{"id":160,"date":"2019-08-20T17:03:30","date_gmt":"2019-08-20T21:03:30","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/parametric-equations\/"},"modified":"2022-06-01T10:39:35","modified_gmt":"2022-06-01T14:39:35","slug":"parametric-equations","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/parametric-equations\/","title":{"raw":"Parametric Equations","rendered":"Parametric Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Parameterize a curve.<\/li>\n \t<li>Eliminate the parameter.<\/li>\n \t<li>Find a rectangular equation for a curve defined parametrically.<\/li>\n \t<li>Find parametric equations for curves defined by rectangular equations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137407242\">Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in <a class=\"autogenerated-content\" href=\"#Figure_08_06_001\">(Figure)<\/a>. At any moment, the moon is located at a particular spot relative to the planet. But how do we write and solve the equation for the position of the moon when the distance from the planet, the speed of the moon\u2019s orbit around the planet, and the speed of rotation around the sun are all unknowns? We can solve only for one variable at a time.<\/p>\n\n<div id=\"Figure_08_06_001\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152657\/CNX_Precalc_Figure_08_06_001.jpg\" alt=\"Illustration of a planet's circular orbit around the sun.\" width=\"487\" height=\"383\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165134167450\">In this section, we will consider sets of equations given by [latex]\\,x\\left(t\\right)\\,[\/latex] and [latex]\\,y\\left(t\\right)\\,[\/latex] where [latex]t[\/latex] is the independent variable of time. We can use these parametric equations in a number of applications when we are looking for not only a particular position but also the direction of the movement. As we trace out successive values of[latex]\\,t,\\,[\/latex]the orientation of the curve becomes clear. This is one of the primary advantages of using <span class=\"no-emphasis\">parametric equations<\/span>: we are able to trace the movement of an object along a path according to time. We begin this section with a look at the basic components of parametric equations and what it means to parameterize a curve. Then we will learn how to eliminate the parameter, translate the equations of a curve defined parametrically into rectangular equations, and find the parametric equations for curves defined by rectangular equations.<\/p>\n\n<div id=\"fs-id1165135367548\" class=\"bc-section section\">\n<h3>Parameterizing a Curve<\/h3>\n<p id=\"fs-id1165137660656\">When an object moves along a curve\u2014or <span class=\"no-emphasis\">curvilinear path<\/span>\u2014in a given direction and in a given amount of time, the position of the object in the plane is given by the <em>x-<\/em>coordinate and the <em>y-<\/em>coordinate. However, both[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]\nvary over time and so are functions of time. For this reason, we add another variable, the parameter, upon which both[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]are dependent functions. In the example in the section opener, the parameter is time,[latex]\\,t.\\,[\/latex]The[latex]\\,x\\,[\/latex]position of the moon at time,[latex]\\,t,\\,[\/latex]is represented as the function[latex]\\,x\\left(t\\right),\\,[\/latex]and the[latex]\\,y\\,[\/latex]position of the moon at time,[latex]\\,t,\\,[\/latex]is represented as the function[latex]\\,y\\left(t\\right).\\,[\/latex]Together,[latex]\\,x\\left(t\\right)\\,[\/latex] and [latex]\\,y\\left(t\\right)\\,[\/latex] are called parametric equations, and generate an ordered pair[latex]\\,\\left(x\\left(t\\right),\\,y\\left(t\\right)\\right).\\,[\/latex]Parametric equations primarily describe motion and direction.<\/p>\n<p id=\"fs-id1165135299186\">When we parameterize a curve, we are translating a single equation in two variables, such as[latex]\\,x\\,[\/latex]and[latex]\\,y\u2009,[\/latex]into an equivalent pair of equations in three variables,[latex]\\,x,y,\\,[\/latex]and[latex]\\,t.\\,[\/latex]One of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object\u2019s motion over time.<\/p>\n<p id=\"fs-id1165137481161\">When we graph parametric equations, we can observe the individual behaviors of[latex]\\,x\\,[\/latex]and of[latex]\\,y.\\,[\/latex]There are a number of shapes that cannot be represented in the form[latex]\\,y=f\\left(x\\right),\\,[\/latex]meaning that they are not functions. For example, consider the graph of a circle, given as[latex]\\,{r}^{2}={x}^{2}+{y}^{2}.\\,[\/latex]Solving for[latex]\\,y\\,[\/latex]gives[latex]\\,y=\u00b1\\sqrt{{r}^{2}-{x}^{2}},\\,[\/latex]or two equations:[latex]\\,{y}_{1}=\\sqrt{{r}^{2}-{x}^{2}}\\,[\/latex]and[latex]\\,{y}_{2}=-\\sqrt{{r}^{2}-{x}^{2}}.\\,[\/latex]If we graph[latex]\\,{y}_{1}\\,[\/latex]and[latex]\\,{y}_{2}\\,[\/latex]together, the graph will not pass the vertical line test, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_002\">(Figure)<\/a>. Thus, the equation for the graph of a circle is not a function.<\/p>\n\n<div id=\"Figure_08_06_002\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152710\/CNX_Precalc_Figure_08_06_002.jpg\" alt=\"Graph of a circle in the rectangular coordinate system - the vertical line test shows that the circle r^2 = x^2 + y^2 is not a function. The dotted red vertical line intersects the function in two places - it should only intersect in one place to be a function.\" width=\"487\" height=\"291\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137675600\">However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.<\/p>\n\n<div id=\"fs-id1165131990671\" class=\"textbox key-takeaways\">\n<h3>Parametric Equations<\/h3>\n<p id=\"fs-id1165137473349\">Suppose[latex]\\,t\\,[\/latex]is a number on an interval,[latex]\\,I.\\,[\/latex]The set of ordered pairs,[latex]\\,\\left(x\\left(t\\right),\\,\\,y\\left(t\\right)\\right),\\,[\/latex]where[latex]\\,x=f\\left(t\\right)\\,[\/latex]and[latex]\\,y=g\\left(t\\right),[\/latex]forms a plane curve based on the parameter[latex]\\,t.\\,[\/latex]The equations[latex]\\,x=f\\left(t\\right)\\,[\/latex]and[latex]\\,y=g\\left(t\\right)\\,[\/latex]are the parametric equations.<\/p>\n\n<\/div>\n<div id=\"Example_08_06_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137784440\">\n<div id=\"fs-id1165137640139\">\n<h3>Parameterizing a Curve<\/h3>\n<p id=\"fs-id1165137422846\">Parameterize the curve[latex]\\,y={x}^{2}-1\\,[\/latex]letting[latex]\\,x\\left(t\\right)=t.\\,[\/latex]Graph both equations.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137460496\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137460496\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137460496\"]\n<p id=\"fs-id1165135344083\">If[latex]\\,x\\left(t\\right)=t,\\,[\/latex]then to find[latex]\\,y\\left(t\\right)\\,[\/latex]we replace the variable[latex]\\,x\\,[\/latex]with the expression given in[latex]\\,x\\left(t\\right).\\,[\/latex]In other words,[latex]\\,y\\left(t\\right)={t}^{2}-1.[\/latex] Make a table of values similar to <a class=\"autogenerated-content\" href=\"#Table_08_06_001\">(Figure)<\/a>, and sketch the graph.<\/p>\n\n<table id=\"Table_08_06_001\" summary=\"Ten rows and three columns. First column is labeled t, second column is labeled x(t), third column is labeled y(t). The table has ordered triples of each of these row values: (-4,-4, y(-4)=(-4)^2 - 1 = 15), (-3,-3, y(-3)= (-3)^2 -1 = 8), (-2,-2, y(-2) = (-2)^2 -1 = 3), (-1,-1, y(-1)= (-1)^2 -1 = 0), (0,0, y(0) = (0)^2 -1 = -1), (1,1, y(1) = (1)^2 -1 = 0), (2,2, y(2) = (2)^2 -1 =3), (3,3, y(3) = (3)^2 - 1 = 8), (4,4, y(4) = (4)^2 - 1 = 15).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x\\left(t\\right)[\/latex]<\/th>\n<th>[latex]y\\left(t\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]y\\left(-4\\right)={\\left(-4\\right)}^{2}-1=15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y\\left(-3\\right)={\\left(-3\\right)}^{2}-1=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y\\left(-2\\right)={\\left(-2\\right)}^{2}-1=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y\\left(-1\\right)={\\left(-1\\right)}^{2}-1=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y\\left(0\\right)={\\left(0\\right)}^{2}-1=-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y\\left(1\\right)={\\left(1\\right)}^{2}-1=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y\\left(2\\right)={\\left(2\\right)}^{2}-1=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y\\left(3\\right)={\\left(3\\right)}^{2}-1=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]y\\left(4\\right)={\\left(4\\right)}^{2}-1=15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nSee the graphs in <a class=\"autogenerated-content\" href=\"#Figure_08_06_015\">(Figure)<\/a>. It may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated as[latex]\\,t\\,[\/latex]increases.\n<div id=\"Figure_08_06_015\" class=\"medium\">[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152724\/CNX_Precalc_Figure_08_06_015.jpg\" alt=\"Graph of a parabola in two forms: a parametric equation and rectangular coordinates. It is the same function, just different ways of writing it.\" width=\"731\" height=\"291\"> <strong>Figure 3. <\/strong>(a) Parametric[latex]\\,y\\left(t\\right)={t}^{2}-1\\,[\/latex](b) Rectangular[latex]\\,y={x}^{2}-1[\/latex][\/caption]<span id=\"fs-id1165137585230\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165134380377\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137539827\">The arrows indicate the direction in which the curve is generated. Notice the curve is identical to the curve of[latex]\\,y={x}^{2}-1.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137404794\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137668350\">\n<div id=\"fs-id1165137611328\">\n<p id=\"fs-id1165135504960\">Construct a table of values and plot the parametric equations:[latex]\\,x\\left(t\\right)=t-3,\\,\\,y\\left(t\\right)=2t+4;\\,\\,\\,-1\\le t\\le 2.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135485790\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135485790\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135485790\"]\n<table id=\"fs-id1165137810323\" class=\"unnumbered\" summary=\"Five rows and three columns. First column is labeled t, second column is labeled x(t), third column is labeled y(t). The table has ordered triples of each of these row values: (-1, -4, 2), (0,-3,4), (1,-2,6), (2,-1,8).\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]t[\/latex]<\/td>\n<td>[latex]x\\left(t\\right)[\/latex]<\/td>\n<td>[latex]y\\left(t\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span id=\"fs-id1165135632065\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152727\/CNX_Precalc_Figure_08_06_006.jpg\" alt=\"\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_06_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135528574\">\n<div id=\"fs-id1165137755555\">\n<h3>Finding a Pair of Parametric Equations<\/h3>\n<p id=\"fs-id1165137757860\">Find a pair of parametric equations that models the graph of[latex]\\,y=1-{x}^{2},\\,[\/latex]using the parameter[latex]\\,x\\left(t\\right)=t.\\,[\/latex]Plot some points and sketch the graph.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137665205\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137665205\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137665205\"]\n<p id=\"fs-id1165137445624\">If[latex]\\,x\\left(t\\right)=t\\,[\/latex]and we substitute[latex]\\,t\\,[\/latex]for[latex]\\,x\\,[\/latex]into the[latex]\\,y\\,[\/latex]equation, then[latex]\\,y\\left(t\\right)=1-{t}^{2}.\\,[\/latex]Our pair of parametric equations is<\/p>\n\n<div id=\"fs-id1165137416849\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x\\left(t\\right)=t\\\\ y\\left(t\\right)=1-{t}^{2}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137735863\">To graph the equations, first we construct a table of values like that in <a class=\"autogenerated-content\" href=\"#Table_08_06_02\">(Figure)<\/a>. We can choose values around[latex]\\,t=0,\\,[\/latex]from[latex]\\,t=-3\\,[\/latex]to[latex]\\,t=3.\\,[\/latex]The values in the[latex]\\,x\\left(t\\right)\\,[\/latex]column will be the same as those in the[latex]\\,t\\,[\/latex]column because[latex]\\,x\\left(t\\right)=t.\\,[\/latex]Calculate values for the column[latex]\\,y\\left(t\\right).\\,[\/latex]<\/p>\n\n<table id=\"Table_08_06_02\" summary=\"Eight rows and three columns. First column is labeled t, second column is labeled x(t)=t, third column is labeled y(t)=1-t^2. The table has ordered triples of each of these row values: (-3,-3, y(-3) = 1 - (-3)^2 = -8 ), (-2,-2, y(-2) = 1 - (-2)^2 = -3), (-1, -1, y(-1) = 1 - (-1)^2 = 0), (0,0, y(0) = 1 - 0 = 1), (1,1, y(1) = 1 - (1)^2 = 0), (2,2, y(2) = 1 - (2)^2 = -3), (3,3, y(3) = 1 - (3)^2 = -8).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x\\left(t\\right)=t[\/latex]<\/th>\n<th>[latex]y\\left(t\\right)=1-{t}^{2}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y\\left(-3\\right)=1-{\\left(-3\\right)}^{2}=-8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y\\left(-2\\right)=1-{\\left(-2\\right)}^{2}=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y\\left(-1\\right)=1-{\\left(-1\\right)}^{2}=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y\\left(0\\right)=1-0=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y\\left(1\\right)=1-{\\left(1\\right)}^{2}=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y\\left(2\\right)=1-{\\left(2\\right)}^{2}=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y\\left(3\\right)=1-{\\left(3\\right)}^{2}=-8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nThe graph of[latex]\\,y=1-{t}^{2}\\,[\/latex]is a parabola facing downward, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_007\">(Figure)<\/a>. We have mapped the curve over the interval[latex]\\,\\left[-3,\\,3\\right],[\/latex] shown as a solid line with arrows indicating the orientation of the curve according to[latex]\\,t.\\,[\/latex]Orientation refers to the path traced along the curve in terms of increasing values of[latex]\\,t.\\,[\/latex]As this parabola is symmetric with respect to the line[latex]\\,x=0,\\,[\/latex]the values of[latex]\\,x\\,[\/latex]are reflected across the <em>y<\/em>-axis.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152729\/CNX_Precalc_Figure_08_06_007.jpg\" alt=\"Graph of given downward facing parabola.\" width=\"487\" height=\"516\"> <strong>Figure 4.<\/strong>[\/caption]\n\n[\/hidden-answer]<span id=\"fs-id1165135187072\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135421668\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_06_01\">\n<div id=\"fs-id1165134237134\">\n<p id=\"fs-id1165134237135\">Parameterize the curve given by[latex]\\,x={y}^{3}-2y.