{"id":1594,"date":"2025-07-25T04:00:53","date_gmt":"2025-07-25T04:00:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1594"},"modified":"2026-03-25T04:58:35","modified_gmt":"2026-03-25T04:58:35","slug":"hyperbolas-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/hyperbolas-fresh-take\/","title":{"raw":"Hyperbolas: Fresh Take","rendered":"Hyperbolas: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write equations of hyperbolas in standard form.<\/li>\r\n \t<li>Graph hyperbolas.<\/li>\r\n \t<li>Solve applied problems involving hyperbolas.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Hyperbolas<\/h2>\r\nA 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.\r\n\r\nNotice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the <em>difference<\/em> of two distances, whereas the ellipse is defined in terms of the <em>sum<\/em> of two distances.\r\n\r\nAs with the ellipse, every hyperbola has two <strong>axes of symmetry<\/strong>. The <strong>transverse axis<\/strong> is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The <strong>conjugate axis<\/strong> is perpendicular to the transverse axis and has the co-vertices as its endpoints. The <strong>center of a hyperbola<\/strong> is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two <strong>asymptotes<\/strong> that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The <strong>central rectangle<\/strong> of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle.\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03203416\/CNX_Precalc_Figure_10_02_0032.jpg\" alt=\"\" width=\"731\" height=\"437\" \/> Hyperbola with key features labeled[\/caption]\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hgaababe-i6vM82SNAUk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/i6vM82SNAUk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hgaababe-i6vM82SNAUk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851281&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hgaababe-i6vM82SNAUk&amp;vembed=0&amp;video_id=i6vM82SNAUk&amp;video_target=tpm-plugin-hgaababe-i6vM82SNAUk\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Conic+Sections+-++The+Hyperbola+part+1+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConic Sections: The Hyperbola part 1 of 2\u201d here (opens in new window).<\/a>\r\n\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-chbfhaac-6Xahrwp6LkI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/6Xahrwp6LkI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-chbfhaac-6Xahrwp6LkI\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851282&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-chbfhaac-6Xahrwp6LkI&amp;vembed=0&amp;video_id=6Xahrwp6LkI&amp;video_target=tpm-plugin-chbfhaac-6Xahrwp6LkI\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Conic+Sections+-+The+Hyperbola+part+2+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConic Sections: The Hyperbola part 2 of 2\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Standard Form of the Equation of a Hyperbola Centered at the Origin<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Transverse Axis: [latex]\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Transverse Axis: [latex]\\frac{y^2}{a^2} - \\frac{x^2}{b^2} = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Features:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertices: [latex](\u00b1a, 0)[\/latex] or [latex](0, \u00b1a)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Co-vertices: [latex](0, \u00b1b)[\/latex] or [latex](\u00b1b, 0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Foci: [latex](\u00b1c, 0)[\/latex] or [latex](0, \u00b1c)[\/latex], where [latex]c^2 = a^2 + b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Transverse axis length: [latex]2a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Conjugate axis length: [latex]2b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal: [latex]y=\\pm \\dfrac{b}{a}x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical: [latex]y=\\pm \\dfrac{a}{b}x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Identifying Features from Equation<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Determine transverse axis orientation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x^2[\/latex] positive: horizontal<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y^2[\/latex] positive: vertical<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find vertices:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a = \\sqrt{\\text{term with positive coefficient}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find co-vertices:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]b = \\sqrt{\\text{term with negative coefficient}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate c:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]c = \\sqrt{a^2 + b^2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Slope = [latex]\\pm\\frac{b}{a}[\/latex] (horizontal) or [latex]\\pm\\frac{a}{b}[\/latex] (vertical)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">Let [latex]\\left(-c,0\\right)[\/latex] and [latex]\\left(c,0\\right)[\/latex] be the <strong>foci<\/strong> of a hyperbola centered at the origin. The hyperbola is the set of all points [latex]\\left(x,y\\right)[\/latex] such that the difference of the distances from [latex]\\left(x,y\\right)[\/latex] to the foci is constant.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03203418\/CNX_Precalc_Figure_10_02_0132.jpg\" alt=\"\" width=\"487\" height=\"389\" \/> Hyperbola with certain features labeled[\/caption]\r\n\r\nIf [latex]\\left(a,0\\right)[\/latex] is a vertex of the hyperbola, the distance from [latex]\\left(-c,0\\right)[\/latex] to [latex]\\left(a,0\\right)[\/latex] is [latex]a-\\left(-c\\right)=a+c[\/latex]. The distance from [latex]\\left(c,0\\right)[\/latex] to [latex]\\left(a,0\\right)[\/latex] is [latex]c-a[\/latex]. The difference of the distances from the foci to the vertex is\r\n<p style=\"text-align: center;\">[latex]\\left(a+c\\right)-\\left(c-a\\right)=2a[\/latex]<\/p>\r\nIf [latex]\\left(x,y\\right)[\/latex] is a point on the hyperbola, we can define the following variables:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{d}_{2}=\\text{the distance from }\\left(-c,0\\right)\\text{ to }\\left(x,y\\right)\\\\ &amp;{d}_{1}=\\text{the distance from }\\left(c,0\\right)\\text{ to }\\left(x,y\\right)\\end{align}[\/latex]<\/p>\r\nBy definition of a hyperbola, [latex]\\lvert{d}_{2}-{d}_{1}\\rvert[\/latex] is constant for any point [latex]\\left(x,y\\right)[\/latex] on the hyperbola. We know that the difference of these distances is [latex]2a[\/latex] for the vertex [latex]\\left(a,0\\right)[\/latex]. It follows that [latex]\\lvert{d}_{2}-{d}_{1}\\rvert=2a[\/latex] for any point on the hyperbola. The derivation of the equation of a hyperbola is based on applying the <strong>distance formula<\/strong>, but is again beyond the scope of this text. The standard form of an equation of a hyperbola centered at the origin with vertices [latex]\\left(\\pm a,0\\right)[\/latex] and co-vertices [latex]\\left(0\\pm b\\right)[\/latex] is [latex]\\dfrac{{x}^{2}}{{a}^{2}}-\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex].\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Identify the vertices and foci of the hyperbola with equation [latex]\\dfrac{{x}^{2}}{9}-\\dfrac{{y}^{2}}{25}=1[\/latex].[reveal-answer q=\"22796\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"22796\"]Vertices: [latex]\\left(\\pm 3,0\\right)[\/latex]; Foci: [latex]\\left(\\pm \\sqrt{34},0\\right)[\/latex][\/hidden-answer]<\/section>\r\n<h2>Writing Equations of Hyperbolas Centered at the Origin<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Standard Forms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal: [latex]\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical: [latex]\\frac{y^2}{a^2} - \\frac{x^2}{b^2} = 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Relationship: [latex]c^2 = a^2 + b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify transverse axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine [latex]a[\/latex] from vertices<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find [latex]c[\/latex] from foci<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]b [\/latex] using [latex]c^2 = a^2 + b^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Steps to Write Equation<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Determine transverse axis orientation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-coordinates different: horizontal<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-coordinates different: vertical<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify [latex]a[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Distance from center to vertex<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify [latex]c[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Distance from center to focus<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]b[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]{b}^{2}={c}^{2}-{a}^{2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">What is the standard form equation of the hyperbola that has vertices [latex]\\left(0,\\pm 2\\right)[\/latex] and foci [latex]\\left(0,\\pm 2\\sqrt{5}\\right)?