{"id":33697,"date":"2021-04-23T13:00:31","date_gmt":"2021-04-23T13:00:31","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33697"},"modified":"2025-07-30T19:44:23","modified_gmt":"2025-07-30T19:44:23","slug":"aequatio-parabolae-definitiones-et-proprietates","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/aequatio-parabolae-definitiones-et-proprietates\/","title":{"rendered":"\u00c6quatio Parabol\u00e6: Definitiones et Proprietates"},"content":{"rendered":"<p><center><\/p>\n<h1>\u00c6quatio Parabol\u00e6: Definitiones et Proprietates<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\n   Hac lectione exploratur definitio et deductio \u00e6quationis parabol\u00e6, illustrando eius originem tamquam collectionem punctorum \u00e6quidistantium a foco et directrice. Ex hoc conceptu, recognoscuntur notiones priores sicut distantia inter puncta in plano cartesiano et translatio graphorum, quod permittit introductionem \u00e6quationis fundamentalis parabol\u00e6 eiusque relationem cum polynomialibus secundi gradus. Denique deducitur \u00e6quatio generalis parabol\u00e6 cum vertice in quovis puncto, atque transformatur in formam canonicam polynomialis quadratici.<br \/>\n   <\/em><\/p>\n<p><strong>Met\u00e6 Discendi:<\/strong><br \/>\n   Expleta hac lectione, discipulus poterit<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Intelligere<\/strong> definitionem geometricam parabol\u00e6 ut collectionem punctorum \u00e6quidistantium a foco et directrice.<\/li>\n<li><strong>Deducere<\/strong> \u00e6quationem fundamentalem parabol\u00e6 utens relatione inter focum et directricem.<\/li>\n<li><strong>Intelligere<\/strong> relationem inter parabolam et polynomia secundi gradus.<\/li>\n<li><strong>Derivare<\/strong> \u00e6quationem generalem parabol\u00e6 cum vertice in quovis puncto (h,k).<\/li>\n<\/ol>\n<p>   <strong>INDEX CONTENTORUM<\/strong><br \/>\n   <a href=\"#1\"><strong>Pr\u00e6notiones ad \u00e6quationem parabol\u00e6 obtinendam<\/strong><\/a><br \/>\n   <a href=\"#2\">Notio geometrica parabol\u00e6<\/a><br \/>\n   <a href=\"#3\">Distantia inter duo puncta plani cartesiani<\/a><br \/>\n   <a href=\"#4\">Translatio Graphorum<\/a><br \/>\n   <a href=\"#5\"><strong>Definitio Parabol\u00e6<\/strong><\/a><br \/>\n   <a href=\"#6\">Deductio \u00c6quationis Fundamentalis Parabol\u00e6<\/a><br \/>\n   <a href=\"#7\">\u00c6quatio Generalis Parabol\u00e6<\/a><br \/>\n   <a href=\"#8\">\u00c6quatio Canonica Parabol\u00e6 et Polynomia Secundi Gradus<\/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><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Pr\u00e6notiones ad \u00c6quationem Parabol\u00e6 Obtinendam<\/h2>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Notio Geometrica Parabol\u00e6<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=130s\" target=\"_blank\" rel=\"noopener\"><strong>Parabola est curva <\/strong><\/a>qu\u00e6 obtinetur ut collectio omnium punctorum \u00e6quidistantium a puncto fixo, quod vocatur focus, et recta fixa qu\u00e6 dicitur directrix. Ad hanc definitionem intellegendam et ad exprimendum eam per formulam algebraicam tractabilem, scilicet \u00e6quationem parabol\u00e6, oportet prius quasdam notiones priores recognoscere.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Distantia inter Duo Puncta Plani Cartesiani<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=199s\" target=\"_blank\" rel=\"noopener\"><strong>Consideremus duo puncta<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">p_1 = (x_1, y_1)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">p_2 = (x_2, y_2).<\/span> Distantia inter h\u00e6c puncta est longitudo segmenti recti quod ea coniungit.<\/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=\"Distancia entre puntos\" 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=\"Distancia entre puntos\" class=\"aligncenter lazyload\" width=\"409\" height=\"263\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Hanc distantiam metiri possumus per <a href=\"https:\/\/youtu.be\/grjSzVl8Acw?t=553\" target=\"_blank\" rel=\"noopener\">theorema Pythagoricum<\/a> componendo figuram sequentem.<\/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=\"teorema de pitagoras\" 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=\"teorema de pitagoras\" class=\" aligncenter lazyload\" width=\"404\" height=\"255\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Itaque distantia <span class=\"katex-eq\" data-katex-display=\"false\">d<\/span> inter duo puncta erit<\/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>Translatio Graphorum<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=390s\" target=\"_blank\" rel=\"noopener\"><strong>Consideremus functionem <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">y(x) = x^2<\/span>. Si hanc functionem depingimus, apparebit figura similis huic:<\/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=\"Traslaci\u00f3n de gr\u00e1ficos\" 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=\"Traslaci\u00f3n de gr\u00e1ficos\" class=\" aligncenter lazyload\" width=\"325\" height=\"321\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Si in hac functione <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> substituimus per <span class=\"katex-eq\" data-katex-display=\"false\">x-1<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> per <span class=\"katex-eq\" data-katex-display=\"false\">y-1,<\/span>, tum observabimus hanc transformationem in graphico:<\/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=\"Traslaci\u00f3n de Gr\u00e1ficos\" 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=\"Traslaci\u00f3n de Gr\u00e1ficos\" class=\" aligncenter lazyload\" width=\"352\" height=\"288\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">In genere, qu\u00e6libet substitutio huius generis efficit transformationem translationis, scilicet:<\/p>\n<ul style=\"text-align: justify;\">\n<li><strong><span class=\"katex-eq\" data-katex-display=\"false\">x\\longmapsto x-a<\/span>:<\/strong> si <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> est positivum, graphum movetur <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> unitatibus ad dextram; si negativum, ad sinistram.<\/li>\n<li><strong><span class=\"katex-eq\" data-katex-display=\"false\">y\\longmapsto y-b<\/span>: <\/strong>si <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> est positivum, graphum movetur <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> unitatibus sursum; si negativum, deorsum.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">H\u00e6 sunt transformationes translationis, quarum effectus generalis in figura sequente ostenditur:<\/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=\"Traslaci\u00f3n general de gr\u00e1ficos\" 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=\"Traslaci\u00f3n general de gr\u00e1ficos\" class=\"aligncenter lazyload\" width=\"522\" height=\"314\" \/><\/noscript><br \/>\n<a name=\"5\"><\/a><\/p>\n<h2>Definitio Parabol\u00e6<\/h2>\n<p style=\"text-align: justify;\">Parabola est collectio omnium punctorum qu\u00e6 sunt \u00e6quidistantia a puncto fixo et linea recta.<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=\"Definici\u00f3n de Par\u00e1bola\" 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=\"Definici\u00f3n de Par\u00e1bola\" class=\"aligncenter lazyload\" width=\"426\" height=\"302\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Punctum fixum dicitur <strong>focus<\/strong>, et linea recta est <strong>directrix.<\/strong> Si attente consideramus, videbimus notionem distanti\u00e6 esse fundamentalem ad definitionem parabol\u00e6; ideo, ad analysin eius profundiorem peragendam, necesse est recognoscere quomodo distantiae in plano cartesiano metiantur et algebraice exprimantur.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Deductio \u00c6quationis Fundamentalis Parabol\u00e6<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=604s\" target=\"_blank\" rel=\"noopener\"><strong>Propter simplicitatem, consideremus punctum focale<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">p_f= (0,f)<\/span> et directricem ut rectam <span class=\"katex-eq\" data-katex-display=\"false\">L<\/span> cuius \u00e6quatio est <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=\"La Ecuaci\u00f3n de las Par\u00e1bolas\" 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=\"La Ecuaci\u00f3n de las Par\u00e1bolas\" class=\" aligncenter lazyload\" width=\"334\" height=\"249\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Si sumamus punctum quodlibet in parabola, cuius coordinat\u00e6 sunt <span class=\"katex-eq\" data-katex-display=\"false\">(x,y)<\/span>, tum hoc punctum erit \u00e6quidistans tam a foco quam a directrice. Hoc algebraice exprimere possumus hoc modo:<\/p>\n<p style=\"text-align: center;\">Distantia Focus-Punctum(x,y) <span class=\"katex-eq\" data-katex-display=\"false\">= \\sqrt{x^2 + (f-y)^2}= y+f =<\/span> Distantia Punctum(x,y)-Directrix<\/p>\n<p style=\"text-align: justify;\">Et ex hoc nascitur sequens ratiocinatio:<\/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\">\\sqrt{x^2 + (f-y)^2}= y+f<\/span><\/td>\n<td>; Distantia puncti a foco = distantia puncti a directrice, Definitio parabol\u00e6<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">x^2 + (f-y)^2= (y+f)^2<\/span><\/td>\n<td>; Ex (1), quadratum utriusque lateris sumitur<\/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;\">Hoc ultimum est quod vocamus <strong>\u00c6quatio Fundamentalis Parabol\u00e6.<\/strong><\/p>\n<p style=\"text-align: justify;\">Si attente consideremus hanc parabolam, videbimus exstare punctum in ea quod est proximum foco (vel \u00e6quivalenter, directrici). Hoc punctum vocatur <strong>vertex<\/strong> et in hoc casu particulari habet coordinatas <span class=\"katex-eq\" data-katex-display=\"false\">(0,0)<\/span>; distantia inter focum et verticem vocatur <strong>distantia focalis,<\/strong> et eius valor <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> esse potest quivis numerus realis excepto nullo.