{"id":93755,"date":"2017-01-10T10:38:57","date_gmt":"2017-01-10T12:38:57","guid":{"rendered":"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/?p=93755"},"modified":"2026-05-19T19:40:16","modified_gmt":"2026-05-19T21:40:16","slug":"funcao-do-2o-grau","status":"publish","type":"post","link":"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/matematica\/funcao-do-2o-grau","title":{"rendered":"Fun\u00e7\u00e3o do 2\u00ba Grau"},"content":{"rendered":"<p>Uma\u00a0<b>fun\u00e7\u00e3o de segundo grau<\/b>, ou fun\u00e7\u00e3o quadr\u00e1tica, \u00e9 uma fun\u00e7\u00e3o polinomial da forma f(x) = ax ^ 2 + bx + c), onde a, b, e c s\u00e3o n\u00fameros reais e # 0\\). O maior expoente da vari\u00e1vel \\(x\\) \u00e9 2, e seu gr\u00e1fico \u00e9 uma par\u00e1bola que abre para cima se a \\&gt; 0\\ ou para baixo se a &lt; 0.<\/p>\n<p><b>Aplica\u00e7\u00f5es:\u00a0<\/b>As fun\u00e7\u00f5es quadr\u00e1ticas s\u00e3o usadas para modelar cen\u00e1rios com formas parab\u00f3licas, como calcular alturas m\u00e1ximas em f\u00edsica, determinar lucro\/custo em economia ou encontrar \u00e1reas m\u00e1ximas.<\/p>\n<p><b>Fun\u00e7\u00e3o do 2\u00ba Grau &#8211; O que \u00e9<\/b><\/p>\n<p>Uma\u00a0<strong>Fun\u00e7\u00e3o do 2\u00ba Grau<\/strong>\u00a0\u00e9 a fun\u00e7\u00e3o cuja vari\u00e1vel independente est\u00e1 em grau 2 em rela\u00e7\u00e3o a vari\u00e1vel dependente.<\/p>\n<p>Matematicamente esta representa\u00e7\u00e3o ocorre pela equa\u00e7\u00e3o generalizada\u00a0<strong>y(x) = ax\u00b2 + bx + c<\/strong>, em que\u00a0<strong>x<\/strong>\u00a0\u00e9 a vari\u00e1vel independente,\u00a0<strong>y<\/strong>\u00a0\u00e9 a vari\u00e1vel dependente e\u00a0<strong>a<\/strong>,\u00a0<strong>b, c<\/strong>\u00a0pertencem ao conjunto dos n\u00fameros Reais e\u00a0<strong>a<\/strong>\u00a0\u00e9 necessariamente diferente de zero.<\/p>\n<p>O comportamento da fun\u00e7\u00e3o de segundo grau \u00e9\u00a0<strong>parab\u00f3lico\u00a0<\/strong>e pode variar em concavidade para\u00a0<strong>cima<\/strong>\u00a0ou para\u00a0<strong>baixo<\/strong>\u00a0se o coeficiente\u00a0<strong>a<\/strong>\u00a0for\u00a0<strong>positivo\u00a0<\/strong>ou\u00a0<strong>negativo<\/strong>, respectivamente.<\/p>\n<p>Vamos observar os gr\u00e1ficos que podemos construir em ambas configura\u00e7\u00f5es.<\/p>\n<p><strong>Ex:<\/strong>\u00a0y(x) = 1x\u00b2 \u2013 5x + 6<\/p>\n<p>Vamos analisar o comportamento desta fun\u00e7\u00e3o atrav\u00e9s do gr\u00e1fico.<\/p>\n<p class=\"imagem\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-127871\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2o-grau-1.png\" alt=\"funcao-do-2o-grau-1\" width=\"450\" height=\"320\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2o-grau-1.png 450w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2o-grau-1-300x213.png 300w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/p>\n<p>Vemos que ela possui <strong>concavidade para cima<\/strong>, pois <strong>a &gt; 0 <\/strong>e consequentemente, esta fun\u00e7\u00e3o possui um m\u00ednimo definido.\u00a0Vamos encontr\u00e1-lo.<\/p>\n<p>Para isso, fazemos a f\u00f3rmula de <strong>Bhaskara <\/strong>para encontrar as ra\u00edzes da fun\u00e7\u00e3o, isto \u00e9, quando <strong>y(x) = 0<\/strong>.