[\/latex][reveal-answer q=\"327389\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"327389\"][latex]\\dfrac{{y}^{2}}{4}-\\dfrac{{x}^{2}}{16}=1[\/latex][\/hidden-answer]<\/section>\r\n<h2>Hyperbolas Not Centered at the Origin<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Standard Forms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal transverse axis: [latex]\\frac{(x-h)^2}{a^2} - \\frac{(y-k)^2}{b^2} = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical transverse axis: [latex]\\frac{(y-k)^2}{a^2} - \\frac{(x-h)^2}{b^2} = 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Features:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Center: [latex](h, k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertices: [latex](h \u00b1 a, k)[\/latex] or [latex](h, k \u00b1 a)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Co-vertices: [latex](h, k \u00b1 b)[\/latex] or [latex](h \u00b1 b, k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Foci: [latex](h \u00b1 c, k)[\/latex] or [latex](h, k \u00b1 c)[\/latex], where [latex]c^2 = a^2 + b^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Transverse Axis:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Length: [latex]2a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Parallel to [latex]x[\/latex]-axis if [latex]y[\/latex]-coordinates of vertices and foci are the same<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Parallel to [latex]y[\/latex]-axis if [latex]x[\/latex]-coordinates of vertices and foci are the same<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>From Points to Equation<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Determine transverse axis orientation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find center using midpoint formula<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]a[\/latex] (half the distance between vertices)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find [latex]c[\/latex] (distance from center to focus)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]b[\/latex] using [latex]b^2 = c^2 - a^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute values into appropriate standard form<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">What is the standard form equation of the hyperbola that has vertices [latex]\\left(1,-2\\right)[\/latex] and [latex]\\left(1,\\text{8}\\right)[\/latex] and foci [latex]\\left(1,-10\\right)[\/latex] and [latex]\\left(1,16\\right)?[\/latex][reveal-answer q=\"59300\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"59300\"][latex]\\dfrac{{\\left(y - 3\\right)}^{2}}{25}-\\dfrac{{\\left(x - 1\\right)}^{2}}{144}=1[\/latex][\/hidden-answer]<\/section>\r\n<h2 aria-label=\"Try It\">Graphing Hyperbolas<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Standard Forms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal, origin: [latex]\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical, origin: [latex]\\frac{y^2}{a^2} - \\frac{x^2}{b^2} = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal, center (h,k): [latex]\\frac{(x-h)^2}{a^2} - \\frac{(y-k)^2}{b^2} = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical, center (h,k): [latex]\\frac{(y-k)^2}{a^2} - \\frac{(x-h)^2}{b^2} = 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Features:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Center: [latex](0,0)[\/latex] or [latex](h,k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertices: [latex](\\pm a,0)[\/latex], [latex](0,\\pm a)[\/latex], [latex](h\\pm a,k)[\/latex], or [latex](h,k\\pm a)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Co-vertices: [latex](0,\\pm b)[\/latex], [latex](\\pm b,0)[\/latex], [latex](h,k\\pm b)[\/latex], or [latex](h\\pm b,k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Foci: [latex](\\pm c,0)[\/latex], [latex](0,\\pm c)[\/latex], [latex](h\\pm c,k)[\/latex], or [latex](h,k\\pm c)[\/latex], where [latex]c^2 = a^2 + b^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Centered at origin: [latex]y = \\pm\\frac{b}{a}x[\/latex] or [latex]y = \\pm\\frac{a}{b}x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Centered at [latex](h,k)[\/latex]: [latex]y = \\pm\\frac{b}{a}(x-h) + k[\/latex] or [latex]y = \\pm\\frac{a}{b}(x-h) + k[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Graphing Process<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the standard form and orientation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine center, vertices, and co-vertices<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate foci using [latex]c = \\sqrt{a^2 + b^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find asymptote equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot center, vertices, co-vertices, and foci<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sketch asymptotes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Draw hyperbola curves approaching asymptotes<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>For Non-Standard Form<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Group like terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complete the square for [latex]x[\/latex] and [latex]y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rewrite as perfect squares<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Divide by constant to get standard form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Follow steps for graphing standard form<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Graph the hyperbola given by the equation [latex]\\dfrac{{x}^{2}}{144}-\\dfrac{{y}^{2}}{81}=1[\/latex]. Identify and label the vertices, co-vertices, foci, and asymptotes.[reveal-answer q=\"168390\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"168390\"]vertices: [latex]\\left(\\pm 12,0\\right)[\/latex]; co-vertices: [latex]\\left(0,\\pm 9\\right)[\/latex]; foci: [latex]\\left(\\pm 15,0\\right)[\/latex]; asymptotes: [latex]y=\\pm \\frac{3}{4}x[\/latex];\r\n\r\n[caption id=\"attachment_3268\" align=\"aligncenter\" width=\"731\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02170323\/CNX_Precalc_Figure_10_02_0072.jpg\"><img class=\"wp-image-3268 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02170323\/CNX_Precalc_Figure_10_02_0072.jpg\" alt=\"\" width=\"731\" height=\"361\" \/><\/a> Hyperbola with key features labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph the hyperbola given by the standard form of an equation [latex]\\dfrac{{\\left(y+4\\right)}^{2}}{100}-\\dfrac{{\\left(x - 3\\right)}^{2}}{64}=1[\/latex]. Identify and label the center, vertices, co-vertices, foci, and asymptotes.[reveal-answer q=\"563114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"563114\"]center: [latex]\\left(3,-4\\right)[\/latex]; vertices: [latex]\\left(3,-14\\right)[\/latex] and [latex]\\left(3,6\\right)[\/latex]; co-vertices: [latex]\\left(-5,-4\\right)[\/latex]; and [latex]\\left(11,-4\\right)[\/latex]; foci: [latex]\\left(3,-4 - 2\\sqrt{41}\\right)[\/latex] and [latex]\\left(3,-4+2\\sqrt{41}\\right)[\/latex]; asymptotes: [latex]y=\\pm \\frac{5}{4}\\left(x - 3\\right)-4[\/latex]\r\n\r\n[caption id=\"attachment_3269\" align=\"aligncenter\" width=\"487\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02170636\/CNX_Precalc_Figure_10_02_0092.jpg\"><img class=\"wp-image-3269 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02170636\/CNX_Precalc_Figure_10_02_0092.jpg\" alt=\"\" width=\"487\" height=\"514\" \/><\/a> Hyperbola with key features labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-chfgfedg-W0IXOdsna9A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/W0IXOdsna9A?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-chfgfedg-W0IXOdsna9A\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851283&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-chfgfedg-W0IXOdsna9A&amp;vembed=0&amp;video_id=W0IXOdsna9A&amp;video_target=tpm-plugin-chfgfedg-W0IXOdsna9A\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Conic+Section+-+Graph+a+Hyperbola+with+Center+at+the+Origin+(Horizontal)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Conic Section - Graph a Hyperbola with Center at the Origin (Horizontal)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hecaffeg-_ssQfu8N6XE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_ssQfu8N6XE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hecaffeg-_ssQfu8N6XE\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851284&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hecaffeg-_ssQfu8N6XE&amp;vembed=0&amp;video_id=_ssQfu8N6XE&amp;video_target=tpm-plugin-hecaffeg-_ssQfu8N6XE\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+3+-+Conic+Section+-+Graph+a+Hyperbola+with+Center+NOT+at+the+Origin+(Horizontal)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Conic Section - Graph a Hyperbola with Center NOT at the Origin (Horizontal)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write equations of hyperbolas in standard form.<\/li>\n<li>Graph hyperbolas.<\/li>\n<li>Solve applied problems involving hyperbolas.<\/li>\n<\/ul>\n<\/section>\n<h2>Hyperbolas<\/h2>\n<p>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.