{"id":348,"date":"2026-02-16T20:57:01","date_gmt":"2026-02-16T20:57:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&#038;p=348"},"modified":"2026-02-18T20:14:09","modified_gmt":"2026-02-18T20:14:09","slug":"parametric-functions-and-vectors-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/parametric-functions-and-vectors-get-stronger-answer-key\/","title":{"raw":"Parametric Functions and Vectors: Get Stronger Answer Key","rendered":"Parametric Functions and Vectors: Get Stronger Answer Key"},"content":{"raw":"<h1>Parametric Equations<\/h1>\r\n1.\u00a0A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, [latex]x=f\\left(t\\right)[\/latex] and [latex]y=f\\left(t\\right)[\/latex].\r\n\r\n3.\u00a0Choose one equation to solve for [latex]t[\/latex], substitute into the other equation and simplify.\r\n\r\n7.\u00a0[latex]y=-2+2x[\/latex]\r\n\r\n9.\u00a0[latex]y=3\\sqrt{\\frac{x - 1}{2}}[\/latex]\r\n\r\n11.\u00a0[latex]x=2{e}^{\\frac{1-y}{5}}[\/latex] or [latex]y=1 - 5ln\\left(\\frac{x}{2}\\right)[\/latex]\r\n\r\n13.\u00a0[latex]x=4\\mathrm{log}\\left(\\frac{y - 3}{2}\\right)[\/latex]\r\n\r\n15.\u00a0[latex]x={\\left(\\frac{y}{2}\\right)}^{3}-\\frac{y}{2}[\/latex]\r\n\r\n17.\u00a0[latex]y={x}^{3}[\/latex]\r\n\r\n19.\u00a0[latex]{\\left(\\frac{x}{4}\\right)}^{2}+{\\left(\\frac{y}{5}\\right)}^{2}=1[\/latex]\r\n\r\n21.\u00a0[latex]{y}^{2}=1-\\frac{1}{2}x[\/latex]\r\n\r\n23.\u00a0[latex]y={x}^{2}+2x+1[\/latex]\r\n\r\n25.\u00a0[latex]y={\\left(\\frac{x+1}{2}\\right)}^{3}-2[\/latex]\r\n\r\n27.\u00a0[latex]y=-3x+14[\/latex]\r\n\r\n29.\u00a0[latex]y=x+3[\/latex]\r\n\r\n31.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=t\\hfill \\\\ y\\left(t\\right)=2\\sin t+1\\hfill \\end{array}[\/latex]\r\n\r\n33.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=\\sqrt{t}+2t\\hfill \\\\ y\\left(t\\right)=t\\hfill \\end{array}[\/latex]\r\n\r\n35.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=4\\cos t\\hfill \\\\ y\\left(t\\right)=6\\sin t\\hfill \\end{array}[\/latex]; Ellipse\r\n\r\n37.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=\\sqrt{10}\\cos t\\hfill \\\\ y\\left(t\\right)=\\sqrt{10}\\sin t\\hfill \\end{array}[\/latex]; Circle\r\n\r\n39.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=-1+4t\\hfill \\\\ y\\left(t\\right)=-2t\\hfill \\end{array}[\/latex]\r\n\r\n41.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=4+2t\\hfill \\\\ y\\left(t\\right)=1 - 3t\\hfill \\end{array}[\/latex]\r\n\r\n45.\r\n<table id=\"fs-id1165137737864\" class=\"unnumbered\" summary=\"..\">\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>-3<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>0<\/td>\r\n<td>7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>5<\/td>\r\n<td>17<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n47.\u00a0answers may vary: [latex]\\begin{array}{l}x\\left(t\\right)=t - 1\\hfill \\\\ y\\left(t\\right)={t}^{2}\\hfill \\end{array}\\text{ and }\\begin{array}{l}x\\left(t\\right)=t+1\\hfill \\\\ y\\left(t\\right)={\\left(t+2\\right)}^{2}\\hfill \\end{array}[\/latex]\r\n\r\n49.\u00a0answers may vary: , [latex]\\begin{array}{l}x\\left(t\\right)=t\\hfill \\\\ y\\left(t\\right)={t}^{2}-4t+4\\hfill \\end{array}\\text{ and }\\begin{array}{l}x\\left(t\\right)=t+2\\hfill \\\\ y\\left(t\\right)={t}^{2}\\hfill \\end{array}[\/latex]\r\n<h1>Parametric Equations: Graphs<\/h1>\r\n1.\u00a0plotting points with the orientation arrow and a graphing calculator\r\n\r\n3.\u00a0The arrows show the orientation, the direction of motion according to increasing values of [latex]t[\/latex].\r\n\r\n5.