex]and the square of[latex]\\,z\\,[\/latex]and inversely as the cube of[latex]\\,w.\\,[\/latex]If when[latex]\\,x=3,\\,[\/latex][latex]z=4,\\,[\/latex]and[latex]\\,w=2,\\,[\/latex][latex]y=48,\\,[\/latex]find[latex]\\,y\\,[\/latex]if[latex]\\,x=4,\\,[\/latex][latex]z=5,\\,[\/latex]and[latex]\\,w=3.[\/latex]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137435024\">For the following exercises, solve the application problem.<\/p>\n\n<div id=\"fs-id1165137435027\">\n<div id=\"fs-id1165137435029\">\n<p id=\"fs-id1165135169106\">The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135698641\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135698641\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135698641\"]148.5 pounds[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135698649\">\n<div id=\"fs-id1165135536256\">\n\nThe volume[latex]\\,V\\,[\/latex]of an ideal gas varies directly with the temperature[latex]\\,T\\,[\/latex]and inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137602034\" class=\"practice-test\">\n<h3>Chapter Test<\/h3>\n<p id=\"fs-id1165134262509\">Give the degree and leading coefficient of the following polynomial function.<\/p>\n\n<div id=\"fs-id1165134262512\">\n<div id=\"fs-id1165134211394\">\n<p id=\"fs-id1165134211396\">[latex]f\\left(x\\right)={x}^{3}\\left(3-6{x}^{2}-2{x}^{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135424704\">Determine the end behavior of the polynomial function.<\/p>\n\n<div id=\"fs-id1165137394106\">\n<div id=\"fs-id1165137394108\">\n<p id=\"fs-id1165137394110\">[latex]f\\left(x\\right)=8{x}^{3}-3{x}^{2}+2x-4[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135192394\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135192394\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135192394\"]\n<p id=\"fs-id1165135192396\">[latex]As\\,x\\to -\\infty ,\\,f\\left(x\\right)\\to -\\infty ,\\,as\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135389015\">\n<p id=\"fs-id1165135503719\">[latex]f\\left(x\\right)=-2{x}^{2}\\left(4-3x-5{x}^{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\nWrite the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.\n<div id=\"fs-id1165137732249\">\n<div id=\"fs-id1165137732251\">\n<p id=\"fs-id1165137732253\">[latex]f\\left(x\\right)={x}^{2}+2x-8[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135503809\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135503809\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135503809\"]\n<p id=\"fs-id1165135694562\">[latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-9[\/latex], vertex[latex]\\,\\left(-1,-9\\right)[\/latex], intercepts[latex]\\,\\left(2,0\\right);\\left(-4,0\\right);\\,\\left(0,-8\\right)[\/latex]<\/p>\n<img src=\"https:\/\/cnx.org\/resources\/c70d329ef7c037f7c148c4c90b777d9b283be80e\/CNX_Precalc_Figure_03_09_214.jpg\" alt=\"Graph of f(x)=x^2+2x-8.\">[\/hidden-answer]<span id=\"fs-id1165137653687\"><\/span>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135354880\">Given information about the graph of a quadratic function, find its equation.<\/p>\n\n<div id=\"fs-id1165137736986\">\n<div id=\"fs-id1165137736988\">\n<p id=\"fs-id1165137736990\">Vertex[latex]\\,\\left(2,0\\right)\\,[\/latex]and point on graph[latex]\\,\\left(4,12\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\nSolve the following application problem.\n<div id=\"fs-id1165131797287\">\n<div id=\"fs-id1165131797290\">\n<p id=\"fs-id1165131797292\">A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137673750\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137673750\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137673750\"]\n<p id=\"fs-id1165137673752\">60,000 square feet<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135454021\">Find all zeros of the following polynomial functions, noting multiplicities.<\/p>\n\n<div id=\"fs-id1165135454024\">\n<div id=\"fs-id1165137726459\">\n<p id=\"fs-id1165137726461\">[latex]f\\left(x\\right)={\\left(x-3\\right)}^{3}\\left(3x-1\\right){\\left(x-1\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135453256\">\n<div id=\"fs-id1165135453258\">\n<p id=\"fs-id1165135453260\">[latex]f\\left(x\\right)=2{x}^{6}-12{x}^{5}+18{x}^{4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134040553\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134040553\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134040553\"]\n<p id=\"fs-id1165134040555\">0 with multiplicity 4, 3 with multiplicity 2<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134040560\">Based on the graph, determine the zeros of the function and multiplicities.<\/p>\n\n<div id=\"fs-id1165137871376\">\n<div id=\"fs-id1165137871378\"><span id=\"fs-id1165135264841\"><img src=\"https:\/\/cnx.org\/resources\/ea0a3097aab1ecce6abbb2dbeb500e1f428d1cb5\/CNX_Precalc_Figure_03_09_215.jpg\" alt=\"Graph of an odd-degree polynomial with two turning points.\"><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165135191491\">Use long division to find the quotient.<\/p>\n\n<div id=\"fs-id1165135191495\">\n<div id=\"fs-id1165135191497\">\n<p id=\"fs-id1165134122806\">[latex]\\frac{2{x}^{3}+3x-4}{x+2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137427782\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137427782\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137427782\"]\n<p id=\"fs-id1165137427784\">[latex]2{x}^{2}-4x+11-\\frac{26}{x+2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135342123\">Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.<\/p>\n\n<div id=\"fs-id1165135342127\">\n<div id=\"fs-id1165135342129\">\n<p id=\"fs-id1165137732538\">[latex]\\frac{{x}^{4}+3{x}^{2}-4}{x-2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137833008\">\n<div id=\"fs-id1165137833010\">\n<p id=\"fs-id1165137827866\">[latex]\\frac{2{x}^{3}+5{x}^{2}-7x-12}{x+3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135152117\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135152117\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135152117\"]\n<p id=\"fs-id1165135152119\">[latex]2{x}^{2}-x-4[\/latex]. So factored form is[latex]\\,\\left(x+3\\right)\\left(2{x}^{2}-x-4\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135694398\">Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.<\/p>\n\n<div id=\"fs-id1165135694401\">\n<div id=\"fs-id1165135694403\">\n<p id=\"fs-id1165137407530\">[latex]f\\left(x\\right)=2{x}^{3}+5{x}^{2}-6x-9[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135161247\">\n<div id=\"fs-id1165135161249\">\n<p id=\"fs-id1165135161251\">[latex]f\\left(x\\right)=4{x}^{4}+8{x}^{3}+21{x}^{2}+17x+4[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135188171\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135188171\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135188171\"]\n<p id=\"fs-id1165135188173\">[latex]-\\frac{1}{2}\\,[\/latex]\n(has multiplicity 2),[latex]\\,\\frac{-1\u00b1i\\sqrt{15}}{2}\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192046\">\n<div id=\"fs-id1165135192048\">[latex]f\\left(x\\right)=4{x}^{4}+16{x}^{3}+13{x}^{2}-15x-18[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137758031\">\n<div id=\"fs-id1165137758034\">\n<p id=\"fs-id1165134190707\">[latex]f\\left(x\\right)={x}^{5}+6{x}^{4}+13{x}^{3}+14{x}^{2}+12x+8[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137427321\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137427321\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137427321\"]\n<p id=\"fs-id1165137427323\">[latex]\\,-2\\,[\/latex](has multiplicity 3),[latex]\\,\u00b1i[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135689439\">Given the following information about a polynomial function, find the function.<\/p>\n\n<div id=\"fs-id1165135689443\">\n<div id=\"fs-id1165135689445\">\n\nIt has a double zero at[latex]\\,x=3\\,[\/latex]and zeros at[latex]\\,x=1\\,[\/latex]and[latex]\\,x=-2\\,[\/latex]. Its <em>y<\/em>-intercept is[latex]\\,\\left(0,12\\right).\\,[\/latex]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132011285\">\n<div id=\"fs-id1165135205666\">\n<p id=\"fs-id1165135205668\">It has a zero of multiplicity 3 at[latex]\\,x=\\frac{1}{2}\\,[\/latex]and another zero at[latex]\\,x=-3\\,[\/latex]. It contains the point[latex]\\,\\left(1,8\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137595278\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137595278\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137595278\"]\n<p id=\"fs-id1165137595280\">[latex]f\\left(x\\right)=2{\\left(2x-1\\right)}^{3}\\left(x+3\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135177605\">Use Descartes\u2019 Rule of Signs to determine the possible number of positive and negative solutions.<\/p>\n\n<div id=\"fs-id1165134065115\">\n<div id=\"fs-id1165134065117\">[latex]8{x}^{3}-21{x}^{2}+6=0[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1165137722354\">For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.<\/p>\n\n<div id=\"fs-id1165137722359\">\n<div id=\"fs-id1165137722361\">\n<p id=\"fs-id1165135188256\">[latex]f\\left(x\\right)=\\frac{x+4}{{x}^{2}-2x-3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135208461\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135208461\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135208461\"]\n<p id=\"fs-id1165135208464\">Intercepts[latex]\\,\\left(-4,0\\right),\\,\\left(0,-\\frac{4}{3}\\right)\\,[\/latex], Asymptotes[latex]\\,x=3,\\text{ }x\\text{ }=-1,\\text{ }y=0[\/latex].<\/p>\n\n<div><\/div>\n<img src=\"https:\/\/cnx.org\/resources\/ecbf795c970a120f915fddb3190eccf9c0d013a6\/CNX_Precalc_Figure_03_09_216.jpg\" alt=\"Graph of f(x)=(x+4)\/(x^2-2x-3).\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135650721\">\n<div id=\"fs-id1165135650723\">\n<p id=\"fs-id1165135650725\">[latex]f\\left(x\\right)=\\frac{{x}^{2}+2x-3}{{x}^{2}-4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137567083\">Find the slant asymptote of the rational function.<\/p>\n\n<div id=\"fs-id1165137567086\">\n<div id=\"fs-id1165137567088\">\n<p id=\"fs-id1165137567090\">[latex]f\\left(x\\right)=\\frac{{x}^{2}+3x-3}{x-1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135151319\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135151319\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135151319\"]\n[latex]y=x+4[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1165137570255\">Find the inverse of the function.