<\/p>\n<p>Notice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the <em>difference<\/em> of two distances, whereas the ellipse is defined in terms of the <em>sum<\/em> of two distances.<\/p>\n<p>As with the ellipse, every hyperbola has two <strong>axes of symmetry<\/strong>. The <strong>transverse axis<\/strong> is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The <strong>conjugate axis<\/strong> is perpendicular to the transverse axis and has the co-vertices as its endpoints. The <strong>center of a hyperbola<\/strong> is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two <strong>asymptotes<\/strong> that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The <strong>central rectangle<\/strong> of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle.<\/p>\n<figure style=\"width: 731px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03203416\/CNX_Precalc_Figure_10_02_0032.jpg\" alt=\"\" width=\"731\" height=\"437\" \/><figcaption class=\"wp-caption-text\">Hyperbola with key features labeled<\/figcaption><\/figure>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hgaababe-i6vM82SNAUk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/i6vM82SNAUk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hgaababe-i6vM82SNAUk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851281&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hgaababe-i6vM82SNAUk&amp;vembed=0&amp;video_id=i6vM82SNAUk&amp;video_target=tpm-plugin-hgaababe-i6vM82SNAUk\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Conic+Sections+-++The+Hyperbola+part+1+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConic Sections: The Hyperbola part 1 of 2\u201d here (opens in new window).<\/a><\/p>\n<p><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-chbfhaac-6Xahrwp6LkI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/6Xahrwp6LkI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-chbfhaac-6Xahrwp6LkI\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851282&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-chbfhaac-6Xahrwp6LkI&amp;vembed=0&amp;video_id=6Xahrwp6LkI&amp;video_target=tpm-plugin-chbfhaac-6Xahrwp6LkI\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Conic+Sections+-+The+Hyperbola+part+2+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConic Sections: The Hyperbola part 2 of 2\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Standard Form of the Equation of a Hyperbola Centered at the Origin<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Horizontal Transverse Axis: [latex]\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Transverse Axis: [latex]\\frac{y^2}{a^2} - \\frac{x^2}{b^2} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key Features:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertices: [latex](\u00b1a, 0)[\/latex] or [latex](0, \u00b1a)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Co-vertices: [latex](0, \u00b1b)[\/latex] or [latex](\u00b1b, 0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Foci: [latex](\u00b1c, 0)[\/latex] or [latex](0, \u00b1c)[\/latex], where [latex]c^2 = a^2 + b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Transverse axis length: [latex]2a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Conjugate axis length: [latex]2b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal: [latex]y=\\pm \\dfrac{b}{a}x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical: [latex]y=\\pm \\dfrac{a}{b}x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Identifying Features from Equation<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determine transverse axis orientation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x^2[\/latex] positive: horizontal<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y^2[\/latex] positive: vertical<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Find vertices:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a = \\sqrt{\\text{term with positive coefficient}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Find co-vertices:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]b = \\sqrt{\\text{term with negative coefficient}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Calculate c:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]c = \\sqrt{a^2 + b^2}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Slope = [latex]\\pm\\frac{b}{a}[\/latex] (horizontal) or [latex]\\pm\\frac{a}{b}[\/latex] (vertical)<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">Let [latex]\\left(-c,0\\right)[\/latex] and [latex]\\left(c,0\\right)[\/latex] be the <strong>foci<\/strong> of a hyperbola centered at the origin. The hyperbola is the set of all points [latex]\\left(x,y\\right)[\/latex] such that the difference of the distances from [latex]\\left(x,y\\right)[\/latex] to the foci is constant.