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137843239\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137843239\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137843239\"]\n[latex]\\begin{array}{l}x\\left(t\\right)={t}^{3}-2t\\\\ y\\left(t\\right)=t\\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_06_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135502946\">\n<div id=\"fs-id1165137473933\">\n<h3>Finding Parametric Equations That Model Given Criteria<\/h3>\n<p id=\"fs-id1165135560806\">An object travels at a steady rate along a straight path [latex]\\,\\left(-5,\\,3\\right)\\,[\/latex]to[latex]\\,\\left(3,\\,-1\\right)\\,[\/latex]in the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137563061\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137563061\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137563061\"]\n<p id=\"fs-id1165137693678\">The parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The <em>x<\/em>-value of the object starts at[latex]\\,-5\\,[\/latex]meters and goes to 3 meters. This means the distance <em>x<\/em> has changed by 8 meters in 4 seconds, which is a rate of[latex]\\,\\frac{\\text{8 m}}{4\\text{ s}},[\/latex] or[latex]\\,2\\,\\text{m}\/\\text{s}.\\,[\/latex]We can write the <em>x<\/em>-coordinate as a linear function with respect to time as[latex]\\,x\\left(t\\right)=2t-5.\\,[\/latex]In the linear function template[latex]\\,y=mx+b,2t=mx\\,[\/latex]and[latex]\\,-5=b.[\/latex]<\/p>\n<p id=\"fs-id1165135389890\">Similarly, the <em>y<\/em>-value of the object starts at 3 and goes to[latex]\\,-1,\\,[\/latex]which is a change in the distance <em>y<\/em> of \u22124 meters in 4 seconds, which is a rate of[latex]\\,\\frac{-4\\text{ m}}{4\\text{ s}},[\/latex] or[latex]\\,-1\\text{m}\/\\text{s}.\\,[\/latex]We can also write the <em>y<\/em>-coordinate as the linear function[latex]\\,y\\left(t\\right)=-t+3.\\,[\/latex]Together, these are the parametric equations for the position of the object, where[latex]\\,x\\,[\/latex]\nand[latex]\\,y\\,[\/latex]\nare expressed in meters and[latex]\\,t\\,[\/latex]\nrepresents time:<\/p>\n\n<div id=\"fs-id1165135251333\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x\\left(t\\right)=2t-5\\hfill \\\\ y\\left(t\\right)=-t+3\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135209011\">Using these equations, we can build a table of values for [latex]\\,t,x,\\,[\/latex]and[latex]\\,y[\/latex] (see <a class=\"autogenerated-content\" href=\"#Table_08_06_03\">(Figure)<\/a>). In this example, we limited values of[latex]\\,t\\,[\/latex]to non-negative numbers. In general, any value of[latex]\\,t\\,[\/latex]can be used.<\/p>\n\n<table id=\"Table_08_06_03\" summary=\"Six rows and three columns. First column is labeled t, second column is labeled x(t)=2t-5, third column is labeled y(t)=-t+3. The table has ordered triples of each of these row values: (0, x=2(0)-5 = -5, y=-(0) +3 = 3), (1, x=2(1)-5 = -3, y=-(1) + 3 = 2), (2, x=2(2) - 5 = -1, y=-(2) + 3 = 1), (3, x=2(3) - 5 = 1, y = -(3) + 3 =0), (4, x=2(4) -5 = 3, y=-(4) + 3 = -1).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x\\left(t\\right)=2t-5[\/latex]<\/th>\n<th>[latex]y\\left(t\\right)=-t+3[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]x=2\\left(0\\right)-5=-5[\/latex]<\/td>\n<td>[latex]y=-\\left(0\\right)+3=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]x=2\\left(1\\right)-5=-3[\/latex]<\/td>\n<td>[latex]y=-\\left(1\\right)+3=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]x=2\\left(2\\right)-5=-1[\/latex]<\/td>\n<td>[latex]y=-\\left(2\\right)+3=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]x=2\\left(3\\right)-5=1[\/latex]<\/td>\n<td>[latex]y=-\\left(3\\right)+3=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]x=2\\left(4\\right)-5=3[\/latex]<\/td>\n<td>[latex]y=-\\left(4\\right)+3=-1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137422635\">From this table, we can create three graphs, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_003\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_06_003\" class=\"wp-caption aligncenter\">\n\n&nbsp;\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152733\/CNX_Precalc_Figure_08_06_003.jpg\" alt=\"Three graphs side by side. (A) has the horizontal position over time, (B) has the vertical position over time, and (C) has the position of the object in the plane at time t. See caption for more information.\" width=\"975\" height=\"514\"> <strong>Figure 5. <\/strong>(a) A graph of[latex]\\,x\\,[\/latex]vs.[latex]\\,t,\\,[\/latex]representing the horizontal position over time. (b) A graph of [latex]y[\/latex] vs. [latex]\\,t,\\,[\/latex]representing the vertical position over time. (c) A graph of [latex]\\,y\\,[\/latex] vs. [latex]\\,x,\\,[\/latex]representing the position of the object in the plane at time[latex]\\,t.[\/latex][\/caption]<span id=\"fs-id1165137793974\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137834887\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137446680\">Again, we see that, in <a class=\"autogenerated-content\" href=\"#Figure_08_06_003\">(Figure)<\/a>(c), when the parameter represents time, we can indicate the movement of the object along the path with arrows.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137812303\" class=\"bc-section section\">\n<h3>Eliminating the Parameter<\/h3>\n<p id=\"fs-id1165137404720\">In many cases, we may have a pair of parametric equations but find that it is simpler to draw a curve if the equation involves only two variables, such as[latex]\\,x\\,[\/latex]and[latex]\\,y.\\,[\/latex]Eliminating the parameter is a method that may make graphing some curves easier. However, if we are concerned with the mapping of the equation according to time, then it will be necessary to indicate the orientation of the curve as well. There are various methods for eliminating the parameter[latex]\\,t\\,[\/latex]from a set of parametric equations; not every method works for every type of equation. Here we will review the methods for the most common types of equations.<\/p>\n\n<div id=\"fs-id1165137634344\" class=\"bc-section section\">\n<h4>Eliminating the Parameter from Polynomial, Exponential, and Logarithmic Equations<\/h4>\n<p id=\"fs-id1165137598578\">For polynomial, exponential, or logarithmic equations expressed as two parametric equations, we choose the equation that is most easily manipulated and solve for[latex]\\,t.\\,[\/latex]We substitute the resulting expression for[latex]\\,t\\,[\/latex]\ninto the second equation. This gives one equation in[latex]\\,x\\,[\/latex]and[latex]\\,y.\\,[\/latex]<\/p>\n\n<div id=\"Example_08_06_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135188471\">\n<div id=\"fs-id1165131892611\">\n<h3>Eliminating the Parameter in Polynomials<\/h3>\n<p id=\"fs-id1165134171257\">Given[latex]\\,x\\left(t\\right)={t}^{2}+1\\,[\/latex]and[latex]\\,y\\left(t\\right)=2+t,\\,[\/latex]eliminate the parameter, and write the parametric equations as a Cartesian equation.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135250612\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135250612\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135250612\"]\n<p id=\"fs-id1165135250614\">We will begin with the equation for[latex]\\,y\\,[\/latex]because the linear equation is easier to solve for[latex]\\,t.[\/latex]<\/p>\n\n<div id=\"fs-id1165137659863\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y=2+t\\hfill \\\\ y-2=t\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137400857\">Next, substitute[latex]\\,y-2\\,[\/latex]for[latex]\\,t\\,[\/latex]in[latex]\\,x\\left(t\\right).[\/latex]<\/p>\n\n<div id=\"fs-id1165134177104\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}x={t}^{2}+1\\hfill &amp; \\hfill \\\\ x={\\left(y-2\\right)}^{2}+1\\hfill &amp; \\text{Substitute the expression for }t\\text{ into }x.\\hfill \\\\ x={y}^{2}-4y+4+1\\hfill &amp; \\hfill \\\\ x={y}^{2}-4y+5\\hfill &amp; \\hfill \\\\ x={y}^{2}-4y+5\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134384572\">The Cartesian form is[latex]\\,x={y}^{2}-4y+5.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137863319\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137846373\">This is an equation for a parabola in which, in rectangular terms,[latex]\\,x\\,[\/latex]is dependent on[latex]\\,y.\\,[\/latex]From the curve\u2019s vertex at[latex]\\,\\left(1,2\\right),\\,[\/latex]the graph sweeps out to the right. See <a class=\"autogenerated-content\" href=\"#Figure_08_06_008\">(Figure)<\/a>. In this section, we consider sets of equations given by the functions[latex]\\,x\\left(t\\right)\\,[\/latex]and[latex]\\,y\\left(t\\right),\\,[\/latex]where[latex]\\,t\\,[\/latex]is the independent variable of time. Notice, both[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]are functions of time; so in general[latex]\\,y\\,[\/latex]is not a function of[latex]\\,x.[\/latex]<\/p>\n\n<div id=\"Figure_08_06_008\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152743\/CNX_Precalc_Figure_08_06_008.jpg\" alt=\"Graph of given sideways (extending to the right) parabola.\" width=\"731\" height=\"366\"> <strong>Figure 6.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137573569\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_06_02\">\n<div id=\"fs-id1165134069178\">\n<p id=\"fs-id1165134069179\">Given the equations below, eliminate the parameter and write as a rectangular equation for[latex]\\,y\\,[\/latex]as a function<\/p>\n[latex]\\begin{array}{l}\\\\ \\begin{array}{l}x\\left(t\\right)=2{t}^{2}+6\\hfill \\\\ y\\left(t\\right)=5-t\\hfill \\end{array}\\end{array}[\/latex]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133001900\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133001900\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133001900\"]\n<p id=\"fs-id1165135260755\">[latex]y=5-\\sqrt{\\frac{1}{2}x-3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_06_05\" class=\"textbox examples\">\n<div id=\"fs-id1165135553475\">\n<div id=\"fs-id1165135553477\">\n<h3>Eliminating the Parameter in Exponential Equations<\/h3>\n<p id=\"fs-id1165137742672\">Eliminate the parameter and write as a Cartesian equation:[latex]\\,x\\left(t\\right)={e}^{-t}\\,[\/latex] and [latex]\\,y\\left(t\\right)=3{e}^{t},\\,\\,t&gt;0.\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137419904\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137419904\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137419904\"]\n<p id=\"fs-id1165134401529\">Isolate[latex]\\,{e}^{t}.\\,[\/latex]<\/p>\n\n<div id=\"fs-id1165137445892\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,x={e}^{-t}\\hfill \\\\ {e}^{t}=\\frac{1}{x}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135499458\">Substitute the expression into[latex]\\,y\\left(t\\right).[\/latex]<\/p>\n\n<div id=\"fs-id1165135176328\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=3{e}^{t}\\hfill \\\\ y=3\\left(\\frac{1}{x}\\right)\\hfill \\\\ y=\\frac{3}{x}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135240893\">The Cartesian form is[latex]\\,y=\\frac{3}{x}.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137408682\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137892461\">The graph of the parametric equation is shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_009\">(Figure)<\/a><strong>(a)<\/strong>. The domain is restricted to[latex]\\,t&gt;0.\\,[\/latex]The Cartesian equation,[latex]\\,y=\\frac{3}{x}\\,[\/latex]is shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_009\">(Figure)<\/a><strong>(b)<\/strong> and has only one restriction on the domain,[latex]\\,x\\ne 0.[\/latex]<\/p>\n\n<div id=\"Figure_08_06_009\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152755\/CNX_Precalc_Figure_08_06_009n.jpg\" alt=\"\" width=\"975\" height=\"553\"> <strong>Figure 7.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_06_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135330577\">\n<div id=\"fs-id1165135347350\">\n<h3>Eliminating the Parameter in Logarithmic Equations<\/h3>\n<p id=\"fs-id1165135408279\">Eliminate the parameter and write as a Cartesian equation:[latex]\\,x\\left(t\\right)=\\sqrt{t}+2\\,[\/latex]and[latex]\\,y\\left(t\\right)=\\mathrm{log}\\left(t\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137482606\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137482606\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137482606\"]\n<p id=\"fs-id1165137693741\">Solve the first equation for[latex]\\,t.[\/latex]<\/p>\n\n<div id=\"fs-id1165132949297\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }x=\\sqrt{t}+2\\hfill &amp; \\hfill \\\\ \\text{ }x-2=\\sqrt{t}\\hfill &amp; \\hfill \\\\ {\\left(x-2\\right)}^{2}=t\\hfill &amp; \\text{Square both sides}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135400273\">Then, substitute the expression for [latex]t[\/latex] into the [latex]y[\/latex] equation.<\/p>\n\n<div id=\"fs-id1165137629216\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=\\mathrm{log}\\left(t\\right)\\\\ y=\\mathrm{log}{\\left(x-2\\right)}^{2}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137930332\">The Cartesian form is[latex]\\,y=\\mathrm{log}{\\left(x-2\\right)}^{2}.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135541906\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134486691\">To be sure that the parametric equations are equivalent to the Cartesian equation, check the domains. The parametric equations restrict the domain on[latex]\\,x=\\sqrt{t}+2\\,[\/latex]to[latex]\\,t&gt;0;[\/latex] we restrict the domain on[latex]\\,x\\,[\/latex]to[latex]\\,x&gt;2.\\,[\/latex]The domain for the parametric equation[latex]\\,y=\\mathrm{log}\\left(t\\right)\\,[\/latex]is restricted to[latex]\\,t&gt;0;[\/latex] we limit the domain on[latex]\\,y=\\mathrm{log}{\\left(x-2\\right)}^{2}\\,[\/latex]to[latex]\\,x&gt;2.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137407279\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_06_03\">\n<div id=\"fs-id1165137424937\">\n<p id=\"fs-id1165137424938\">Eliminate the parameter and write as a <span class=\"no-emphasis\">rectangular equation<\/span>.<\/p>\n\n<div id=\"fs-id1165137535506\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}x\\left(t\\right)={t}^{2}\\hfill \\\\ y\\left(t\\right)=\\mathrm{ln}\\,t\\,\\,\\,\\,\\,\\,\\,\\,t&gt;0\\hfill \\end{array}\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137408218\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137408218\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137408218\"]\n<p id=\"fs-id1165137749294\">[latex]y=\\mathrm{ln}\\sqrt{x}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135530475\" class=\"bc-section section\">\n<h4>Eliminating the Parameter from Trigonometric Equations<\/h4>\n<p id=\"fs-id1165135347435\">Eliminating the parameter from trigonometric equations is a straightforward substitution. We can use a few of the familiar trigonometric identities and the Pythagorean Theorem.