{"id":28892,"date":"2021-04-23T13:00:40","date_gmt":"2021-04-23T13:00:40","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28892"},"modified":"2024-09-22T02:10:41","modified_gmt":"2024-09-22T02:10:41","slug":"the-equation-of-parabolas-definitions-and-properties","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/the-equation-of-parabolas-definitions-and-properties\/","title":{"rendered":"The Equation of Parabolas: Definitions and Properties"},"content":{"rendered":"<p><center><\/p>\n<h1>The Equation of Parabolas: Definitions and Properties<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\n   This class explores the definition and derivation of the equation of a parabola, highlighting its origin as the set of points equidistant from a focus and a directrix. Based on this concept, previous notions such as the distance between points in the Cartesian plane and the translation of graphs are reviewed, allowing the introduction of the fundamental equation of parabolas and its relationship with second-degree polynomials. Finally, the general equation of parabolas with vertices at any point is deduced and transformed into the canonical form of a quadratic polynomial.<br \/>\n   <\/em><\/p>\n<p><strong>Learning Objectives:<\/strong><br \/>\n   By the end of this class, the student will be able to:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Understand<\/strong> the geometric definition of a parabola as the set of points equidistant from a focus and a directrix.<\/li>\n<li><strong>Derive<\/strong> the fundamental equation of the parabola using the focus-directrix distance relationship.<\/li>\n<li><strong>Understand<\/strong> the relationship between parabolas and second-degree polynomials.<\/li>\n<li><strong>Derive<\/strong> the general equation of parabolas with a vertex at any point (h,k).<\/li>\n<\/ol>\n<p>   <strong>CONTENTS INDEX<\/strong><br \/>\n   <a href=\"#1\"><strong>Preliminary ideas for obtaining the equation of parabolas<\/strong><\/a><br \/>\n   <a href=\"#2\">Geometric notion of parabolas<\/a><br \/>\n   <a href=\"#3\">Distance between two points in the Cartesian plane<\/a><br \/>\n   <a href=\"#4\">Graph Translation<\/a><br \/>\n   <a href=\"#5\"><strong>Definition of a Parabola<\/strong><\/a><br \/>\n   <a href=\"#6\">Derivation of the Fundamental Equation of Parabolas<\/a><br \/>\n   <a href=\"#7\">The General Equation of Parabolas<\/a><br \/>\n   <a href=\"#8\">Canonical Equation of Parabolas and Second-Degree Polynomials<\/a>\n   <\/p>\n<p>   <\/center><\/p>\n<p>   <center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/O2EYxxK03jU\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>   <a name=\"1\"><\/a><\/p>\n<h2>Preliminary ideas for obtaining the equation of parabolas<\/h2>\n<p>   <a name=\"2\"><\/a><\/p>\n<h3>Geometric notion of parabolas<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=130s\" target=\"_blank\" rel=\"noopener\"><strong>A parabola is the curve <\/strong><\/a>that is obtained as the collection of all points equidistant from a fixed point, called the focus, and a fixed line called the directrix. To understand this definition and transform it into an algebraic expression we can manipulate, the equation of parabolas, we first need to review some preliminary concepts<\/p>\n<p>   <a name=\"3\"><\/a><\/p>\n<h3>Distance between two points in the Cartesian plane<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=199s\" target=\"_blank\" rel=\"noopener\"><strong>Consider two points<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">p_1 = (x_1, y_1)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">p_2 = (x_2, y_2).