{"id":1597,"date":"2025-07-25T04:01:32","date_gmt":"2025-07-25T04:01:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1597"},"modified":"2026-03-25T05:05:38","modified_gmt":"2026-03-25T05:05:38","slug":"parabolas-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/parabolas-fresh-take\/","title":{"raw":"Parabolas: Fresh Take","rendered":"Parabolas: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write equations of parabolas in standard form.<\/li>\r\n \t<li>Graph parabolas.<\/li>\r\n \t<li>Solve applied problems involving parabolas.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Parabolas<\/h2>\r\nTo work with parabolas in the <strong>coordinate plane<\/strong>, we consider two cases: those with a vertex at the origin and those with a <strong>vertex<\/strong> at a point other than the origin. We begin with the former.\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\/03204538\/CNX_Precalc_Figure_10_03_0182.jpg\" alt=\"\" width=\"487\" height=\"292\" \/> Parabola with a vertex at the origin and key features labeled[\/caption]\r\n\r\nLet [latex]\\left(x,y\\right)[\/latex] be a point on the parabola with vertex [latex]\\left(0,0\\right)[\/latex], focus [latex]\\left(0,p\\right)[\/latex], and directrix [latex]y= -p[\/latex]\u00a0as shown in Figure 4. The distance [latex]d[\/latex] from point [latex]\\left(x,y\\right)[\/latex] to point [latex]\\left(x,-p\\right)[\/latex]\u00a0on the directrix is the difference of the <em>y<\/em>-values: [latex]d=y+p[\/latex]. The distance from the focus [latex]\\left(0,p\\right)[\/latex] to the point [latex]\\left(x,y\\right)[\/latex] is also equal to [latex]d[\/latex] and can be expressed using the <strong>distance formula<\/strong>.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}d&amp;=\\sqrt{{\\left(x - 0\\right)}^{2}+{\\left(y-p\\right)}^{2}} \\\\ &amp;=\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}} \\end{align}[\/latex]<\/p>\r\nSet the two expressions for [latex]d[\/latex] equal to each other and solve for [latex]y[\/latex] to derive the equation of the parabola. We do this because the distance from [latex]\\left(x,y\\right)[\/latex] to [latex]\\left(0,p\\right)[\/latex] equals the distance from [latex]\\left(x,y\\right)[\/latex] to [latex]\\left(x, -p\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}}=y+p[\/latex]<\/p>\r\nWe then square both sides of the equation, expand the squared terms, and simplify by combining like terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}{x}^{2}+{\\left(y-p\\right)}^{2}={\\left(y+p\\right)}^{2} \\\\ {x}^{2}+{y}^{2}-2py+{p}^{2}={y}^{2}+2py+{p}^{2}\\\\ {x}^{2}-2py=2py \\\\ {x}^{2}=4py\\end{gathered}[\/latex]<\/p>\r\nThe equations of parabolas with vertex [latex]\\left(0,0\\right)[\/latex] are [latex]{y}^{2}=4px[\/latex] when the <em>x<\/em>-axis is the axis of symmetry and [latex]{x}^{2}=4py[\/latex] when the <em>y<\/em>-axis is the axis of symmetry.\r\n<h2 data-type=\"title\">Graphing Parabolas with Vertices 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\">Vertical parabolas: [latex]x^2 = 4py[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal parabolas: [latex]y^2 = 4px[\/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\">Vertex: Always at [latex](0, 0)[\/latex] for these standard forms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Axis of symmetry: [latex]x[\/latex]-axis for horizontal parabolas, [latex]y[\/latex]-axis for vertical parabolas<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Focus: [latex](p, 0)[\/latex] for horizontal parabolas, [latex](0, p)[\/latex] for vertical parabolas<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Directrix: [latex]x = -p[\/latex] for horizontal parabolas, [latex]y = -p[\/latex] for vertical parabolas<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Focal diameter endpoints: [latex](p, \u00b12p)[\/latex] for horizontal parabolas, [latex](\u00b12p, p)[\/latex] for vertical parabolas<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Direction of Opening:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal parabolas: Right if [latex]p &gt; 0[\/latex], Left if [latex]p &lt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical parabolas: Up if [latex]p &gt; 0[\/latex], Down if [latex]p &lt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Tangent Lines:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Tangent lines at the endpoints of the focal diameter intersect on the axis of symmetry<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]{y}^{2}=-16x[\/latex]. Identify and label the focus, directrix, and endpoints of the focal diameter.[reveal-answer q=\"277220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"277220\"]Focus: [latex]\\left(-4,0\\right)[\/latex]; Directrix: [latex]x=4[\/latex]; Endpoints of the latus rectum: [latex]\\left(-4,\\pm 8\\right)[\/latex]\r\n\r\n[caption id=\"attachment_3276\" align=\"aligncenter\" width=\"487\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02180919\/leftopen.jpg\"><img class=\"wp-image-3276 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02180919\/leftopen.jpg\" alt=\"\" width=\"487\" height=\"366\" \/><\/a> Parabola with a vertex at the origin and key features labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph [latex]{x}^{2}=8y[\/latex]. Identify and label the focus, directrix, and endpoints of the focal diameter.