{"id":93828,"date":"2017-01-10T11:17:43","date_gmt":"2017-01-10T13:17:43","guid":{"rendered":"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/?p=93828"},"modified":"2026-05-20T19:57:04","modified_gmt":"2026-05-20T21:57:04","slug":"funcao-quadratica","status":"publish","type":"post","link":"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/matematica\/funcao-quadratica","title":{"rendered":"Fun\u00e7\u00e3o Quadr\u00e1tica"},"content":{"rendered":"<p><b>Fun\u00e7\u00e3o Quadr\u00e1tica &#8211; O\u200b \u200bque\u200b \u200b\u00e9\u200b<\/b><\/p>\n<p><span class=\"HwtZe\" lang=\"pt\"><span class=\"jCAhz ChMk0b\"><span class=\"ryNqvb\">Uma\u00a0<b>fun\u00e7\u00e3o quadr\u00e1tica<\/b>\u00a0\u00e9 um tipo de\u00a0<b>fun\u00e7\u00e3o polinomial matem\u00e1tica<\/b>\u00a0onde o maior expoente da vari\u00e1vel \u00e9 2.<\/span><\/span><\/span><\/p>\n<p><b>Vejamos alguns exemplos:<\/b><\/p>\n<p><strong>Exemplo:<\/strong>\u200b \u200bx\u00b2\u200b \u200b+\u200b \u200b45x\u200b \u200b+\u200b \u200b3<\/p>\n<p><strong>Exemplo:<\/strong>\u200b \u200b- 87x\u00b2\u200b \u200b+\u200b \u200b123x\u200b \u200b-\u200b \u200b4801<\/p>\n<p>Escrevemos as fun\u00e7\u00f5es quadr\u00e1ticas de forma geral como a\u200bx\u00b2\u200b \u200b+\u200b \u200bb\u200bx\u200b \u200b+\u200b \u200bc\u200bem que a\u200b, b\u200b,\u200b \u200bc pertencem ao conjunto dos n\u00fameros Reais e a\u200b \u00e9 necessariamente diferente de zero.<\/p>\n<p><b>Vamos estudar o comportamento destas fun\u00e7\u00f5es:<\/b><\/p>\n<p class=\"imagem\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-128040\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-1.png\" alt=\"funcao-quadratica-1\" width=\"450\" height=\"250\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-1.png 450w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-1-300x167.png 300w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/p>\n<p><strong>Ra\u00edzes\u200b \u200bdas\u200b \u200bfun\u00e7\u00f5es\u200b \u200bquadr\u00e1ticas<\/strong><\/p>\n<p>A raiz da fun\u00e7\u00e3o quadr\u00e1tica pode ser calculada com a f\u00f3rmula de Bhaskara como<\/p>\n<p class=\"imagem\"><img decoding=\"async\" class=\"alignnone size-full wp-image-128041\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-2.png\" alt=\"funcao-quadratica-2\" width=\"214\" height=\"57\" \/><\/p>\n<p>Veja que essa f\u00f3rmula nos d\u00e1 2 respostas! Isso se deve ao fato de que temos uma equa\u00e7\u00e3o de segundo\u200b \u200bgrau\u200b, ent\u00e3o existem 2\u200b \u200bvalores\u200b \u200bque podem ser:<\/p>\n<p><strong>Ra\u00edzes Reais iguais:<\/strong> Voc\u00ea deve estar pensando como podemos dizer que s\u00e3o dois valores iguais?\u200b \u200bPodemos dizer que s\u00e3o dois valores reais iguais pois o comportamento da fun\u00e7\u00e3o continua sendo quadr\u00e1tico. Em fun\u00e7\u00f5es que temos apenas uma solu\u00e7\u00e3o, o comportamento \u00e9 linear, como em x + 2. Para comparar com uma fun\u00e7\u00e3o quadr\u00e1tica, vamos utilizar x\u00b2\u200b \u200b+\u200b \u200b4x\u200b \u200b+\u200b \u200b4\u200b.<\/p>\n<p class=\"imagem\"><img decoding=\"async\" class=\"alignnone size-full wp-image-128042\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-3.png\" alt=\"funcao-quadratica-3\" width=\"450\" height=\"212\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-3.png 450w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-3-300x141.png 300w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/p>\n<p><strong>Exemplo:<\/strong>\u200b \u200bx\u00b2 + 4x + 4, a f\u00f3rmula de Bhaskara para esse caso fica<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128043\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-4.png\" alt=\"funcao-quadratica-4\" width=\"432\" height=\"63\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-4.png 432w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-4-300x44.png 300w\" sizes=\"(max-width: 432px) 100vw, 432px\" \/><\/p>\n<p>Como sabemos as ra\u00edzes, podemos simplificar a fun\u00e7\u00e3o quadr\u00e1tica como <strong>(x+2)(x+2)<\/strong>\u200b. Fa\u00e7a a multiplica\u00e7\u00e3o distributiva e confirme que resulta na mesma fun\u00e7\u00e3o quadr\u00e1tica! Perceba tamb\u00e9m que a multiplica\u00e7\u00e3o dos termos <strong>(x+2)(x+2)<\/strong> tamb\u00e9m possui como raiz o valor -2.<\/p>\n<p><strong>Ra\u00edzes Reais distintas: <\/strong><\/p>\n<p><strong>Exemplo:<\/strong>\u200b \u200bx\u00b2 &#8211; 8x + 7 a f\u00f3rmula de Bhaskara para esse caso fica<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128044\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-5.png\" alt=\"funcao-quadratica-5\" width=\"201\" height=\"59\" \/><\/p>\n<p>Realizando as opera\u00e7\u00f5es devidas, vemos que os valores de x raiz da fun\u00e7\u00e3o quadr\u00e1tica s\u00e3o x<sub>1<\/sub> = 7 e\u00a0 x<sub>2<\/sub> = 1.