\u00a0The parametric equations show the different vertical and horizontal motions over time.\r\n\r\n7.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181019\/CNX_Precalc_Figure_08_07_2022.jpg\" alt=\"Graph of the given equations - looks like an upward opening parabola.\" \/>\r\n\r\n9.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181022\/CNX_Precalc_Figure_08_07_2042.jpg\" alt=\"Graph of the given equations - a line, negative slope.\" \/>\r\n\r\n11.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181024\/CNX_Precalc_Figure_08_07_2062.jpg\" alt=\"Graph of the given equations - looks like a sideways parabola, opening to the right.\" \/>\r\n\r\n13.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181026\/CNX_Precalc_Figure_08_07_2082.jpg\" alt=\"Graph of the given equations - looks like the left half of an upward opening parabola.\" \/>\r\n\r\n15.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181028\/CNX_Precalc_Figure_08_07_2102.jpg\" alt=\"Graph of the given equations - looks like a downward opening absolute value function.\" \/>\r\n\r\n17.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181031\/CNX_Precalc_Figure_08_07_2122.jpg\" alt=\"Graph of the given equations - a vertical ellipse.\" \/>\r\n\r\n19.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181034\/CNX_Precalc_Figure_08_07_2142.jpg\" alt=\"Graph of the given equations- line from (0, -3) to (3,0). It is traversed in both directions, positive and negative slope.\" \/>\r\n\r\n21.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181037\/CNX_Precalc_Figure_08_07_2162.jpg\" alt=\"Graph of the given equations- looks like an upward opening parabola.\" \/>\r\n\r\n23.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181039\/CNX_Precalc_Figure_08_07_2182.jpg\" alt=\"Graph of the given equations- looks like a downward opening parabola.\" \/>\r\n\r\n25.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181041\/CNX_Precalc_Figure_08_07_2202.jpg\" alt=\"Graph of the given equations- horizontal ellipse.\" \/>\r\n\r\n27.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181043\/CNX_Precalc_Figure_08_07_2222.jpg\" alt=\"Graph of the given equations- looks like the lower half of a sideways parabola opening to the right\" \/>\r\n\r\n29.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181045\/CNX_Precalc_Figure_08_07_2242.jpg\" alt=\"Graph of the given equations- looks like an upwards opening parabola\" \/>\r\n\r\n31.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181048\/CNX_Precalc_Figure_08_07_2262.jpg\" alt=\"Graph of the given equations- looks like the upper half of a sideways parabola opening to the left\" \/>\r\n\r\n41. Take the opposite of the [latex]x\\left(t\\right)[\/latex] equation.\r\n\r\n63.\u00a0[latex]y\\left(x\\right)=-16{\\left(\\frac{x}{15}\\right)}^{2}+20\\left(\\frac{x}{15}\\right)[\/latex]\r\n\r\n65.\u00a0[latex]\\begin{cases}x\\left(t\\right)=64t\\cos \\left(52^\\circ \\right)\\\\ y\\left(t\\right)=-16{t}^{2}+64t\\sin \\left(52^\\circ \\right)\\end{cases}[\/latex]\r\n\r\n67.\u00a0approximately 3.2 seconds\r\n\r\n69.\u00a01.6 seconds\r\n<h1>Vectors<\/h1>\r\n3.\u00a0They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.\r\n\r\n4. Component form is a way to write a vector using its horizontal and vertical distances.\r\n\r\n5.\u00a0The first number always represents the coefficient of the <strong><em>i<\/em><\/strong>, and the second represents the <em><strong>j<\/strong><\/em>.\r\n\r\n7.