{"id":3329,"date":"2018-07-30T18:13:39","date_gmt":"2018-07-30T18:13:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/?post_type=chapter&#038;p=3329"},"modified":"2018-08-06T18:07:05","modified_gmt":"2018-08-06T18:07:05","slug":"parametric-equations-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/parametric-equations-graphs\/","title":{"raw":"Parametric Equations: Graphs","rendered":"Parametric Equations: Graphs"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section you will:\r\n<ul>\r\n \t<li>Graph plane curves described by parametric equations by plotting points.<\/li>\r\n \t<li>Graph parametric equations.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165137758339\">It is the bottom of the ninth inning, with two outs and two men on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately[latex]\\,45\u00b0\\,[\/latex]to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using <span class=\"no-emphasis\">parametric equations<\/span>. In this section, we\u2019ll discuss parametric equations and some common applications, such as projectile motion problems.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/cnx.org\/resources\/4c0775a0076c07a6585617c97ae7cf0ffaff107d\/CNX_Precalc_Figure_08_07_001.jpg\" alt=\"Photo of a baseball batter swinging.\" width=\"488\" height=\"333\" \/> <strong>Figure 1.<\/strong> Parametric equations can model the path of a projectile. (credit: Paul Kreher, Flickr)[\/caption]\r\n\r\n<div id=\"fs-id1165137566996\" class=\"bc-section section\">\r\n<h3>Graphing Parametric Equations by Plotting Points<\/h3>\r\n<p id=\"fs-id1165137470035\">In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.<\/p>\r\n\r\n<div id=\"fs-id1165135361790\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165132005266\"><strong>Given a pair of parametric equations, sketch a graph by plotting points.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137428125\" type=\"1\">\r\n \t<li>Construct a table with three columns:[latex]\\,t,x\\left(t\\right),\\text{and}\\,\\,y\\left(t\\right).[\/latex]<\/li>\r\n \t<li>Evaluate [latex]x[\/latex] and [latex]y[\/latex] for values of [latex]t[\/latex] over the interval for which the functions are defined.<\/li>\r\n \t<li>Plot the resulting pairs[latex]\\,\\left(x,y\\right).[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_08_07_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134532923\">\r\n<div id=\"fs-id1165137854808\">\r\n<h3>Sketching the Graph of a Pair of Parametric Equations by Plotting Points<\/h3>\r\n<p id=\"fs-id1165137769779\">Sketch the graph of the <span class=\"no-emphasis\">parametric equations<\/span> [latex]x\\left(t\\right)={t}^{2}+1,\\,\\,y\\left(t\\right)=2+t.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137938383\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137938383\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137938383\"]\r\n<p id=\"fs-id1165135181426\">Construct a table of values for[latex]\\,t,x\\left(t\\right),\\,[\/latex]and[latex]\\,y\\left(t\\right),\\,[\/latex]as in <a class=\"autogenerated-content\" href=\"#Table_08_07_01\">(Figure)<\/a>, and plot the points in a plane.<\/p>\r\n\r\n<table id=\"Table_08_07_01\" summary=\"Twelve rows and three columns. First column is labeled t, second column is labeled x(t)=t^2 + 1, third column is labeled y(t) = 2 + t. The table has ordered triples of each of these row values: (-5, 26, -3), (-4, 17, -2), (-3, 10, -1), (-2, 5, 0), (-1, 2, 1), (0, 1, 2), (1, 2, 3), (2, 5, 4), (3, 10, 5), (4, 17, 6), (5, 26, 7).\">\r\n<thead>\r\n<tr>\r\n<th>[latex]t[\/latex]<\/th>\r\n<th>[latex]x\\left(t\\right)={t}^{2}+1[\/latex]<\/th>\r\n<th>[latex]y\\left(t\\right)=2+t[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<td>[latex]26[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-4[\/latex]<\/td>\r\n<td>[latex]17[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]17[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]26[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/551e6c2de6d75d9ab3297287596f3ddfbf881ca7\/CNX_Precalc_Figure_08_07_002.jpg\" alt=\"Graph of the given parabola opening to the right.\" width=\"487\" height=\"366\" \/> <strong>Figure 2.<\/strong>[\/caption]\r\n<p id=\"fs-id1165137628360\">The graph is a <span class=\"no-emphasis\">parabola<\/span> with vertex at the point[latex]\\,\\left(1,2\\right),[\/latex]opening to the right. See <a class=\"autogenerated-content\" href=\"#Figure_08_07_002\">(Figure)<\/a>.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135515789\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1165137758642\">As values for[latex]\\,t\\,[\/latex]progress in a positive direction from 0 to 5, the plotted points trace out the top half of the parabola. As values of[latex]\\,t\\,[\/latex]become negative, they trace out the lower half of the parabola. There are no restrictions on the domain. The arrows indicate direction according to increasing values of [latex]\\,t.\\,[\/latex]The graph does not represent a function, as it will fail the vertical line test. The graph is drawn in two parts: the positive values for [latex]t,[\/latex] and the negative values for [latex]t.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134485685\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div>\r\n<div id=\"fs-id1165133210758\">\r\n\r\nSketch the graph of the parametric equations[latex]\\,x=\\sqrt{t},\\,\\,y=2t+3,\\,\\,\\,0\\le t\\le 3.[\/latex]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137849525\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137849525\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137849525\"]<img src=\"https:\/\/cnx.org\/resources\/5a648144babb4dd4e08a1611cead7856231ff4e7\/CNX_Precalc_Figure_08_07_003.jpg\" alt=\"Graph of the given parametric equations with the restricted domain - it looks like the right half of an upward opening parabola.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_08_07_02\" class=\"textbox examples\">\r\n<div>\r\n<div id=\"fs-id1165135305906\">\r\n<h3>Sketching the Graph of Trigonometric Parametric Equations<\/h3>\r\n<p id=\"fs-id1165137942382\">Construct a table of values for the given parametric equations and sketch the graph:<\/p>\r\n\r\n<div id=\"eip-id2721082\" class=\"unnumbered\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}x=2\\mathrm{cos}\\,t\\hfill \\\\ y=4\\mathrm{sin}\\,t\\hfill \\end{array}\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137932663\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137932663\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137932663\"]\r\n<p id=\"fs-id1165137665416\">Construct a table like that in <a class=\"autogenerated-content\" href=\"#Table_08_07_02\">(Figure)<\/a> using angle measure in radians as inputs for[latex]\\,t,\\,[\/latex]and evaluating[latex]\\,x\\,[\/latex]and[latex]\\,y.\\,[\/latex]Using angles with known sine and cosine values for[latex]\\,t\\,[\/latex]makes calculations easier.<\/p>\r\n\r\n<table id=\"Table_08_07_02\" summary=\"Fourteen rows and three columns. First column is labeled t, second column is labeled x(t)=2cos(1), third column is labeled y(t)=4sin(1). The table has ordered triples of each of these row values: (0, x=2cos(0)=2, y=4sin(0)=0), (pi\/6, x=2cos(pi\/6)=rad3, y=4sin(pi\/6)=2), (pi\/3, x=2cos(pi\/3)=1, y=4sin(pi\/3)=2rad3), (pi\/2, x=2cos(pi\/2)=0, y=4sin(pi\/2)=4), (2pi\/3, x=2cos(2pi\/3)=-1, y=4sin(2pi\/3)=2rad3), (5pi\/6, x=2cos(5pi\/6)=-rad3, y=4sin(5pi\/6)=2), (pi, x=2cos(pi)=-2, y=4sin(pi)=0), (7pi\/6, x=2cos(7pi\/6) = -rad3, y=4sin(7pi\/6)=-2), (4pi\/3, x=2cos(4pi\/3)=-1, y=4sin(4pi\/3)=-2rad3), (3pi\/2, x=2cos(3pi\/2)=0, y=4sin(3pi\/2)=-4), (5pi\/3, x=2cos(5pi\/3)=1, y=4sin(5pi\/3)=-2rad3), (11pi\/6, x=2cos(11pi\/6)=rad3, y=4sin(11pi\/6)=-2), (2pi, x=2cos(2pi)=2, y=4sin(2pi)=0).\">\r\n<thead>\r\n<tr>\r\n<th>[latex]t[\/latex]<\/th>\r\n<th>[latex]x=2\\mathrm{cos}\\,t[\/latex]<\/th>\r\n<th>[latex]y=4\\mathrm{sin}\\,t[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(0\\right)=2[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(0\\right)=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{\\pi }{6}[\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)=\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{\\pi }{3}[\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{\\pi }{3}\\right)=1[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{\\pi }{3}\\right)=2\\sqrt{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)=0[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{\\pi }{2}\\right)=4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{2\\pi }{3}\\right)=-1[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{2\\pi }{3}\\right)=2\\sqrt{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{5\\pi }{6}\\right)=-\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{5\\pi }{6}\\right)=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\pi [\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\pi \\right)=-2[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\pi \\right)=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{7\\pi }{6}[\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{7\\pi }{6}\\right)=-\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{7\\pi }{6}\\right)=-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{4\\pi }{3}[\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{4\\pi }{3}\\right)=-1[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{4\\pi }{3}\\right)=-2\\sqrt{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{3\\pi }{2}\\right)=0[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{3\\pi }{2}\\right)=-4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{5\\pi }{3}[\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{5\\pi }{3}\\right)=1[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{5\\pi }{3}\\right)=-2\\sqrt{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{11\\pi }{6}[\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{11\\pi }{6}\\right)=\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{11\\pi }{6}\\right)=-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\pi [\/latex]<\/td>\r\n<td>[latex]x=2\\mathrm{cos}\\left(2\\pi \\right)=2[\/latex]<\/td>\r\n<td>[latex]y=4\\mathrm{sin}\\left(2\\pi \\right)=0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137581718\"><a class=\"autogenerated-content\" href=\"#Figure_08_07_004\">(Figure)<\/a> shows the graph.<span id=\"fs-id1165137644320\"><\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/3332600db5000513af59f82a010b36b8aba0e29c\/CNX_Precalc_Figure_08_07_004.jpg\" alt=\"Graph of the given equations - a vertical ellipse.\" width=\"487\" height=\"441\" \/> <strong>Figure 3.<\/strong>[\/caption]\r\n<p id=\"fs-id1165135569879\">By the symmetry shown in the values of [latex]x[\/latex] and [latex]\\,y,\\,[\/latex]we see that the parametric equations represent an <span class=\"no-emphasis\">ellipse<\/span>. The <span class=\"no-emphasis\">ellipse<\/span> is mapped in a counterclockwise direction as shown by the arrows indicating increasing[latex]\\,t\\,[\/latex]values.<span id=\"fs-id1165137644320\"><\/span>[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<div id=\"eip-id1551946\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1165137811673\">We have seen that parametric equations can be graphed by plotting points. However, a graphing calculator will save some time and reveal nuances in a graph that may be too tedious to discover using only hand calculations.<\/p>\r\n<p id=\"eip-157\">Make sure to change the mode on the calculator to parametric (PAR). To confirm, the[latex]\\,Y=\\,[\/latex]window should show<\/p>\r\n\r\n<div id=\"fs-id1165137824587\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}{X}_{1T}=\\\\ {Y}_{1T}=\\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165135168174\">instead of[latex]\\,{Y}_{1}=.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137934405\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"fs-id1165137714934\">\r\n<div id=\"fs-id1165137935182\">\r\n<p id=\"fs-id1165137935183\">Graph the parametric equations:[latex]\\,x=5\\mathrm{cos}\\,t,\\,\\,y=3\\mathrm{sin}\\,t.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137408223\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137408223\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137408223\"]<img src=\"https:\/\/cnx.org\/resources\/2f2ed4dddafe78c5d1c0e38d229a62b6fc0673ea\/CNX_Precalc_Figure_08_07_005.jpg\" alt=\"Graph of the given equations - a horizontal ellipse.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137610977\" class=\"bc-section section\">\r\n<div id=\"Example_08_07_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135192637\">\r\n<div id=\"fs-id1165135181483\">\r\n<h3>Graphing Parametric Equations and Rectangular Form Together<\/h3>\r\n<p id=\"fs-id1165137466433\">Graph the parametric equations[latex]\\,x=5\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,y=2\\mathrm{sin}\\,t.\\,[\/latex]First, construct the graph using data points generated from the <span class=\"no-emphasis\">parametric form<\/span>. Then graph the <span class=\"no-emphasis\">rectangular form<\/span> of the equation. Compare the two graphs.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137387636\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137387636\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137387636\"]\r\n<p id=\"fs-id1165137807398\">Construct a table of values like that in <a class=\"autogenerated-content\" href=\"#Table_08_07_03\">(Figure)<\/a>.<\/p>\r\n\r\n<table id=\"Table_08_07_03\" summary=\"Twelve rows and three columns. First column is labeled t, second column is labeled x(t)=5cos(t), third column is labeled y(t) = 2sin(t). The table has ordered triples of each of these row values: (0, x=5cos(0)=5, y=2sin(0)=0), (1, x=5cos(1) =approx 2.7, y=2sin(1) =approx 1.7), (2, x=5cos(2) =approx -2.1, y=2sin(2) =approx 1.8), (3, x=5cos(3) =approx -4.95, y=2sin(3) =approx 0.28), (4, x=5cos(4) =approx -3.3, y=2sin(4) =approx -1.5), (5, x=5cos(5) =approx 1.4, y=2sin(5) =approx -1.9), (-1, x=5cos(-1) =approx 2.7, y=2sin(-1) =approx -1.7), (-2, x=5cos(-2) =approx -2.1, y=2sin(-2) =approx -1.8), (-3, x=5cos(-3) =approx -4.95, y=2sin(-3) =approx -0.28), (-4, x=5cos(-4) =approx -3.3, y=2sin(-4) =approx 1.5), (-5, x=5cos(-5) =approx 1.4, y=2sin(-5) =approx 1.9).\">\r\n<thead>\r\n<tr>\r\n<th>[latex]t[\/latex]<\/th>\r\n<th>[latex]x=5\\mathrm{cos}\\,t[\/latex]<\/th>\r\n<th>[latex]y=2\\mathrm{sin}\\,t[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\text{0}[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(0\\right)=5[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(0\\right)=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{1}[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(1\\right)\\approx 2.7[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(1\\right)\\approx 1.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{2}[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(2\\right)\\approx -2.1[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(2\\right)\\approx 1.8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{3}[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(3\\right)\\approx -4.95[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(3\\right)\\approx 0.28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{4}[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(4\\right)\\approx -3.3[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(4\\right)\\approx -1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{5}[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(5\\right)\\approx 1.4[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(5\\right)\\approx -1.9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(-1\\right)\\approx 2.7[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(-1\\right)\\approx -1.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(-2\\right)\\approx -2.1[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(-2\\right)\\approx -1.8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(-3\\right)\\approx -4.95[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(-3\\right)\\approx -0.28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-4[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(-4\\right)\\approx -3.3[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(-4\\right)\\approx 1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<td>[latex]x=5\\mathrm{cos}\\left(-5\\right)\\approx 1.4[\/latex]<\/td>\r\n<td>[latex]y=2\\mathrm{sin}\\left(-5\\right)\\approx 1.9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137910988\">Plot the[latex]\\,\\left(x,y\\right)\\,[\/latex]values from the table. See <a class=\"autogenerated-content\" href=\"#Figure_08_07_006\">(Figure)<\/a>.