<\/p>\n<p id=\"fs-id1165137849553\">First, we use the identities:<\/p>\n\n<div id=\"fs-id1165135407250\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x\\left(t\\right)=a\\mathrm{cos}\\,t\\\\ y\\left(t\\right)=b\\mathrm{sin}\\,t\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135523374\">Solving for[latex]\\,\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,\\mathrm{sin}\\,t,\\,[\/latex]we have<\/p>\n\n<div id=\"fs-id1165135194180\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{x}{a}=\\mathrm{cos}\\,t\\\\ \\frac{y}{b}=\\mathrm{sin}\\,t\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135593591\">Then, use the Pythagorean Theorem:<\/p>\n\n<div id=\"fs-id1165133347599\" class=\"unnumbered aligncenter\">[latex]{\\mathrm{cos}}^{2}t+{\\mathrm{sin}}^{2}t=1[\/latex]<\/div>\n<p id=\"fs-id1165137501242\">Substituting gives<\/p>\n\n<div id=\"fs-id1165137749996\" class=\"unnumbered aligncenter\">[latex]{\\mathrm{cos}}^{2}t+{\\mathrm{sin}}^{2}t={\\left(\\frac{x}{a}\\right)}^{2}+{\\left(\\frac{y}{b}\\right)}^{2}=1[\/latex]<\/div>\n<div id=\"Example_08_06_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137736705\">\n<div id=\"fs-id1165137736707\">\n<h3>Eliminating the Parameter from a Pair of Trigonometric Parametric Equations<\/h3>\nEliminate the parameter from the given pair of <span class=\"no-emphasis\">trigonometric equations<\/span> where[latex]\\,0\\le t\\le 2\\pi \\,[\/latex]and sketch the graph.\n<div class=\"unnumbered\">[latex]\\begin{array}{l}x\\left(t\\right)=4\\mathrm{cos}\\,t\\\\ y\\left(t\\right)=3\\mathrm{sin}\\,t\\end{array}[\/latex]<\/div>\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165135198527\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135198527\"]\n<p id=\"fs-id1165135198527\">Solving for[latex]\\,\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,\\mathrm{sin}\\,t,[\/latex] we have<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\,x=4\\mathrm{cos}\\,t\\hfill \\\\ \\frac{x}{4}=\\mathrm{cos}\\,t\\hfill \\\\ \\,y=3\\mathrm{sin}\\,t\\hfill \\\\ \\frac{y}{3}=\\mathrm{sin}\\,t\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135176411\">Next, use the Pythagorean identity and make the substitutions.<\/p>\n\n<div id=\"fs-id1165135708088\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{r}\\hfill {\\mathrm{cos}}^{2}t+{\\mathrm{sin}}^{2}t=1\\\\ \\hfill {\\left(\\frac{x}{4}\\right)}^{2}+{\\left(\\frac{y}{3}\\right)}^{2}=1\\\\ \\hfill \\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1\\end{array}[\/latex]<\/div>\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152758\/CNX_Precalc_Figure_08_06_011.jpg\" alt=\"Graph of given ellipse centered at (0,0).\" width=\"487\" height=\"366\"> <strong>Figure 8.<\/strong>[\/caption]\n<p id=\"fs-id1165134267966\">The graph for the equation is shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_011\">(Figure)<\/a>.<span id=\"fs-id1165137728240\"><\/span>[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137425336\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137668705\">Applying the general equations for <span class=\"no-emphasis\">conic sections<\/span> (introduced in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/introduction-to-analytic-geometry\/\">Analytic Geometry<\/a>, we can identify[latex]\\,\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1\\,[\/latex]as an ellipse centered at[latex]\\,\\left(0,0\\right).\\,[\/latex]Notice that when[latex]\\,t=0\\,[\/latex]the coordinates are[latex]\\,\\left(4,0\\right),\\,[\/latex]and when[latex]\\,t=\\frac{\\pi }{2}\\,[\/latex]the coordinates are[latex]\\,\\left(0,3\\right).\\,[\/latex]This shows the orientation of the curve with increasing values of[latex]\\,t.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135307870\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_06_04\">\n<div id=\"fs-id1165133088121\">\n<p id=\"fs-id1165133088122\">Eliminate the parameter from the given pair of parametric equations and write as a Cartesian equation:[latex]\\,x\\left(t\\right)=2\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,y\\left(t\\right)=3\\mathrm{sin}\\,t.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137456226\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137456226\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137456226\"]\n<p id=\"fs-id1165137456227\">[latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137761679\" class=\"bc-section section\">\n<h3>Finding Cartesian Equations from Curves Defined Parametrically<\/h3>\nWhen we are given a set of parametric equations and need to find an equivalent Cartesian equation, we are essentially \u201celiminating the parameter.\u201d However, there are various methods we can use to rewrite a set of parametric equations as a Cartesian equation. The simplest method is to set one equation equal to the parameter, such as[latex]\\,x\\left(t\\right)=t.\\,[\/latex]In this case, [latex]\\,y\\left(t\\right)\\,[\/latex]can be any expression. For example, consider the following pair of equations.\n<div id=\"fs-id1165135407372\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x\\left(t\\right)=t\\\\ y\\left(t\\right)={t}^{2}-3\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137570641\">Rewriting this set of parametric equations is a matter of substituting[latex]\\,x\\,[\/latex]for[latex]\\,t.\\,[\/latex]Thus, the Cartesian equation is[latex]\\,y={x}^{2}-3.[\/latex]<\/p>\n\n<div id=\"Example_08_06_08\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137675888\">\n<h3>Finding a Cartesian Equation Using Alternate Methods<\/h3>\n<p id=\"fs-id1165137445188\">Use two different methods to find the Cartesian equation equivalent to the given set of parametric equations.<\/p>\n\n<div id=\"fs-id1165135575262\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}x\\left(t\\right)=3t-2\\hfill \\\\ y\\left(t\\right)=t+1\\hfill \\end{array}\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135390989\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135390989\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135390989\"]\n<p id=\"fs-id1165137871635\"><em>Method 1<\/em>. First, let\u2019s solve the[latex]\\,x\\,[\/latex]equation for[latex]\\,t.\\,[\/latex]Then we can substitute the result into the [latex]y[\/latex] equation.<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\text{ }x=3t-2\\hfill \\\\ \\text{ }x+2=3t\\hfill \\\\ \\frac{x+2}{3}=t\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137558751\">Now substitute the expression for[latex]\\,t\\,[\/latex]into the[latex]\\,y\\,[\/latex]equation.<\/p>\n\n<div id=\"fs-id1165135207505\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=t+1\\hfill \\\\ y=\\left(\\frac{x+2}{3}\\right)+1\\hfill \\\\ y=\\frac{x}{3}+\\frac{2}{3}+1\\hfill \\\\ y=\\frac{1}{3}x+\\frac{5}{3}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137434265\"><em>Method 2<\/em>. Solve the[latex]\\,y\\,[\/latex]equation for[latex]\\,t\\,[\/latex]and substitute this expression in the[latex]\\,x\\,[\/latex]equation.<\/p>\n\n<div id=\"fs-id1165135404813\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }y=t+1\\hfill \\\\ y-1=t\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137461585\">Make the substitution and then solve for[latex]\\,y.[\/latex]<\/p>\n\n<div id=\"fs-id1165137655884\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }x=3\\left(y-1\\right)-2\\hfill \\\\ \\text{ }x=3y-3-2\\hfill \\\\ \\text{ }x=3y-5\\hfill \\\\ \\,x+5=3y\\hfill \\\\ \\frac{x+5}{3}=y\\hfill \\\\ \\text{ }y=\\frac{1}{3}x+\\frac{5}{3}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137566016\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_06_05\">\n<div id=\"fs-id1165134239722\">\n<p id=\"fs-id1165134239724\">Write the given parametric equations as a Cartesian equation:[latex]\\,x\\left(t\\right)={t}^{3}\\,[\/latex] and [latex]\\,y\\left(t\\right)={t}^{6}.[\/latex]<\/p>\n\n<\/div>\n<div>[reveal-answer q=\"fs-id1165137437198\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137437198\"]\n<p id=\"fs-id1165137437198\">[latex]y={x}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137771822\" class=\"bc-section section\">\n<h3>Finding Parametric Equations for Curves Defined by Rectangular Equations<\/h3>\n<p id=\"fs-id1165137660044\">Although we have just shown that there is only one way to interpret a set of parametric equations as a rectangular equation, there are multiple ways to interpret a rectangular equation as a set of parametric equations. Any strategy we may use to find the parametric equations is valid if it produces equivalency. In other words, if we choose an expression to represent[latex]\\,x,\\,[\/latex]and then substitute it into the[latex]\\,y\\,[\/latex]equation, and it produces the same graph over the same domain as the rectangular equation, then the set of parametric equations is valid. If the domain becomes restricted in the set of parametric equations, and the function does not allow the same values for[latex]\\,x\\,[\/latex]as the domain of the rectangular equation, then the graphs will be different.<\/p>\n\n<div id=\"Example_08_06_09\" class=\"textbox examples\">\n<div id=\"fs-id1165135424595\">\n<div id=\"fs-id1165135424597\">\n<h3>Finding a Set of Parametric Equations for Curves Defined by Rectangular Equations<\/h3>\n<p id=\"fs-id1165134086132\">Find a set of equivalent parametric equations for[latex]\\,y={\\left(x+3\\right)}^{2}+1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135671968\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135671968\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135671968\"]\n<p id=\"fs-id1165135671970\">An obvious choice would be to let[latex]\\,x\\left(t\\right)=t.\\,[\/latex]Then[latex]\\,y\\left(t\\right)={\\left(t+3\\right)}^{2}+1.[\/latex] But let\u2019s try something more interesting. What if we let[latex]\\,x=t+3?\\,[\/latex]Then we have<\/p>\n\n<div id=\"fs-id1165137827180\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y={\\left(x+3\\right)}^{2}+1\\hfill \\\\ y={\\left(\\left(t+3\\right)+3\\right)}^{2}+1\\hfill \\\\ y={\\left(t+6\\right)}^{2}+1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137871010\">The set of parametric equations is<\/p>\n\n<div id=\"fs-id1165137443605\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ x\\left(t\\right)=t+3\\hfill \\\\ y\\left(t\\right)={\\left(t+6\\right)}^{2}+1\\hfill \\end{array}[\/latex]<\/div>\nSee <a class=\"autogenerated-content\" href=\"#Figure_08_06_012\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152814\/CNX_Precalc_Figure_08_06_012.jpg\" alt=\"Graph of parametric and rectangular coordinate versions of the same parabola - they are the same!\" width=\"731\" height=\"402\"> <strong>Figure 6.<\/strong>[\/caption]\n<p id=\"fs-id1165134401525\">[\/hidden-answer]<\/p>\n\n<div id=\"Figure_08_06_012\" class=\"medium\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137930360\" class=\"precalculus media\">\n<p id=\"fs-id1165137740782\">Access these online resources for additional instruction and practice with parametric equations.<\/p>\n\n<ul id=\"fs-id1165137841610\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/introparametric\">Introduction to Parametric Equations<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/convertpara\">Converting Parametric Equations to Rectangular Form<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134548875\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137472078\">\n \t<li>Parameterizing a curve involves translating a rectangular equation in two variables,[latex]\\,x\\,[\/latex]and[latex]\\,y,\\,[\/latex]into two equations in three variables, <em>x<\/em>, <em>y<\/em>, and <em>t<\/em>. Often, more information is obtained from a set of parametric equations. See <a class=\"autogenerated-content\" href=\"#Example_08_06_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_08_06_02\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_08_06_03\">(Figure)<\/a>.<\/li>\n \t<li>Sometimes equations are simpler to graph when written in rectangular form. By eliminating[latex]\\,t,\\,[\/latex]an equation in[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]is the result.<\/li>\n \t<li>To eliminate[latex]\\,t,\\,[\/latex]solve one of the equations for[latex]\\,t,\\,[\/latex]and substitute the expression into the second equation. See <a class=\"autogenerated-content\" href=\"#Example_08_06_04\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_08_06_05\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_08_06_06\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_08_06_07\">(Figure)<\/a>.<\/li>\n \t<li>Finding the rectangular equation for a curve defined parametrically is basically the same as eliminating the parameter. Solve for[latex]\\,t\\,[\/latex]in one of the equations, and substitute the expression into the second equation. See <a class=\"autogenerated-content\" href=\"#Example_08_06_08\">(Figure)<\/a>.<\/li>\n \t<li>There are an infinite number of ways to choose a set of parametric equations for a curve defined as a rectangular equation.<\/li>\n \t<li>Find an expression for[latex]\\,x\\,[\/latex]such that the domain of the set of parametric equations remains the same as the original rectangular equation. See <a class=\"autogenerated-content\" href=\"#Example_08_06_09\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134068973\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137726497\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137452582\">\n<div id=\"fs-id1165137452584\">\n<p id=\"fs-id1165137430646\">What is a system of parametric equations?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135457745\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135457745\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135457745\"]\n<p id=\"fs-id1165135457747\">A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example,[latex]\\,x=f\\left(t\\right)\\,[\/latex]and[latex]\\,y=f\\left(t\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135515786\">\n<div id=\"fs-id1165135149809\">\n<p id=\"fs-id1165135149811\">Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137526957\">Explain how to eliminate a parameter given a set of parametric equations.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137768026\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137768026\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137768026\"]\n<p id=\"fs-id1165137768028\">Choose one equation to solve for[latex]\\,t,\\,[\/latex]substitute into the other equation and simplify.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137596332\">\n<div id=\"fs-id1165137596334\">\n<p id=\"fs-id1165133434649\">What is a benefit of writing a system of parametric equations as a Cartesian equation?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137655519\">\n<div id=\"fs-id1165137426830\">\n<p id=\"fs-id1165137426832\">What is a benefit of using parametric equations?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135250841\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135250841\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135250841\"]\n<p id=\"fs-id1165137461006\">Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137482276\">\n<div id=\"fs-id1165137465476\">\n<p id=\"fs-id1165137465478\">Why are there many sets of parametric equations to represent on Cartesian function?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451350\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137453740\">For the following exercises, eliminate the parameter[latex]\\,t\\,[\/latex]to rewrite the parametric equation as a Cartesian equation.