<\/span> The distance between these points is the length of the straight line segment connecting them<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-Nm00_Mr0PJE\/YIH_4rGiO0I\/AAAAAAAAE-s\/gLJ6gjIGUuwsDhwVt6pa3MDZ0YQXEgJVgCLcBGAsYHQ\/s0\/distancia.PNG\" alt=\"Distance between points\" class=\"aligncenter lazyload\" width=\"409\" height=\"263\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-Nm00_Mr0PJE\/YIH_4rGiO0I\/AAAAAAAAE-s\/gLJ6gjIGUuwsDhwVt6pa3MDZ0YQXEgJVgCLcBGAsYHQ\/s0\/distancia.PNG\" alt=\"Distance between points\" class=\"aligncenter lazyload\" width=\"409\" height=\"263\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">We can measure this distance through the <a href=\"https:\/\/youtu.be\/grjSzVl8Acw?t=553\" target=\"_blank\" rel=\"noopener\">Pythagorean theorem<\/a> by making the following figure<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-xOGFNRBeUU0\/YIIAohgMqNI\/AAAAAAAAE-0\/LCiSnSMHVwMpTGOnTchVFtkVL0izESn3ACLcBGAsYHQ\/s0\/teorema%2Bde%2Bpitagoras.PNG\" alt=\"Pythagorean theorem\" class=\" aligncenter lazyload\" width=\"404\" height=\"255\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-xOGFNRBeUU0\/YIIAohgMqNI\/AAAAAAAAE-0\/LCiSnSMHVwMpTGOnTchVFtkVL0izESn3ACLcBGAsYHQ\/s0\/teorema%2Bde%2Bpitagoras.PNG\" alt=\"Pythagorean theorem\" class=\" aligncenter lazyload\" width=\"404\" height=\"255\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Thus, the distance <span class=\"katex-eq\" data-katex-display=\"false\">d<\/span> between the two points will be<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">d= \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}<\/span>\n<p>   <a name=\"4\"><\/a><\/p>\n<h3>Graph Translation<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=390s\" target=\"_blank\" rel=\"noopener\"><strong>Consider a function <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">y(x) = x^2<\/span>. If we graph this, it will look like what is shown in the figure<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-hf1AvDmNoYc\/YIIhv8_zkjI\/AAAAAAAAE_E\/kedKQoFwJSIkoIZgev9cQS--frmDbwTOwCLcBGAsYHQ\/s0\/traslacionDeGraficos1.PNG\" alt=\"Graph Translation\" class=\" aligncenter lazyload\" width=\"325\" height=\"321\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-hf1AvDmNoYc\/YIIhv8_zkjI\/AAAAAAAAE_E\/kedKQoFwJSIkoIZgev9cQS--frmDbwTOwCLcBGAsYHQ\/s0\/traslacionDeGraficos1.PNG\" alt=\"Graph Translation\" class=\" aligncenter lazyload\" width=\"325\" height=\"321\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">If in this function we replace <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> by <span class=\"katex-eq\" data-katex-display=\"false\">x-1<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> by <span class=\"katex-eq\" data-katex-display=\"false\">y-1,<\/span>, we will then see the following transformation in the graph<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-TRHquCBRoB4\/YIIjbaFx5WI\/AAAAAAAAE_M\/ngYFLBH4y6AS4ywUaSwo2sscmAVT6viHgCLcBGAsYHQ\/s0\/traslacionDeGraficos2.PNG\" alt=\"Graph Translation\" class=\" aligncenter lazyload\" width=\"352\" height=\"288\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-TRHquCBRoB4\/YIIjbaFx5WI\/AAAAAAAAE_M\/ngYFLBH4y6AS4ywUaSwo2sscmAVT6viHgCLcBGAsYHQ\/s0\/traslacionDeGraficos2.PNG\" alt=\"Graph Translation\" class=\" aligncenter lazyload\" width=\"352\" height=\"288\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">In general, each such replacement produces a translation transformation, namely<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong><span class=\"katex-eq\" data-katex-display=\"false\">x\\longmapsto x-a<\/span>:<\/strong> if <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> is positive, it moves <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> units to the right; if it is negative, it moves to the left.<\/li>\n<li><strong><span class=\"katex-eq\" data-katex-display=\"false\">y\\longmapsto y-b<\/span>: <\/strong>if <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> is positive, it moves <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> units up; if negative, it moves down.