{"id":1915,"date":"2015-11-12T18:30:43","date_gmt":"2015-11-12T18:30:43","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1915"},"modified":"2015-11-12T18:30:43","modified_gmt":"2015-11-12T18:30:43","slug":"writing-equations-of-parabolas-in-standard-form","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/writing-equations-of-parabolas-in-standard-form\/","title":{"raw":"Writing Equations of Parabolas in Standard Form","rendered":"Writing Equations of Parabolas in Standard Form"},"content":{"raw":"<p>In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.\n<\/p><div class=\"textbox\">\n<h3>How To: Given its focus and directrix, write the equation for a parabola in standard form.<\/h3>\n<ul><li>Determine whether the axis of symmetry is the <em>x<\/em>- or <em>y<\/em>-axis.\n<ul><li>If the given coordinates of the focus have the form [latex]\\left(p,0\\right)[\/latex], then the axis of symmetry is the <em>x<\/em>-axis. Use the standard form [latex]{y}^{2}=4px[\/latex].<\/li>\n\t<li>If the given coordinates of the focus have the form [latex]\\left(0,p\\right)[\/latex], then the axis of symmetry is the <em>y<\/em>-axis. Use the standard form [latex]{x}^{2}=4py[\/latex].<\/li>\n<\/ul><\/li>\n\t<li>Multiply [latex]4p[\/latex].<\/li>\n\t<li>Substitute the value from Step 2 into the equation determined in Step 1.<\/li>\n<\/ul><\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix<\/h3>\nWhat is the equation for the <strong>parabola<\/strong> with <strong>focus<\/strong> [latex]\\left(-\\frac{1}{2},0\\right)[\/latex] and <strong>directrix<\/strong> [latex]x=\\frac{1}{2}?[\/latex]\n\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Solution<\/h3>\nThe focus has the form [latex]\\left(p,0\\right)[\/latex], so the equation will have the form [latex]{y}^{2}=4px[\/latex].\n<div>\n<ul><li>Multiplying [latex]4p[\/latex], we have [latex]4p=4\\left(-\\frac{1}{2}\\right)=-2[\/latex].<\/li>\n\t<li>Substituting for [latex]4p[\/latex], we have [latex]{y}^{2}=4px=-2x[\/latex].<\/li>\n<\/ul><\/div>\nTherefore, the equation for the parabola is [latex]{y}^{2}=-2x[\/latex].\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 4<\/h3>\nWhat is the equation for the parabola with focus [latex]\\left(0,\\frac{7}{2}\\right)[\/latex] and directrix [latex]y=-\\frac{7}{2}?[\/latex]\n\n<a href=\"https:\/\/courses.candelalearning.com\/precalctwo1xmaster\/chapter\/solutions-25\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>","rendered":"<p>In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.\n<\/p>\n<div class=\"textbox\">\n<h3>How To: Given its focus and directrix, write the equation for a parabola in standard form.<\/h3>\n<ul>\n<li>Determine whether the axis of symmetry is the <em>x<\/em>&#8211; or <em>y<\/em>-axis.\n<ul>\n<li>If the given coordinates of the focus have the form [latex]\\left(p,0\\right)[\/latex], then the axis of symmetry is the <em>x<\/em>-axis. Use the standard form [latex]{y}^{2}=4px[\/latex].<\/li>\n<li>If the given coordinates of the focus have the form [latex]\\left(0,p\\right)[\/latex], then the axis of symmetry is the <em>y<\/em>-axis. Use the standard form [latex]{x}^{2}=4py[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>Multiply [latex]4p[\/latex].<\/li>\n<li>Substitute the value from Step 2 into the equation determined in Step 1.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix<\/h3>\n<p>What is the equation for the <strong>parabola<\/strong> with <strong>focus<\/strong> [latex]\\left(-\\frac{1}{2},0\\right)[\/latex] and <strong>directrix<\/strong> [latex]x=\\frac{1}{2}?[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Solution<\/h3>\n<p>The focus has the form [latex]\\left(p,0\\right)[\/latex], so the equation will have the form [latex]{y}^{2}=4px[\/latex].<\/p>\n<div>\n<ul>\n<li>Multiplying [latex]4p[\/latex], we have [latex]4p=4\\left(-\\frac{1}{2}\\right)=-2[\/latex].<\/li>\n<li>Substituting for [latex]4p[\/latex], we have [latex]{y}^{2}=4px=-2x[\/latex].<\/li>\n<\/ul>\n<\/div>\n<p>Therefore, the equation for the parabola is [latex]{y}^{2}=-2x[\/latex].<\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 4<\/h3>\n<p>What is the equation for the parabola with focus [latex]\\left(0,\\frac{7}{2}\\right)[\/latex] and directrix [latex]y=-\\frac{7}{2}?[\/latex]<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/precalctwo1xmaster\/chapter\/solutions-25\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\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-1915\">\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":276,"menu_order":3,"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-1915","chapter","type-chapter","status-publish","hentry"],"part":1904,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1915","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1915\/revisions"}],"predecessor-version":[{"id":2206,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1915\/revisions\/2206"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1904"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1915\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1915"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1915"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1915"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1915"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}