{"id":28930,"date":"2021-04-27T13:00:20","date_gmt":"2021-04-27T13:00:20","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28930"},"modified":"2024-09-22T02:14:52","modified_gmt":"2024-09-22T02:14:52","slug":"characterization-of-parabolas-and-their-graphs","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/","title":{"rendered":"Characterization of Parabolas and their Graphs"},"content":{"rendered":"<p><center><\/p>\n<h1>Characterization of Parabolas and their Graphs<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\n   In this class, we will review the characterization of parabolas based on their general and canonical forms, explaining how to identify key elements such as the vertex, focus, directrix, axis of symmetry, and possible intersections with the x-axis.<br \/>\n   <\/em><br \/>\n   <strong>Learning Objectives:<\/strong><br \/>\n   By the end of this class, students will be able to:<\/p>\n<ol style=\"text-align: left;\">\n<li><strong>Calculate<\/strong> the position of the vertex, focus, and directrix of the parabola from its general and canonical forms.<\/li>\n<li><strong>Transform<\/strong> the canonical equation into the general form to extract geometric information.<\/li>\n<li><strong>Sketch<\/strong> the graph of the parabola using the obtained information.<\/li>\n<\/ol>\n<p>   <strong>CONTENTS INDEX<\/strong><br \/>\n   <a href=\"#1\">General and Canonical Forms of Parabolas<\/a><br \/>\n   <a href=\"#2\">Characterization of Parabolas from the General Equation<\/a><br \/>\n   <a href=\"#3\">Characterization of Parabolas from the Canonical Equation<\/a><br \/>\n   <a href=\"#4\">Automatic Characterization with Excel<\/a>\n   <\/p>\n<p>   <\/center><br \/>\n   <center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/C6DbrJDiZTM\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n   <a name=\"1\"><\/a><\/p>\n<h2>General and Canonical Forms of Parabolas<\/h2>\n<p style=\"text-align: justify;\">In the previous class, we saw that parabolas can be expressed algebraically through the general equation of parabolas as follows.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(x-x_0)^2 =4f(y-y_0)<\/span>\n<p style=\"text-align: justify;\">Where the pair <span class=\"katex-eq\" data-katex-display=\"false\">(x_0,y_0)<\/span> represents the position of the vertex and <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> is the focal distance. If <span class=\"katex-eq\" data-katex-display=\"false\">f \\gt 0<\/span>, the focus is at a distance <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> above the vertex, and if <span class=\"katex-eq\" data-katex-display=\"false\">f \\lt 0,<\/span> the focus is at a distance <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> below the vertex.<\/p>\n<p style=\"text-align: justify;\">We have also seen that the equation of parabolas in its canonical form is equivalent to a second-degree polynomial.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x) = ax^2 + bx + c,<\/span> with <span class=\"katex-eq\" data-katex-display=\"false\">a \\neq 0<\/span>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=C6DbrJDiZTM&amp;t=228s\" target=\"_blank\" rel=\"noopener\"><strong>Characterizing a parabola<\/strong><\/a> consists of revealing the following information:<\/p>\n<ul style=\"text-align: justify;\">\n<li>The coordinates of the vertex<\/li>\n<li>The coordinates of the focus<\/li>\n<li>The equation of the directrix<\/li>\n<li>The equation of the axis of symmetry<\/li>\n<li>The intersections with the x-axis (if they exist)<\/li>\n<li>Finally, constructing a sketch of the graph using the collected information.<\/li>\n<\/ul>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-xAUdvfTRbjw\/YIbmIXDdT-I\/AAAAAAAAFAI\/8NH0t_EWbH0KIuFDnsRu2IyHdyN4WU54wCLcBGAsYHQ\/s0\/caracterizaci%25C3%25B3n%2Bde%2Bpar%25C3%25A1bolas.PNG\" alt=\"Characterization of Parabolas\" class=\"aligncenter lazyload\" width=\"537\" height=\"414\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-xAUdvfTRbjw\/YIbmIXDdT-I\/AAAAAAAAFAI\/8NH0t_EWbH0KIuFDnsRu2IyHdyN4WU54wCLcBGAsYHQ\/s0\/caracterizaci%25C3%25B3n%2Bde%2Bpar%25C3%25A1bolas.PNG\" alt=\"Characterization of Parabolas\" class=\"aligncenter lazyload\" width=\"537\" height=\"414\" \/><\/noscript><br \/>\n   <a name=\"2\"><\/a><\/p>\n<h2>Characterization of Parabolas from the General Equation<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=C6DbrJDiZTM&amp;t=316s\" target=\"_blank\" rel=\"noopener\"><strong>If you have the parabola described through <\/strong><\/a>the general equation, then you already have almost all the information necessary to complete the characterization, only the intersections with the x-axis will require additional analysis.