{"id":4616,"date":"2022-01-14T21:01:46","date_gmt":"2022-01-14T21:01:46","guid":{"rendered":"https:\/\/blogs.ugto.mx\/rea\/?p=4616"},"modified":"2022-02-08T18:59:50","modified_gmt":"2022-02-08T18:59:50","slug":"clase-digital-4-funciones-algebraicas-polinomiales-y-racionales-funciones-trascendentes-trigonometricas-logaritmicas-y-exponenciales-operaciones-con-funciones-adicion-multipl","status":"publish","type":"post","link":"https:\/\/blogs.ugto.mx\/rea\/clase-digital-4-funciones-algebraicas-polinomiales-y-racionales-funciones-trascendentes-trigonometricas-logaritmicas-y-exponenciales-operaciones-con-funciones-adicion-multipl\/","title":{"rendered":"Clase digital 4: Funciones algebraicas: polinomiales y racionales. Funciones trascendentes: trigonome\u0301tricas, logari\u0301tmicas y exponenciales. Operaciones con funciones: adicio\u0301n, multiplicacio\u0301n, divisio\u0301n y composicio\u0301n"},"content":{"rendered":"\n\n\n<div class=\"wp-block-cover is-light\" style=\"min-height:284px;aspect-ratio:unset;\"><span aria-hidden=\"true\" class=\"has-background-dim-40 wp-block-cover__gradient-background has-background-dim\"><\/span><img decoding=\"async\" class=\"wp-block-cover__image-background wp-image-4606\" alt=\"\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/zsl306fdck.jpg\" style=\"object-position:59% 64%\" data-object-fit=\"cover\" data-object-position=\"59% 64%\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1600\" height=\"1067\" class=\"wp-block-cover__image-background wp-image-4606\" alt=\"\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/zsl306fdck.jpg\" style=\"object-position:59% 64%\" data-object-fit=\"cover\" data-object-position=\"59% 64%\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/zsl306fdck.jpg 1600w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/zsl306fdck-300x200.jpg 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/zsl306fdck-1024x683.jpg 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/zsl306fdck-768x512.jpg 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/zsl306fdck-1536x1024.jpg 1536w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/zsl306fdck-272x182.jpg 272w\" sizes=\"auto, (max-width: 1600px) 100vw, 1600px\" \/><\/noscript><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<p class=\"has-text-align-center has-base-3-color has-text-color has-large-font-size wp-block-paragraph\">Funciones algebraicas: polinomiales y racionales. Funciones trascendentes: trigonome\u0301tricas, logari\u0301tmicas y exponenciales. Operaciones con funciones: adicio\u0301n, multiplicacio\u0301n, divisio\u0301n y composicio\u0301n<\/p>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"introduccion\">Introducci\u00f3n<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u00a1Hola!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Siempre es un gusto saludarte y saber que tienes el \u00e1nimo para continuar, te invito a seguir en este camino formativo en tu cuarta clase titulada Funciones algebraicas: polinomiales y racionales del curso <strong>C\u00e1lculo Diferencial<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En esta clase comprender\u00e1s el concepto de funci\u00f3n y las nociones fundamentales de la misma en las matem\u00e1ticas.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Como se vio en la clase 3, las funciones tienen su propia clasificaci\u00f3n, las cuales abordaremos con m\u00e1s detalle, as\u00ed como las transformaciones que pueden tener. Transformaci\u00f3n r\u00edgida y no r\u00edgida, se abordar\u00e1 el concepto de simetr\u00eda de una funci\u00f3n.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Dependiendo del tipo de funci\u00f3n, en la gr\u00e1fica se pueden observar as\u00edntotas, las cuales se determinan de manera anal\u00edtica, as\u00ed mismo, podr\u00e1s obtener funciones inversas.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">De acuerdo con lo anterior, te invito a continuar la sesi\u00f3n, \u00a1mucho \u00e9xito!<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"desarrollo-del-tema\">Desarrollo del tema<\/h2>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"combinacion-de-funciones\">Combinaci\u00f3n de funciones<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Dos funciones <em>f<\/em> y <em>g<\/em> pueden combinarse en varias formas para obtener nuevas funciones. Analicemos dos formas en que es posible combinar funciones: mediante operaciones aritme\u0301ticas y a trave\u0301s de la operacio\u0301n de composicio\u0301n de funciones.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"funcion-potencia\">Funci\u00f3n potencia<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Una funcio\u0301n de la forma <em>f(x) = x<sup>n<\/sup><\/em> se denomina funcio\u0301n potencia.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">El dominio de la funcio\u0301n potencia depende de la potencia n. Por ejemplo, para n = 2, n = 1\/2 y n = -1, respectivamente,<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>el dominio de <em>f(x) =x<sup>2<\/sup><\/em> es el conjunto R de nu\u0301meros reales,<\/li><li>el dominio de<\/li><\/ul>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM1.png\" alt=\"\" class=\"wp-image-4650\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"597\" height=\"110\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM1.png\" alt=\"\" class=\"wp-image-4650\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM1.png 597w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM1-300x55.png 300w\" sizes=\"auto, (max-width: 597px) 100vw, 597px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<ul class=\"wp-block-list\"><li>el dominio de<\/li><\/ul>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM2.png\" alt=\"\" class=\"wp-image-4651\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"395\" height=\"99\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM2.png\" alt=\"\" class=\"wp-image-4651\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM2.png 395w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM2-300x75.png 300w\" sizes=\"auto, (max-width: 395px) 100vw, 395px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Es el conjunto R de nu\u0301meros reales excepto <em>x = 0<\/em>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"grafica-de-funciones-potencia\">Gr\u00e1fica de funciones potencia<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM3.png\" alt=\"\" class=\"wp-image-4655\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"843\" height=\"590\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM3.png\" alt=\"\" class=\"wp-image-4655\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM3.png 843w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM3-300x210.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM3-768x538.png 768w\" sizes=\"auto, (max-width: 843px) 100vw, 843px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"combinaciones-aritmeticas\">Combinaciones aritm\u00e9ticas<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Dos funciones pueden combinarse por medio de las cuatro conocidas operaciones aritme\u0301ticas de suma, resta, multiplicacio\u0301n y divisio\u0301n.