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/cnx.org\/resources\/a069f98cd1e9673cef2e6d699842fbf33e6c98b1\/CNX_Precalc_Figure_08_07_006.jpg\" alt=\"Graph of the given ellipse in parametric and rectangular coordinates - it is the same thing in both images.\" width=\"975\" height=\"290\" \/> <strong>Figure 4.<\/strong>[\/caption]\r\n\r\nNext, translate the parametric equations to rectangular form. To do this, we solve for[latex]\\,t\\,[\/latex]in either[latex]\\,x\\left(t\\right)\\,[\/latex]or[latex]\\,y\\left(t\\right),\\,[\/latex]and then substitute the expression for[latex]\\,t\\,[\/latex]in the other equation. The result will be a function [latex]y\\left(x\\right)[\/latex] if solving for[latex]\\,t\\,[\/latex]as a function of[latex]\\,x,\\,[\/latex]or[latex]\\,x\\left(y\\right)[\/latex] if solving for[latex]\\,t\\,[\/latex]as a function of[latex]\\,y.\\,[\/latex]\r\n<div id=\"fs-id1165137565254\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\,x=5\\mathrm{cos}\\,t\\hfill &amp; \\hfill \\\\ \\frac{x}{5}=\\mathrm{cos}\\,t\\hfill &amp; \\text{Solve for }\\mathrm{cos}\\,t.\\hfill \\\\ \\,y=2\\mathrm{sin}\\,t\\begin{array}{cccc}&amp; &amp; &amp; \\end{array} \\hfill &amp; \\text{Solve for }\\mathrm{sin}\\,t.\\hfill \\\\ \\frac{y}{2}=\\mathrm{sin}\\,t\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\r\nThen, use the <span class=\"no-emphasis\">Pythagorean Theorem<\/span>.\r\n<div id=\"fs-id1165137653844\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{r}\\hfill {\\mathrm{cos}}^{2}t+{\\mathrm{sin}}^{2}t=1\\\\ \\hfill {\\left(\\frac{x}{5}\\right)}^{2}+{\\left(\\frac{y}{2}\\right)}^{2}=1\\\\ \\hfill \\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{4}=1\\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137470251\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1165137424263\">In <a class=\"autogenerated-content\" href=\"#Figure_08_07_007\">(Figure)<\/a>, the data from the parametric equations and the rectangular equation are plotted together. The parametric equations are plotted in blue; the graph for the rectangular equation is drawn on top of the parametric in a dashed style colored red. Clearly, both forms produce the same graph.<span id=\"fs-id1165134032313\"><\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/cb601162053c131dacd113d2d6e6db5d15dc482d\/CNX_Precalc_Figure_08_07_007.jpg\" alt=\"Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.\" width=\"487\" height=\"290\" \/> <strong>Figure 5.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_08_07_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134152557\">\r\n<div id=\"fs-id1165135208839\">\r\n<h3>Graphing Parametric Equations and Rectangular Equations on the Coordinate System<\/h3>\r\n<p id=\"fs-id1165137898083\">Graph the parametric equations[latex]\\,x=t+1\\,[\/latex] and[latex]\\,y=\\sqrt{t},\\,\\,t\\ge 0,\\,[\/latex]and the rectangular equivalent [latex]y=\\sqrt{x-1}\\,[\/latex]on the same coordinate system.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134248714\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134248714\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134248714\"]Construct a table of values for the parametric equations, as we did in the previous example, and graph[latex]\\,y=\\sqrt{t},\\,\\,t\\ge 0\\,[\/latex]on the same grid, as in <a class=\"autogenerated-content\" href=\"#Figure_08_07_008\">(Figure)<\/a>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/cnx.org\/resources\/39ca8a7fc4de08dc9bb71f0097f9be82c4a06f8f\/CNX_Precalc_Figure_08_07_008.jpg\" alt=\"Overlayed graph of the two versions of the given function, showing that they are the same whether they are given in parametric or rectangular coordinates.\" width=\"488\" height=\"291\" \/> <strong>Figure 6.<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135347348\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1165134254196\">With the domain on[latex]\\,t\\,[\/latex]restricted, we only plot positive values of[latex]\\,t.\\,[\/latex]The parametric data is graphed in blue and the graph of the rectangular equation is dashed in red. Once again, we see that the two forms overlap.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137823949\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_08_07_03\">\r\n<div id=\"fs-id1165137551494\">\r\n<p id=\"fs-id1165134347381\">Sketch the graph of the parametric equations[latex]\\,x=2\\mathrm{cos}\\,\\theta \\,\\,\\,\\text{and}\\,\\,y=4\\mathrm{sin}\\,\\theta ,\\,[\/latex]along with the rectangular equation on the same grid.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137807092\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137807092\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137807092\"]\r\n<p id=\"fs-id1165137807093\">The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations.<\/p>\r\n<img src=\"https:\/\/cnx.org\/resources\/994011c9d17beb64f9dc4af835f756ba8beb1b37\/CNX_Precalc_Figure_08_07_009.jpg\" alt=\"Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134181759\" class=\"bc-section section\">\r\n<h3>Applications of Parametric Equations<\/h3>\r\n<p id=\"fs-id1165135393365\">Many of the advantages of parametric equations become obvious when applied to solving real-world problems. Although rectangular equations in <em>x<\/em> and <em>y<\/em> give an overall picture of an object's path, they do not reveal the position of an object at a specific time. Parametric equations, however, illustrate how the values of <em>x<\/em> and <em>y<\/em> change depending on <em>t<\/em>, as the location of a moving object at a particular time.<\/p>\r\n<p id=\"fs-id1165137927248\">A common application of parametric equations is solving problems involving projectile motion. In this type of motion, an object is propelled forward in an upward direction forming an angle of [latex]\\theta [\/latex] to the horizontal, with an initial speed of [latex]{v}_{0},\\,[\/latex]and at a height [latex]h[\/latex] above the horizontal.<\/p>\r\nThe path of an object propelled at an inclination of [latex]\\theta [\/latex] to the horizontal, with initial speed [latex]{v}_{0},\\,[\/latex]and at a height [latex]h[\/latex] above the horizontal, is given by\r\n<div class=\"unnumbered\">[latex]\\begin{array}{l}x=\\left({v}_{0}\\mathrm{cos}\\theta \\right)t\\text{ }\\hfill \\\\ y=-\\frac{1}{2}g{t}^{2}+\\left({v}_{0}\\mathrm{sin}\\theta \\right)t+h\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137723578\">where[latex]\\,g\\,[\/latex]accounts for the effects of <span class=\"no-emphasis\">gravity<\/span> and [latex]h[\/latex] is the initial height of the object. Depending on the units involved in the problem, use[latex]\\,g=32\\,\\text{ft}\\,\\text{\/}\\,{\\text{s}}^{2}\\,[\/latex]or[latex]\\,g=9.8\\,\\text{m}\\,\\text{\/}\\,{\\text{s}}^{2}.\\,[\/latex]The equation for[latex]\\,x\\,[\/latex]gives horizontal distance, and the equation for[latex]\\,y\\,[\/latex] gives the vertical distance.<\/p>\r\n\r\n<div id=\"fs-id1165135469031\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137761099\"><strong>Given a projectile motion problem, use parametric equations to solve.<\/strong><\/p>\r\n\r\n<ol type=\"1\">\r\n \t<li>The horizontal distance is given by[latex]\\,x=\\left({v}_{0}\\mathrm{cos}\\,\\theta \\right)t.\\,[\/latex]Substitute the initial speed of the object for[latex]\\,{v}_{0}.[\/latex]<\/li>\r\n \t<li>The expression[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]indicates the angle at which the object is propelled. Substitute that angle in degrees for[latex]\\,\\mathrm{cos}\\,\\theta .[\/latex]<\/li>\r\n \t<li>The vertical distance is given by the formula[latex]\\,y=-\\frac{1}{2}g{t}^{2}+\\left({v}_{0}\\mathrm{sin}\\,\\theta \\right)t+h.\\,[\/latex]The term[latex]\\,-\\frac{1}{2}g{t}^{2}\\,[\/latex]represents the effect of gravity. Depending on units involved, use[latex]\\,g=32\\,{\\text{ft\/s}}^{2}\\,[\/latex]or[latex]\\,g=9.8\\,{\\text{m\/s}}^{2}.\\,[\/latex]Again, substitute the initial speed for[latex]\\,{v}_{0},\\,[\/latex]and the height at which the object was propelled for[latex]\\,h.[\/latex]<\/li>\r\n \t<li>Proceed by calculating each term to solve for[latex]\\,t.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_08_07_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137836910\">\r\n<div id=\"fs-id1165137501592\">\r\n<h3>Finding the Parametric Equations to Describe the Motion of a Baseball<\/h3>\r\n<p id=\"fs-id1165135570049\">Solve the problem presented at the beginning of this section. Does the batter hit the game-winning home run? Assume that the ball is hit with an initial velocity of 140 feet per second at an angle of[latex]\\,45\u00b0\\,[\/latex]to the horizontal, making contact 3 feet above the ground.<\/p>\r\n\r\n<ol type=\"a\">\r\n \t<li>Find the parametric equations to model the path of the baseball.<\/li>\r\n \t<li>Where is the ball after 2 seconds?<\/li>\r\n \t<li>How long is the ball in the air?<\/li>\r\n \t<li>Is it a home run?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165137453386\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137453386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137453386\"]\r\n<ol id=\"fs-id1165135445817\" type=\"a\">\r\n \t<li>\r\n<p id=\"fs-id1165133299084\">Use the formulas to set up the equations. The horizontal position is found using the parametric equation for[latex]\\,x.\\,[\/latex]Thus,<\/p>\r\n\r\n<div class=\"unnumbered\">[latex]\\begin{array}{l}x=\\left({v}_{0}\\mathrm{cos}\\,\\theta \\right)t\\hfill \\\\ x=\\left(140\\mathrm{cos}\\left(45\u00b0\\right)\\right)t\\hfill \\end{array}[\/latex<\/div>\r\n<p id=\"fs-id1165135192687\">The vertical position is found using the parametric equation for[latex]\\,y.\\,[\/latex]Thus,<\/p>\r\n\r\n<div id=\"fs-id1165137662399\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}y=-16{t}^{2}+\\left({v}_{0}\\mathrm{sin}\\,\\theta \\right)t+h\\hfill \\\\ y=-16{t}^{2}+\\left(140\\mathrm{sin}\\left(45\u00b0\\right)\\right)t+3\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>\r\n<p id=\"fs-id1165137643152\">Substitute 2 into the equations to find the horizontal and vertical positions of the ball.<\/p>\r\n\r\n<div id=\"fs-id1165133052900\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ x=\\left(140\\mathrm{cos}\\left(45\u00b0\\right)\\right)\\left(2\\right)\\hfill \\\\ x=198\\text{ feet}\\hfill \\\\ \\hfill \\\\ y=-16{\\left(2\\right)}^{2}+\\left(140\\mathrm{sin}\\left(45\u00b0\\right)\\right)\\left(2\\right)+3\\hfill \\\\ y=137\\text{ feet}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165134063228\">After 2 seconds, the ball is 198 feet away from the batter\u2019s box and 137 feet above the ground.<\/p>\r\n<\/li>\r\n \t<li>\r\n<p id=\"fs-id1165134347441\">To calculate how long the ball is in the air, we have to find out when it will hit ground, or when[latex]\\,y=0.\\,[\/latex]Thus,<\/p>\r\n\r\n<div id=\"fs-id1165137667988\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}y=-16{t}^{2}+\\left(140\\mathrm{sin}\\left({45}^{\\circ }\\right)\\right)t+3\\hfill &amp; \\hfill \\\\ y=0\\hfill &amp; \\text{Set }y\\left(t\\right)=0\\text{ and solve the quadratic}.\\hfill \\\\ t=6.2173\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137666794\">When[latex]\\,t=6.2173\\,[\/latex]seconds, the ball has hit the ground. (The quadratic equation can be solved in various ways, but this problem was solved using a computer math program.)<\/p>\r\n<\/li>\r\n \t<li>\r\n<p id=\"fs-id1165137462832\">We cannot confirm that the hit was a home run without considering the size of the outfield, which varies from field to field. However, for simplicity\u2019s sake, let\u2019s assume that the outfield wall is 400 feet from home plate in the deepest part of the park. Let\u2019s also assume that the wall is 10 feet high. In order to determine whether the ball clears the wall, we need to calculate how high the ball is when <em>x<\/em> = 400 feet. So we will set <em>x<\/em> = 400, solve for [latex]t,\\,[\/latex]and input [latex]t[\/latex] into[latex]\\,y.\\,[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137921676\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\text{ }x=\\left(140\\mathrm{cos}\\left(45\u00b0\\right)\\right)t\\hfill \\\\ 400=\\left(140\\mathrm{cos}\\left(45\u00b0\\right)\\right)t\\hfill \\\\ \\text{ }t=4.04\\hfill \\end{array}\\hfill \\\\ \\hfill \\\\ \\hfill \\\\ \\begin{array}{l}\\text{ }y=-16{\\left(4.04\\right)}^{2}+\\left(140\\mathrm{sin}\\left(45\u00b0\\right)\\right)\\left(4.04\\right)+3\\hfill \\\\ \\text{ }y=141.8\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\r\nThe ball is 141.8 feet in the air when it soars out of the ballpark. It was indeed a home run. See <a class=\"autogenerated-content\" href=\"#Figure_08_07_010\">(Figure)<\/a>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/cnx.org\/resources\/36e3aba935a97ade9ab59a5fa21a6c81e6a83a7c\/CNX_Precalc_Figure_08_07_010n.jpg\" alt=\"Plotted trajectory of a hit ball, showing the position of the batter at the origin, the ball's path in the shape of a wide downward facing parabola, and the outfield wall as a vertical line segment rising to 10 ft under the ball's path.\" width=\"731\" height=\"310\" \/> <strong>Figure 7.<\/strong>[\/caption]\r\n<p id=\"fs-id1165134148366\">[\/hidden-answer]<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137842378\" class=\"precalculus media\">\r\n<p id=\"fs-id1165134323598\">Access the following online resource for additional instruction and practice with graphs of parametric equations.<\/p>\r\n\r\n<ul id=\"fs-id1165134323602\">\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphpara84\">Graphing Parametric Equations on the TI-84<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137470285\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165135209598\">\r\n \t<li>When there is a third variable, a third parameter on which[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]depend, parametric equations can be used.<\/li>\r\n \t<li>To graph parametric equations by plotting points, make a table with three columns labeled[latex]\\,t,x\\left(t\\right),\\,[\/latex]and[latex]\\,y\\left(t\\right).\\,[\/latex]Choose values for[latex]\\,t\\,[\/latex]in increasing order. Plot the last two columns for[latex]\\,x\\,[\/latex]and[latex]\\,y.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_07_01\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_07_02\">(Figure)<\/a>.<\/li>\r\n \t<li>When graphing a parametric curve by plotting points, note the associated <em>t<\/em>-values and show arrows on the graph indicating the orientation of the curve. See <a class=\"autogenerated-content\" href=\"#Example_08_07_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_07_04\">(Figure)<\/a>.<\/li>\r\n \t<li>Parametric equations allow the direction or the orientation of the curve to be shown on the graph. Equations that are not functions can be graphed and used in many applications involving motion. See <a class=\"autogenerated-content\" href=\"#Example_08_07_05\">(Figure)<\/a>.<\/li>\r\n \t<li>Projectile motion depends on two parametric equations:[latex]\\,x=\\left({v}_{0}\\mathrm{cos}\\,\\theta \\right)t\\,[\/latex]and[latex]\\,y=-16{t}^{2}+\\left({v}_{0}\\mathrm{sin}\\,\\theta \\right)t+h.\\,[\/latex]Initial velocity is symbolized as[latex]\\,{v}_{0}.\\,\\theta [\/latex] represents the initial angle of the object when thrown, and [latex]\\,h\\,[\/latex] represents the height at which the object is propelled.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165134283619\" class=\"textbox exercises\">\r\n<h3>Section Exercises<\/h3>\r\n<div id=\"fs-id1165137851723\" class=\"bc-section section\">\r\n<h4>Verbal<\/h4>\r\n<div id=\"fs-id1165137851729\">\r\n<div id=\"fs-id1165135194722\">\r\n<p id=\"fs-id1165135194723\">What are two methods used to graph parametric equations?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135194726\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135194726\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135194726\"]\r\n<p id=\"fs-id1165135194727\">plotting points with the orientation arrow and a graphing calculator<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135347318\">\r\n<div id=\"fs-id1165135347320\">\r\n\r\nWhat is one difference in point-plotting parametric equations compared to Cartesian equations?\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134113856\">\r\n<div>\r\n<p id=\"fs-id1165134113858\">Why are some graphs drawn with arrows?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137758269\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137758269\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137758269\"]\r\n<p id=\"fs-id1165137758270\">The arrows show the orientation, the direction of motion according to increasing values of[latex]\\,t.