{"id":87895,"date":"2023-02-21T18:00:00","date_gmt":"2023-02-21T21:00:00","guid":{"rendered":"https:\/\/vestibulares.estrategia.com\/portal\/?p=87895"},"modified":"2024-04-02T15:56:21","modified_gmt":"2024-04-02T18:56:21","slug":"funcao-quadratica","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcao-quadratica\/","title":{"rendered":"Fun\u00e7\u00e3o quadr\u00e1tica: o que \u00e9, gr\u00e1fico e f\u00f3rmulas&nbsp;"},"content":{"rendered":"<p class=\"wp-block-paragraph\">A <span id=\"urn:enhancement-a1808434-0120-404c-8127-4984a645a5c4\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcao-quadratica\">fun&ccedil;&atilde;o quadr&aacute;tica<\/span> est&aacute; entre as principais <span id=\"urn:local-annotation-410640\" class=\"textannotation disambiguated\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcoes-matematicas\">fun&ccedil;&otilde;es matem&aacute;ticas<\/span> no estudo pr&eacute;-vestibular.&nbsp; &Eacute; importante tanto no sentido de resolver <span id=\"urn:enhancement-49fd9f4c-77c9-4bff-baf3-59295f593acd\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span>, como para entender problemas e tamb&eacute;m pode ser explorada em outras disciplinas exatas, como na <span id=\"urn:enhancement-bf59ffc2-d93a-4131-a443-17e990e27de5\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/fisica\">f&iacute;sica<\/span>.<p class=\"wp-block-paragraph\">Conhe&ccedil;a agora sobre a <span id=\"urn:enhancement-929b2350-b974-4d81-b326-75f30e952c80\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcao-quadratica\">fun&ccedil;&atilde;o quadr&aacute;tica<\/span>, qual &eacute; sua estrutura geral, qual gr&aacute;fico ela constr&oacute;i no <span id=\"urn:enhancement-417831e5-b6d4-4efe-85b5-6529c0c08c08\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/plano-cartesiano\">plano cartesiano<\/span> e quais as caracter&iacute;sticas desse tra&ccedil;ado. Veja tamb&eacute;m <span id=\"urn:local-annotation-351095\" class=\"textannotation disambiguated\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/formulas-matematicas\">f&oacute;rmulas matem&aacute;ticas<\/span> que est&atilde;o envolvidas nesse tema, como a f&oacute;rmula de Bh&aacute;skara, de <span id=\"urn:enhancement-f0421702-46e3-4ff6-9cac-772746530b12\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> e produto, c&aacute;lculos para encontrar o x e o y do v&eacute;rtice do gr&aacute;fico, entre outras informa&ccedil;&otilde;es importantes.<\/p><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcao-quadratica\/#O-que-e-a-funcao-quadratica\" >O que &eacute; a fun&ccedil;&atilde;o quadr&aacute;tica?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcao-quadratica\/#Como-encontrar-a-raiz-de-uma-funcao-quadratica\" >Como encontrar a raiz de uma fun&ccedil;&atilde;o quadr&aacute;tica?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcao-quadratica\/#Grafico-de-funcao-quadratica\" >Gr&aacute;fico de fun&ccedil;&atilde;o quadr&aacute;tica<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcao-quadratica\/#Estude-com-o-Estrategia-Vestibulares\" >Estude com o Estrat&eacute;gia Vestibulares!<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-o-que-e-a-funcao-quadratica\"><span class=\"ez-toc-section\" id=\"O-que-e-a-funcao-quadratica\"><\/span>O que &eacute; a fun&ccedil;&atilde;o quadr&aacute;tica?