{"id":15747,"date":"2019-09-05T18:47:09","date_gmt":"2019-09-05T18:47:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=15747"},"modified":"2025-02-05T05:24:22","modified_gmt":"2025-02-05T05:24:22","slug":"problem-set-62-parametric-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-62-parametric-equations\/","title":{"raw":"Problem Set 62: Parametric Equations","rendered":"Problem Set 62: Parametric Equations"},"content":{"raw":"1. What is a system of parametric equations?\r\n\r\n2.\u00a0Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.\r\n\r\n3. Explain how to eliminate a parameter given a set of parametric equations.\r\n\r\n4.\u00a0What is a benefit of writing a system of parametric equations as a Cartesian equation?\r\n\r\n5. What is a benefit of using parametric equations?\r\n\r\n6.\u00a0Why are there many sets of parametric equations to represent on Cartesian function?\r\n\r\nFor the following exercises, eliminate the parameter [latex]t[\/latex] to rewrite the parametric equation as a Cartesian equation.\r\n\r\n7. [latex]\\begin{cases}x\\left(t\\right)=5-t\\hfill \\\\ y\\left(t\\right)=8 - 2t\\hfill \\end{cases}[\/latex]\r\n\r\n8.\u00a0[latex]\\begin{cases}x\\left(t\\right)=6 - 3t\\hfill \\\\ y\\left(t\\right)=10-t\\hfill \\end{cases}[\/latex]\r\n\r\n9. [latex]\\begin{cases}x\\left(t\\right)=2t+1\\hfill \\\\ y\\left(t\\right)=3\\sqrt{t}\\hfill \\end{cases}[\/latex]\r\n\r\n10.\u00a0[latex]\\begin{cases}x\\left(t\\right)=3t - 1\\hfill \\\\ y\\left(t\\right)=2{t}^{2}\\hfill \\end{cases}[\/latex]\r\n\r\n11. [latex]\\begin{cases}x\\left(t\\right)=2{e}^{t}\\hfill \\\\ y\\left(t\\right)=1 - 5t\\hfill \\end{cases}[\/latex]\r\n\r\n12.\u00a0[latex]\\begin{cases}x\\left(t\\right)={e}^{-2t}\\hfill \\\\ y\\left(t\\right)=2{e}^{-t}\\hfill \\end{cases}[\/latex]\r\n\r\n13. [latex]\\begin{cases}x\\left(t\\right)=4\\text{log}\\left(t\\right)\\hfill \\\\ y\\left(t\\right)=3+2t\\hfill \\end{cases}[\/latex]\r\n\r\n14.\u00a0[latex]\\begin{cases}x\\left(t\\right)=\\text{log}\\left(2t\\right)\\hfill \\\\ y\\left(t\\right)=\\sqrt{t - 1}\\hfill \\end{cases}[\/latex]\r\n\r\n15. [latex]\\begin{cases}x\\left(t\\right)={t}^{3}-t\\hfill \\\\ y\\left(t\\right)=2t\\hfill \\end{cases}[\/latex]\r\n\r\n16.\u00a0[latex]\\begin{cases}x\\left(t\\right)=t-{t}^{4}\\hfill \\\\ y\\left(t\\right)=t+2\\hfill \\end{cases}[\/latex]\r\n\r\n17. [latex]\\begin{cases}x\\left(t\\right)={e}^{2t}\\hfill \\\\ y\\left(t\\right)={e}^{6t}\\hfill \\end{cases}[\/latex]\r\n\r\n18.\u00a0[latex]\\begin{cases}x\\left(t\\right)={t}^{5}\\hfill \\\\ y\\left(t\\right)={t}^{10}\\hfill \\end{cases}[\/latex]\r\n\r\n19. [latex]\\begin{cases}x\\left(t\\right)=4\\text{cos}t\\hfill \\\\ y\\left(t\\right)=5\\sin t \\hfill \\end{cases}[\/latex]\r\n\r\n20.\u00a0[latex]\\begin{cases}x\\left(t\\right)=3\\sin t\\hfill \\\\ y\\left(t\\right)=6\\cos t\\hfill \\end{cases}[\/latex]\r\n\r\n21. [latex]\\begin{cases}x\\left(t\\right)=2{\\text{cos}}^{2}t\\hfill \\\\ y\\left(t\\right)=-\\sin t \\hfill \\end{cases}[\/latex]\r\n\r\n22.\u00a0[latex]\\begin{cases}x\\left(t\\right)=\\cos t+4\\\\ y\\left(t\\right)=2{\\sin }^{2}t\\end{cases}[\/latex]\r\n\r\n23. [latex]\\begin{cases}x\\left(t\\right)=t - 1\\\\ y\\left(t\\right)={t}^{2}\\end{cases}[\/latex]\r\n\r\n24.\u00a0[latex]\\begin{cases}x\\left(t\\right)=-t\\\\ y\\left(t\\right)={t}^{3}+1\\end{cases}[\/latex]\r\n\r\n25. [latex]\\begin{cases}x\\left(t\\right)=2t - 1\\\\ y\\left(t\\right)={t}^{3}-2\\end{cases}[\/latex]\r\n\r\nFor the following exercises, rewrite the parametric equation as a Cartesian equation by building an [latex]x\\text{-}y[\/latex] table.