{"id":1420,"date":"2023-06-05T14:51:33","date_gmt":"2023-06-05T14:51:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-polar-coordinates-graphs\/"},"modified":"2023-06-05T14:51:33","modified_gmt":"2023-06-05T14:51:33","slug":"solutions-for-polar-coordinates-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-polar-coordinates-graphs\/","title":{"raw":"Solutions 60: Polar Coordinates: Graphs","rendered":"Solutions 60: Polar Coordinates: Graphs"},"content":{"raw":"\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\n1.&nbsp;Symmetry with respect to the polar axis is similar to symmetry about the [latex]x[\/latex] -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line [latex]\\theta =\\frac{\\pi }{2}[\/latex] is similar to symmetry about the [latex]y[\/latex] -axis.\n\n3.&nbsp;Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, lima\u00e7on, lemniscate, etc., then plot points at [latex]\\theta =0,\\frac{\\pi }{2},\\pi \\text{and }\\frac{3\\pi }{2}[\/latex], and sketch the graph.\n\n5.&nbsp;The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.\n\n7.&nbsp;symmetric with respect to the polar axis\n\n9.&nbsp;symmetric with respect to the polar axis, symmetric with respect to the line [latex]\\theta =\\frac{\\pi }{2}[\/latex], symmetric with respect to the pole\n\n11.&nbsp;no symmetry\n\n13.&nbsp;no symmetry\n\n15.&nbsp;symmetric with respect to the pole\n\n17.&nbsp;circle\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165607\/CNX_Precalc_Figure_08_04_202.jpg\" alt=\"Graph of given circle.\">\n\n19.&nbsp;cardioid\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165610\/CNX_Precalc_Figure_08_04_204.jpg\" alt=\"Graph of given cardioid.\">\n\n21.&nbsp;cardioid\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165613\/CNX_Precalc_Figure_08_04_206.jpg\" alt=\"Graph of given cardioid.\">\n\n23.&nbsp;one-loop\/dimpled lima\u00e7on\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165615\/CNX_Precalc_Figure_08_04_208.jpg\" alt=\"Graph of given one-loop\/dimpled lima\u00e7on \">\n\n25.&nbsp;one-loop\/dimpled lima\u00e7on\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165617\/CNX_Precalc_Figure_08_04_210.jpg\" alt=\"Graph of given one-loop\/dimpled lima\u00e7on \">\n\n27.&nbsp;inner loop\/two-loop lima\u00e7on\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165620\/CNX_Precalc_Figure_08_04_212.jpg\" alt=\"Graph of given inner loop\/two-loop lima\u00e7on \">\n\n29.&nbsp;inner loop\/two-loop lima\u00e7on\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165622\/CNX_Precalc_Figure_08_04_214.jpg\" alt=\"Graph of given inner loop\/two-loop lima\u00e7on \">\n\n31.&nbsp;inner loop\/two-loop lima\u00e7on\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165624\/CNX_Precalc_Figure_08_04_216.jpg\" alt=\"Graph of given inner loop\/two-loop lima\u00e7on \">\n\n33.&nbsp;lemniscate\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165627\/CNX_Precalc_Figure_08_04_218.jpg\" alt=\"Graph of given lemniscate (along horizontal axis)\">\n\n35.&nbsp;lemniscate\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165629\/CNX_Precalc_Figure_08_04_220.jpg\" alt=\"Graph of given lemniscate (along y=x)\">\n\n37.&nbsp;rose curve\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165631\/CNX_Precalc_Figure_08_04_222.jpg\" alt=\"Graph of given rose curve - four petals.\">\n\n39.&nbsp;rose curve\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165634\/CNX_Precalc_Figure_08_04_224.jpg\" alt=\"Graph of given rose curve - eight petals.\">\n\n41.&nbsp;Archimedes\u2019 spiral\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165636\/CNX_Precalc_Figure_08_04_226.jpg\" alt=\"Graph of given Archimedes' spiral\">\n\n43.&nbsp;Archimedes\u2019 spiral\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165639\/CNX_Precalc_Figure_08_04_228.jpg\" alt=\"Graph of given Archimedes' spiral\">\n\n45.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165641\/CNX_Precalc_Figure_08_04_231.jpg\" alt=\"Graph of given equation.\">\n\n47.