[reveal-answer q=\"824847\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"824847\"]Focus: [latex]\\left(0,2\\right)[\/latex]; Directrix: [latex]y=-2[\/latex]; Endpoints of the latus rectum: [latex]\\left(\\pm 4,2\\right)[\/latex].\r\n\r\n[caption id=\"attachment_3277\" align=\"aligncenter\" width=\"487\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02181513\/CNX_Precalc_Figure_10_03_0082.jpg\"><img class=\"wp-image-3277 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02181513\/CNX_Precalc_Figure_10_03_0082.jpg\" alt=\"\" width=\"487\" height=\"365\" \/><\/a> Parabola with a vertex at the origin and 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-aeffbbbc-k7wSPisQQYs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/k7wSPisQQYs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aeffbbbc-k7wSPisQQYs\" 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=12851286&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aeffbbbc-k7wSPisQQYs&amp;vembed=0&amp;video_id=k7wSPisQQYs&amp;video_target=tpm-plugin-aeffbbbc-k7wSPisQQYs\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College%2BAlgebra%2BCorequisite\/Transcripts\/Conic+Sections+-++The+Parabola+part+1+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConic Sections: The Parabola 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-hhacfbgc-CKepZr52G6Y\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/CKepZr52G6Y?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hhacfbgc-CKepZr52G6Y\" 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=12851287&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hhacfbgc-CKepZr52G6Y&amp;vembed=0&amp;video_id=CKepZr52G6Y&amp;video_target=tpm-plugin-hhacfbgc-CKepZr52G6Y\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College%2BAlgebra%2BCorequisite\/Transcripts\/Conic+Sections+-++The+Parabola+part+2+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConic Sections: The Parabola part 2 of 2\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Writing Equations of Parabolas in Standard Form<\/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 parabolas: [latex]y^2 = 4px[\/latex] (focus form: [latex](p, 0)[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical parabolas: [latex]x^2 = 4py[\/latex] (focus form: [latex](0, p)[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Features Used:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Focus coordinates<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Directrix equation<\/li>\r\n<\/ul>\r\n<\/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 axis of symmetry based on focus coordinates<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]4p[\/latex] using focus coordinates<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute [latex]4p[\/latex] into the appropriate standard form equation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Distinction:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">These are equations of parabolas as geometric objects, not quadratic functions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Specific language is used to describe these geometric forms<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">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]?[reveal-answer q=\"886076\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"886076\"][latex]{x}^{2}=14y[\/latex][\/hidden-answer]<\/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-fbdgdaec-dj0gCTc5Bug\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/dj0gCTc5Bug?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fbdgdaec-dj0gCTc5Bug\" 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=12851285&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbdgdaec-dj0gCTc5Bug&amp;vembed=0&amp;video_id=dj0gCTc5Bug&amp;video_target=tpm-plugin-fbdgdaec-dj0gCTc5Bug\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College%2BAlgebra%2BCorequisite\/Transcripts\/Ex+2+-+Conic+Section+-+Parabola+with+Vertical+Axis+and+Vertex+at+the+Origin+(Down)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Conic Section: Parabola with Vertical Axis and Vertex at the Origin (Down)\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-hffeffcb-f_vhsg5sspU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/f_vhsg5sspU?