<\/p>\n\n<div id=\"fs-id1165137834208\">\n<div>\n<p id=\"fs-id1165137834212\">[latex]f\\left(x\\right)=\\sqrt{x-2}+4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723709\">\n<div id=\"fs-id1165137723711\">\n<p id=\"fs-id1165135194622\">[latex]f\\left(x\\right)=3{x}^{3}-4[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137920721\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137920721\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137920721\"]\n<p id=\"fs-id1165137805519\">[latex]{f}^{-1}\\left(x\\right)=\\sqrt[3]{\\frac{x+4}{3}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137502317\">\n<div id=\"fs-id1165137502320\">\n<p id=\"fs-id1165135521251\">[latex]f\\left(x\\right)=\\frac{2x+3}{3x-1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\nFind the unknown value.\n<div id=\"fs-id1165137401576\">\n<div id=\"fs-id1165137401578\">\n<p id=\"fs-id1165137848951\">[latex]\\,y\\,[\/latex]varies inversely as the square of[latex]\\,x\\,[\/latex]and when[latex]\\,x=3,\\,[\/latex][latex]y=2.\\,[\/latex]Find[latex]\\,y\\,[\/latex]if[latex]\\,x=1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137722658\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137722658\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137722658\"]\n<p id=\"fs-id1165137722660\">[latex]y=18[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135559531\">\n<div>\n<p id=\"fs-id1165137425335\">[latex]\\,y\\,[\/latex]varies jointly with[latex]\\,x\\,[\/latex]and the cube root of[latex]\\,z.\\,[\/latex]If when[latex]\\,x=2\\,[\/latex]and[latex]\\,z=27,\\,[\/latex][latex]y=12,\\,[\/latex]find[latex]\\,y\\,[\/latex]if[latex]\\,x=5\\,[\/latex]and[latex]\\,z=8.[\/latex]<\/p>\n\n<\/div>\n<\/div>\nSolve the following application problem.\n<div id=\"fs-id1165137427522\">\n<div id=\"fs-id1165137427524\">\n\nThe distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?\n\n<\/div>\n<div id=\"fs-id1165137734636\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137734636\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137734636\"]\n<p id=\"fs-id1165137734638\">4 seconds<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137735724\">\n \t<dt>constant of variation<\/dt>\n \t<dd id=\"fs-id1165137735729\">the non-zero value[latex]\\,k\\,[\/latex]\nthat helps define the relationship between variables in direct or inverse variation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137762202\">\n \t<dt>direct variation<\/dt>\n \t<dd>the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137462046\">\n \t<dt>inverse variation<\/dt>\n \t<dd>the relationship between two variables in which the product of the variables is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135501040\">\n \t<dt>inversely proportional<\/dt>\n \t<dd id=\"fs-id1165137874542\">a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137874546\">\n \t<dt>joint variation<\/dt>\n \t<dd id=\"fs-id1165135696715\">a relationship where a variable varies directly or inversely with multiple variables<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135696718\">\n \t<dt>varies directly<\/dt>\n \t<dd id=\"fs-id1165137432955\">a relationship where one quantity is a constant multiplied by the other quantity<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137432958\">\n \t<dt>varies inversely<\/dt>\n \t<dd id=\"fs-id1165135439853\">a relationship where one quantity is a constant divided by the other quantity<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\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<\/div>\n<p id=\"fs-id1165135356540\">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<div id=\"fs-id1165137730306\" class=\"bc-section section\">\n<h3>Solving Direct Variation Problems<\/h3>\n<p id=\"fs-id1165137823230\">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. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. See <a class=\"autogenerated-content\" href=\"#Table_03_09_01\">(Figure)<\/a>.<\/p>\n<table id=\"Table_03_09_01\" summary=\"..\">\n<thead>\n<tr>\n<th>[latex]\\,s\\,[\/latex], sales price<\/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\\left(4,600\\right)=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\\left(9,200\\right)=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\\left(18,400\\right)=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 id=\"fs-id1165135188294\">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 id=\"fs-id1165137937533\"><a class=\"autogenerated-content\" href=\"#Figure_03_09_001\">(Figure)<\/a> 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]and[latex]\\,n=1.\\,[\/latex]We saw functions like this one when we discussed power functions.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/8bfb65f3c8025c36830c5246429f7688276ed66d\/CNX_Precalc_Figure_03_09_001.jpg\" alt=\"Graph of y=(0.16)x where the horizontal axis is labeled, \u201cs, Sales Price in Dollars\u201d, and the vertical axis is labeled, \u201ce, Earnings, $\u201d.\" width=\"487\" height=\"459\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<div id=\"fs-id1165137730075\" class=\"textbox key-takeaways\">\n<h3>Direct Variation<\/h3>\n<p id=\"fs-id1165137827458\">If[latex]\\,x\\,\\text{and}\\,y\\,[\/latex]are related by an equation of the form<\/p>\n<div id=\"fs-id1165135437156\">[latex]\\,y=k{x}^{n}\\,[\/latex]<\/div>\n<p id=\"fs-id1165133155266\">then we say that the relationship is direct variation and [latex]\\,y\\,[\/latex] varies directly with, or is proportional to, the[latex]\\,n\\text{th}\\,[\/latex]power of[latex]\\,x.\\,[\/latex]In direct variation relationships, there is a nonzero constant ratio[latex]\\,k=\\frac{y}{{x}^{n}},\\,[\/latex]where[latex]\\,k\\,[\/latex]is called the constant of variation, which help defines the relationship between the variables.<\/p>\n<\/div>\n<div id=\"fs-id1165137550958\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137723932\"><strong>Given a description of a direct variation problem, solve for an unknown.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1165137724401\" type=\"1\">\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]by the specified power of[latex]\\,x\\,[\/latex]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 id=\"Example_03_09_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137676066\">\n<div id=\"fs-id1165137434564\">\n<h3>Solving a Direct Variation Problem<\/h3>\n<p id=\"fs-id1165137849016\">The quantity[latex]\\,y\\,[\/latex]varies directly with the cube of[latex]\\,x.\\,[\/latex]If[latex]\\,y=25\\,[\/latex]when[latex]\\,x=2,\\,[\/latex]find[latex]\\,y\\,[\/latex]when[latex]\\,x\\,[\/latex]is 6.<\/p>\n<\/div>\n<div id=\"fs-id1165137642960\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137659713\">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]by the cube of[latex]\\,x.\\,[\/latex]<\/p>\n<div id=\"eip-id1165134084945\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill k& =& \\frac{y}{{x}^{3}}\\hfill \\\\ & =& \\frac{25}{{2}^{3}}\\hfill \\\\ & =& \\frac{25}{8}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137628102\">Now use the constant to write an equation that represents this relationship.<\/p>\n<div id=\"eip-id1165135440091\" class=\"unnumbered\">[latex]y=\\frac{25}{8}{x}^{3}[\/latex]<\/div>\n<p id=\"fs-id1165135432964\">Substitute[latex]\\,x=6\\,[\/latex]and solve for[latex]\\,y.[\/latex]<\/p>\n<div id=\"eip-id1165135207297\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill y& =& \\frac{25}{8}{\\left(6\\right)}^{3}\\hfill \\\\ & =& 675\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135533140\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134557390\">The graph of this equation is a simple cubic, as shown in <a class=\"autogenerated-content\" href=\"#Figure_03_09_002\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/b4093a150bd577af57b691e9c4b951e271614aa9\/CNX_Precalc_Figure_03_09_002.jpg\" alt=\"Graph of y=25\/8(x^3) with the labeled points (2, 25) and (6, 675).\" width=\"487\" height=\"367\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736204\" class=\"precalculus qa textbox shaded\">\n<p id=\"eip-id1165137772190\"><strong>Do the graphs of all direct variation equations look like <a class=\"autogenerated-content\" href=\"#Example_03_09_01\">(Figure)<\/a>?<\/strong><\/p>\n<p id=\"fs-id1165137596402\"><em>No. Direct variation equations are power functions\u2014they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through[latex]\\,\\left(0,0\\right).[\/latex]<br \/>\n<\/em><\/p>\n<\/div>\n<div id=\"fs-id1165137783766\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_09_01\">\n<div id=\"fs-id1165137645447\">\n<p id=\"fs-id1165135160334\">The quantity[latex]\\,y\\,[\/latex]varies directly with the square of[latex]\\,x.\\,[\/latex]If[latex]\\,y=24\\,[\/latex]when[latex]\\,x=3,\\,[\/latex]find[latex]\\,y\\,[\/latex]when[latex]\\,x\\,[\/latex]is 4.<\/p>\n<\/div>\n<div id=\"fs-id1165137659774\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137558700\">[latex]\\frac{128}{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137758211\" class=\"bc-section section\">\n<h3>Solving Inverse Variation Problems<\/h3>\n<p id=\"fs-id1165137734583\">Water temperature in an ocean varies inversely to the water\u2019s depth. The formula[latex]\\,T=\\frac{14,000}{d}\\,[\/latex]gives us the temperature in degrees Fahrenheit at a depth in feet below Earth\u2019s surface. Consider the Atlantic Ocean, which covers 22% of Earth\u2019s surface. At a certain location, at the depth of 500 feet, the temperature may be 28\u00b0F.<\/p>\n<p id=\"fs-id1165137761800\">If we create <a class=\"autogenerated-content\" href=\"#Table_03_09_02\">(Figure)<\/a>, we observe that, as the depth increases, the water temperature decreases.<\/p>\n<table id=\"Table_03_09_02\" summary=\"..\">\n<thead>\n<tr>\n<th>[latex]d,\\,[\/latex]depth<\/th>\n<th>[latex]T=\\frac{\\text{14,000}}{d}[\/latex]<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>500 ft<\/td>\n<td>[latex]\\frac{14,000}{500}=28[\/latex]<\/td>\n<td>At a depth of 500 ft, the water temperature is 28\u00b0 F.<\/td>\n<\/tr>\n<tr>\n<td>1000 ft<\/td>\n<td>[latex]\\frac{14,000}{1000}=14[\/latex]<\/td>\n<td>At a depth of 1,000 ft, the water temperature is 14\u00b0 F.<\/td>\n<\/tr>\n<tr>\n<td>2000 ft<\/td>\n<td>[latex]\\frac{14,000}{2000}=7[\/latex]<\/td>\n<td>At a depth of 2,000 ft, the water temperature is 7\u00b0 F.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137645896\">We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be <strong>inversely proportional<\/strong> and each term <strong>varies inversely<\/strong> with the other. Inversely proportional relationships are also called <strong>inverse variations<\/strong>.<\/p>\n<p id=\"fs-id1165137805474\">For our example, <a class=\"autogenerated-content\" href=\"#Figure_03_09_003\">(Figure)<\/a> depicts the <span class=\"no-emphasis\">inverse variation<\/span>. