faça um estudo do sinal da função y= -x²+4
Soluções para a tarefa
Respondido por
19
y = -x² + 4
-x² + 4 = 0
-x² = -4
x² = 4
x = +/- √4
x = +/- 2
y < 0, se x < -2 ou x > 2
y = 0, se x = -2 ou x = 2
y > 0, se -2 < x < 2
-x² + 4 = 0
-x² = -4
x² = 4
x = +/- √4
x = +/- 2
y < 0, se x < -2 ou x > 2
y = 0, se x = -2 ou x = 2
y > 0, se -2 < x < 2
Respondido por
16
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![\large\begin{array}{l} \textsf{Estudar o sinal da fun\c{c}\~ao }\mathsf{y=-x^2+4.}\\\\ \mathsf{y=4-x^2}\\\\ \mathsf{y=2^2-x^2}\\\\ \mathsf{y=2^2+2x-2x-x^2}\\\\ \mathsf{y=2(2+x)-x(2+x)}\\\\ \mathsf{y=(2+x)(2-x)} \end{array} \large\begin{array}{l} \textsf{Estudar o sinal da fun\c{c}\~ao }\mathsf{y=-x^2+4.}\\\\ \mathsf{y=4-x^2}\\\\ \mathsf{y=2^2-x^2}\\\\ \mathsf{y=2^2+2x-2x-x^2}\\\\ \mathsf{y=2(2+x)-x(2+x)}\\\\ \mathsf{y=(2+x)(2-x)} \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D+%5Ctextsf%7BEstudar+o+sinal+da+fun%5Cc%7Bc%7D%5C%7Eao+%7D%5Cmathsf%7By%3D-x%5E2%2B4.%7D%5C%5C%5C%5C+%5Cmathsf%7By%3D4-x%5E2%7D%5C%5C%5C%5C+%5Cmathsf%7By%3D2%5E2-x%5E2%7D%5C%5C%5C%5C+%5Cmathsf%7By%3D2%5E2%2B2x-2x-x%5E2%7D%5C%5C%5C%5C+%5Cmathsf%7By%3D2%282%2Bx%29-x%282%2Bx%29%7D%5C%5C%5C%5C+%5Cmathsf%7By%3D%282%2Bx%29%282-x%29%7D+%5Cend%7Barray%7D)
![\large\begin{array}{l} \mathsf{y=(x+2)(-x+2)\qquad\quad(i)} \end{array} \large\begin{array}{l} \mathsf{y=(x+2)(-x+2)\qquad\quad(i)} \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D+%5Cmathsf%7By%3D%28x%2B2%29%28-x%2B2%29%5Cqquad%5Cquad%28i%29%7D+%5Cend%7Barray%7D)
![\large\begin{array}{l} \textsf{Fica f\'acil analisar o sinal de y, se olharmos para o}\\\textsf{sinal de cada fator envolvido.}\\\\\\ \textsf{As ra\'izes de y s\~ao}\\\\ \mathsf{x_1=-2~~e~~x_2=2}\\\\ \textsf{(basta fazer cada fator igual a zero para encontr\'a-las).} \end{array} \large\begin{array}{l} \textsf{Fica f\'acil analisar o sinal de y, se olharmos para o}\\\textsf{sinal de cada fator envolvido.}\\\\\\ \textsf{As ra\'izes de y s\~ao}\\\\ \mathsf{x_1=-2~~e~~x_2=2}\\\\ \textsf{(basta fazer cada fator igual a zero para encontr\'a-las).} \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D+%5Ctextsf%7BFica+f%5C%27acil+analisar+o+sinal+de+y%2C+se+olharmos+para+o%7D%5C%5C%5Ctextsf%7Bsinal+de+cada+fator+envolvido.%7D%5C%5C%5C%5C%5C%5C+%5Ctextsf%7BAs+ra%5C%27izes+de+y+s%5C%7Eao%7D%5C%5C%5C%5C+%5Cmathsf%7Bx_1%3D-2%7E%7Ee%7E%7Ex_2%3D2%7D%5C%5C%5C%5C+%5Ctextsf%7B%28basta+fazer+cada+fator+igual+a+zero+para+encontr%5C%27a-las%29.%7D+%5Cend%7Barray%7D)
![\large\begin{array}{l} \textsf{Fazendo o quadro de sinais, temos}\\\\ \begin{array}{cc} \mathsf{-x+2}\quad&\mathsf{\underline{~+++}\underset{-2}{\bullet}\underline{++++}\underset{2}{\bullet}\underline{---~\quad}_{\blacktriangleright}}\\\\ \mathsf{x+2}\quad&\mathsf{\underline{~---}\underset{-2}{\bullet}\underline{++++}\underset{2}{\bullet}\underline{+++\quad}_{\blacktriangleright}}\\\\\\ \mathsf{(-x+2)(x+2)}\quad&\mathsf{\underline{~---}\underset{-2}{\bullet}\underline{++++}\underset{2}{\bullet}\underline{---\quad}_{\blacktriangleright}} \end{array} \end{array} \large\begin{array}{l} \textsf{Fazendo o quadro de sinais, temos}\\\\ \begin{array}{cc} \mathsf{-x+2}\quad&\mathsf{\underline{~+++}\underset{-2}{\bullet}\underline{++++}\underset{2}{\bullet}\underline{---~\quad}_{\blacktriangleright}}\\\\ \mathsf{x+2}\quad&\mathsf{\underline{~---}\underset{-2}{\bullet}\underline{++++}\underset{2}{\bullet}\underline{+++\quad}_{\blacktriangleright}}\\\\\\ \mathsf{(-x+2)(x+2)}\quad&\mathsf{\underline{~---}\underset{-2}{\bullet}\underline{++++}\underset{2}{\bullet}\underline{---\quad}_{\blacktriangleright}} \end{array} \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D+%5Ctextsf%7BFazendo+o+quadro+de+sinais%2C+temos%7D%5C%5C%5C%5C+%5Cbegin%7Barray%7D%7Bcc%7D+%5Cmathsf%7B-x%2B2%7D%5Cquad%26amp%3B%5Cmathsf%7B%5Cunderline%7B%7E%2B%2B%2B%7D%5Cunderset%7B-2%7D%7B%5Cbullet%7D%5Cunderline%7B%2B%2B%2B%2B%7D%5Cunderset%7B2%7D%7B%5Cbullet%7D%5Cunderline%7B---%7E%5Cquad%7D_%7B%5Cblacktriangleright%7D%7D%5C%5C%5C%5C+%5Cmathsf%7Bx%2B2%7D%5Cquad%26amp%3B%5Cmathsf%7B%5Cunderline%7B%7E---%7D%5Cunderset%7B-2%7D%7B%5Cbullet%7D%5Cunderline%7B%2B%2B%2B%2B%7D%5Cunderset%7B2%7D%7B%5Cbullet%7D%5Cunderline%7B%2B%2B%2B%5Cquad%7D_%7B%5Cblacktriangleright%7D%7D%5C%5C%5C%5C%5C%5C+%5Cmathsf%7B%28-x%2B2%29%28x%2B2%29%7D%5Cquad%26amp%3B%5Cmathsf%7B%5Cunderline%7B%7E---%7D%5Cunderset%7B-2%7D%7B%5Cbullet%7D%5Cunderline%7B%2B%2B%2B%2B%7D%5Cunderset%7B2%7D%7B%5Cbullet%7D%5Cunderline%7B---%5Cquad%7D_%7B%5Cblacktriangleright%7D%7D+%5Cend%7Barray%7D+%5Cend%7Barray%7D)
![\large\textsf{Pela \'ultima linha acima, conclu\'imos que} \large\textsf{Pela \'ultima linha acima, conclu\'imos que}](https://tex.z-dn.net/?f=%5Clarge%5Ctextsf%7BPela+%5C%27ultima+linha+acima%2C+conclu%5C%27imos+que%7D)
•![\large\begin{array}{l} \mathsf{y<0,}\textsf{ quando }\mathsf{x<-2~ou~x>2;} \end{array} \large\begin{array}{l} \mathsf{y<0,}\textsf{ quando }\mathsf{x<-2~ou~x>2;} \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D+%5Cmathsf%7By%26lt%3B0%2C%7D%5Ctextsf%7B+quando+%7D%5Cmathsf%7Bx%26lt%3B-2%7Eou%7Ex%26gt%3B2%3B%7D+%5Cend%7Barray%7D)
•![\large\begin{array}{l} \mathsf{y=0,}\textsf{ quando }\mathsf{x=-2~ou~x=2;} \end{array} \large\begin{array}{l} \mathsf{y=0,}\textsf{ quando }\mathsf{x=-2~ou~x=2;} \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D+%5Cmathsf%7By%3D0%2C%7D%5Ctextsf%7B+quando+%7D%5Cmathsf%7Bx%3D-2%7Eou%7Ex%3D2%3B%7D+%5Cend%7Barray%7D)
•![\large\begin{array}{l} \mathsf{y>0,}\textsf{ quando }\mathsf{-2<x<2;} \end{array} \large\begin{array}{l} \mathsf{y>0,}\textsf{ quando }\mathsf{-2<x<2;} \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D+%5Cmathsf%7By%26gt%3B0%2C%7D%5Ctextsf%7B+quando+%7D%5Cmathsf%7B-2%26lt%3Bx%26lt%3B2%3B%7D+%5Cend%7Barray%7D)
![\large\begin{array}{l} \textsf{D\'uvidas? Comente.}\\\\\\ \textsf{Bons estudos! :-)} \end{array} \large\begin{array}{l} \textsf{D\'uvidas? Comente.}\\\\\\ \textsf{Bons estudos! :-)} \end{array}](https://tex.z-dn.net/?f=%5Clarge%5Cbegin%7Barray%7D%7Bl%7D+%5Ctextsf%7BD%5C%27uvidas%3F+Comente.%7D%5C%5C%5C%5C%5C%5C+%5Ctextsf%7BBons+estudos%21+%3A-%29%7D+%5Cend%7Barray%7D)
Tags: estudo do sinal função quadrática segundo grau fatoração produto raízes álgebra
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Tags: estudo do sinal função quadrática segundo grau fatoração produto raízes álgebra
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