{"id":1961,"date":"2023-10-12T00:36:14","date_gmt":"2023-10-12T00:36:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/parametric-equations-graphs\/"},"modified":"2023-10-23T19:09:48","modified_gmt":"2023-10-23T19:09:48","slug":"parametric-equations-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/parametric-equations-graphs\/","title":{"raw":"Parametric Equations: Graphs","rendered":"Parametric Equations: Graphs"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\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\nIt 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^\\circ [\/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 <strong>parametric equations<\/strong>. In this section, we\u2019ll discuss parametric equations and some common applications, such as projectile motion problems.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180943\/CNX_Precalc_Figure_08_07_0012.jpg\" alt=\"Photo of a baseball batter swinging.\" width=\"488\" height=\"333\" \/> <b>Figure 1.<\/b> Parametric equations can model the path of a projectile. (credit: Paul Kreher, Flickr)[\/caption]\r\n\r\n&nbsp;\r\n<h2>Graphing Parametric Equations by Plotting Points<\/h2>\r\nIn 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.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a pair of parametric equations, sketch a graph by plotting points.<\/h3>\r\n<ol>\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 class=\"textbox shaded\">\r\n<h3>Example 1: Sketching the Graph of a Pair of Parametric Equations by Plotting Points<\/h3>\r\nSketch the graph of the <strong>parametric equations<\/strong> [latex]x\\left(t\\right)={t}^{2}+1,y\\left(t\\right)=2+t[\/latex].\r\n\r\n[reveal-answer q=\"34813\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"34813\"]\r\n\r\nConstruct a table of values for [latex]t,x\\left(t\\right)[\/latex], and [latex]y\\left(t\\right)[\/latex], as in the table below, and plot the points in a plane.\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\nThe graph is a <strong>parabola<\/strong> with vertex at the point [latex]\\left(1,2\\right)[\/latex], opening to the right. See Figure 2.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180945\/CNX_Precalc_Figure_08_07_0022.jpg\" alt=\"Graph of the given parabola opening to the right.\" width=\"487\" height=\"366\" \/> <b>Figure 2<\/b>[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\nAs 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].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\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[reveal-answer q=\"858141\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"858141\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181012\/CNX_Precalc_Figure_08_07_0032.jpg\" alt=\"Graph of the given parametric equations with the restricted domain - it looks like the right half of an upward opening parabola.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Sketching the Graph of Trigonometric Parametric Equations<\/h3>\r\nConstruct a table of values for the given parametric equations and sketch the graph:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;x=2\\cos t \\\\ &amp;y=4\\sin t\\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"177517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"177517\"]\r\n\r\nConstruct a table like the one below\u00a0using 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.\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\\cos t[\/latex]<\/th>\r\n<th>[latex]y=4\\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\\cos \\left(0\\right)=2[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\frac{\\pi }{6}\\right)=\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\frac{\\pi }{3}\\right)=1[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\frac{\\pi }{2}\\right)=0[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\frac{2\\pi }{3}\\right)=-1[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\frac{5\\pi }{6}\\right)=-\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\pi \\right)=-2[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\frac{7\\pi }{6}\\right)=-\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\frac{4\\pi }{3}\\right)=-1[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\frac{3\\pi }{2}\\right)=0[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\frac{5\\pi }{3}\\right)=1[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(\\frac{11\\pi }{6}\\right)=\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]y=4\\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\\cos \\left(2\\pi \\right)=2[\/latex]<\/td>\r\n<td>[latex]y=4\\sin \\left(2\\pi \\right)=0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFigure 3\u00a0shows the graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180948\/CNX_Precalc_Figure_08_07_0042.jpg\" alt=\"Graph of the given equations - a vertical ellipse.\" width=\"487\" height=\"441\" \/> <b>Figure 3<\/b>[\/caption]\r\n\r\nBy the symmetry shown in the values of [latex]x[\/latex] and [latex]y[\/latex], we see that the parametric equations represent an <strong>ellipse<\/strong>. The <strong>ellipse<\/strong> is mapped in a counterclockwise direction as shown by the arrows indicating increasing [latex]t[\/latex] values.