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137641293\">\r\n<div id=\"fs-id1165134205947\">\r\n<p id=\"fs-id1165134205948\">Name a few common types of graphs of parametric equations.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134205952\">\r\n<div id=\"fs-id1165134205953\">\r\n<p id=\"fs-id1165137715144\">Why are parametric graphs important in understanding projectile motion?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137715148\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137715148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137715148\"]\r\n<p id=\"fs-id1165137715149\">The parametric equations show the different vertical and horizontal motions over time.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\">\r\n<h4>Graphical<\/h4>\r\n<p id=\"fs-id1165135511376\">For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.<\/p>\r\n\r\n<div id=\"fs-id1165135581156\">\r\n<div id=\"fs-id1165135581157\">\r\n<p id=\"fs-id1165135581158\">[latex]\\{\\begin{array}{l}x(t)=t\\hfill \\\\ y(t)={t}^{2}-1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<table id=\"eip-id3065206\" class=\"unnumbered\" summary=\"Three columns and eight rows. The first row is labeled t, the second is labeled x, and the third is labeled y. The first columns contains the numbers -3, -2, -1, 0, 1, 2, 3. The other two columns are left blank for completion.\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134139687\">\r\n<div id=\"fs-id1165133361432\">\r\n<p id=\"fs-id1165133361433\">[latex]\\{\\begin{array}{l}x(t)=t-1\\hfill \\\\ y(t)={t}^{2}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<table id=\"eip-id2869464\" class=\"unnumbered\" summary=\"Three rows and eight columns. The first row is labeled t, the second is labeled x, and the third is labeled y. The first row contains the numbers -3, -2, -1, 0, 1, 2. The other two columns are left blank for completion.\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165135188326\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135188326\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135188326\"]<img src=\"https:\/\/cnx.org\/resources\/7802eb9e10c849cdfd0a571920cde1e460a71ac4\/CNX_Precalc_Figure_08_07_202.jpg\" alt=\"Graph of the given equations - looks like an upward opening parabola.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135172250\">\r\n<div id=\"fs-id1165135172252\">\r\n<p id=\"fs-id1165135172253\">[latex]\\{\\begin{array}{l}x(t)=2+t\\hfill \\\\ y(t)=3-2t\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<table id=\"eip-id2018706\" class=\"unnumbered\" summary=\"Three rows and eight columns. The first row is labeled t, the second is labeled x, and the third is labeled y. The first row contains the numbers -2, -1, 0, 1, 2, 3. The other two columns are left blank for completion.\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137457020\">\r\n<div id=\"fs-id1165137457021\">\r\n<p id=\"fs-id1165137457022\">[latex]\\{\\begin{array}{l}x(t)=-2-2t\\hfill \\\\ y(t)=3+t\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<table id=\"eip-id2478452\" class=\"unnumbered\" summary=\"Three rows and eight columns. The first row is labeled t, the second is labeled x, and the third is labeled y. The first row contains the numbers -3, -2, -1, 0, 1. The other two columns are left blank for completion.\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165134583390\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134583390\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134583390\"]<img src=\"https:\/\/cnx.org\/resources\/5f55cc1b94c9db2f3254f69f4f5429b15a4d2569\/CNX_Precalc_Figure_08_07_204.jpg\" alt=\"Graph of the given equations - a line, negative slope.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137643624\">\r\n<div id=\"fs-id1165137643625\">\r\n<p id=\"fs-id1165137643626\">[latex]\\{\\begin{array}{l}x(t)={t}^{3}\\hfill \\\\ y(t)=t+2\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<table id=\"eip-id1838664\" class=\"unnumbered\" summary=\"Three rows and eight columns. The first row is labeled t, the second is labeled x, and the third is labeled y. The first row contains the numbers - -2, -1, 0, 1, 2. The other two columns are left blank for completion.\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137465652\">\r\n<div id=\"fs-id1165137465653\">\r\n<p id=\"fs-id1165137465654\">[latex]\\{\\begin{array}{l}x(t)={t}^{2}\\hfill \\\\ y(t)=t+3\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<table id=\"eip-id2086049\" class=\"unnumbered\" summary=\"Three rows and eight columns. The first row is labeled t, the second is labeled x, and the third is labeled y. The first row contains the numbers - -2, -1, 0, 1, 2. The other two columns are left blank for completion.\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165135237047\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135237047\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135237047\"]<img src=\"https:\/\/cnx.org\/resources\/f59bfc5343493b9d924228f66e3c63e0411f8e30\/CNX_Precalc_Figure_08_07_206.jpg\" alt=\"Graph of the given equations - looks like a sideways parabola, opening to the right.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137698486\">For the following exercises, sketch the curve and include the orientation.<\/p>\r\n\r\n<div id=\"fs-id1165135155255\">\r\n<div id=\"fs-id1165135155256\">\r\n<p id=\"fs-id1165135155257\">[latex]\\{\\begin{array}{l}x(t)=t\\\\ y(t)=\\sqrt{t}\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137427635\">\r\n<div id=\"fs-id1165137427636\">\r\n<p id=\"fs-id1165135181824\">[latex]\\{\\begin{array}{l}x(t)=-\\,\\sqrt{t}\\\\ y(t)=t\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134069188\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134069188\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134069188\"]<img src=\"https:\/\/cnx.org\/resources\/cf987e35edf5471fa2b92fa9450b29c4cda88a54\/CNX_Precalc_Figure_08_07_208.jpg\" alt=\"Graph of the given equations - looks like the left half of an upward opening parabola.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135404698\">\r\n<div id=\"fs-id1165135404699\">\r\n<p id=\"fs-id1165137811212\">[latex]\\{\\begin{array}{l}x(t)=5-|t|\\\\ y(t)=t+2\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137771695\">\r\n<div id=\"fs-id1165137771696\">\r\n<p id=\"fs-id1165137771697\">[latex]\\{\\begin{array}{l}x(t)=-t+2\\\\ y(t)=5-|t|\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137417002\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137417002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137417002\"]<img src=\"https:\/\/cnx.org\/resources\/615da221012d753baf879541277b194b2b4d4072\/CNX_Precalc_Figure_08_07_210.jpg\" alt=\"Graph of the given equations - looks like a downward opening absolute value function.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137737228\">\r\n<div id=\"fs-id1165137737229\">\r\n<p id=\"fs-id1165137737230\">[latex]\\{\\begin{array}{l}x(t)=4\\text{sin}\\,t\\hfill \\\\ y(t)=2\\mathrm{cos}\\,t\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135168304\">\r\n<div id=\"fs-id1165135237110\">\r\n<p id=\"fs-id1165135237111\">[latex]\\{\\begin{array}{l}x(t)=2\\text{sin}\\,t\\hfill \\\\ y(t)=4\\text{cos}\\,t\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137936702\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137936702\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137936702\"]<img src=\"https:\/\/cnx.org\/resources\/2cdb23093c6b20df36ba6b001ca3558ee48c6ae2\/CNX_Precalc_Figure_08_07_212.jpg\" alt=\"Graph of the given equations - a vertical ellipse.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div>\r\n<div id=\"fs-id1165137589850\">[latex]\\{\\begin{array}{l}x(t)=3{\\mathrm{cos}}^{2}t\\\\ y(t)=-3\\mathrm{sin}\\,t\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137410365\">\r\n<div id=\"fs-id1165137410366\">\r\n<p id=\"fs-id1165137410367\">[latex]\\{\\begin{array}{l}x(t)=3{\\mathrm{cos}}^{2}t\\\\ y(t)=-3{\\mathrm{sin}}^{2}t\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134081386\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134081386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134081386\"]<img src=\"https:\/\/cnx.org\/resources\/83a62a440e0dfc37568cf86c50fb48d17f49c19d\/CNX_Precalc_Figure_08_07_214.jpg\" alt=\"Graph of the given equations- line from (0, -3) to (3,0). It is traversed in both directions, positive and negative slope.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137601187\">\r\n<div id=\"fs-id1165137601188\">\r\n<p id=\"fs-id1165137601189\">[latex]\\{\\begin{array}{l}x(t)=\\mathrm{sec}\\,t\\\\ y(t)=\\mathrm{tan}\\,t\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135209757\">\r\n<div id=\"fs-id1165137431122\">\r\n<p id=\"fs-id1165137431123\">[latex]\\{\\begin{array}{l}x(t)=\\mathrm{sec}\\,t\\\\ y(t)={\\mathrm{tan}}^{2}t\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137805783\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137805783\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137805783\"]<img src=\"https:\/\/cnx.org\/resources\/2722c324b807d246f21e1dadde6b56fb1db0842a\/CNX_Precalc_Figure_08_07_216.jpg\" alt=\"Graph of the given equations- looks like an upward opening parabola.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137896941\">\r\n<div id=\"fs-id1165137896942\">[latex]\\{\\begin{array}{l}x(t)=\\frac{1}{{e}^{2t}}\\\\ y(t)={e}^{-\\,t}\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137925361\">For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.<\/p>\r\n\r\n<div id=\"fs-id1165137925365\">\r\n<div id=\"fs-id1165134149986\">\r\n<p id=\"fs-id1165134149987\">[latex]\\{\\begin{array}{l}x(t)=t-1\\hfill \\\\ y(t)=-{t}^{2}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135149887\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135149887\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135149887\"]<img src=\"https:\/\/cnx.org\/resources\/6fa45950ce2f0303bc9c40a2a8aa197972da1315\/CNX_Precalc_Figure_08_07_218.jpg\" alt=\"Graph of the given equations- looks like a downward opening parabola.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135190005\">\r\n<div id=\"fs-id1165135190006\">\r\n<p id=\"fs-id1165135190007\">[latex]\\{\\begin{array}{l}x(t)={t}^{3}\\hfill \\\\ y(t)=t+3\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135597699\">\r\n<div id=\"fs-id1165135597700\">\r\n<p id=\"fs-id1165135597701\">[latex]\\{\\begin{array}{l}x(t)=2\\mathrm{cos}\\,t\\\\ y(t)=-\\mathrm{sin}\\,t\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135512730\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135512730\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135512730\"]<img src=\"https:\/\/cnx.org\/resources\/4101f0f0d1a9cae3a869e1d7b58b90f249afc8a4\/CNX_Precalc_Figure_08_07_220.jpg\" alt=\"Graph of the given equations- horizontal ellipse.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div>\r\n<div id=\"fs-id1165135209077\">\r\n<p id=\"fs-id1165135209078\">[latex]\\{\\begin{array}{l}x(t)=7\\mathrm{cos}\\,t\\\\ y(t)=7\\mathrm{sin}\\,t\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135192319\">\r\n<div id=\"fs-id1165135192320\">\r\n<p id=\"fs-id1165135192322\">[latex]\\{\\begin{array}{l}x(t)={e}^{2t}\\\\ y(t)=-{e}^{\\,t}\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165132912723\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132912723\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132912723\"]<img src=\"https:\/\/cnx.org\/resources\/64f7a2670f4176ecaaa6e972c9be6d011620219f\/CNX_Precalc_Figure_08_07_222.jpg\" alt=\"Graph of the given equations- looks like the lower half of a sideways parabola opening to the right\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135185162\">For the following exercises, graph the equation and include the orientation.<\/p>\r\n\r\n<div id=\"fs-id1165135185165\">\r\n<div id=\"fs-id1165135185166\">[latex]x={t}^{2},\\,y\\,=\\,3t,\\,0\\le t\\le 5[\/latex]<\/div>\r\n<\/div>\r\n<div>\r\n<div id=\"fs-id1165135186345\">\r\n<p id=\"fs-id1165135186346\">[latex]x=2t,\\,y\\,=\\,\\,{t}^{2},\\,-5\\le t\\le 5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"211246\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"211246\"]<img src=\"https:\/\/cnx.org\/resources\/2204c70368902e6f3c19e5a486a9adb776d4dda2\/CNX_Precalc_Figure_08_07_224.jpg\" alt=\"Graph of the given equations- looks like an upwards opening parabola\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133065688\">\r\n<div id=\"fs-id1165133065689\">\r\n<p id=\"fs-id1165135553506\">[latex]x=t,\\,y=\\sqrt{25-{t}^{2}},\\,0&lt;t\\le 5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137742391\">\r\n<div id=\"fs-id1165137742392\">[latex]x\\left(t\\right)=-t,y\\left(t\\right)=\\sqrt{t},\\,t\\ge 0[\/latex]<\/div>\r\n<div id=\"fs-id1165135252162\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135252162\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135252162\"]<img src=\"https:\/\/cnx.org\/resources\/ea3577e0761b0bfb69aae5a9cadbdae6463870a1\/CNX_Precalc_Figure_08_07_226.jpg\" alt=\"Graph of the given equations- looks like the upper half of a sideways parabola opening to the left\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134575412\">\r\n<div id=\"fs-id1165134575413\">[latex]x=-2\\mathrm{cos}\\,t,\\,y=6\\,\\mathrm{sin}\\,t,\\,0\\le t\\le \\pi [\/latex]<\/div>\r\n<\/div>\r\n<div>\r\n<div id=\"fs-id1165137897084\">\r\n<p id=\"fs-id1165137897085\">[latex]x=-\\mathrm{sec}\\,t,\\,y=\\mathrm{tan}\\,t,\\,-\\frac{\\,\\pi }{2}&lt;t&lt;\\frac{\\pi }{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137680409\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137680409\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137680409\"]<img src=\"https:\/\/cnx.org\/resources\/ede6e666e5b9e83204d913926164721c52312d83\/CNX_Precalc_Figure_08_07_228.jpg\" alt=\"Graph of the given equations- the left half of a hyperbola with diagonal asymptotes\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165134032307\">For the following exercises, use the parametric equations for integers <em>a <\/em>and <em>b<\/em>:<\/p>\r\n\r\n<div id=\"eip-155\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x\\left(t\\right)=a\\mathrm{cos}\\left(\\left(a+b\\right)t\\right)\\\\ y\\left(t\\right)=a\\mathrm{cos}\\left(\\left(a-b\\right)t\\right)\\end{array}[\/latex]<\/div>\r\n<div>\r\n<div id=\"fs-id1165135160277\">\r\n<p id=\"fs-id1165135160278\">Graph on the domain[latex]\\,\\left[-\\pi ,0\\right],\\,[\/latex]where[latex]\\,a=2\\,[\/latex]and[latex]\\,b=1,\\,[\/latex]and include the orientation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137844152\">\r\n<div id=\"fs-id1165137844153\">\r\n<p id=\"fs-id1165137844154\">Graph on the domain[latex]\\,\\left[-\\pi ,0\\right],\\,[\/latex]where[latex]\\,a=3\\,[\/latex]and[latex]\\,b=2[\/latex], and include the orientation.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137409296\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137409296\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137409296\"]<img src=\"https:\/\/cnx.org\/resources\/1fe8c8d9d88000e5c27471bf752630a946e7562b\/CNX_Precalc_Figure_08_07_230.jpg\" alt=\"Graph of the given equations - vertical periodic trajectory\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137619318\">\r\n<div id=\"fs-id1165137619319\">\r\n<p id=\"fs-id1165137619320\">Graph on the domain[latex]\\,\\left[-\\pi ,0\\right],\\,[\/latex]where[latex]\\,a=4\\,[\/latex]and[latex]\\,b=3[\/latex], and include the orientation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137404658\">\r\n<div id=\"fs-id1165137404659\">\r\n<p id=\"fs-id1165137404660\">Graph on the domain[latex]\\,\\left[-\\pi ,0\\right],\\,[\/latex]where[latex]\\,a=5\\,[\/latex]and[latex]\\,b=4[\/latex], and include the orientation.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137551247\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137551247\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137551247\"]<img src=\"https:\/\/cnx.org\/resources\/892611bcff1d279af76db753ed3548573dce74fc\/CNX_Precalc_Figure_08_07_232.jpg\" alt=\"Graph of the given equations - vertical periodic trajectory\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133233024\">\r\n<div id=\"fs-id1165133233025\">\r\n<p id=\"fs-id1165133233026\">If[latex]\\,a\\,[\/latex]is 1 more than[latex]\\,b,\\,[\/latex]describe the effect the values of[latex]\\,a\\,[\/latex]and[latex]\\,b\\,[\/latex]have on the graph of the parametric equations.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135638542\">\r\n<div id=\"fs-id1165135638543\">\r\n<p id=\"fs-id1165137767042\">Describe the graph if[latex]\\,a=100\\,[\/latex]and[latex]\\,b=99.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135189746\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135189746\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135189746\"]\r\n<p id=\"fs-id1165135189747\">There will be 100 back-and-forth motions.