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si <em>f<\/em> y <em>g<\/em> son dos funciones, entonces la suma <em>f + g<\/em>, la diferencia <em>f \u2013 g<\/em>, el producto <em>fg<\/em> y el cociente <em>f\/g<\/em> se definen como sigue:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM4.png\" alt=\"\" class=\"wp-image-4659\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"425\" height=\"291\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM4.png\" alt=\"\" class=\"wp-image-4659\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM4.png 425w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM4-300x205.png 300w\" sizes=\"auto, (max-width: 425px) 100vw, 425px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"dominio-de-una-combinacion-aritmetica\">Dominio de una combinaci\u00f3n aritm\u00e9tica<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Al combinar dos funciones aritme\u0301ticamente es necesario que ambas <em>f<\/em> y <em>g<\/em> este\u0301n definidas en el mismo nu\u0301mero <em>x<\/em>. Por tanto, el dominio de las funciones <em>f + g, f \u2013 g <\/em>y<em> fg<\/em> es el conjunto de nu\u0301meros reales que son comunes a ambos dominios; es decir, el dominio es la interseccio\u0301n del dominio de <em>f<\/em> con el dominio de <em>g<\/em>. En el caso del cociente <em>f\/g<\/em>, el dominio tambie\u0301n es la interseccio\u0301n de los dos dominios, pero tambie\u0301n es necesario excluir cualquier valor de <em>x<\/em> para el que el denominador <em>g(x)<\/em> sea cero.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Determina el dominio de la funci\u00f3n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM5.png\" alt=\"\" class=\"wp-image-4666\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"951\" height=\"126\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM5.png\" alt=\"\" class=\"wp-image-4666\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM5.png 951w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM5-300x40.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM5-768x102.png 768w\" sizes=\"auto, (max-width: 951px) 100vw, 951px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Planteando la combinaci\u00f3n de funciones<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM6.png\" alt=\"\" class=\"wp-image-4669\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"507\" height=\"87\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM6.png\" alt=\"\" class=\"wp-image-4669\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM6.png 507w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM6-300x51.png 300w\" sizes=\"auto, (max-width: 507px) 100vw, 507px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Como se mencion\u00f3 el dominio de la combinaci\u00f3n de funciones es la intersecci\u00f3n de sus dominios.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">El dominio de la funci\u00f3n f(x) son todos los n\u00fameros reales, sin embargo, para la funci\u00f3n g(x) se debe plantear la desigualdad para conocer el dominio.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM7.png\" alt=\"\" class=\"wp-image-4673\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"248\" height=\"135\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM7.png\" alt=\"\" class=\"wp-image-4673\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Por lo tanto el dominio de<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM8.png\" alt=\"\" class=\"wp-image-4675\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"547\" height=\"107\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM8.png\" alt=\"\" class=\"wp-image-4675\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM8.png 547w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM8-300x59.png 300w\" sizes=\"auto, (max-width: 547px) 100vw, 547px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ahora hay que buscar la intersecci\u00f3n de ambos dominios<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM9.png\" alt=\"\" class=\"wp-image-4679\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"966\" height=\"98\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM9.png\" alt=\"\" class=\"wp-image-4679\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM9.png 966w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM9-300x30.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM9-768x78.png 768w\" sizes=\"auto, (max-width: 966px) 100vw, 966px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"funciones-racionales\">Funciones racionales<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Las funciones racionales tienen la forma:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM10.png\" alt=\"\" class=\"wp-image-4684\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"309\" height=\"112\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM10.png\" alt=\"\" class=\"wp-image-4684\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM10.png 309w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM10-300x109.png 300w\" sizes=\"auto, (max-width: 309px) 100vw, 309px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Donde <em>q(x)<\/em> debe ser diferente de cero, en general <em>p(x)<\/em> y <em>q(x)<\/em> son funciones polinomiales.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"composicion-de-funciones\">Composici\u00f3n de funciones<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM11-1024x147.png\" alt=\"\" class=\"wp-image-4692\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"147\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM11-1024x147.png\" alt=\"\" class=\"wp-image-4692\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM11-1024x147.png 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM11-300x43.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM11-768x111.png 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM11.png 1056w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM12-1024x127.png\" alt=\"\" class=\"wp-image-4694\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"127\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM12-1024x127.png\" alt=\"\" class=\"wp-image-4694\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM12-1024x127.png 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM12-300x37.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM12-768x95.png 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM12.png 1031w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Si<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM13.png\" alt=\"\" class=\"wp-image-4701\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"802\" height=\"100\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM13.png\" alt=\"\" class=\"wp-image-4701\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM13.png 802w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM13-300x37.