<\/p>\n\n<div>\n<div>\n<p id=\"fs-id1165137453513\">[latex]\\{\\begin{array}{l}x(t)=5-t\\hfill \\\\ y(t)=8-2t\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135389966\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135389966\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135389966\"]\n<p id=\"fs-id1165137706604\">[latex]y=-2+2x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134036798\">\n<div id=\"fs-id1165134036800\">\n<p id=\"fs-id1165134041404\">[latex]\\{\\begin{array}{l}x(t)=6-3t\\hfill \\\\ y(t)=10-t\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137475359\">\n<div id=\"fs-id1165137639024\">\n<p id=\"fs-id1165137639026\">[latex]\\{\\begin{array}{l}x(t)=2t+1\\hfill \\\\ y(t)=3\\sqrt{t}\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137463947\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137463947\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137463947\"]\n<p id=\"fs-id1165137463950\">[latex]y=3\\sqrt{\\frac{x-1}{2}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137401054\">\n<div id=\"fs-id1165137401056\">\n<p id=\"fs-id1165134050546\">[latex\\]\\{\\begin{array}{l}x(t)=3t-1\\hfill \\\\ y(t)=2{t}^{2}\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135581966\">\n<div id=\"fs-id1165137401665\">\n<p id=\"fs-id1165137401667\">[latex]\\{\\begin{array}{l}x(t)=2{e}^{t}\\hfill \\\\ y(t)=1-5t\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135496291\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135496291\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135496291\"]\n<p id=\"fs-id1165135496294\">[latex]x=2{e}^{\\frac{1-y}{5}}\\,[\/latex]or[latex]\\,y=1-5ln\\left(\\frac{x}{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134090131\">\n<p id=\"fs-id1165137659602\">[latex]\\{\\begin{array}{l}x(t)={e}^{-2t}\\hfill \\\\ y(t)=2{e}^{-t}\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571890\">\n<div id=\"fs-id1165135571892\">\n<p id=\"fs-id1165135541958\">[latex]\\{\\begin{array}{l}x(t)=4\\text{log}(t)\\hfill \\\\ y(t)=3+2t\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135353727\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135353727\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135353727\"]\n<p id=\"fs-id1165135353729\">[latex]x=4\\mathrm{log}\\left(\\frac{y-3}{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135185154\">\n<div id=\"fs-id1165135185157\">[latex]\\{\\begin{array}{l}x(t)=\\text{log}(2t)\\hfill \\\\ y(t)=\\sqrt{t-1}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137767323\">\n<div id=\"fs-id1165137841698\">[latex]\\{\\begin{array}{l}x(t)={t}^{3}-t\\hfill \\\\ y(t)=2t\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1165137803762\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137803762\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137803762\"]\n<p id=\"fs-id1165137803764\">[latex]x={\\left(\\frac{y}{2}\\right)}^{3}-\\frac{y}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137693705\">\n<div id=\"fs-id1165135543530\">\n<p id=\"fs-id1165135543532\">[latex]\\{\\begin{array}{l}x(t)=t-{t}^{4}\\hfill \\\\ y(t)=t+2\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137827977\">\n<div id=\"fs-id1165132944934\">\n<p id=\"fs-id1165132944936\">[latex]\\{\\begin{array}{l}x(t)={e}^{2t}\\hfill \\\\ y(t)={e}^{6t}\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135414214\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135414214\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135414214\"]\n<p id=\"fs-id1165135414216\">[latex]y={x}^{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137410877\">\n<div>\n<p id=\"fs-id1165135543509\">[latex]\\{\\begin{array}{l}x(t)={t}^{5}\\hfill \\\\ y(t)={t}^{10}\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137668139\">\n<div id=\"fs-id1165135667825\">\n<p id=\"fs-id1165135667827\">[latex]\\{\\begin{array}{l}x(t)=4\\text{cos}\\,t\\hfill \\\\ y(t)=5\\mathrm{sin}\\,t \\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137762341\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137762341\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137762341\"]\n<p id=\"fs-id1165137762343\">[latex]{\\left(\\frac{x}{4}\\right)}^{2}+{\\left(\\frac{y}{5}\\right)}^{2}=1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137585726\">\n<div id=\"fs-id1165137585728\">\n<p id=\"fs-id1165131895934\">[latex]\\{\\begin{array}{l}x(t)=3\\mathrm{sin}\\,t\\hfill \\\\ y(t)=6\\mathrm{cos}\\,t\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135384981\">\n<div id=\"fs-id1165135384983\">\n<p id=\"fs-id1165133210007\">[latex]\\{\\begin{array}{l}x(t)=2{\\text{cos}}^{2}t\\hfill \\\\ y(t)=-\\mathrm{sin}\\,t \\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137602808\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137602808\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137602808\"]\n[latex]{y}^{2}=1-\\frac{1}{2}x[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135189886\">\n<div id=\"fs-id1165135189889\">\n<p id=\"fs-id1165135189891\">[latex]\\{\\begin{array}{l}x(t)=\\mathrm{cos}\\,t+4\\\\ y(t)=2{\\mathrm{sin}}^{2}t\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193646\">\n<div id=\"fs-id1165135193648\">\n<p id=\"fs-id1165135316171\">[latex]\\{\\begin{array}{l}x(t)=t-1\\\\ y(t)={t}^{2}\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133213863\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133213863\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133213863\"]\n<p id=\"fs-id1165133454382\">[latex]y={x}^{2}+2x+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134037467\">\n<p id=\"fs-id1165134037469\">[latex]\\{\\begin{array}{l}x(t)=-t\\\\ y(t)={t}^{3}+1\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134319713\">\n<div id=\"fs-id1165134319715\">\n<p id=\"fs-id1165135341173\">[latex]\\{\\begin{array}{l}x(t)=2t-1\\\\ y(t)={t}^{3}-2\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137704906\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137704906\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137704906\"]\n<p id=\"fs-id1165137704908\">[latex]y={\\left(\\frac{x+1}{2}\\right)}^{3}-2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137548472\">For the following exercises, rewrite the parametric equation as a Cartesian equation by building an [latex]x\\text{-}y[\/latex] table.<\/p>\n\n<div id=\"fs-id1165137582353\">\n<div id=\"fs-id1165137582355\">\n<p id=\"fs-id1165137582357\">[latex]\\{\\begin{array}{l}x(t)=2t-1\\\\ y(t)=t+4\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134239737\">\n<div id=\"fs-id1165137735773\">\n<p id=\"fs-id1165137735775\">[latex]\\{\\begin{array}{l}x(t)=4-t\\\\ y(t)=3t+2\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134495094\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134495094\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134495094\"]\n<p id=\"fs-id1165134495096\">[latex]y=-3x+14[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132947372\">\n<div id=\"fs-id1165132947374\">\n<p id=\"fs-id1165134109660\">[latex]\\{\\begin{array}{l}x(t)=2t-1\\\\ y(t)=5t\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134116716\">\n<div id=\"fs-id1165134116719\">\n<p id=\"fs-id1165135173427\">[latex]\\{\\begin{array}{l}x(t)=4t-1\\\\ y(t)=4t+2\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134187207\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134187207\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134187207\"]\n<p id=\"fs-id1165135528264\">[latex]y=x+3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135456667\">For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting [latex]x\\left(t\\right)=t[\/latex] or by setting[latex]\\,y\\left(t\\right)=t.[\/latex]<\/p>\n\n<div id=\"fs-id1165137595902\">\n<div id=\"fs-id1165135329750\">[latex]y\\left(x\\right)=3{x}^{2}+3[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134148489\">\n<div>[latex]y\\left(x\\right)=2\\mathrm{sin}\\,x+1[\/latex]<\/div>\n<div id=\"fs-id1165137925225\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137925225\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137925225\"]\n<p id=\"fs-id1165137925228\">[latex]\\{\\begin{array}{l}x(t)=t\\hfill \\\\ y(t)=2\\mathrm{sin}t+1\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134373445\">\n<div id=\"fs-id1165134373447\">\n<p id=\"fs-id1165135641460\">[latex]x\\left(y\\right)=3\\mathrm{log}\\left(y\\right)+y[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135582390\">\n<div id=\"fs-id1165135582392\">\n<p id=\"fs-id1165135582394\">[latex]x\\left(y\\right)=\\sqrt{y}+2y[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137575928\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137575928\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137575928\"]\n<p id=\"fs-id1165137575930\">[latex]\\{\\begin{array}{l}x(t)=\\sqrt{t}+2t\\hfill \\\\ y(t)=t\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\nFor the following exercises, parameterize (write parametric equations for) each Cartesian equation by using [latex]x\\left(t\\right)=a\\mathrm{cos}\\,t[\/latex] and[latex]\\,y\\left(t\\right)=b\\mathrm{sin}\\,t.\\,[\/latex]Identify the curve.\n<div id=\"fs-id1165137548806\">\n<div id=\"fs-id1165137548808\">\n<p id=\"fs-id1165137548810\">[latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134149924\">\n<div id=\"fs-id1165134149926\">\n<p id=\"fs-id1165137854890\">[latex]\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{36}=1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137849389\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137849389\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137849389\"]\n<p id=\"fs-id1165137849392\">[latex]\\{\\begin{array}{l}x(t)=4\\mathrm{cos}\\,t\\hfill \\\\ y(t)=6\\mathrm{sin}\\,t\\hfill \\end{array};\\,[\/latex]Ellipse<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135575985\">\n<div id=\"fs-id1165135575987\">\n<p id=\"fs-id1165137833111\">[latex]{x}^{2}+{y}^{2}=16[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135160120\">\n<div id=\"fs-id1165137444304\">\n<p id=\"fs-id1165137444306\">[latex]{x}^{2}+{y}^{2}=10[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137559264\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137559264\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137559264\"]\n<p id=\"fs-id1165137559266\">[latex]\\{\\begin{array}{l}x(t)=\\sqrt{10}\\mathrm{cos}t\\hfill \\\\ y(t)=\\sqrt{10}\\mathrm{sin}t\\hfill \\end{array};\\,[\/latex]\nCircle<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137893366\">\n<div id=\"fs-id1165137893368\">\n<p id=\"fs-id1165137893478\">Parameterize the line from[latex]\\,\\left(3,0\\right)\\,[\/latex]to[latex]\\,\\left(-2,-5\\right)\\,[\/latex]so that the line is at[latex]\\,\\left(3,0\\right)\\,[\/latex]at[latex]\\,t=0,\\,[\/latex]and at[latex]\\,\\left(-2,-5\\right)\\,[\/latex]at[latex]\\,t=1.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137651086\">\n<div id=\"fs-id1165137651088\">\n<p id=\"fs-id1165137651090\">Parameterize the line from[latex]\\,\\left(-1,0\\right)\\,[\/latex]to[latex]\\,\\left(3,-2\\right)\\,[\/latex]so that the line is at[latex]\\,\\left(-1,0\\right)\\,[\/latex]at[latex]\\,t=0,\\,[\/latex]and at[latex]\\,\\left(3,-2\\right)\\,[\/latex]at[latex]\\,t=1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137400302\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137400302\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137400302\"]\n<p id=\"fs-id1165137705382\">[latex]\\{\\begin{array}{l}x(t)=-1+4t\\hfill \\\\ y(t)=-2t\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137547147\">\n<div id=\"fs-id1165137547149\">\n<p id=\"fs-id1165137547151\">Parameterize the line from[latex]\\,\\left(-1,5\\right)\\,[\/latex]to[latex]\\,\\left(2,3\\right)[\/latex]so that the line is at[latex]\\,\\left(-1,5\\right)\\,[\/latex]at[latex]\\,t=0,\\,[\/latex]and at[latex]\\,\\left(2,3\\right)\\,[\/latex]at[latex]\\,t=1.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135452295\">\n<div id=\"fs-id1165137663434\">\n<p id=\"fs-id1165137663436\">Parameterize the line from[latex]\\,\\left(4,1\\right)\\,[\/latex]to[latex]\\,\\left(6,-2\\right)\\,[\/latex]so that the line is at[latex]\\,\\left(4,1\\right)\\,[\/latex]at[latex]\\,t=0,\\,[\/latex]and at[latex]\\,\\left(6,-2\\right)\\,[\/latex]at[latex]\\,t=1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134164960\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134164960\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134164960\"]\n<p id=\"fs-id1165134164962\">[latex]\\{\\begin{array}{l}x(t)=4+2t\\hfill \\\\ y(t)=1-3t\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134211398\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137408704\">For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.<\/p>\n\n<div id=\"fs-id1165131937984\">\n<div id=\"fs-id1165131937986\">\n<p id=\"fs-id1165131937988\">[latex]\\{\\begin{array}{l}{x}_{1}(t)=3t\\hfill \\\\ {y}_{1}(t)=2t-1\\hfill \\end{array}\\text{ and }\\{\\begin{array}{l}{x}_{2}(t)=t+3\\hfill \\\\ {y}_{2}(t)=4t-4\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137470483\">\n<div id=\"fs-id1165137939491\">\n<p id=\"fs-id1165137939493\">[latex]\\{\\begin{array}{l}{x}_{1}(t)={t}^{2}\\hfill \\\\ {y}_{1}(t)=2t-1\\hfill \\end{array}\\text{ and }\\{\\begin{array}{l}{x}_{2}(t)=-t+6\\hfill \\\\ {y}_{2}(t)=t+1\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137453625\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137453625\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137453625\"]\n<p id=\"fs-id1165137453627\">yes, at [latex]t=2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137400113\">For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.<\/p>\n\n<div id=\"fs-id1165137730286\">\n<div id=\"fs-id1165137730288\">\n<p id=\"fs-id1165135181639\">[latex]\\{\\begin{array}{l}{x}_{1}(t)=3{t}^{2}-3t+7\\hfill \\\\ {y}_{1}(t)=2t+3\\hfill \\end{array}[\/latex]<\/p>\n\n<table id=\"fs-id1165137565962\" class=\"unnumbered\" summary=\"Four rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains -1, 0, 1. The rest of the values in columns x and y are blank.\"><caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u20131<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137667655\">\n<div id=\"fs-id1165137647616\">\n<p id=\"fs-id1165137647618\">[latex]\\{\\begin{array}{l}{x}_{1}(t)={t}^{2}-4\\hfill \\\\ {y}_{1}(t)=2{t}^{2}-1\\hfill \\end{array}[\/latex]<\/p>\n\n<table id=\"fs-id1165135407032\" class=\"unnumbered\" summary=\"Four rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains 1, 2, 3. The rest of the values in columns x and y are blank.\"><caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137737862\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137737862\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137737862\"]\n<table id=\"fs-id1165137737864\" class=\"unnumbered\" summary=\"..\"><caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>-3<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>0<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>5<\/td>\n<td>17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135640921\">\n<div id=\"fs-id1165135640922\">\n<p id=\"fs-id1165135640923\">[latex]\\{\\begin{array}{l}{x}_{1}(t)={t}^{4}\\hfill \\\\ {y}_{1}(t)={t}^{3}+4\\hfill \\end{array}[\/latex]<\/p>\n\n<table id=\"fs-id1165137921660\" class=\"unnumbered\" summary=\"Five rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains -1, 0, 1, 2. The rest of the values in columns x and y are blank.