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135189750\">\r\n<div id=\"fs-id1165135189751\">\r\n<p id=\"fs-id1165135530296\">What happens if[latex]\\,b\\,[\/latex]is 1 more than[latex]\\,a?\\,[\/latex]Describe the graph.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134122767\">\r\n<div id=\"fs-id1165137823080\">\r\n<p id=\"fs-id1165137823081\">If the parametric equations[latex]\\,x\\left(t\\right)={t}^{2}\\,[\/latex]and[latex]\\,y\\left(t\\right)=6-3t\\,[\/latex]have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135499919\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135499919\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135499919\"]\r\n<p id=\"fs-id1165134272702\">Take the opposite of the[latex]\\,x\\left(t\\right)\\,[\/latex]equation.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137911683\">For the following exercises, describe the graph of the set of parametric equations.<\/p>\r\n\r\n<div id=\"fs-id1165137911686\">\r\n<div id=\"fs-id1165137911687\">\r\n<p id=\"fs-id1165137911688\">[latex]x\\left(t\\right)=-{t}^{2}\\,[\/latex]and[latex]\\,y\\left(t\\right)\\,[\/latex]is linear<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135251241\">\r\n<div id=\"fs-id1165135251242\">\r\n<p id=\"fs-id1165135251243\">[latex]y\\left(t\\right)={t}^{2}\\,[\/latex]and[latex]\\,x\\left(t\\right)\\,[\/latex]is linear<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135609231\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135609231\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135609231\"]\r\n<p id=\"fs-id1165135609232\">The parabola opens up.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135609236\">\r\n<div id=\"fs-id1165135609237\">\r\n\r\n[latex]y\\left(t\\right)=-{t}^{2}\\,[\/latex]and[latex]\\,x\\left(t\\right)\\,[\/latex]is linear\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135453880\">\r\n<div id=\"fs-id1165135453881\">\r\n<p id=\"fs-id1165135453882\">Write the parametric equations of a circle with center[latex]\\,\\left(0,0\\right),[\/latex]radius 5, and a counterclockwise orientation.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135352460\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135352460\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135352460\"]\r\n<p id=\"fs-id1165135352461\">[latex]\\{\\begin{array}{l}x(t)=5\\mathrm{cos}t\\\\ y(t)=5\\mathrm{sin}t\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133141349\">\r\n<div id=\"fs-id1165133141350\">\r\n<p id=\"fs-id1165133141351\">Write the parametric equations of an ellipse with center[latex]\\,\\left(0,0\\right),[\/latex]major axis of length 10, minor axis of length 6, and a counterclockwise orientation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137838292\">For the following exercises, use a graphing utility to graph on the window[latex]\\,\\left[-3,3\\right]\\,[\/latex]by[latex]\\,\\left[-3,3\\right]\\,[\/latex]on the domain[latex]\\,\\left[0,2\\pi \\right)\\,[\/latex]for the following values of[latex]\\,a\\,[\/latex]and[latex]\\,b[\/latex], and include the orientation.<\/p>\r\n\r\n<div id=\"eip-169\" class=\"unnumbered aligncenter\">[latex]\\{\\begin{array}{l}x(t)=\\mathrm{sin}(at)\\\\ y(t)=\\mathrm{sin}(bt)\\end{array}[\/latex]<\/div>\r\n<div id=\"fs-id1165137761928\">\r\n<div id=\"fs-id1165134053964\">\r\n<p id=\"fs-id1165134053965\">[latex]a=1,b=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165132960728\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132960728\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132960728\"]<img src=\"https:\/\/cnx.org\/resources\/2bab68aaea09ee9ec930461b83bf06e0b6d239ff\/CNX_Precalc_Figure_08_07_233.jpg\" alt=\"Graph of the given equations \" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133276223\">\r\n<div id=\"fs-id1165133276224\">\r\n<p id=\"fs-id1165133276225\">[latex]a=2,b=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135415664\">\r\n<div id=\"fs-id1165135415665\">\r\n<p id=\"fs-id1165135415666\">[latex]a=3,b=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134386554\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134386554\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134386554\"]<img src=\"https:\/\/cnx.org\/resources\/ab11cb1a439b9d52ab738299a39b8a1670f42fd4\/CNX_Precalc_Figure_08_07_235.jpg\" alt=\"Graph of the given equations - lines extending into Q1 and Q3 (in both directions) from the origin to 1 unit.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135572012\">\r\n<div id=\"fs-id1165135572013\">\r\n<p id=\"fs-id1165135572014\">[latex]a=5,b=5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137832502\">\r\n<div id=\"fs-id1165134118436\">\r\n<p id=\"fs-id1165134118437\">[latex]a=2,b=5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165133349420\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133349420\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133349420\"]<img src=\"https:\/\/cnx.org\/resources\/d44fdde2ddd0a3966c80c1c63537633ce034197a\/CNX_Precalc_Figure_08_07_237.jpg\" alt=\"Graph of the given equations - lines extending into Q1 and Q3 (in both directions) from the origin to 3 units.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134357489\">\r\n<div id=\"fs-id1165134357490\">\r\n<p id=\"fs-id1165134357491\">[latex]a=5,b=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133315097\" class=\"bc-section section\">\r\n<h4>Technology<\/h4>\r\n<p id=\"fs-id1165134589497\">For the following exercises, look at the graphs that were created by parametric equations of the form[latex]\\,\\{\\begin{array}{l}x(t)=a\\text{cos}(bt)\\hfill \\\\ y(t)=c\\text{sin}(dt)\\hfill \\end{array}.\\,[\/latex]Use the parametric mode on the graphing calculator to find the values of [latex]a,b,c,[\/latex] and [latex]d[\/latex] to achieve each graph.<\/p>\r\n\r\n<div id=\"fs-id1165137701974\">\r\n<div id=\"fs-id1165137701975\"><span id=\"fs-id1165137701981\"><img src=\"https:\/\/cnx.org\/resources\/59cdfbe8b34369000a5c7ac21765f879c03c26eb\/CNX_Precalc_Figure_08_07_239.jpg\" alt=\"Graph of the given equations \" \/><\/span><\/div>\r\n<div id=\"fs-id1165137651752\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137651752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137651752\"]\r\n<p id=\"fs-id1165137651754\">[latex]a=4,\\,b=3,\\,c=6,\\,d=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165132032531\">\r\n<div id=\"fs-id1165132032532\"><img src=\"https:\/\/cnx.org\/resources\/b88749b1f3f254204ac4570264e80f43871159bc\/CNX_Precalc_Figure_08_07_240.jpg\" alt=\"Graph of the given equations \" \/><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134237259\">\r\n<div id=\"fs-id1165134237260\"><span id=\"fs-id1165134237266\"><img src=\"https:\/\/cnx.org\/resources\/3d75d7eeb5e2a2fac17b80720c7408be76e46d84\/CNX_Precalc_Figure_08_07_241.jpg\" alt=\"Graph of the given equations \" \/><\/span><\/div>\r\n<div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137693607\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137693607\"]\r\n<p id=\"fs-id1165137693607\">[latex]a=4,\\,b=2,\\,c=3,\\,d=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133393487\">\r\n<div id=\"fs-id1165133393488\"><img src=\"https:\/\/cnx.org\/resources\/5eaae6a03845596ad92f25282fac6aaf2a25182f\/CNX_Precalc_Figure_08_07_242.jpg\" alt=\"Graph of the given equations \" \/><\/div>\r\n<\/div>\r\n<p id=\"fs-id1165134211299\">For the following exercises, use a graphing utility to graph the given parametric equations.<\/p>\r\n\r\n<ol id=\"fs-id1165134211302\" type=\"a\">\r\n \t<li>[latex]\\{\\begin{array}{l}x(t)=\\mathrm{cos}t-1\\\\ y(t)=\\mathrm{sin}t+t\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\{\\begin{array}{l}x(t)=\\mathrm{cos}t+t\\\\ y(t)=\\mathrm{sin}t-1\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\{\\begin{array}{l}x(t)=t-\\mathrm{sin}t\\\\ y(t)=\\mathrm{cos}t-1\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1165137729163\">\r\n<div id=\"fs-id1165135453820\">\r\n<p id=\"fs-id1165135453821\">Graph all three sets of parametric equations on the domain[latex]\\,\\left[0,\\,2\\pi \\right].[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137419669\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137419669\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137419669\"]<img src=\"https:\/\/cnx.org\/resources\/b09dc27fcd7c07eb5526c596e83f201413dc38b1\/CNX_Precalc_Figure_08_07_243.jpg\" alt=\"Graph of the given equations \" \/><img src=\"https:\/\/cnx.org\/resources\/c68224c477101bf9f8515e4966e3bdb0c21849ee\/CNX_Precalc_Figure_08_07_244.jpg\" alt=\"Graph of the given equations \" \/><img src=\"https:\/\/cnx.org\/resources\/583e7ce2f6f87bd60fa0d6bc065eec7c8a4ed0f3\/CNX_Precalc_Figure_08_07_245.jpg\" alt=\"Graph of the given equations \" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134422229\">\r\n<div id=\"fs-id1165135253195\">\r\n<p id=\"fs-id1165135253196\">Graph all three sets of parametric equations on the domain[latex]\\,\\left[0,4\\pi \\right].[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134546248\">\r\n<div id=\"fs-id1165134546249\">\r\n<p id=\"fs-id1165134546250\">Graph all three sets of parametric equations on the domain[latex]\\,\\left[-4\\pi ,6\\pi \\right].[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165131962224\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131962224\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165131962224\"]<img src=\"https:\/\/cnx.org\/resources\/849a48802af36c85319d64bc349c1ff273ee703b\/CNX_Precalc_Figure_08_07_249.jpg\" alt=\"Graph of the given equations \" \/><img src=\"https:\/\/cnx.org\/resources\/4f6094a56c565941ba80d7c1f42accddfb4e92c0\/CNX_Precalc_Figure_08_07_250.jpg\" alt=\"Graph of the given equations \" \/><img src=\"https:\/\/cnx.org\/resources\/798c654d12b96561c2b2343b5be7fcbb3e2f3228\/CNX_Precalc_Figure_08_07_251.jpg\" alt=\"Graph of the given equations \" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135515798\">\r\n<div id=\"fs-id1165135515799\">\r\n<p id=\"fs-id1165135515800\">The graph of each set of parametric equations appears to \u201ccreep\u201d along one of the axes. What controls which axis the graph creeps along?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div id=\"fs-id1165135662545\">\r\n\r\nExplain the effect on the graph of the parametric equation when we switched[latex]\\,\\mathrm{sin}\\,t\\,[\/latex]and[latex]\\,\\mathrm{cos}\\,t[\/latex].\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135252078\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135252078\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135252078\"]\r\n<p id=\"fs-id1165135252079\">The[latex]\\,y[\/latex]-intercept changes.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135606062\">\r\n<div id=\"fs-id1165135606063\">\r\n<p id=\"fs-id1165135606064\">Explain the effect on the graph of the parametric equation when we changed the domain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135606068\" class=\"bc-section section\">\r\n<h4>Extensions<\/h4>\r\n<div id=\"fs-id1165134040395\">\r\n<div id=\"fs-id1165134040396\">\r\n<p id=\"fs-id1165134040397\">An object is thrown in the air with vertical velocity of 20 ft\/s and horizontal velocity of 15 ft\/s. The object\u2019s height can be described by the equation[latex]\\,y\\left(t\\right)=-16{t}^{2}+20t[\/latex], while the object moves horizontally with constant velocity 15 ft\/s. Write parametric equations for the object\u2019s position, and then eliminate time to write height as a function of horizontal position.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165132079269\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132079269\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132079269\"]\r\n<p id=\"fs-id1165132079270\">[latex]y\\left(x\\right)=-16{\\left(\\frac{x}{15}\\right)}^{2}+20\\left(\\frac{x}{15}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137642896\">\r\n<div id=\"fs-id1165137642897\">\r\n<p id=\"fs-id1165137642898\">A skateboarder riding on a level surface at a constant speed of 9 ft\/s throws a ball in the air, the height of which can be described by the equation[latex]\\,y\\left(t\\right)=-16{t}^{2}+10t+5.\\text{}[\/latex]Write parametric equations for the ball\u2019s position, and then eliminate time to write height as a function of horizontal position.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165132035992\">For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft\/s at an angle of elevation of 52\u00b0. Consider the position of the dart at any time[latex]\\,t.\\,[\/latex]Neglect air resistance.<\/p>\r\n\r\n<div id=\"fs-id1165131938001\">\r\n<div id=\"fs-id1165131938002\">\r\n<p id=\"fs-id1165134371096\">Find parametric equations that model the problem situation.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134371099\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134371099\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134371099\"]\r\n<p id=\"fs-id1165134371100\">[latex]\\{\\begin{array}{l}x(t)=64t\\mathrm{cos}(52\u00b0)\\\\ y(t)=-16{t}^{2}+64t\\mathrm{sin}(52\u00b0)\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137401314\">\r\n<div id=\"fs-id1165137401315\">\r\n<p id=\"fs-id1165133087397\">Find all possible values of[latex]\\,x\\,[\/latex]that represent the situation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135445716\">\r\n<div id=\"fs-id1165135445718\">\r\n<p id=\"fs-id1165135445719\">When will the dart hit the ground?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135445722\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135445722\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135445722\"]\r\n<p id=\"fs-id1165133311063\">approximately 3.2 seconds<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div id=\"fs-id1165133311067\">\r\n\r\nFind the maximum height of the dart.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133311071\">\r\n<div id=\"fs-id1165133311072\">\r\n<p id=\"fs-id1165133311073\">At what time will the dart reach maximum height?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135444043\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135444043\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135444043\"]\r\n<p id=\"fs-id1165135444044\">1.6 seconds<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135444047\">For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.<\/p>\r\n\r\n<div id=\"fs-id1165137723597\">\r\n<div id=\"fs-id1165137723598\">\r\n<p id=\"fs-id1165137723599\">An epicycloid:[latex]\\,\\{\\begin{array}{l}x(t)=14\\mathrm{cos}\\,t-\\mathrm{cos}(14t)\\hfill \\\\ y(t)=14\\mathrm{sin}\\,t+\\mathrm{sin}(14t)\\hfill \\end{array}\\,[\/latex]on the domain[latex]\\,[0,2\\pi ][\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135596508\">\r\n<div>\r\n<p id=\"fs-id1165135596510\">A hypocycloid:[latex]\\{\\begin{array}{l}x(t)=6\\mathrm{sin}\\,t+2\\mathrm{sin}(6t)\\hfill \\\\ y(t)=6\\mathrm{cos}\\,t-2\\mathrm{cos}(6t)\\hfill \\end{array}\\,[\/latex]on the domain[latex]\\,[0,2\\pi ][\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134569130\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134569130\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134569130\"]<img src=\"https:\/\/cnx.org\/resources\/2bf25ddb8ab79c6544edc7beba2d7ec4b72d514b\/CNX_Precalc_Figure_08_07_253.jpg\" alt=\"Graph of the given equations - a hypocycloid\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165131958330\">\r\n<div id=\"fs-id1165131958331\">\r\n<p id=\"fs-id1165131958332\">A hypotrochoid:[latex]\\{\\begin{array}{l}x(t)=2\\mathrm{sin}\\,t+5\\mathrm{cos}(6t)\\hfill \\\\ y(t)=5\\mathrm{cos}\\,t-2\\mathrm{sin}(6t)\\hfill \\end{array}\\,[\/latex]on the domain[latex]\\,\\left[0,2\\pi \\right][\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137854188\">\r\n<div id=\"fs-id1165137854189\">\r\n<p id=\"fs-id1165137854190\">A rose:[latex]\\,\\{\\begin{array}{l}x(t)=5\\mathrm{sin}(2t)\\mathrm{sin}t\\hfill \\\\ y(t)=5\\mathrm{sin}(2t)\\mathrm{cos}t\\hfill \\end{array}\\,[\/latex]on the domain[latex]\\,\\left[0,2\\pi \\right][\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134134026\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134134026\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134134026\"]<img src=\"https:\/\/cnx.org\/resources\/100968f7788e3205fd1623f96ef4e3e16dcb1010\/CNX_Precalc_Figure_08_07_255.jpg\" alt=\"Graph of the given equations - a four petal rose\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\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>Graph plane curves described by parametric equations by plotting points.