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(x-x_0)^2 =4f(y-y_0)<\/span>\n<p style=\"text-align: justify;\">From this, you already have:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Vertex:<\/strong> The point with coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(x_0,y_0)<\/span><\/li>\n<li><strong>Focal position:<\/strong> At <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> units above the vertex<\/li>\n<li><strong>Focus:<\/strong> The point with coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(x_0,y_0 + f)<\/span><\/li>\n<li><strong>Directrix:<\/strong> The line with the equation <span class=\"katex-eq\" data-katex-display=\"false\">y= y_0 - f<\/span><\/li>\n<li><strong>Axis of symmetry:<\/strong> The line with the equation <span class=\"katex-eq\" data-katex-display=\"false\">x= x_0<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">To find the intersections with the x-axis, you will need to convert the general equation to its canonical form, set the resulting quadratic polynomial equal to zero. If there are solutions, these will be the intersections with the x-axis.<\/p>\n<p>   <a name=\"3\"><\/a><\/p>\n<h2>Characterization of Parabolas from the Canonical Equation<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=C6DbrJDiZTM&amp;t=393s\" target=\"_blank\" rel=\"noopener\"><strong>When the equation of the parabola<\/strong><\/a> is presented in its canonical form, you have two options: 1) Characterize by transforming into the general equation, or 2) Use the symmetry and the intersections with the x-axis. Both methods have their merits. The second is generally faster, but parabolas do not always intersect the x-axis, the first is more laborious, but as we will see later, it is easy to automate. We will examine both alternatives so you can choose according to your preferences and needs which path to take.<\/p>\n<h3>Transforming into the General Equation<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=C6DbrJDiZTM&amp;t=475s\" target=\"_blank\" rel=\"noopener\"><strong>The transformation to the general form<\/strong><\/a> is done through the following reasoning, where <span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{R}<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0.<\/span>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td width=\"50\">(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=ax^2 + bx + c<\/span><\/td>\n<td>; Canonical equation of parabolas<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=a\\left[x^2 + \\dfrac{b}{a}x + \\dfrac{c}{a}\\right]<\/span><\/td>\n<td>; Factoring by <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=a\\left[ \\left(x + \\dfrac{b}{2a}\\right)^2 - \\dfrac{b^2}{4a^2} + \\dfrac{c}{a}\\right]<\/span><\/td>\n<td>; Because <span class=\"katex-eq\" data-katex-display=\"false\">\\left(x + \\dfrac{b}{2a}\\right)^2 = x^2 + \\dfrac{b}{a}x + \\dfrac{b^2}{4a^2}<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=a\\left[ \\left(x + \\dfrac{b}{2a}\\right)^2 + \\dfrac{4ac - b^2}{4a^2} \\right]<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=a \\left(x + \\dfrac{b}{2a}\\right)^2 + \\dfrac{4ac - b^2}{4a}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=a \\left(x + \\dfrac{b}{2a}\\right)^2 + \\left(c - \\dfrac{b^2}{4a}\\right)<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\left[x - \\left(- \\dfrac{b}{2a}\\right)\\right]^2 = \\dfrac{1}{a} \\left[y - \\left(c - \\dfrac{b^2}{4a}\\right)\\right]<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\left[x - \\left( -\\dfrac{b}{2a}\\right)\\right]^2 = 4\\left(\\dfrac{1}{4a}\\right) \\left[y - \\left(c - \\dfrac{b^2}{4a}\\right)\\right]<\/span><\/td>\n<td>; General form of parabolas<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">From this, we can extract all the information we had from the general equation by relating its parameters with those of the canonical equation, thus we have:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Vertex:<\/strong> The point with coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(x_0,y_0) = \\left(-\\dfrac{b}{2a}, c -\\dfrac{b^2}{4a} \\right)<\/span><\/li>\n<li><strong>Focal position:<\/strong> At <span class=\"katex-eq\" data-katex-display=\"false\">f = \\dfrac{1}{4a}<\/span> units above the vertex<\/li>\n<li><strong>Focus:<\/strong> The point with coordinates <span class=\"katex-eq\" data-katex-display=\"false\">(x_0,y_0 + f) = \\left(-\\dfrac{b}{2a}, c -\\dfrac{b^2}{4a} + \\dfrac{1}{4a}\\right) =\\left(-\\dfrac{b}{2a}, c +\\dfrac{1-b^2}{4a}\\right) <\/span><\/li>\n<li><strong>Directrix:<\/strong> The line with the equation <span class=\"katex-eq\" data-katex-display=\"false\">y=y_0 - f= c -\\dfrac{b^2}{4a} - \\dfrac{1}{4a} = c -\\dfrac{1 + b^2}{4a}<\/span><\/li>\n<li><strong>Axis of symmetry:<\/strong> The line with the equation <span class=\"katex-eq\" data-katex-display=\"false\">x= x_0 = -\\dfrac{b}{2a}<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">And from here, the characterization of parabolas is done as we have already seen using the general equation.