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM13-768x96.png 768w\" sizes=\"auto, (max-width: 802px) 100vw, 802px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Encuentre <em>f(g(x))<\/em> y <em>g(f(x))<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituyendo para obtener <em>f(g(x))<\/em>, coloque las funciones para que pueda identificarlas<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM14.png\" alt=\"\" class=\"wp-image-4703\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"646\" height=\"76\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM14.png\" alt=\"\" class=\"wp-image-4703\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM14.png 646w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM14-300x35.png 300w\" sizes=\"auto, (max-width: 646px) 100vw, 646px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM15.png\" alt=\"\" class=\"wp-image-4704\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"453\" height=\"77\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM15.png\" alt=\"\" class=\"wp-image-4704\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM15.png 453w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM15-300x51.png 300w\" sizes=\"auto, (max-width: 453px) 100vw, 453px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituyendo para obtener <em>g(f(x))<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM16.png\" alt=\"\" class=\"wp-image-4711\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"545\" height=\"109\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM16.png\" alt=\"\" class=\"wp-image-4711\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM16.png 545w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM16-300x60.png 300w\" sizes=\"auto, (max-width: 545px) 100vw, 545px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Se puede observar que las funciones obtenidas son diferentes.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"transformaciones-rigidas\">Transformaciones r\u00edgidas<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Una transformacio\u0301n ri\u0301gida de una gra\u0301fica es una transformacio\u0301n que cambia so\u0301lo la posicio\u0301n de la gra\u0301fica en el plano <em>xy<\/em>, pero no su forma. Para la gra\u0301fica de una funcio\u0301n <em>y=f(x)<\/em> se analizan cuatro tipos de desplazamientos o traslaciones.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM17.png\" alt=\"\" class=\"wp-image-4716\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"850\" height=\"312\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM17.png\" alt=\"\" class=\"wp-image-4716\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM17.png 850w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM17-300x110.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM17-768x282.png 768w\" sizes=\"auto, (max-width: 850px) 100vw, 850px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Para visualizar un poco esta transformaci\u00f3n observa la siguiente imagen, la primera imagen es la funci\u00f3n original y continuando se pueden observar todas las traslaciones.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM18.png\" alt=\"\" class=\"wp-image-4720\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"461\" height=\"257\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM18.png\" alt=\"\" class=\"wp-image-4720\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM18.png 461w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM18-300x167.png 300w\" sizes=\"auto, (max-width: 461px) 100vw, 461px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM19.png\" alt=\"\" class=\"wp-image-4732\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"933\" height=\"252\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM19.png\" alt=\"\" class=\"wp-image-4732\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM19.png 933w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM19-300x81.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM19-768x207.png 768w\" sizes=\"auto, (max-width: 933px) 100vw, 933px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Otra forma de transformar ri\u0301gidamente la gra\u0301fica de una funcio\u0301n es por medio de una reflexio\u0301n en un eje de coordenadas.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM21-1024x249.png\" alt=\"\" class=\"wp-image-4785\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"249\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM21-1024x249.png\" alt=\"\" class=\"wp-image-4785\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM21-1024x249.png 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM21-300x73.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM21-768x187.png 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM21.png 1051w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"transformaciones-no-rigidas\">Transformaciones no r\u00edgidas<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Si una funcio\u0301n <em>f<\/em> se multiplica por una constante <em>c &gt; 0<\/em>, la forma de la gra\u0301fica cambia pero retiene, aproximadamente, su forma original. La gra\u0301fica de <em>y = cf(x)<\/em> es la gra\u0301fica de <em>y = f(x)<\/em> distorsionada verticalmente; la gra\u0301fica de <em>f<\/em> se estira (o elonga) verticalmente o se comprime (o aplana) verticalmente, dependiendo del valor de <em>c<\/em>. En otros te\u0301rminos, un estiramiento vertical es un estiramiento de la gra\u0301fica de <em>y = f(x)<\/em> alej\u00e1ndose del eje <em>x<\/em>, mientras que una compresio\u0301n vertical es una compresio\u0301n de la gra\u0301fica de <em>y = f(x)<\/em> hacia el eje <em>x<\/em>. La gra\u0301fica de la funcio\u0301n <em>y = f(cx)<\/em> esta\u0301 distorsionada horizontalmente, ya sea por un estiramiento de la gra\u0301fica de <em>y = f(x)<\/em> alej\u00e1ndose del eje y o por una compresio\u0301n de la gra\u0301fica de <em>y = f(x)<\/em> hacia el eje <em>y<\/em>. El estiramiento o la compresio\u0301n de una gra\u0301fica constituyen ejemplos de transformaciones no ri\u0301gidas.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM22-1024x398.png\" alt=\"\" class=\"wp-image-4791\" width=\"768\" height=\"299\" \/><noscript><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM22-1024x398.png\" alt=\"\" class=\"wp-image-4791\" width=\"768\" height=\"299\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM22-1024x398.png 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM22-300x117.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM22-768x298.png 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM22.png 1027w\" sizes=\"auto, (max-width: 768px) 100vw, 768px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sea la funci\u00f3n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM23-1.png\" alt=\"\" class=\"wp-image-4813\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"250\" height=\"93\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM23-1.png\" alt=\"\" class=\"wp-image-4813\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">realiza la traslaci\u00f3n:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>Dos unidades hacia arriba<\/li><li>Dos unidades hacia abajo<\/li><li>Dos unidades a la izquierda<\/li><li>Dos unidades a la derecha<\/li><li>Refleja la funci\u00f3n en el eje de las x<\/li><\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">a) Para deslizar dos unidades hacia arriba se debe utilizar <em>y = f(x) + c<\/em>, en este caso <em>c = 2<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM24-1.png\" alt=\"\" class=\"wp-image-4829\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"984\" height=\"537\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM24-1.png\" alt=\"\" class=\"wp-image-4829\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM24-1.png 984w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM24-1-300x164.