<\/li>\n<li>Graph parametric equations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137758339\">It is the bottom of the ninth inning, with two outs and two men on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately[latex]\\,45\u00b0\\,[\/latex]to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using <span class=\"no-emphasis\">parametric equations<\/span>. In this section, we\u2019ll discuss parametric equations and some common applications, such as projectile motion problems.<\/p>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/4c0775a0076c07a6585617c97ae7cf0ffaff107d\/CNX_Precalc_Figure_08_07_001.jpg\" alt=\"Photo of a baseball batter swinging.\" width=\"488\" height=\"333\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1.<\/strong> Parametric equations can model the path of a projectile. (credit: Paul Kreher, Flickr)<\/p>\n<\/div>\n<div id=\"fs-id1165137566996\" class=\"bc-section section\">\n<h3>Graphing Parametric Equations by Plotting Points<\/h3>\n<p id=\"fs-id1165137470035\">In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.<\/p>\n<div id=\"fs-id1165135361790\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165132005266\"><strong>Given a pair of parametric equations, sketch a graph by plotting points.<\/strong><\/p>\n<ol id=\"fs-id1165137428125\" type=\"1\">\n<li>Construct a table with three columns:[latex]\\,t,x\\left(t\\right),\\text{and}\\,\\,y\\left(t\\right).[\/latex]<\/li>\n<li>Evaluate [latex]x[\/latex] and [latex]y[\/latex] for values of [latex]t[\/latex] over the interval for which the functions are defined.<\/li>\n<li>Plot the resulting pairs[latex]\\,\\left(x,y\\right).[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_08_07_01\" class=\"textbox examples\">\n<div id=\"fs-id1165134532923\">\n<div id=\"fs-id1165137854808\">\n<h3>Sketching the Graph of a Pair of Parametric Equations by Plotting Points<\/h3>\n<p id=\"fs-id1165137769779\">Sketch the graph of the <span class=\"no-emphasis\">parametric equations<\/span> [latex]x\\left(t\\right)={t}^{2}+1,\\,\\,y\\left(t\\right)=2+t.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137938383\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137938383\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137938383\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135181426\">Construct a table of values for[latex]\\,t,x\\left(t\\right),\\,[\/latex]and[latex]\\,y\\left(t\\right),\\,[\/latex]as in <a class=\"autogenerated-content\" href=\"#Table_08_07_01\">(Figure)<\/a>, and plot the points in a plane.<\/p>\n<table id=\"Table_08_07_01\" summary=\"Twelve rows and three columns. First column is labeled t, second column is labeled x(t)=t^2 + 1, third column is labeled y(t) = 2 + t. The table has ordered triples of each of these row values: (-5, 26, -3), (-4, 17, -2), (-3, 10, -1), (-2, 5, 0), (-1, 2, 1), (0, 1, 2), (1, 2, 3), (2, 5, 4), (3, 10, 5), (4, 17, 6), (5, 26, 7).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x\\left(t\\right)={t}^{2}+1[\/latex]<\/th>\n<th>[latex]y\\left(t\\right)=2+t[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]26[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]17[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]17[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]26[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/551e6c2de6d75d9ab3297287596f3ddfbf881ca7\/CNX_Precalc_Figure_08_07_002.jpg\" alt=\"Graph of the given parabola opening to the right.\" width=\"487\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/p>\n<\/div>\n<p id=\"fs-id1165137628360\">The graph is a <span class=\"no-emphasis\">parabola<\/span> with vertex at the point[latex]\\,\\left(1,2\\right),[\/latex]opening to the right. See <a class=\"autogenerated-content\" href=\"#Figure_08_07_002\">(Figure)<\/a>.<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135515789\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137758642\">As values for[latex]\\,t\\,[\/latex]progress in a positive direction from 0 to 5, the plotted points trace out the top half of the parabola. As values of[latex]\\,t\\,[\/latex]become negative, they trace out the lower half of the parabola. There are no restrictions on the domain. The arrows indicate direction according to increasing values of [latex]\\,t.\\,[\/latex]The graph does not represent a function, as it will fail the vertical line test. The graph is drawn in two parts: the positive values for [latex]t,[\/latex] and the negative values for [latex]t.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134485685\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165133210758\">\n<p>Sketch the graph of the parametric equations[latex]\\,x=\\sqrt{t},\\,\\,y=2t+3,\\,\\,\\,0\\le t\\le 3.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137849525\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137849525\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137849525\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/5a648144babb4dd4e08a1611cead7856231ff4e7\/CNX_Precalc_Figure_08_07_003.jpg\" alt=\"Graph of the given parametric equations with the restricted domain - it looks like the right half of an upward opening parabola.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_07_02\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165135305906\">\n<h3>Sketching the Graph of Trigonometric Parametric Equations<\/h3>\n<p id=\"fs-id1165137942382\">Construct a table of values for the given parametric equations and sketch the graph:<\/p>\n<div id=\"eip-id2721082\" class=\"unnumbered\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}x=2\\mathrm{cos}\\,t\\hfill \\\\ y=4\\mathrm{sin}\\,t\\hfill \\end{array}\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137932663\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137932663\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137932663\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137665416\">Construct a table like that in <a class=\"autogenerated-content\" href=\"#Table_08_07_02\">(Figure)<\/a> using angle measure in radians as inputs for[latex]\\,t,\\,[\/latex]and evaluating[latex]\\,x\\,[\/latex]and[latex]\\,y.\\,[\/latex]Using angles with known sine and cosine values for[latex]\\,t\\,[\/latex]makes calculations easier.<\/p>\n<table id=\"Table_08_07_02\" summary=\"Fourteen rows and three columns. First column is labeled t, second column is labeled x(t)=2cos(1), third column is labeled y(t)=4sin(1). The table has ordered triples of each of these row values: (0, x=2cos(0)=2, y=4sin(0)=0), (pi\/6, x=2cos(pi\/6)=rad3, y=4sin(pi\/6)=2), (pi\/3, x=2cos(pi\/3)=1, y=4sin(pi\/3)=2rad3), (pi\/2, x=2cos(pi\/2)=0, y=4sin(pi\/2)=4), (2pi\/3, x=2cos(2pi\/3)=-1, y=4sin(2pi\/3)=2rad3), (5pi\/6, x=2cos(5pi\/6)=-rad3, y=4sin(5pi\/6)=2), (pi, x=2cos(pi)=-2, y=4sin(pi)=0), (7pi\/6, x=2cos(7pi\/6) = -rad3, y=4sin(7pi\/6)=-2), (4pi\/3, x=2cos(4pi\/3)=-1, y=4sin(4pi\/3)=-2rad3), (3pi\/2, x=2cos(3pi\/2)=0, y=4sin(3pi\/2)=-4), (5pi\/3, x=2cos(5pi\/3)=1, y=4sin(5pi\/3)=-2rad3), (11pi\/6, x=2cos(11pi\/6)=rad3, y=4sin(11pi\/6)=-2), (2pi, x=2cos(2pi)=2, y=4sin(2pi)=0).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x=2\\mathrm{cos}\\,t[\/latex]<\/th>\n<th>[latex]y=4\\mathrm{sin}\\,t[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(0\\right)=2[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(0\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)=\\sqrt{3}[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{\\pi }{3}\\right)=1[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{\\pi }{3}\\right)=2\\sqrt{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)=0[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{\\pi }{2}\\right)=4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{2\\pi }{3}\\right)=-1[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{2\\pi }{3}\\right)=2\\sqrt{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{5\\pi }{6}\\right)=-\\sqrt{3}[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{5\\pi }{6}\\right)=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\pi[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\pi \\right)=-2[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\pi \\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{7\\pi }{6}[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{7\\pi }{6}\\right)=-\\sqrt{3}[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{7\\pi }{6}\\right)=-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{4\\pi }{3}[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{4\\pi }{3}\\right)=-1[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{4\\pi }{3}\\right)=-2\\sqrt{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{3\\pi }{2}\\right)=0[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{3\\pi }{2}\\right)=-4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{5\\pi }{3}[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{5\\pi }{3}\\right)=1[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{5\\pi }{3}\\right)=-2\\sqrt{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{11\\pi }{6}[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(\\frac{11\\pi }{6}\\right)=\\sqrt{3}[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(\\frac{11\\pi }{6}\\right)=-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\pi[\/latex]<\/td>\n<td>[latex]x=2\\mathrm{cos}\\left(2\\pi \\right)=2[\/latex]<\/td>\n<td>[latex]y=4\\mathrm{sin}\\left(2\\pi \\right)=0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137581718\"><a class=\"autogenerated-content\" href=\"#Figure_08_07_004\">(Figure)<\/a> shows the graph.<span id=\"fs-id1165137644320\"><\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/3332600db5000513af59f82a010b36b8aba0e29c\/CNX_Precalc_Figure_08_07_004.jpg\" alt=\"Graph of the given equations - a vertical ellipse.\" width=\"487\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/p>\n<\/div>\n<p id=\"fs-id1165135569879\">By the symmetry shown in the values of [latex]x[\/latex] and [latex]\\,y,\\,[\/latex]we see that the parametric equations represent an <span class=\"no-emphasis\">ellipse<\/span>. The <span class=\"no-emphasis\">ellipse<\/span> is mapped in a counterclockwise direction as shown by the arrows indicating increasing[latex]\\,t\\,[\/latex]values.<span id=\"fs-id1165137644320\"><\/span><\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1551946\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137811673\">We have seen that parametric equations can be graphed by plotting points. However, a graphing calculator will save some time and reveal nuances in a graph that may be too tedious to discover using only hand calculations.<\/p>\n<p id=\"eip-157\">Make sure to change the mode on the calculator to parametric (PAR). To confirm, the[latex]\\,Y=\\,[\/latex]window should show<\/p>\n<div id=\"fs-id1165137824587\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}{X}_{1T}=\\\\ {Y}_{1T}=\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135168174\">instead of[latex]\\,{Y}_{1}=.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137934405\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137714934\">\n<div id=\"fs-id1165137935182\">\n<p id=\"fs-id1165137935183\">Graph the parametric equations:[latex]\\,x=5\\mathrm{cos}\\,t,\\,\\,y=3\\mathrm{sin}\\,t.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137408223\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137408223\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137408223\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/2f2ed4dddafe78c5d1c0e38d229a62b6fc0673ea\/CNX_Precalc_Figure_08_07_005.jpg\" alt=\"Graph of the given equations - a horizontal ellipse.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137610977\" class=\"bc-section section\">\n<div id=\"Example_08_07_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135192637\">\n<div id=\"fs-id1165135181483\">\n<h3>Graphing Parametric Equations and Rectangular Form Together<\/h3>\n<p id=\"fs-id1165137466433\">Graph the parametric equations[latex]\\,x=5\\mathrm{cos}\\,t\\,[\/latex]and[latex]\\,y=2\\mathrm{sin}\\,t.\\,[\/latex]First, construct the graph using data points generated from the <span class=\"no-emphasis\">parametric form<\/span>. Then graph the <span class=\"no-emphasis\">rectangular form<\/span> of the equation. Compare the two graphs.<\/p>\n<\/div>\n<div id=\"fs-id1165137387636\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137387636\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137387636\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137807398\">Construct a table of values like that in <a class=\"autogenerated-content\" href=\"#Table_08_07_03\">(Figure)<\/a>.<\/p>\n<table id=\"Table_08_07_03\" summary=\"Twelve rows and three columns. First column is labeled t, second column is labeled x(t)=5cos(t), third column is labeled y(t) = 2sin(t). The table has ordered triples of each of these row values: (0, x=5cos(0)=5, y=2sin(0)=0), (1, x=5cos(1) =approx 2.7, y=2sin(1) =approx 1.7), (2, x=5cos(2) =approx -2.1, y=2sin(2) =approx 1.8), (3, x=5cos(3) =approx -4.95, y=2sin(3) =approx 0.28), (4, x=5cos(4) =approx -3.3, y=2sin(4) =approx -1.5), (5, x=5cos(5) =approx 1.4, y=2sin(5) =approx -1.9), (-1, x=5cos(-1) =approx 2.7, y=2sin(-1) =approx -1.7), (-2, x=5cos(-2) =approx -2.1, y=2sin(-2) =approx -1.8), (-3, x=5cos(-3) =approx -4.95, y=2sin(-3) =approx -0.28), (-4, x=5cos(-4) =approx -3.3, y=2sin(-4) =approx 1.5), (-5, x=5cos(-5) =approx 1.4, y=2sin(-5) =approx 1.9).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x=5\\mathrm{cos}\\,t[\/latex]<\/th>\n<th>[latex]y=2\\mathrm{sin}\\,t[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\text{0}[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(0\\right)=5[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(0\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{1}[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(1\\right)\\approx 2.7[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(1\\right)\\approx 1.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{2}[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(2\\right)\\approx -2.1[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(2\\right)\\approx 1.8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{3}[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(3\\right)\\approx -4.95[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(3\\right)\\approx 0.28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{4}[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(4\\right)\\approx -3.3[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(4\\right)\\approx -1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{5}[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(5\\right)\\approx 1.4[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(5\\right)\\approx -1.9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(-1\\right)\\approx 2.7[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(-1\\right)\\approx -1.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(-2\\right)\\approx -2.1[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(-2\\right)\\approx -1.8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(-3\\right)\\approx -4.95[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(-3\\right)\\approx -0.28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(-4\\right)\\approx -3.3[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(-4\\right)\\approx 1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]x=5\\mathrm{cos}\\left(-5\\right)\\approx 1.4[\/latex]<\/td>\n<td>[latex]y=2\\mathrm{sin}\\left(-5\\right)\\approx 1.9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137910988\">Plot the[latex]\\,\\left(x,y\\right)\\,[\/latex]values from the table. See <a class=\"autogenerated-content\" href=\"#Figure_08_07_006\">(Figure)<\/a>.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/a069f98cd1e9673cef2e6d699842fbf33e6c98b1\/CNX_Precalc_Figure_08_07_006.jpg\" alt=\"Graph of the given ellipse in parametric and rectangular coordinates - it is the same thing in both images.\" width=\"975\" height=\"290\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/p>\n<\/div>\n<p>Next, translate the parametric equations to rectangular form. To do this, we solve for[latex]\\,t\\,[\/latex]in either[latex]\\,x\\left(t\\right)\\,[\/latex]or[latex]\\,y\\left(t\\right),\\,[\/latex]and then substitute the expression for[latex]\\,t\\,[\/latex]in the other equation. The result will be a function [latex]y\\left(x\\right)[\/latex] if solving for[latex]\\,t\\,[\/latex]as a function of[latex]\\,x,\\,[\/latex]or[latex]\\,x\\left(y\\right)[\/latex] if solving for[latex]\\,t\\,[\/latex]as a function of[latex]\\,y.\\,[\/latex]<\/p>\n<div id=\"fs-id1165137565254\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\,x=5\\mathrm{cos}\\,t\\hfill & \\hfill \\\\ \\frac{x}{5}=\\mathrm{cos}\\,t\\hfill & \\text{Solve for }\\mathrm{cos}\\,t.