\"><caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>-1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193961\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137588584\">\n<div id=\"fs-id1165137588586\">\n<p id=\"fs-id1165134389802\">Find two different sets of parametric equations for[latex]\\,y={\\left(x+1\\right)}^{2}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137437388\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137437388\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137437388\"]\n<p id=\"fs-id1165137437390\">answers may vary:[latex]\\,\\{\\begin{array}{l}x(t)=t-1\\hfill \\\\ y(t)={t}^{2}\\hfill \\end{array}\\text{ and }\\{\\begin{array}{l}x(t)=t+1\\hfill \\\\ y(t)={(t+2)}^{2}\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135189893\">\n<div id=\"fs-id1165135189896\">\n<p id=\"fs-id1165135189898\">Find two different sets of parametric equations for[latex]\\,y=3x-2.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134042368\">\n<div id=\"fs-id1165137471410\">\n<p id=\"fs-id1165137471412\">Find two different sets of parametric equations for[latex]\\,y={x}^{2}-4x+4.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137834340\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137834340\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137834340\"]\n<p id=\"fs-id1165137834342\">answers may vary: ,[latex]\\,\\{\\begin{array}{l}x(t)=t\\hfill \\\\ y(t)={t}^{2}-4t+4\\hfill \\end{array}\\text{ and }\\{\\begin{array}{l}x(t)=t+2\\hfill \\\\ y(t)={t}^{2}\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135186874\">\n \t<dt>parameter<\/dt>\n \t<dd id=\"fs-id1165134357588\">a variable, often representing time, upon which[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]are both dependent<\/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>Parameterize a curve.<\/li>\n<li>Eliminate the parameter.<\/li>\n<li>Find a rectangular equation for a curve defined parametrically.<\/li>\n<li>Find parametric equations for curves defined by rectangular equations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137407242\">Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in <a class=\"autogenerated-content\" href=\"#Figure_08_06_001\">(Figure)<\/a>. At any moment, the moon is located at a particular spot relative to the planet. But how do we write and solve the equation for the position of the moon when the distance from the planet, the speed of the moon\u2019s orbit around the planet, and the speed of rotation around the sun are all unknowns? We can solve only for one variable at a time.<\/p>\n<div id=\"Figure_08_06_001\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152657\/CNX_Precalc_Figure_08_06_001.jpg\" alt=\"Illustration of a planet's circular orbit around the sun.\" width=\"487\" height=\"383\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165134167450\">In this section, we will consider sets of equations given by [latex]\\,x\\left(t\\right)\\,[\/latex] and [latex]\\,y\\left(t\\right)\\,[\/latex] where [latex]t[\/latex] is the independent variable of time. We can use these parametric equations in a number of applications when we are looking for not only a particular position but also the direction of the movement. As we trace out successive values of[latex]\\,t,\\,[\/latex]the orientation of the curve becomes clear. This is one of the primary advantages of using <span class=\"no-emphasis\">parametric equations<\/span>: we are able to trace the movement of an object along a path according to time. We begin this section with a look at the basic components of parametric equations and what it means to parameterize a curve. Then we will learn how to eliminate the parameter, translate the equations of a curve defined parametrically into rectangular equations, and find the parametric equations for curves defined by rectangular equations.<\/p>\n<div id=\"fs-id1165135367548\" class=\"bc-section section\">\n<h3>Parameterizing a Curve<\/h3>\n<p id=\"fs-id1165137660656\">When an object moves along a curve\u2014or <span class=\"no-emphasis\">curvilinear path<\/span>\u2014in a given direction and in a given amount of time, the position of the object in the plane is given by the <em>x-<\/em>coordinate and the <em>y-<\/em>coordinate. However, both[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]<br \/>\nvary over time and so are functions of time. For this reason, we add another variable, the parameter, upon which both[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]are dependent functions. In the example in the section opener, the parameter is time,[latex]\\,t.\\,[\/latex]The[latex]\\,x\\,[\/latex]position of the moon at time,[latex]\\,t,\\,[\/latex]is represented as the function[latex]\\,x\\left(t\\right),\\,[\/latex]and the[latex]\\,y\\,[\/latex]position of the moon at time,[latex]\\,t,\\,[\/latex]is represented as the function[latex]\\,y\\left(t\\right).\\,[\/latex]Together,[latex]\\,x\\left(t\\right)\\,[\/latex] and [latex]\\,y\\left(t\\right)\\,[\/latex] are called parametric equations, and generate an ordered pair[latex]\\,\\left(x\\left(t\\right),\\,y\\left(t\\right)\\right).\\,[\/latex]Parametric equations primarily describe motion and direction.<\/p>\n<p id=\"fs-id1165135299186\">When we parameterize a curve, we are translating a single equation in two variables, such as[latex]\\,x\\,[\/latex]and[latex]\\,y\u2009,[\/latex]into an equivalent pair of equations in three variables,[latex]\\,x,y,\\,[\/latex]and[latex]\\,t.\\,[\/latex]One of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object\u2019s motion over time.<\/p>\n<p id=\"fs-id1165137481161\">When we graph parametric equations, we can observe the individual behaviors of[latex]\\,x\\,[\/latex]and of[latex]\\,y.\\,[\/latex]There are a number of shapes that cannot be represented in the form[latex]\\,y=f\\left(x\\right),\\,[\/latex]meaning that they are not functions. For example, consider the graph of a circle, given as[latex]\\,{r}^{2}={x}^{2}+{y}^{2}.\\,[\/latex]Solving for[latex]\\,y\\,[\/latex]gives[latex]\\,y=\u00b1\\sqrt{{r}^{2}-{x}^{2}},\\,[\/latex]or two equations:[latex]\\,{y}_{1}=\\sqrt{{r}^{2}-{x}^{2}}\\,[\/latex]and[latex]\\,{y}_{2}=-\\sqrt{{r}^{2}-{x}^{2}}.\\,[\/latex]If we graph[latex]\\,{y}_{1}\\,[\/latex]and[latex]\\,{y}_{2}\\,[\/latex]together, the graph will not pass the vertical line test, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_002\">(Figure)<\/a>. Thus, the equation for the graph of a circle is not a function.<\/p>\n<div id=\"Figure_08_06_002\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152710\/CNX_Precalc_Figure_08_06_002.jpg\" alt=\"Graph of a circle in the rectangular coordinate system - the vertical line test shows that the circle r^2 = x^2 + y^2 is not a function. The dotted red vertical line intersects the function in two places - it should only intersect in one place to be a function.\" width=\"487\" height=\"291\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137675600\">However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.<\/p>\n<div id=\"fs-id1165131990671\" class=\"textbox key-takeaways\">\n<h3>Parametric Equations<\/h3>\n<p id=\"fs-id1165137473349\">Suppose[latex]\\,t\\,[\/latex]is a number on an interval,[latex]\\,I.\\,[\/latex]The set of ordered pairs,[latex]\\,\\left(x\\left(t\\right),\\,\\,y\\left(t\\right)\\right),\\,[\/latex]where[latex]\\,x=f\\left(t\\right)\\,[\/latex]and[latex]\\,y=g\\left(t\\right),[\/latex]forms a plane curve based on the parameter[latex]\\,t.\\,[\/latex]The equations[latex]\\,x=f\\left(t\\right)\\,[\/latex]and[latex]\\,y=g\\left(t\\right)\\,[\/latex]are the parametric equations.<\/p>\n<\/div>\n<div id=\"Example_08_06_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137784440\">\n<div id=\"fs-id1165137640139\">\n<h3>Parameterizing a Curve<\/h3>\n<p id=\"fs-id1165137422846\">Parameterize the curve[latex]\\,y={x}^{2}-1\\,[\/latex]letting[latex]\\,x\\left(t\\right)=t.\\,[\/latex]Graph both equations.<\/p>\n<\/div>\n<div id=\"fs-id1165137460496\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135344083\">If[latex]\\,x\\left(t\\right)=t,\\,[\/latex]then to find[latex]\\,y\\left(t\\right)\\,[\/latex]we replace the variable[latex]\\,x\\,[\/latex]with the expression given in[latex]\\,x\\left(t\\right).\\,[\/latex]In other words,[latex]\\,y\\left(t\\right)={t}^{2}-1.[\/latex] Make a table of values similar to <a class=\"autogenerated-content\" href=\"#Table_08_06_001\">(Figure)<\/a>, and sketch the graph.<\/p>\n<table id=\"Table_08_06_001\" summary=\"Ten rows and three columns. First column is labeled t, second column is labeled x(t), third column is labeled y(t). The table has ordered triples of each of these row values: (-4,-4, y(-4)=(-4)^2 - 1 = 15), (-3,-3, y(-3)= (-3)^2 -1 = 8), (-2,-2, y(-2) = (-2)^2 -1 = 3), (-1,-1, y(-1)= (-1)^2 -1 = 0), (0,0, y(0) = (0)^2 -1 = -1), (1,1, y(1) = (1)^2 -1 = 0), (2,2, y(2) = (2)^2 -1 =3), (3,3, y(3) = (3)^2 - 1 = 8), (4,4, y(4) = (4)^2 - 1 = 15).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x\\left(t\\right)[\/latex]<\/th>\n<th>[latex]y\\left(t\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]y\\left(-4\\right)={\\left(-4\\right)}^{2}-1=15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y\\left(-3\\right)={\\left(-3\\right)}^{2}-1=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y\\left(-2\\right)={\\left(-2\\right)}^{2}-1=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y\\left(-1\\right)={\\left(-1\\right)}^{2}-1=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y\\left(0\\right)={\\left(0\\right)}^{2}-1=-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y\\left(1\\right)={\\left(1\\right)}^{2}-1=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y\\left(2\\right)={\\left(2\\right)}^{2}-1=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y\\left(3\\right)={\\left(3\\right)}^{2}-1=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]y\\left(4\\right)={\\left(4\\right)}^{2}-1=15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>See the graphs in <a class=\"autogenerated-content\" href=\"#Figure_08_06_015\">(Figure)<\/a>. It may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated as[latex]\\,t\\,[\/latex]increases.<\/p>\n<div id=\"Figure_08_06_015\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152724\/CNX_Precalc_Figure_08_06_015.jpg\" alt=\"Graph of a parabola in two forms: a parametric equation and rectangular coordinates. It is the same function, just different ways of writing it.\" width=\"731\" height=\"291\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3. <\/strong>(a) Parametric[latex]\\,y\\left(t\\right)={t}^{2}-1\\,[\/latex](b) Rectangular[latex]\\,y={x}^{2}-1[\/latex]<\/figcaption><\/figure>\n<p><span id=\"fs-id1165137585230\"><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134380377\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137539827\">The arrows indicate the direction in which the curve is generated. Notice the curve is identical to the curve of[latex]\\,y={x}^{2}-1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137404794\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137668350\">\n<div id=\"fs-id1165137611328\">\n<p id=\"fs-id1165135504960\">Construct a table of values and plot the parametric equations:[latex]\\,x\\left(t\\right)=t-3,\\,\\,y\\left(t\\right)=2t+4;\\,\\,\\,-1\\le t\\le 2.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135485790\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<table id=\"fs-id1165137810323\" class=\"unnumbered\" summary=\"Five rows and three columns. First column is labeled t, second column is labeled x(t), third column is labeled y(t). The table has ordered triples of each of these row values: (-1, -4, 2), (0,-3,4), (1,-2,6), (2,-1,8).\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]t[\/latex]<\/td>\n<td>[latex]x\\left(t\\right)[\/latex]<\/td>\n<td>[latex]y\\left(t\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span id=\"fs-id1165135632065\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152727\/CNX_Precalc_Figure_08_06_006.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_06_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135528574\">\n<div id=\"fs-id1165137755555\">\n<h3>Finding a Pair of Parametric Equations<\/h3>\n<p id=\"fs-id1165137757860\">Find a pair of parametric equations that models the graph of[latex]\\,y=1-{x}^{2},\\,[\/latex]using the parameter[latex]\\,x\\left(t\\right)=t.\\,[\/latex]Plot some points and sketch the graph.<\/p>\n<\/div>\n<div id=\"fs-id1165137665205\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137445624\">If[latex]\\,x\\left(t\\right)=t\\,[\/latex]and we substitute[latex]\\,t\\,[\/latex]for[latex]\\,x\\,[\/latex]into the[latex]\\,y\\,[\/latex]equation, then[latex]\\,y\\left(t\\right)=1-{t}^{2}.\\,[\/latex]Our pair of parametric equations is<\/p>\n<div id=\"fs-id1165137416849\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x\\left(t\\right)=t\\\\ y\\left(t\\right)=1-{t}^{2}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137735863\">To graph the equations, first we construct a table of values like that in <a class=\"autogenerated-content\" href=\"#Table_08_06_02\">(Figure)<\/a>. We can choose values around[latex]\\,t=0,\\,[\/latex]from[latex]\\,t=-3\\,[\/latex]to[latex]\\,t=3.\\,[\/latex]The values in the[latex]\\,x\\left(t\\right)\\,[\/latex]column will be the same as those in the[latex]\\,t\\,[\/latex]column because[latex]\\,x\\left(t\\right)=t.\\,[\/latex]Calculate values for the column[latex]\\,y\\left(t\\right).\\,[\/latex]<\/p>\n<table id=\"Table_08_06_02\" summary=\"Eight rows and three columns. First column is labeled t, second column is labeled x(t)=t, third column is labeled y(t)=1-t^2. The table has ordered triples of each of these row values: (-3,-3, y(-3) = 1 - (-3)^2 = -8 ), (-2,-2, y(-2) = 1 - (-2)^2 = -3), (-1, -1, y(-1) = 1 - (-1)^2 = 0), (0,0, y(0) = 1 - 0 = 1), (1,1, y(1) = 1 - (1)^2 = 0), (2,2, y(2) = 1 - (2)^2 = -3), (3,3, y(3) = 1 - (3)^2 = -8).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x\\left(t\\right)=t[\/latex]<\/th>\n<th>[latex]y\\left(t\\right)=1-{t}^{2}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y\\left(-3\\right)=1-{\\left(-3\\right)}^{2}=-8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y\\left(-2\\right)=1-{\\left(-2\\right)}^{2}=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y\\left(-1\\right)=1-{\\left(-1\\right)}^{2}=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y\\left(0\\right)=1-0=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y\\left(1\\right)=1-{\\left(1\\right)}^{2}=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y\\left(2\\right)=1-{\\left(2\\right)}^{2}=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y\\left(3\\right)=1-{\\left(3\\right)}^{2}=-8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The graph of[latex]\\,y=1-{t}^{2}\\,[\/latex]is a parabola facing downward, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_007\">(Figure)<\/a>. We have mapped the curve over the interval[latex]\\,\\left[-3,\\,3\\right],[\/latex] shown as a solid line with arrows indicating the orientation of the curve according to[latex]\\,t.\\,[\/latex]Orientation refers to the path traced along the curve in terms of increasing values of[latex]\\,t.