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM24-1-768x419.png 768w\" sizes=\"auto, (max-width: 984px) 100vw, 984px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">b) Para deslizar dos unidades hacia abajo se debe utilizar <em>y = f(x) &#8211; c<\/em>, en este caso <em>c = 2<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM27.png\" alt=\"\" class=\"wp-image-4832\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"441\" height=\"473\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM27.png\" alt=\"\" class=\"wp-image-4832\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM27.png 441w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM27-280x300.png 280w\" sizes=\"auto, (max-width: 441px) 100vw, 441px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">c) Para deslizar dos unidades a la izquierda utilizar <em>y = f(x) + c<\/em>, en este caso <em>c = 2<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM29.png\" alt=\"\" class=\"wp-image-4833\" width=\"127\" height=\"50\" \/><noscript><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM29.png\" alt=\"\" class=\"wp-image-4833\" width=\"127\" height=\"50\" \/><\/noscript><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM28.png\" alt=\"\" class=\"wp-image-4834\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"604\" height=\"533\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM28.png\" alt=\"\" class=\"wp-image-4834\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM28.png 604w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM28-300x265.png 300w\" sizes=\"auto, (max-width: 604px) 100vw, 604px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">d) Para deslizar dos unidades a la derecha utilizar <em>y = f(x) &#8211; c<\/em>, en este caso <em>c = 2<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM30.png\" alt=\"\" class=\"wp-image-4835\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"417\" height=\"465\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM30.png\" alt=\"\" class=\"wp-image-4835\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM30.png 417w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM30-269x300.png 269w\" sizes=\"auto, (max-width: 417px) 100vw, 417px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">e) Para reflejar la funci\u00f3n en el eje de las <em>x<\/em>, se utiliza <em>y = &#8211; f(x)<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM31.png\" alt=\"\" class=\"wp-image-4836\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"134\" height=\"67\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM31.png\" alt=\"\" class=\"wp-image-4836\" \/><\/noscript><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM32.png\" alt=\"\" class=\"wp-image-4838\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"534\" height=\"528\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM32.png\" alt=\"\" class=\"wp-image-4838\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM32.png 534w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM32-300x297.png 300w\" sizes=\"auto, (max-width: 534px) 100vw, 534px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"simetria\">Simetr\u00eda<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Si la gra\u0301fica de una funcio\u0301n es sime\u0301trica con respecto al eje y, decimos que f es una funcio\u0301n par. Se dice que una funcio\u0301n cuya gra\u0301fica es sime\u0301trica con respecto al origen es una funcio\u0301n impar. Contamos con las siguientes pruebas para simetri\u0301a.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM33.png\" alt=\"\" class=\"wp-image-4837\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"817\" height=\"167\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM33.png\" alt=\"\" class=\"wp-image-4837\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM33.png 817w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM33-300x61.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM33-768x157.png 768w\" sizes=\"auto, (max-width: 817px) 100vw, 817px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Determina la simetr\u00eda de la funci\u00f3n:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM34.png\" alt=\"\" class=\"wp-image-4842\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"508\" height=\"93\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM34.png\" alt=\"\" class=\"wp-image-4842\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM34.png 508w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM34-300x55.png 300w\" sizes=\"auto, (max-width: 508px) 100vw, 508px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Verificando simetr\u00eda con el eje y <em>f (- x) = f(x)<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Realizando <em>f(-x)<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM35.png\" alt=\"\" class=\"wp-image-4843\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"797\" height=\"173\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM35.png\" alt=\"\" class=\"wp-image-4843\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM35.png 797w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM35-300x65.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM35-768x167.png 768w\" sizes=\"auto, (max-width: 797px) 100vw, 797px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Al cumplirse la igualdad se dice que la funci\u00f3n tiene simetr\u00eda con el eje y, ya que <em>f(-x) = f(x)<\/em>, al graficar la funci\u00f3n se puede observar esa simetr\u00eda<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM36.png\" alt=\"\" class=\"wp-image-4844\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"328\" height=\"486\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM36.png\" alt=\"\" class=\"wp-image-4844\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM36.png 328w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM36-202x300.png 202w\" sizes=\"auto, (max-width: 328px) 100vw, 328px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Determina la simetr\u00eda de la funci\u00f3n:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM38.png\" alt=\"\" class=\"wp-image-4846\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"196\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM38.png\" alt=\"\" class=\"wp-image-4846\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM38.png 620w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM38-300x95.