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula[latex]\\,y=\\frac{k}{x}\\,[\/latex]for inverse variation in this case uses[latex]\\,k=14,000.\\,[\/latex]<span id=\"fs-id1165135161265\"><\/span><\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/9e5691ee3d78b06ea7ff5f12e359208ead427ef2\/CNX_Precalc_Figure_03_09_003.jpg\" alt=\"Graph of y=(14000)\/x where the horizontal axis is labeled, \u201cDepth, d (ft)\u201d, and the vertical axis is labeled, \u201cTemperature, T (Degrees Fahrenheit)\u201d.\" width=\"487\" height=\"312\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<div id=\"fs-id1165135397976\" class=\"textbox key-takeaways\">\n<h3>Inverse Variation<\/h3>\n<p id=\"fs-id1165137536242\">If[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]are related by an equation of the form<\/p>\n<div id=\"fs-id1165137571596\">[latex]y=\\frac{k}{{x}^{n}}[\/latex]<\/div>\n<p id=\"fs-id1165137843973\">where[latex]\\,k\\,[\/latex]is a nonzero constant, then we say that[latex]\\,y[\/latex] varies inversely with the[latex]\\,n\\text{th}\\,[\/latex]power of[latex]\\,x.\\,[\/latex]In inversely proportional relationships, or inverse variations, there is a constant multiple[latex]\\,k={x}^{n}y.\\,[\/latex]<\/p>\n<\/div>\n<div id=\"Example_03_09_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137641735\">\n<div id=\"fs-id1165137658061\">\n<h3>Writing a Formula for an Inversely Proportional Relationship<\/h3>\n<p id=\"fs-id1165131797298\">A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.<\/p>\n<\/div>\n<div id=\"fs-id1165137654947\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137827766\">Recall that multiplying speed by time gives distance. If we let[latex]\\,t\\,[\/latex]represent the drive time in hours, and[latex]\\,v\\,[\/latex]represent the velocity (speed or rate) at which the tourist drives, then[latex]\\,vt=\\text{distance}\\text{.}\\,[\/latex]Because the distance is fixed at 100 miles,[latex]\\,vt=100\\,[\/latex] so[latex]t=100\/v.\\,[\/latex]Because time is a function of velocity, we can write[latex]\\,t\\left(v\\right).[\/latex]<\/p>\n<div id=\"eip-id1165134094568\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill t\\left(v\\right)& =& \\frac{100}{v}\\hfill \\\\ & =& 100{v}^{-1}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137748472\">We can see that the constant of variation is 100 and, although we can write the relationship using the negative exponent, it is more common to see it written as a fraction. We say that time varies inversely with velocity.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135187117\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137677962\"><strong>Given a description of an indirect variation problem, solve for an unknown.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1165137662822\" type=\"1\">\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 multiply[latex]\\,y\\,[\/latex]by the specified power of[latex]\\,x\\,[\/latex]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 id=\"Example_03_09_03\" class=\"textbox examples\">\n<div id=\"fs-id1165134328944\">\n<div>\n<h3>Solving an Inverse Variation Problem<\/h3>\n<p id=\"fs-id1165135209804\">A quantity[latex]\\,y\\,[\/latex]varies inversely with the cube of[latex]\\,x.\\,[\/latex]If[latex]\\,y=25\\,[\/latex]when[latex]\\,x=2,\\,[\/latex]find[latex]\\,y\\,[\/latex]when[latex]\\,x\\,[\/latex]is 6.<\/p>\n<\/div>\n<div id=\"fs-id1165137547532\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137627457\">The general formula for inverse variation with a cube is[latex]\\,y=\\frac{k}{{x}^{3}}.\\,[\/latex]The constant can be found by multiplying[latex]\\,y\\,[\/latex]by the cube of[latex]\\,x.\\,[\/latex]<\/p>\n<div id=\"eip-id1165132213474\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill k& =& {x}^{3}y\\hfill \\\\ & =& {2}^{3}\\cdot 25\\hfill \\\\ & =& 200\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135188786\">Now we use the constant to write an equation that represents this relationship.<\/p>\n<div id=\"eip-id1165133333885\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill y& =& \\frac{k}{{x}^{3}},\\,\\,k=200\\hfill \\\\ y\\hfill & =& \\frac{200}{{x}^{3}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137653904\">Substitute[latex]\\,x=6\\,[\/latex]and solve for[latex]\\,y.[\/latex]<\/p>\n<div id=\"eip-id1165131878567\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill y& =& \\frac{200}{{6}^{3}}\\hfill \\\\ & =& \\frac{25}{27}\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137573081\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137852181\">The graph of this equation is a rational function, as shown in <a class=\"autogenerated-content\" href=\"#Figure_03_09_004\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/66fddde286812d980caabaca4f75faa30f86dd71\/CNX_Precalc_Figure_03_09_004.jpg\" alt=\"Graph of y=25\/(x^3) with the labeled points (2, 25) and (6, 25\/27).\" width=\"488\" height=\"292\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135178098\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_09_02\">\n<div id=\"fs-id1165137810877\">\n<p id=\"fs-id1165137810878\">A quantity[latex]\\,y\\,[\/latex]varies inversely with the square of[latex]\\,x.\\,[\/latex]If[latex]\\,y=8\\,[\/latex]when[latex]\\,x=3,\\,[\/latex]find[latex]\\,y\\,[\/latex]when[latex]\\,x\\,[\/latex]is 4.<\/p>\n<\/div>\n<div id=\"fs-id1165137683037\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137683038\">[latex]\\,\\frac{9}{2}\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134054044\" class=\"bc-section section\">\n<h3>Solving Problems Involving Joint Variation<\/h3>\n<p id=\"fs-id1165137558033\">Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable[latex]\\,c,[\/latex]cost, varies jointly with the number of students,[latex]\\,n,[\/latex]and the distance,[latex]\\,d.\\,[\/latex]<\/p>\n<div id=\"fs-id1165135177639\" class=\"textbox key-takeaways\">\n<h3>Joint Variation<\/h3>\n<p id=\"fs-id1165135195246\">Joint variation occurs when a variable varies directly or inversely with multiple variables.<\/p>\n<p id=\"fs-id1165137678943\">For instance, if[latex]\\,x\\,[\/latex]varies directly with both[latex]\\,y\\,[\/latex]and[latex]\\,z,\\,[\/latex] we have[latex]\\,x=kyz.\\,[\/latex]If[latex]\\,x\\,[\/latex]varies directly with[latex]\\,y\\,[\/latex]and inversely with[latex]z,[\/latex]we have[latex]\\,x=\\frac{ky}{z}.\\,[\/latex]Notice that we only use one constant in a joint variation equation.<\/p>\n<\/div>\n<div id=\"Example_03_09_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137673524\">\n<div id=\"fs-id1165135394333\">\n<h3>Solving Problems Involving Joint Variation<\/h3>\n<p id=\"fs-id1165137452990\">A quantity[latex]\\,x\\,[\/latex]varies directly with the square of[latex]\\,y\\,[\/latex]and inversely with the cube root of[latex]\\,z.\\,[\/latex]If[latex]\\,x=6\\,[\/latex]when[latex]\\,y=2\\,[\/latex] and[latex]\\,z=8,\\,[\/latex]find[latex]\\,x\\,[\/latex]when[latex]\\,y=1\\,[\/latex] and [latex]\\,z=27.\\,[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div id=\"eip-id1165131924522\" class=\"unnumbered\">\n<details>\n<summary>Show Solution<\/summary>\n<p>Begin by writing an equation to show the relationship between the variables.<\/p>\n<div id=\"eip-id1165133305360\" class=\"unnumbered\">[latex]x=\\frac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/div>\n<p>Substitute[latex]\\,x=6,\\,[\/latex][latex]y=2,\\,[\/latex]and[latex]\\,z=8\\,[\/latex]to find the value of the constant[latex]\\,k.\\,[\/latex]<\/p>\n<div id=\"eip-id1165133420152\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill 6& =& \\frac{k{2}^{2}}{\\sqrt[3]{8}}\\hfill \\\\ \\hfill 6& =& \\frac{4k}{2}\\hfill \\\\ \\hfill 3& =& k\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137863719\">Now we can substitute the value of the constant into the equation for the relationship.<\/p>\n<div id=\"eip-id1165134503062\" class=\"unnumbered\">[latex]x=\\frac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/div>\n<p id=\"fs-id1165137742401\">To find[latex]\\,x\\,[\/latex]when[latex]\\,y=1\\,[\/latex]and[latex]\\,z=27,\\,[\/latex]we will substitute values for[latex]\\,y\\,[\/latex]and[latex]\\,z\\,[\/latex]into our equation.<\/p>\n<div id=\"eip-id1165131924522\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill x& =& \\hfill \\frac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}}\\\\ & =& 1\\hfill \\end{array}[\/latex]<\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135508625\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_09_03\">\n<div id=\"fs-id1165137588085\">\n<p id=\"fs-id1165137588086\">A quantity[latex]\\,x\\,[\/latex]varies directly with the square of[latex]\\,y\\,[\/latex]and inversely with[latex]\\,z.\\,[\/latex]If[latex]\\,x=40\\,[\/latex]when[latex]\\,y=4\\,[\/latex]and[latex]\\,z=2,\\,[\/latex]find[latex]\\,x\\,[\/latex]when[latex]\\,y=10\\,[\/latex]<br \/>\nand[latex]\\,z=25.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137755838\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137755840\">[latex]\\,x=20\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137425403\" class=\"precalculus media\">\n<p id=\"fs-id1165137844171\">Access these online resources for additional instruction and practice with direct and inverse variation.<\/p>\n<ul id=\"fs-id1165137854996\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/directvariation\">Direct Variation<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/inversevariatio\">Inverse Variation<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/directinverse\">Direct and Inverse Variation<\/a><\/li>\n<\/ul>\n<\/div>\n<p id=\"eip-251\">Visit <a href=\"http:\/\/openstaxcollege.org\/l\/PreCalcLPC03\">this website<\/a> for additional practice questions from Learningpod.<\/p>\n<\/div>\n<div id=\"fs-id1165137898092\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165133094986\" summary=\"..\">\n<tbody>\n<tr>\n<td>Direct variation<\/td>\n<td>[latex]y=k{x}^{n},k\\text{ is a nonzero constant}.[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Inverse variation<\/td>\n<td>[latex]y=\\frac{k}{{x}^{n}},k\\text{ is a nonzero constant}.[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137419773\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137723142\">\n<li>A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See <a class=\"autogenerated-content\" href=\"#Example_03_09_01\">(Figure)<\/a>.<\/li>\n<li>Two variables that are directly proportional to one another will have a constant ratio.<\/li>\n<li>A relationship where one quantity is a constant divided by another quantity is called inverse variation. See <a class=\"autogenerated-content\" href=\"#Example_03_09_02\">(Figure)<\/a>.