\r\n<div>\r\n<h4>Analysis of the Solution<\/h4>\r\nWe 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.\r\n\r\nMake sure to change the mode on the calculator to parametric (PAR). To confirm, the [latex]Y=[\/latex] window should show\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{X}_{1T}=\\\\ &amp;{Y}_{1T}=\\end{align}[\/latex]<\/p>\r\ninstead of [latex]{Y}_{1}=[\/latex].\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGraph the parametric equations: [latex]x=5\\cos t,y=3\\sin t[\/latex].\r\n\r\n[reveal-answer q=\"10248\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"10248\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181014\/CNX_Precalc_Figure_08_07_0052.jpg\" alt=\"Graph of the given equations - a horizontal ellipse.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Graphing Parametric Equations and Rectangular Form Together<\/h3>\r\nGraph the parametric equations [latex]x=5\\cos t[\/latex] and [latex]y=2\\sin t[\/latex]. First, construct the graph using data points generated from the <strong>parametric form<\/strong>. Then graph the <strong>rectangular form<\/strong> of the equation. Compare the two graphs.\r\n\r\n[reveal-answer q=\"651424\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"651424\"]\r\n\r\nConstruct a table of values like the table below.\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\\cos t[\/latex]<\/th>\r\n<th>[latex]y=2\\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\\cos \\left(0\\right)=5[\/latex]<\/td>\r\n<td>[latex]y=2\\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\\cos \\left(1\\right)\\approx 2.7[\/latex]<\/td>\r\n<td>[latex]y=2\\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\\cos \\left(2\\right)\\approx -2.1[\/latex]<\/td>\r\n<td>[latex]y=2\\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\\cos \\left(3\\right)\\approx -4.95[\/latex]<\/td>\r\n<td>[latex]y=2\\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\\cos \\left(4\\right)\\approx -3.3[\/latex]<\/td>\r\n<td>[latex]y=2\\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\\cos \\left(5\\right)\\approx 1.4[\/latex]<\/td>\r\n<td>[latex]y=2\\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\\cos \\left(-1\\right)\\approx 2.7[\/latex]<\/td>\r\n<td>[latex]y=2\\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\\cos \\left(-2\\right)\\approx -2.1[\/latex]<\/td>\r\n<td>[latex]y=2\\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\\cos \\left(-3\\right)\\approx -4.95[\/latex]<\/td>\r\n<td>[latex]y=2\\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\\cos \\left(-4\\right)\\approx -3.3[\/latex]<\/td>\r\n<td>[latex]y=2\\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\\cos \\left(-5\\right)\\approx 1.4[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(-5\\right)\\approx 1.9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the [latex]\\left(x,y\\right)[\/latex] values from the table. See Figure 4.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180951\/CNX_Precalc_Figure_08_07_0062.jpg\" alt=\"Graph of the given ellipse in parametric and rectangular coordinates - it is the same thing in both images.\" width=\"975\" height=\"290\" \/> <b>Figure 4<\/b>[\/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<p style=\"text-align: center;\">[latex]\\begin{align}&amp;x=5\\cos t \\\\ &amp;\\frac{x}{5}=\\cos t&amp;&amp; \\text{Solve for }\\cos t. \\\\ &amp;y=2\\sin t&amp;&amp; \\text{Solve for }\\sin t. \\\\ &amp;\\frac{y}{2}=\\sin t \\end{align}[\/latex]<\/p>\r\nThen, use the <strong>Pythagorean Theorem<\/strong>.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} {\\cos }^{2}t+{\\sin }^{2}t=1\\\\ {\\left(\\frac{x}{5}\\right)}^{2}+{\\left(\\frac{y}{2}\\right)}^{2}=1\\\\ \\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{4}=1\\end{gathered}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nIn Figure 5, 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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180953\/CNX_Precalc_Figure_08_07_0072.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\" \/> <b>Figure 5<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 4: Graphing Parametric Equations and Rectangular Equations on the Coordinate System<\/h3>\r\nGraph 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.\r\n\r\n[reveal-answer q=\"771276\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"771276\"]\r\n\r\nConstruct 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 Figure 6.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180957\/CNX_Precalc_Figure_08_07_0082.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\" \/> <b>Figure 6<\/b>[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\nWith 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.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nSketch the graph of the parametric equations [latex]x=2\\cos \\theta \\text{ and }y=4\\sin \\theta [\/latex], along with the rectangular equation on the same grid.