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165643\/CNX_Precalc_Figure_08_04_233.jpg\" alt=\"Graph of given hippopede (two circles that are centered along the x-axis and meet at the origin)\">\n\n49.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165645\/CNX_Precalc_Figure_08_04_235.jpg\" alt=\"Graph of given equation.\">\n\n51.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165648\/CNX_Precalc_Figure_08_04_237.jpg\" alt=\"Graph of given equation. Similar to original Archimedes' spiral.\">\n\n53.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165651\/CNX_Precalc_Figure_08_04_239.jpg\" alt=\"Graph of given equation.\">\n\n55.&nbsp;They are both spirals, but not quite the same.\n\n57.&nbsp;Both graphs are curves with 2 loops. The equation with a coefficient of [latex]\\theta [\/latex] has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to [latex]4\\pi [\/latex] to get a better picture.\n\n59.&nbsp;When the width of the domain is increased, more petals of the flower are visible.\n\n61.&nbsp;The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve\u2019s distance from the pole.\n\n63.&nbsp;The graphs are spirals. The smaller the coefficient, the tighter the spiral.\n\n65.&nbsp;[latex]\\left(4,\\frac{\\pi }{3}\\right),\\left(4,\\frac{5\\pi }{3}\\right)[\/latex]\n\n67.&nbsp;[latex]\\left(\\frac{3}{2},\\frac{\\pi }{3}\\right),\\left(\\frac{3}{2},\\frac{5\\pi }{3}\\right)[\/latex]\n\n69.&nbsp;[latex]\\left(0,\\frac{\\pi }{2}\\right),\\left(0,\\pi \\right),\\left(0,\\frac{3\\pi }{2}\\right),\\left(0,2\\pi \\right)[\/latex]\n\n71.&nbsp;[latex]\\left(\\frac{\\sqrt[4]{8}}{2},\\frac{\\pi }{4}\\right),\\left(\\frac{\\sqrt[4]{8}}{2},\\frac{5\\pi }{4}\\right)[\/latex]\nand at [latex]\\theta =\\frac{3\\pi }{4},\\frac{7\\pi }{4}[\/latex]&nbsp;since [latex]r[\/latex] is squared\n","rendered":"<h2>Solutions to Odd-Numbered Exercises<\/h2>\n<p>1.&nbsp;Symmetry with respect to the polar axis is similar to symmetry about the [latex]x[\/latex] -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line [latex]\\theta =\\frac{\\pi }{2}[\/latex] is similar to symmetry about the [latex]y[\/latex] -axis.<\/p>\n<p>3.&nbsp;Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, lima\u00e7on, lemniscate, etc., then plot points at [latex]\\theta =0,\\frac{\\pi }{2},\\pi \\text{and }\\frac{3\\pi }{2}[\/latex], and sketch the graph.<\/p>\n<p>5.&nbsp;The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.<\/p>\n<p>7.&nbsp;symmetric with respect to the polar axis<\/p>\n<p>9.&nbsp;symmetric with respect to the polar axis, symmetric with respect to the line [latex]\\theta =\\frac{\\pi }{2}[\/latex], symmetric with respect to the pole<\/p>\n<p>11.&nbsp;no symmetry<\/p>\n<p>13.&nbsp;no symmetry<\/p>\n<p>15.&nbsp;symmetric with respect to the pole<\/p>\n<p>17.&nbsp;circle<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165607\/CNX_Precalc_Figure_08_04_202.jpg\" alt=\"Graph of given circle.\" \/><\/p>\n<p>19.&nbsp;cardioid<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165610\/CNX_Precalc_Figure_08_04_204.jpg\" alt=\"Graph of given cardioid.\" \/><\/p>\n<p>21.&nbsp;cardioid<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165613\/CNX_Precalc_Figure_08_04_206.jpg\" alt=\"Graph of given cardioid.\" \/><\/p>\n<p>23.&nbsp;one-loop\/dimpled lima\u00e7on<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165615\/CNX_Precalc_Figure_08_04_208.jpg\" alt=\"Graph of given one-loop\/dimpled lima\u00e7on\" \/><\/p>\n<p>25.&nbsp;one-loop\/dimpled lima\u00e7on<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165617\/CNX_Precalc_Figure_08_04_210.jpg\" alt=\"Graph of given one-loop\/dimpled lima\u00e7on\" \/><\/p>\n<p>27.&nbsp;inner loop\/two-loop lima\u00e7on<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165620\/CNX_Precalc_Figure_08_04_212.jpg\" alt=\"Graph of given inner loop\/two-loop lima\u00e7on\" \/><\/p>\n<p>29.&nbsp;inner loop\/two-loop lima\u00e7on<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165622\/CNX_Precalc_Figure_08_04_214.jpg\" alt=\"Graph of given inner loop\/two-loop lima\u00e7on\" \/><\/p>\n<p>31.