<\/li>\n<li>Two variables that are inversely proportional to one another will have a constant multiple. See <a class=\"autogenerated-content\" href=\"#Example_03_09_03\">(Figure)<\/a>.<\/li>\n<li>In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See <a class=\"autogenerated-content\" href=\"#Example_03_09_04\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165131958288\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137430832\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165134389954\">\n<div id=\"fs-id1165134389956\">\n<p id=\"fs-id1165137612339\">What is true of the appearance of graphs that reflect a direct variation between two variables?<\/p>\n<\/div>\n<div id=\"fs-id1165137759992\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137646694\">The graph will have the appearance of a power function.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137386819\">\n<div id=\"fs-id1165137386821\">\n<p id=\"fs-id1165137581165\">If two variables vary inversely, what will an equation representing their relationship look like?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137641727\">\n<div id=\"fs-id1165135522169\">\n<p id=\"fs-id1165135522171\">Is there a limit to the number of variables that can vary jointly? Explain.<\/p>\n<\/div>\n<div id=\"fs-id1165134192951\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137598495\">No. Multiple variables may jointly vary.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131959628\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137507782\">For the following exercises, write an equation describing the relationship of the given variables.<\/p>\n<div id=\"fs-id1165137611285\">\n<div id=\"fs-id1165137611287\">\n<p id=\"fs-id1165137535129\">[latex]\\,y\\,[\/latex]varies directly as[latex]\\,x\\,[\/latex]and when[latex]\\,x=6, y=12.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176447\">\n<div>\n<p id=\"fs-id1165135250657\">[latex]\\,y\\,[\/latex]varies directly as the square of[latex]\\,x\\,[\/latex]and when[latex]\\,x=4,\\, y=80.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134381645\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134381647\">[latex]y=5{x}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134108533\">\n<div id=\"fs-id1697066\">\n<p id=\"fs-id1165135332727\">[latex]\\,y\\,[\/latex]varies directly as the square root of[latex]\\,x\\,[\/latex]and when[latex]\\,x=36, \\,y=24.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134192961\">\n<div id=\"fs-id1165135496491\">\n<p id=\"fs-id1165135496493\">[latex]\\,y\\,[\/latex]varies directly as the cube of[latex]\\,x\\,[\/latex]and when[latex]\\,x=36,\\,y=24.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137760761\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137760763\">[latex]y=10{x}^{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137809885\">\n<div id=\"fs-id1165137679904\">\n<p id=\"fs-id1165137679906\">[latex]\\,y\\,[\/latex]varies directly as the cube root of[latex]\\,x\\,[\/latex]and when[latex]\\,x=27,\\, y=15.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137924366\">\n<div id=\"fs-id1165137924368\">\n<p id=\"fs-id1165137748522\">[latex]\\,y\\,[\/latex]varies directly as the fourth power of[latex]\\,x\\,[\/latex]and when[latex]\\,x=1,\\, y=6.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137667538\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137740851\">[latex]y=6{x}^{4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137465246\">\n<p id=\"fs-id1165137465248\">[latex]\\,y\\,[\/latex]varies inversely as[latex]\\,x\\,[\/latex]and when[latex]\\,x=4,\\, y=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447334\">\n<div id=\"fs-id1165137447336\">\n<p id=\"fs-id1165133030852\">[latex]\\,y\\,[\/latex]varies inversely as the square of[latex]\\,x\\,[\/latex]and when[latex]\\,x=3,\\, y=2.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137662476\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137706986\">[latex]y=\\frac{18}{{x}^{2}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137766904\">\n<div id=\"fs-id1165135528413\">\n<p id=\"fs-id1165135528415\">[latex]\\,y\\,[\/latex]varies inversely as the cube of[latex]\\,x\\,[\/latex]and when[latex]\\,x=2,\\, y=5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137805102\">\n<div id=\"fs-id1165137805103\">\n<p id=\"fs-id1165137805104\">[latex]\\,y\\,[\/latex]varies inversely as the fourth power of[latex]\\,x\\,[\/latex]and when[latex]\\,x=3,\\, y=1.\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135524553\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135524556\">[latex]y=\\frac{81}{{x}^{4}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137898971\">\n<div id=\"fs-id1165137656213\">\n<p id=\"fs-id1165137656216\">[latex]\\,y\\,[\/latex]varies inversely as the square root of[latex]\\,x\\,[\/latex]and when[latex]\\,x=25, \\,y=3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627360\">\n<div id=\"fs-id1165137627362\">\n<p id=\"fs-id1165137843982\">[latex]\\,y\\,[\/latex]varies inversely as the cube root of[latex]\\,x\\,[\/latex]and when[latex]\\,x=64, \\,y=5.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137664262\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137664264\">[latex]y=\\frac{20}{\\sqrt[3]{x}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137541589\">[latex]\\,y\\,[\/latex]varies jointly with[latex]\\,x\\,[\/latex]and[latex]\\,z\\,[\/latex]and when[latex]\\,x=2\\,[\/latex]and[latex]\\,z=3,\\, y=36.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137658781\">\n<div id=\"fs-id1165137507651\">\n<p id=\"fs-id1165137507654\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x, z,\\,[\/latex]and[latex]\\,w\\,[\/latex]and when[latex]\\,x=1,\\, z=2,\\, w=5,\\,[\/latex]then[latex]\\,y=100.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137563934\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137563936\">[latex]y=10xzw[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135686723\">\n<p id=\"fs-id1165135237103\">[latex]\\,y\\,[\/latex]varies jointly as the square of[latex]\\,x\\,[\/latex]and the square of[latex]\\,z\\,[\/latex]and when[latex]\\,x=3\\,[\/latex]and[latex]\\,z=4,\\,[\/latex]then[latex]\\,y=72.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203444\">\n<div id=\"fs-id1165135408490\">\n<p id=\"fs-id1165135408493\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x\\,[\/latex]and the square root of[latex]\\,z\\,[\/latex]and when[latex]\\,x=2\\,[\/latex]and[latex]\\,z=25,\\,[\/latex]then[latex]\\,y=100.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134323767\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134323770\">[latex]y=10x\\sqrt{z}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137675512\">\n<div id=\"fs-id1165135445761\">\n<p id=\"fs-id1165135445763\">[latex]\\,y[\/latex]varies jointly as the square of[latex]\\,x[\/latex]the cube of[latex]\\,z[\/latex]and the square root of[latex]\\,W.\\,[\/latex]When[latex]\\,x=1,z=2,\\,[\/latex]and[latex]\\,w=36,\\,[\/latex]then[latex]\\,y=48.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134047625\">\n<div id=\"fs-id1165134047627\">\n<p id=\"fs-id1165137758007\">[latex]\\,y\\,[\/latex]<br \/>\nvaries jointly as[latex]\\,x \\text{and} z\\,[\/latex]and inversely as[latex]\\,w.\\,[\/latex]When[latex]\\,x=3, \\,z=5,\\,[\/latex]and[latex]\\,w=6,\\,[\/latex]then[latex]\\,y=10.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137723403\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]y=4\\frac{xz}{w}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661782\">\n<div id=\"fs-id1165137661783\">\n<p>[latex]\\,y\\,[\/latex]varies jointly as the square of[latex]\\,x\\,[\/latex]and the square root of[latex]\\,z\\,[\/latex]and inversely as the cube of[latex]\\,w\\text{.\\hspace{0.17em}}[\/latex]When[latex]\\,x=3,z=4,\\,[\/latex]and[latex]\\,w=3,\\,[\/latex]then[latex]\\,y=6.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"eip-387\">\n<div>\n<p id=\"eip-646\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x\\,[\/latex]and[latex]\\,z\\,[\/latex]and inversely as the square root of[latex]\\,w\\,[\/latex]and the square of[latex]\\,t\\text{\\hspace{0.17em}.}[\/latex]When[latex]\\,x=3,z=1,w=25,\\,[\/latex]and[latex]\\,t=2,\\,[\/latex]then[latex]\\,y=6.[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]y=40\\frac{xz}{\\sqrt{w}{t}^{2}}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137741107\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165137476436\">For the following exercises, use the given information to find the unknown value.<\/p>\n<div id=\"fs-id1165137476439\">\n<div id=\"fs-id1165135511345\">\n<p id=\"fs-id1165135511347\">[latex]\\,y\\,[\/latex]varies directly as[latex]\\,x.\\,[\/latex]When[latex]\\,x=3,\\,[\/latex]then[latex]\\,y=12.\\,[\/latex]Find[latex]\\,y\\,[\/latex]wneh[latex]\\,x=20.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137619342\">\n<div id=\"fs-id1165137619344\">\n<p id=\"fs-id1165137677831\">[latex]\\,y\\,[\/latex]varies directly as the square of[latex]\\,x.\\,[\/latex]When[latex]\\,x=2,\\,[\/latex]then[latex]\\,y=16.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]x=8.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135194956\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137476412\">[latex]y=256[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137638624\">\n<div id=\"fs-id1165137638626\">\n<p id=\"fs-id1165137805207\">[latex]\\,y\\,[\/latex]varies directly as the cube of[latex]\\,x.\\,[\/latex]When[latex]\\,x=3,\\,[\/latex]then[latex]\\,y=5. \\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137722717\">\n<div>\n<p id=\"fs-id1165137737434\">[latex]\\,y\\,[\/latex]varies directly as the square root of[latex]\\,x.\\,[\/latex]When[latex]\\,x=16,\\,[\/latex]then[latex]\\,y=4.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=36.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137571788\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]y=6[\/latex]<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137812576\">\n<p id=\"fs-id1165137693975\">[latex]\\,y\\,[\/latex]varies directly as the cube root of[latex]\\,x.\\,[\/latex]When[latex]\\,x=125,\\,[\/latex]then[latex]\\,y=15.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=1,000.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137476513\">\n<div id=\"fs-id1165137476515\">\n<p id=\"fs-id1165137476517\">[latex]\\,y\\,[\/latex]varies inversely with[latex]\\,x.\\,[\/latex]When[latex]\\,x=3,\\,[\/latex]then[latex]\\,y=2.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137445714\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137651705\">[latex]y=6[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627408\">\n<div id=\"fs-id1165137627410\">\n<p id=\"fs-id1165134306730\">[latex]\\,y\\,[\/latex]varies inversely with the square of[latex]\\,x.