{"id":7,"date":"2023-09-17T11:16:54","date_gmt":"2023-09-17T11:16:54","guid":{"rendered":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/"},"modified":"2023-09-17T11:16:54","modified_gmt":"2023-09-17T11:16:54","slug":"funcao-de-parabola-quadratica","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/","title":{"rendered":"Fun\u00e7\u00e3o quadr\u00e1tica ou par\u00e1bola"},"content":{"rendered":"<p>Esta p\u00e1gina explica o que \u00e9 uma fun\u00e7\u00e3o quadr\u00e1tica bem como todas as suas caracter\u00edsticas: curvatura, v\u00e9rtice, pontos de intersec\u00e7\u00e3o com os eixos, etc. Voc\u00ea tamb\u00e9m aprender\u00e1 como representar uma fun\u00e7\u00e3o quadr\u00e1tica em um gr\u00e1fico. E por fim, voc\u00ea pode praticar com exemplos, exerc\u00edcios passo a passo e problemas sobre fun\u00e7\u00f5es quadr\u00e1ticas. <\/p>\n<p><strong><\/strong><\/p>\n<p><strong><\/strong><\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-104\"><strong> <\/strong><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-una-funcion-cuadratica\"><\/span> O que \u00e9 uma fun\u00e7\u00e3o quadr\u00e1tica? <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong><\/strong><\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><strong> <\/strong><\/div>\n<\/div>\n<p>A defini\u00e7\u00e3o de uma fun\u00e7\u00e3o quadr\u00e1tica \u00e9 a seguinte: <\/p>\n<div style=\"padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px;\">\n<p style=\"text-align:left\"> Em matem\u00e1tica, uma <strong>fun\u00e7\u00e3o quadr\u00e1tica (ou parab\u00f3lica)<\/strong> \u00e9 uma fun\u00e7\u00e3o polinomial de grau 2, ou seja, uma fun\u00e7\u00e3o em que o termo de maior grau \u00e9 de segundo grau. Portanto, a f\u00f3rmula para uma fun\u00e7\u00e3o quadr\u00e1tica \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-576d742ea12f780a76fa5809f5112086_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=ax^2+bx+c\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"154\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Ouro:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0dfe37f98696a04358a5783f12ff4d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"ax^2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 o termo quadr\u00e1tico.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e925448cc849aa310b3d4a7b77594e4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"bx\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 o termo linear.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"c\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 o termo independente.<\/li>\n<\/ul>\n<\/div>\n<p> O dom\u00ednio de uma fun\u00e7\u00e3o quadr\u00e1tica sempre consiste em n\u00fameros reais. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-91d461485d0f02bb14db6855a3774878_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f=\\mathbff{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"concavidad-y-convexidad-de-una-funcion-cuadratica\"><\/span> Concavidade e convexidade de uma fun\u00e7\u00e3o quadr\u00e1tica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Analisar a curvatura de uma fun\u00e7\u00e3o quadr\u00e1tica ou parab\u00f3lica \u00e9 muito simples, pois depende apenas do coeficiente quadr\u00e1tico.<\/p>\n<ul>\n<li> Se o coeficiente\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 positiva, a fun\u00e7\u00e3o quadr\u00e1tica \u00e9 <strong>convexa<\/strong> (na forma<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f5ebc563dbe58138d1de6b7fe99e8d31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\cup}\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> ). A cimeira \u00e9, portanto, um m\u00ednimo.<\/li>\n<li> Se o coeficiente\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 negativo, a fun\u00e7\u00e3o quadr\u00e1tica \u00e9 <strong>c\u00f4ncava<\/strong> (em forma<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dccbfcebef91876585ebd365457c3d24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\cap}\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> ). O pico \u00e9, portanto, um m\u00e1ximo. <\/li>\n<\/ul>\n<div class=\"wp-block-columns is-layout-flex wp-container-349\">\n<div class=\"wp-block-column is-layout-flow\">\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/fonction-quadratique-ou-parabole-convexe.webp\" alt=\"fun\u00e7\u00e3o quadr\u00e1tica ou par\u00e1bola convexa\" class=\"wp-image-132\" width=\"254\" height=\"259\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/fonction-quadratique-ou-parabole-concave.webp\" alt=\"fun\u00e7\u00e3o quadr\u00e1tica ou par\u00e1bola c\u00f4ncava\" class=\"wp-image-133\" width=\"255\" height=\"260\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<p> <strong>Nota:<\/strong> A comunidade matem\u00e1tica ainda n\u00e3o concorda plenamente e, por isso, alguns professores dizem o contr\u00e1rio: chamam uma fun\u00e7\u00e3o c\u00f4ncava aquela que tem a forma de um<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f5ebc563dbe58138d1de6b7fe99e8d31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\cup}\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> , e uma fun\u00e7\u00e3o convexa que tem a forma de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dccbfcebef91876585ebd365457c3d24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\cap}\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Em todo caso, o importante \u00e9 qual forma tem a fun\u00e7\u00e3o, seja qual for o nome. <\/p>\n<p><strong><\/strong><\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><strong> <\/strong><\/div>\n<p><strong><br \/><\/strong><\/p>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"vertice-de-una-funcion-cuadratica\"><\/span> V\u00e9rtice de uma fun\u00e7\u00e3o quadr\u00e1tica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para representar graficamente uma fun\u00e7\u00e3o