{"id":425,"date":"2019-07-15T22:44:55","date_gmt":"2019-07-15T22:44:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/summary-the-hyperbola\/"},"modified":"2019-07-15T22:44:55","modified_gmt":"2019-07-15T22:44:55","slug":"summary-the-hyperbola","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/summary-the-hyperbola\/","title":{"raw":"Summary: The Hyperbola","rendered":"Summary: The Hyperbola"},"content":{"raw":"\n<h2>Key Equations<\/h2>\n<table style=\"height: 92px\" summary=\"..\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Hyperbola, center at origin, transverse axis on <em>x<\/em>-axis<\/td>\n<td style=\"height: 15px\">[latex]\\dfrac{{x}^{2}}{{a}^{2}}-\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Hyperbola, center at origin, transverse axis on <em>y<\/em>-axis<\/td>\n<td style=\"height: 15px\">[latex]\\dfrac{{y}^{2}}{{a}^{2}}-\\dfrac{{x}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 31px\">\n<td style=\"height: 31px\">Hyperbola, center at [latex]\\left(h,k\\right)[\/latex], transverse axis parallel to <em>x<\/em>-axis<\/td>\n<td style=\"height: 31px\">[latex]\\dfrac{{\\left(x-h\\right)}^{2}}{{a}^{2}}-\\dfrac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 31px\">\n<td style=\"height: 31px\">Hyperbola, center at [latex]\\left(h,k\\right)[\/latex], transverse axis parallel to <em>y<\/em>-axis<\/td>\n<td style=\"height: 31px\">[latex]\\dfrac{{\\left(y-k\\right)}^{2}}{{a}^{2}}-\\dfrac{{\\left(x-h\\right)}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>A hyperbola is the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane such that the difference of the distances between [latex]\\left(x,y\\right)[\/latex] and the foci is a positive constant.<\/li>\n \t<li>The standard form of a hyperbola can be used to locate its vertices and foci.<\/li>\n \t<li>When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form.<\/li>\n \t<li>When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola.<\/li>\n \t<li>Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<strong>center of a hyperbola<\/strong> the midpoint of both the transverse and conjugate axes of a hyperbola\n\n<strong>conjugate axis<\/strong> the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints\n\n<strong>hyperbola<\/strong> the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane such that the difference of the distances between [latex]\\left(x,y\\right)[\/latex] and the foci is a positive constant\n\n<strong>transverse axis<\/strong> the axis of a hyperbola that includes the foci and has the vertices as its endpoints\n","rendered":"<h2>Key Equations<\/h2>\n<table style=\"height: 92px\" summary=\"..\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Hyperbola, center at origin, transverse axis on <em>x<\/em>-axis<\/td>\n<td style=\"height: 15px\">[latex]\\dfrac{{x}^{2}}{{a}^{2}}-\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Hyperbola, center at origin, transverse axis on <em>y<\/em>-axis<\/td>\n<td style=\"height: 15px\">[latex]\\dfrac{{y}^{2}}{{a}^{2}}-\\dfrac{{x}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 31px\">\n<td style=\"height: 31px\">Hyperbola, center at [latex]\\left(h,k\\right)[\/latex], transverse axis parallel to <em>x<\/em>-axis<\/td>\n<td style=\"height: 31px\">[latex]\\dfrac{{\\left(x-h\\right)}^{2}}{{a}^{2}}-\\dfrac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 31px\">\n<td style=\"height: 31px\">Hyperbola, center at [latex]\\left(h,k\\right)[\/latex], transverse axis parallel to <em>y<\/em>-axis<\/td>\n<td style=\"height: 31px\">[latex]\\dfrac{{\\left(y-k\\right)}^{2}}{{a}^{2}}-\\dfrac{{\\left(x-h\\right)}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>A hyperbola is the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane such that the difference of the distances between [latex]\\left(x,y\\right)[\/latex] and the foci is a positive constant.<\/li>\n<li>The standard form of a hyperbola can be used to locate its vertices and foci.<\/li>\n<li>When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form.<\/li>\n<li>When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola.<\/li>\n<li>Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>center of a hyperbola<\/strong> the midpoint of both the transverse and conjugate axes of a hyperbola<\/p>\n<p><strong>conjugate axis<\/strong> the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints<\/p>\n<p><strong>hyperbola<\/strong> the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane such that the difference of the distances between [latex]\\left(x,y\\right)[\/latex] and the foci is a positive constant<\/p>\n<p><strong>transverse axis<\/strong> the axis of a hyperbola that includes the foci and has the vertices as its endpoints<\/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-425\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><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":17533,"menu_order":12,"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\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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