{"id":1882,"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=1882"},"modified":"2015-11-12T18:30:43","modified_gmt":"2015-11-12T18:30:43","slug":"key-concepts-glossary-24","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/key-concepts-glossary-24\/","title":{"raw":"Key Concepts &amp; Glossary","rendered":"Key Concepts &amp; Glossary"},"content":{"raw":"<h2>Key Equations<\/h2>\n<table id=\"fs-id2238445\" summary=\"..\"><tbody><tr><td>Hyperbola, center at origin, transverse axis on <em data-effect=\"italics\">x<\/em>-axis<\/td>\n<td>[latex]\\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr><tr><td>Hyperbola, center at origin, transverse axis on <em data-effect=\"italics\">y<\/em>-axis<\/td>\n<td>[latex]\\frac{{y}^{2}}{{a}^{2}}-\\frac{{x}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr><tr><td>Hyperbola, center at [latex]\\left(h,k\\right)[\/latex], transverse axis parallel to <em data-effect=\"italics\">x<\/em>-axis<\/td>\n<td>[latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr><tr><td>Hyperbola, center at [latex]\\left(h,k\\right)[\/latex], transverse axis parallel to <em data-effect=\"italics\">y<\/em>-axis<\/td>\n<td>[latex]\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr><\/tbody><\/table><h2>Key Concepts<\/h2>\n<ul><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><h2>Glossary<\/h2>\n<dl id=\"fs-id2590473\" class=\"definition\"><dt>center of a hyperbola<\/dt><dd id=\"fs-id2590476\">the midpoint of both the transverse and conjugate axes of a hyperbola<\/dd><\/dl><dl id=\"fs-id2590479\" class=\"definition\"><dt>conjugate axis<\/dt><dd id=\"fs-id2590483\">the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints<\/dd><\/dl><dl id=\"fs-id2590486\" class=\"definition\"><dt>hyperbola<\/dt><dd id=\"fs-id2590489\">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<\/dd><\/dl><dl id=\"fs-id2571756\" class=\"definition\"><dt>transverse axis<\/dt><dd id=\"fs-id2571760\">the axis of a hyperbola that includes the foci and has the vertices as its endpoints<\/dd><\/dl>","rendered":"<h2>Key Equations<\/h2>\n<table id=\"fs-id2238445\" summary=\"..\">\n<tbody>\n<tr>\n<td>Hyperbola, center at origin, transverse axis on <em data-effect=\"italics\">x<\/em>-axis<\/td>\n<td>[latex]\\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Hyperbola, center at origin, transverse axis on <em data-effect=\"italics\">y<\/em>-axis<\/td>\n<td>[latex]\\frac{{y}^{2}}{{a}^{2}}-\\frac{{x}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Hyperbola, center at [latex]\\left(h,k\\right)[\/latex], transverse axis parallel to <em data-effect=\"italics\">x<\/em>-axis<\/td>\n<td>[latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Hyperbola, center at [latex]\\left(h,k\\right)[\/latex], transverse axis parallel to <em data-effect=\"italics\">y<\/em>-axis<\/td>\n<td>[latex]\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}-\\frac{{\\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<dl id=\"fs-id2590473\" class=\"definition\">\n<dt>center of a hyperbola<\/dt>\n<dd id=\"fs-id2590476\">the midpoint of both the transverse and conjugate axes of a hyperbola<\/dd>\n<\/dl>\n<dl id=\"fs-id2590479\" class=\"definition\">\n<dt>conjugate axis<\/dt>\n<dd id=\"fs-id2590483\">the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints<\/dd>\n<\/dl>\n<dl id=\"fs-id2590486\" class=\"definition\">\n<dt>hyperbola<\/dt>\n<dd id=\"fs-id2590489\">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<\/dd>\n<\/dl>\n<dl id=\"fs-id2571756\" class=\"definition\">\n<dt>transverse axis<\/dt>\n<dd id=\"fs-id2571760\">the axis of a hyperbola that includes the foci and has the vertices as its endpoints<\/dd>\n<\/dl>\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-1882\">\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":7,"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-1882","chapter","type-chapter","status-publish","hentry"],"part":1863,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1882","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\/1882\/revisions"}],"predecessor-version":[{"id":2211,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1882\/revisions\/2211"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1863"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1882\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1882"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1882"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1882"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1882"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}