quadr\u00e1tica, \u00e9 necess\u00e1rio conhecer as coordenadas do v\u00e9rtice da par\u00e1bola. <\/p>\n<div style=\"padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px;\">\n<p style=\"text-align:left\"> Para encontrar o v\u00e9rtice de uma fun\u00e7\u00e3o quadr\u00e1tica, precisamos calcular a coordenada X do ponto usando a seguinte f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-93ad079a75f2c2a452e1aaf0654aaaf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x=\\frac{-b}{2a}\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"58\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<p> Ent\u00e3o podemos encontrar a outra coordenada do v\u00e9rtice calculando a imagem da fun\u00e7\u00e3o naquele ponto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8ad3822f13f081a20e55c52589118288_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f\\left(\\frac{-b}{2a}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"62\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Assim, as coordenadas do v\u00e9rtice de uma fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola) s\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-37777c8a435f1df10f87842c8bd463d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left(\\frac{-b}{2a} \\ , \\ f\\left(\\frac{-b}{2a}\\right)\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"130\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"puntos-de-corte-con-los-ejes-de-una-funcion-cuadratica\"><\/span> Cortando pontos com os eixos de uma fun\u00e7\u00e3o quadr\u00e1tica <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong><\/strong><\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-107\"><strong> <\/strong><\/div>\n<\/div>\n<div style=\"padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px;\">\n<p style=\"text-align:left\"> Uma par\u00e1bola sempre intercepta o eixo y (eixo Y), e isso acontece quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<p> Portanto, para calcular o ponto de corte de uma fun\u00e7\u00e3o quadr\u00e1tica com eixo Y, deve-se resolver<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d449eebd1f011aebdf90931f3a66a3b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<p> Por exemplo, o ponto de intersec\u00e7\u00e3o com o eixo OY da seguinte fun\u00e7\u00e3o quadr\u00e1tica \u00e9: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7a32a12ecec8c6b0617daf9e84bd4d07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"146\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9d17bae0af24be9eaa8833d402a1709c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=0^2-2\\cdot 0+1 = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"188\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-308d0aeca611c20e62ee56db1ced4618_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div style=\"padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px;\">\n<p style=\"text-align:left\"> Por outro lado, o ponto de corte de uma fun\u00e7\u00e3o quadr\u00e1tica com o eixo x (eixo X) ocorre quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ae937c042e388c1f5126fceae07be7ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<p> Ent\u00e3o para calcular o ponto de intersec\u00e7\u00e3o com o eixo X voc\u00ea tem que resolver a equa\u00e7\u00e3o<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ae937c042e388c1f5126fceae07be7ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<p> A t\u00edtulo de exemplo, segue abaixo o c\u00e1lculo do ponto de corte com o eixo OX da mesma fun\u00e7\u00e3o quadr\u00e1tica:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7a32a12ecec8c6b0617daf9e84bd4d07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"146\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-62b9becef089451225ec98e56e46fb45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"0=x^2-2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"121\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Resolvemos a equa\u00e7\u00e3o quadr\u00e1tica com a f\u00f3rmula geral:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd0f7a0d21b888b4e5016eb027b534a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{-b\\pm \\sqrt{b^2-4ac}}{2a}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"165\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c8fa2c0e690ee979b162d63c9ee18904_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{-(-2)\\pm \\sqrt{(-2)^2-4\\cdot 1\\cdot1}}{2\\cdot 1} =\\cfrac{2\\pm 0}{2} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"348\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> O ponto de intersec\u00e7\u00e3o da fun\u00e7\u00e3o quadr\u00e1tica com o eixo X \u00e9, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-84f3677bb18b0fdcb23f9f99693d049e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste caso, obtivemos apenas uma solu\u00e7\u00e3o para a equa\u00e7\u00e3o quadr\u00e1tica, mas poder\u00edamos ter obtido duas solu\u00e7\u00f5es. Neste caso, isso significa que a fun\u00e7\u00e3o quadr\u00e1tica intercepta o eixo X em dois pontos diferentes. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-representar-una-funcion-cuadratica-o-parabola\"><\/span> Exemplo de representa\u00e7\u00e3o de uma fun\u00e7\u00e3o quadr\u00e1tica ou parab\u00f3lica <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong><\/strong><\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-108\"><strong> <\/strong><\/div>\n<p><strong><br \/><\/strong><\/p>\n<\/div>\n<p>Vamos ver <strong>como representar uma fun\u00e7\u00e3o quadr\u00e1tica<\/strong> em um gr\u00e1fico usando um exemplo.<\/p>\n<ul>\n<li> Fa\u00e7a um gr\u00e1fico da seguinte fun\u00e7\u00e3o:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-daee208a4b18c835d0b39c0a1bbc1ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"146\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A primeira coisa a fazer \u00e9 <strong>calcular o v\u00e9rtice da par\u00e1bola.