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Verificando simetr\u00eda con <em>f(-x) = f(x)<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM39.png\" alt=\"\" class=\"wp-image-4848\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"496\" height=\"188\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM39.png\" alt=\"\" class=\"wp-image-4848\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM39.png 496w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM39-300x114.png 300w\" sizes=\"auto, (max-width: 496px) 100vw, 496px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Al no cumplirse la igualdad se dice que la funci\u00f3n no tiene simetr\u00eda par o con el eje de las y<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Verificando simetr\u00eda con <em>f(-x) = -f(x)<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM40.png\" alt=\"\" class=\"wp-image-4847\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"890\" height=\"121\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM40.png\" alt=\"\" class=\"wp-image-4847\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM40.png 890w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM40-300x41.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM40-768x104.png 768w\" sizes=\"auto, (max-width: 890px) 100vw, 890px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Al cumplirse <em>f(-x) = -f(x)<\/em>, se dice que la funci\u00f3n es impar y tiene simetr\u00eda con el eje <em>x<\/em><\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"funciones-polinomiales-y-racionales\">Funciones polinomiales y racionales<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Una funci\u00f3n polinomial tiene la forma:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM41.png\" alt=\"\" class=\"wp-image-4849\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"301\" height=\"126\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM41.png\" alt=\"\" class=\"wp-image-4849\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Donde p(x) y q(x) son funciones polinomiales.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En las funciones polinomiales <em><strong>a<sub>0<\/sub>, a<sub>1<\/sub>, a<sub>2<\/sub>, \u2026, a<sub>n<\/sub><\/strong><\/em> son constantes que se denominan coeficientes el nu\u0301mero <em><strong>a<sub>n<\/sub><\/strong><\/em> se llama coeficiente principal y a0 se denomina te\u0301rmino constante del polinomio. Se dice que la mayor potencia de x en un polinomio es el grado de e\u0301ste.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM42.png\" alt=\"\" class=\"wp-image-4850\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"996\" height=\"221\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM42.png\" alt=\"\" class=\"wp-image-4850\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM42.png 996w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM42-300x67.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM42-768x170.png 768w\" sizes=\"auto, (max-width: 996px) 100vw, 996px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"rectas\">Rectas<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">En el plano <em>xy<\/em> hay tres tipos de rectas: rectas horizontales, rectas verticales y rectas inclinadas u oblicuas. Cabe mencionar que la funci\u00f3n lineal es una recta. La pendiente es conocida como el ascenso o descenso, es decir, el cambio de posici\u00f3n en el eje <em>x<\/em> y <em>y<\/em>, la ecuaci\u00f3n de la pendiente es:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM43.png\" alt=\"\" class=\"wp-image-4851\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"412\" height=\"135\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM43.png\" alt=\"\" class=\"wp-image-4851\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM43.png 412w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM43-300x98.png 300w\" sizes=\"auto, (max-width: 412px) 100vw, 412px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">en la figura se observa su ascenso vertical y recorrido horizontal.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM44.png\" alt=\"\" class=\"wp-image-4852\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"386\" height=\"295\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM44.png\" alt=\"\" class=\"wp-image-4852\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM44.png 386w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM44-300x229.png 300w\" sizes=\"auto, (max-width: 386px) 100vw, 386px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Dependiendo del valor de la pendiente es el sentido que la recta tomar\u00e1, en la siguiente imagen podr\u00e1 observar esa direcci\u00f3n.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM45-1024x373.png\" alt=\"\" class=\"wp-image-4853\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"373\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM45-1024x373.png\" alt=\"\" class=\"wp-image-4853\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM45-1024x373.png 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM45-300x109.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM45-768x280.png 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM45.png 1030w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM46.png\" alt=\"\" class=\"wp-image-4854\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"935\" height=\"353\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM46.png\" alt=\"\" class=\"wp-image-4854\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM46.png 935w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM46-300x113.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM46-768x290.png 768w\" sizes=\"auto, (max-width: 935px) 100vw, 935px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">La ecuaci\u00f3n de la recta es <em>y = mx + b<\/em>; donde <em>m<\/em> es la pendiente, <em>b<\/em> la ordenada al origen<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Encuentre la ecuaci\u00f3n de la recta que pasa por los puntos A(-1,8) y B(1,4)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En los pares ordenados identificar <em>x<\/em> y <em>y<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM47.png\" alt=\"\" class=\"wp-image-4855\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"421\" height=\"188\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM47.png\" alt=\"\" class=\"wp-image-4855\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM47.png 421w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM47-300x134.png 300w\" sizes=\"auto, (max-width: 421px) 100vw, 421px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Calculando la pendiente<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM48.png\" alt=\"\" class=\"wp-image-4856\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"631\" height=\"133\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM48.png\" alt=\"\" class=\"wp-image-4856\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM48.png 631w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM48-300x63.png 300w\" sizes=\"auto, (max-width: 631px) 100vw, 631px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Para calcular la ordenada al origen se toma un par ordenado y se despeja b de la ecuaci\u00f3n de la recta tangente<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM49.png\" alt=\"\" class=\"wp-image-4857\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"638\" height=\"134\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM49.png\" alt=\"\" class=\"wp-image-4857\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM49.png 638w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM49-300x63.png 300w\" sizes=\"auto, (max-width: 638px) 100vw, 638px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">La ecuaci\u00f3n de la recta que pasa por los puntos A(-1,8) y B(1,4)<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM50.png\" alt=\"\" class=\"wp-image-4858\" width=\"276\" height=\"48\" \/><noscript><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM50.png\" alt=\"\" class=\"wp-image-4858\" width=\"276\" height=\"48\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM50.png 368w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM50-300x52.png 300w\" sizes=\"auto, (max-width: 276px) 100vw, 276px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM51.png\" alt=\"\" class=\"wp-image-4859\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"813\" height=\"546\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM51.png\" alt=\"\" class=\"wp-image-4859\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM51.png 813w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM51-300x201.