\r\n\r\n[reveal-answer q=\"857454\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"857454\"]\r\n\r\nThe 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.<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181016\/CNX_Precalc_Figure_08_07_0092.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 class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]173887[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>\u00a0Applications of Parametric Equations<\/h2>\r\nMany 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.\r\n\r\nA 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.\r\n\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 style=\"text-align: center;\">[latex]\\begin{align}x&amp;=\\left({v}_{0}\\cos \\theta \\right)t \\\\ y&amp;=-\\frac{1}{2}g{t}^{2}+\\left({v}_{0}\\sin \\theta \\right)t+h \\end{align}[\/latex]<\/div>\r\nwhere [latex]g[\/latex] accounts for the effects of <strong>gravity<\/strong> 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.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a projectile motion problem, use parametric equations to solve.<\/h3>\r\n<ol>\r\n \t<li>The horizontal distance is given by [latex]x=\\left({v}_{0}\\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]\\cos \\theta [\/latex] indicates the angle at which the object is propelled. Substitute that angle in degrees for [latex]\\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}\\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 class=\"textbox shaded\">\r\n<h3>Example 5: Finding the Parametric Equations to Describe the Motion of a Baseball<\/h3>\r\nSolve 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^\\circ [\/latex] to the horizontal, making contact 3 feet above the ground.\r\n<ol>\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[reveal-answer q=\"730987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"730987\"]\r\n<ol>\r\n \t<li>Use the formulas to set up the equations. The horizontal position is found using the parametric equation for [latex]x[\/latex]. Thus,\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;x=\\left({v}_{0}\\cos \\theta \\right)t\\\\ &amp;x=\\left(140\\cos \\left(45^\\circ \\right)\\right)t \\end{align}[\/latex]<\/div>\r\nThe vertical position is found using the parametric equation for [latex]y[\/latex]. Thus,\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;y=-16{t}^{2}+\\left({v}_{0}\\sin \\theta \\right)t+h \\\\ &amp;y=-16{t}^{2}+\\left(140\\sin \\left(45^\\circ \\right)\\right)t+3 \\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 2 into the equations to find the horizontal and vertical positions of the ball.\r\n<div style=\"text-align: center;\">[latex]\\begin{align} &amp;x=\\left(140\\cos \\left(45^\\circ \\right)\\right)\\left(2\\right) \\\\ &amp;x=198\\text{ feet} \\\\ \\text{ } \\\\ &amp;y=-16{\\left(2\\right)}^{2}+\\left(140\\sin \\left(45^\\circ \\right)\\right)\\left(2\\right)+3 \\\\ &amp;y=137\\text{ feet} \\end{align}[\/latex]<\/div>\r\nAfter 2 seconds, the ball is 198 feet away from the batter\u2019s box and 137 feet above the ground.<\/li>\r\n \t<li>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,\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;y=-16{t}^{2}+\\left(140\\sin \\left({45}^{\\circ }\\right)\\right)t+3 \\\\ &amp;y=0&amp;&amp; \\text{Set }y\\left(t\\right)=0\\text{ and solve the quadratic}.\\\\ &amp;t=6.2173 \\end{align}[\/latex]<\/div>\r\nWhen [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.)<\/li>\r\n \t<li>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].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;x=\\left(140\\cos \\left(45^\\circ \\right)\\right)t \\\\ &amp;400=\\left(140\\cos \\left(45^\\circ \\right)\\right)t \\\\ &amp;t=4.04 \\\\ \\text{ } \\\\ &amp;y=-16{\\left(4.04\\right)}^{2}+\\left(140\\sin \\left(45^\\circ \\right)\\right)\\left(4.04\\right)+3 \\\\ &amp;y=141.8\\end{align}[\/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 Figure 7.<\/li>\r\n<\/ol>\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180959\/CNX_Precalc_Figure_08_07_010n2.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\" \/> <b>Figure 7<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\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].<\/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.<\/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.<\/li>\r\n \t<li>Projectile motion depends on two parametric equations: [latex]x=\\left({v}_{0}\\cos \\theta \\right)t[\/latex] and [latex]y=-16{t}^{2}+\\left({v}_{0}\\sin \\theta \\right)t+h[\/latex]. Initial velocity is symbolized as [latex]{v}_{0}[\/latex]. [latex]\\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>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\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>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^\\circ[\/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 <strong>parametric equations<\/strong>. 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:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180943\/CNX_Precalc_Figure_08_07_0012.jpg\" alt=\"Photo of a baseball batter swinging.