\u00a0[latex]\\langle 7,\u22125\\rangle[\/latex]\r\n\r\n9. not equal\r\n\r\n11. equal\r\n\r\n13. equal\r\n\r\n15. [latex]7\\boldsymbol{i}\u22123\\boldsymbol{j}[\/latex]\r\n\r\n17. [latex]\u22126\\boldsymbol{i}\u22122\\boldsymbol{j}[\/latex]\r\n\r\n19. [latex]\\boldsymbol{u}+\\boldsymbol{v}=\\langle\u22125,5\\rangle,\\boldsymbol{u}\u2212\\boldsymbol{v}=\\langle\u22121,3\\rangle,2\\boldsymbol{u}\u22123\\boldsymbol{v}=\\langle 0,5\\rangle[\/latex]\r\n\r\n21. [latex]\u221210\\boldsymbol{i}\u20134\\boldsymbol{j}[\/latex]\r\n\r\n23. [latex]\u2212\\frac{2\\sqrt{29}}{29}\\boldsymbol{i}+\\frac{5\\sqrt{29}}{29}\\boldsymbol{j}[\/latex]\r\n\r\n25. [latex]\u2013\\frac{2\\sqrt{229}}{229}\\boldsymbol{i}+\\frac{15\\sqrt{229}}{229}\\boldsymbol{j}[\/latex]\r\n\r\n27. [latex]\u2013\\frac{7\\sqrt{2}}\\boldsymbol{i}+\\frac{\\sqrt{2}}{10}\\boldsymbol{j}[\/latex]\r\n\r\n29. [latex]|\\boldsymbol{v}|=7.810,\\theta=39.806^{\\circ}[\/latex]\r\n\r\n31. [latex]|\\boldsymbol{v}|=7.211,\\theta=236.310^{\\circ}[\/latex]\r\n\r\n33.\u00a0\u22126\r\n\r\n35.\u00a0\u221212\r\n\r\n37.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181238\/CNX_Precalc_Figure_08_08_253.jpg\" alt=\"\" \/>\r\n\r\n39.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181240\/CNX_Precalc_Figure_08_08_205.jpg\" alt=\"Plot of u+v, u-v, and 2u based on the above vectors. In relation to the same origin point, u+v goes to (0,3), u-v goes to (2,-1), and 2u goes to (2,2).\" \/>\r\n\r\n41.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181242\/CNX_Precalc_Figure_08_08_209.jpg\" alt=\"Plot of vectors u+v, u-v, and 2u based on the above vectors.Given that u's start point was the origin, u+v starts at the origin and goes to (2,-3); u-v starts at the origin and goes to (4,-1); 2u goes from the origin to (6,-4).\" \/>\r\n\r\n43.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181245\/CNX_Precalc_Figure_08_08_213.jpg\" alt=\"Plot of a single vector. Taking the start point of the vector as (0,0) from the above set up, the vector goes from the origin to (-1,-6).\" \/>\r\n\r\n45.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181247\/CNX_Precalc_Figure_08_08_217.jpg\" alt=\"Vector extending from the origin to (7,5), taking the base as the origin.\" \/>\r\n\r\n47. [latex]\\langle 4,1\\rangle[\/latex]\r\n\r\n49. [latex]\\boldsymbol{v}=\u22127\\boldsymbol{i}+3\\boldsymbol{j}[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181250\/CNX_Precalc_Figure_08_08_221.jpg\" alt=\"Vector going from (4,-1) to (-3,2).\" \/>\r\n\r\n51. [latex]3\\sqrt{2}\\boldsymbol{i}+3\\sqrt{2}\\boldsymbol{j}[\/latex]\r\n\r\n53. [latex]\\boldsymbol{i}\u2212\\sqrt{3}\\boldsymbol{j}[\/latex]\r\n\r\n55. a. 58.7; b. 12.5\r\n\r\n57. [latex]x=7.13[\/latex] pounds, [latex]y=3.63[\/latex] pounds\r\n\r\n59.\u00a0[latex]x=2.87[\/latex] pounds, [latex]y=4.10[\/latex] pounds\r\n\r\n61. 4.635 miles, [latex]17.764^{\\circ}[\/latex] N of E\r\n\r\n63.\u00a017 miles. 10.318 miles\r\n\r\n65.\u00a0Distance: 2.868. Direction: [latex]86.474^{\\circ}[\/latex] North of West, or [latex]3.526^{\\circ}[\/latex] West of North\r\n\r\n67. [latex]4.924^{\\circ}[\/latex]. 659 km\/hr\r\n\r\n69. [latex]4.424^{\\circ}[\/latex]\r\n\r\n71. (0.081, 8.602)\r\n\r\n73. [latex]21.801^{\\circ}[\/latex], relative to the car\u2019s forward direction\r\n\r\n75.\u00a0parallel: 16.28, perpendicular: 47.28 pounds\r\n\r\n77.\u00a019.35 pounds, [latex]231.54^{\\circ}[\/latex] from the horizontal\r\n\r\n79.\u00a05.1583 pounds, [latex]75.8^{\\circ}[\/latex] from the horizontal","rendered":"<h1>Parametric Equations<\/h1>\n<p>1.\u00a0A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, [latex]x=f\\left(t\\right)[\/latex] and [latex]y=f\\left(t\\right)[\/latex].<\/p>\n<p>3.