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM51-768x516.png 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM51-272x182.png 272w\" sizes=\"auto, (max-width: 813px) 100vw, 813px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">En la imagen anterior se muestra como la ecuaci\u00f3n obtenida pasa por los puntos.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"funciones-crecientes-y-decrecientes\">Funciones crecientes y decrecientes<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Los valores de una funcio\u0301n lineal<em> f(x) = ax + b<\/em> crecen cuando <em>x<\/em> crece, mientras que para <em>a&lt;0<\/em>, los valores de <em>f(x)<\/em> disminuyen cuando <em>x<\/em> crece. Los conceptos creciente y decreciente pueden extenderse a cualquier funcio\u0301n. Se dice que una funcio\u0301n <em>f<\/em> es:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Creciente sobre un intervalo <em>f(x1) &lt; f(x2)<\/em><\/li><li>Decreciente sobre un intervalo&nbsp;<em>f(x1) &gt; f(x2)<\/em><\/li><\/ul>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM52.png\" alt=\"\" class=\"wp-image-4860\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"932\" height=\"356\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM52.png\" alt=\"\" class=\"wp-image-4860\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM52.png 932w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM52-300x115.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM52-768x293.png 768w\" sizes=\"auto, (max-width: 932px) 100vw, 932px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"asintotas\">As\u00edntotas<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">La gra\u0301fica de una funcio\u0301n racional<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM53.png\" alt=\"\" class=\"wp-image-4861\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"308\" height=\"111\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM53.png\" alt=\"\" class=\"wp-image-4861\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM53.png 308w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM53-300x108.png 300w\" sizes=\"auto, (max-width: 308px) 100vw, 308px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">puede tener asi\u0301ntotas. Para los objetivos de este libro, las asi\u0301ntotas pueden ser una recta horizontal, una recta vertical o una recta inclinada. En un nivel pra\u0301ctico, las asi\u0301ntotas vertical y horizontal de la gra\u0301fica de una funcio\u0301n racional f pueden determinarse por inspeccio\u0301n. Asi\u0301, por el bien del ana\u0301lisis se supondra\u0301 que<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM54-1024x120.png\" alt=\"\" class=\"wp-image-4862\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"120\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM54-1024x120.png\" alt=\"\" class=\"wp-image-4862\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM54-1024x120.png 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM54-300x35.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM54-768x90.png 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM54.png 1310w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Criterios para encontrar as\u00edntotas en una funci\u00f3n polinomial<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM55.png\" alt=\"\" class=\"wp-image-4863\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"968\" height=\"268\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM55.png\" alt=\"\" class=\"wp-image-4863\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM55.png 968w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM55-300x83.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM55-768x213.png 768w\" sizes=\"auto, (max-width: 968px) 100vw, 968px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">a) Encuentra las intersecciones y as\u00edntotas, despu\u00e9s grafica<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM56.png\" alt=\"\" class=\"wp-image-4864\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"314\" height=\"125\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM56.png\" alt=\"\" class=\"wp-image-4864\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM56.png 314w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM56-300x119.png 300w\" sizes=\"auto, (max-width: 314px) 100vw, 314px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Soluci\u00f3n<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Encontrando intersecci\u00f3n con el eje <em>y<\/em><br>Sustituyendo <em>x = 0<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM57.png\" alt=\"\" class=\"wp-image-4865\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"236\" height=\"96\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM57.png\" alt=\"\" class=\"wp-image-4865\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">La intersecci\u00f3n con el eje y es en el punto (0,0)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Encontrando la intersecci\u00f3n con el eje <em>x<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituyendo <em>y = 0<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM58.png\" alt=\"\" class=\"wp-image-4866\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"175\" height=\"143\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM58.png\" alt=\"\" class=\"wp-image-4866\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">La intersecci\u00f3n con el eje <em>x<\/em> es en el punto (0,0)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Encontrando las as\u00edntotas<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM59.png\" alt=\"\" class=\"wp-image-4867\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"368\" height=\"110\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM59.png\" alt=\"\" class=\"wp-image-4867\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM59.png 368w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM59-300x90.png 300w\" sizes=\"auto, (max-width: 368px) 100vw, 368px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Encontrando la as\u00edntota Vertical, significa igualar q(x) a