\\,[\/latex]As this parabola is symmetric with respect to the line[latex]\\,x=0,\\,[\/latex]the values of[latex]\\,x\\,[\/latex]are reflected across the <em>y<\/em>-axis.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152729\/CNX_Precalc_Figure_08_06_007.jpg\" alt=\"Graph of given downward facing parabola.\" width=\"487\" height=\"516\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/details>\n<p><span id=\"fs-id1165135187072\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135421668\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_06_01\">\n<div id=\"fs-id1165134237134\">\n<p id=\"fs-id1165134237135\">Parameterize the curve given by[latex]\\,x={y}^{3}-2y.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137843239\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]\\begin{array}{l}x\\left(t\\right)={t}^{3}-2t\\\\ y\\left(t\\right)=t\\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_06_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135502946\">\n<div id=\"fs-id1165137473933\">\n<h3>Finding Parametric Equations That Model Given Criteria<\/h3>\n<p id=\"fs-id1165135560806\">An object travels at a steady rate along a straight path [latex]\\,\\left(-5,\\,3\\right)\\,[\/latex]to[latex]\\,\\left(3,\\,-1\\right)\\,[\/latex]in the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.<\/p>\n<\/div>\n<div id=\"fs-id1165137563061\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137693678\">The parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The <em>x<\/em>-value of the object starts at[latex]\\,-5\\,[\/latex]meters and goes to 3 meters. This means the distance <em>x<\/em> has changed by 8 meters in 4 seconds, which is a rate of[latex]\\,\\frac{\\text{8 m}}{4\\text{ s}},[\/latex] or[latex]\\,2\\,\\text{m}\/\\text{s}.\\,[\/latex]We can write the <em>x<\/em>-coordinate as a linear function with respect to time as[latex]\\,x\\left(t\\right)=2t-5.\\,[\/latex]In the linear function template[latex]\\,y=mx+b,2t=mx\\,[\/latex]and[latex]\\,-5=b.[\/latex]<\/p>\n<p id=\"fs-id1165135389890\">Similarly, the <em>y<\/em>-value of the object starts at 3 and goes to[latex]\\,-1,\\,[\/latex]which is a change in the distance <em>y<\/em> of \u22124 meters in 4 seconds, which is a rate of[latex]\\,\\frac{-4\\text{ m}}{4\\text{ s}},[\/latex] or[latex]\\,-1\\text{m}\/\\text{s}.\\,[\/latex]We can also write the <em>y<\/em>-coordinate as the linear function[latex]\\,y\\left(t\\right)=-t+3.\\,[\/latex]Together, these are the parametric equations for the position of the object, where[latex]\\,x\\,[\/latex]<br \/>\nand[latex]\\,y\\,[\/latex]<br \/>\nare expressed in meters and[latex]\\,t\\,[\/latex]<br \/>\nrepresents time:<\/p>\n<div id=\"fs-id1165135251333\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x\\left(t\\right)=2t-5\\hfill \\\\ y\\left(t\\right)=-t+3\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135209011\">Using these equations, we can build a table of values for [latex]\\,t,x,\\,[\/latex]and[latex]\\,y[\/latex] (see <a class=\"autogenerated-content\" href=\"#Table_08_06_03\">(Figure)<\/a>). In this example, we limited values of[latex]\\,t\\,[\/latex]to non-negative numbers. In general, any value of[latex]\\,t\\,[\/latex]can be used.<\/p>\n<table id=\"Table_08_06_03\" summary=\"Six rows and three columns. First column is labeled t, second column is labeled x(t)=2t-5, third column is labeled y(t)=-t+3. The table has ordered triples of each of these row values: (0, x=2(0)-5 = -5, y=-(0) +3 = 3), (1, x=2(1)-5 = -3, y=-(1) + 3 = 2), (2, x=2(2) - 5 = -1, y=-(2) + 3 = 1), (3, x=2(3) - 5 = 1, y = -(3) + 3 =0), (4, x=2(4) -5 = 3, y=-(4) + 3 = -1).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x\\left(t\\right)=2t-5[\/latex]<\/th>\n<th>[latex]y\\left(t\\right)=-t+3[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]x=2\\left(0\\right)-5=-5[\/latex]<\/td>\n<td>[latex]y=-\\left(0\\right)+3=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]x=2\\left(1\\right)-5=-3[\/latex]<\/td>\n<td>[latex]y=-\\left(1\\right)+3=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]x=2\\left(2\\right)-5=-1[\/latex]<\/td>\n<td>[latex]y=-\\left(2\\right)+3=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]x=2\\left(3\\right)-5=1[\/latex]<\/td>\n<td>[latex]y=-\\left(3\\right)+3=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]x=2\\left(4\\right)-5=3[\/latex]<\/td>\n<td>[latex]y=-\\left(4\\right)+3=-1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137422635\">From this table, we can create three graphs, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_06_003\" class=\"wp-caption aligncenter\">\n<p>&nbsp;<\/p>\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152733\/CNX_Precalc_Figure_08_06_003.jpg\" alt=\"Three graphs side by side. (A) has the horizontal position over time, (B) has the vertical position over time, and (C) has the position of the object in the plane at time t. See caption for more information.\" width=\"975\" height=\"514\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5. <\/strong>(a) A graph of[latex]\\,x\\,[\/latex]vs.[latex]\\,t,\\,[\/latex]representing the horizontal position over time. (b) A graph of [latex]y[\/latex] vs. [latex]\\,t,\\,[\/latex]representing the vertical position over time. (c) A graph of [latex]\\,y\\,[\/latex] vs. [latex]\\,x,\\,[\/latex]representing the position of the object in the plane at time[latex]\\,t.[\/latex]<\/figcaption><\/figure>\n<p><span id=\"fs-id1165137793974\"><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137834887\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137446680\">Again, we see that, in <a class=\"autogenerated-content\" href=\"#Figure_08_06_003\">(Figure)<\/a>(c), when the parameter represents time, we can indicate the movement of the object along the path with arrows.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137812303\" class=\"bc-section section\">\n<h3>Eliminating the Parameter<\/h3>\n<p id=\"fs-id1165137404720\">In many cases, we may have a pair of parametric equations but find that it is simpler to draw a curve if the equation involves only two variables, such as[latex]\\,x\\,[\/latex]and[latex]\\,y.\\,[\/latex]Eliminating the parameter is a method that may make graphing some curves easier. However, if we are concerned with the mapping of the equation according to time, then it will be necessary to indicate the orientation of the curve as well. There are various methods for eliminating the parameter[latex]\\,t\\,[\/latex]from a set of parametric equations; not every method works for every type of equation. Here we will review the methods for the most common types of equations.<\/p>\n<div id=\"fs-id1165137634344\" class=\"bc-section section\">\n<h4>Eliminating the Parameter from Polynomial, Exponential, and Logarithmic Equations<\/h4>\n<p id=\"fs-id1165137598578\">For polynomial, exponential, or logarithmic equations expressed as two parametric equations, we choose the equation that is most easily manipulated and solve for[latex]\\,t.\\,[\/latex]We substitute the resulting expression for[latex]\\,t\\,[\/latex]<br \/>\ninto the second equation. This gives one equation in[latex]\\,x\\,[\/latex]and[latex]\\,y.\\,[\/latex]<\/p>\n<div id=\"Example_08_06_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135188471\">\n<div id=\"fs-id1165131892611\">\n<h3>Eliminating the Parameter in Polynomials<\/h3>\n<p id=\"fs-id1165134171257\">Given[latex]\\,x\\left(t\\right)={t}^{2}+1\\,[\/latex]and[latex]\\,y\\left(t\\right)=2+t,\\,[\/latex]eliminate the parameter, and write the parametric equations as a Cartesian equation.<\/p>\n<\/div>\n<div id=\"fs-id1165135250612\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135250614\">We will begin with the equation for[latex]\\,y\\,[\/latex]because the linear equation is easier to solve for[latex]\\,t.[\/latex]<\/p>\n<div id=\"fs-id1165137659863\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y=2+t\\hfill \\\\ y-2=t\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137400857\">Next, substitute[latex]\\,y-2\\,[\/latex]for[latex]\\,t\\,[\/latex]in[latex]\\,x\\left(t\\right).[\/latex]<\/p>\n<div id=\"fs-id1165134177104\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}x={t}^{2}+1\\hfill & \\hfill \\\\ x={\\left(y-2\\right)}^{2}+1\\hfill & \\text{Substitute the expression for }t\\text{ into }x.\\hfill \\\\ x={y}^{2}-4y+4+1\\hfill & \\hfill \\\\ x={y}^{2}-4y+5\\hfill & \\hfill \\\\ x={y}^{2}-4y+5\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134384572\">The Cartesian form is[latex]\\,x={y}^{2}-4y+5.[\/latex]<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165137863319\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137846373\">This is an equation for a parabola in which, in rectangular terms,[latex]\\,x\\,[\/latex]is dependent on[latex]\\,y.\\,[\/latex]From the curve\u2019s vertex at[latex]\\,\\left(1,2\\right),\\,[\/latex]the graph sweeps out to the right. See <a class=\"autogenerated-content\" href=\"#Figure_08_06_008\">(Figure)<\/a>. In this section, we consider sets of equations given by the functions[latex]\\,x\\left(t\\right)\\,[\/latex]and[latex]\\,y\\left(t\\right),\\,[\/latex]where[latex]\\,t\\,[\/latex]is the independent variable of time. Notice, both[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]are functions of time; so in general[latex]\\,y\\,[\/latex]is not a function of[latex]\\,x.[\/latex]<\/p>\n<div id=\"Figure_08_06_008\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152743\/CNX_Precalc_Figure_08_06_008.jpg\" alt=\"Graph of given sideways (extending to the right) parabola.\" width=\"731\" height=\"366\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137573569\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_06_02\">\n<div id=\"fs-id1165134069178\">\n<p id=\"fs-id1165134069179\">Given the equations below, eliminate the parameter and write as a rectangular equation for[latex]\\,y\\,[\/latex]as a function<\/p>\n<p>[latex]\\begin{array}{l}\\\\ \\begin{array}{l}x\\left(t\\right)=2{t}^{2}+6\\hfill \\\\ y\\left(t\\right)=5-t\\hfill \\end{array}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133001900\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135260755\">[latex]y=5-\\sqrt{\\frac{1}{2}x-3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_06_05\" class=\"textbox examples\">\n<div id=\"fs-id1165135553475\">\n<div id=\"fs-id1165135553477\">\n<h3>Eliminating the Parameter in Exponential Equations<\/h3>\n<p id=\"fs-id1165137742672\">Eliminate the parameter and write as a Cartesian equation:[latex]\\,x\\left(t\\right)={e}^{-t}\\,[\/latex] and [latex]\\,y\\left(t\\right)=3{e}^{t},\\,\\,t>0.\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137419904\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134401529\">Isolate[latex]\\,{e}^{t}.\\,[\/latex]<\/p>\n<div id=\"fs-id1165137445892\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,x={e}^{-t}\\hfill \\\\ {e}^{t}=\\frac{1}{x}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135499458\">Substitute the expression into[latex]\\,y\\left(t\\right).[\/latex]<\/p>\n<div id=\"fs-id1165135176328\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=3{e}^{t}\\hfill \\\\ y=3\\left(\\frac{1}{x}\\right)\\hfill \\\\ y=\\frac{3}{x}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135240893\">The Cartesian form is[latex]\\,y=\\frac{3}{x}.[\/latex]<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165137408682\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137892461\">The graph of the parametric equation is shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_009\">(Figure)<\/a><strong>(a)<\/strong>. The domain is restricted to[latex]\\,t>0.\\,[\/latex]The Cartesian equation,[latex]\\,y=\\frac{3}{x}\\,[\/latex]is shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_009\">(Figure)<\/a><strong>(b)<\/strong> and has only one restriction on the domain,[latex]\\,x\\ne 0.[\/latex]<\/p>\n<div id=\"Figure_08_06_009\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152755\/CNX_Precalc_Figure_08_06_009n.jpg\" alt=\"\" width=\"975\" height=\"553\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_06_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135330577\">\n<div id=\"fs-id1165135347350\">\n<h3>Eliminating the Parameter in Logarithmic Equations<\/h3>\n<p id=\"fs-id1165135408279\">Eliminate the parameter and write as a Cartesian equation:[latex]\\,x\\left(t\\right)=\\sqrt{t}+2\\,[\/latex]and[latex]\\,y\\left(t\\right)=\\mathrm{log}\\left(t\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137482606\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137693741\">Solve the first equation for[latex]\\,t.[\/latex]<\/p>\n<div id=\"fs-id1165132949297\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }x=\\sqrt{t}+2\\hfill & \\hfill \\\\ \\text{ }x-2=\\sqrt{t}\\hfill & \\hfill \\\\ {\\left(x-2\\right)}^{2}=t\\hfill & \\text{Square both sides}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135400273\">Then, substitute the expression for [latex]t[\/latex] into the [latex]y[\/latex] equation.<\/p>\n<div id=\"fs-id1165137629216\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=\\mathrm{log}\\left(t\\right)\\\\ y=\\mathrm{log}{\\left(x-2\\right)}^{2}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137930332\">The Cartesian form is[latex]\\,y=\\mathrm{log}{\\left(x-2\\right)}^{2}.[\/latex]<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165135541906\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134486691\">To be sure that the parametric equations are equivalent to the Cartesian equation, check the domains. The parametric equations restrict the domain on[latex]\\,x=\\sqrt{t}+2\\,[\/latex]to[latex]\\,t>0;[\/latex] we restrict the domain on[latex]\\,x\\,[\/latex]to[latex]\\,x>2.\\,[\/latex]The domain for the parametric equation[latex]\\,y=\\mathrm{log}\\left(t\\right)\\,[\/latex]is restricted to[latex]\\,t>0;[\/latex] we limit the domain on[latex]\\,y=\\mathrm{log}{\\left(x-2\\right)}^{2}\\,[\/latex]to[latex]\\,x>2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137407279\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_06_03\">\n<div id=\"fs-id1165137424937\">\n<p id=\"fs-id1165137424938\">Eliminate the parameter and write as a <span class=\"no-emphasis\">rectangular equation<\/span>.<\/p>\n<div id=\"fs-id1165137535506\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}x\\left(t\\right)={t}^{2}\\hfill \\\\ y\\left(t\\right)=\\mathrm{ln}\\,t\\,\\,\\,\\,\\,\\,\\,\\,t>0\\hfill \\end{array}\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137408218\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137749294\">[latex]y=\\mathrm{ln}\\sqrt{x}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135530475\" class=\"bc-section section\">\n<h4>Eliminating the Parameter from Trigonometric Equations<\/h4>\n<p id=\"fs-id1165135347435\">Eliminating the parameter from trigonometric equations is a straightforward substitution. We can use a few of the familiar trigonometric identities and the Pythagorean Theorem.