<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Tamb&eacute;m chamada de <a href=\"https:\/\/www.youtube.com\/watch?v=sgPURsxjxZ8\" target=\"_blank\">fun&ccedil;&atilde;o do segundo grau<\/a>, a <span id=\"urn:enhancement-d73492f5-1db1-4b13-be73-23905ec34cf7\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcao-quadratica\">fun&ccedil;&atilde;o quadr&aacute;tica<\/span> &eacute; um c&aacute;lculo matem&aacute;tico em fun&ccedil;&atilde;o de uma inc&oacute;gnita, em <span id=\"urn:enhancement-5fd85988-bfa0-4b7f-9596-afb7a7da3a2b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> de <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/polinomios\/\" target=\"_blank\">polin&ocirc;mio <\/a>de grau dois. Ou seja, a inc&oacute;gnita est&aacute; distribu&iacute;da de <span id=\"urn:enhancement-f82fc8a0-83a6-4866-8979-e7287c718772\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> que aparece elevada ao quadrado e tamb&eacute;m pode aparecer elevada ao grau um.<\/p><p class=\"wp-block-paragraph\">A estrutura geral de uma <span id=\"urn:enhancement-0db89aec-632e-4484-b83f-13922853fe81\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcao-quadratica\">fun&ccedil;&atilde;o quadr&aacute;tica<\/span> &eacute; determinada por:&nbsp;<\/p><p class=\"wp-block-paragraph\">f(x) = ax<sup>2<\/sup> + bx + c<\/p><p class=\"wp-block-paragraph\">a, b e c s&atilde;o <span id=\"urn:enhancement-375faf7c-c669-4e91-9341-f735f53886d8\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> do c&aacute;lculo&nbsp;<\/p><p class=\"wp-block-paragraph\">a &ne; 0&nbsp;<\/p><p class=\"wp-block-paragraph\">Perceba que a precisa ser diferente de zero, porque, caso a = 0, o elemento &ldquo;x<sup>2<\/sup>&rdquo; &eacute; eliminado da <span id=\"urn:enhancement-7a36a400-bad6-4c34-bb69-8e93e35d5496\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressao-matematica\">express&atilde;o matem&aacute;tica<\/span>, o que impede a classifica&ccedil;&atilde;o como fun&ccedil;&atilde;o do segundo grau.<\/p><p class=\"wp-block-paragraph\">No estudo alg&eacute;brico, est&aacute; determinado que o grau da <span id=\"urn:enhancement-838a42c4-b663-4057-a84b-2a10128deb05\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> &eacute; igual ao n&uacute;mero de ra&iacute;zes (<span id=\"urn:enhancement-f6f40200-d387-4e49-aaf9-1071ea8410b9\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/valores\">valores<\/span>) que podem ser encontradas para essa <span id=\"urn:enhancement-73226a3b-deb1-48b7-a12a-a55383cb6978\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span>. Lembre-se que uma raiz &eacute; o valor que, ao ser substitu&iacute;do na <span id=\"urn:enhancement-62cd64e9-e372-47b5-ae44-eadd9616acfa\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span>, faz com que o resultado da conta seja 0.&nbsp;<\/p><p class=\"wp-block-paragraph\">Conforme essa informa&ccedil;&atilde;o, basta lembrar que a <span id=\"urn:enhancement-46eb219e-665c-42c5-bfdf-eed863c3128f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcao-quadratica\">fun&ccedil;&atilde;o quadr&aacute;tica<\/span> possui grau 2 para saber que, em geral, as fun&ccedil;&otilde;es de segundo grau possuem duas ra&iacute;zes, que s&atilde;o chamadas por x<sub>1<\/sub> e x<sub>2<\/sub>, de maneira que f(x<sub>1<\/sub>) = 0 e f(x<sub>2<\/sub>)=0.<\/p><figure class=\"wp-block-embed aligncenter is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"cs-embed cs-embed-responsive\"><iframe title=\"Revis&atilde;o de Fun&ccedil;&otilde;es (Parte 1) - Matem&aacute;tica para Vestibulares e ENEM 2022 - Prof. Andrew Cazemiro\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/yx_wV2eiil8?