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03203418\/CNX_Precalc_Figure_10_02_0132.jpg\" alt=\"\" width=\"487\" height=\"389\" \/><figcaption class=\"wp-caption-text\">Hyperbola with certain features labeled<\/figcaption><\/figure>\n<p>If [latex]\\left(a,0\\right)[\/latex] is a vertex of the hyperbola, the distance from [latex]\\left(-c,0\\right)[\/latex] to [latex]\\left(a,0\\right)[\/latex] is [latex]a-\\left(-c\\right)=a+c[\/latex]. The distance from [latex]\\left(c,0\\right)[\/latex] to [latex]\\left(a,0\\right)[\/latex] is [latex]c-a[\/latex]. The difference of the distances from the foci to the vertex is<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+c\\right)-\\left(c-a\\right)=2a[\/latex]<\/p>\n<p>If [latex]\\left(x,y\\right)[\/latex] is a point on the hyperbola, we can define the following variables:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{d}_{2}=\\text{the distance from }\\left(-c,0\\right)\\text{ to }\\left(x,y\\right)\\\\ &{d}_{1}=\\text{the distance from }\\left(c,0\\right)\\text{ to }\\left(x,y\\right)\\end{align}[\/latex]<\/p>\n<p>By definition of a hyperbola, [latex]\\lvert{d}_{2}-{d}_{1}\\rvert[\/latex] is constant for any point [latex]\\left(x,y\\right)[\/latex] on the hyperbola. We know that the difference of these distances is [latex]2a[\/latex] for the vertex [latex]\\left(a,0\\right)[\/latex]. It follows that [latex]\\lvert{d}_{2}-{d}_{1}\\rvert=2a[\/latex] for any point on the hyperbola. The derivation of the equation of a hyperbola is based on applying the <strong>distance formula<\/strong>, but is again beyond the scope of this text. The standard form of an equation of a hyperbola centered at the origin with vertices [latex]\\left(\\pm a,0\\right)[\/latex] and co-vertices [latex]\\left(0\\pm b\\right)[\/latex] is [latex]\\dfrac{{x}^{2}}{{a}^{2}}-\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Identify the vertices and foci of the hyperbola with equation [latex]\\dfrac{{x}^{2}}{9}-\\dfrac{{y}^{2}}{25}=1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q22796\">Show Solution<\/button><\/p>\n<div id=\"q22796\" class=\"hidden-answer\" style=\"display: none\">Vertices: [latex]\\left(\\pm 3,0\\right)[\/latex]; Foci: [latex]\\left(\\pm \\sqrt{34},0\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Writing Equations of Hyperbolas Centered at the Origin<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Standard Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal: [latex]\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical: [latex]\\frac{y^2}{a^2} - \\frac{x^2}{b^2} = 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relationship: [latex]c^2 = a^2 + b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify transverse axis<\/li>\n<li class=\"whitespace-normal break-words\">Determine [latex]a[\/latex] from vertices<\/li>\n<li class=\"whitespace-normal break-words\">Find [latex]c[\/latex] from foci<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]b[\/latex] using [latex]c^2 = a^2 + b^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Steps to Write Equation<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determine transverse axis orientation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-coordinates different: horizontal<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-coordinates different: vertical<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identify [latex]a[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Distance from center to vertex<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identify [latex]c[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Distance from center to focus<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]b[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]{b}^{2}={c}^{2}-{a}^{2}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">What is the standard form equation of the hyperbola that has vertices [latex]\\left(0,\\pm 2\\right)[\/latex] and foci [latex]\\left(0,\\pm 2\\sqrt{5}\\right)?