<\/p>\n<p id=\"fs-id1165137849553\">First, we use the identities:<\/p>\n<div id=\"fs-id1165135407250\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x\\left(t\\right)=a\\mathrm{cos}\\,t\\\\ y\\left(t\\right)=b\\mathrm{sin}\\,t\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135523374\">Solving for[latex]\\,\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,\\mathrm{sin}\\,t,\\,[\/latex]we have<\/p>\n<div id=\"fs-id1165135194180\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{x}{a}=\\mathrm{cos}\\,t\\\\ \\frac{y}{b}=\\mathrm{sin}\\,t\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135593591\">Then, use the Pythagorean Theorem:<\/p>\n<div id=\"fs-id1165133347599\" class=\"unnumbered aligncenter\">[latex]{\\mathrm{cos}}^{2}t+{\\mathrm{sin}}^{2}t=1[\/latex]<\/div>\n<p id=\"fs-id1165137501242\">Substituting gives<\/p>\n<div id=\"fs-id1165137749996\" class=\"unnumbered aligncenter\">[latex]{\\mathrm{cos}}^{2}t+{\\mathrm{sin}}^{2}t={\\left(\\frac{x}{a}\\right)}^{2}+{\\left(\\frac{y}{b}\\right)}^{2}=1[\/latex]<\/div>\n<div id=\"Example_08_06_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137736705\">\n<div id=\"fs-id1165137736707\">\n<h3>Eliminating the Parameter from a Pair of Trigonometric Parametric Equations<\/h3>\n<p>Eliminate the parameter from the given pair of <span class=\"no-emphasis\">trigonometric equations<\/span> where[latex]\\,0\\le t\\le 2\\pi \\,[\/latex]and sketch the graph.<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}x\\left(t\\right)=4\\mathrm{cos}\\,t\\\\ y\\left(t\\right)=3\\mathrm{sin}\\,t\\end{array}[\/latex]<\/div>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135198527\">Solving for[latex]\\,\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,\\mathrm{sin}\\,t,[\/latex] we have<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\,x=4\\mathrm{cos}\\,t\\hfill \\\\ \\frac{x}{4}=\\mathrm{cos}\\,t\\hfill \\\\ \\,y=3\\mathrm{sin}\\,t\\hfill \\\\ \\frac{y}{3}=\\mathrm{sin}\\,t\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135176411\">Next, use the Pythagorean identity and make the substitutions.<\/p>\n<div id=\"fs-id1165135708088\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{r}\\hfill {\\mathrm{cos}}^{2}t+{\\mathrm{sin}}^{2}t=1\\\\ \\hfill {\\left(\\frac{x}{4}\\right)}^{2}+{\\left(\\frac{y}{3}\\right)}^{2}=1\\\\ \\hfill \\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1\\end{array}[\/latex]<\/div>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152758\/CNX_Precalc_Figure_08_06_011.jpg\" alt=\"Graph of given ellipse centered at (0,0).\" width=\"487\" height=\"366\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165134267966\">The graph for the equation is shown in <a class=\"autogenerated-content\" href=\"#Figure_08_06_011\">(Figure)<\/a>.<span id=\"fs-id1165137728240\"><\/span><\/details>\n<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137425336\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137668705\">Applying the general equations for <span class=\"no-emphasis\">conic sections<\/span> (introduced in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/introduction-to-analytic-geometry\/\">Analytic Geometry<\/a>, we can identify[latex]\\,\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1\\,[\/latex]as an ellipse centered at[latex]\\,\\left(0,0\\right).\\,[\/latex]Notice that when[latex]\\,t=0\\,[\/latex]the coordinates are[latex]\\,\\left(4,0\\right),\\,[\/latex]and when[latex]\\,t=\\frac{\\pi }{2}\\,[\/latex]the coordinates are[latex]\\,\\left(0,3\\right).\\,[\/latex]This shows the orientation of the curve with increasing values of[latex]\\,t.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135307870\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_06_04\">\n<div id=\"fs-id1165133088121\">\n<p id=\"fs-id1165133088122\">Eliminate the parameter from the given pair of parametric equations and write as a Cartesian equation:[latex]\\,x\\left(t\\right)=2\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,y\\left(t\\right)=3\\mathrm{sin}\\,t.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137456226\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137456227\">[latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137761679\" class=\"bc-section section\">\n<h3>Finding Cartesian Equations from Curves Defined Parametrically<\/h3>\n<p>When we are given a set of parametric equations and need to find an equivalent Cartesian equation, we are essentially \u201celiminating the parameter.\u201d However, there are various methods we can use to rewrite a set of parametric equations as a Cartesian equation. The simplest method is to set one equation equal to the parameter, such as[latex]\\,x\\left(t\\right)=t.\\,[\/latex]In this case, [latex]\\,y\\left(t\\right)\\,[\/latex]can be any expression. For example, consider the following pair of equations.<\/p>\n<div id=\"fs-id1165135407372\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x\\left(t\\right)=t\\\\ y\\left(t\\right)={t}^{2}-3\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137570641\">Rewriting this set of parametric equations is a matter of substituting[latex]\\,x\\,[\/latex]for[latex]\\,t.\\,[\/latex]Thus, the Cartesian equation is[latex]\\,y={x}^{2}-3.[\/latex]<\/p>\n<div id=\"Example_08_06_08\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137675888\">\n<h3>Finding a Cartesian Equation Using Alternate Methods<\/h3>\n<p id=\"fs-id1165137445188\">Use two different methods to find the Cartesian equation equivalent to the given set of parametric equations.<\/p>\n<div id=\"fs-id1165135575262\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}x\\left(t\\right)=3t-2\\hfill \\\\ y\\left(t\\right)=t+1\\hfill \\end{array}\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135390989\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137871635\"><em>Method 1<\/em>. First, let\u2019s solve the[latex]\\,x\\,[\/latex]equation for[latex]\\,t.\\,[\/latex]Then we can substitute the result into the [latex]y[\/latex] equation.<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\text{ }x=3t-2\\hfill \\\\ \\text{ }x+2=3t\\hfill \\\\ \\frac{x+2}{3}=t\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137558751\">Now substitute the expression for[latex]\\,t\\,[\/latex]into the[latex]\\,y\\,[\/latex]equation.<\/p>\n<div id=\"fs-id1165135207505\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=t+1\\hfill \\\\ y=\\left(\\frac{x+2}{3}\\right)+1\\hfill \\\\ y=\\frac{x}{3}+\\frac{2}{3}+1\\hfill \\\\ y=\\frac{1}{3}x+\\frac{5}{3}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137434265\"><em>Method 2<\/em>. Solve the[latex]\\,y\\,[\/latex]equation for[latex]\\,t\\,[\/latex]and substitute this expression in the[latex]\\,x\\,[\/latex]equation.<\/p>\n<div id=\"fs-id1165135404813\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }y=t+1\\hfill \\\\ y-1=t\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137461585\">Make the substitution and then solve for[latex]\\,y.[\/latex]<\/p>\n<div id=\"fs-id1165137655884\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }x=3\\left(y-1\\right)-2\\hfill \\\\ \\text{ }x=3y-3-2\\hfill \\\\ \\text{ }x=3y-5\\hfill \\\\ \\,x+5=3y\\hfill \\\\ \\frac{x+5}{3}=y\\hfill \\\\ \\text{ }y=\\frac{1}{3}x+\\frac{5}{3}\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137566016\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_06_05\">\n<div id=\"fs-id1165134239722\">\n<p id=\"fs-id1165134239724\">Write the given parametric equations as a Cartesian equation:[latex]\\,x\\left(t\\right)={t}^{3}\\,[\/latex] and [latex]\\,y\\left(t\\right)={t}^{6}.[\/latex]<\/p>\n<\/div>\n<div>\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137437198\">[latex]y={x}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137771822\" class=\"bc-section section\">\n<h3>Finding Parametric Equations for Curves Defined by Rectangular Equations<\/h3>\n<p id=\"fs-id1165137660044\">Although we have just shown that there is only one way to interpret a set of parametric equations as a rectangular equation, there are multiple ways to interpret a rectangular equation as a set of parametric equations. Any strategy we may use to find the parametric equations is valid if it produces equivalency. In other words, if we choose an expression to represent[latex]\\,x,\\,[\/latex]and then substitute it into the[latex]\\,y\\,[\/latex]equation, and it produces the same graph over the same domain as the rectangular equation, then the set of parametric equations is valid. If the domain becomes restricted in the set of parametric equations, and the function does not allow the same values for[latex]\\,x\\,[\/latex]as the domain of the rectangular equation, then the graphs will be different.<\/p>\n<div id=\"Example_08_06_09\" class=\"textbox examples\">\n<div id=\"fs-id1165135424595\">\n<div id=\"fs-id1165135424597\">\n<h3>Finding a Set of Parametric Equations for Curves Defined by Rectangular Equations<\/h3>\n<p id=\"fs-id1165134086132\">Find a set of equivalent parametric equations for[latex]\\,y={\\left(x+3\\right)}^{2}+1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135671968\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135671970\">An obvious choice would be to let[latex]\\,x\\left(t\\right)=t.\\,[\/latex]Then[latex]\\,y\\left(t\\right)={\\left(t+3\\right)}^{2}+1.[\/latex] But let\u2019s try something more interesting. What if we let[latex]\\,x=t+3?\\,[\/latex]Then we have<\/p>\n<div id=\"fs-id1165137827180\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y={\\left(x+3\\right)}^{2}+1\\hfill \\\\ y={\\left(\\left(t+3\\right)+3\\right)}^{2}+1\\hfill \\\\ y={\\left(t+6\\right)}^{2}+1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137871010\">The set of parametric equations is<\/p>\n<div id=\"fs-id1165137443605\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ x\\left(t\\right)=t+3\\hfill \\\\ y\\left(t\\right)={\\left(t+6\\right)}^{2}+1\\hfill \\end{array}[\/latex]<\/div>\n<p>See <a class=\"autogenerated-content\" href=\"#Figure_08_06_012\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152814\/CNX_Precalc_Figure_08_06_012.jpg\" alt=\"Graph of parametric and rectangular coordinate versions of the same parabola - they are the same!\" width=\"731\" height=\"402\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165134401525\"><\/details>\n<\/p>\n<div id=\"Figure_08_06_012\" class=\"medium\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137930360\" class=\"precalculus media\">\n<p id=\"fs-id1165137740782\">Access these online resources for additional instruction and practice with parametric equations.<\/p>\n<ul id=\"fs-id1165137841610\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/introparametric\">Introduction to Parametric Equations<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/convertpara\">Converting Parametric Equations to Rectangular Form<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134548875\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137472078\">\n<li>Parameterizing a curve involves translating a rectangular equation in two variables,[latex]\\,x\\,[\/latex]and[latex]\\,y,\\,[\/latex]into two equations in three variables, <em>x<\/em>, <em>y<\/em>, and <em>t<\/em>. Often, more information is obtained from a set of parametric equations. See <a class=\"autogenerated-content\" href=\"#Example_08_06_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_08_06_02\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_08_06_03\">(Figure)<\/a>.<\/li>\n<li>Sometimes equations are simpler to graph when written in rectangular form. By eliminating[latex]\\,t,\\,[\/latex]an equation in[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]is the result.<\/li>\n<li>To eliminate[latex]\\,t,\\,[\/latex]solve one of the equations for[latex]\\,t,\\,[\/latex]and substitute the expression into the second equation. See <a class=\"autogenerated-content\" href=\"#Example_08_06_04\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_08_06_05\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_08_06_06\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_08_06_07\">(Figure)<\/a>.<\/li>\n<li>Finding the rectangular equation for a curve defined parametrically is basically the same as eliminating the parameter. Solve for[latex]\\,t\\,[\/latex]in one of the equations, and substitute the expression into the second equation. See <a class=\"autogenerated-content\" href=\"#Example_08_06_08\">(Figure)<\/a>.<\/li>\n<li>There are an infinite number of ways to choose a set of parametric equations for a curve defined as a rectangular equation.<\/li>\n<li>Find an expression for[latex]\\,x\\,[\/latex]such that the domain of the set of parametric equations remains the same as the original rectangular equation. See <a class=\"autogenerated-content\" href=\"#Example_08_06_09\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134068973\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137726497\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137452582\">\n<div id=\"fs-id1165137452584\">\n<p id=\"fs-id1165137430646\">What is a system of parametric equations?<\/p>\n<\/div>\n<div id=\"fs-id1165135457745\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135457747\">A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example,[latex]\\,x=f\\left(t\\right)\\,[\/latex]and[latex]\\,y=f\\left(t\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135515786\">\n<div id=\"fs-id1165135149809\">\n<p id=\"fs-id1165135149811\">Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137526957\">Explain how to eliminate a parameter given a set of parametric equations.<\/p>\n<\/div>\n<div id=\"fs-id1165137768026\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137768028\">Choose one equation to solve for[latex]\\,t,\\,[\/latex]substitute into the other equation and simplify.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137596332\">\n<div id=\"fs-id1165137596334\">\n<p id=\"fs-id1165133434649\">What is a benefit of writing a system of parametric equations as a Cartesian equation?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137655519\">\n<div id=\"fs-id1165137426830\">\n<p id=\"fs-id1165137426832\">What is a benefit of using parametric equations?<\/p>\n<\/div>\n<div id=\"fs-id1165135250841\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137461006\">Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137482276\">\n<div id=\"fs-id1165137465476\">\n<p id=\"fs-id1165137465478\">Why are there many sets of parametric equations to represent on Cartesian function?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451350\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137453740\">For the following exercises, eliminate the parameter[latex]\\,t\\,[\/latex]to rewrite the parametric equation as a Cartesian equation.