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/div>\n<\/div><\/figure><h2 class=\"wp-block-heading\" id=\"h-como-encontrar-a-raiz-de-uma-funcao-quadratica\"><span class=\"ez-toc-section\" id=\"Como-encontrar-a-raiz-de-uma-funcao-quadratica\"><\/span>Como encontrar a raiz de uma fun&ccedil;&atilde;o quadr&aacute;tica?<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Existem dois principais caminhos para encontrar as ra&iacute;zes de uma <span id=\"urn:enhancement-298522f9-1122-41d4-a4de-acc3ffeea2bd\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcao-quadratica\">fun&ccedil;&atilde;o quadr&aacute;tica<\/span>. A primeira, muito conhecida, &eacute; a f&oacute;rmula de Bh&aacute;skara. Outro modo seria utilizar a f&oacute;rmula de <span id=\"urn:enhancement-0e1f6c95-8746-42fd-bfbe-585fcdc154f4\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> e produto. Veja, a seguir, como elas funcionam, com exemplos.<\/p><h3 class=\"wp-block-heading\" id=\"h-formula-de-bhaskara\">F&oacute;rmula de Bh&aacute;skara<\/h3><p class=\"wp-block-paragraph\">A f&oacute;rmula de Bh&aacute;skara &eacute; um c&aacute;lculo que re&uacute;ne os <span id=\"urn:enhancement-da9b43fe-4ce1-43fe-85fc-7e26bfe162de\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficientes<\/span> de uma <span id=\"urn:enhancement-dec5be12-7dcf-4e23-972a-948557a3cc8d\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao-do-segundo-grau\">equa&ccedil;&atilde;o do segundo grau<\/span>, para determinar quais <span id=\"urn:enhancement-0a83f57e-6920-400e-b369-ab391ed0be02\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/valores\">valores<\/span> de x determinar um resultado igual a 0.<\/p><figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/x9TRxxDtfWg7oNXh29wUFMjVZjIBOLB-qblqkXY2i6wNgaTM-CeyECQ-cVyz-PpEND2oDRhlytfgHoE6f2Myy6o79TgguS2YaALeLmV4JBkQ7ylxlQyv0watMBB6wvNTAYPKlZdMRprIfV0NN1EOVb0\" alt=\"F&oacute;rmula de Bh&aacute;skara: fun&ccedil;&atilde;o quadr&aacute;tica\"><\/figure><p class=\"wp-block-paragraph\">Seja a fun&ccedil;&atilde;o f(x) = x<sup>2<\/sup>+ 3x &ndash; 10&nbsp;<\/p><p class=\"wp-block-paragraph\">a = 1&nbsp;<br>b = 3<br>c = -10&nbsp;<\/p><p class=\"wp-block-paragraph\">x<sub>1<\/sub> = {-b + [&radic;b<sup>2<\/sup>&ndash; (4.a.c)]}\/ 2.a<br>x<sub>1<\/sub> = {-3+ [&radic;3<sup>2<\/sup> &ndash; (4.1.-10)]} \/ 2<br>x<sub>1<\/sub> = {-3+ [&radic;9 &ndash; (-40)]} \/ 2<br>x<sub>1<\/sub> = {-3+ [&radic;9 + 40]} \/ 2<br>x<sub>1<\/sub> = {-3+ [&radic;49]} \/ 2<br>x<sub>1<\/sub> = {-3 + 7} \/ 2<br>x<sub>1<\/sub> = 4 \/ 2<br>x<sub>1<\/sub> = 2<\/p><p class=\"wp-block-paragraph\">x<sub>2<\/sub> = {-b &ndash; [&radic;b<sup>2<\/sup>&ndash; (4.a.c)]}\/ 2.a<br>x<sub>2<\/sub> = {-3 &ndash; [&radic;3<sup>2<\/sup> &ndash; (4.1.-10)]} \/ 2<br>x<sub>2<\/sub> = {-3 &ndash; [&radic;9 &ndash; (-40)]} \/ 2}<br>x<sub>2<\/sub> = {-3 &ndash; [&radic;9 + 40]} \/ 2<br>x<sub>2<\/sub> = {-3 &ndash; [&radic;49]} \/ 2<br>x<sub>2<\/sub> = {-3 &ndash; 7} \/ 2<br>x<sub>2<\/sub> = &ndash;10 \/ 2<br>x<sub>2<\/sub> = -5<\/p><h3 class=\"wp-block-heading\" id=\"h-soma-e-produto-nbsp\">Soma e produto&nbsp;<\/h3><p class=\"wp-block-paragraph\">A f&oacute;rmula para <span id=\"urn:enhancement-bed37dfa-46f6-49db-871c-47dd3b077009\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> e produto &eacute; mais simples, mas &eacute; necess&aacute;rio desenvolver um sistema de <span id=\"urn:enhancement-5685fcd9-306d-4464-99ed-40e33479ef25\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> para chegar &agrave;s ra&iacute;zes.