\\,[\/latex]When[latex]\\,x=4,\\,[\/latex]then[latex]\\,y=3.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545751\">\n<div id=\"fs-id1165135545753\">\n<p id=\"fs-id1165135545756\">[latex]\\,y\\,[\/latex]varies inversely with the cube of[latex]\\,x.\\,[\/latex]When[latex]\\,x=3,\\,[\/latex]then[latex]\\,y=1.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137431582\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135535029\">[latex]y=27[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137932431\">\n<div id=\"fs-id1165137932433\">\n<p id=\"fs-id1165135403502\">[latex]\\,y\\,[\/latex]varies inversely with the square root of[latex]\\,x.\\,[\/latex]When[latex]\\,x=64,\\,[\/latex]then[latex]\\,y=12.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=36.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696700\">\n<div id=\"fs-id1165137579619\">\n<p id=\"fs-id1165137579622\">[latex]\\,y\\,[\/latex]varies inversely with the cube root of[latex]\\,x.\\,[\/latex]When[latex]\\,x=27,\\,[\/latex]then[latex]\\,y=5.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=125.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137542372\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137542374\">[latex]y=3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137706810\">\n<div>\n<p id=\"fs-id1165137885945\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x \\text{and} z.[\/latex]When[latex]\\,x=4\\,[\/latex]and[latex]\\,z=2,\\,[\/latex]then[latex]\\,y=16.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=3\\,[\/latex]and[latex]\\,z=3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134401069\">\n<div id=\"fs-id1165134401071\">\n<p id=\"fs-id1165137863353\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x, z, \\text{and} w.[\/latex]When[latex]\\,x=2,\\,[\/latex][latex]z=1,\\,[\/latex]and[latex]\\,w=12,\\,[\/latex]then[latex]\\,y=72.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=1,\\,[\/latex][latex]z=2,\\,[\/latex] and[latex]\\,w=3.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137406882\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137444724\">[latex]y=18[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135411328\">\n<div id=\"fs-id1165135411330\">\n<p id=\"fs-id1165137393586\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x\\,[\/latex]and the square of[latex]\\,\\mathrm{z.}\\,[\/latex]When[latex]\\,x=2\\,[\/latex]and[latex]\\,z=4,\\,[\/latex]then[latex]\\,y=144.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=4\\,[\/latex]and[latex]\\,z=5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737695\">\n<div>\n<p id=\"fs-id1165137662667\">[latex]\\,y\\,[\/latex]varies jointly as the square of[latex]\\,x\\,[\/latex]and the square root of[latex]\\,z.\\,[\/latex]When[latex]\\,x=2\\,[\/latex]and[latex]\\,z=9,\\,[\/latex]then[latex]\\,y=24.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=3\\,[\/latex]and[latex]\\,z=25.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135543455\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135543457\">[latex]y=90[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137637576\">\n<div>\n<p id=\"fs-id1165137637581\">[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x\\,[\/latex]and[latex]\\,z\\,[\/latex]and inversely as[latex]\\,w.\\,[\/latex]When[latex]\\,x=5,\\,[\/latex][latex]z=2, \\,[\/latex]and[latex]\\,w=20,\\,[\/latex]then[latex]y=4.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=3\\,[\/latex]and[latex]\\,z=8, \\,[\/latex]and[latex]\\, w=48.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440221\">\n<div id=\"fs-id1165135194226\">\n<p id=\"fs-id1165135194229\">[latex]\\,y\\,[\/latex]varies jointly as the square of[latex]\\,x\\,[\/latex]and the cube of[latex]\\,z\\,[\/latex]and inversely as the square root of[latex]\\,w\\.[\/latex]When[latex]\\,x=2,\\,[\/latex][latex]z=2,\\,[\/latex]and[latex]\\,w=64,\\,[\/latex]then[latex]\\,y=12.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=1,\\,[\/latex][latex]z=3,\\,[\/latex]and[latex]\\,w=4.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137470711\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137470713\">[latex]y=\\frac{81}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451415\">\n<div id=\"fs-id1165137663674\">\n<p id=\"fs-id1165137663676\">[latex]\\,y\\,[\/latex]varies jointly as the square of[latex]\\,x\\,[\/latex]and of[latex]\\,z\\,[\/latex]and inversely as the square root of[latex]\\,w\\,[\/latex]and of[latex]\\,t\\.[\/latex]When[latex]\\,x=2,\\,[\/latex][latex]z=3,\\,[\/latex][latex]w=16,\\,[\/latex]and[latex]\\,t=3,\\,[\/latex]then[latex]\\,y=1.\\,[\/latex]Find[latex]\\,y\\,[\/latex]when[latex]\\,x=3,\\,[\/latex][latex]z=2,\\,[\/latex][latex]w=36,\\,[\/latex]and[latex]\\,t=5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135496464\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137758637\">For the following exercises, use a calculator to graph the equation implied by the given variation.<\/p>\n<div>\n<div id=\"fs-id1165137560300\">\n<p id=\"fs-id1165137560302\">[latex]\\,y\\,[\/latex]varies directly with the square of[latex]\\,x\\,[\/latex]and when[latex]\\,x=2, y=3.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137832227\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137748367\">[latex]y=\\frac{3}{4}{x}^{2}[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/56c4b015811357d76fcb67eea800b48dbc5db5bc\/CNX_Precalc_Figure_03_09_201.jpg\" alt=\"Graph of y=3\/4(x^2).\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137785175\">\n<div id=\"fs-id1165137785177\">\n<p id=\"fs-id1165137811898\">[latex]\\,y\\,[\/latex]varies directly as the cube of[latex]\\,x\\,[\/latex]and when[latex]\\,x=2, y=4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137481150\">\n<div id=\"fs-id1165137735571\">\n<p id=\"fs-id1165137735573\">[latex]\\,y\\,[\/latex]varies directly as the square root of[latex]\\,x\\,[\/latex]and when[latex]\\,x=36, y=2.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137936864\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137936866\">[latex]y=\\frac{1}{3}\\sqrt{x}[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/fc13afa0b8f2db0496a47e9bc91b52dc008cc086\/CNX_Precalc_Figure_03_09_203.jpg\" alt=\"Graph of y=1\/3sqrt(x).\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137460988\">\n<div id=\"fs-id1165137460990\">\n<p id=\"fs-id1165137460992\">[latex]\\,y\\,[\/latex]varies inversely with[latex]\\,x\\,[\/latex]and when[latex]\\,x=6, y=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137472903\">\n<div id=\"fs-id1165137472905\">\n<p id=\"fs-id1165137638128\">[latex]\\,y\\,[\/latex]varies inversely as the square of[latex]\\,x\\,[\/latex]and when[latex]\\,x=1, y=4.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134248054\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134248055\">[latex]y=\\frac{4}{{x}^{2}}[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/53473051e0a2f6be7fe18c2acd62a8912b7a4961\/CNX_Precalc_Figure_03_09_205.jpg\" alt=\"Graph of y=4\/(x^2).\" \/><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193362\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165137762704\">For the following exercises, use Kepler\u2019s Law, which states that the square of the time,[latex]\\,T,\\,[\/latex]required for a planet to orbit the Sun varies directly with the cube of the mean distance,[latex]\\,a,\\,[\/latex]that the planet is from the Sun.<\/p>\n<div id=\"fs-id1165137401766\">\n<div id=\"fs-id1165137805286\">\n<p id=\"fs-id1165137805288\">Using Earth\u2019s time of 1 year and mean distance of 93 million miles, find the equation relating [latex]\\,T\\,[\/latex] and [latex]\\,a.\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135433006\">\n<div id=\"fs-id1165137812556\">\n<p id=\"fs-id1165137677528\">Use the result from the previous exercise to determine the time required for Mars to orbit the Sun if its mean distance is 142 million miles.<\/p>\n<\/div>\n<div id=\"fs-id1165137871569\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137871571\">1.89 years<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137417733\">\n<div id=\"fs-id1165137417735\">\n<p id=\"fs-id1165137417737\">Using Earth\u2019s distance of 150 million kilometers, find the equation relating[latex]\\,T\\,[\/latex]and[latex]\\,a.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137552664\">\n<div id=\"fs-id1165137552667\">\n<p id=\"fs-id1165137552669\">Use the result from the previous exercise to determine the time required for Venus to orbit the Sun if its mean distance is 108 million kilometers.<\/p>\n<\/div>\n<div id=\"fs-id1165137668810\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137807323\">0.61 years<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503720\">\n<div id=\"fs-id1165135500737\">\n<p id=\"fs-id1165135500739\">Using Earth\u2019s distance of 1 astronomical unit (A.U.), determine the time for Saturn to orbit the Sun if its mean distance is 9.54 A.U.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137407207\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id1165137407212\">For the following exercises, use the given information to answer the questions.<\/p>\n<div id=\"fs-id1165137662862\">\n<div id=\"fs-id1165137662864\">\n<p id=\"fs-id1165137843829\">The distance[latex]\\,s\\,[\/latex]that an object falls varies directly with the square of the time,[latex]\\,t,\\,[\/latex]of the fall. If an object falls 16 feet in one se<strong>c<\/strong>ond, how long for it to fall 144 feet?<\/p>\n<\/div>\n<div id=\"fs-id1165133101738\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133101740\">3 seconds<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810169\">\n<div id=\"fs-id1165137810171\">\n<p id=\"fs-id1165134374051\">The velocity[latex]\\,v\\,[\/latex]of a falling object varies directly to the time,[latex]\\,t,\\,[\/latex]of the fall. If after 2 seconds, the velocity of the object is 64 feet per second, what is the velocity after 5 seconds?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731890\">\n<div id=\"fs-id1165137731892\">\n<p id=\"fs-id1165137731894\">The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?<\/p>\n<\/div>\n<div id=\"fs-id1165137728416\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137728418\">48 inches<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137619719\">\n<div id=\"fs-id1165137619721\">\n<p id=\"fs-id1165137658372\">The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135613164\">\n<div id=\"fs-id1165135613167\">\n<p id=\"fs-id1165135613169\">The weight of an object above the surface of Earth varies inversely with the square of the distance from the center of Earth. If a body weighs 50 pounds when it is 3960 miles from Earth\u2019s center, what would it weigh it were 3970 miles from Earth\u2019s center?