<\/p>\n<p>Para facilitar e quando \u00e9 f\u00e1cil fazer de cabe\u00e7a, podemos usar um pequeno truque.<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128045\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-6.png\" alt=\"funcao-quadratica-6\" width=\"167\" height=\"86\" \/><\/p>\n<p>No nosso exemplo, podemos fazer de cabe\u00e7a o seguinte racioc\u00ednio: quais dois n\u00fameros resultam na soma de 8 e quando multiplicados um pelo outro temos o valor 7?<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128046\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-7.png\" alt=\"funcao-quadratica-7\" width=\"121\" height=\"83\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-7.png 121w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-7-110x75.png 110w\" sizes=\"(max-width: 121px) 100vw, 121px\" \/><\/p>\n<p>Aten\u00e7\u00e3o:\u200b \u200bNem todos os casos possibilitam essa estrat\u00e9gia! Essa ideia \u00e9 para agilizar as contas e evitar as opera\u00e7\u00f5es da f\u00f3rmula de Bhaskara, mas por vezes as ra\u00edzes s\u00e3o bem dif\u00edceis de acertar.<\/p>\n<p><strong>Ra\u00edzes Complexas:<\/strong> As ra\u00edzes complexas possuem a forma a\u200b \u200b+\u200b \u200bbi\u200b \u200b \u200be\u200b \u200b \u200ba\u200b \u200b-\u200b \u200bbi\u200b, em que i \u00e9 a unidade imagin\u00e1ria i\u00b2\u200b \u200b=\u200b \u200b-1\u200b. Dizemos que s\u00e3o ra\u00edzes <strong>complexas\u200b \u200bconjugadas<\/strong>\u200b pois possuem parte real igual e a parte imagin\u00e1ria \u00e9 uma a negativa da outra.<\/p>\n<p><strong>Exemplo:<\/strong>\u200b \u200bx\u00b2 &#8211; 3x + 10\u00a0a f\u00f3rmula de Bhaskara para esse caso fica<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128047\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-8.png\" alt=\"funcao-quadratica-8\" width=\"360\" height=\"63\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-8.png 360w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-8-300x53.png 300w\" sizes=\"(max-width: 360px) 100vw, 360px\" \/><\/p>\n<p>As ra\u00edzes que iremos obter s\u00e3o x<sub>1<\/sub> = 1, 5 + i\u221a31 e x<sub>2<\/sub> = 1 5 \u2212 i\u221a31.<\/p>\n<p>Note que como a parte imagin\u00e1ria vem da raiz quadrada, realmente sempre teremos ra\u00edzes complexas conjugadas.<\/p>\n<p>O que isso significa, em termos do gr\u00e1fico da fun\u00e7\u00e3o, uma vez que a raiz \u00e9 o ponto de cruzamento da fun\u00e7\u00e3o com o eixo x?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128050\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-91.png\" alt=\"funcao-quadratica-9\" width=\"157\" height=\"400\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-91.png 157w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-quadratica-91-118x300.png 118w\" sizes=\"(max-width: 157px) 100vw, 157px\" \/><br \/>\nSignifica que a fun\u00e7\u00e3o n\u00e3o cruza o eixo, pois, como nesse caso o coeficiente <strong>a&gt;0<\/strong>\u200b, ela est\u00e1 localizada acima do eixo x.<\/p>\n<p><strong>Fonte:<\/strong>\u00a0Col\u00e9gio S\u00e3o Francisco\/Luisa Boccardo Burini<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Fun\u00e7\u00e3o Quadr\u00e1tica &#8211; O\u200b \u200bque\u200b \u200b\u00e9\u200b Uma\u00a0fun\u00e7\u00e3o quadr\u00e1tica\u00a0\u00e9 um tipo de\u00a0fun\u00e7\u00e3o polinomial matem\u00e1tica\u00a0onde o maior expoente da vari\u00e1vel \u00e9 2. Vejamos alguns exemplos: Exemplo:\u200b \u200bx\u00b2\u200b \u200b+\u200b \u200b45x\u200b \u200b+\u200b \u200b3 Exemplo:\u200b \u200b- 87x\u00b2\u200b \u200b+\u200b \u200b123x\u200b \u200b-\u200b \u200b4801 Escrevemos as fun\u00e7\u00f5es quadr\u00e1ticas de forma geral como a\u200bx\u00b2\u200b \u200b+\u200b \u200bb\u200bx\u200b \u200b+\u200b \u200bc\u200bem que a\u200b, b\u200b,\u200b \u200bc pertencem ao [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[414],"tags":[],"class_list":["post-93828","post","type-post","status-publish","format-standard","hentry","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Fun\u00e7\u00e3o Quadr\u00e1tica - Portal S\u00e3o Francisco<\/title>\n<meta name=\"description\" content=\"Fun\u00e7\u00e3o Quadr\u00e1tica, Matem\u00e1tica, Gr\u00e1fico, Par\u00e1bola, Concavidade, Ra\u00edzes, Iguais, Distintas, Exemplo, Comportamento, Fun\u00e7\u00e3o Quadr\u00e1tica\" \/>\n<meta name=\"robots\" content=\"noindex, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Fun\u00e7\u00e3o Quadr\u00e1tica - 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