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hffeffcb-f_vhsg5sspU\" 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=12851288&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hffeffcb-f_vhsg5sspU&amp;vembed=0&amp;video_id=f_vhsg5sspU&amp;video_target=tpm-plugin-hffeffcb-f_vhsg5sspU\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College%2BAlgebra%2BCorequisite\/Transcripts\/Ex+3+-+Conic+Section+-+Parabola+with+Horizontal+Axis+and+Vertex+at+the+Origin+(Right)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Conic Section: Parabola with Horizontal Axis and Vertex at the Origin (Right)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Parabolas with Vertices Not 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\">Translated 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 parabolas: [latex]{\\left(y-k\\right)}^{2}=4p\\left(x-h\\right)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical parabolas: [latex]{\\left(x-h\\right)}^{2}=4p\\left(y-k\\right)[\/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\">Vertex: [latex](h, k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Axis of symmetry: [latex]x = h[\/latex] (vertical) or [latex]y = k[\/latex] (horizontal)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Focus: [latex](h+p, k)[\/latex] for horizontal, [latex](h, k+p)[\/latex] for vertical<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Directrix: [latex]x = h-p[\/latex] for horizontal, [latex]y = k-p[\/latex] for vertical<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Opening Direction:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal: Right if [latex]p &gt; 0[\/latex], Left if [latex]p &lt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical: Up if [latex]p &gt; 0[\/latex], Down if [latex]p &lt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Completing the Square:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used to convert non-standard equations into standard form<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]{\\left(y+1\\right)}^{2}=4\\left(x - 8\\right)[\/latex]. Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter.[reveal-answer q=\"626564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"626564\"]Vertex: [latex]\\left(8,-1\\right)[\/latex]; Axis of symmetry: [latex]y=-1[\/latex]; Focus: [latex]\\left(9,-1\\right)[\/latex]; Directrix: [latex]x=7[\/latex]; Endpoints of the latus rectum: [latex]\\left(9,-3\\right)[\/latex] and [latex]\\left(9,1\\right)[\/latex].\r\n\r\n[caption id=\"attachment_3280\" align=\"aligncenter\" width=\"487\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02182612\/CNX_Precalc_Figure_10_03_0112.jpg\"><img class=\"wp-image-3280 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02182612\/CNX_Precalc_Figure_10_03_0112.jpg\" alt=\"\" width=\"487\" height=\"522\" \/><\/a> Parabola with a vertex not at the origin and key features labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph [latex]{\\left(x+2\\right)}^{2}=-20\\left(y - 3\\right)[\/latex]. Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter.[reveal-answer q=\"665189\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"665189\"]Vertex: [latex]\\left(-2,3\\right)[\/latex]; Axis of symmetry: [latex]x=-2[\/latex]; Focus: [latex]\\left(-2,-2\\right)[\/latex]; Directrix: [latex]y=8[\/latex]; Endpoints of the latus rectum: [latex]\\left(-12,-2\\right)[\/latex] and [latex]\\left(8,-2\\right)[\/latex].\r\n\r\n[caption id=\"attachment_3281\" align=\"aligncenter\" width=\"731\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02182942\/CNX_Precalc_Figure_10_03_0132.jpg\"><img class=\"wp-image-3281 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02182942\/CNX_Precalc_Figure_10_03_0132.jpg\" alt=\"\" width=\"731\" height=\"438\" \/><\/a> Parabola with a vertex not at the origin and key features labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write equations of parabolas in standard form.<\/li>\n<li>Graph parabolas.<\/li>\n<li>Solve applied problems involving parabolas.<\/li>\n<\/ul>\n<\/section>\n<h2>Parabolas<\/h2>\n<p>To work with parabolas in the <strong>coordinate plane<\/strong>, we consider two cases: those with a vertex at the origin and those with a <strong>vertex<\/strong> at a point other than the origin. We begin with the former.<\/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\/03204538\/CNX_Precalc_Figure_10_03_0182.jpg\" alt=\"\" width=\"487\" height=\"292\" \/><figcaption class=\"wp-caption-text\">Parabola with a vertex at the origin and key features labeled<\/figcaption><\/figure>\n<p>Let [latex]\\left(x,y\\right)[\/latex] be a point on the parabola with vertex [latex]\\left(0,0\\right)[\/latex], focus [latex]\\left(0,p\\right)[\/latex], and directrix [latex]y= -p[\/latex]\u00a0as shown in Figure 4. The distance [latex]d[\/latex] from point [latex]\\left(x,y\\right)[\/latex] to point [latex]\\left(x,-p\\right)[\/latex]\u00a0on the directrix is the difference of the <em>y<\/em>-values: [latex]d=y+p[\/latex]. The distance from the focus [latex]\\left(0,p\\right)[\/latex] to the point [latex]\\left(x,y\\right)[\/latex] is also equal to [latex]d[\/latex] and can be expressed using the <strong>distance formula<\/strong>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}d&=\\sqrt{{\\left(x - 0\\right)}^{2}+{\\left(y-p\\right)}^{2}} \\\\ &=\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}} \\end{align}[\/latex]<\/p>\n<p>Set the two expressions for [latex]d[\/latex] equal to each other and solve for [latex]y[\/latex] to derive the equation of the parabola. We do this because the