&nbsp;inner loop\/two-loop lima\u00e7on<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165624\/CNX_Precalc_Figure_08_04_216.jpg\" alt=\"Graph of given inner loop\/two-loop lima\u00e7on\" \/><\/p>\n<p>33.&nbsp;lemniscate<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165627\/CNX_Precalc_Figure_08_04_218.jpg\" alt=\"Graph of given lemniscate (along horizontal axis)\" \/><\/p>\n<p>35.&nbsp;lemniscate<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165629\/CNX_Precalc_Figure_08_04_220.jpg\" alt=\"Graph of given lemniscate (along y=x)\" \/><\/p>\n<p>37.&nbsp;rose curve<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165631\/CNX_Precalc_Figure_08_04_222.jpg\" alt=\"Graph of given rose curve - four petals.\" \/><\/p>\n<p>39.&nbsp;rose curve<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165634\/CNX_Precalc_Figure_08_04_224.jpg\" alt=\"Graph of given rose curve - eight petals.\" \/><\/p>\n<p>41.&nbsp;Archimedes\u2019 spiral<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165636\/CNX_Precalc_Figure_08_04_226.jpg\" alt=\"Graph of given Archimedes' spiral\" \/><\/p>\n<p>43.&nbsp;Archimedes\u2019 spiral<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165639\/CNX_Precalc_Figure_08_04_228.jpg\" alt=\"Graph of given Archimedes' spiral\" \/><\/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\/27165641\/CNX_Precalc_Figure_08_04_231.jpg\" alt=\"Graph of given equation.\" \/><\/p>\n<p>47.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165643\/CNX_Precalc_Figure_08_04_233.jpg\" alt=\"Graph of given hippopede (two circles that are centered along the x-axis and meet at the origin)\" \/><\/p>\n<p>49.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165645\/CNX_Precalc_Figure_08_04_235.jpg\" alt=\"Graph of given equation.\" \/><\/p>\n<p>51.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165648\/CNX_Precalc_Figure_08_04_237.jpg\" alt=\"Graph of given equation. Similar to original Archimedes' spiral.\" \/><\/p>\n<p>53.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165651\/CNX_Precalc_Figure_08_04_239.jpg\" alt=\"Graph of given equation.\" \/><\/p>\n<p>55.&nbsp;They are both spirals, but not quite the same.<\/p>\n<p>57.&nbsp;Both graphs are curves with 2 loops. The equation with a coefficient of [latex]\\theta[\/latex] has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to [latex]4\\pi[\/latex] to get a better picture.<\/p>\n<p>59.&nbsp;When the width of the domain is increased, more petals of the flower are visible.<\/p>\n<p>61.&nbsp;The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve\u2019s distance from the pole.<\/p>\n<p>63.&nbsp;The graphs are spirals. The smaller the coefficient, the tighter the spiral.<\/p>\n<p>65.&nbsp;[latex]\\left(4,\\frac{\\pi }{3}\\right),\\left(4,\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\n<p>67.&nbsp;[latex]\\left(\\frac{3}{2},\\frac{\\pi }{3}\\right),\\left(\\frac{3}{2},\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\n<p>69.&nbsp;[latex]\\left(0,\\frac{\\pi }{2}\\right),\\left(0,\\pi \\right),\\left(0,\\frac{3\\pi }{2}\\right),\\left(0,2\\pi \\right)[\/latex]<\/p>\n<p>71.&nbsp;[latex]\\left(\\frac{\\sqrt[4]{8}}{2},\\frac{\\pi }{4}\\right),\\left(\\frac{\\sqrt[4]{8}}{2},\\frac{5\\pi }{4}\\right)[\/latex]<br \/>\nand at [latex]\\theta =\\frac{3\\pi }{4},\\frac{7\\pi }{4}[\/latex]&nbsp;since [latex]r[\/latex] is squared<\/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-1420\">\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":16,"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-1420","chapter","type-chapter","status-publish","hentry"],"part":1404,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1420","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\/1420\/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\/1420\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/media?parent=1420"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapter-type?post=1420"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/contributor?post=1420"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/license?post=1420"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}