<\/p><p class=\"wp-block-paragraph\">x<sub>1<\/sub>+ x<sub>2<\/sub> = -b\/a<br>x<sub>1<\/sub>.x<sub>2<\/sub> = c\/a<\/p><p class=\"wp-block-paragraph\">Com a mesma fun&ccedil;&atilde;o do exemplo acima, vamos explorar esse segundo m&eacute;todo.<\/p><p class=\"wp-block-paragraph\">f(x) = x<sup>2<\/sup>+ 3x &ndash; 10&nbsp;<\/p><p class=\"wp-block-paragraph\">a = 1&nbsp;<br>b = 3<br>c = -10&nbsp;<\/p><p class=\"wp-block-paragraph\">x<sub>1<\/sub>+ x<sub>2<\/sub> = -b\/a<br>x<sub>1<\/sub>+ x<sub>2<\/sub> = -3\/1<br><strong>(<span id=\"urn:enhancement-f29d5f26-6069-464a-b871-8ac8e0d572a8\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> I)<\/strong> x<sub>1<\/sub>+ x<sub>2<\/sub> = -3&nbsp;<\/p><p class=\"wp-block-paragraph\">x<sub>1<\/sub>.x<sub>2<\/sub> = c\/a<br>x<sub>1<\/sub>.x<sub>2<\/sub> = -10\/1<br><strong>(<span id=\"urn:enhancement-4c728b04-3aaa-4762-ae40-998d993eaffc\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> II)<\/strong> x<sub>1<\/sub>.x<sub>2<\/sub> = -10<\/p><p class=\"wp-block-paragraph\">Uma das melhores maneiras de desenvolver o m&eacute;todo de <span id=\"urn:enhancement-1ddad078-e8b8-4baf-95d5-9f7f5ab15d71\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> e produto &eacute; por tentativa e erro. Primeiro, observe a <span id=\"urn:enhancement-09d4c86c-c52e-4af6-893f-e6e94bcca98d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> apresentada e pense em <span id=\"urn:enhancement-18e2b215-e8f3-42be-9234-deeb94f2c0c5\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/valores\">valores<\/span> que podem ser multiplicados para chegar a esse produto.<\/p><p class=\"wp-block-paragraph\">1 . 10 = 10<br>2 . 5 = 10<br>2,5 . 4 = 10<\/p><p class=\"wp-block-paragraph\">Depois, leve em considera&ccedil;&atilde;o o sinal do resultado da <span id=\"urn:enhancement-17071106-a344-489d-8d1c-3c14dfd2db2d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span> de ra&iacute;zes, bem como a <span id=\"urn:enhancement-bd901f77-33ff-4662-a38a-ef83dd47afd2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> apresentada na primeira f&oacute;rmula.<\/p><p class=\"wp-block-paragraph\">Dentre as op&ccedil;&otilde;es apresentadas, os fatores 2 e 5 resultam em 10. Ao mesmo tempo, a <span id=\"urn:enhancement-fb099d76-6a43-42f1-b146-fd8caa4898c2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/subtracao\">diferen&ccedil;a<\/span> entre esses <span id=\"urn:enhancement-7c117d9a-4329-4ba7-af75-7dd982a81138\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/valores\">valores<\/span> &eacute; igual a 3. Sim, a primeira f&oacute;rmula parte da <span id=\"urn:enhancement-41e923e2-d779-4642-8e35-1624147d6876\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span>, mas qual configura&ccedil;&atilde;o de sinais pode fazer com que 2 e 5 resultem em -3? Basicamente, &ndash; 5 + 2 = -3.&nbsp;<\/p><p class=\"wp-block-paragraph\">Ao mesmo tempo, -5.2 = -10, respeitando o valor e o sinal da <span id=\"urn:enhancement-5c02aeb0-80fa-4f75-a463-6ba38b373bff\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/multiplicacao\">multiplica&ccedil;&atilde;o<\/span> de ra&iacute;zes. Ent&atilde;o, -5 e 2 s&atilde;o <span id=\"urn:enhancement-e70be227-5111-4a70-bf89-55a34b71315e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/valores\">valores<\/span> que correspondem &agrave;s necessidades apresentadas na f&oacute;rmula e s&atilde;o as ra&iacute;zes da fun&ccedil;&atilde;o apresentada.