{"id":33710,"date":"2021-04-27T13:00:10","date_gmt":"2021-04-27T13:00:10","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33710"},"modified":"2025-07-30T20:31:19","modified_gmt":"2025-07-30T20:31:19","slug":"characterizatio-parabolarum-et-eorum-graphica-repraesentatio","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/characterizatio-parabolarum-et-eorum-graphica-repraesentatio\/","title":{"rendered":"Characterizatio Parabolarum et Eorum Graphica Repraesentatio"},"content":{"rendered":"<p><center><\/p>\n<h1>Characterizatio Parabolarum et Earum Graphica Repraesentatio<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\n   In hac lectione tractabitur characterizatio parabolarum ex aequatione generali et forma canonica, explanantes quomodo elementa clavis recognoscantur, ut verticem, focum, directricem, axem symmetriae, et sectiones cum axe X.<br \/>\n   <\/em><br \/>\n   <strong>Propositi Discendi:<\/strong><br \/>\n   Expleta hac lectione discipulus poterit:<\/p>\n<ol style=\"text-align: left;\">\n<li><strong>Computare<\/strong> situm verticis, foci et directricis parabolae ex forma generali et canonica.<\/li>\n<li><strong>Transformare<\/strong> aequationem canonicam in formam generalem ad informationem geometricam extrahendam.<\/li>\n<li><strong>Designare<\/strong> schematice graphice parabolam ex informatione obtenta.<\/li>\n<\/ol>\n<p>   <strong>INDEX RERUM<\/strong><br \/>\n   <a href=\"#1\">Forma generalis et canonica parabolarum<\/a><br \/>\n   <a href=\"#2\">Characterizatio Parabolarum ex Aequatione Generali<\/a><br \/>\n   <a href=\"#3\">Characterizatio Parabolarum ex Aequatione Canonica<\/a><br \/>\n   <a href=\"#4\">Characterizatio automatica per 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>Forma generalis et canonica parabolarum<\/h2>\n<p style=\"text-align: justify;\">In praeterita lectione vidimus parabolas algebrice exprimi posse per aequationem generalem parabolarum ut sequitur.<\/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;\">Ubi par <span class=\"katex-eq\" data-katex-display=\"false\">(x_0,y_0)<\/span> est positio verticis et <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> est distantia focalis. Si <span class=\"katex-eq\" data-katex-display=\"false\">f \\gt 0<\/span>, tunc focus est ad distantiam <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> supra verticem; si vero <span class=\"katex-eq\" data-katex-display=\"false\">f \\lt 0<\/span>, tunc focus est ad distantiam <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> infra verticem.<\/p>\n<p style=\"text-align: justify;\">Vidimus etiam aequationem parabolarum ad formam canonicam redactam esse aequivalentem polynomio secundi gradus.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x) = ax^2 + bx + c,<\/span> cum <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>Characterizare parabolam significat<\/strong><\/a> sequentia data detegere.<\/p>\n<ul style=\"text-align: justify;\">\n<li>Coordinatae verticis<\/li>\n<li>Coordinatae foci<\/li>\n<li>Aequatio directricis<\/li>\n<li>Aequatio axis symmetriae<\/li>\n<li>Intersectiones cum axe x (si existant)<\/li>\n<li>Denique, delineatio schematica graphica ex collecta informatione.<\/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=\"Characterizatio Parabolarum\" 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=\"Characterizatio Parabolarum\" class=\" aligncenter lazyload\" width=\"537\" height=\"414\" \/><\/noscript><br \/>\n<a name=\"2\"><\/a><\/p>\n<h2>Characterizatio Parabolarum ex Aequatione Generali<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=C6DbrJDiZTM&amp;t=316s\" target=\"_blank\" rel=\"noopener\"><strong>Si parabola describitur per<\/strong><\/a> aequationem generalem, iam fere omnem informationem necessariam ad characterisationem conficiendam habes; solum intersectiones cum axe x ampliorem analysin exigent.<\/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;\">Ex hac forma iam habes:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Vertex:<\/strong> Punctum cum coordinatis <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0)<\/span><\/li>\n<li><strong>Positio focalis:<\/strong> ad <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> unitates supra verticem<\/li>\n<li><strong>Focus:<\/strong> punctum cum coordinatis <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, y_0 + f)<\/span><\/li>\n<li><strong>Directrix:<\/strong> recta aequationis <span class=\"katex-eq\" data-katex-display=\"false\">y = y_0 - f<\/span><\/li>\n<li><strong>Axis symmetriae:<\/strong> recta aequationis <span class=\"katex-eq\" data-katex-display=\"false\">x = x_0<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Ut sectiones cum axe x invenias, aequationem generalem in formam canonicam convertere debes, deinde aequare polynomium secundi gradus ad zero. Si solutiones exsistunt, illae sunt sectiones cum axe x.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Characterizatio Parabolarum ex Aequatione Canonica<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=C6DbrJDiZTM&amp;t=393s\" target=\"_blank\" rel=\"noopener\"><strong>Cum aequatio parabolarum<\/strong><\/a> exhibetur in forma canonica, duae optiones habentur: 1) Characterisatio per conversionem ad aequationem generalem aut 2) Usus symmetriae et intersectionum cum axe x. Utraque methodus suas utilitates habet. Secunda plerumque celerior est, sed parabolae non semper axem x secant; prima longior est sed, ut videbimus postea, facile automatizari potest. Utrasque vias examinabimus ut eligere possis secundum tuae necessitates et praelationes.<\/p>\n<h3>Conversio ad aequationem generalem<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=C6DbrJDiZTM&amp;t=475s\" target=\"_blank\" rel=\"noopener\"><strong>Conversio ad formam generalem<\/strong><\/a> fit secundum hanc rationem, ubi <span class=\"katex-eq\" data-katex-display=\"false\">a,b,c \\in \\mathbb{R}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">a \\neq 0<\/span>.