distance from [latex]\\left(x,y\\right)[\/latex] to [latex]\\left(0,p\\right)[\/latex] equals the distance from [latex]\\left(x,y\\right)[\/latex] to [latex]\\left(x, -p\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}}=y+p[\/latex]<\/p>\n<p>We then square both sides of the equation, expand the squared terms, and simplify by combining like terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}{x}^{2}+{\\left(y-p\\right)}^{2}={\\left(y+p\\right)}^{2} \\\\ {x}^{2}+{y}^{2}-2py+{p}^{2}={y}^{2}+2py+{p}^{2}\\\\ {x}^{2}-2py=2py \\\\ {x}^{2}=4py\\end{gathered}[\/latex]<\/p>\n<p>The equations of parabolas with vertex [latex]\\left(0,0\\right)[\/latex] are [latex]{y}^{2}=4px[\/latex] when the <em>x<\/em>-axis is the axis of symmetry and [latex]{x}^{2}=4py[\/latex] when the <em>y<\/em>-axis is the axis of symmetry.<\/p>\n<h2 data-type=\"title\">Graphing Parabolas with Vertices 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\">Vertical parabolas: [latex]x^2 = 4py[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal parabolas: [latex]y^2 = 4px[\/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\">Vertex: Always at [latex](0, 0)[\/latex] for these standard forms<\/li>\n<li class=\"whitespace-normal break-words\">Axis of symmetry: [latex]x[\/latex]-axis for horizontal parabolas, [latex]y[\/latex]-axis for vertical parabolas<\/li>\n<li class=\"whitespace-normal break-words\">Focus: [latex](p, 0)[\/latex] for horizontal parabolas, [latex](0, p)[\/latex] for vertical parabolas<\/li>\n<li class=\"whitespace-normal break-words\">Directrix: [latex]x = -p[\/latex] for horizontal parabolas, [latex]y = -p[\/latex] for vertical parabolas<\/li>\n<li class=\"whitespace-normal break-words\">Focal diameter endpoints: [latex](p, \u00b12p)[\/latex] for horizontal parabolas, [latex](\u00b12p, p)[\/latex] for vertical parabolas<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Direction of Opening:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal parabolas: Right if [latex]p > 0[\/latex], Left if [latex]p < 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical parabolas: Up if [latex]p > 0[\/latex], Down if [latex]p < 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Tangent Lines:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Tangent lines at the endpoints of the focal diameter intersect on the axis of symmetry<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]{y}^{2}=-16x[\/latex]. Identify and label the focus, directrix, and endpoints of the focal diameter.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q277220\">Show Solution<\/button><\/p>\n<div id=\"q277220\" class=\"hidden-answer\" style=\"display: none\">Focus: [latex]\\left(-4,0\\right)[\/latex]; Directrix: [latex]x=4[\/latex]; Endpoints of the latus rectum: [latex]\\left(-4,\\pm 8\\right)[\/latex]<\/p>\n<figure id=\"attachment_3276\" aria-describedby=\"caption-attachment-3276\" 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\/02180919\/leftopen.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3276 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02180919\/leftopen.jpg\" alt=\"\" width=\"487\" height=\"366\" \/><\/a><figcaption id=\"caption-attachment-3276\" class=\"wp-caption-text\">Parabola with a vertex at the origin and key features labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]{x}^{2}=8y[\/latex]. Identify and label the focus, directrix, and endpoints of the focal diameter.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q824847\">Show Solution<\/button><\/p>\n<div id=\"q824847\" class=\"hidden-answer\" style=\"display: none\">Focus: [latex]\\left(0,2\\right)[\/latex]; Directrix: [latex]y=-2[\/latex]; Endpoints of the latus rectum: [latex]\\left(\\pm 4,2\\right)[\/latex].<\/p>\n<figure id=\"attachment_3277\" aria-describedby=\"caption-attachment-3277\" 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\/02181513\/CNX_Precalc_Figure_10_03_0082.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3277 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02181513\/CNX_Precalc_Figure_10_03_0082.jpg\" alt=\"\" width=\"487\" height=\"365\" \/><\/a><figcaption id=\"caption-attachment-3277\" class=\"wp-caption-text\">Parabola with a vertex at the origin and 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-aeffbbbc-k7wSPisQQYs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/k7wSPisQQYs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aeffbbbc-k7wSPisQQYs\" 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=12851286&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aeffbbbc-k7wSPisQQYs&amp;vembed=0&amp;video_id=k7wSPisQQYs&amp;video_target=tpm-plugin-aeffbbbc-k7wSPisQQYs\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College%2BAlgebra%2BCorequisite\/Transcripts\/Conic+Sections+-++The+Parabola+part+1+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConic Sections: The Parabola 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-hhacfbgc-CKepZr52G6Y\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/CKepZr52G6Y?