<\/p><figure class=\"wp-block-embed aligncenter is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"cs-embed cs-embed-responsive\"><iframe title=\"UTI da Matem&aacute;tica para o ENEM: Fun&ccedil;&otilde;es Afins e Quadr&aacute;ticas\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/sZf0dj0PrFQ?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/div>\n<\/div><\/figure><h2 class=\"wp-block-heading\" id=\"h-grafico-de-funcao-quadratica\"><span class=\"ez-toc-section\" id=\"Grafico-de-funcao-quadratica\"><\/span>Gr&aacute;fico de fun&ccedil;&atilde;o quadr&aacute;tica<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">O gr&aacute;fico de uma <span id=\"urn:enhancement-7026056c-5dfe-4051-a039-c7f7d1290e7c\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/funcao-quadratica\">fun&ccedil;&atilde;o quadr&aacute;tica<\/span> &eacute; sempre uma par&aacute;bola. A <span id=\"urn:enhancement-1c2bf63d-79cb-4267-a8bc-efb2b4e60a60\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/subtracao\">diferen&ccedil;a<\/span> est&aacute; em outros fatores, por exemplo, se a boca est&aacute; virada para cima ou para baixo, se o v&eacute;rtice representa um <span id=\"urn:enhancement-f4382aef-7f32-41e2-8bc0-4f63453e273b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">ponto<\/span> m&iacute;nimo ou m&aacute;ximo, entre outras caracter&iacute;sticas.&nbsp;<\/p><p class=\"wp-block-paragraph\">Quando o <span id=\"urn:enhancement-46eb4ee8-2fab-45ed-949a-62b35b4ed2f5\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficiente<\/span> a&gt;0, a par&aacute;bola est&aacute; voltada para cima.&nbsp;<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/LvihXDidm5Z3WWCkayUQdd-aEjjdQ1o791DGTJWedQ4XSuvCihSdxq9y-UB2GDU_PZOlVOE5C8JbNnwCjNAeWbcCOuVKgF3RVLtu3UYPh95g9652B33ruhChhvFHHxwpiAcBlrxT0gu2bH2jyl_h1CE\" alt=\"par&aacute;bola: gr&aacute;fico de fun&ccedil;&atilde;o quadr&aacute;tica\" style=\"width:-244px;height:-207px\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Por outro lado, quando o <span id=\"urn:enhancement-e256ab07-3a3a-49d8-b85c-7d6fcb0bcbe6\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/coeficientes\">coeficiente<\/span> a&lt;0, sua concavidade &eacute; para baixo.<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/Rjew0H8NCIxIoWCRgLUqYs8VSyrWq_5onAdNQqTKCADiu7zZmIWmHP_sNtQnyLkxx-tB91qErlaMylOyu2XPUpkW-g7KnQugWx6cOVozkuyuI--NBnBF4wn-khmc2nb4UVHORoMmOTFzecPZkrkqyB0\" alt=\"par&aacute;bola: gr&aacute;fico de fun&ccedil;&atilde;o quadr&aacute;tica decrescente\" style=\"width:235px;height:220px\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">Outro <span id=\"urn:enhancement-bc51ed88-43ac-40a5-9c95-f2186b891ba2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/fatoracao\">fator<\/span> importante &eacute; o discriminante da f&oacute;rmula de Bh&aacute;skara. Representa todo o c&aacute;lculo que fica dentro da raiz quadrada (b<sup>2<\/sup> &ndash; 4.a.c). Ele tamb&eacute;m pode ser chamado de delta (&Delta; = b<sup>2<\/sup> &ndash; 4.a.c) e &eacute; considerado para entender como s&atilde;o as ra&iacute;zes dessa fun&ccedil;&atilde;o.