\r\n\r\n26. [latex]\\begin{cases}x\\left(t\\right)=2t - 1\\\\ y\\left(t\\right)=t+4\\end{cases}[\/latex]\r\n\r\n27. [latex]\\begin{cases}x\\left(t\\right)=4-t\\\\ y\\left(t\\right)=3t+2\\end{cases}[\/latex]\r\n\r\n28.\u00a0[latex]\\begin{cases}x\\left(t\\right)=2t - 1\\\\ y\\left(t\\right)=5t\\end{cases}[\/latex]\r\n\r\n29. [latex]\\begin{cases}x\\left(t\\right)=4t - 1\\\\ y\\left(t\\right)=4t+2\\end{cases}[\/latex]\r\n\r\nFor the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting [latex]x\\left(t\\right)=t[\/latex] or by setting [latex]y\\left(t\\right)=t[\/latex].\r\n\r\n30. [latex]y\\left(x\\right)=3{x}^{2}+3[\/latex]\r\n\r\n31. [latex]y\\left(x\\right)=2\\sin x+1[\/latex]\r\n\r\n32.\u00a0[latex]x\\left(y\\right)=3\\mathrm{log}\\left(y\\right)+y[\/latex]\r\n\r\n33. [latex]x\\left(y\\right)=\\sqrt{y}+2y[\/latex]\r\n\r\nFor the following exercises, parameterize (write parametric equations for) each Cartesian equation by using [latex]x\\left(t\\right)=a\\cos t[\/latex] and [latex]y\\left(t\\right)=b\\sin t[\/latex]. Identify the curve.\r\n\r\n34. [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]\r\n\r\n35. [latex]\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{36}=1[\/latex]\r\n\r\n36.\u00a0[latex]{x}^{2}+{y}^{2}=16[\/latex]\r\n\r\n37. [latex]{x}^{2}+{y}^{2}=10[\/latex]\r\n\r\n38.\u00a0Parameterize the line from [latex]\\left(3,0\\right)[\/latex] to [latex]\\left(-2,-5\\right)[\/latex] so that the line is at [latex]\\left(3,0\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(-2,-5\\right)[\/latex] at [latex]t=1[\/latex].\r\n\r\n39. Parameterize the line from [latex]\\left(-1,0\\right)[\/latex] to [latex]\\left(3,-2\\right)[\/latex] so that the line is at [latex]\\left(-1,0\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(3,-2\\right)[\/latex] at [latex]t=1[\/latex].\r\n\r\n40.\u00a0Parameterize the line from [latex]\\left(-1,5\\right)[\/latex] to [latex]\\left(2,3\\right)[\/latex] so that the line is at [latex]\\left(-1,5\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(2,3\\right)[\/latex] at [latex]t=1[\/latex].\r\n\r\n41. Parameterize the line from [latex]\\left(4,1\\right)[\/latex] to [latex]\\left(6,-2\\right)[\/latex] so that the line is at [latex]\\left(4,1\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(6,-2\\right)[\/latex] at [latex]t=1[\/latex].\r\n\r\nFor the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.\r\n\r\n42. [latex]\\begin{cases}{x}_{1}\\left(t\\right)=3t\\hfill \\\\ {y}_{1}\\left(t\\right)=2t - 1\\hfill \\end{cases}\\text{ and }\\begin{cases}{x}_{2}\\left(t\\right)=t+3\\hfill \\\\ {y}_{2}\\left(t\\right)=4t - 4\\hfill \\end{cases}[\/latex]\r\n\r\n43. [latex]\\begin{cases}{x}_{1}\\left(t\\right)={t}^{2}\\hfill \\\\ {y}_{1}\\left(t\\right)=2t - 1\\hfill \\end{cases}\\text{ and }\\begin{cases}{x}_{2}\\left(t\\right)=-t+6\\hfill \\\\ {y}_{2}\\left(t\\right)=t+1\\hfill \\end{cases}[\/latex]\r\n\r\nFor the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.\r\n\r\n44. [latex]\\begin{cases}{x}_{1}\\left(t\\right)=3{t}^{2}-3t+7\\hfill \\\\ {y}_{1}\\left(t\\right)=2t+3\\hfill \\end{cases}[\/latex]\r\n<table id=\"fs-id1165137565962\" class=\"unnumbered\" summary=\"Four rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains -1, 0, 1. The rest of the values in columns x and y are blank.