\\hfill \\\\ \\,y=2\\mathrm{sin}\\,t\\begin{array}{cccc}& & & \\end{array} \\hfill & \\text{Solve for }\\mathrm{sin}\\,t.\\hfill \\\\ \\frac{y}{2}=\\mathrm{sin}\\,t\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<p>Then, use the <span class=\"no-emphasis\">Pythagorean Theorem<\/span>.<\/p>\n<div id=\"fs-id1165137653844\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{r}\\hfill {\\mathrm{cos}}^{2}t+{\\mathrm{sin}}^{2}t=1\\\\ \\hfill {\\left(\\frac{x}{5}\\right)}^{2}+{\\left(\\frac{y}{2}\\right)}^{2}=1\\\\ \\hfill \\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{4}=1\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137470251\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137424263\">In <a class=\"autogenerated-content\" href=\"#Figure_08_07_007\">(Figure)<\/a>, the data from the parametric equations and the rectangular equation are plotted together. The parametric equations are plotted in blue; the graph for the rectangular equation is drawn on top of the parametric in a dashed style colored red. Clearly, both forms produce the same graph.<span id=\"fs-id1165134032313\"><\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/cb601162053c131dacd113d2d6e6db5d15dc482d\/CNX_Precalc_Figure_08_07_007.jpg\" alt=\"Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.\" width=\"487\" height=\"290\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_07_04\" class=\"textbox examples\">\n<div id=\"fs-id1165134152557\">\n<div id=\"fs-id1165135208839\">\n<h3>Graphing Parametric Equations and Rectangular Equations on the Coordinate System<\/h3>\n<p id=\"fs-id1165137898083\">Graph the parametric equations[latex]\\,x=t+1\\,[\/latex] and[latex]\\,y=\\sqrt{t},\\,\\,t\\ge 0,\\,[\/latex]and the rectangular equivalent [latex]y=\\sqrt{x-1}\\,[\/latex]on the same coordinate system.<\/p>\n<\/div>\n<div id=\"fs-id1165134248714\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134248714\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134248714\" class=\"hidden-answer\" style=\"display: none\">Construct a table of values for the parametric equations, as we did in the previous example, and graph[latex]\\,y=\\sqrt{t},\\,\\,t\\ge 0\\,[\/latex]on the same grid, as in <a class=\"autogenerated-content\" href=\"#Figure_08_07_008\">(Figure)<\/a>.<\/p>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/39ca8a7fc4de08dc9bb71f0097f9be82c4a06f8f\/CNX_Precalc_Figure_08_07_008.jpg\" alt=\"Overlayed graph of the two versions of the given function, showing that they are the same whether they are given in parametric or rectangular coordinates.\" width=\"488\" height=\"291\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135347348\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134254196\">With the domain on[latex]\\,t\\,[\/latex]restricted, we only plot positive values of[latex]\\,t.\\,[\/latex]The parametric data is graphed in blue and the graph of the rectangular equation is dashed in red. Once again, we see that the two forms overlap.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823949\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_07_03\">\n<div id=\"fs-id1165137551494\">\n<p id=\"fs-id1165134347381\">Sketch the graph of the parametric equations[latex]\\,x=2\\mathrm{cos}\\,\\theta \\,\\,\\,\\text{and}\\,\\,y=4\\mathrm{sin}\\,\\theta ,\\,[\/latex]along with the rectangular equation on the same grid.<\/p>\n<\/div>\n<div id=\"fs-id1165137807092\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137807092\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137807092\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137807093\">The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/994011c9d17beb64f9dc4af835f756ba8beb1b37\/CNX_Precalc_Figure_08_07_009.jpg\" alt=\"Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134181759\" class=\"bc-section section\">\n<h3>Applications of Parametric Equations<\/h3>\n<p id=\"fs-id1165135393365\">Many of the advantages of parametric equations become obvious when applied to solving real-world problems. Although rectangular equations in <em>x<\/em> and <em>y<\/em> give an overall picture of an object&#8217;s path, they do not reveal the position of an object at a specific time. Parametric equations, however, illustrate how the values of <em>x<\/em> and <em>y<\/em> change depending on <em>t<\/em>, as the location of a moving object at a particular time.<\/p>\n<p id=\"fs-id1165137927248\">A common application of parametric equations is solving problems involving projectile motion. In this type of motion, an object is propelled forward in an upward direction forming an angle of [latex]\\theta[\/latex] to the horizontal, with an initial speed of [latex]{v}_{0},\\,[\/latex]and at a height [latex]h[\/latex] above the horizontal.<\/p>\n<p>The path of an object propelled at an inclination of [latex]\\theta[\/latex] to the horizontal, with initial speed [latex]{v}_{0},\\,[\/latex]and at a height [latex]h[\/latex] above the horizontal, is given by<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}x=\\left({v}_{0}\\mathrm{cos}\\theta \\right)t\\text{ }\\hfill \\\\ y=-\\frac{1}{2}g{t}^{2}+\\left({v}_{0}\\mathrm{sin}\\theta \\right)t+h\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137723578\">where[latex]\\,g\\,[\/latex]accounts for the effects of <span class=\"no-emphasis\">gravity<\/span> and [latex]h[\/latex] is the initial height of the object. Depending on the units involved in the problem, use[latex]\\,g=32\\,\\text{ft}\\,\\text{\/}\\,{\\text{s}}^{2}\\,[\/latex]or[latex]\\,g=9.8\\,\\text{m}\\,\\text{\/}\\,{\\text{s}}^{2}.\\,[\/latex]The equation for[latex]\\,x\\,[\/latex]gives horizontal distance, and the equation for[latex]\\,y\\,[\/latex] gives the vertical distance.<\/p>\n<div id=\"fs-id1165135469031\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137761099\"><strong>Given a projectile motion problem, use parametric equations to solve.<\/strong><\/p>\n<ol type=\"1\">\n<li>The horizontal distance is given by[latex]\\,x=\\left({v}_{0}\\mathrm{cos}\\,\\theta \\right)t.\\,[\/latex]Substitute the initial speed of the object for[latex]\\,{v}_{0}.[\/latex]<\/li>\n<li>The expression[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]indicates the angle at which the object is propelled. Substitute that angle in degrees for[latex]\\,\\mathrm{cos}\\,\\theta .[\/latex]<\/li>\n<li>The vertical distance is given by the formula[latex]\\,y=-\\frac{1}{2}g{t}^{2}+\\left({v}_{0}\\mathrm{sin}\\,\\theta \\right)t+h.\\,[\/latex]The term[latex]\\,-\\frac{1}{2}g{t}^{2}\\,[\/latex]represents the effect of gravity. Depending on units involved, use[latex]\\,g=32\\,{\\text{ft\/s}}^{2}\\,[\/latex]or[latex]\\,g=9.8\\,{\\text{m\/s}}^{2}.\\,[\/latex]Again, substitute the initial speed for[latex]\\,{v}_{0},\\,[\/latex]and the height at which the object was propelled for[latex]\\,h.[\/latex]<\/li>\n<li>Proceed by calculating each term to solve for[latex]\\,t.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_08_07_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137836910\">\n<div id=\"fs-id1165137501592\">\n<h3>Finding the Parametric Equations to Describe the Motion of a Baseball<\/h3>\n<p id=\"fs-id1165135570049\">Solve the problem presented at the beginning of this section. Does the batter hit the game-winning home run? Assume that the ball is hit with an initial velocity of 140 feet per second at an angle of[latex]\\,45\u00b0\\,[\/latex]to the horizontal, making contact 3 feet above the ground.<\/p>\n<ol type=\"a\">\n<li>Find the parametric equations to model the path of the baseball.<\/li>\n<li>Where is the ball after 2 seconds?<\/li>\n<li>How long is the ball in the air?<\/li>\n<li>Is it a home run?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137453386\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137453386\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137453386\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165135445817\" type=\"a\">\n<li>\n<p id=\"fs-id1165133299084\">Use the formulas to set up the equations. The horizontal position is found using the parametric equation for[latex]\\,x.\\,[\/latex]Thus,<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}x=\\left({v}_{0}\\mathrm{cos}\\,\\theta \\right)t\\hfill \\\\ x=\\left(140\\mathrm{cos}\\left(45\u00b0\\right)\\right)t\\hfill \\end{array}[\/latex<\/div>\n<p id=\"fs-id1165135192687\">The vertical position is found using the parametric equation for[latex]\\,y.\\,[\/latex]Thus,<\/p>\n<div id=\"fs-id1165137662399\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}y=-16{t}^{2}+\\left({v}_{0}\\mathrm{sin}\\,\\theta \\right)t+h\\hfill \\\\ y=-16{t}^{2}+\\left(140\\mathrm{sin}\\left(45\u00b0\\right)\\right)t+3\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>\n<p id=\"fs-id1165137643152\">Substitute 2 into the equations to find the horizontal and vertical positions of the ball.<\/p>\n<div id=\"fs-id1165133052900\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ x=\\left(140\\mathrm{cos}\\left(45\u00b0\\right)\\right)\\left(2\\right)\\hfill \\\\ x=198\\text{ feet}\\hfill \\\\ \\hfill \\\\ y=-16{\\left(2\\right)}^{2}+\\left(140\\mathrm{sin}\\left(45\u00b0\\right)\\right)\\left(2\\right)+3\\hfill \\\\ y=137\\text{ feet}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134063228\">After 2 seconds, the ball is 198 feet away from the batter\u2019s box and 137 feet above the ground.<\/p>\n<\/li>\n<li>\n<p id=\"fs-id1165134347441\">To calculate how long the ball is in the air, we have to find out when it will hit ground, or when[latex]\\,y=0.\\,[\/latex]Thus,<\/p>\n<div id=\"fs-id1165137667988\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}y=-16{t}^{2}+\\left(140\\mathrm{sin}\\left({45}^{\\circ }\\right)\\right)t+3\\hfill & \\hfill \\\\ y=0\\hfill & \\text{Set }y\\left(t\\right)=0\\text{ and solve the quadratic}.\\hfill \\\\ t=6.2173\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137666794\">When[latex]\\,t=6.2173\\,[\/latex]seconds, the ball has hit the ground. (The quadratic equation can be solved in various ways, but this problem was solved using a computer math program.)<\/p>\n<\/li>\n<li>\n<p id=\"fs-id1165137462832\">We cannot confirm that the hit was a home run without considering the size of the outfield, which varies from field to field. However, for simplicity\u2019s sake, let\u2019s assume that the outfield wall is 400 feet from home plate in the deepest part of the park. Let\u2019s also assume that the wall is 10 feet high. In order to determine whether the ball clears the wall, we need to calculate how high the ball is when <em>x<\/em> = 400 feet. So we will set <em>x<\/em> = 400, solve for [latex]t,\\,[\/latex]and input [latex]t[\/latex] into[latex]\\,y.\\,[\/latex]<\/p>\n<div id=\"fs-id1165137921676\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\text{ }x=\\left(140\\mathrm{cos}\\left(45\u00b0\\right)\\right)t\\hfill \\\\ 400=\\left(140\\mathrm{cos}\\left(45\u00b0\\right)\\right)t\\hfill \\\\ \\text{ }t=4.04\\hfill \\end{array}\\hfill \\\\ \\hfill \\\\ \\hfill \\\\ \\begin{array}{l}\\text{ }y=-16{\\left(4.04\\right)}^{2}+\\left(140\\mathrm{sin}\\left(45\u00b0\\right)\\right)\\left(4.04\\right)+3\\hfill \\\\ \\text{ }y=141.8\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p>The ball is 141.8 feet in the air when it soars out of the ballpark. It was indeed a home run. See <a class=\"autogenerated-content\" href=\"#Figure_08_07_010\">(Figure)<\/a>.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/36e3aba935a97ade9ab59a5fa21a6c81e6a83a7c\/CNX_Precalc_Figure_08_07_010n.jpg\" alt=\"Plotted trajectory of a hit ball, showing the position of the batter at the origin, the ball's path in the shape of a wide downward facing parabola, and the outfield wall as a vertical line segment rising to 10 ft under the ball's path.\" width=\"731\" height=\"310\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/p>\n<\/div>\n<p id=\"fs-id1165134148366\"><\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137842378\" class=\"precalculus media\">\n<p id=\"fs-id1165134323598\">Access the following online resource for additional instruction and practice with graphs of parametric equations.<\/p>\n<ul id=\"fs-id1165134323602\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphpara84\">Graphing Parametric Equations on the TI-84<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137470285\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135209598\">\n<li>When there is a third variable, a third parameter on which[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]depend, parametric equations can be used.<\/li>\n<li>To graph parametric equations by plotting points, make a table with three columns labeled[latex]\\,t,x\\left(t\\right),\\,[\/latex]and[latex]\\,y\\left(t\\right).\\,[\/latex]Choose values for[latex]\\,t\\,[\/latex]in increasing order. Plot the last two columns for[latex]\\,x\\,[\/latex]and[latex]\\,y.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_07_01\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_07_02\">(Figure)<\/a>.<\/li>\n<li>When graphing a parametric curve by plotting points, note the associated <em>t<\/em>-values and show arrows on the graph indicating the orientation of the curve. See <a class=\"autogenerated-content\" href=\"#Example_08_07_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_07_04\">(Figure)<\/a>.<\/li>\n<li>Parametric equations allow the direction or the orientation of the curve to be shown on the graph. Equations that are not functions can be graphed and used in many applications involving motion. See <a class=\"autogenerated-content\" href=\"#Example_08_07_05\">(Figure)<\/a>.<\/li>\n<li>Projectile motion depends on two parametric equations:[latex]\\,x=\\left({v}_{0}\\mathrm{cos}\\,\\theta \\right)t\\,[\/latex]and[latex]\\,y=-16{t}^{2}+\\left({v}_{0}\\mathrm{sin}\\,\\theta \\right)t+h.\\,[\/latex]Initial velocity is symbolized as[latex]\\,{v}_{0}.\\,\\theta[\/latex] represents the initial angle of the object when thrown, and [latex]\\,h\\,[\/latex] represents the height at which the object is propelled.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134283619\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137851723\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137851729\">\n<div id=\"fs-id1165135194722\">\n<p id=\"fs-id1165135194723\">What are two methods used to graph parametric equations?<\/p>\n<\/div>\n<div id=\"fs-id1165135194726\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135194726\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135194726\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135194727\">plotting points with the orientation arrow and a graphing calculator<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135347318\">\n<div id=\"fs-id1165135347320\">\n<p>What is one difference in point-plotting parametric equations compared to Cartesian equations?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134113856\">\n<div>\n<p id=\"fs-id1165134113858\">Why are some graphs drawn with arrows?<\/p>\n<\/div>\n<div id=\"fs-id1165137758269\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137758269\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137758269\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137758270\">The arrows show the orientation, the direction of motion according to increasing values of[latex]\\,t.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137641293\">\n<div id=\"fs-id1165134205947\">\n<p id=\"fs-id1165134205948\">Name a few common types of graphs of parametric equations.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134205952\">\n<div id=\"fs-id1165134205953\">\n<p id=\"fs-id1165137715144\">Why are parametric graphs important in understanding projectile motion?<\/p>\n<\/div>\n<div id=\"fs-id1165137715148\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137715148\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137715148\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137715149\">The parametric equations show the different vertical and horizontal motions over time.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135511376\">For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.<\/p>\n<div id=\"fs-id1165135581156\">\n<div id=\"fs-id1165135581157\">\n<p id=\"fs-id1165135581158\">[latex]\\{\\begin{array}{l}x(t)=t\\hfill \\\\ y(t)={t}^{2}-1\\hfill \\end{array}[\/latex]<\/p>\n<table id=\"eip-id3065206\" class=\"unnumbered\" summary=\"Three columns and eight rows. The first row is labeled t, the second is labeled x, and the third is labeled y. The first columns contains the numbers -3, -2, -1, 0, 1, 2, 3. The other two columns are left blank for completion.