\" width=\"488\" height=\"333\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> Parametric equations can model the path of a projectile. (credit: Paul Kreher, Flickr)<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Graphing Parametric Equations by Plotting Points<\/h2>\n<p>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 class=\"textbox\">\n<h3>How To: Given a pair of parametric equations, sketch a graph by plotting points.<\/h3>\n<ol>\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 class=\"textbox shaded\">\n<h3>Example 1: Sketching the Graph of a Pair of Parametric Equations by Plotting Points<\/h3>\n<p>Sketch the graph of the <strong>parametric equations<\/strong> [latex]x\\left(t\\right)={t}^{2}+1,y\\left(t\\right)=2+t[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q34813\">Show Solution<\/span><\/p>\n<div id=\"q34813\" class=\"hidden-answer\" style=\"display: none\">\n<p>Construct a table of values for [latex]t,x\\left(t\\right)[\/latex], and [latex]y\\left(t\\right)[\/latex], as in the table below, 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<p>The graph is a <strong>parabola<\/strong> with vertex at the point [latex]\\left(1,2\\right)[\/latex], opening to the right. See Figure 2.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180945\/CNX_Precalc_Figure_08_07_0022.jpg\" alt=\"Graph of the given parabola opening to the right.\" width=\"487\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>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 class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q858141\">Show Solution<\/span><\/p>\n<div id=\"q858141\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181012\/CNX_Precalc_Figure_08_07_0032.jpg\" alt=\"Graph of the given parametric equations with the restricted domain - it looks like the right half of an upward opening parabola.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Sketching the Graph of Trigonometric Parametric Equations<\/h3>\n<p>Construct a table of values for the given parametric equations and sketch the graph:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&x=2\\cos t \\\\ &y=4\\sin t\\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q177517\">Show Solution<\/span><\/p>\n<div id=\"q177517\" class=\"hidden-answer\" style=\"display: none\">\n<p>Construct a table like the one below\u00a0using 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\\cos t[\/latex]<\/th>\n<th>[latex]y=4\\sin t[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>[latex]x=2\\cos \\left(0\\right)=2[\/latex]<\/td>\n<td>[latex]y=4\\sin \\left(0\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td>[latex]x=2\\cos \\left(\\frac{\\pi }{6}\\right)=\\sqrt{3}[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(\\frac{\\pi }{3}\\right)=1[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(\\frac{\\pi }{2}\\right)=0[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(\\frac{2\\pi }{3}\\right)=-1[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(\\frac{5\\pi }{6}\\right)=-\\sqrt{3}[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(\\pi \\right)=-2[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(\\frac{7\\pi }{6}\\right)=-\\sqrt{3}[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(\\frac{4\\pi }{3}\\right)=-1[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(\\frac{3\\pi }{2}\\right)=0[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(\\frac{5\\pi }{3}\\right)=1[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(\\frac{11\\pi }{6}\\right)=\\sqrt{3}[\/latex]<\/td>\n<td>[latex]y=4\\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\\cos \\left(2\\pi \\right)=2[\/latex]<\/td>\n<td>[latex]y=4\\sin \\left(2\\pi \\right)=0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Figure 3\u00a0shows the graph.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180948\/CNX_Precalc_Figure_08_07_0042.jpg\" alt=\"Graph of the given equations - a vertical ellipse.\" width=\"487\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<p>By the symmetry shown in the values of [latex]x[\/latex] and [latex]y[\/latex], we see that the parametric equations represent an <strong>ellipse<\/strong>. The <strong>ellipse<\/strong> is mapped in a counterclockwise direction as shown by the arrows indicating increasing [latex]t[\/latex] values.<\/p>\n<div>\n<h4>Analysis of the Solution<\/h4>\n<p>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>Make sure to change the mode on the calculator to parametric (PAR). To confirm, the [latex]Y=[\/latex] window should show<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{X}_{1T}=\\\\ &{Y}_{1T}=\\end{align}[\/latex]<\/p>\n<p>instead of [latex]{Y}_{1}=[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Graph the parametric equations: [latex]x=5\\cos t,y=3\\sin t[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q10248\">Show Solution<\/span><\/p>\n<div id=\"q10248\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181014\/CNX_Precalc_Figure_08_07_0052.jpg\" alt=\"Graph of the given equations - a horizontal ellipse.