<\/p>\n<p class=\"imagem\"><img decoding=\"async\" class=\"alignnone size-full wp-image-127873\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/formula-de-bhaskara.png\" alt=\"formula-de-bhaskara\" width=\"338\" height=\"100\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/formula-de-bhaskara.png 338w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/formula-de-bhaskara-300x89.png 300w\" sizes=\"(max-width: 338px) 100vw, 338px\" \/><\/p>\n<p>Ent\u00e3o, obteremos os valores de<strong> x<\/strong> que fazem <strong>y(x) = 0<\/strong> O valor m\u00ednimo de <strong>y(x) <\/strong>ocorre no meio destas ra\u00edzes, logo, precisamos calcular o <strong>x<\/strong> m\u00e9dio entre as ra\u00edzes e obter o <strong>y(x) <\/strong>correspondente.<\/p>\n<p>Agora vamos seguir estes passos para a equa\u00e7\u00e3o do exemplo, em que<\/p>\n<p><strong>a = 1<\/strong>,\u00a0 <strong>b = -5,\u00a0\u00a0 c = 6, <\/strong>encontrando ambos os valores de <em>x<sub>1<\/sub><\/em> e <em>x<sub>2<\/sub><\/em><\/p>\n<p class=\"imagem\"><img decoding=\"async\" class=\"alignnone size-full wp-image-127877\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2o-grau-3.png\" alt=\"funcao-do-2o-grau-3\" width=\"450\" height=\"61\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2o-grau-3.png 450w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2o-grau-3-300x41.png 300w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/p>\n<p>Ao realizarmos as opera\u00e7\u00f5es necess\u00e1rias, descobriremos que x<sub>1<\/sub> = 3 e x<sub>2<\/sub> = 2 e x<sub>1<\/sub> + x<sub>2<\/sub> \/ 2 = 2,5.<\/p>\n<p>Agora vamos calcular y (2,5) = 1 (2,5)\u00b2 &#8211; 5 (2,5) + 6 = 24,75 \u00e9 o m\u00ednimo da\u00a0fun\u00e7\u00e3o de segundo grau.<\/p>\n<p><strong>Ex:<\/strong> y (x) = -1 x\u00b2 &#8211; 10 x + 40<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-127953\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2-grau-1.png\" alt=\"funcao-do-2-grau-1\" width=\"450\" height=\"221\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2-grau-1.png 450w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2-grau-1-300x147.png 300w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/p>\n<p>Nesse exemplo, percebemos que a concavidade\u200b \u200bpara\u200b \u200bbaixo\u200b, isto \u00e9, a par\u00e1bola possui um valor m\u00e1ximo\u200b definido\u200b, mas n\u00e3o um valor m\u00ednimo definido pois o coeficiente do termo ao quadrado \u00e9 menor que zero, a\u200b \u200b&lt;\u200b \u200b0.<\/p>\n<p>Como descobrir esse m\u00e1ximo?<\/p>\n<p>Vamos avaliar primeiro as ra\u00edzes da equa\u00e7\u00e3o, obtendo assim 2 valores de\u200b \u200bx\u200b que satisfazem y\u200b(x)\u200b \u200b=\u200b \u200b0\u200b, uma vez que estamos trabalhando com uma fun\u00e7\u00e3o de segundo grau.