<\/p>\n<h3>Using Symmetry and x-axis Intersections<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=C6DbrJDiZTM&amp;t=769s\" target=\"_blank\" rel=\"noopener\"><strong>When we have the equation <\/strong><\/a>of parabolas written in canonical form <span class=\"katex-eq\" data-katex-display=\"false\">y=ax^2 + bx+c<\/span>, we see that it is relatively simple to calculate its intersections with the x-axis, just solve the equation:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">ax^2 + bx + c = 0<\/span>\n<p style=\"text-align: justify;\">When this is possible, we get intersections <span class=\"katex-eq\" data-katex-display=\"false\">x_1<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">x_2<\/span> given by:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1 = \\dfrac{-b + \\sqrt{b^2-4ac}}{2a}<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">x_2 = \\dfrac{-b - \\sqrt{b^2-4ac}}{2a}<\/span>\n<p style=\"text-align: justify;\">As parabolas are symmetrical, the axis of symmetry will have the equation:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">x = x_0 = \\dfrac{x_1 + x_2}{2}= -\\dfrac{b}{2a}<\/span>\n<p style=\"text-align: justify;\">The axis of symmetry necessarily passes through the vertex of the parabola, whose coordinates are:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0) = (x_0, y(x_0)) = \\left( -\\dfrac{b}{2a}, y\\left(-\\dfrac{b}{2a}\\right) \\right)<\/span>\n<p style=\"text-align: justify;\">Where:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y_0 = y\\left(-\\dfrac{b}{2a} \\right) = a\\left(-\\dfrac{b}{2a}\\right)^2 + b\\left(-\\dfrac{b}{2a}\\right) + c = \\dfrac{b^2}{4a} - \\dfrac{b^2}{2a} + c = c - \\dfrac{b^2}{4a}<\/span>\n<p style=\"text-align: justify;\">This is how we arrive at the coordinates of the vertex that we already knew by other means:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0) = \\left( -\\dfrac{b}{2a},c - \\dfrac{b^2}{4a} \\right)<\/span>\n<p style=\"text-align: justify;\">The focal position is, as we already saw <span class=\"katex-eq\" data-katex-display=\"false\">f=\\dfrac{1}{4a},<\/span> and from this, we can now calculate the position of the directrix, the focus, and all the information we already had from the general equation.<\/p>\n<p>   <a name=\"4\"><\/a><\/p>\n<h2>Automatic Characterization with Excel<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=C6DbrJDiZTM&amp;t=1086s\" target=\"_blank\" rel=\"noopener\"><strong>Having made<\/strong><\/a> all these reasonings, it is now very easy to automate the characterization of any parabola using Excel. You can find an example <a href=\"https:\/\/drive.google.com\/file\/d\/1LbNOKHHfzlPgHI3_NSzuB_6_b7KmXlTd\/view?usp=sharing\" rel=\"noopener\" target=\"_blank\">here.<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Characterization of Parabolas and their Graphs Summary: In this class, we will review the characterization of parabolas based on their general and canonical forms, explaining how to identify key elements such as the vertex, focus, directrix, axis of symmetry, and possible intersections with the x-axis. Learning Objectives: By the end of this class, students will [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28929,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":7,"footnotes":""},"categories":[583,567],"tags":[],"class_list":["post-28930","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>Characterization of Parabolas and their Graphs - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Learn to characterize parabolas by identifying the vertex, focus, directrix, axis of symmetry, and intersections with the x-axis.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Characterization of Parabolas and their Graphs\" \/>\n<meta property=\"og:description\" content=\"Learn to characterize parabolas by identifying the vertex, focus, directrix, axis of symmetry, and intersections with the x-axis.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-04-27T13:00:20+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-09-22T02:14:52+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/focodirectriz-1024x467.png\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Characterization of Parabolas and their Graphs\" \/>\n<meta name=\"twitter:description\" content=\"Learn to characterize parabolas by identifying the vertex, focus, directrix, axis of symmetry, and intersections with the x-axis.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/focodirectriz.png\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Characterization of Parabolas and their Graphs\",\"datePublished\":\"2021-04-27T13:00:20+00:00\",\"dateModified\":\"2024-09-22T02:14:52+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/\"},\"wordCount\":1164,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/focodirectriz.png\",\"articleSection\":[\"Algebra and Geometry\",\"Mathematics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/\",\"name\":\"Characterization of Parabolas and their Graphs - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/#primaryimage\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/focodirectriz.png\",\"datePublished\":\"2021-04-27T13:00:20+00:00\",\"dateModified\":\"2024-09-22T02:14:52+00:00\",\"description\":\"Learn to characterize parabolas by identifying the vertex, focus, directrix, axis of symmetry, and intersections with the x-axis.