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Graphing Parametric Equations and Rectangular Form Together<\/h3>\n<p>Graph the parametric equations [latex]x=5\\cos t[\/latex] and [latex]y=2\\sin t[\/latex]. First, construct the graph using data points generated from the <strong>parametric form<\/strong>. Then graph the <strong>rectangular form<\/strong> of the equation. Compare the two graphs.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q651424\">Show Solution<\/span><\/p>\n<div id=\"q651424\" class=\"hidden-answer\" style=\"display: none\">\n<p>Construct a table of values like the table below.<\/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\\cos t[\/latex]<\/th>\n<th>[latex]y=2\\sin t[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\text{0}[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(0\\right)=5[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(0\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{1}[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(1\\right)\\approx 2.7[\/latex]<\/td>\n<td>[latex]y=2\\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\\cos \\left(2\\right)\\approx -2.1[\/latex]<\/td>\n<td>[latex]y=2\\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\\cos \\left(3\\right)\\approx -4.95[\/latex]<\/td>\n<td>[latex]y=2\\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\\cos \\left(4\\right)\\approx -3.3[\/latex]<\/td>\n<td>[latex]y=2\\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\\cos \\left(5\\right)\\approx 1.4[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(5\\right)\\approx -1.9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(-1\\right)\\approx 2.7[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(-1\\right)\\approx -1.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(-2\\right)\\approx -2.1[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(-2\\right)\\approx -1.8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(-3\\right)\\approx -4.95[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(-3\\right)\\approx -0.28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(-4\\right)\\approx -3.3[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(-4\\right)\\approx 1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(-5\\right)\\approx 1.4[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(-5\\right)\\approx 1.9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the [latex]\\left(x,y\\right)[\/latex] values from the table. See Figure 4.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180951\/CNX_Precalc_Figure_08_07_0062.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\"><b>Figure 4<\/b><\/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<p style=\"text-align: center;\">[latex]\\begin{align}&x=5\\cos t \\\\ &\\frac{x}{5}=\\cos t&& \\text{Solve for }\\cos t. \\\\ &y=2\\sin t&& \\text{Solve for }\\sin t. \\\\ &\\frac{y}{2}=\\sin t \\end{align}[\/latex]<\/p>\n<p>Then, use the <strong>Pythagorean Theorem<\/strong>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} {\\cos }^{2}t+{\\sin }^{2}t=1\\\\ {\\left(\\frac{x}{5}\\right)}^{2}+{\\left(\\frac{y}{2}\\right)}^{2}=1\\\\ \\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{4}=1\\end{gathered}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>In Figure 5, 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.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180953\/CNX_Precalc_Figure_08_07_0072.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\"><b>Figure 5<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Graphing Parametric Equations and Rectangular Equations on the Coordinate System<\/h3>\n<p>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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q771276\">Show Solution<\/span><\/p>\n<div id=\"q771276\" class=\"hidden-answer\" style=\"display: none\">\n<p>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 Figure 6.<\/p>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180957\/CNX_Precalc_Figure_08_07_0082.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\"><b>Figure 6<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>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 class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Sketch the graph of the parametric equations [latex]x=2\\cos \\theta \\text{ and }y=4\\sin \\theta[\/latex], along with the rectangular equation on the same grid.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q857454\">Show Solution<\/span><\/p>\n<div id=\"q857454\" class=\"hidden-answer\" style=\"display: none\">\n<p>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.<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181016\/CNX_Precalc_Figure_08_07_0092.