<\/strong> Para fazer isso, usamos a f\u00f3rmula que vimos acima:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c7b3ed3c7bf6661b563290769c1b3f75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = \\cfrac{-b}{2a} = \\cfrac{-(-4)}{2\\cdot 1}= \\cfrac{4}{2}= 2\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"215\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Depois de sabermos onde estar\u00e1 o v\u00e9rtice, precisamos construir uma tabela de valores: <strong>&nbsp;<\/strong> Calculamos o valor da fun\u00e7\u00e3o no v\u00e9rtice e nos pontos que o rodeiam:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-daee208a4b18c835d0b39c0a1bbc1ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"146\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-352\">\n<div class=\"wp-block-column is-layout-flow\">\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-20fbda178bf7ed0b06d87f8b9558352b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 2 \\ \\longrightarrow \\ f(2) = 2^2-4\\cdot2+5=1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"281\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2561b16ae2d9dc6ea4f9734bedbcdf81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 1 \\ \\longrightarrow \\ f(1)=1^2-4\\cdot1+5=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"281\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-790d7ae514d18f03db8e9c373adbf050_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 3 \\ \\longrightarrow \\ f(3) = 3^2-4\\cdot3+5=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"281\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6820340ec293e65b9be9ed3e85ce8308_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 0 \\ \\longrightarrow \\ f(0) = 0^2-4\\cdot0+5=5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"281\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b61ea6ec8c7982dc52816f3e27e1637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 4 \\ \\longrightarrow \\ f(4) = 4^2-4\\cdot4+5=5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"281\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-361c835cbcad63310e504c58d5d603c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; f(x) \\\\ \\hline 2 &amp; 1 \\\\ 1 &amp; 2 \\\\ 3 &amp; 2 \\\\ 0 &amp; 5 \\\\ 4 &amp; 5 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"78\" style=\"vertical-align: -62px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> Voc\u00ea tamb\u00e9m pode calcular os pontos de corte da fun\u00e7\u00e3o quadr\u00e1tica com os eixos cartesianos para desenhar melhor a par\u00e1bola, mas isso n\u00e3o \u00e9 estritamente necess\u00e1rio.<\/p>\n<p> Representamos agora os pontos obtidos em um gr\u00e1fico <strong>:<\/strong> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-comment-representer-une-fonction-quadratique-ou-parabole.webp\" alt=\"exemplo de como representar uma fun\u00e7\u00e3o quadr\u00e1tica ou parab\u00f3lica\" class=\"wp-image-136\" width=\"260\" height=\"300\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> E por fim, juntamos os pontos formando a par\u00e1bola. Depois alongamos os ramos da par\u00e1bola para indicar que ela continua para cima: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/representation-d-une-fonction-quadratique-ou-parabole.webp\" alt=\"representa\u00e7\u00e3o de uma fun\u00e7\u00e3o quadr\u00e1tica ou parab\u00f3lica\" class=\"wp-image-137\" width=\"260\" height=\"293\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-funciones-cuadraticas\"><\/span> Exerc\u00edcios resolvidos sobre fun\u00e7\u00f5es quadr\u00e1ticas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Encontre o v\u00e9rtice da seguinte fun\u00e7\u00e3o quadr\u00e1tica: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d138d20f23c7aa6367c85907671a073b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2+8x+4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro calculamos a coordenada X do v\u00e9rtice usando a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f393ada7f0e57fd87a582657183d4f2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = \\cfrac{-b}{2a} = \\cfrac{-8}{2\\cdot 2} = \\cfrac{-8}{4} = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"223\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora calculamos a outra coordenada avaliando a fun\u00e7\u00e3o no ponto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d6067325564a5af06f7384d76157f3aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} f(-2) &amp; =2(-2)^2+8(-2)+4 \\\\[1.7ex] &amp; = 2 \\cdot 4 - 16 +4 \\\\[1.7ex] &amp; = 8-16+4 \\\\[1.7ex] &amp; = -4 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"136\" width=\"221\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O v\u00e9rtice da fun\u00e7\u00e3o quadr\u00e1tica \u00e9, portanto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-331c4c81f99a9749f15d457746a5f745_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-2,-4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2 <\/h3>\n<p><strong><\/strong><\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-110\"><strong> <\/strong><\/div>\n<p><strong><br \/><\/strong><\/p>\n<\/div>\n<p>Encontre os pontos de corte da seguinte fun\u00e7\u00e3o com os eixos: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09640c22a09e153b538a5aff018e02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-4x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para calcular o ponto de corte com o eixo Y, precisamos calcular <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f92d7beea0ed3a053927c2d429d3450_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b60958618b8dddf8706873a570491dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=0^2-4\\cdot 0+3 = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"189\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o, portanto, passa