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">These are the translation transformations, and their general effect is summarized in the following figure<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-anQa6JMVTyA\/YIIm1AgBVbI\/AAAAAAAAE_U\/7aKD9WSMVawEwJXOnpqpr-cskL-NxoPUgCLcBGAsYHQ\/s0\/traslacionDeGraficos3.PNG\" alt=\"General graph translation\" class=\"aligncenter lazyload\" width=\"522\" height=\"314\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-anQa6JMVTyA\/YIIm1AgBVbI\/AAAAAAAAE_U\/7aKD9WSMVawEwJXOnpqpr-cskL-NxoPUgCLcBGAsYHQ\/s0\/traslacionDeGraficos3.PNG\" alt=\"General graph translation\" class=\"aligncenter lazyload\" width=\"522\" height=\"314\" \/><\/noscript><br \/>\n   <a name=\"5\"><\/a><\/p>\n<h2>Definition of a Parabola<\/h2>\n<p style=\"text-align: justify;\">A parabola is the set of all points that are equidistant from a fixed point and a line.<br \/>\n   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-16fe-qg141I\/YIH4NpKtc4I\/AAAAAAAAE-k\/ILLGwpB_IQQZbUBJvXOIl5HChpo8Noi_gCLcBGAsYHQ\/s0\/Parabola.PNG\" alt=\"Definition of Parabola\" class=\"aligncenter lazyload\" width=\"426\" height=\"302\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-16fe-qg141I\/YIH4NpKtc4I\/AAAAAAAAE-k\/ILLGwpB_IQQZbUBJvXOIl5HChpo8Noi_gCLcBGAsYHQ\/s0\/Parabola.PNG\" alt=\"Definition of Parabola\" class=\"aligncenter lazyload\" width=\"426\" height=\"302\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">The fixed point is called the <strong>focus<\/strong>, and the line is the <strong>directrix. <\/strong> If we pay attention, we will see that the notion of distance is fundamental in defining parabolas, so to delve deeper into their analysis, it will be necessary to review how distances are measured in the Cartesian plane and how they are algebraically obtained.<\/p>\n<p>   <a name=\"6\"><\/a><\/p>\n<h3>Derivation of the Fundamental Equation of Parabolas<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=604s\" target=\"_blank\" rel=\"noopener\"><strong>For simplicity, consider the focal point<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">p_f= (0,f)<\/span> and the directrix as the line <span class=\"katex-eq\" data-katex-display=\"false\">L<\/span> with the equation <span class=\"katex-eq\" data-katex-display=\"false\">y=-p<\/span>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-XfDceUwcaFM\/YIIZEUT6DrI\/AAAAAAAAE-8\/keal2s_uSooVvkKsa3Sw1roANTDH8592QCLcBGAsYHQ\/s0\/analisisParabola.PNG\" alt=\"The Equation of Parabolas\" class=\" aligncenter lazyload\" width=\"334\" height=\"249\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-XfDceUwcaFM\/YIIZEUT6DrI\/AAAAAAAAE-8\/keal2s_uSooVvkKsa3Sw1roANTDH8592QCLcBGAsYHQ\/s0\/analisisParabola.PNG\" alt=\"The Equation of Parabolas\" class=\" aligncenter lazyload\" width=\"334\" height=\"249\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">If we take any point on the parabola with coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(x,y)<\/span>, then it will be equidistant from both the focus and the directrix. This can be described algebraically as follows:<\/p>\n<p style=\"text-align: center;\">Distance Focus-Point(x,y) <span class=\"katex-eq\" data-katex-display=\"false\">= \\sqrt{x^2 + (f-y)^2}= y+f =<\/span> Distance Point(x,y)-Directrix<\/p>\n<p style=\"text-align: justify;\">And from this, the following reasoning develops:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{x^2 + (f-y)^2}= y+f<\/span><\/td>\n<td>; Distance point-focus = distance point-directrix, Def. of parabola<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">x^2 + (f-y)^2= (y+f)^2<\/span><\/td>\n<td>; From <span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span>, squaring both sides<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">x^2 + \\cancel{f^2} - 2fy + \\cancel{y^2}= \\cancel{y^2} + 2fy + \\cancel{f^2}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">x^2 - 2fy = 2fy <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{y=\\dfrac{x^2}{4f}}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">This is what we call the <strong>Fundamental Equation of Parabolas.<\/strong><\/p>\n<p style=\"text-align: justify;\">If we pay attention to this parabola, we will notice that there is a point with the property of being closest to the focus (or equivalently to the directrix). This point is called the <strong>vertex,<\/strong> and for this particular case, it has coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(0,0)<\/span>; the distance between the focus and the vertex is called the <strong>focal distance,<\/strong> and its value <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> can be any real number except zero.