\u00a0Choose one equation to solve for [latex]t[\/latex], substitute into the other equation and simplify.<\/p>\n<p>7.\u00a0[latex]y=-2+2x[\/latex]<\/p>\n<p>9.\u00a0[latex]y=3\\sqrt{\\frac{x - 1}{2}}[\/latex]<\/p>\n<p>11.\u00a0[latex]x=2{e}^{\\frac{1-y}{5}}[\/latex] or [latex]y=1 - 5ln\\left(\\frac{x}{2}\\right)[\/latex]<\/p>\n<p>13.\u00a0[latex]x=4\\mathrm{log}\\left(\\frac{y - 3}{2}\\right)[\/latex]<\/p>\n<p>15.\u00a0[latex]x={\\left(\\frac{y}{2}\\right)}^{3}-\\frac{y}{2}[\/latex]<\/p>\n<p>17.\u00a0[latex]y={x}^{3}[\/latex]<\/p>\n<p>19.\u00a0[latex]{\\left(\\frac{x}{4}\\right)}^{2}+{\\left(\\frac{y}{5}\\right)}^{2}=1[\/latex]<\/p>\n<p>21.\u00a0[latex]{y}^{2}=1-\\frac{1}{2}x[\/latex]<\/p>\n<p>23.\u00a0[latex]y={x}^{2}+2x+1[\/latex]<\/p>\n<p>25.\u00a0[latex]y={\\left(\\frac{x+1}{2}\\right)}^{3}-2[\/latex]<\/p>\n<p>27.\u00a0[latex]y=-3x+14[\/latex]<\/p>\n<p>29.\u00a0[latex]y=x+3[\/latex]<\/p>\n<p>31.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=t\\hfill \\\\ y\\left(t\\right)=2\\sin t+1\\hfill \\end{array}[\/latex]<\/p>\n<p>33.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=\\sqrt{t}+2t\\hfill \\\\ y\\left(t\\right)=t\\hfill \\end{array}[\/latex]<\/p>\n<p>35.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=4\\cos t\\hfill \\\\ y\\left(t\\right)=6\\sin t\\hfill \\end{array}[\/latex]; Ellipse<\/p>\n<p>37.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=\\sqrt{10}\\cos t\\hfill \\\\ y\\left(t\\right)=\\sqrt{10}\\sin t\\hfill \\end{array}[\/latex]; Circle<\/p>\n<p>39.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=-1+4t\\hfill \\\\ y\\left(t\\right)=-2t\\hfill \\end{array}[\/latex]<\/p>\n<p>41.\u00a0[latex]\\begin{array}{l}x\\left(t\\right)=4+2t\\hfill \\\\ y\\left(t\\right)=1 - 3t\\hfill \\end{array}[\/latex]<\/p>\n<p>45.<\/p>\n<table id=\"fs-id1165137737864\" class=\"unnumbered\" summary=\"..\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>-3<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>0<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>5<\/td>\n<td>17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>47.\u00a0answers may vary: [latex]\\begin{array}{l}x\\left(t\\right)=t - 1\\hfill \\\\ y\\left(t\\right)={t}^{2}\\hfill \\end{array}\\text{ and }\\begin{array}{l}x\\left(t\\right)=t+1\\hfill \\\\ y\\left(t\\right)={\\left(t+2\\right)}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<p>49.\u00a0answers may vary: , [latex]\\begin{array}{l}x\\left(t\\right)=t\\hfill \\\\ y\\left(t\\right)={t}^{2}-4t+4\\hfill \\end{array}\\text{ and }\\begin{array}{l}x\\left(t\\right)=t+2\\hfill \\\\ y\\left(t\\right)={t}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<h1>Parametric Equations: Graphs<\/h1>\n<p>1.\u00a0plotting points with the orientation arrow and a graphing calculator<\/p>\n<p>3.\u00a0The arrows show the orientation, the direction of motion according to increasing values of [latex]t[\/latex].<\/p>\n<p>5.\u00a0The parametric equations show the different vertical and horizontal motions over time.<\/p>\n<p>7.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181019\/CNX_Precalc_Figure_08_07_2022.jpg\" alt=\"Graph of the given equations - looks like an upward opening parabola.\" \/><\/p>\n<p>9.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181022\/CNX_Precalc_Figure_08_07_2042.jpg\" alt=\"Graph of the given equations - a line, negative slope.\" \/><\/p>\n<p>11.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181024\/CNX_Precalc_Figure_08_07_2062.jpg\" alt=\"Graph of the given equations - looks like a sideways parabola, opening to the right.\" \/><\/p>\n<p>13.