pelo eixo Y no ponto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fce94a71978b3cf055b18dce85b4746f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E para encontrar os pontos de corte com o eixo X precisamos resolver <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-05bb421b504b7ae4aa483574cd6f28d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0:\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09640c22a09e153b538a5aff018e02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-4x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6d78e40252b245c2ea862b8de892b646_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"0=x^2-4x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Calculamos as ra\u00edzes da equa\u00e7\u00e3o quadr\u00e1tica com a f\u00f3rmula: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd0f7a0d21b888b4e5016eb027b534a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{-b\\pm \\sqrt{b^2-4ac}}{2a}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"165\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0d909ba6581faf5916f0b1c0df7e471f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x=\\cfrac{-(-4)\\pm \\sqrt{(-4)^2-4\\cdot 1\\cdot 3}}{2\\cdot 1} =\\cfrac{4\\pm 2}{2} = \\begin{cases} 3 \\\\[2ex] 1 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"365\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o, portanto, corta o eixo X em dois pontos: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-30411245eaebedd735c4c804372a58e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,0) \\qquad (3,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3<\/h3>\n<p> Fa\u00e7a um gr\u00e1fico da seguinte fun\u00e7\u00e3o quadr\u00e1tica: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2dea5d0b581ec4e3a7e39ff51b8a25b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=-x^2+4x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"160\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Esta \u00e9 uma fun\u00e7\u00e3o quadr\u00e1tica. conseq\u00fcentemente, para represent\u00e1-la deve-se primeiro calcular a abcissa do v\u00e9rtice da par\u00e1bola com a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e0d4a78b32998cdd3b2bdff7d143e870_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = \\cfrac{-b}{2a} = \\cfrac{-4}{2\\cdot (-1)} = \\cfrac{-4}{-2} = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"236\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora criamos a tabela de valores. Para fazer isso, calculamos o valor de<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> no topo e ao redor do topo: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-355\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-44c68320944b905e1f9bdc40e71de785_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 2 \\ \\longrightarrow \\ f(2)=-2^2+4\\cdot2+1 = 5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"294\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b8a73224d11487e6153b9fce73715cbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 1 \\ \\longrightarrow \\ f(1)=-1^2+4\\cdot1+1 = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"295\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1db5d9c522aa5f555daed75308f9f5d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 3 \\ \\longrightarrow \\ f(3)=-3^2+4\\cdot3+1 =4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"295\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8011dabdadc4055e9bb1d225a5ce0630_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 0 \\ \\longrightarrow \\ f(0)=-0^2+4\\cdot0+1 = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"294\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce78a26ab2e830c17be7b66a4d1092a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 4 \\ \\longrightarrow \\ f(4)=-4^2+4\\cdot4+1 = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"294\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32fd0aec4a82615193ca7a894555a8fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; f(x) \\\\ \\hline 2 &amp; 5 \\\\ 1 &amp; 4 \\\\ 3 &amp; 4 \\\\ 0 &amp; 1 \\\\ 4 &amp; 1 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"78\" style=\"vertical-align: -62px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> E por fim, plotamos os pontos no gr\u00e1fico e desenhamos a par\u00e1bola: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-fonction-quadratique.webp\" alt=\"exemplo de fun\u00e7\u00e3o quadr\u00e1tica\" class=\"wp-image-138\" width=\"267\" height=\"273\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 4<\/h3>\n<p> Fa\u00e7a um gr\u00e1fico da seguinte fun\u00e7\u00e3o quadr\u00e1tica: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-743023f52acec123a62e655d6e2d954d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=-2x^2-8x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"169\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Esta \u00e9 uma fun\u00e7\u00e3o de segunda ordem. conseq\u00fcentemente, para represent\u00e1-la voc\u00ea deve primeiro encontrar a abcissa do v\u00e9rtice da par\u00e1bola com a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7b29662b7e8b1efcbc7ccdccbb082a2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = \\cfrac{-b}{2a} = \\cfrac{-(-8)}{2\\cdot (-2)} = \\cfrac{+8}{-4} = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"250\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora constru\u00edmos a tabela de valores. Para fazer isso, calculamos o valor de<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> no topo e ao redor do topo: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-358\">\n<div class=\"wp-block-column is-layout-flow\" style=\"flex-basis:66.66%\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e77f3b13fa69e6004d7032fec18a3904_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -2 \\ \\longrightarrow \\ f(-2)=-2(-2)^2-8\\cdot(-2)-1 =7\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"387\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48e126275f3f7f7c497b49cdf8370100_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1 \\ \\longrightarrow \\ f(-1)=-2(-1)^2-8\\cdot(-1)-1= 5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"386\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5a182ace189ad944ee9650df34fc695d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -3 \\ \\longrightarrow \\ f(-3)=-2(-3)^2-8\\cdot(-3)-1= 5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"386\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-336ae2592fa21f948dc76ddb165dbf7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 0 \\ \\longrightarrow \\ f(0)=-2\\cdot0^2-8\\cdot0-1=  -1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"330\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aafee3fbd7edddaff9f3a0f72676a3b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -4 \\ \\longrightarrow \\ f(-4)=-2(-4)^2-8\\cdot(-4)-1= -1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"400\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\" style=\"flex-basis:33.33%\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7a2a1163e24c8a1302015eaff2bd1503_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; f(x) \\\\ \\hline -2 &amp; 7 \\\\ -1 &amp; 5 \\\\ -3 &amp; 5 \\\\ 0 &amp; -1 \\\\ -4 &amp; -1 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"90\" style=\"vertical-align: -62px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Por fim, plotamos os pontos no gr\u00e1fico e desenhamos a par\u00e1bola: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-etape-par-etape-de-la-fonction-quadratique.webp\" alt=\"exerc\u00edcio resolvido passo a passo da fun\u00e7\u00e3o quadr\u00e1tica\" class=\"wp-image-139\" width=\"281\" height=\"336\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 5<\/h3>\n<div id=\"ezoic-pub-ad-placeholder-111\"><\/div>\n<p> Trace a seguinte fun\u00e7\u00e3o quadr\u00e1tica incompleta em um gr\u00e1fico: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2e5bac9d65f6cadab9818bc92de4830f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 uma fun\u00e7\u00e3o polinomial de grau dois. conseq\u00fcentemente, para represent\u00e1-la deve-se primeiro calcular a abcissa do v\u00e9rtice da par\u00e1bola com a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a07f94b2c2b30a834d1292de34ae82ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = \\cfrac{-b}{2a} = \\cfrac{-0}{2\\cdot1} = \\cfrac{0}{2} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"188\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nesse caso, a fun\u00e7\u00e3o est\u00e1 incompleta, pois n\u00e3o possui termo de primeiro grau. Por isso<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dad334f6e000c51bc6691844f1a7ffab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=0 .\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora fazemos a tabela de valores. Para fazer isso, calculamos o valor de<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> no topo e ao redor do topo: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-361\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e178a1ad103666d3b960f843389f3fce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 0 \\ \\longrightarrow \\ f(0)=0^2+2=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"228\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-93f9a7aaa75802bd85717ec9ebdd5ed0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 1 \\ \\longrightarrow \\ f(1)=1^2+2=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"229\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41993cef6c11ca85064167dec57c0e9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1 \\ \\longrightarrow \\ f(-1)=(-1)^2+2=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"285\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e6135721b6b23419b1fa746b6183acc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 2 \\ \\longrightarrow \\ f(2)=2^2+2=6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"229\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3160882279942b03e20689c0764f2135_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -2 \\ \\longrightarrow \\ f(-2)=(-2)^2+2=6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"285\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-71b2750173c31f2e7a31f9aeff6ac45f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; f(x) \\\\ \\hline 0 &amp; 2 \\\\ 1 &amp; 3 \\\\ -1 &amp; 3 \\\\ 2 &amp; 6 \\\\ -2 &amp; 6 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"90\" style=\"vertical-align: -62px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Por fim, plotamos os pontos no gr\u00e1fico e desenhamos a par\u00e1bola: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercices-resolus-pour-representer-une-fonction-quadratique-incomplete.webp\" alt=\"exerc\u00edcios resolvidos para representar uma fun\u00e7\u00e3o quadr\u00e1tica incompleta\" class=\"wp-image-140\" width=\"255\" height=\"285\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 6<\/h3>\n<p> Resolva o seguinte problema relacionado a fun\u00e7\u00f5es quadr\u00e1ticas:<\/p>\n<p> O custo de produ\u00e7\u00e3o de um produto \u00e9 definido pela seguinte fun\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c3e866f3ef954385f97070c37aeda9ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-12x+76\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"165\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Ouro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> s\u00e3o as unidades produzidas (em milhares) e<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u00e9 o custo de produ\u00e7\u00e3o das unidades (em milhares de euros).<\/p>\n<ul>\n<li> Representa a fun\u00e7\u00e3o de custo de produ\u00e7\u00e3o em um gr\u00e1fico.