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q327389\">Show Solution<\/button><\/p>\n<div id=\"q327389\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{{y}^{2}}{4}-\\dfrac{{x}^{2}}{16}=1[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Hyperbolas Not Centered at the Origin<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Standard Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal transverse axis: [latex]\\frac{(x-h)^2}{a^2} - \\frac{(y-k)^2}{b^2} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical transverse axis: [latex]\\frac{(y-k)^2}{a^2} - \\frac{(x-h)^2}{b^2} = 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Features:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Center: [latex](h, k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertices: [latex](h \u00b1 a, k)[\/latex] or [latex](h, k \u00b1 a)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Co-vertices: [latex](h, k \u00b1 b)[\/latex] or [latex](h \u00b1 b, k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Foci: [latex](h \u00b1 c, k)[\/latex] or [latex](h, k \u00b1 c)[\/latex], where [latex]c^2 = a^2 + b^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Transverse Axis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Length: [latex]2a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Parallel to [latex]x[\/latex]-axis if [latex]y[\/latex]-coordinates of vertices and foci are the same<\/li>\n<li class=\"whitespace-normal break-words\">Parallel to [latex]y[\/latex]-axis if [latex]x[\/latex]-coordinates of vertices and foci are the same<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>From Points to Equation<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determine transverse axis orientation<\/li>\n<li class=\"whitespace-normal break-words\">Find center using midpoint formula<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]a[\/latex] (half the distance between vertices)<\/li>\n<li class=\"whitespace-normal break-words\">Find [latex]c[\/latex] (distance from center to focus)<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]b[\/latex] using [latex]b^2 = c^2 - a^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Substitute values into appropriate standard form<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">What is the standard form equation of the hyperbola that has vertices [latex]\\left(1,-2\\right)[\/latex] and [latex]\\left(1,\\text{8}\\right)[\/latex] and foci [latex]\\left(1,-10\\right)[\/latex] and [latex]\\left(1,16\\right)?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q59300\">Show Solution<\/button><\/p>\n<div id=\"q59300\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{{\\left(y - 3\\right)}^{2}}{25}-\\dfrac{{\\left(x - 1\\right)}^{2}}{144}=1[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2 aria-label=\"Try It\">Graphing Hyperbolas<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Standard Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal, origin: [latex]\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical, origin: [latex]\\frac{y^2}{a^2} - \\frac{x^2}{b^2} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal, center (h,k): [latex]\\frac{(x-h)^2}{a^2} - \\frac{(y-k)^2}{b^2} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical, center (h,k): [latex]\\frac{(y-k)^2}{a^2} - \\frac{(x-h)^2}{b^2} = 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Features:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Center: [latex](0,0)[\/latex] or [latex](h,k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertices: [latex](\\pm a,0)[\/latex], [latex](0,\\pm a)[\/latex], [latex](h\\pm a,k)[\/latex], or [latex](h,k\\pm a)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Co-vertices: [latex](0,\\pm b)[\/latex], [latex](\\pm b,0)[\/latex], [latex](h,k\\pm b)[\/latex], or [latex](h\\pm b,k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Foci: [latex](\\pm c,0)[\/latex], [latex](0,\\pm c)[\/latex], [latex](h\\pm c,k)[\/latex], or [latex](h,k\\pm c)[\/latex], where [latex]c^2 = a^2 + b^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Centered at origin: [latex]y = \\pm\\frac{b}{a}x[\/latex] or [latex]y = \\pm\\frac{a}{b}x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Centered at [latex](h,k)[\/latex]: [latex]y = \\pm\\frac{b}{a}(x-h) + k[\/latex] or [latex]y = \\pm\\frac{a}{b}(x-h) + k[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Graphing Process<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the standard form and orientation<\/li>\n<li class=\"whitespace-normal break-words\">Determine center, vertices, and co-vertices<\/li>\n<li class=\"whitespace-normal break-words\">Calculate foci using [latex]c = \\sqrt{a^2 + b^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Find asymptote equations<\/li>\n<li class=\"whitespace-normal break-words\">Plot center, vertices, co-vertices, and foci<\/li>\n<li class=\"whitespace-normal break-words\">Sketch asymptotes<\/li>\n<li class=\"whitespace-normal break-words\">Draw hyperbola curves approaching asymptotes<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>For Non-Standard Form<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Group like terms<\/li>\n<li class=\"whitespace-normal break-words\">Complete the square for [latex]x[\/latex] and [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Rewrite as perfect squares<\/li>\n<li class=\"whitespace-normal break-words\">Divide by constant to get standard form<\/li>\n<li class=\"whitespace-normal break-words\">Follow steps for graphing standard form<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Graph the hyperbola given by the equation [latex]\\dfrac{{x}^{2}}{144}-\\dfrac{{y}^{2}}{81}=1[\/latex]. Identify and label the vertices, co-vertices, foci, and asymptotes.