<\/p>\n<p style=\"text-align: justify;\">When <span class=\"katex-eq\" data-katex-display=\"false\">f\\gt 0<\/span>, the parabola opens upwards, and conversely, when <span class=\"katex-eq\" data-katex-display=\"false\">f\\lt 0 <\/span>, it opens downwards. As <span class=\"katex-eq\" data-katex-display=\"false\">f\\to 0<\/span>, the parabola flattens, keeping the vertex in position and the directrix moves closer to the vertex, until the parabola and the directrix merge into a single line. When <span class=\"katex-eq\" data-katex-display=\"false\">f=0<\/span>, the graph disappears as divisions by zero do not exist.<\/p>\n<p>   <a name=\"7\"><\/a><\/p>\n<h3>The General Equation of Parabolas<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=1007s\" target=\"_blank\" rel=\"noopener\"><strong>From the fundamental equation of parabolas<\/strong><\/a> and graph translation, by substituting <span class=\"katex-eq\" data-katex-display=\"false\">x\\longmapsto (x-h)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">y\\longmapsto (y-k),<\/span> we obtain the <strong>General Equation of Parabolas<\/strong> with a vertex at <span class=\"katex-eq\" data-katex-display=\"false\">(h,k)<\/span>.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(y-k) = \\dfrac{(x-h)^2}{4f}<\/span>\n<p>   <a name=\"8\"><\/a><\/p>\n<h3>Canonical Equation of Parabolas and Second-Degree Polynomials<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=1116s\" target=\"_blank\" rel=\"noopener\"><strong>If we expand the general equation<\/strong><\/a> of parabolas, the following reasoning arises:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(y-k) = \\dfrac{(x-h)^2}{4f}<\/span><\/td>\n<td>; General equation of parabolas<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> 4f(y-k) = (x-h)^2<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> 4fy-4fk = x^2 - 2hx + h^2<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> 4fy = x^2 - 2hx + h^2 + 4fk<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> y = \\dfrac{1}{4f}x^2 - \\dfrac{h}{2f}x + \\dfrac{h^2 + 4fk}{4f}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">If we substitute <span class=\"katex-eq\" data-katex-display=\"false\">a=\\dfrac{1}{4f},<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">b=-\\dfrac{2h}{4f}<\/span>, and <span class=\"katex-eq\" data-katex-display=\"false\">c=\\dfrac{h^2 + 4fk}{4f},<\/span>, then the general equation of parabolas transforms into the canonical equation, which turns out to be the <a href=\"https:\/\/toposuranos.com\/factorizacion-del-polinomio-cuadratico-y-2n-cuadratico\/\" rel=\"noopener\" target=\"_blank\">quadratic polynomial<\/a>.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{y=ax^2 + bx + c}<\/span>\n","protected":false},"excerpt":{"rendered":"<p>The Equation of Parabolas: Definitions and Properties Summary: This class explores the definition and derivation of the equation of a parabola, highlighting its origin as the set of points equidistant from a focus and a directrix. Based on this concept, previous notions such as the distance between points in the Cartesian plane and the translation [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28890,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":13,"footnotes":""},"categories":[583,567],"tags":[],"class_list":["post-28892","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-and-geometry","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>The Equation of Parabolas: Definitions and Properties - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"In this class on the equation of parabolas, you will learn to derive the fundamental equation and its relationship with second-degree polynomials.\" \/>\n<meta name=\"robots\" content=\"index, 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