&nbsp;<\/p><div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/Sg7_xAMA1a1pGTjNZp3oscnXYz6HlHmoF1lUGGbwwUGswPPDdKy9aSInY2gTTz4E_F8ircbE98EOrOkzm64O6Wuz74gmCjllBp0d67ccApH93JOTqfFywbiNo3CF4u8HXyJ7yS3JjJdoymxqES178KE\" alt=\"par&aacute;bola: n&uacute;mero de ra&iacute;zes\" style=\"width:324px;height:324px\"><\/figure>\n<\/div><p class=\"wp-block-paragraph\">A imagem resume as tr&ecirc;s informa&ccedil;&otilde;es relevantes a respeito da rela&ccedil;&atilde;o entre par&aacute;bola e discriminante:<\/p><ul class=\"wp-block-list\">\n<li>Toda vez que &Delta;&gt;0, a par&aacute;bola possui duas ra&iacute;zes reais. Ou seja, cruza o eixo x duas vezes, como est&aacute; demonstrado na figura em azul;<\/li>\n\n\n\n<li>Se o &Delta; = 0, apenas um valor &eacute; ser&aacute; a raiz dupla da fun&ccedil;&atilde;o quadr&aacute;tica. Com isso, somente um ponto da par&aacute;bola tocar&aacute; o eixo x, exemplo mostrado em vermelho; e<\/li>\n\n\n\n<li>Em amarelo, por fim, trata-se de quando &Delta;&lt;0. Como n&atilde;o existe uma raiz real para valores negativos, a f&oacute;rmula de Bh&aacute;skara ficaria incompleta e, por isso, nenhum n&uacute;mero real pode fazer essa fun&ccedil;&atilde;o resultar em zero. Por isso, a par&aacute;bola n&atilde;o toca o eixo x.<br><\/li>\n<\/ul><p class=\"wp-block-paragraph\">Outro <span id=\"urn:enhancement-ef0d2521-8756-41c5-9fe5-47b182535321\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">ponto<\/span> importante da par&aacute;bola &eacute; o v&eacute;rtice, <span id=\"urn:enhancement-20c28979-d325-4dd3-8d73-7fe97aaf5010\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">ponto<\/span> que divide os &ldquo;dois bracinhos&rdquo; do desenho. Para as par&aacute;bolas que est&atilde;o c&ocirc;ncavas para baixo, &eacute; o <span id=\"urn:enhancement-90e475e9-f223-4d36-a4ae-7be6ce67a0b2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">ponto<\/span> m&aacute;ximo. Para os desenhos voltados para cima, &eacute; o local de valor m&iacute;nimo. Para encontrar esses <span id=\"urn:enhancement-c0eaa4ec-164a-4efe-87ca-1f15be0fa1cd\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/valores\">valores<\/span>, utilizam-se as f&oacute;rmulas:<\/p><p class=\"wp-block-paragraph\">x<sub>v <\/sub>= -b\/2a<\/p><p class=\"wp-block-paragraph\">y<sub>v<\/sub> = -&Delta;\/4a&nbsp;<\/p><p class=\"wp-block-paragraph\">Veja uma quest&atilde;o do Enem 2013 que utiliza esse conceito:<\/p><p class=\"wp-block-paragraph\">A parte interior de uma ta&ccedil;a foi gerada pela rota&ccedil;&atilde;o de uma par&aacute;bola em torno de um eixo z, conforme mostra a figura.<\/p><figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/LTRuQgz1gYQkhTIq_4E9kqelzZLZOirCiZYPWqPL-34TbfgW0Bsjed3wmOhKbedg9N2-rf_rOx5S_q0AiEkwb4BqTsKA0nznhvrnC4FXUwHEUOPEQKkt_MbVS08rpHBmT3yhq5falruZHNDm36lRYgs\" alt=\"quest&atilde;o de fun&ccedil;&atilde;o quadr&aacute;tica\"><\/figure><p class=\"wp-block-paragraph\">A fun&ccedil;&atilde;o real que expressa a par&aacute;bola, no <span id=\"urn:enhancement-accf4e10-a343-4d36-9682-36178972d957\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/plano-cartesiano\">plano cartesiano<\/span> da figura, &eacute; dada pela lei f(x) = (3\/2).x<sup>2<\/sup> &ndash; 6x + C, onde C &eacute; a medida da altura do l&iacute;quido contido na ta&ccedil;a, em cent&iacute;metros. Sabe-se que o <span id=\"urn:enhancement-c400a672-d4c4-44ae-a1c4-a96d76fdeae2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">ponto<\/span> V, na figura, representa o v&eacute;rtice da par&aacute;bola, localizado sobre o eixo x. Nessas condi&ccedil;&otilde;es, a altura do l&iacute;quido contido na ta&ccedil;a, em cent&iacute;metros, &eacute;<\/p><p class=\"wp-block-paragraph\">a) 1.<br>b) 2.<br>c) 4.<br>d) 5.<br>e) 6.