cero<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM60.png\" alt=\"\" class=\"wp-image-4868\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"283\" height=\"78\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM60.png\" alt=\"\" class=\"wp-image-4868\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Despejando <em>x<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM61.png\" alt=\"\" class=\"wp-image-4869\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"288\" height=\"273\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM61.png\" alt=\"\" class=\"wp-image-4869\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Existen dos as\u00edntotas verticales que son <em>x = -1<\/em> y <em>x = 1<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Encontrando las as\u00edntotas horizontales<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Identificando n y m de los polinomios <em>n = 1<\/em>, <em>m = 2<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Como <em>n &lt; m<\/em>, la as\u00edntota horizontal es <em>y=0<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM62.png\" alt=\"\" class=\"wp-image-4870\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"616\" height=\"309\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM62.png\" alt=\"\" class=\"wp-image-4870\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM62.png 616w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM62-300x150.png 300w\" sizes=\"auto, (max-width: 616px) 100vw, 616px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">b) Encuentra las intersecciones y as\u00edntotas, despu\u00e9s grafica<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM63.png\" alt=\"\" class=\"wp-image-4871\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"371\" height=\"123\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM63.png\" alt=\"\" class=\"wp-image-4871\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM63.png 371w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM63-300x99.png 300w\" sizes=\"auto, (max-width: 371px) 100vw, 371px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Encontrando intersecci\u00f3n con el eje <em>y<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituir <em>x = 0<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM64.png\" alt=\"\" class=\"wp-image-4872\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"464\" height=\"131\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM64.png\" alt=\"\" class=\"wp-image-4872\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM64.png 464w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM64-300x85.png 300w\" sizes=\"auto, (max-width: 464px) 100vw, 464px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">La intersecci\u00f3n con el eje <em>y<\/em> es<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM65.png\" alt=\"\" class=\"wp-image-4873\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"186\" height=\"133\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM65.png\" alt=\"\" class=\"wp-image-4873\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Encontrando intersecci\u00f3n con el eje <em>x<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Sustituir <em>y = 0<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM66.png\" alt=\"\" class=\"wp-image-4874\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"323\" height=\"112\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM66.png\" alt=\"\" class=\"wp-image-4874\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM66.png 323w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM66-300x104.png 300w\" sizes=\"auto, (max-width: 323px) 100vw, 323px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Factorizando<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM69.png\" alt=\"\" class=\"wp-image-4876\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"668\" height=\"546\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM69.png\" alt=\"\" class=\"wp-image-4876\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM69.png 668w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM69-300x245.png 300w\" sizes=\"auto, (max-width: 668px) 100vw, 668px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">La intersecciones con x son (-2,0) y (3,0)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Buscando as\u00edntotas<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">As\u00edntota vertical<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM68.png\" alt=\"\" class=\"wp-image-4877\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"364\" height=\"146\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM68.png\" alt=\"\" class=\"wp-image-4877\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM68.png 364w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM68-300x120.png 300w\" sizes=\"auto, (max-width: 364px) 100vw, 364px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Igualando <em>q(x) = 0<\/em><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM70.png\" alt=\"\" class=\"wp-image-4878\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"280\" height=\"138\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM70.png\" alt=\"\" class=\"wp-image-4878\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">As\u00edntota horizontal<br>Identificando <em>n<\/em> y <em>m<\/em><br><em>n = 2<\/em>, <em>m = 1<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Como n&gt;m no hay as\u00edntota horizontal, entonces si hay as\u00edntota oblicua<br><em>n = m + 1<br>2 = 1 + 1<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Realizar el cociente de la funci\u00f3n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM71.png\" alt=\"\" class=\"wp-image-4879\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"649\" height=\"285\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM71.png\" alt=\"\" class=\"wp-image-4879\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM71.png 649w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM71-300x132.png 300w\" sizes=\"auto, (max-width: 649px) 100vw, 649px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM72.png\" alt=\"\" class=\"wp-image-4880\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"706\" height=\"428\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM72.png\" alt=\"\" class=\"wp-image-4880\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM72.png 706w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM72-300x182.png 300w\" sizes=\"auto, (max-width: 706px) 100vw, 706px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Funciones trascendentes<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Una funcio\u0301n trascendente puede ser tan simple como la funcio\u0301n potencia y=xn, donde la potencia es un nu\u0301mero irracional, pero las conocidas funciones trascendentes de preca\u0301lculo en matema\u0301ticas son las funciones trigonome\u0301tricas, las funciones trigonome\u0301tricas inversas y las funciones exponencial y logari\u0301tmica.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"funciones-trigonometricas\">Funciones trigonom\u00e9tricas<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Las funciones trigonom\u00e9tricas presentan periodicidad, lo que permite que sean empleadas para modelar fen\u00f3menos f\u00edsicos que presentan cierta repetitividad.