Question

faça um estudo do sinal da função y= -x²+4

Answer 1

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

Answer 2

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$\large\begin{array}{l} \textsf{Estudar o sinal da fun\c{c}\~ao }\mathsf{y=-x^2+4.}\\\\ \mathsf{y=4-x^2}\\\\ \mathsf{y=2^2-x^2}\\\\ \mathsf{y=2^2+2x-2x-x^2}\\\\ \mathsf{y=2(2+x)-x(2+x)}\\\\ \mathsf{y=(2+x)(2-x)} \end{array}$

$\large\begin{array}{l} \mathsf{y=(x+2)(-x+2)\qquad\quad(i)} \end{array}$


$\large\begin{array}{l} \textsf{Fica f\'acil analisar o sinal de y, se olharmos para o}\\\textsf{sinal de cada fator envolvido.}\\\\\\ \textsf{As ra\'izes de y s\~ao}\\\\ \mathsf{x_1=-2~~e~~x_2=2}\\\\ \textsf{(basta fazer cada fator igual a zero para encontr\'a-las).} \end{array}$


$\large\begin{array}{l} \textsf{Fazendo o quadro de sinais, temos}\\\\ \begin{array}{cc} \mathsf{-x+2}\quad&\mathsf{\underline{~+++}\underset{-2}{\bullet}\underline{++++}\underset{2}{\bullet}\underline{---~\quad}_{\blacktriangleright}}\\\\ \mathsf{x+2}\quad&\mathsf{\underline{~---}\underset{-2}{\bullet}\underline{++++}\underset{2}{\bullet}\underline{+++\quad}_{\blacktriangleright}}\\\\\\ \mathsf{(-x+2)(x+2)}\quad&\mathsf{\underline{~---}\underset{-2}{\bullet}\underline{++++}\underset{2}{\bullet}\underline{---\quad}_{\blacktriangleright}} \end{array} \end{array}$


$\large\textsf{Pela \'ultima linha acima, conclu\'imos que}$

•   $\large\begin{array}{l} \mathsf{y<0,}\textsf{ quando }\mathsf{x<-2~ou~x>2;} \end{array}$

•   $\large\begin{array}{l} \mathsf{y=0,}\textsf{ quando }\mathsf{x=-2~ou~x=2;} \end{array}$

•   $\large\begin{array}{l} \mathsf{y>0,}\textsf{ quando }\mathsf{-2<x<2;} \end{array}$


$\large\begin{array}{l} \textsf{D\'uvidas? Comente.}\\\\\\ \textsf{Bons estudos! :-)} \end{array}$


Tags:  estudo do sinal função quadrática segundo grau fatoração produto raízes álgebra