\">\r\n<thead>\r\n<tr>\r\n<th>[latex]t[\/latex]<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>\u20131<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n45. [latex]\\begin{cases}{x}_{1}\\left(t\\right)={t}^{2}-4\\hfill \\\\ {y}_{1}\\left(t\\right)=2{t}^{2}-1\\hfill \\end{cases}[\/latex]\r\n<table id=\"fs-id1165135407032\" class=\"unnumbered\" summary=\"Four rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains 1, 2, 3. The rest of the values in columns x and y are blank.\">\r\n<thead>\r\n<tr>\r\n<th>[latex]t[\/latex]<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n46. [latex]\\begin{cases}{x}_{1}\\left(t\\right)={t}^{4}\\hfill \\\\ {y}_{1}\\left(t\\right)={t}^{3}+4\\hfill \\end{cases}[\/latex]\r\n<table id=\"fs-id1165137921660\" class=\"unnumbered\" summary=\"Five rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains -1, 0, 1, 2. The rest of the values in columns x and y are blank.\">\r\n<thead>\r\n<tr>\r\n<th>[latex]t[\/latex]<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>-1<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n47.\u00a0Find two different sets of parametric equations for [latex]y={\\left(x+1\\right)}^{2}[\/latex].\r\n\r\n48.\u00a0Find two different sets of parametric equations for [latex]y=3x - 2[\/latex].\r\n\r\n49. Find two different sets of parametric equations for [latex]y={x}^{2}-4x+4[\/latex].","rendered":"<p>1. What is a system of parametric equations?<\/p>\n<p>2.\u00a0Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.<\/p>\n<p>3. Explain how to eliminate a parameter given a set of parametric equations.<\/p>\n<p>4.\u00a0What is a benefit of writing a system of parametric equations as a Cartesian equation?<\/p>\n<p>5. What is a benefit of using parametric equations?<\/p>\n<p>6.\u00a0Why are there many sets of parametric equations to represent on Cartesian function?<\/p>\n<p>For the following exercises, eliminate the parameter [latex]t[\/latex] to rewrite the parametric equation as a Cartesian equation.<\/p>\n<p>7. [latex]\\begin{cases}x\\left(t\\right)=5-t\\hfill \\\\ y\\left(t\\right)=8 - 2t\\hfill \\end{cases}[\/latex]<\/p>\n<p>8.\u00a0[latex]\\begin{cases}x\\left(t\\right)=6 - 3t\\hfill \\\\ y\\left(t\\right)=10-t\\hfill \\end{cases}[\/latex]<\/p>\n<p>9. [latex]\\begin{cases}x\\left(t\\right)=2t+1\\hfill \\\\ y\\left(t\\right)=3\\sqrt{t}\\hfill \\end{cases}[\/latex]<\/p>\n<p>10.\u00a0[latex]\\begin{cases}x\\left(t\\right)=3t - 1\\hfill \\\\ y\\left(t\\right)=2{t}^{2}\\hfill \\end{cases}[\/latex]<\/p>\n<p>11. [latex]\\begin{cases}x\\left(t\\right)=2{e}^{t}\\hfill \\\\ y\\left(t\\right)=1 - 5t\\hfill \\end{cases}[\/latex]<\/p>\n<p>12.\u00a0[latex]\\begin{cases}x\\left(t\\right)={e}^{-2t}\\hfill \\\\ y\\left(t\\right)=2{e}^{-t}\\hfill \\end{cases}[\/latex]<\/p>\n<p>13. [latex]\\begin{cases}x\\left(t\\right)=4\\text{log}\\left(t\\right)\\hfill \\\\ y\\left(t\\right)=3+2t\\hfill \\end{cases}[\/latex]<\/p>\n<p>14.\u00a0[latex]\\begin{cases}x\\left(t\\right)=\\text{log}\\left(2t\\right)\\hfill \\\\ y\\left(t\\right)=\\sqrt{t - 1}\\hfill \\end{cases}[\/latex]<\/p>\n<p>15. [latex]\\begin{cases}x\\left(t\\right)={t}^{3}-t\\hfill \\\\ y\\left(t\\right)=2t\\hfill \\end{cases}[\/latex]<\/p>\n<p>16.\u00a0[latex]\\begin{cases}x\\left(t\\right)=t-{t}^{4}\\hfill \\\\ y\\left(t\\right)=t+2\\hfill \\end{cases}[\/latex]<\/p>\n<p>17. [latex]\\begin{cases}x\\left(t\\right)={e}^{2t}\\hfill \\\\ y\\left(t\\right)={e}^{6t}\\hfill \\end{cases}[\/latex]<\/p>\n<p>18.