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Las funciones trigonom\u00e9tricas son tambi\u00e9n conocidas como funciones circulares, ya que extienden la definici\u00f3n de las razones trigonom\u00e9tricas a todos los n\u00fameros reales.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"funcion-seno-y-coseno\">Funci\u00f3n seno y coseno<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Sea x un \u00e1ngulo en radianes. Para graficar la funci\u00f3n seno se asignan valores a <em>x<\/em> entre&nbsp;<em>0 \u2264 x \u2264 2\u03c0<\/em> y determina el correspondiente valor del seno. Las parejas ordenadas se ubican en el plano cartesiano y se unen con una curva suave, completando un ciclo de la gr\u00e1fica de la funci\u00f3n. Por ser peri\u00f3dica <em>f(x) = sin sin x<\/em> se traza de manera repetida el ciclo.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En la imagen se muestra la gr\u00e1fica de la funci\u00f3n seno de <em>0 \u2264 x \u2264 2\u03c0<\/em>, esta parte representa solo un ciclo de la funci\u00f3n, si se desean graficar m\u00e1s ciclos, se puede realizar:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM73.png\" alt=\"\" class=\"wp-image-4881\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"449\" height=\"261\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM73.png\" alt=\"\" class=\"wp-image-4881\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM73.png 449w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM73-300x174.png 300w\" sizes=\"auto, (max-width: 449px) 100vw, 449px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">A continuaci\u00f3n se muestra la gr\u00e1fica de la funci\u00f3n seno con m\u00e1s de un ciclo.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM74.png\" alt=\"\" class=\"wp-image-4882\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"867\" height=\"307\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM74.png\" alt=\"\" class=\"wp-image-4882\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM74.png 867w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM74-300x106.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM74-768x272.png 768w\" sizes=\"auto, (max-width: 867px) 100vw, 867px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">De las gr\u00e1ficas de las funciones seno y coseno se sabe que el dominio de ambas son todos los n\u00fameros reales y el rango oscila entre [-1,1]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"conclusion\">Conclusi\u00f3n<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">En conclusi\u00f3n, las funciones polinomiales constituyen los objetos fundamentales de una clase conocida como funciones algebraicas. En esta clase se abord\u00f3 que una funci\u00f3n racional es el cociente de dos funciones polinomiales. En general, una funci\u00f3n algebraica implica un n\u00famero finito de sumas, restas, multiplicaciones, divisiones y ra\u00edces cuadradas de funciones polinomiales. As\u00ed<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM25-1024x113.png\" alt=\"\" class=\"wp-image-4812\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"113\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM25-1024x113.png\" alt=\"\" class=\"wp-image-4812\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM25-1024x113.png 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM25-300x33.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM25-768x85.png 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2022\/01\/CD4_IM25.png 1048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Son funciones algebraicas.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Una funci\u00f3n trascendente f se define como una funci\u00f3n que no es algebraica. Las seis funciones trigonom\u00e9tricas y las funciones exponencial y logar\u00edtmica son ejemplos de funciones trascendentes.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hasta aqu\u00ed se concluye la clase. \u00a1Te felicito, vas muy bien! Te recuerdo que depende mucho de tu entusiasmo por aprender. No olvides hacer y mandar como corresponde la tarea asignada. Te espero en tu pr\u00f3xima clase, hasta entonces.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"fuentes-de-informacion\">Fuentes de informaci\u00f3n<\/h2>\n\n\n\n<ul class=\"wp-block-list\"><li>Zill, D. G., &amp; Wrigth. W. S. (2011). Matem\u00e1ticas 1: C\u00e1lculo Diferencial. M\u00e9xico: MC Graw Hill.<\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Introducci\u00f3n \u00a1Hola! Siempre es un gusto saludarte y saber que tienes el \u00e1nimo para continuar, te invito a seguir en este camino formativo en tu cuarta clase titulada Funciones algebraicas: polinomiales y racionales del curso C\u00e1lculo Diferencial. En esta clase comprender\u00e1s el concepto de funci\u00f3n y las nociones fundamentales de la misma en las matem\u00e1ticas. &#8230; <a title=\"Clase digital 4: Funciones algebraicas: polinomiales y racionales. Funciones trascendentes: trigonome\u0301tricas, logari\u0301tmicas y exponenciales. Operaciones con funciones: adicio\u0301n, multiplicacio\u0301n, divisio\u0301n y composicio\u0301n\" class=\"read-more\" href=\"https:\/\/blogs.ugto.mx\/rea\/clase-digital-4-funciones-algebraicas-polinomiales-y-racionales-funciones-trascendentes-trigonometricas-logaritmicas-y-exponenciales-operaciones-con-funciones-adicion-multipl\/\" aria-label=\"Leer m\u00e1s sobre Clase digital 4: Funciones algebraicas: polinomiales y racionales. Funciones trascendentes: trigonome\u0301tricas, logari\u0301tmicas y exponenciales. Operaciones con funciones: adicio\u0301n, multiplicacio\u0301n, divisio\u0301n y composicio\u0301n\">Leer m\u00e1s<\/a><\/p>\n","protected":false},"author":142,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_crdt_document":"","episode_type":"","audio_file":"","podmotor_file_id":"","podmotor_episode_id":"","cover_image":"","cover_image_id":"","duration":"","filesize":"","filesize_raw":"","date_recorded":"","explicit":"","block":"","itunes_episode_number":"","itunes_title":"","itunes_season_number":"","itunes_episode_type":"","footnotes":""},"categories":[10,12],"tags":[41,149,44],"class_list":["post-4616","post","type-post","status-publish","format-standard","hentry","category-ingenieria-civil","category-uda-calculo-diferencial","tag-clase-digital","tag-marcela-guadalupe-tellez-martinez","tag-neli04036"],"acf":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/4616","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/users\/142"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/comments?post=4616"}],"version-history":[{"count":9,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/4616\/revisions"}],"predecessor-version":[{"id":6957,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/4616\/revisions\/6957"}],"wp:attachment":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/media?parent=4616"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/categories?post=4616"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/tags?post=4616"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}