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q168390\">Show Solution<\/button><\/p>\n<div id=\"q168390\" class=\"hidden-answer\" style=\"display: none\">vertices: [latex]\\left(\\pm 12,0\\right)[\/latex]; co-vertices: [latex]\\left(0,\\pm 9\\right)[\/latex]; foci: [latex]\\left(\\pm 15,0\\right)[\/latex]; asymptotes: [latex]y=\\pm \\frac{3}{4}x[\/latex];<\/p>\n<figure id=\"attachment_3268\" aria-describedby=\"caption-attachment-3268\" style=\"width: 731px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02170323\/CNX_Precalc_Figure_10_02_0072.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3268 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02170323\/CNX_Precalc_Figure_10_02_0072.jpg\" alt=\"\" width=\"731\" height=\"361\" \/><\/a><figcaption id=\"caption-attachment-3268\" class=\"wp-caption-text\">Hyperbola with key features labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph the hyperbola given by the standard form of an equation [latex]\\dfrac{{\\left(y+4\\right)}^{2}}{100}-\\dfrac{{\\left(x - 3\\right)}^{2}}{64}=1[\/latex]. Identify and label the center, vertices, co-vertices, foci, and asymptotes.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q563114\">Show Solution<\/button><\/p>\n<div id=\"q563114\" class=\"hidden-answer\" style=\"display: none\">center: [latex]\\left(3,-4\\right)[\/latex]; vertices: [latex]\\left(3,-14\\right)[\/latex] and [latex]\\left(3,6\\right)[\/latex]; co-vertices: [latex]\\left(-5,-4\\right)[\/latex]; and [latex]\\left(11,-4\\right)[\/latex]; foci: [latex]\\left(3,-4 - 2\\sqrt{41}\\right)[\/latex] and [latex]\\left(3,-4+2\\sqrt{41}\\right)[\/latex]; asymptotes: [latex]y=\\pm \\frac{5}{4}\\left(x - 3\\right)-4[\/latex]<\/p>\n<figure id=\"attachment_3269\" aria-describedby=\"caption-attachment-3269\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02170636\/CNX_Precalc_Figure_10_02_0092.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3269 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02170636\/CNX_Precalc_Figure_10_02_0092.jpg\" alt=\"\" width=\"487\" height=\"514\" \/><\/a><figcaption id=\"caption-attachment-3269\" class=\"wp-caption-text\">Hyperbola with key features labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-chfgfedg-W0IXOdsna9A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/W0IXOdsna9A?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-chfgfedg-W0IXOdsna9A\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851283&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-chfgfedg-W0IXOdsna9A&amp;vembed=0&amp;video_id=W0IXOdsna9A&amp;video_target=tpm-plugin-chfgfedg-W0IXOdsna9A\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Conic+Section+-+Graph+a+Hyperbola+with+Center+at+the+Origin+(Horizontal)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Conic Section &#8211; Graph a Hyperbola with Center at the Origin (Horizontal)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hecaffeg-_ssQfu8N6XE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_ssQfu8N6XE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hecaffeg-_ssQfu8N6XE\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851284&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hecaffeg-_ssQfu8N6XE&amp;vembed=0&amp;video_id=_ssQfu8N6XE&amp;video_target=tpm-plugin-hecaffeg-_ssQfu8N6XE\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+3+-+Conic+Section+-+Graph+a+Hyperbola+with+Center+NOT+at+the+Origin+(Horizontal)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Conic Section &#8211; Graph a Hyperbola with Center NOT at the Origin (Horizontal)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Conic Sections: The Hyperbola part 1 of 2\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/i6vM82SNAUk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Conic Sections: The Hyperbola part 2 of 2\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/6Xahrwp6LkI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Conic Section - 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