<\/li>\n<li> Determine quantos milhares de unidades devem ser produzidas para minimizar custos. <\/li>\n<\/ul>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Esta \u00e9 uma fun\u00e7\u00e3o quadr\u00e1tica. conseq\u00fcentemente, para represent\u00e1-la voc\u00ea deve primeiro encontrar a abcissa do v\u00e9rtice da par\u00e1bola com a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a237afcda60e57e87029cd7bca6d4134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = \\cfrac{-b}{2a} = \\cfrac{-(-12)}{2\\cdot 1} = \\cfrac{12}{2} = 6\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"233\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora fazemos a tabela de valores. Para fazer isso, calculamos o valor de<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> no topo e ao redor do topo: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-364\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a354ded9ac2e79841029224a3e6d062a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 6 \\ \\longrightarrow \\ f(6)=6^2-12\\cdot6+76 = 40\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"308\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fad9733a751d03644ed384789ac0287c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 5 \\ \\longrightarrow \\ f(5)=5^2-12\\cdot5+76 = 41\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"307\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0e801ae3dc944bbd7d5638bbba43c3eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 7 \\ \\longrightarrow \\ f(7)=7^2-12\\cdot7+76 = 41\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"307\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-750d34e08cf06f200b0beb142402fbd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 4 \\ \\longrightarrow \\ f(4)=4^2-12\\cdot4+76 =  44\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"308\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a8c9a488faf10289c77c6c79465bee25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 8 \\ \\longrightarrow \\ f(8)=8^2-12\\cdot8+76 = 44\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"308\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa20ac17af14b4726a38ec315091fc8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; f(x) \\\\ \\hline 6 &amp; 40 \\\\ 5 &amp; 41 \\\\ 7 &amp; 41 \\\\ 4 &amp; 44 \\\\ 8 &amp; 44 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"78\" style=\"vertical-align: -62px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Agora plotamos os pontos no gr\u00e1fico e desenhamos a par\u00e1bola: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/probleme-des-fonctions-quadratiques-ou-des-paraboles.webp\" alt=\"problema de fun\u00e7\u00e3o quadr\u00e1tica ou parab\u00f3lica\" class=\"wp-image-141\" width=\"440\" height=\"536\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Uma vez representada a fun\u00e7\u00e3o, veremos o quanto os custos s\u00e3o minimizados.<\/p>\n<p class=\"has-text-align-left\"> Como mostra o gr\u00e1fico, os custos m\u00ednimos ser\u00e3o alcan\u00e7ados no topo da par\u00e1bola. Porque \u00e9 a\u00ed que a fun\u00e7\u00e3o assume o menor valor.<\/p>\n<p class=\"has-text-align-left\"> Concluindo, os custos ser\u00e3o minimizados com a produ\u00e7\u00e3o de 6.000 unidades.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 7<\/h3>\n<p> Resolva o seguinte problema de fun\u00e7\u00e3o quadr\u00e1tica:<\/p>\n<p> Um atleta realiza um lan\u00e7amento de dardo cuja trajet\u00f3ria pode ser representada pela seguinte fun\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9124ad4bc986e46f99678ea39abbd596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"h(x) = -0,025x^2+2x+2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"203\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Ouro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> s\u00e3o os metros percorridos pelo dardo e<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"h\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> sua altura (tamb\u00e9m em metros).<\/p>\n<p> Qual \u00e9 a altura m\u00e1xima que o dardo pode atingir? <\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Esta \u00e9 uma fun\u00e7\u00e3o quadr\u00e1tica, portanto a trajet\u00f3ria do dardo ser\u00e1 uma par\u00e1bola.<\/p>\n<p class=\"has-text-align-left\"> Al\u00e9m disso, como o coeficiente do termo quadr\u00e1tico \u00e9 negativo (-0,025), a par\u00e1bola ter\u00e1 formato de U invertido e seus ramos ser\u00e3o descendentes. Assim o dardo atingir\u00e1 a altura m\u00e1xima no topo, pois este ser\u00e1 o ponto mais alto da par\u00e1bola.<\/p>\n<p class=\"has-text-align-left\"> Portanto, calculamos a abcissa do v\u00e9rtice da par\u00e1bola com a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-07e71d291cd16f48045ef0f2f184fd32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = \\cfrac{-b}{2a} = \\cfrac{-2}{2\\cdot (-0,025)} = \\cfrac{-2}{-0,05} = 40\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"299\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E ent\u00e3o calculamos a altura do dardo naquele ponto, avaliando a fun\u00e7\u00e3o em <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a547927c07de2d21e35a28a03a13ad2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=40:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"60\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e03890524ba1ae9c5c40bc3a07f49511_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"h(40) = -0,025\\cdot (40)^2+2\\cdot 40+2 = 42\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"307\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A altura m\u00e1xima que o dardo pode atingir \u00e9, portanto, de 42 metros.