{"id":1424,"date":"2023-06-05T14:51:36","date_gmt":"2023-06-05T14:51:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-parametric-equations\/"},"modified":"2023-06-05T14:51:36","modified_gmt":"2023-06-05T14:51:36","slug":"solutions-for-parametric-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-parametric-equations\/","title":{"raw":"Solutions 62: Parametric Equations","rendered":"Solutions 62: Parametric Equations"},"content":{"raw":"\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\n1.&nbsp;A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, [latex]x=f\\left(t\\right)[\/latex] and [latex]y=f\\left(t\\right)[\/latex].\n\n3.&nbsp;Choose one equation to solve for [latex]t[\/latex], substitute into the other equation and simplify.\n\n5.&nbsp;Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.\n\n7.&nbsp;[latex]y=-2+2x[\/latex]\n\n9.&nbsp;[latex]y=3\\sqrt{\\frac{x - 1}{2}}[\/latex]\n\n11.&nbsp;[latex]x=2{e}^{\\frac{1-y}{5}}[\/latex] or [latex]y=1 - 5ln\\left(\\frac{x}{2}\\right)[\/latex]\n\n13.&nbsp;[latex]x=4\\mathrm{log}\\left(\\frac{y - 3}{2}\\right)[\/latex]\n\n15.&nbsp;[latex]x={\\left(\\frac{y}{2}\\right)}^{3}-\\frac{y}{2}[\/latex]\n\n17.&nbsp;[latex]y={x}^{3}[\/latex]\n\n19.&nbsp;[latex]{\\left(\\frac{x}{4}\\right)}^{2}+{\\left(\\frac{y}{5}\\right)}^{2}=1[\/latex]\n\n21.&nbsp;[latex]{y}^{2}=1-\\frac{1}{2}x[\/latex]\n\n23.&nbsp;[latex]y={x}^{2}+2x+1[\/latex]\n\n25.&nbsp;[latex]y={\\left(\\frac{x+1}{2}\\right)}^{3}-2[\/latex]\n\n27.&nbsp;[latex]y=-3x+14[\/latex]\n\n29.&nbsp;[latex]y=x+3[\/latex]\n\n31.&nbsp;[latex]\\begin{array}{l}x\\left(t\\right)=t\\hfill \\\\ y\\left(t\\right)=2\\sin t+1\\hfill \\end{array}[\/latex]\n\n33.&nbsp;[latex]\\begin{array}{l}x\\left(t\\right)=\\sqrt{t}+2t\\hfill \\\\ y\\left(t\\right)=t\\hfill \\end{array}[\/latex]\n\n35.&nbsp;[latex]\\begin{array}{l}x\\left(t\\right)=4\\cos t\\hfill \\\\ y\\left(t\\right)=6\\sin t\\hfill \\end{array}[\/latex]; Ellipse\n\n37.&nbsp;[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\n\n39.&nbsp;[latex]\\begin{array}{l}x\\left(t\\right)=-1+4t\\hfill \\\\ y\\left(t\\right)=-2t\\hfill \\end{array}[\/latex]\n\n41.&nbsp;[latex]\\begin{array}{l}x\\left(t\\right)=4+2t\\hfill \\\\ y\\left(t\\right)=1 - 3t\\hfill \\end{array}[\/latex]\n\n43.&nbsp;yes, at [latex]t=2[\/latex]\n\n45.\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>\n47.&nbsp;answers 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]\n\n49.&nbsp;answers 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]\n","rendered":"<h2>Solutions to Odd-Numbered Exercises<\/h2>\n<p>1.&nbsp;A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, [latex]x=f\\left(t\\right)[\/latex] and [latex]y=f\\left(t\\right)[\/latex].<\/p>\n<p>3.&nbsp;Choose one equation to solve for [latex]t[\/latex], substitute into the other equation and simplify.<\/p>\n<p>5.&nbsp;Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.<\/p>\n<p>7.&nbsp;[latex]y=-2+2x[\/latex]<\/p>\n<p>9.&nbsp;[latex]y=3\\sqrt{\\frac{x - 1}{2}}[\/latex]<\/p>\n<p>11.&nbsp;[latex]x=2{e}^{\\frac{1-y}{5}}[\/latex] or [latex]y=1 - 5ln\\left(\\frac{x}{2}\\right)[\/latex]<\/p>\n<p>13.&nbsp;[latex]x=4\\mathrm{log}\\left(\\frac{y - 3}{2}\\right)[\/latex]<\/p>\n<p>15.&nbsp;[latex]x={\\left(\\frac{y}{2}\\right)}^{3}-\\frac{y}{2}[\/latex]<\/p>\n<p>17.&nbsp;[latex]y={x}^{3}[\/latex]<\/p>\n<p>19.&nbsp;[latex]{\\left(\\frac{x}{4}\\right)}^{2}+{\\left(\\frac{y}{5}\\right)}^{2}=1[\/latex]<\/p>\n<p>21.&nbsp;[latex]{y}^{2}=1-\\frac{1}{2}x[\/latex]<\/p>\n<p>23.&nbsp;[latex]y={x}^{2}+2x+1[\/latex]<\/p>\n<p>25.&nbsp;[latex]y={\\left(\\frac{x+1}{2}\\right)}^{3}-2[\/latex]<\/p>\n<p>27.&nbsp;[latex]y=-3x+14[\/latex]<\/p>\n<p>29.&nbsp;[latex]y=x+3[\/latex]<\/p>\n<p>31.&nbsp;[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.&nbsp;[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.&nbsp;[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.&nbsp;[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.&nbsp;[latex]\\begin{array}{l}x\\left(t\\right)=-1+4t\\hfill \\\\ y\\left(t\\right)=-2t\\hfill \\end{array}[\/latex]<\/p>\n<p>41.&nbsp;[latex]\\begin{array}{l}x\\left(t\\right)=4+2t\\hfill \\\\ y\\left(t\\right)=1 - 3t\\hfill \\end{array}[\/latex]<\/p>\n<p>43.&nbsp;yes, at [latex]t=2[\/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.&nbsp;answers 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.&nbsp;answers 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\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-1424\">\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":503070,"menu_order":20,"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-1424","chapter","type-chapter","status-publish","hentry"],"part":1404,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1424","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/users\/503070"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1424\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/parts\/1404"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1424\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/media?parent=1424"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapter-type?post=1424"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/contributor?post=1424"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/license?post=1424"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}