<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137760890\">49.75 pounds<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545895\">\n<div id=\"fs-id1165135545898\">\n<p id=\"fs-id1165137598943\">The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot-candles at a distance of 3 meters. Find the intensity level at 8 meters.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137836546\">\n<div id=\"fs-id1165137836548\">\n<p id=\"fs-id1165134118360\">The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms.<\/p>\n<\/div>\n<div id=\"fs-id1165135194664\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135194666\">33.33 amperes<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137535610\">\n<div id=\"fs-id1165137535612\">\n<p id=\"fs-id1165137535614\">The force exerted by the wind on a plane surface varies jointly with the square of the velocity of the wind and with the area of the plane surface. If the area of the surface is 40 square feet surface and the wind velocity is 20 miles per hour, the resulting force is 15 pounds. Find the force on a surface of 65 square feet with a velocity of 30 miles per hour.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137676536\">\n<div id=\"fs-id1165137676537\">\n<p id=\"fs-id1165137676538\">The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?<\/p>\n<\/div>\n<div id=\"fs-id1165137641112\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137676540\">2.88 inches<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137605795\">\n<div id=\"fs-id1165137605797\">\n<p id=\"fs-id1165137462259\">The kinetic energy[latex]\\,K\\,[\/latex]of a moving object varies jointly with its mass[latex]\\,m\\,[\/latex]and the square of its velocity[latex]\\,v.\\,[\/latex]If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135208752\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id1165137548191\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f929bd85-ed50-4a67-a528-40008aa14d0c\">Quadratic Functions<\/a><\/h4>\n<p id=\"fs-id1165137639859\">For the following exercises, write the quadratic function in standard form. Then give the vertex and axes intercepts. Finally, graph the function.<\/p>\n<div id=\"fs-id1165137477136\">\n<div id=\"fs-id1165137477138\">\n<p id=\"fs-id1165137477140\">[latex]f\\left(x\\right)={x}^{2}-4x-5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137935589\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137811941\">[latex]f\\left(x\\right)={\\left(x-2\\right)}^{2}-9\\,\\text{vertex} \\left(2,\u20139\\right), \\text{intercepts} \\left(5,0\\right); \\left(\u20131,0\\right); \\left(0,\u20135\\right)[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/aa64c0c5507f70d1697a35a9c5ce60ec4ba29f5f\/CNX_Precalc_Figure_03_09_206.jpg\" alt=\"Graph of f(x)=x^2-4x-5.\" \/><\/p>\n<div><\/details>\n<p><span id=\"fs-id1165137460981\"><\/span><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137863438\">\n<div id=\"fs-id1165137626743\">\n<p id=\"fs-id1165137626745\">[latex]f\\left(x\\right)=-2{x}^{2}-4x[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137923495\">For the following exercises, find the equation of the quadratic function using the given information.<\/p>\n<div id=\"fs-id1165137923498\">\n<div id=\"fs-id1165135445905\">\n<p id=\"fs-id1165135445907\">The vertex is[latex]\\left(\u20132,3\\right)[\/latex]and a point on the graph is[latex]\\,\\left(3,6\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137724885\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137398179\">[latex]f\\left(x\\right)=\\frac{3}{25}{\\left(x+2\\right)}^{2}+3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135457900\">\n<div id=\"fs-id1165135457902\">\n<p id=\"fs-id1165135457904\">The vertex is[latex]\\,\\left(\u20133,6.5\\right)\\,[\/latex]and a point on the graph is[latex]\\,\\left(2,6\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135443862\">For the following exercises, complete the task.<\/p>\n<div id=\"fs-id1165135443865\">\n<div id=\"fs-id1165135443867\">\n<p id=\"fs-id1165137767424\">A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.<\/p>\n<\/div>\n<div id=\"fs-id1165137725075\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137725077\">300 meters by 150 meters, the longer side parallel to river.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137422734\">\n<div id=\"fs-id1165137422736\">\n<p id=\"fs-id1165135706774\">An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height,[latex]\\,h,\\,[\/latex]in terms of horizontal distance traveled,[latex]\\,x,\\,[\/latex]given by[latex]\\,h\\left(x\\right)=\\frac{-32}{{\\left(120\\right)}^{2}}{x}^{2}+x.\\,[\/latex]Find the maximum height the object attains.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133233027\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/e017cb02-d494-4d95-8d09-966d0968a606\">Power Functions and Polynomial Functions<\/a><\/h4>\n<p id=\"fs-id1165137596106\">For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.<\/p>\n<div id=\"fs-id1165137596111\">\n<div id=\"fs-id1165137927725\">\n<p id=\"fs-id1165137927727\">[latex]f\\left(x\\right)=4{x}^{5}-3{x}^{3}+2x-1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137451016\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137430842\">Yes, degree = 5, leading coefficient = 4<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137786551\">\n<div id=\"fs-id1165137786554\">\n<p id=\"fs-id1165137595265\">[latex]f\\left(x\\right)={5}^{x+1}-{x}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135407052\">\n<div id=\"fs-id1165135407054\">\n<p id=\"fs-id1165135407057\">[latex]f\\left(x\\right)={x}^{2}\\left(3-6x+{x}^{2}\\right)[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>Yes, degree = 4, leading coefficient = 1<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135181695\">For the following exercises, determine end behavior of the polynomial function.<\/p>\n<div id=\"fs-id1165135181698\">\n<div id=\"fs-id1165137735839\">\n<p id=\"fs-id1165137735841\">[latex]f\\left(x\\right)=2{x}^{4}+3{x}^{3}-5{x}^{2}+7[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137643403\">\n<div id=\"fs-id1165137643405\">\n<p id=\"fs-id1165137643407\">[latex]f\\left(x\\right)=4{x}^{3}-6{x}^{2}+2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134155145\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137386933\">[latex]\\text{As}\\,x\\to -\\infty ,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137601687\">\n<div id=\"fs-id1165137601688\">\n<p id=\"fs-id1165137601689\">[latex]f\\left(x\\right)=2{x}^{2}\\left(1+3x-{x}^{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135169092\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/644f6a93-7429-4768-8752-eccf5fb8e0bb\">Graphs of Polynomial Functions<\/a><\/h4>\n<p id=\"fs-id1165137451676\">For the following exercises, find all zeros of the polynomial function, noting multiplicities.<\/p>\n<div id=\"fs-id1165137451679\">\n<div id=\"fs-id1165137557555\">\n<p id=\"fs-id1165137557556\">[latex]f\\left(x\\right)={\\left(x+3\\right)}^{2}\\left(2x-1\\right){\\left(x+1\\right)}^{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137432758\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137806236\">\u20133 with multiplicity 2,[latex]-\\frac{1}{2}[\/latex]with multiplicity 1, \u20131 with multiplicity 3<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135683516\">\n<div id=\"fs-id1165135683518\">\n<p id=\"fs-id1165137402103\">[latex]f\\left(x\\right)={x}^{5}+4{x}^{4}+4{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137645405\">\n<div id=\"fs-id1165137404519\">\n<p id=\"fs-id1165137404521\">[latex]f\\left(x\\right)={x}^{3}-4{x}^{2}+x-4[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137749873\">4 with multiplicity 1<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137846483\">For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.<\/p>\n<div id=\"fs-id1165137846487\">\n<div id=\"fs-id1165137697060\"><span id=\"fs-id1165135368421\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/2bfb5f803ebe608bc71ee1abd0f111c8a6d38ded\/CNX_Precalc_Figure_03_09_208.jpg\" alt=\"Graph of an odd-degree polynomial with two turning points.\" \/><\/span><\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137742630\"><span id=\"fs-id1165134313923\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/dc5c225da94ba479e35a0d1bee5759184b6f78a5\/CNX_Precalc_Figure_03_09_209.jpg\" alt=\"Graph of an even-degree polynomial with two turning points.\" \/><\/span><\/div>\n<div id=\"fs-id1165137539727\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137648165\">[latex]\\frac{1}{2}\\,[\/latex]with multiplicity 1, 3 with multiplicity 3<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137557643\">\n<div id=\"fs-id1165135694414\">\n<p id=\"fs-id1165135694416\">Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the function[latex]\\,f\\left(x\\right)={x}^{3}-5x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137920803\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/11f2bd05-8eff-4fb8-9e06-02a657fbb031\">Dividing Polynomials<\/a><\/h4>\n<p id=\"fs-id1165137749390\">For the following exercises, use long division to find the quotient and remainder.<\/p>\n<div id=\"fs-id1165137696726\">\n<div id=\"fs-id1165137696728\">\n<p id=\"fs-id1165137696730\">[latex]\\frac{{x}^{3}-2{x}^{2}+4x+4}{x-2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137423736\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137423738\">[latex]\\,{x}^{2}+4\\,[\/latex] with remainder 12<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135446689\">\n<div id=\"fs-id1165135446691\">\n<p id=\"fs-id1165135446693\">[latex]\\frac{3{x}^{4}-4{x}^{2}+4x+8}{x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165132944971\">For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.