\",\"breadcrumb\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/#primaryimage\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/focodirectriz.png\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/focodirectriz.png\",\"width\":1183,\"height\":539},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/characterization-of-parabolas-and-their-graphs\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Characterization of Parabolas and their Graphs\"}]},{\"@type\":\"WebSite\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/\",\"name\":\"toposuranos.com\\\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\",\"name\":\"toposuranos.com\\\/material\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\\\/material\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/www.facebook.com\\\/groups\\\/toposuranos\",\"https:\\\/\\\/x.com\\\/topuranos\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\\\/\\\/www.linkedin.com\\\/company\\\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\\\/\\\/toposuranos.com\\\/material\"],\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/author\\\/giorgio-reveco\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Characterization of Parabolas and their Graphs - toposuranos.com\/material","description":"Learn to characterize parabolas by identifying the vertex, focus, directrix, axis of symmetry, and intersections with the x-axis.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/","og_locale":"es_ES","og_type":"article","og_title":"Characterization of Parabolas and their Graphs","og_description":"Learn to characterize parabolas by identifying the vertex, focus, directrix, axis of symmetry, and intersections with the x-axis.","og_url":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-04-27T13:00:20+00:00","article_modified_time":"2024-09-22T02:14:52+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/focodirectriz-1024x467.png","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Characterization of Parabolas and their Graphs","twitter_description":"Learn to characterize parabolas by identifying the vertex, focus, directrix, axis of symmetry, and intersections with the x-axis.","twitter_image":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/focodirectriz.png","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"5 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/#article","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/"},"author":{"name":"giorgio.reveco","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Characterization of Parabolas and their Graphs","datePublished":"2021-04-27T13:00:20+00:00","dateModified":"2024-09-22T02:14:52+00:00","mainEntityOfPage":{"@id":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/"},"wordCount":1164,"commentCount":0,"publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/focodirectriz.png","articleSection":["Algebra and Geometry","Mathematics"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/#respond"]}]},{"@type":"WebPage","@id":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/","url":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/","name":"Characterization of Parabolas and their Graphs - toposuranos.com\/material","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/#primaryimage"},"image":{"@id":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/focodirectriz.png","datePublished":"2021-04-27T13:00:20+00:00","dateModified":"2024-09-22T02:14:52+00:00","description":"Learn to characterize parabolas by identifying the vertex, focus, directrix, axis of symmetry, and intersections with the x-axis.","breadcrumb":{"@id":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/#primaryimage","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/focodirectriz.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/focodirectriz.png","width":1183,"height":539},{"@type":"BreadcrumbList","@id":"http:\/\/toposuranos.com\/material\/en\/characterization-of-parabolas-and-their-graphs\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Characterization of Parabolas and their Graphs"}]},{"@type":"WebSite","@id":"http:\/\/toposuranos.com\/material\/#website","url":"http:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"http:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"http:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"http:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/28930","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=28930"}],"version-history":[{"count":0,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/28930\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/28929"}],"wp:attachment":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=28930"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=28930"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=28930"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}