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181026\/CNX_Precalc_Figure_08_07_2082.jpg\" alt=\"Graph of the given equations - looks like the left half of an upward opening parabola.\" \/><\/p>\n<p>15.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181028\/CNX_Precalc_Figure_08_07_2102.jpg\" alt=\"Graph of the given equations - looks like a downward opening absolute value function.\" \/><\/p>\n<p>17.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181031\/CNX_Precalc_Figure_08_07_2122.jpg\" alt=\"Graph of the given equations - a vertical ellipse.\" \/><\/p>\n<p>19.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181034\/CNX_Precalc_Figure_08_07_2142.jpg\" alt=\"Graph of the given equations- line from (0, -3) to (3,0). It is traversed in both directions, positive and negative slope.\" \/><\/p>\n<p>21.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181037\/CNX_Precalc_Figure_08_07_2162.jpg\" alt=\"Graph of the given equations- looks like an upward opening parabola.\" \/><\/p>\n<p>23.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181039\/CNX_Precalc_Figure_08_07_2182.jpg\" alt=\"Graph of the given equations- looks like a downward opening parabola.\" \/><\/p>\n<p>25.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181041\/CNX_Precalc_Figure_08_07_2202.jpg\" alt=\"Graph of the given equations- horizontal ellipse.\" \/><\/p>\n<p>27.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181043\/CNX_Precalc_Figure_08_07_2222.jpg\" alt=\"Graph of the given equations- looks like the lower half of a sideways parabola opening to the right\" \/><\/p>\n<p>29.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181045\/CNX_Precalc_Figure_08_07_2242.jpg\" alt=\"Graph of the given equations- looks like an upwards opening parabola\" \/><\/p>\n<p>31.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181048\/CNX_Precalc_Figure_08_07_2262.jpg\" alt=\"Graph of the given equations- looks like the upper half of a sideways parabola opening to the left\" \/><\/p>\n<p>41. Take the opposite of the [latex]x\\left(t\\right)[\/latex] equation.<\/p>\n<p>63.\u00a0[latex]y\\left(x\\right)=-16{\\left(\\frac{x}{15}\\right)}^{2}+20\\left(\\frac{x}{15}\\right)[\/latex]<\/p>\n<p>65.\u00a0[latex]\\begin{cases}x\\left(t\\right)=64t\\cos \\left(52^\\circ \\right)\\\\ y\\left(t\\right)=-16{t}^{2}+64t\\sin \\left(52^\\circ \\right)\\end{cases}[\/latex]<\/p>\n<p>67.\u00a0approximately 3.2 seconds<\/p>\n<p>69.\u00a01.6 seconds<\/p>\n<h1>Vectors<\/h1>\n<p>3.\u00a0They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.<\/p>\n<p>4. Component form is a way to write a vector using its horizontal and vertical distances.<\/p>\n<p>5.\u00a0The first number always represents the coefficient of the <strong><em>i<\/em><\/strong>, and the second represents the <em><strong>j<\/strong><\/em>.<\/p>\n<p>7.\u00a0[latex]\\langle 7,\u22125\\rangle[\/latex]<\/p>\n<p>9. not equal<\/p>\n<p>11. equal<\/p>\n<p>13. equal<\/p>\n<p>15. [latex]7\\boldsymbol{i}\u22123\\boldsymbol{j}[\/latex]<\/p>\n<p>17. [latex]\u22126\\boldsymbol{i}\u22122\\boldsymbol{j}[\/latex]<\/p>\n<p>19. [latex]\\boldsymbol{u}+\\boldsymbol{v}=\\langle\u22125,5\\rangle,\\boldsymbol{u}\u2212\\boldsymbol{v}=\\langle\u22121,3\\rangle,2\\boldsymbol{u}\u22123\\boldsymbol{v}=\\langle 0,5\\rangle[\/latex]<\/p>\n<p>21. [latex]\u221210\\boldsymbol{i}\u20134\\boldsymbol{j}[\/latex]<\/p>\n<p>23. [latex]\u2212\\frac{2\\sqrt{29}}{29}\\boldsymbol{i}+\\frac{5\\sqrt{29}}{29}\\boldsymbol{j}[\/latex]<\/p>\n<p>25. [latex]\u2013\\frac{2\\sqrt{229}}{229}\\boldsymbol{i}+\\frac{15\\sqrt{229}}{229}\\boldsymbol{j}[\/latex]<\/p>\n<p>27. [latex]\u2013\\frac{7\\sqrt{2}}\\boldsymbol{i}+\\frac{\\sqrt{2}}{10}\\boldsymbol{j}[\/latex]<\/p>\n<p>29. [latex]|\\boldsymbol{v}|=7.810,\\theta=39.806^{\\circ}[\/latex]<\/p>\n<p>31. [latex]|\\boldsymbol{v}|=7.211,\\theta=236.310^{\\circ}[\/latex]<\/p>\n<p>33.