Question

Determine o vértice (Xv e Yv) e as raízes da função. Use 5 valores para X.
a)Y= x²+6x+8
b)Y= x²-2x-8
c)Y= x²+8x-15
d)Y= -4x²+6x
e)Y= -x²+36

Answer 1

$Fo\´rmulas\to~~ x_v= \dfrac{-b}{2a} ~~~~e~~~~ y_v= \dfrac{-\Delta}{4a}$



$a)~~ y=x^{2} +6x+8\\\\x^{2} +6x+8=0\\\\ a=1~;~b=6~;~c=8\\\\\\ \Delta=b^2-4ac\to~~\Delta=6^2-4.1.8\to~~\Delta=36-32\to~~ \boxed{\Delta=4} \\\\\\ x_v= \dfrac{-6}{2.1}\to~~ x_v= \dfrac{-6}{2}\to~~ \boxed{x_v=-3} \\\\\\y_v= \dfrac{-4}{4.1}\to~~ y_v= \dfrac{-4}{4}\to~~ \boxed{y_v=- 1} \\\\\\\\ \large\boxed{\boxed{V=\left\{- 3~;~-1\right\} }}$




$b)~~ y=x^{2} -2x-8\\\\x^{2} -2x-8=0\\\\ a=1~;~b=-2~;~c=-8\\\\\\ \Delta=(-2)^2-4.1.(-8)\to~~ \Delta=4+32\to~~ \boxed{\Delta=36} \\\\\\ x_v= \dfrac{-(-2)}{2.1}\to~~ x_v= \dfrac{2}{2}\to~~ \boxed{x_v= 1} \\\\\\y_v= \dfrac{-36}{4.1}\to~~ y_v= \dfrac{-36}{4}\to~~ \boxed{y_v=- 9} \\\\\\\\ \large\boxed{\boxed{V=\left\{1~;~-9\right\} }}$




$c)~~ y=x^{2} +8x-15\\\\x^{2} +8x-15=0\\\\ a=1~;~b=8~;~c=-15\\\\\\ \Delta=8^2-4.1.(-15)\to~~ \Delta=64+60\to~~ \boxed{\Delta=124} \\\\\\ x_v= \dfrac{-8}{2.1}\to~~ x_v= \dfrac{-8}{2}\to~~ \boxed{x_v= -4} \\\\\\y_v= \dfrac{-124}{4.1}\to~~ y_v= \dfrac{-124}{4}\to~~ \boxed{y_v=- 31} \\\\\\\\ \large\boxed{\boxed{V=\left\{-4~;~-31\right\} }}$




$d)~~ y=-4x^{2} +6x\\\\-4x^{2} +6x=0\\\\ a=-4~;~b=6~;~c=0\\\\\\ \Delta=6^2-4.(-4).0\to~~ \Delta=36+0\to~~ \boxed{\Delta=36} \\x_v= \dfrac{-6}{2.(-4)}\to~~ x_v= \dfrac{-6}{-8}\to~~ \boxed{x_v= \frac{3}{4} } \\\\\\y_v= \dfrac{-36}{4.(-4)}\to~~ y_v= \dfrac{-36}{-16}\to~~ \boxed{y_v= \frac{9}{4} } \\\\\\\\ \large\boxed{\boxed{V=\left\{ \frac{3}{4}~;~\frac{9}{4} \right\} }}$




$e)~~ y=-x^{2} +36\\\\-x^{2} +36=0\\\\ a=-1~;~b=0~;~c=36\\\\\\ \Delta=0^2-4.(-1).36\to~~ \Delta=0+144\to~~ \boxed{\Delta=144} \\\\\\ x_v= \dfrac{-0}{2.(-1)}\to~~ x_v= \dfrac{0}{-2}\to~~ \boxed{x_v= 0} \\\\\\y_v= \dfrac{-144}{4.(-1)}\to~~ y_v= \dfrac{-144}{-4}\to~~ \boxed{y_v=36} \\\\\\\\ \large\boxed{\boxed{V=\left\{0~;~36\right\} }}$

Answer 2

Determine o vértice (Xv e Yv) e as raízes da função. Use 5 valores para X.a)Y= x²+6x+8
b)Y= x²-2x-8
c)Y= x²+8x-15
d)Y= -4x²+6x
e)Y= -x²+36