<\/p>\n<p style=\"text-align: justify;\">Cum <span class=\"katex-eq\" data-katex-display=\"false\">f\\gt 0<\/span>, parabola aperitur sursum, et si contra <span class=\"katex-eq\" data-katex-display=\"false\">f\\lt 0<\/span>, aperitur deorsum. Cum <span class=\"katex-eq\" data-katex-display=\"false\">f\\to 0<\/span>, parabola deprimetur, vertice in eadem positione manente, et directrix vertici appropinquabit, ita ut videatur parabolam et directricem in unam rectam coalescere; cum <span class=\"katex-eq\" data-katex-display=\"false\">f=0<\/span>, graphice evanescit quia divisionibus per nihilum carere non possumus.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>\u00c6quatio Generalis Parabol\u00e6<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=1007s\" target=\"_blank\" rel=\"noopener\"><strong>Ex \u00e6quatione fundamentali parabol\u00e6<\/strong><\/a> et translatione graphorum, obtinetur, substituendo <span class=\"katex-eq\" data-katex-display=\"false\">x\\longmapsto (x-h)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">y\\longmapsto (y-k),<\/span>, <strong>\u00c6quatio Generalis Parabol\u00e6<\/strong> cum vertice in <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>\u00c6quatio Canonica Parabol\u00e6 et Polynomia Secundi Gradus<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=O2EYxxK03jU&amp;t=1116s\" target=\"_blank\" rel=\"noopener\"><strong>Si evolvamus \u00e6quationem<\/strong><\/a> generalem parabol\u00e6, consequemur ratiocinium sequentem:<\/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-k) = \\dfrac{(x-h)^2}{4f}<\/span><\/td>\n<td>; \u00c6quatio generalis parabol\u00e6<\/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;\">Si in hac \u00e6quatione substitutionem facimus <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> et <span class=\"katex-eq\" data-katex-display=\"false\">c=\\dfrac{h^2 + 4fk}{4f},<\/span>, tum \u00e6quatio generalis parabol\u00e6 transformatur in \u00e6quationem canonicam, qu\u00e6 revera est <a href=\"https:\/\/toposuranos.com\/factorizacion-del-polinomio-cuadratico-y-2n-cuadratico\/\" rel=\"noopener\" target=\"_blank\">polynomia secundi gradus<\/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>\u00c6quatio Parabol\u00e6: Definitiones et Proprietates Summarium: Hac lectione exploratur definitio et deductio \u00e6quationis parabol\u00e6, illustrando eius originem tamquam collectionem punctorum \u00e6quidistantium a foco et directrice. Ex hoc conceptu, recognoscuntur notiones priores sicut distantia inter puncta in plano cartesiano et translatio graphorum, quod permittit introductionem \u00e6quationis fundamentalis parabol\u00e6 eiusque relationem cum polynomialibus secundi gradus. Denique deducitur [&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":2,"footnotes":""},"categories":[1304,1298],"tags":[],"class_list":["post-33697","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u00c6quatio Parabol\u00e6: Definitiones et Proprietates - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"In hac lectione de \u00e6quatione parabol\u00e6 disces \u00e6quationem fundamentalem deducere eiusque nexum cum polynomialibus secundi gradus intellegere.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/aequatio-parabolae-definitiones-et-proprietates\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u00c6quatio Parabol\u00e6: Definitiones et Proprietates\" \/>\n<meta property=\"og:description\" content=\"In hac lectione de \u00e6quatione parabol\u00e6 disces \u00e6quationem fundamentalem deducere eiusque nexum cum polynomialibus secundi gradus intellegere.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/aequatio-parabolae-definitiones-et-proprietates\/\" \/>\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=\"2021-04-23T13:00:31+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-30T19:44:23+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/parabolas-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"\u00c6quatio Parabol\u00e6: Definitiones et Proprietates\" \/>\n<meta name=\"twitter:description\" content=\"In hac lectione de \u00e6quatione parabol\u00e6 disces \u00e6quationem fundamentalem deducere eiusque nexum cum polynomialibus secundi gradus intellegere.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/09\/parabolas.jpg\" \/>\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\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-parabolae-definitiones-et-proprietates\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-parabolae-definitiones-et-proprietates\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"\u00c6quatio Parabol\u00e6: Definitiones et Proprietates\",\"datePublished\":\"2021-04-23T13:00:31+00:00\",\"dateModified\":\"2025-07-30T19:44:23+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-parabolae-definitiones-et-proprietates\\\/\"},\"wordCount\":831,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-parabolae-definitiones-et-proprietates\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/09\\\/parabolas.jpg\",\"articleSection\":[\"Algebra et Geometria\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-parabolae-definitiones-et-proprietates\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-parabolae-definitiones-et-proprietates\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-parabolae-definitiones-et-proprietates\\\/\",\"name\":\"\u00c6quatio Parabol\u00e6: Definitiones et Proprietates - 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