\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr>\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134139687\">\n<div id=\"fs-id1165133361432\">\n<p id=\"fs-id1165133361433\">[latex]\\{\\begin{array}{l}x(t)=t-1\\hfill \\\\ y(t)={t}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<table id=\"eip-id2869464\" class=\"unnumbered\" summary=\"Three rows and eight columns. The first row is labeled t, the second is labeled x, and the third is labeled y. The first row contains the numbers -3, -2, -1, 0, 1, 2. The other two columns are left blank for completion.\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr>\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135188326\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135188326\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135188326\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/7802eb9e10c849cdfd0a571920cde1e460a71ac4\/CNX_Precalc_Figure_08_07_202.jpg\" alt=\"Graph of the given equations - looks like an upward opening parabola.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135172250\">\n<div id=\"fs-id1165135172252\">\n<p id=\"fs-id1165135172253\">[latex]\\{\\begin{array}{l}x(t)=2+t\\hfill \\\\ y(t)=3-2t\\hfill \\end{array}[\/latex]<\/p>\n<table id=\"eip-id2018706\" class=\"unnumbered\" summary=\"Three rows and eight columns. The first row is labeled t, the second is labeled x, and the third is labeled y. The first row contains the numbers -2, -1, 0, 1, 2, 3. The other two columns are left blank for completion.\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr>\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137457020\">\n<div id=\"fs-id1165137457021\">\n<p id=\"fs-id1165137457022\">[latex]\\{\\begin{array}{l}x(t)=-2-2t\\hfill \\\\ y(t)=3+t\\hfill \\end{array}[\/latex]<\/p>\n<table id=\"eip-id2478452\" class=\"unnumbered\" summary=\"Three rows and eight columns. The first row is labeled t, the second is labeled x, and the third is labeled y. The first row contains the numbers -3, -2, -1, 0, 1. The other two columns are left blank for completion.\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr>\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165134583390\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134583390\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134583390\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/5f55cc1b94c9db2f3254f69f4f5429b15a4d2569\/CNX_Precalc_Figure_08_07_204.jpg\" alt=\"Graph of the given equations - a line, negative slope.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137643624\">\n<div id=\"fs-id1165137643625\">\n<p id=\"fs-id1165137643626\">[latex]\\{\\begin{array}{l}x(t)={t}^{3}\\hfill \\\\ y(t)=t+2\\hfill \\end{array}[\/latex]<\/p>\n<table id=\"eip-id1838664\" class=\"unnumbered\" summary=\"Three rows and eight columns. The first row is labeled t, the second is labeled x, and the third is labeled y. The first row contains the numbers - -2, -1, 0, 1, 2. The other two columns are left blank for completion.\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr>\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137465652\">\n<div id=\"fs-id1165137465653\">\n<p id=\"fs-id1165137465654\">[latex]\\{\\begin{array}{l}x(t)={t}^{2}\\hfill \\\\ y(t)=t+3\\hfill \\end{array}[\/latex]<\/p>\n<table id=\"eip-id2086049\" class=\"unnumbered\" summary=\"Three rows and eight columns. The first row is labeled t, the second is labeled x, and the third is labeled y. The first row contains the numbers - -2, -1, 0, 1, 2. The other two columns are left blank for completion.\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr>\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135237047\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135237047\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135237047\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/f59bfc5343493b9d924228f66e3c63e0411f8e30\/CNX_Precalc_Figure_08_07_206.jpg\" alt=\"Graph of the given equations - looks like a sideways parabola, opening to the right.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137698486\">For the following exercises, sketch the curve and include the orientation.<\/p>\n<div id=\"fs-id1165135155255\">\n<div id=\"fs-id1165135155256\">\n<p id=\"fs-id1165135155257\">[latex]\\{\\begin{array}{l}x(t)=t\\\\ y(t)=\\sqrt{t}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137427635\">\n<div id=\"fs-id1165137427636\">\n<p id=\"fs-id1165135181824\">[latex]\\{\\begin{array}{l}x(t)=-\\,\\sqrt{t}\\\\ y(t)=t\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134069188\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134069188\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134069188\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/cf987e35edf5471fa2b92fa9450b29c4cda88a54\/CNX_Precalc_Figure_08_07_208.jpg\" alt=\"Graph of the given equations - looks like the left half of an upward opening parabola.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135404698\">\n<div id=\"fs-id1165135404699\">\n<p id=\"fs-id1165137811212\">[latex]\\{\\begin{array}{l}x(t)=5-|t|\\\\ y(t)=t+2\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137771695\">\n<div id=\"fs-id1165137771696\">\n<p id=\"fs-id1165137771697\">[latex]\\{\\begin{array}{l}x(t)=-t+2\\\\ y(t)=5-|t|\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137417002\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137417002\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137417002\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/615da221012d753baf879541277b194b2b4d4072\/CNX_Precalc_Figure_08_07_210.jpg\" alt=\"Graph of the given equations - looks like a downward opening absolute value function.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737228\">\n<div id=\"fs-id1165137737229\">\n<p id=\"fs-id1165137737230\">[latex]\\{\\begin{array}{l}x(t)=4\\text{sin}\\,t\\hfill \\\\ y(t)=2\\mathrm{cos}\\,t\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135168304\">\n<div id=\"fs-id1165135237110\">\n<p id=\"fs-id1165135237111\">[latex]\\{\\begin{array}{l}x(t)=2\\text{sin}\\,t\\hfill \\\\ y(t)=4\\text{cos}\\,t\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137936702\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137936702\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137936702\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/2cdb23093c6b20df36ba6b001ca3558ee48c6ae2\/CNX_Precalc_Figure_08_07_212.jpg\" alt=\"Graph of the given equations - a vertical ellipse.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137589850\">[latex]\\{\\begin{array}{l}x(t)=3{\\mathrm{cos}}^{2}t\\\\ y(t)=-3\\mathrm{sin}\\,t\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137410365\">\n<div id=\"fs-id1165137410366\">\n<p id=\"fs-id1165137410367\">[latex]\\{\\begin{array}{l}x(t)=3{\\mathrm{cos}}^{2}t\\\\ y(t)=-3{\\mathrm{sin}}^{2}t\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134081386\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134081386\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134081386\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/83a62a440e0dfc37568cf86c50fb48d17f49c19d\/CNX_Precalc_Figure_08_07_214.jpg\" alt=\"Graph of the given equations- line from (0, -3) to (3,0). It is traversed in both directions, positive and negative slope.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137601187\">\n<div id=\"fs-id1165137601188\">\n<p id=\"fs-id1165137601189\">[latex]\\{\\begin{array}{l}x(t)=\\mathrm{sec}\\,t\\\\ y(t)=\\mathrm{tan}\\,t\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135209757\">\n<div id=\"fs-id1165137431122\">\n<p id=\"fs-id1165137431123\">[latex]\\{\\begin{array}{l}x(t)=\\mathrm{sec}\\,t\\\\ y(t)={\\mathrm{tan}}^{2}t\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137805783\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137805783\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137805783\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/2722c324b807d246f21e1dadde6b56fb1db0842a\/CNX_Precalc_Figure_08_07_216.jpg\" alt=\"Graph of the given equations- looks like an upward opening parabola.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137896941\">\n<div id=\"fs-id1165137896942\">[latex]\\{\\begin{array}{l}x(t)=\\frac{1}{{e}^{2t}}\\\\ y(t)={e}^{-\\,t}\\end{array}[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1165137925361\">For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.<\/p>\n<div id=\"fs-id1165137925365\">\n<div id=\"fs-id1165134149986\">\n<p id=\"fs-id1165134149987\">[latex]\\{\\begin{array}{l}x(t)=t-1\\hfill \\\\ y(t)=-{t}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135149887\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135149887\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135149887\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/6fa45950ce2f0303bc9c40a2a8aa197972da1315\/CNX_Precalc_Figure_08_07_218.jpg\" alt=\"Graph of the given equations- looks like a downward opening parabola.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190005\">\n<div id=\"fs-id1165135190006\">\n<p id=\"fs-id1165135190007\">[latex]\\{\\begin{array}{l}x(t)={t}^{3}\\hfill \\\\ y(t)=t+3\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135597699\">\n<div id=\"fs-id1165135597700\">\n<p id=\"fs-id1165135597701\">[latex]\\{\\begin{array}{l}x(t)=2\\mathrm{cos}\\,t\\\\ y(t)=-\\mathrm{sin}\\,t\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135512730\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135512730\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135512730\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/4101f0f0d1a9cae3a869e1d7b58b90f249afc8a4\/CNX_Precalc_Figure_08_07_220.jpg\" alt=\"Graph of the given equations- horizontal ellipse.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135209077\">\n<p id=\"fs-id1165135209078\">[latex]\\{\\begin{array}{l}x(t)=7\\mathrm{cos}\\,t\\\\ y(t)=7\\mathrm{sin}\\,t\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192319\">\n<div id=\"fs-id1165135192320\">\n<p id=\"fs-id1165135192322\">[latex]\\{\\begin{array}{l}x(t)={e}^{2t}\\\\ y(t)=-{e}^{\\,t}\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165132912723\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165132912723\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165132912723\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/64f7a2670f4176ecaaa6e972c9be6d011620219f\/CNX_Precalc_Figure_08_07_222.jpg\" alt=\"Graph of the given equations- looks like the lower half of a sideways parabola opening to the right\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135185162\">For the following exercises, graph the equation and include the orientation.<\/p>\n<div id=\"fs-id1165135185165\">\n<div id=\"fs-id1165135185166\">[latex]x={t}^{2},\\,y\\,=\\,3t,\\,0\\le t\\le 5[\/latex]<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135186345\">\n<p id=\"fs-id1165135186346\">[latex]x=2t,\\,y\\,=\\,\\,{t}^{2},\\,-5\\le t\\le 5[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q211246\">Show Solution<\/span><\/p>\n<div id=\"q211246\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/2204c70368902e6f3c19e5a486a9adb776d4dda2\/CNX_Precalc_Figure_08_07_224.jpg\" alt=\"Graph of the given equations- looks like an upwards opening parabola\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133065688\">\n<div id=\"fs-id1165133065689\">\n<p id=\"fs-id1165135553506\">[latex]x=t,\\,y=\\sqrt{25-{t}^{2}},\\,0<t\\le 5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137742391\">\n<div id=\"fs-id1165137742392\">[latex]x\\left(t\\right)=-t,y\\left(t\\right)=\\sqrt{t},\\,t\\ge 0[\/latex]<\/div>\n<div id=\"fs-id1165135252162\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135252162\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135252162\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/ea3577e0761b0bfb69aae5a9cadbdae6463870a1\/CNX_Precalc_Figure_08_07_226.jpg\" alt=\"Graph of the given equations- looks like the upper half of a sideways parabola opening to the left\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134575412\">\n<div id=\"fs-id1165134575413\">[latex]x=-2\\mathrm{cos}\\,t,\\,y=6\\,\\mathrm{sin}\\,t,\\,0\\le t\\le \\pi[\/latex]<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137897084\">\n<p id=\"fs-id1165137897085\">[latex]x=-\\mathrm{sec}\\,t,\\,y=\\mathrm{tan}\\,t,\\,-\\frac{\\,\\pi }{2}<t<\\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137680409\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137680409\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137680409\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/ede6e666e5b9e83204d913926164721c52312d83\/CNX_Precalc_Figure_08_07_228.jpg\" alt=\"Graph of the given equations- the left half of a hyperbola with diagonal asymptotes\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134032307\">For the following exercises, use the parametric equations for integers <em>a <\/em>and <em>b<\/em>:<\/p>\n<div id=\"eip-155\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x\\left(t\\right)=a\\mathrm{cos}\\left(\\left(a+b\\right)t\\right)\\\\ y\\left(t\\right)=a\\mathrm{cos}\\left(\\left(a-b\\right)t\\right)\\end{array}[\/latex]<\/div>\n<div>\n<div id=\"fs-id1165135160277\">\n<p id=\"fs-id1165135160278\">Graph on the domain[latex]\\,\\left[-\\pi ,0\\right],\\,[\/latex]where[latex]\\,a=2\\,[\/latex]and[latex]\\,b=1,\\,[\/latex]and include the orientation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137844152\">\n<div id=\"fs-id1165137844153\">\n<p id=\"fs-id1165137844154\">Graph on the domain[latex]\\,\\left[-\\pi ,0\\right],\\,[\/latex]where[latex]\\,a=3\\,[\/latex]and[latex]\\,b=2[\/latex], and include the orientation.<\/p>\n<\/div>\n<div id=\"fs-id1165137409296\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137409296\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137409296\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/1fe8c8d9d88000e5c27471bf752630a946e7562b\/CNX_Precalc_Figure_08_07_230.jpg\" alt=\"Graph of the given equations - vertical periodic trajectory\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137619318\">\n<div id=\"fs-id1165137619319\">\n<p id=\"fs-id1165137619320\">Graph on the domain[latex]\\,\\left[-\\pi ,0\\right],\\,[\/latex]where[latex]\\,a=4\\,[\/latex]and[latex]\\,b=3[\/latex], and include the orientation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137404658\">\n<div id=\"fs-id1165137404659\">\n<p id=\"fs-id1165137404660\">Graph on the domain[latex]\\,\\left[-\\pi ,0\\right],\\,[\/latex]where[latex]\\,a=5\\,[\/latex]and[latex]\\,b=4[\/latex], and include the orientation.<\/p>\n<\/div>\n<div id=\"fs-id1165137551247\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137551247\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137551247\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/892611bcff1d279af76db753ed3548573dce74fc\/CNX_Precalc_Figure_08_07_232.jpg\" alt=\"Graph of the given equations - vertical periodic trajectory\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133233024\">\n<div id=\"fs-id1165133233025\">\n<p id=\"fs-id1165133233026\">If[latex]\\,a\\,[\/latex]is 1 more than[latex]\\,b,\\,[\/latex]describe the effect the values of[latex]\\,a\\,[\/latex]and[latex]\\,b\\,[\/latex]have on the graph of the parametric equations.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135638542\">\n<div id=\"fs-id1165135638543\">\n<p id=\"fs-id1165137767042\">Describe the graph if[latex]\\,a=100\\,[\/latex]and[latex]\\,b=99.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135189746\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135189746\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135189746\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135189747\">There will be 100 back-and-forth motions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135189750\">\n<div id=\"fs-id1165135189751\">\n<p id=\"fs-id1165135530296\">What happens if[latex]\\,b\\,[\/latex]is 1 more than[latex]\\,a?\\,[\/latex]Describe the graph.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134122767\">\n<div id=\"fs-id1165137823080\">\n<p id=\"fs-id1165137823081\">If the parametric equations[latex]\\,x\\left(t\\right)={t}^{2}\\,[\/latex]and[latex]\\,y\\left(t\\right)=6-3t\\,[\/latex]have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?<\/p>\n<\/div>\n<div id=\"fs-id1165135499919\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135499919\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135499919\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134272702\">Take the opposite of the[latex]\\,x\\left(t\\right)\\,[\/latex]equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137911683\">For the following exercises, describe the graph of the set of parametric equations.