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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm173887\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173887&theme=oea&iframe_resize_id=ohm173887\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>\u00a0Applications of Parametric Equations<\/h2>\n<p>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>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 style=\"text-align: center;\">[latex]\\begin{align}x&=\\left({v}_{0}\\cos \\theta \\right)t \\\\ y&=-\\frac{1}{2}g{t}^{2}+\\left({v}_{0}\\sin \\theta \\right)t+h \\end{align}[\/latex]<\/div>\n<p>where [latex]g[\/latex] accounts for the effects of <strong>gravity<\/strong> 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 class=\"textbox\">\n<h3>How To: Given a projectile motion problem, use parametric equations to solve.<\/h3>\n<ol>\n<li>The horizontal distance is given by [latex]x=\\left({v}_{0}\\cos \\theta \\right)t[\/latex]. Substitute the initial speed of the object for [latex]{v}_{0}[\/latex].<\/li>\n<li>The expression [latex]\\cos \\theta[\/latex] indicates the angle at which the object is propelled. Substitute that angle in degrees for [latex]\\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}\\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 class=\"textbox shaded\">\n<h3>Example 5: Finding the Parametric Equations to Describe the Motion of a Baseball<\/h3>\n<p>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^\\circ[\/latex] to the horizontal, making contact 3 feet above the ground.<\/p>\n<ol>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q730987\">Show Solution<\/span><\/p>\n<div id=\"q730987\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Use the formulas to set up the equations. The horizontal position is found using the parametric equation for [latex]x[\/latex]. Thus,\n<div style=\"text-align: center;\">[latex]\\begin{align}&x=\\left({v}_{0}\\cos \\theta \\right)t\\\\ &x=\\left(140\\cos \\left(45^\\circ \\right)\\right)t \\end{align}[\/latex]<\/div>\n<p>The vertical position is found using the parametric equation for [latex]y[\/latex]. Thus,<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}&y=-16{t}^{2}+\\left({v}_{0}\\sin \\theta \\right)t+h \\\\ &y=-16{t}^{2}+\\left(140\\sin \\left(45^\\circ \\right)\\right)t+3 \\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 2 into the equations to find the horizontal and vertical positions of the ball.\n<div style=\"text-align: center;\">[latex]\\begin{align} &x=\\left(140\\cos \\left(45^\\circ \\right)\\right)\\left(2\\right) \\\\ &x=198\\text{ feet} \\\\ \\text{ } \\\\ &y=-16{\\left(2\\right)}^{2}+\\left(140\\sin \\left(45^\\circ \\right)\\right)\\left(2\\right)+3 \\\\ &y=137\\text{ feet} \\end{align}[\/latex]<\/div>\n<p>After 2 seconds, the ball is 198 feet away from the batter\u2019s box and 137 feet above the ground.<\/li>\n<li>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,\n<div style=\"text-align: center;\">[latex]\\begin{align}&y=-16{t}^{2}+\\left(140\\sin \\left({45}^{\\circ }\\right)\\right)t+3 \\\\ &y=0&& \\text{Set }y\\left(t\\right)=0\\text{ and solve the quadratic}.\\\\ &t=6.2173 \\end{align}[\/latex]<\/div>\n<p>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.)<\/li>\n<li>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].\n<div style=\"text-align: center;\">[latex]\\begin{align}&x=\\left(140\\cos \\left(45^\\circ \\right)\\right)t \\\\ &400=\\left(140\\cos \\left(45^\\circ \\right)\\right)t \\\\ &t=4.04 \\\\ \\text{ } \\\\ &y=-16{\\left(4.04\\right)}^{2}+\\left(140\\sin \\left(45^\\circ \\right)\\right)\\left(4.04\\right)+3 \\\\ &y=141.8\\end{align}[\/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 Figure 7.<\/li>\n<\/ol>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180959\/CNX_Precalc_Figure_08_07_010n2.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\"><b>Figure 7<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\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].<\/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.<\/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.<\/li>\n<li>Projectile motion depends on two parametric equations: [latex]x=\\left({v}_{0}\\cos \\theta \\right)t[\/latex] and [latex]y=-16{t}^{2}+\\left({v}_{0}\\sin \\theta \\right)t+h[\/latex]. Initial velocity is symbolized as [latex]{v}_{0}[\/latex]. [latex]\\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\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-1961\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":708740,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1961","chapter","type-chapter","status-publish","hentry"],"part":1875,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1961","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/users\/708740"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1961\/revisions"}],"predecessor-version":[{"id":2223,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1961\/revisions\/2223"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/parts\/1875"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1961\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/media?parent=1961"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1961"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/contributor?post=1961"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/license?post=1961"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}