<\/p>\n<div>\n<div>\n<p id=\"fs-id1165137453513\">[latex]\\{\\begin{array}{l}x(t)=5-t\\hfill \\\\ y(t)=8-2t\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135389966\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137706604\">[latex]y=-2+2x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134036798\">\n<div id=\"fs-id1165134036800\">\n<p id=\"fs-id1165134041404\">[latex]\\{\\begin{array}{l}x(t)=6-3t\\hfill \\\\ y(t)=10-t\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137475359\">\n<div id=\"fs-id1165137639024\">\n<p id=\"fs-id1165137639026\">[latex]\\{\\begin{array}{l}x(t)=2t+1\\hfill \\\\ y(t)=3\\sqrt{t}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137463947\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137463950\">[latex]y=3\\sqrt{\\frac{x-1}{2}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137401054\">\n<div id=\"fs-id1165137401056\">\n<p id=\"fs-id1165134050546\">[latex]\\{\\begin{array}{l}x(t)=3t-1\\hfill \\\\ y(t)=2{t}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135581966\">\n<div id=\"fs-id1165137401665\">\n<p id=\"fs-id1165137401667\">[latex]\\{\\begin{array}{l}x(t)=2{e}^{t}\\hfill \\\\ y(t)=1-5t\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135496291\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135496294\">[latex]x=2{e}^{\\frac{1-y}{5}}\\,[\/latex]or[latex]\\,y=1-5ln\\left(\\frac{x}{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134090131\">\n<p id=\"fs-id1165137659602\">[latex]\\{\\begin{array}{l}x(t)={e}^{-2t}\\hfill \\\\ y(t)=2{e}^{-t}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571890\">\n<div id=\"fs-id1165135571892\">\n<p id=\"fs-id1165135541958\">[latex]\\{\\begin{array}{l}x(t)=4\\text{log}(t)\\hfill \\\\ y(t)=3+2t\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135353727\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135353729\">[latex]x=4\\mathrm{log}\\left(\\frac{y-3}{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135185154\">\n<div id=\"fs-id1165135185157\">[latex]\\{\\begin{array}{l}x(t)=\\text{log}(2t)\\hfill \\\\ y(t)=\\sqrt{t-1}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137767323\">\n<div id=\"fs-id1165137841698\">[latex]\\{\\begin{array}{l}x(t)={t}^{3}-t\\hfill \\\\ y(t)=2t\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1165137803762\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137803764\">[latex]x={\\left(\\frac{y}{2}\\right)}^{3}-\\frac{y}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137693705\">\n<div id=\"fs-id1165135543530\">\n<p id=\"fs-id1165135543532\">[latex]\\{\\begin{array}{l}x(t)=t-{t}^{4}\\hfill \\\\ y(t)=t+2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137827977\">\n<div id=\"fs-id1165132944934\">\n<p id=\"fs-id1165132944936\">[latex]\\{\\begin{array}{l}x(t)={e}^{2t}\\hfill \\\\ y(t)={e}^{6t}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135414214\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135414216\">[latex]y={x}^{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137410877\">\n<div>\n<p id=\"fs-id1165135543509\">[latex]\\{\\begin{array}{l}x(t)={t}^{5}\\hfill \\\\ y(t)={t}^{10}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137668139\">\n<div id=\"fs-id1165135667825\">\n<p id=\"fs-id1165135667827\">[latex]\\{\\begin{array}{l}x(t)=4\\text{cos}\\,t\\hfill \\\\ y(t)=5\\mathrm{sin}\\,t \\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137762341\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137762343\">[latex]{\\left(\\frac{x}{4}\\right)}^{2}+{\\left(\\frac{y}{5}\\right)}^{2}=1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137585726\">\n<div id=\"fs-id1165137585728\">\n<p id=\"fs-id1165131895934\">[latex]\\{\\begin{array}{l}x(t)=3\\mathrm{sin}\\,t\\hfill \\\\ y(t)=6\\mathrm{cos}\\,t\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135384981\">\n<div id=\"fs-id1165135384983\">\n<p id=\"fs-id1165133210007\">[latex]\\{\\begin{array}{l}x(t)=2{\\text{cos}}^{2}t\\hfill \\\\ y(t)=-\\mathrm{sin}\\,t \\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137602808\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]{y}^{2}=1-\\frac{1}{2}x[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135189886\">\n<div id=\"fs-id1165135189889\">\n<p id=\"fs-id1165135189891\">[latex]\\{\\begin{array}{l}x(t)=\\mathrm{cos}\\,t+4\\\\ y(t)=2{\\mathrm{sin}}^{2}t\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193646\">\n<div id=\"fs-id1165135193648\">\n<p id=\"fs-id1165135316171\">[latex]\\{\\begin{array}{l}x(t)=t-1\\\\ y(t)={t}^{2}\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133213863\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133454382\">[latex]y={x}^{2}+2x+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134037467\">\n<p id=\"fs-id1165134037469\">[latex]\\{\\begin{array}{l}x(t)=-t\\\\ y(t)={t}^{3}+1\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134319713\">\n<div id=\"fs-id1165134319715\">\n<p id=\"fs-id1165135341173\">[latex]\\{\\begin{array}{l}x(t)=2t-1\\\\ y(t)={t}^{3}-2\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137704906\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137704908\">[latex]y={\\left(\\frac{x+1}{2}\\right)}^{3}-2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137548472\">For the following exercises, rewrite the parametric equation as a Cartesian equation by building an [latex]x\\text{-}y[\/latex] table.<\/p>\n<div id=\"fs-id1165137582353\">\n<div id=\"fs-id1165137582355\">\n<p id=\"fs-id1165137582357\">[latex]\\{\\begin{array}{l}x(t)=2t-1\\\\ y(t)=t+4\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134239737\">\n<div id=\"fs-id1165137735773\">\n<p id=\"fs-id1165137735775\">[latex]\\{\\begin{array}{l}x(t)=4-t\\\\ y(t)=3t+2\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134495094\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134495096\">[latex]y=-3x+14[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132947372\">\n<div id=\"fs-id1165132947374\">\n<p id=\"fs-id1165134109660\">[latex]\\{\\begin{array}{l}x(t)=2t-1\\\\ y(t)=5t\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134116716\">\n<div id=\"fs-id1165134116719\">\n<p id=\"fs-id1165135173427\">[latex]\\{\\begin{array}{l}x(t)=4t-1\\\\ y(t)=4t+2\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134187207\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135528264\">[latex]y=x+3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135456667\">For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting [latex]x\\left(t\\right)=t[\/latex] or by setting[latex]\\,y\\left(t\\right)=t.[\/latex]<\/p>\n<div id=\"fs-id1165137595902\">\n<div id=\"fs-id1165135329750\">[latex]y\\left(x\\right)=3{x}^{2}+3[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134148489\">\n<div>[latex]y\\left(x\\right)=2\\mathrm{sin}\\,x+1[\/latex]<\/div>\n<div id=\"fs-id1165137925225\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137925228\">[latex]\\{\\begin{array}{l}x(t)=t\\hfill \\\\ y(t)=2\\mathrm{sin}t+1\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134373445\">\n<div id=\"fs-id1165134373447\">\n<p id=\"fs-id1165135641460\">[latex]x\\left(y\\right)=3\\mathrm{log}\\left(y\\right)+y[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135582390\">\n<div id=\"fs-id1165135582392\">\n<p id=\"fs-id1165135582394\">[latex]x\\left(y\\right)=\\sqrt{y}+2y[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137575928\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137575930\">[latex]\\{\\begin{array}{l}x(t)=\\sqrt{t}+2t\\hfill \\\\ y(t)=t\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p>For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using [latex]x\\left(t\\right)=a\\mathrm{cos}\\,t[\/latex] and[latex]\\,y\\left(t\\right)=b\\mathrm{sin}\\,t.\\,[\/latex]Identify the curve.<\/p>\n<div id=\"fs-id1165137548806\">\n<div id=\"fs-id1165137548808\">\n<p id=\"fs-id1165137548810\">[latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134149924\">\n<div id=\"fs-id1165134149926\">\n<p id=\"fs-id1165137854890\">[latex]\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{36}=1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137849389\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137849392\">[latex]\\{\\begin{array}{l}x(t)=4\\mathrm{cos}\\,t\\hfill \\\\ y(t)=6\\mathrm{sin}\\,t\\hfill \\end{array};\\,[\/latex]Ellipse<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135575985\">\n<div id=\"fs-id1165135575987\">\n<p id=\"fs-id1165137833111\">[latex]{x}^{2}+{y}^{2}=16[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135160120\">\n<div id=\"fs-id1165137444304\">\n<p id=\"fs-id1165137444306\">[latex]{x}^{2}+{y}^{2}=10[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137559264\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137559266\">[latex]\\{\\begin{array}{l}x(t)=\\sqrt{10}\\mathrm{cos}t\\hfill \\\\ y(t)=\\sqrt{10}\\mathrm{sin}t\\hfill \\end{array};\\,[\/latex]<br \/>\nCircle<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137893366\">\n<div id=\"fs-id1165137893368\">\n<p id=\"fs-id1165137893478\">Parameterize the line from[latex]\\,\\left(3,0\\right)\\,[\/latex]to[latex]\\,\\left(-2,-5\\right)\\,[\/latex]so that the line is at[latex]\\,\\left(3,0\\right)\\,[\/latex]at[latex]\\,t=0,\\,[\/latex]and at[latex]\\,\\left(-2,-5\\right)\\,[\/latex]at[latex]\\,t=1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137651086\">\n<div id=\"fs-id1165137651088\">\n<p id=\"fs-id1165137651090\">Parameterize the line from[latex]\\,\\left(-1,0\\right)\\,[\/latex]to[latex]\\,\\left(3,-2\\right)\\,[\/latex]so that the line is at[latex]\\,\\left(-1,0\\right)\\,[\/latex]at[latex]\\,t=0,\\,[\/latex]and at[latex]\\,\\left(3,-2\\right)\\,[\/latex]at[latex]\\,t=1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137400302\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137705382\">[latex]\\{\\begin{array}{l}x(t)=-1+4t\\hfill \\\\ y(t)=-2t\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137547147\">\n<div id=\"fs-id1165137547149\">\n<p id=\"fs-id1165137547151\">Parameterize the line from[latex]\\,\\left(-1,5\\right)\\,[\/latex]to[latex]\\,\\left(2,3\\right)[\/latex]so that the line is at[latex]\\,\\left(-1,5\\right)\\,[\/latex]at[latex]\\,t=0,\\,[\/latex]and at[latex]\\,\\left(2,3\\right)\\,[\/latex]at[latex]\\,t=1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135452295\">\n<div id=\"fs-id1165137663434\">\n<p id=\"fs-id1165137663436\">Parameterize the line from[latex]\\,\\left(4,1\\right)\\,[\/latex]to[latex]\\,\\left(6,-2\\right)\\,[\/latex]so that the line is at[latex]\\,\\left(4,1\\right)\\,[\/latex]at[latex]\\,t=0,\\,[\/latex]and at[latex]\\,\\left(6,-2\\right)\\,[\/latex]at[latex]\\,t=1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134164960\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134164962\">[latex]\\{\\begin{array}{l}x(t)=4+2t\\hfill \\\\ y(t)=1-3t\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134211398\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137408704\">For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.<\/p>\n<div id=\"fs-id1165131937984\">\n<div id=\"fs-id1165131937986\">\n<p id=\"fs-id1165131937988\">[latex]\\{\\begin{array}{l}{x}_{1}(t)=3t\\hfill \\\\ {y}_{1}(t)=2t-1\\hfill \\end{array}\\text{ and }\\{\\begin{array}{l}{x}_{2}(t)=t+3\\hfill \\\\ {y}_{2}(t)=4t-4\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137470483\">\n<div id=\"fs-id1165137939491\">\n<p id=\"fs-id1165137939493\">[latex]\\{\\begin{array}{l}{x}_{1}(t)={t}^{2}\\hfill \\\\ {y}_{1}(t)=2t-1\\hfill \\end{array}\\text{ and }\\{\\begin{array}{l}{x}_{2}(t)=-t+6\\hfill \\\\ {y}_{2}(t)=t+1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137453625\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137453627\">yes, at [latex]t=2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137400113\">For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.<\/p>\n<div id=\"fs-id1165137730286\">\n<div id=\"fs-id1165137730288\">\n<p id=\"fs-id1165135181639\">[latex]\\{\\begin{array}{l}{x}_{1}(t)=3{t}^{2}-3t+7\\hfill \\\\ {y}_{1}(t)=2t+3\\hfill \\end{array}[\/latex]<\/p>\n<table id=\"fs-id1165137565962\" class=\"unnumbered\" summary=\"Four rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains -1, 0, 1. The rest of the values in columns x and y are blank.\">\n<caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u20131<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137667655\">\n<div id=\"fs-id1165137647616\">\n<p id=\"fs-id1165137647618\">[latex]\\{\\begin{array}{l}{x}_{1}(t)={t}^{2}-4\\hfill \\\\ {y}_{1}(t)=2{t}^{2}-1\\hfill \\end{array}[\/latex]<\/p>\n<table id=\"fs-id1165135407032\" class=\"unnumbered\" summary=\"Four rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains 1, 2, 3. The rest of the values in columns x and y are blank.\">\n<caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137737862\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<table id=\"fs-id1165137737864\" class=\"unnumbered\" summary=\"..\">\n<caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>-3<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>0<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>5<\/td>\n<td>17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135640921\">\n<div id=\"fs-id1165135640922\">\n<p id=\"fs-id1165135640923\">[latex]\\{\\begin{array}{l}{x}_{1}(t)={t}^{4}\\hfill \\\\ {y}_{1}(t)={t}^{3}+4\\hfill \\end{array}[\/latex]<\/p>\n<table id=\"fs-id1165137921660\" class=\"unnumbered\" summary=\"Five rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains -1, 0, 1, 2. The rest of the values in columns x and y are blank.\">\n<caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>-1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193961\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137588584\">\n<div id=\"fs-id1165137588586\">\n<p id=\"fs-id1165134389802\">Find two different sets of parametric equations for[latex]\\,y={\\left(x+1\\right)}^{2}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137437388\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137437390\">answers may vary:[latex]\\,\\{\\begin{array}{l}x(t)=t-1\\hfill \\\\ y(t)={t}^{2}\\hfill \\end{array}\\text{ and }\\{\\begin{array}{l}x(t)=t+1\\hfill \\\\ y(t)={(t+2)}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135189893\">\n<div id=\"fs-id1165135189896\">\n<p id=\"fs-id1165135189898\">Find two different sets of parametric equations for[latex]\\,y=3x-2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134042368\">\n<div id=\"fs-id1165137471410\">\n<p id=\"fs-id1165137471412\">Find two different sets of parametric equations for[latex]\\,y={x}^{2}-4x+4.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137834340\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137834342\">answers may vary: ,[latex]\\,\\{\\begin{array}{l}x(t)=t\\hfill \\\\ y(t)={t}^{2}-4t+4\\hfill \\end{array}\\text{ and }\\{\\begin{array}{l}x(t)=t+2\\hfill \\\\ y(t)={t}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135186874\">\n<dt>parameter<\/dt>\n<dd id=\"fs-id1165134357588\">a variable, often representing time, upon which[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]are both dependent<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":7,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-160","chapter","type-chapter","status-publish","hentry"],"part":147,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/160","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\/160\/revisions"}],"predecessor-version":[{"id":161,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/160\/revisions\/161"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/147"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/160\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=160"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=160"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=160"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}