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 8<\/h3>\n<p> Resolva o seguinte problema relativo a fun\u00e7\u00f5es quadr\u00e1ticas:<\/p>\n<p> Os custos de produ\u00e7\u00e3o (em euros) de uma empresa s\u00e3o definidos pela seguinte fun\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-56d12a49461c15a22fb08ddaebb62e9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C(q)=40000+20q+q^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"189\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Ouro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\"><\/p>\n<p> s\u00e3o as unidades produzidas.<\/p>\n<p> E o pre\u00e7o de venda de cada unidade \u00e9 de 520\u20ac.<\/p>\n<ul>\n<li> Quanto lucro a empresa ter\u00e1 se vender 150 unidades?<\/li>\n<li> Quantas unidades devem ser vendidas para obter lucro m\u00e1ximo? <\/li>\n<\/ul>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A empresa ganha 520\u20ac por cada unidade vendida. Portanto, a fun\u00e7\u00e3o que define a renda \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-63d0ddca6dbabb334734ab4237be3e8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"I(q)=520\\cdot q = 520q\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"162\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ouro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\"><\/p>\n<p> s\u00e3o as unidades vendidas.<\/p>\n<p class=\"has-text-align-left\"> Mas nos perguntam sobre o lucro, ou seja, a renda menos os custos. Subtra\u00edmos, portanto, a receita menos os custos para obter a fun\u00e7\u00e3o que descreve o lucro da empresa: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8ddf6cb21f2e6c8ff0e6f7f8bc156b7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(q)=I(q)-C(q)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-846ac3b3a0fe227a1753a0eebc5eb5ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(q)=520q - (40000+20q+q^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"260\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-402c374248bad2289ed95378a3a40fa1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(q)=520q - 40000-20q-q^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"247\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6409f4ec7006749f7bcce2f9f7dab978_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(q)=-q^2 + 500q - 40000\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"214\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Uma vez conhecida a fun\u00e7\u00e3o que descreve o lucro da empresa, basta substituir 150 na express\u00e3o da fun\u00e7\u00e3o para calcular o lucro que a empresa obter\u00e1 com a venda de 150 unidades:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f5e6f3101145bcf1a2ece4db3e07c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} B(150) &amp; =-(150)^2 + 500\\cdot 150 - 40000 \\\\[2ex] &amp; =  -22500+75000 - 40000 \\\\[2ex] &amp; = \\bm{12500} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"294\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, ao vender 150 unidades, a empresa ter\u00e1 um lucro de 12.500\u20ac.<\/p>\n<p class=\"has-text-align-left\"> O comunicado tamb\u00e9m nos pede para calcular em quantas unidades o lucro m\u00e1ximo \u00e9 alcan\u00e7ado.<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o que descreve o lucro \u00e9 uma fun\u00e7\u00e3o quadr\u00e1tica, portanto ter\u00e1 o formato de uma par\u00e1bola. E como o coeficiente do termo quadr\u00e1tico \u00e9 negativo (-1), a par\u00e1bola ter\u00e1 formato de U invertido e seus ramos ir\u00e3o para baixo. Portanto, os ganhos m\u00e1ximos ser\u00e3o obtidos no topo, j\u00e1 que este \u00e9 o ponto mais alto da par\u00e1bola.<\/p>\n<p class=\"has-text-align-left\"> Portanto, calculamos a abcissa do v\u00e9rtice da par\u00e1bola com a f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c2a67803a02dce4f27a39a9a39319734_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = \\cfrac{-b}{2a} = \\cfrac{-500}{2\\cdot(-1)} = \\cfrac{-500}{-2} = 250\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"273\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, a empresa obter\u00e1 o lucro m\u00e1ximo com a venda de 250 unidades.<\/p>\n<p class=\"has-text-align-left\"> Por outro lado, mesmo que o comunicado n\u00e3o o solicite, podemos determinar o lucro que ser\u00e1 obtido com a venda destas 250 unidades:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2823628ca7484d5baaa4e1b7b003cc41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(250) =-(250)^2 + 500\\cdot250- 40000= 22500\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"362\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u20ac <\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p><strong><\/strong><\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-109\"><strong> <\/strong><\/div>\n<p><strong><br \/><\/strong><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Esta p\u00e1gina explica o que \u00e9 uma fun\u00e7\u00e3o quadr\u00e1tica bem como todas as suas caracter\u00edsticas: curvatura, v\u00e9rtice, pontos de intersec\u00e7\u00e3o com os eixos, etc. Voc\u00ea tamb\u00e9m aprender\u00e1 como representar uma fun\u00e7\u00e3o quadr\u00e1tica em um gr\u00e1fico. E por fim, voc\u00ea pode praticar com exemplos, exerc\u00edcios passo a passo e problemas sobre fun\u00e7\u00f5es quadr\u00e1ticas. O que \u00e9 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/\"> <span class=\"screen-reader-text\">Fun\u00e7\u00e3o quadr\u00e1tica ou par\u00e1bola<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[22],"tags":[],"class_list":["post-7","post","type-post","status-publish","format-standard","hentry","category-representacao-de-funcao"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 O que \u00e9 e como representar uma fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola)<\/title>\n<meta name=\"description\" content=\"\u2705 TUDO sobre a fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola): f\u00f3rmula, dom\u00ednio, curvatura, v\u00e9rtice, pontos de intersec\u00e7\u00e3o com os eixos, representa\u00e7\u00e3o gr\u00e1fica, exerc\u00edcios,...