<\/p><p class=\"wp-block-paragraph\">Segundo o enunciado, o <span id=\"urn:enhancement-459377cc-91d0-472e-b888-d56fcbf21cec\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/pontos\">ponto<\/span> V &eacute; o v&eacute;rtice da par&aacute;bola e est&aacute; localizado em cima do eixo das abscissas. Isso significa que o valor de y &eacute; 0, ou seja, y<sub>V<\/sub> = 0. Agora, vamos determinar o x<sub>V<\/sub>:<\/p><p class=\"wp-block-paragraph\">x<sub>v <\/sub>= -b\/2a<\/p><p class=\"wp-block-paragraph\">Na fun&ccedil;&atilde;o f(x) = (3\/2).x<sup>2<\/sup> &ndash; 6x + C, a = 3\/2 = 1,5 e b = -6, ent&atilde;o:<\/p><p class=\"wp-block-paragraph\">x<sub>v <\/sub>= -b\/2a<br>x<sub>v <\/sub>= -(-6)\/2.1,5<br>x<sub>v <\/sub>= 6\/3<br>x<sub>v <\/sub>= 2<\/p><p class=\"wp-block-paragraph\">Com as informa&ccedil;&otilde;es descritas at&eacute; aqui, ent&atilde;o, o v&eacute;rtice da fun&ccedil;&atilde;o &eacute; (2,0). Agora, vamos substituir o valor de x e y na <span id=\"urn:enhancement-e76bc7b7-9a3b-4e05-8b92-b8bc5076e1c9\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span>, para encontrar a inc&oacute;gnita C:<\/p><p class=\"wp-block-paragraph\">f(x) = (3\/2).x<sup>2<\/sup> &ndash; 6x + C<br>f(2) = (3\/2).2<sup>2<\/sup> &ndash; 6.2 + C<br>f(2) = 0&nbsp;<\/p><p class=\"wp-block-paragraph\">0 = (3\/2).2<sup>2<\/sup> &ndash; 6.2 + C<br>0 = (1,5).4 &ndash; 6.2 + C<br>0 = 6 &ndash; 6.2 + C<br>0 = 6 &ndash; 12 + C<br>0 = &ndash; 6 + C<br>C = 6, conforme a alternativa E.<br><\/p><p class=\"wp-block-paragraph\">+ Veja tamb&eacute;m: <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-do-segundo-grau\/\" target=\"_blank\">Equa&ccedil;&atilde;o do segundo grau: como resolver, f&oacute;rmulas, exerc&iacute;cios de vestibulares<\/a><\/p><h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Estude-com-o-Estrategia-Vestibulares\"><\/span>Estude com o Estrat&eacute;gia Vestibulares!<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Os cursos da Coruja trazem o essencial para te ajudar na jornada rumo &agrave; aprova&ccedil;&atilde;o: as aulas est&atilde;o dispon&iacute;veis em v&iacute;deos, pdf ou livros digitais. Com isso, o acesso ao conte&uacute;do &eacute; facilitado e voc&ecirc; pode aprender de onde quiser, al&eacute;m disso, suas anota&ccedil;&otilde;es ficam salvas e podem ser conferidas durante as revis&otilde;es. N&atilde;o perca tempo e saiba mais!<\/p><p class=\"wp-block-paragraph\"><a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/vestibulares.estrategia.com\/cursos\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2021\/12\/EV2.jpg\" alt=\"CTA - Estrat\u00e9gia Vestibulares 2\" title=\"CTA - Estrat\u00e9gia Vestibulares 2\">\n        <\/a>\n\n\n\n\n<\/p><\/p>\n","protected":false},"excerpt":{"rendered":"A fun&ccedil;&atilde;o quadr&aacute;tica est&aacute; entre as principais fun&ccedil;&otilde;es matem&aacute;ticas no estudo pr&eacute;-vestibular.&nbsp; &Eacute; importante tanto no sentido de&hellip;\n","protected":false},"author":23,"featured_media":87886,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[],"class_list":["post-87895","post","type-post","status-publish","format-standard","has-post-thumbnail","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v28.2) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Fun\u00e7\u00e3o quadr\u00e1tica: o que \u00e9, gr\u00e1fico e f\u00f3rmulas&nbsp;<\/title>\n<meta name=\"description\" content=\"Conhe\u00e7a agora sobre a fun\u00e7\u00e3o quadr\u00e1tica, qual \u00e9 sua estrutura geral, qual gr\u00e1fico ela constr\u00f3i no plano cartesiano e suas caracter\u00edsticas.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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