<\/p>\n<div id=\"fs-id1165137911686\">\n<div id=\"fs-id1165137911687\">\n<p id=\"fs-id1165137911688\">[latex]x\\left(t\\right)=-{t}^{2}\\,[\/latex]and[latex]\\,y\\left(t\\right)\\,[\/latex]is linear<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135251241\">\n<div id=\"fs-id1165135251242\">\n<p id=\"fs-id1165135251243\">[latex]y\\left(t\\right)={t}^{2}\\,[\/latex]and[latex]\\,x\\left(t\\right)\\,[\/latex]is linear<\/p>\n<\/div>\n<div id=\"fs-id1165135609231\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135609231\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135609231\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135609232\">The parabola opens up.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135609236\">\n<div id=\"fs-id1165135609237\">\n<p>[latex]y\\left(t\\right)=-{t}^{2}\\,[\/latex]and[latex]\\,x\\left(t\\right)\\,[\/latex]is linear<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135453880\">\n<div id=\"fs-id1165135453881\">\n<p id=\"fs-id1165135453882\">Write the parametric equations of a circle with center[latex]\\,\\left(0,0\\right),[\/latex]radius 5, and a counterclockwise orientation.<\/p>\n<\/div>\n<div id=\"fs-id1165135352460\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135352460\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135352460\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135352461\">[latex]\\{\\begin{array}{l}x(t)=5\\mathrm{cos}t\\\\ y(t)=5\\mathrm{sin}t\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133141349\">\n<div id=\"fs-id1165133141350\">\n<p id=\"fs-id1165133141351\">Write the parametric equations of an ellipse with center[latex]\\,\\left(0,0\\right),[\/latex]major axis of length 10, minor axis of length 6, and a counterclockwise orientation.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137838292\">For the following exercises, use a graphing utility to graph on the window[latex]\\,\\left[-3,3\\right]\\,[\/latex]by[latex]\\,\\left[-3,3\\right]\\,[\/latex]on the domain[latex]\\,\\left[0,2\\pi \\right)\\,[\/latex]for the following values of[latex]\\,a\\,[\/latex]and[latex]\\,b[\/latex], and include the orientation.<\/p>\n<div id=\"eip-169\" class=\"unnumbered aligncenter\">[latex]\\{\\begin{array}{l}x(t)=\\mathrm{sin}(at)\\\\ y(t)=\\mathrm{sin}(bt)\\end{array}[\/latex]<\/div>\n<div id=\"fs-id1165137761928\">\n<div id=\"fs-id1165134053964\">\n<p id=\"fs-id1165134053965\">[latex]a=1,b=2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165132960728\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165132960728\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165132960728\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/2bab68aaea09ee9ec930461b83bf06e0b6d239ff\/CNX_Precalc_Figure_08_07_233.jpg\" alt=\"Graph of the given equations\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133276223\">\n<div id=\"fs-id1165133276224\">\n<p id=\"fs-id1165133276225\">[latex]a=2,b=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135415664\">\n<div id=\"fs-id1165135415665\">\n<p id=\"fs-id1165135415666\">[latex]a=3,b=3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134386554\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134386554\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134386554\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/ab11cb1a439b9d52ab738299a39b8a1670f42fd4\/CNX_Precalc_Figure_08_07_235.jpg\" alt=\"Graph of the given equations - lines extending into Q1 and Q3 (in both directions) from the origin to 1 unit.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135572012\">\n<div id=\"fs-id1165135572013\">\n<p id=\"fs-id1165135572014\">[latex]a=5,b=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137832502\">\n<div id=\"fs-id1165134118436\">\n<p id=\"fs-id1165134118437\">[latex]a=2,b=5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133349420\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165133349420\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165133349420\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/d44fdde2ddd0a3966c80c1c63537633ce034197a\/CNX_Precalc_Figure_08_07_237.jpg\" alt=\"Graph of the given equations - lines extending into Q1 and Q3 (in both directions) from the origin to 3 units.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134357489\">\n<div id=\"fs-id1165134357490\">\n<p id=\"fs-id1165134357491\">[latex]a=5,b=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133315097\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165134589497\">For the following exercises, look at the graphs that were created by parametric equations of the form[latex]\\,\\{\\begin{array}{l}x(t)=a\\text{cos}(bt)\\hfill \\\\ y(t)=c\\text{sin}(dt)\\hfill \\end{array}.\\,[\/latex]Use the parametric mode on the graphing calculator to find the values of [latex]a,b,c,[\/latex] and [latex]d[\/latex] to achieve each graph.<\/p>\n<div id=\"fs-id1165137701974\">\n<div id=\"fs-id1165137701975\"><span id=\"fs-id1165137701981\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/59cdfbe8b34369000a5c7ac21765f879c03c26eb\/CNX_Precalc_Figure_08_07_239.jpg\" alt=\"Graph of the given equations\" \/><\/span><\/div>\n<div id=\"fs-id1165137651752\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137651752\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137651752\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137651754\">[latex]a=4,\\,b=3,\\,c=6,\\,d=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132032531\">\n<div id=\"fs-id1165132032532\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/b88749b1f3f254204ac4570264e80f43871159bc\/CNX_Precalc_Figure_08_07_240.jpg\" alt=\"Graph of the given equations\" \/><\/div>\n<\/div>\n<div id=\"fs-id1165134237259\">\n<div id=\"fs-id1165134237260\"><span id=\"fs-id1165134237266\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/3d75d7eeb5e2a2fac17b80720c7408be76e46d84\/CNX_Precalc_Figure_08_07_241.jpg\" alt=\"Graph of the given equations\" \/><\/span><\/div>\n<div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137693607\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137693607\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137693607\">[latex]a=4,\\,b=2,\\,c=3,\\,d=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133393487\">\n<div id=\"fs-id1165133393488\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/5eaae6a03845596ad92f25282fac6aaf2a25182f\/CNX_Precalc_Figure_08_07_242.jpg\" alt=\"Graph of the given equations\" \/><\/div>\n<\/div>\n<p id=\"fs-id1165134211299\">For the following exercises, use a graphing utility to graph the given parametric equations.<\/p>\n<ol id=\"fs-id1165134211302\" type=\"a\">\n<li>[latex]\\{\\begin{array}{l}x(t)=\\mathrm{cos}t-1\\\\ y(t)=\\mathrm{sin}t+t\\end{array}[\/latex]<\/li>\n<li>[latex]\\{\\begin{array}{l}x(t)=\\mathrm{cos}t+t\\\\ y(t)=\\mathrm{sin}t-1\\end{array}[\/latex]<\/li>\n<li>[latex]\\{\\begin{array}{l}x(t)=t-\\mathrm{sin}t\\\\ y(t)=\\mathrm{cos}t-1\\end{array}[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1165137729163\">\n<div id=\"fs-id1165135453820\">\n<p id=\"fs-id1165135453821\">Graph all three sets of parametric equations on the domain[latex]\\,\\left[0,\\,2\\pi \\right].[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137419669\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137419669\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137419669\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/b09dc27fcd7c07eb5526c596e83f201413dc38b1\/CNX_Precalc_Figure_08_07_243.jpg\" alt=\"Graph of the given equations\" \/><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/c68224c477101bf9f8515e4966e3bdb0c21849ee\/CNX_Precalc_Figure_08_07_244.jpg\" alt=\"Graph of the given equations\" \/><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/583e7ce2f6f87bd60fa0d6bc065eec7c8a4ed0f3\/CNX_Precalc_Figure_08_07_245.jpg\" alt=\"Graph of the given equations\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134422229\">\n<div id=\"fs-id1165135253195\">\n<p id=\"fs-id1165135253196\">Graph all three sets of parametric equations on the domain[latex]\\,\\left[0,4\\pi \\right].[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134546248\">\n<div id=\"fs-id1165134546249\">\n<p id=\"fs-id1165134546250\">Graph all three sets of parametric equations on the domain[latex]\\,\\left[-4\\pi ,6\\pi \\right].[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165131962224\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165131962224\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165131962224\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/849a48802af36c85319d64bc349c1ff273ee703b\/CNX_Precalc_Figure_08_07_249.jpg\" alt=\"Graph of the given equations\" \/><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/4f6094a56c565941ba80d7c1f42accddfb4e92c0\/CNX_Precalc_Figure_08_07_250.jpg\" alt=\"Graph of the given equations\" \/><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/798c654d12b96561c2b2343b5be7fcbb3e2f3228\/CNX_Precalc_Figure_08_07_251.jpg\" alt=\"Graph of the given equations\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135515798\">\n<div id=\"fs-id1165135515799\">\n<p id=\"fs-id1165135515800\">The graph of each set of parametric equations appears to \u201ccreep\u201d along one of the axes. What controls which axis the graph creeps along?<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135662545\">\n<p>Explain the effect on the graph of the parametric equation when we switched[latex]\\,\\mathrm{sin}\\,t\\,[\/latex]and[latex]\\,\\mathrm{cos}\\,t[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165135252078\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135252078\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135252078\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135252079\">The[latex]\\,y[\/latex]-intercept changes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135606062\">\n<div id=\"fs-id1165135606063\">\n<p id=\"fs-id1165135606064\">Explain the effect on the graph of the parametric equation when we changed the domain.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135606068\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165134040395\">\n<div id=\"fs-id1165134040396\">\n<p id=\"fs-id1165134040397\">An object is thrown in the air with vertical velocity of 20 ft\/s and horizontal velocity of 15 ft\/s. The object\u2019s height can be described by the equation[latex]\\,y\\left(t\\right)=-16{t}^{2}+20t[\/latex], while the object moves horizontally with constant velocity 15 ft\/s. Write parametric equations for the object\u2019s position, and then eliminate time to write height as a function of horizontal position.<\/p>\n<\/div>\n<div id=\"fs-id1165132079269\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165132079269\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165132079269\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165132079270\">[latex]y\\left(x\\right)=-16{\\left(\\frac{x}{15}\\right)}^{2}+20\\left(\\frac{x}{15}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137642896\">\n<div id=\"fs-id1165137642897\">\n<p id=\"fs-id1165137642898\">A skateboarder riding on a level surface at a constant speed of 9 ft\/s throws a ball in the air, the height of which can be described by the equation[latex]\\,y\\left(t\\right)=-16{t}^{2}+10t+5.\\text{}[\/latex]Write parametric equations for the ball\u2019s position, and then eliminate time to write height as a function of horizontal position.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165132035992\">For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft\/s at an angle of elevation of 52\u00b0. Consider the position of the dart at any time[latex]\\,t.\\,[\/latex]Neglect air resistance.<\/p>\n<div id=\"fs-id1165131938001\">\n<div id=\"fs-id1165131938002\">\n<p id=\"fs-id1165134371096\">Find parametric equations that model the problem situation.<\/p>\n<\/div>\n<div id=\"fs-id1165134371099\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134371099\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134371099\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134371100\">[latex]\\{\\begin{array}{l}x(t)=64t\\mathrm{cos}(52\u00b0)\\\\ y(t)=-16{t}^{2}+64t\\mathrm{sin}(52\u00b0)\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137401314\">\n<div id=\"fs-id1165137401315\">\n<p id=\"fs-id1165133087397\">Find all possible values of[latex]\\,x\\,[\/latex]that represent the situation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135445716\">\n<div id=\"fs-id1165135445718\">\n<p id=\"fs-id1165135445719\">When will the dart hit the ground?<\/p>\n<\/div>\n<div id=\"fs-id1165135445722\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135445722\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135445722\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165133311063\">approximately 3.2 seconds<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165133311067\">\n<p>Find the maximum height of the dart.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133311071\">\n<div id=\"fs-id1165133311072\">\n<p id=\"fs-id1165133311073\">At what time will the dart reach maximum height?<\/p>\n<\/div>\n<div id=\"fs-id1165135444043\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135444043\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135444043\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135444044\">1.6 seconds<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135444047\">For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.<\/p>\n<div id=\"fs-id1165137723597\">\n<div id=\"fs-id1165137723598\">\n<p id=\"fs-id1165137723599\">An epicycloid:[latex]\\,\\{\\begin{array}{l}x(t)=14\\mathrm{cos}\\,t-\\mathrm{cos}(14t)\\hfill \\\\ y(t)=14\\mathrm{sin}\\,t+\\mathrm{sin}(14t)\\hfill \\end{array}\\,[\/latex]on the domain[latex]\\,[0,2\\pi ][\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135596508\">\n<div>\n<p id=\"fs-id1165135596510\">A hypocycloid:[latex]\\{\\begin{array}{l}x(t)=6\\mathrm{sin}\\,t+2\\mathrm{sin}(6t)\\hfill \\\\ y(t)=6\\mathrm{cos}\\,t-2\\mathrm{cos}(6t)\\hfill \\end{array}\\,[\/latex]on the domain[latex]\\,[0,2\\pi ][\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165134569130\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134569130\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134569130\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/2bf25ddb8ab79c6544edc7beba2d7ec4b72d514b\/CNX_Precalc_Figure_08_07_253.jpg\" alt=\"Graph of the given equations - a hypocycloid\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131958330\">\n<div id=\"fs-id1165131958331\">\n<p id=\"fs-id1165131958332\">A hypotrochoid:[latex]\\{\\begin{array}{l}x(t)=2\\mathrm{sin}\\,t+5\\mathrm{cos}(6t)\\hfill \\\\ y(t)=5\\mathrm{cos}\\,t-2\\mathrm{sin}(6t)\\hfill \\end{array}\\,[\/latex]on the domain[latex]\\,\\left[0,2\\pi \\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137854188\">\n<div id=\"fs-id1165137854189\">\n<p id=\"fs-id1165137854190\">A rose:[latex]\\,\\{\\begin{array}{l}x(t)=5\\mathrm{sin}(2t)\\mathrm{sin}t\\hfill \\\\ y(t)=5\\mathrm{sin}(2t)\\mathrm{cos}t\\hfill \\end{array}\\,[\/latex]on the domain[latex]\\,\\left[0,2\\pi \\right][\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165134134026\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134134026\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134134026\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/100968f7788e3205fd1623f96ef4e3e16dcb1010\/CNX_Precalc_Figure_08_07_255.jpg\" alt=\"Graph of the given equations - a four petal rose\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3329\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Algebra and Trigonometry. <strong>Authored by<\/strong>: Jay Abramson, et. al. <strong>Provided by<\/strong>: OpenStax CNX. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\">http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":53384,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Algebra and Trigonometry\",\"author\":\"Jay Abramson, et. al\",\"organization\":\"OpenStax CNX\",\"url\":\"http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3329","chapter","type-chapter","status-publish","hentry"],"part":2978,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/3329","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/users\/53384"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/3329\/revisions"}],"predecessor-version":[{"id":3843,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/3329\/revisions\/3843"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/parts\/2978"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/3329\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/media?parent=3329"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapter-type?post=3329"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/contributor?post=3329"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/license?post=3329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}