<\/p>\n<p>As ra\u00edzes da equa\u00e7\u00e3o podem ser obtidas atrav\u00e9s da f\u00f3rmula de Bhaskara\u200b. Com essa f\u00f3rmula teremos os dois valores de x para os quais\u200b \u200by\u200b(x)\u200b \u200b=\u200b \u200b0\u200b. Para obter o m\u00e1ximo, temos que calcular a m\u00e9dia entre os valores de x, para a soma e subtra\u00e7\u00e3o da f\u00f3rmula e depois encontrar o valor de y\u200b(x)\u200b \u200bcorrespondente. Agora vamos seguir estes passos para a equa\u00e7\u00e3o do exemplo, em que a\u200b \u200b=\u200b \u200b-1\u200b, b\u200b \u200b=\u200b \u200b-10,\u200b \u200b \u200b \u200bc\u200b \u200b=\u200b \u200b40,\u200b \u200bencontrando ambos os valores de x<sub>1<\/sub> e x<sub>2<\/sub><\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-127954\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2-grau-2.png\" alt=\"funcao-do-2-grau-2\" width=\"450\" height=\"87\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2-grau-2.png 450w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2-grau-2-300x58.png 300w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/p>\n<p>Aproximando o valor de \u221a160 por 12,6 teremos os resultados para x<sub>1<\/sub> = 11, 3 e x<sub>2<\/sub> =\u2212 1, 3 . A m\u00e9dia destes valores ser\u00e1 de , aproximadamente. 2 x +x 1 2 = 5.<\/p>\n<p>Agora iremos fazer y\u200b(5)\u200b para obter o valor m\u00e1ximo. \u200b \u200by\u200b(5)\u200b \u200b=\u200b \u200b(\u200b-1)\u200b \u200b5\u00b2\u200b \u200b-\u200b \u200b(\u200b10\u200b)\u200b \u200b5\u200b \u200b+\u200b \u200b40\u200b \u200b=\u200b \u200b-25\u200b \u200b-50\u200b \u200b+\u200b \u200b40\u200b \u200b=\u200b \u200b-35.<\/p>\n<p><strong>M\u00e1ximos\u200b \u200be\u200b \u200bM\u00ednimos<\/strong><\/p>\n<p>Desta forma, podemos ver que para as fun\u00e7\u00f5es de<\/p>\n<p><strong>1) Concavidade\u200b \u200bpara\u200b \u200bcima:\u200b<\/strong> \u200bo m\u00ednimo \u00e9 calculado como<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-127955\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2-grau-3.png\" alt=\"funcao-do-2-grau-3\" width=\"303\" height=\"113\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2-grau-3.png 303w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2-grau-3-300x112.png 300w\" sizes=\"(max-width: 303px) 100vw, 303px\" \/><\/p>\n<p><strong>2) Concavidade\u200b \u200bpara\u200b \u200bbaixo:<\/strong>\u200b \u200bo m\u00e1ximo \u00e9 calculado como<\/p>\n<p class=\"imagem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-127956\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/funcao-do-2-grau-4.png\" alt=\"funcao-do-2-grau-4\" width=\"295\" height=\"107\" \/><\/p>\n<p><strong>Fonte:<\/strong>\u00a0Col\u00e9gio S\u00e3o Francisco\/Luisa Boccardo Burini<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Uma\u00a0fun\u00e7\u00e3o de segundo grau, ou fun\u00e7\u00e3o quadr\u00e1tica, \u00e9 uma fun\u00e7\u00e3o polinomial da forma f(x) = ax ^ 2 + bx + c), onde a, b, e c s\u00e3o n\u00fameros reais e # 0\\). O maior expoente da vari\u00e1vel \\(x\\) \u00e9 2, e seu gr\u00e1fico \u00e9 uma par\u00e1bola que abre para cima se a \\&gt; 0\\ [&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-93755","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 do 2\u00ba Grau, Matem\u00e1tica Fun\u00e7\u00e3o do 2\u00ba Grau<\/title>\n<meta name=\"description\" content=\"Fun\u00e7\u00e3o do 2\u00ba Grau, Matem\u00e1tica, Quadr\u00e1tica, Equa\u00e7\u00e3o, Gr\u00e1fico, Par\u00e1bola, Concavidade, Ra\u00edzes, Comportamento Fun\u00e7\u00e3o do 2\u00ba Grau\" \/>\n<meta name=\"robots\" content=\"noindex, 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