\u00a0\u22126<\/p>\n<p>35.\u00a0\u221212<\/p>\n<p>37.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181238\/CNX_Precalc_Figure_08_08_253.jpg\" alt=\"\" \/><\/p>\n<p>39.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181240\/CNX_Precalc_Figure_08_08_205.jpg\" alt=\"Plot of u+v, u-v, and 2u based on the above vectors. In relation to the same origin point, u+v goes to (0,3), u-v goes to (2,-1), and 2u goes to (2,2).\" \/><\/p>\n<p>41.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181242\/CNX_Precalc_Figure_08_08_209.jpg\" alt=\"Plot of vectors u+v, u-v, and 2u based on the above vectors.Given that u's start point was the origin, u+v starts at the origin and goes to (2,-3); u-v starts at the origin and goes to (4,-1); 2u goes from the origin to (6,-4).\" \/><\/p>\n<p>43.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181245\/CNX_Precalc_Figure_08_08_213.jpg\" alt=\"Plot of a single vector. Taking the start point of the vector as (0,0) from the above set up, the vector goes from the origin to (-1,-6).\" \/><\/p>\n<p>45.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181247\/CNX_Precalc_Figure_08_08_217.jpg\" alt=\"Vector extending from the origin to (7,5), taking the base as the origin.\" \/><\/p>\n<p>47. [latex]\\langle 4,1\\rangle[\/latex]<\/p>\n<p>49. [latex]\\boldsymbol{v}=\u22127\\boldsymbol{i}+3\\boldsymbol{j}[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181250\/CNX_Precalc_Figure_08_08_221.jpg\" alt=\"Vector going from (4,-1) to (-3,2).\" \/><\/p>\n<p>51. [latex]3\\sqrt{2}\\boldsymbol{i}+3\\sqrt{2}\\boldsymbol{j}[\/latex]<\/p>\n<p>53. [latex]\\boldsymbol{i}\u2212\\sqrt{3}\\boldsymbol{j}[\/latex]<\/p>\n<p>55. a. 58.7; b. 12.5<\/p>\n<p>57. [latex]x=7.13[\/latex] pounds, [latex]y=3.63[\/latex] pounds<\/p>\n<p>59.\u00a0[latex]x=2.87[\/latex] pounds, [latex]y=4.10[\/latex] pounds<\/p>\n<p>61. 4.635 miles, [latex]17.764^{\\circ}[\/latex] N of E<\/p>\n<p>63.\u00a017 miles. 10.318 miles<\/p>\n<p>65.\u00a0Distance: 2.868. Direction: [latex]86.474^{\\circ}[\/latex] North of West, or [latex]3.526^{\\circ}[\/latex] West of North<\/p>\n<p>67. [latex]4.924^{\\circ}[\/latex]. 659 km\/hr<\/p>\n<p>69. [latex]4.424^{\\circ}[\/latex]<\/p>\n<p>71. (0.081, 8.602)<\/p>\n<p>73. [latex]21.801^{\\circ}[\/latex], relative to the car\u2019s forward direction<\/p>\n<p>75.\u00a0parallel: 16.28, perpendicular: 47.28 pounds<\/p>\n<p>77.\u00a019.35 pounds, [latex]231.54^{\\circ}[\/latex] from the horizontal<\/p>\n<p>79.\u00a05.1583 pounds, [latex]75.8^{\\circ}[\/latex] from the horizontal<\/p>\n","protected":false},"author":13,"menu_order":18,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":224,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/348"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/348\/revisions"}],"predecessor-version":[{"id":369,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/348\/revisions\/369"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/224"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/348\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=348"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=348"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=348"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=348"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}