<\/p>\n<div id=\"fs-id1165134378622\">\n<div id=\"fs-id1165134378624\">\n<p id=\"fs-id1165134378626\">[latex]\\frac{{x}^{3}-2{x}^{2}+5x-1}{x+3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135365932\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135365934\">[latex]{x}^{2}-5x+20-\\frac{61}{x+3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135560695\">\n<div id=\"fs-id1165135560697\">\n<p id=\"fs-id1165137762397\">[latex]\\frac{{x}^{3}+4x+10}{x-3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137935190\">\n<div id=\"fs-id1165137935192\">\n<p id=\"fs-id1165135176562\">[latex]\\frac{2{x}^{3}+6{x}^{2}-11x-12}{x+4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137772480\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135558432\">[latex]2{x}^{2}-2x-3[\/latex], so factored form is [latex]\\left(x+4\\right)\\left(2{x}^{2}-2x-3\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135414382\">\n<div id=\"fs-id1165135414383\">\n<p id=\"fs-id1165137735052\">[latex]\\frac{3{x}^{4}+3{x}^{3}+2x+2}{x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135194274\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/1bb57d2c-789f-4252-bfe4-edc0873f1e29\">Zeros of Polynomial Functions<\/a><\/h4>\n<p id=\"fs-id1165137653216\">For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.<\/p>\n<div id=\"fs-id1165134248795\">\n<div id=\"fs-id1165134248797\">\n<p id=\"fs-id1165134248799\">[latex]2{x}^{3}-3{x}^{2}-18x-8=0[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137780754\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137780756\">[latex]\\left\\{-2, 4, -\\frac{1}{2}\\right\\}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137901245\">\n<div id=\"fs-id1165137901248\">\n<p id=\"fs-id1165137901250\">[latex]3{x}^{3}+11{x}^{2}+8x-4=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137452024\">\n<div id=\"fs-id1165135640601\">\n<p id=\"fs-id1165135640603\">[latex]2{x}^{4}-17{x}^{3}+46{x}^{2}-43x+12=0[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137654771\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]\\left\\{1, 3, 4, \\frac{1}{2}\\right\\}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135531610\">\n<div id=\"fs-id1165135531612\">\n<p id=\"fs-id1165137770047\">[latex]4{x}^{4}+8{x}^{3}+19{x}^{2}+32x+12=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137407778\">For the following exercises, use Descartes\u2019 Rule of Signs to find the possible number of positive and negative solutions.<\/p>\n<div id=\"fs-id1165134573782\">\n<div id=\"fs-id1165134573784\">\n<p id=\"fs-id1165134573786\">[latex]{x}^{3}-3{x}^{2}-2x+4=0[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134108392\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134108394\">0 or 2 positive, 1 negative<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135613231\">\n<div id=\"fs-id1165135613233\">\n<p id=\"fs-id1165134224004\">[latex]2{x}^{4}-{x}^{3}+4{x}^{2}-5x+1=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134224073\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/28d4cfda-bec3-48cf-9695-2cc71939e22a\">Rational Functions<\/a><\/h4>\n<p id=\"fs-id1165135176618\">For the following exercises, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph of the function.<\/p>\n<div id=\"fs-id1165135176623\">\n<div id=\"fs-id1165135516926\">\n<p id=\"fs-id1165135516928\">[latex]f\\left(x\\right)=\\frac{x+2}{x-5}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137740757\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137740759\">Intercepts [latex]\\left(\u20132,0\\right)\\text{and}\\left(0,-\\frac{2}{5}\\right)[\/latex], Asymptotes[latex]\\,x=5\\,[\/latex]<br \/>\nand[latex]\\,y=1.[\/latex]<\/p>\n<div><\/div>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/37cb1aa7a7578e54135f0defd76697c6ec0236cc\/CNX_Precalc_Figure_03_09_210.jpg\" alt=\"Graph of f(x)=(x+1)\/(x-5).\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135205950\">\n<div id=\"fs-id1165135529121\">\n<p id=\"fs-id1165135529123\">[latex]f\\left(x\\right)=\\frac{{x}^{2}+1}{{x}^{2}-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137863121\">\n<div id=\"fs-id1165137863124\">\n<p id=\"fs-id1165137863126\">[latex]f\\left(x\\right)=\\frac{3{x}^{2}-27}{{x}^{2}+x-2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137452244\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137452246\">Intercepts (3, 0), (-3, 0), and[latex]\\,\\left(0,\\frac{27}{2}\\right)\\,[\/latex], Asymptotes[latex]\\,x=1, x=\u20132, y=3.[\/latex]<\/p>\n<div><\/div>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/1cb79c1497ffbdf7e97bf632533f2d5253e8671c\/CNX_Precalc_Figure_03_09_212.jpg\" alt=\"Graph of f(x)=(3x^2-27)\/(x^2+x-2).\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134226797\">\n<div>[latex]f\\left(x\\right)=\\frac{x+2}{{x}^{2}-9}[\/latex]<\/div>\n<\/div>\n<p>For the following exercises, find the slant asymptote.<\/p>\n<div id=\"fs-id1165137842426\">\n<div id=\"fs-id1165137842428\">[latex]f\\left(x\\right)=\\frac{{x}^{2}-1}{x+2}[\/latex]<\/div>\n<div id=\"fs-id1165137642577\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135252057\">[latex]y=\\text{ }x-2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438219\">\n<div id=\"fs-id1165135508577\">\n<p id=\"fs-id1165135508579\">[latex]f\\left(x\\right)=\\frac{2{x}^{3}-{x}^{2}+4}{{x}^{2}+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/ce233d35-8468-4ded-97dc-b34c3475a26a\">Inverses and Radical Functions<\/a><\/h4>\n<p>For the following exercises, find the inverse of the function with the domain given.<\/p>\n<div>\n<div id=\"fs-id1165135458657\">\n<p id=\"fs-id1165135458659\">[latex]f\\left(x\\right)={\\left(x-2\\right)}^{2},\\,x\\ge 2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135640618\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135593176\">[latex]{f}^{-1}\\left(x\\right)=\\sqrt{x}+2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133103950\">\n<div id=\"fs-id1165133103952\">\n<p id=\"fs-id1165133103954\">[latex]f\\left(x\\right)={\\left(x+4\\right)}^{2}-3,\\,x\\ge -4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137770318\">\n<div id=\"fs-id1165137770321\">\n<p id=\"fs-id1165137770323\">[latex]f\\left(x\\right)={x}^{2}+6x-2,\\,x\\ge -3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135210050\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135210052\">[latex]{f}^{-1}\\left(x\\right)=\\sqrt{x+11}-3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135301653\">\n<div id=\"fs-id1165135301655\">\n<p id=\"fs-id1165137619912\">[latex]f\\left(x\\right)=2{x}^{3}-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137749717\">\n<div id=\"fs-id1165137749719\">\n<p id=\"fs-id1165137692176\">[latex]f\\left(x\\right)=\\sqrt{4x+5}-3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134224016\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134224018\">[latex]{f}^{-1}\\left(x\\right)=\\frac{{\\left(x+3\\right)}^{2}-5}{4},\\,x\\ge -3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135591017\">\n<div id=\"fs-id1165135342998\">\n<p id=\"fs-id1165135343000\">[latex]f\\left(x\\right)=\\frac{x-3}{2x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135245788\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/c941f485-13ab-481a-a029-2160ef470707\">Modeling Using Variation<\/a><\/h4>\n<p id=\"fs-id1165137462872\">For the following exercises, find the unknown value.<\/p>\n<div id=\"fs-id1165137462875\">\n<div id=\"fs-id1165137462877\">\n<p id=\"fs-id1165137939566\">[latex]\\,y\\,[\/latex] varies directly as the square of[latex]\\,x.\\,[\/latex] If when[latex]\\,x=3,\\text{ }y=36,\\,[\/latex]find[latex]\\,y\\,[\/latex]if[latex]\\,x=4.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135319534\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135203742\">[latex]y=64[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137872027\">\n<div id=\"fs-id1165137872029\">\n<p id=\"fs-id1165137696624\">[latex]\\,y\\,[\/latex] varies inversely as the square root of[latex]\\,x\\,[\/latex] If when[latex]\\,x=25,\\text{ }y=2,\\,[\/latex]find[latex]\\,y\\,[\/latex]if[latex]\\,x=4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135209414\">\n<div>\n<p>[latex]\\,y\\,[\/latex]varies jointly as the cube of[latex]\\,x\\,[\/latex]and as[latex]\\,z.\\,[\/latex]If when[latex]\\,x=1\\,[\/latex]and[latex]\\,z=2,\\,[\/latex][latex]y=6,\\,[\/latex]find[latex]\\,y\\,[\/latex]if[latex]\\,x=2\\,[\/latex]and[latex]\\,z=3.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137934401\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137452780\">[latex]y\\text{ }=\\text{ }72[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137456499\">\n<div id=\"fs-id1165137456502\">\n<p>[latex]\\,y\\,[\/latex]varies jointly as[latex]\\,x\\,[\/latex]and the square of[latex]\\,z\\,[\/latex]and inversely as the cube of[latex]\\,w.\\,[\/latex]If when[latex]\\,x=3,\\,[\/latex][latex]z=4,\\,[\/latex]and[latex]\\,w=2,\\,[\/latex][latex]y=48,\\,[\/latex]find[latex]\\,y\\,[\/latex]if[latex]\\,x=4,\\,[\/latex][latex]z=5,\\,[\/latex]and[latex]\\,w=3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137435024\">For the following exercises, solve the application problem.<\/p>\n<div id=\"fs-id1165137435027\">\n<div id=\"fs-id1165137435029\">\n<p id=\"fs-id1165135169106\">The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.<\/p>\n<\/div>\n<div id=\"fs-id1165135698641\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>148.5 pounds<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135698649\">\n<div id=\"fs-id1165135536256\">\n<p>The volume[latex]\\,V\\,[\/latex]of an ideal gas varies directly with the temperature[latex]\\,T\\,[\/latex]and inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137602034\" class=\"practice-test\">\n<h3>Chapter Test<\/h3>\n<p id=\"fs-id1165134262509\">Give the degree and leading coefficient of the following polynomial function.<\/p>\n<div id=\"fs-id1165134262512\">\n<div id=\"fs-id1165134211394\">\n<p id=\"fs-id1165134211396\">[latex]f\\left(x\\right)={x}^{3}\\left(3-6{x}^{2}-2{x}^{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135424704\">Determine the end behavior of the polynomial function.<\/p>\n<div id=\"fs-id1165137394106\">\n<div id=\"fs-id1165137394108\">\n<p id=\"fs-id1165137394110\">[latex]f\\left(x\\right)=8{x}^{3}-3{x}^{2}+2x-4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135192394\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135192396\">[latex]As\\,x\\to -\\infty ,\\,f\\left(x\\right)\\to -\\infty ,\\,as\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135389015\">\n<p id=\"fs-id1165135503719\">[latex]f\\left(x\\right)=-2{x}^{2}\\left(4-3x-5{x}^{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.<\/p>\n<div id=\"fs-id1165137732249\">\n<div id=\"fs-id1165137732251\">\n<p id=\"fs-id1165137732253\">[latex]f\\left(x\\right)={x}^{2}+2x-8[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135503809\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135694562\">[latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-9[\/latex], vertex[latex]\\,\\left(-1,-9\\right)[\/latex], intercepts[latex]\\,\\left(2,0\\right);\\left(-4,0\\right);\\,\\left(0,-8\\right)[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/c70d329ef7c037f7c148c4c90b777d9b283be80e\/CNX_Precalc_Figure_03_09_214.jpg\" alt=\"Graph of f(x)=x^2+2x-8.