\u00a0[latex]\\begin{cases}x\\left(t\\right)={t}^{5}\\hfill \\\\ y\\left(t\\right)={t}^{10}\\hfill \\end{cases}[\/latex]<\/p>\n<p>19. [latex]\\begin{cases}x\\left(t\\right)=4\\text{cos}t\\hfill \\\\ y\\left(t\\right)=5\\sin t \\hfill \\end{cases}[\/latex]<\/p>\n<p>20.\u00a0[latex]\\begin{cases}x\\left(t\\right)=3\\sin t\\hfill \\\\ y\\left(t\\right)=6\\cos t\\hfill \\end{cases}[\/latex]<\/p>\n<p>21. [latex]\\begin{cases}x\\left(t\\right)=2{\\text{cos}}^{2}t\\hfill \\\\ y\\left(t\\right)=-\\sin t \\hfill \\end{cases}[\/latex]<\/p>\n<p>22.\u00a0[latex]\\begin{cases}x\\left(t\\right)=\\cos t+4\\\\ y\\left(t\\right)=2{\\sin }^{2}t\\end{cases}[\/latex]<\/p>\n<p>23. [latex]\\begin{cases}x\\left(t\\right)=t - 1\\\\ y\\left(t\\right)={t}^{2}\\end{cases}[\/latex]<\/p>\n<p>24.\u00a0[latex]\\begin{cases}x\\left(t\\right)=-t\\\\ y\\left(t\\right)={t}^{3}+1\\end{cases}[\/latex]<\/p>\n<p>25. [latex]\\begin{cases}x\\left(t\\right)=2t - 1\\\\ y\\left(t\\right)={t}^{3}-2\\end{cases}[\/latex]<\/p>\n<p>For the following exercises, rewrite the parametric equation as a Cartesian equation by building an [latex]x\\text{-}y[\/latex] table.<\/p>\n<p>26. [latex]\\begin{cases}x\\left(t\\right)=2t - 1\\\\ y\\left(t\\right)=t+4\\end{cases}[\/latex]<\/p>\n<p>27. [latex]\\begin{cases}x\\left(t\\right)=4-t\\\\ y\\left(t\\right)=3t+2\\end{cases}[\/latex]<\/p>\n<p>28.\u00a0[latex]\\begin{cases}x\\left(t\\right)=2t - 1\\\\ y\\left(t\\right)=5t\\end{cases}[\/latex]<\/p>\n<p>29. [latex]\\begin{cases}x\\left(t\\right)=4t - 1\\\\ y\\left(t\\right)=4t+2\\end{cases}[\/latex]<\/p>\n<p>For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting [latex]x\\left(t\\right)=t[\/latex] or by setting [latex]y\\left(t\\right)=t[\/latex].<\/p>\n<p>30. [latex]y\\left(x\\right)=3{x}^{2}+3[\/latex]<\/p>\n<p>31. [latex]y\\left(x\\right)=2\\sin x+1[\/latex]<\/p>\n<p>32.\u00a0[latex]x\\left(y\\right)=3\\mathrm{log}\\left(y\\right)+y[\/latex]<\/p>\n<p>33. [latex]x\\left(y\\right)=\\sqrt{y}+2y[\/latex]<\/p>\n<p>For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using [latex]x\\left(t\\right)=a\\cos t[\/latex] and [latex]y\\left(t\\right)=b\\sin t[\/latex]. Identify the curve.<\/p>\n<p>34. [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n<p>35. [latex]\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{36}=1[\/latex]<\/p>\n<p>36.\u00a0[latex]{x}^{2}+{y}^{2}=16[\/latex]<\/p>\n<p>37. [latex]{x}^{2}+{y}^{2}=10[\/latex]<\/p>\n<p>38.\u00a0Parameterize the line from [latex]\\left(3,0\\right)[\/latex] to [latex]\\left(-2,-5\\right)[\/latex] so that the line is at [latex]\\left(3,0\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(-2,-5\\right)[\/latex] at [latex]t=1[\/latex].<\/p>\n<p>39. Parameterize the line from [latex]\\left(-1,0\\right)[\/latex] to [latex]\\left(3,-2\\right)[\/latex] so that the line is at [latex]\\left(-1,0\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(3,-2\\right)[\/latex] at [latex]t=1[\/latex].<\/p>\n<p>40.\u00a0Parameterize the line from [latex]\\left(-1,5\\right)[\/latex] to [latex]\\left(2,3\\right)[\/latex] so that the line is at [latex]\\left(-1,5\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(2,3\\right)[\/latex] at [latex]t=1[\/latex].<\/p>\n<p>41. Parameterize the line from [latex]\\left(4,1\\right)[\/latex] to [latex]\\left(6,-2\\right)[\/latex] so that the line is at [latex]\\left(4,1\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(6,-2\\right)[\/latex] at [latex]t=1[\/latex].