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hhacfbgc-CKepZr52G6Y\" 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=12851287&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hhacfbgc-CKepZr52G6Y&amp;vembed=0&amp;video_id=CKepZr52G6Y&amp;video_target=tpm-plugin-hhacfbgc-CKepZr52G6Y\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College%2BAlgebra%2BCorequisite\/Transcripts\/Conic+Sections+-++The+Parabola+part+2+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConic Sections: The Parabola part 2 of 2\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Writing Equations of Parabolas in Standard Form<\/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 parabolas: [latex]y^2 = 4px[\/latex] (focus form: [latex](p, 0)[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Vertical parabolas: [latex]x^2 = 4py[\/latex] (focus form: [latex](0, p)[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Features Used:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Focus coordinates<\/li>\n<li class=\"whitespace-normal break-words\">Directrix equation<\/li>\n<\/ul>\n<\/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 axis of symmetry based on focus coordinates<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]4p[\/latex] using focus coordinates<\/li>\n<li class=\"whitespace-normal break-words\">Substitute [latex]4p[\/latex] into the appropriate standard form equation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Distinction:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">These are equations of parabolas as geometric objects, not quadratic functions<\/li>\n<li class=\"whitespace-normal break-words\">Specific language is used to describe these geometric forms<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">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<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q886076\">Show Solution<\/button><\/p>\n<div id=\"q886076\" class=\"hidden-answer\" style=\"display: none\">[latex]{x}^{2}=14y[\/latex]<\/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-fbdgdaec-dj0gCTc5Bug\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/dj0gCTc5Bug?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fbdgdaec-dj0gCTc5Bug\" 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=12851285&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbdgdaec-dj0gCTc5Bug&amp;vembed=0&amp;video_id=dj0gCTc5Bug&amp;video_target=tpm-plugin-fbdgdaec-dj0gCTc5Bug\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College%2BAlgebra%2BCorequisite\/Transcripts\/Ex+2+-+Conic+Section+-+Parabola+with+Vertical+Axis+and+Vertex+at+the+Origin+(Down)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Conic Section: Parabola with Vertical Axis and Vertex at the Origin (Down)\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-hffeffcb-f_vhsg5sspU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/f_vhsg5sspU?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hffeffcb-f_vhsg5sspU\" 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=12851288&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hffeffcb-f_vhsg5sspU&amp;vembed=0&amp;video_id=f_vhsg5sspU&amp;video_target=tpm-plugin-hffeffcb-f_vhsg5sspU\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College%2BAlgebra%2BCorequisite\/Transcripts\/Ex+3+-+Conic+Section+-+Parabola+with+Horizontal+Axis+and+Vertex+at+the+Origin+(Right)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Conic Section: Parabola with Horizontal Axis and Vertex at the Origin (Right)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Parabolas with Vertices Not 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\">Translated Standard Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal parabolas: [latex]{\\left(y-k\\right)}^{2}=4p\\left(x-h\\right)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical parabolas: [latex]{\\left(x-h\\right)}^{2}=4p\\left(y-k\\right)[\/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\">Vertex: [latex](h, k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Axis of symmetry: [latex]x = h[\/latex] (vertical) or [latex]y = k[\/latex] (horizontal)<\/li>\n<li class=\"whitespace-normal break-words\">Focus: [latex](h+p, k)[\/latex] for horizontal, [latex](h, k+p)[\/latex] for vertical<\/li>\n<li class=\"whitespace-normal break-words\">Directrix: [latex]x = h-p[\/latex] for horizontal, [latex]y = k-p[\/latex] for vertical<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Opening Direction:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal: Right if [latex]p > 0[\/latex], Left if [latex]p < 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical: Up if [latex]p > 0[\/latex], Down if [latex]p < 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Completing the Square:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used to convert non-standard equations into standard form<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]{\\left(y+1\\right)}^{2}=4\\left(x - 8\\right)[\/latex]. Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q626564\">Show Solution<\/button><\/p>\n<div id=\"q626564\" class=\"hidden-answer\" style=\"display: none\">Vertex: [latex]\\left(8,-1\\right)[\/latex]; Axis of symmetry: [latex]y=-1[\/latex]; Focus: [latex]\\left(9,-1\\right)[\/latex]; Directrix: [latex]x=7[\/latex]; Endpoints of the latus rectum: [latex]\\left(9,-3\\right)[\/latex] and [latex]\\left(9,1\\right)[\/latex].<\/p>\n<figure id=\"attachment_3280\" aria-describedby=\"caption-attachment-3280\" 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\/02182612\/CNX_Precalc_Figure_10_03_0112.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3280 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02182612\/CNX_Precalc_Figure_10_03_0112.jpg\" alt=\"\" width=\"487\" height=\"522\" \/><\/a><figcaption id=\"caption-attachment-3280\" class=\"wp-caption-text\">Parabola with a vertex not at the origin and key features labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]{\\left(x+2\\right)}^{2}=-20\\left(y - 3\\right)[\/latex]. Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the focal diameter.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q665189\">Show Solution<\/button><\/p>\n<div id=\"q665189\" class=\"hidden-answer\" style=\"display: none\">Vertex: [latex]\\left(-2,3\\right)[\/latex]; Axis of symmetry: [latex]x=-2[\/latex]; Focus: [latex]\\left(-2,-2\\right)[\/latex]; Directrix: [latex]y=8[\/latex]; Endpoints of the latus rectum: [latex]\\left(-12,-2\\right)[\/latex] and [latex]\\left(8,-2\\right)[\/latex].<\/p>\n<figure id=\"attachment_3281\" aria-describedby=\"caption-attachment-3281\" 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\/02182942\/CNX_Precalc_Figure_10_03_0132.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3281 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02182942\/CNX_Precalc_Figure_10_03_0132.jpg\" alt=\"\" width=\"731\" height=\"438\" \/><\/a><figcaption id=\"caption-attachment-3281\" class=\"wp-caption-text\">Parabola with a vertex not at the origin and key features labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":21,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Conic Sections: The Parabola part 1 of 2\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/k7wSPisQQYs\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Conic Sections: The Parabola part 2 of 2\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/CKepZr52G6Y\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Conic Section: Parabola with Vertical Axis and Vertex at the Origin (Down)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/dj0gCTc5Bug\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 3: Conic Section: Parabola with Horizontal Axis and Vertex at the Origin (Right)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/f_vhsg5sspU\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Conic Sections: The Parabola part 1 of 2","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/k7wSPisQQYs","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Conic Sections: The Parabola part 2 of 2","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/CKepZr52G6Y","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 2: Conic Section: Parabola with Vertical Axis and Vertex at the Origin (Down)","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/dj0gCTc5Bug","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 3: Conic Section: Parabola with Horizontal Axis and Vertex at the Origin (Right)","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/f_vhsg5sspU","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851286&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-aeffbbbc-k7wSPisQQYs&vembed=0&video_id=k7wSPisQQYs&video_target=tpm-plugin-aeffbbbc-k7wSPisQQYs'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851287&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hhacfbgc-CKepZr52G6Y&vembed=0&video_id=CKepZr52G6Y&video_target=tpm-plugin-hhacfbgc-CKepZr52G6Y'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851285&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fbdgdaec-dj0gCTc5Bug&vembed=0&video_id=dj0gCTc5Bug&video_target=tpm-plugin-fbdgdaec-dj0gCTc5Bug'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851288&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hffeffcb-f_vhsg5sspU&vembed=0&video_id=f_vhsg5sspU&video_target=tpm-plugin-hffeffcb-f_vhsg5sspU'><\/script>\n","media_targets":["tpm-plugin-aeffbbbc-k7wSPisQQYs","tpm-plugin-hhacfbgc-CKepZr52G6Y","tpm-plugin-fbdgdaec-dj0gCTc5Bug","tpm-plugin-hffeffcb-f_vhsg5sspU"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1597"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1597\/revisions"}],"predecessor-version":[{"id":5641,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1597\/revisions\/5641"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/522"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1597\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1597"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1597"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1597"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1597"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}