\" \/><\/details>\n<p><span id=\"fs-id1165137653687\"><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135354880\">Given information about the graph of a quadratic function, find its equation.<\/p>\n<div id=\"fs-id1165137736986\">\n<div id=\"fs-id1165137736988\">\n<p id=\"fs-id1165137736990\">Vertex[latex]\\,\\left(2,0\\right)\\,[\/latex]and point on graph[latex]\\,\\left(4,12\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Solve the following application problem.<\/p>\n<div id=\"fs-id1165131797287\">\n<div id=\"fs-id1165131797290\">\n<p id=\"fs-id1165131797292\">A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.<\/p>\n<\/div>\n<div id=\"fs-id1165137673750\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137673752\">60,000 square feet<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135454021\">Find all zeros of the following polynomial functions, noting multiplicities.<\/p>\n<div id=\"fs-id1165135454024\">\n<div id=\"fs-id1165137726459\">\n<p id=\"fs-id1165137726461\">[latex]f\\left(x\\right)={\\left(x-3\\right)}^{3}\\left(3x-1\\right){\\left(x-1\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135453256\">\n<div id=\"fs-id1165135453258\">\n<p id=\"fs-id1165135453260\">[latex]f\\left(x\\right)=2{x}^{6}-12{x}^{5}+18{x}^{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134040553\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134040555\">0 with multiplicity 4, 3 with multiplicity 2<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134040560\">Based on the graph, determine the zeros of the function and multiplicities.<\/p>\n<div id=\"fs-id1165137871376\">\n<div id=\"fs-id1165137871378\"><span id=\"fs-id1165135264841\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/ea0a3097aab1ecce6abbb2dbeb500e1f428d1cb5\/CNX_Precalc_Figure_03_09_215.jpg\" alt=\"Graph of an odd-degree polynomial with two turning points.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165135191491\">Use long division to find the quotient.<\/p>\n<div id=\"fs-id1165135191495\">\n<div id=\"fs-id1165135191497\">\n<p id=\"fs-id1165134122806\">[latex]\\frac{2{x}^{3}+3x-4}{x+2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137427782\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137427784\">[latex]2{x}^{2}-4x+11-\\frac{26}{x+2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135342123\">Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.<\/p>\n<div id=\"fs-id1165135342127\">\n<div id=\"fs-id1165135342129\">\n<p id=\"fs-id1165137732538\">[latex]\\frac{{x}^{4}+3{x}^{2}-4}{x-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137833008\">\n<div id=\"fs-id1165137833010\">\n<p id=\"fs-id1165137827866\">[latex]\\frac{2{x}^{3}+5{x}^{2}-7x-12}{x+3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135152117\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135152119\">[latex]2{x}^{2}-x-4[\/latex]. So factored form is[latex]\\,\\left(x+3\\right)\\left(2{x}^{2}-x-4\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135694398\">Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.<\/p>\n<div id=\"fs-id1165135694401\">\n<div id=\"fs-id1165135694403\">\n<p id=\"fs-id1165137407530\">[latex]f\\left(x\\right)=2{x}^{3}+5{x}^{2}-6x-9[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135161247\">\n<div id=\"fs-id1165135161249\">\n<p id=\"fs-id1165135161251\">[latex]f\\left(x\\right)=4{x}^{4}+8{x}^{3}+21{x}^{2}+17x+4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135188171\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135188173\">[latex]-\\frac{1}{2}\\,[\/latex]<br \/>\n(has multiplicity 2),[latex]\\,\\frac{-1\u00b1i\\sqrt{15}}{2}\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192046\">\n<div id=\"fs-id1165135192048\">[latex]f\\left(x\\right)=4{x}^{4}+16{x}^{3}+13{x}^{2}-15x-18[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137758031\">\n<div id=\"fs-id1165137758034\">\n<p id=\"fs-id1165134190707\">[latex]f\\left(x\\right)={x}^{5}+6{x}^{4}+13{x}^{3}+14{x}^{2}+12x+8[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137427321\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137427323\">[latex]\\,-2\\,[\/latex](has multiplicity 3),[latex]\\,\u00b1i[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135689439\">Given the following information about a polynomial function, find the function.<\/p>\n<div id=\"fs-id1165135689443\">\n<div id=\"fs-id1165135689445\">\n<p>It has a double zero at[latex]\\,x=3\\,[\/latex]and zeros at[latex]\\,x=1\\,[\/latex]and[latex]\\,x=-2\\,[\/latex]. Its <em>y<\/em>-intercept is[latex]\\,\\left(0,12\\right).\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132011285\">\n<div id=\"fs-id1165135205666\">\n<p id=\"fs-id1165135205668\">It has a zero of multiplicity 3 at[latex]\\,x=\\frac{1}{2}\\,[\/latex]and another zero at[latex]\\,x=-3\\,[\/latex]. It contains the point[latex]\\,\\left(1,8\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137595278\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137595280\">[latex]f\\left(x\\right)=2{\\left(2x-1\\right)}^{3}\\left(x+3\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135177605\">Use Descartes\u2019 Rule of Signs to determine the possible number of positive and negative solutions.<\/p>\n<div id=\"fs-id1165134065115\">\n<div id=\"fs-id1165134065117\">[latex]8{x}^{3}-21{x}^{2}+6=0[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1165137722354\">For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.<\/p>\n<div id=\"fs-id1165137722359\">\n<div id=\"fs-id1165137722361\">\n<p id=\"fs-id1165135188256\">[latex]f\\left(x\\right)=\\frac{x+4}{{x}^{2}-2x-3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135208461\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135208464\">Intercepts[latex]\\,\\left(-4,0\\right),\\,\\left(0,-\\frac{4}{3}\\right)\\,[\/latex], Asymptotes[latex]\\,x=3,\\text{ }x\\text{ }=-1,\\text{ }y=0[\/latex].<\/p>\n<div><\/div>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/ecbf795c970a120f915fddb3190eccf9c0d013a6\/CNX_Precalc_Figure_03_09_216.jpg\" alt=\"Graph of f(x)=(x+4)\/(x^2-2x-3).\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135650721\">\n<div id=\"fs-id1165135650723\">\n<p id=\"fs-id1165135650725\">[latex]f\\left(x\\right)=\\frac{{x}^{2}+2x-3}{{x}^{2}-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137567083\">Find the slant asymptote of the rational function.<\/p>\n<div id=\"fs-id1165137567086\">\n<div id=\"fs-id1165137567088\">\n<p id=\"fs-id1165137567090\">[latex]f\\left(x\\right)=\\frac{{x}^{2}+3x-3}{x-1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135151319\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]y=x+4[\/latex]<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137570255\">Find the inverse of the function.<\/p>\n<div id=\"fs-id1165137834208\">\n<div>\n<p id=\"fs-id1165137834212\">[latex]f\\left(x\\right)=\\sqrt{x-2}+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723709\">\n<div id=\"fs-id1165137723711\">\n<p id=\"fs-id1165135194622\">[latex]f\\left(x\\right)=3{x}^{3}-4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137920721\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137805519\">[latex]{f}^{-1}\\left(x\\right)=\\sqrt[3]{\\frac{x+4}{3}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137502317\">\n<div id=\"fs-id1165137502320\">\n<p id=\"fs-id1165135521251\">[latex]f\\left(x\\right)=\\frac{2x+3}{3x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Find the unknown value.<\/p>\n<div id=\"fs-id1165137401576\">\n<div id=\"fs-id1165137401578\">\n<p id=\"fs-id1165137848951\">[latex]\\,y\\,[\/latex]varies inversely as the square of[latex]\\,x\\,[\/latex]and when[latex]\\,x=3,\\,[\/latex][latex]y=2.\\,[\/latex]Find[latex]\\,y\\,[\/latex]if[latex]\\,x=1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137722658\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137722660\">[latex]y=18[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135559531\">\n<div>\n<p id=\"fs-id1165137425335\">[latex]\\,y\\,[\/latex]varies jointly with[latex]\\,x\\,[\/latex]and the cube root of[latex]\\,z.\\,[\/latex]If when[latex]\\,x=2\\,[\/latex]and[latex]\\,z=27,\\,[\/latex][latex]y=12,\\,[\/latex]find[latex]\\,y\\,[\/latex]if[latex]\\,x=5\\,[\/latex]and[latex]\\,z=8.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Solve the following application problem.<\/p>\n<div id=\"fs-id1165137427522\">\n<div id=\"fs-id1165137427524\">\n<p>The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?<\/p>\n<\/div>\n<div id=\"fs-id1165137734636\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137734638\">4 seconds<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137735724\">\n<dt>constant of variation<\/dt>\n<dd id=\"fs-id1165137735729\">the non-zero value[latex]\\,k\\,[\/latex]<br \/>\nthat helps define the relationship between variables in direct or inverse variation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137762202\">\n<dt>direct variation<\/dt>\n<dd>the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137462046\">\n<dt>inverse variation<\/dt>\n<dd>the relationship between two variables in which the product of the variables is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135501040\">\n<dt>inversely proportional<\/dt>\n<dd id=\"fs-id1165137874542\">a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137874546\">\n<dt>joint variation<\/dt>\n<dd id=\"fs-id1165135696715\">a relationship where a variable varies directly or inversely with multiple variables<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135696718\">\n<dt>varies directly<\/dt>\n<dd id=\"fs-id1165137432955\">a relationship where one quantity is a constant multiplied by the other quantity<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137432958\">\n<dt>varies inversely<\/dt>\n<dd id=\"fs-id1165135439853\">a relationship where one quantity is a constant divided by the other quantity<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":9,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-93","chapter","type-chapter","status-publish","hentry"],"part":76,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/93","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/93\/revisions"}],"predecessor-version":[{"id":94,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/93\/revisions\/94"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/76"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/93\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=93"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=93"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=93"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=93"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}