\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 O que \u00e9 e como representar uma fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola)\" \/>\n<meta property=\"og:description\" content=\"\u2705 TUDO sobre a fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola): f\u00f3rmula, dom\u00ednio, curvatura, v\u00e9rtice, pontos de intersec\u00e7\u00e3o com os eixos, representa\u00e7\u00e3o gr\u00e1fica, exerc\u00edcios,...\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-17T11:16:54+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-576d742ea12f780a76fa5809f5112086_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"9 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Fun\u00e7\u00e3o quadr\u00e1tica ou par\u00e1bola\",\"datePublished\":\"2023-09-17T11:16:54+00:00\",\"dateModified\":\"2023-09-17T11:16:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/\"},\"wordCount\":1738,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Representa\u00e7\u00e3o de fun\u00e7\u00e3o\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/\",\"url\":\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/\",\"name\":\"\u25b7 O que \u00e9 e como representar uma fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola)\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/#website\"},\"datePublished\":\"2023-09-17T11:16:54+00:00\",\"dateModified\":\"2023-09-17T11:16:54+00:00\",\"description\":\"\u2705 TUDO sobre a fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola): f\u00f3rmula, dom\u00ednio, curvatura, v\u00e9rtice, pontos de intersec\u00e7\u00e3o com os eixos, representa\u00e7\u00e3o gr\u00e1fica, exerc\u00edcios,...\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Fun\u00e7\u00e3o quadr\u00e1tica ou par\u00e1bola\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/pt\/#website\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"name\":\"Mathority\",\"description\":\"Onde a curiosidade encontra o c\u00e1lculo!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/pt\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pt-BR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/pt\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\",\"name\":\"Equipe Mathoridade\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Equipe Mathoridade\"},\"sameAs\":[\"http:\/\/mathority.org\/pt\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\u25b7 O que \u00e9 e como representar uma fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola)","description":"\u2705 TUDO sobre a fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola): f\u00f3rmula, dom\u00ednio, curvatura, v\u00e9rtice, pontos de intersec\u00e7\u00e3o com os eixos, representa\u00e7\u00e3o gr\u00e1fica, exerc\u00edcios,...","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/","og_locale":"pt_BR","og_type":"article","og_title":"\u25b7 O que \u00e9 e como representar uma fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola)","og_description":"\u2705 TUDO sobre a fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola): f\u00f3rmula, dom\u00ednio, curvatura, v\u00e9rtice, pontos de intersec\u00e7\u00e3o com os eixos, representa\u00e7\u00e3o gr\u00e1fica, exerc\u00edcios,...","og_url":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/","article_published_time":"2023-09-17T11:16:54+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-576d742ea12f780a76fa5809f5112086_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"9 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Fun\u00e7\u00e3o quadr\u00e1tica ou par\u00e1bola","datePublished":"2023-09-17T11:16:54+00:00","dateModified":"2023-09-17T11:16:54+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/"},"wordCount":1738,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Representa\u00e7\u00e3o de fun\u00e7\u00e3o"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/","url":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/","name":"\u25b7 O que \u00e9 e como representar uma fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola)","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/#website"},"datePublished":"2023-09-17T11:16:54+00:00","dateModified":"2023-09-17T11:16:54+00:00","description":"\u2705 TUDO sobre a fun\u00e7\u00e3o quadr\u00e1tica (ou par\u00e1bola): f\u00f3rmula, dom\u00ednio, curvatura, v\u00e9rtice, pontos de intersec\u00e7\u00e3o com os eixos, representa\u00e7\u00e3o gr\u00e1fica, exerc\u00edcios,...","breadcrumb":{"@id":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/pt\/funcao-de-parabola-quadratica\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/pt\/"},{"@type":"ListItem","position":2,"name":"Fun\u00e7\u00e3o quadr\u00e1tica ou par\u00e1bola"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/pt\/#website","url":"https:\/\/mathority.org\/pt\/","name":"Mathority","description":"Onde a curiosidade encontra o c\u00e1lculo!","publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/pt\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"pt-BR"},{"@type":"Organization","@id":"https:\/\/mathority.org\/pt\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/pt\/","logo":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00","name":"Equipe Mathoridade","image":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Equipe Mathoridade"},"sameAs":["http:\/\/mathority.org\/pt"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/7","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/comments?post=7"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/7\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/media?parent=7"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/categories?post=7"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/tags?post=7"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}