<\/p>\n<p>For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.<\/p>\n<p>42. [latex]\\begin{cases}{x}_{1}\\left(t\\right)=3t\\hfill \\\\ {y}_{1}\\left(t\\right)=2t - 1\\hfill \\end{cases}\\text{ and }\\begin{cases}{x}_{2}\\left(t\\right)=t+3\\hfill \\\\ {y}_{2}\\left(t\\right)=4t - 4\\hfill \\end{cases}[\/latex]<\/p>\n<p>43. [latex]\\begin{cases}{x}_{1}\\left(t\\right)={t}^{2}\\hfill \\\\ {y}_{1}\\left(t\\right)=2t - 1\\hfill \\end{cases}\\text{ and }\\begin{cases}{x}_{2}\\left(t\\right)=-t+6\\hfill \\\\ {y}_{2}\\left(t\\right)=t+1\\hfill \\end{cases}[\/latex]<\/p>\n<p>For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.<\/p>\n<p>44. [latex]\\begin{cases}{x}_{1}\\left(t\\right)=3{t}^{2}-3t+7\\hfill \\\\ {y}_{1}\\left(t\\right)=2t+3\\hfill \\end{cases}[\/latex]<\/p>\n<table id=\"fs-id1165137565962\" class=\"unnumbered\" summary=\"Four rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains -1, 0, 1. The rest of the values in columns x and y are blank.\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u20131<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>45. [latex]\\begin{cases}{x}_{1}\\left(t\\right)={t}^{2}-4\\hfill \\\\ {y}_{1}\\left(t\\right)=2{t}^{2}-1\\hfill \\end{cases}[\/latex]<\/p>\n<table id=\"fs-id1165135407032\" class=\"unnumbered\" summary=\"Four rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains 1, 2, 3. The rest of the values in columns x and y are blank.\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>46. [latex]\\begin{cases}{x}_{1}\\left(t\\right)={t}^{4}\\hfill \\\\ {y}_{1}\\left(t\\right)={t}^{3}+4\\hfill \\end{cases}[\/latex]<\/p>\n<table id=\"fs-id1165137921660\" class=\"unnumbered\" summary=\"Five rows and 3 columns. First column is labeled t, second is labeled x, and third is labeled y. The first column contains -1, 0, 1, 2. The rest of the values in columns x and y are blank.\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>-1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>47.\u00a0Find two different sets of parametric equations for [latex]y={\\left(x+1\\right)}^{2}[\/latex].<\/p>\n<p>48.\u00a0Find two different sets of parametric equations for [latex]y=3x - 2[\/latex].<\/p>\n<p>49. Find two different sets of parametric equations for [latex]y={x}^{2}-4x+4[\/latex].<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15747\">\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>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <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>. <strong>License Terms<\/strong>: Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/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":169554,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15747","chapter","type-chapter","status-publish","hentry"],"part":14256,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15747","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15747\/revisions"}],"predecessor-version":[{"id":15748,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15747\/revisions\/15748"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/14256"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15747\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=15747"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=15747"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=15747"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=15747"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}