<\/p>\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>; Aequatio canonica parabolarum<\/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>; Per extractionem factoris <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>; Quia <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>; Aequatio parabolarum in forma generali<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Ex hoc, possumus totam informationem ex aequatione generali derivatam extrahere, connectendo eius parametros cum illis aequationis canonicae. Ita habemus:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>Vertex:<\/strong> Punctum cum coordinatis <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>Positio focalis:<\/strong> ad <span class=\"katex-eq\" data-katex-display=\"false\">f = \\dfrac{1}{4a}<\/span> unitates supra verticem<\/li>\n<li><strong>Focus:<\/strong> Punctum cum coordinatis <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> Recta aequationis <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 symmetriae:<\/strong> Recta aequationis <span class=\"katex-eq\" data-katex-display=\"false\">x = x_0 = -\\dfrac{b}{2a}<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Et hinc characterizatio parabolarum fit sicut iam antea vidimus per aequationem generalem.<\/p>\n<h3>Utendo symmetria et intersectionibus cum axe x<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=C6DbrJDiZTM&amp;t=769s\" target=\"_blank\" rel=\"noopener\"><strong>Cum habemus aequationem<\/strong><\/a> parabolarum in forma canonica scriptam <span class=\"katex-eq\" data-katex-display=\"false\">y = ax^2 + bx + c<\/span>, videmus esse relative facile calculare eius intersectiones cum axe x; sufficit solvere aequationem<\/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;\">Cum hoc fieri potest, obtinemus intersectiones <span class=\"katex-eq\" data-katex-display=\"false\">x_1<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">x_2<\/span> datas per<\/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;\">Cum parabolae sint symmetricae, axis symmetriae habebit aequationem:<\/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;\">Axis symmetriae necessario transit per verticem parabolae, cuius coordinatae erunt<\/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;\">Ubi<\/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;\">Ita attingimus coordinatas verticis quas iam aliter cognovimus<\/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;\">Positio focalis est, ut iam vidimus, <span class=\"katex-eq\" data-katex-display=\"false\">f = \\dfrac{1}{4a},<\/span> atque ex hoc iam possumus calculare positionem directricis, foci, et omnem informationem quam ex aequatione generali habebamus.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Characterizatio automatica per 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>His rationibus completis<\/strong><\/a> nunc valde facile est automatizare characterisationem cuiuslibet parabolae per Excel. Exemplum invenire potes <a href=\"https:\/\/drive.google.com\/file\/d\/1LbNOKHHfzlPgHI3_NSzuB_6_b7KmXlTd\/view?usp=sharing\" rel=\"noopener\" target=\"_blank\">hic.<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Characterizatio Parabolarum et Earum Graphica Repraesentatio Summarium: In hac lectione tractabitur characterizatio parabolarum ex aequatione generali et forma canonica, explanantes quomodo elementa clavis recognoscantur, ut verticem, focum, directricem, axem symmetriae, et sectiones cum axe X. Propositi Discendi: Expleta hac lectione discipulus poterit: Computare situm verticis, foci et directricis parabolae ex forma generali et canonica. Transformare [&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":1,"footnotes":""},"categories":[1304,1298],"tags":[],"class_list":["post-33710","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-et-geometria","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Characterizatio Parabolarum et Eorum Graphica Repraesentatio - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce parabolas characterizare, verticem, focum, directricem, axem symmetriae ac sectiones cum axe x agnoscendo.\" \/>\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\/la\/characterizatio-parabolarum-et-eorum-graphica-repraesentatio\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Characterizatio Parabolarum et Eorum Graphica Repraesentatio\" \/>\n<meta property=\"og:description\" content=\"Disce parabolas characterizare, verticem, focum, directricem, axem symmetriae ac sectiones cum axe x agnoscendo.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/characterizatio-parabolarum-et-eorum-graphica-repraesentatio\/\" \/>\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=\"2021-04-27T13:00:10+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-30T20:31:19+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=\"Characterizatio Parabolarum et Eorum Graphica Repraesentatio\" \/>\n<meta name=\"twitter:description\" content=\"Disce parabolas characterizare, verticem, focum, directricem, axem symmetriae ac sectiones cum axe x agnoscendo.\" \/>\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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/characterizatio-parabolarum-et-eorum-graphica-repraesentatio\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/characterizatio-parabolarum-et-eorum-graphica-repraesentatio\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Characterizatio Parabolarum et Eorum Graphica Repraesentatio\",\"datePublished\":\"2021-04-27T13:00:10+00:00\",\"dateModified\":\"2025-07-30T20:31:19+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/characterizatio-parabolarum-et-eorum-graphica-repraesentatio\/\"},\"wordCount\":966,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/characterizatio-parabolarum-et-eorum-graphica-repraesentatio\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/focodirectriz.png\",\"articleSection\":[\"Algebra et Geometria\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/la\/characterizatio-parabolarum-et-eorum-graphica-repraesentatio\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/characterizatio-parabolarum-et-eorum-graphica-repraesentatio\/\",\"url\":\"http:\/\/toposuranos.com\/material\/la\/characterizatio-parabolarum-et-eorum-graphica-repraesentatio\/\",\"name\":\"Characterizatio Parabolarum et Eorum Graphica Repraesentatio - 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