# What is the vertex of y=-2x^2 - 8x + 9?

Dec 15, 2015

Vertex: $\left(- 2 , 17\right)$

#### Explanation:

Our objective will be to convert the given equation into "vertex form":
$\textcolor{w h i t e}{\text{XXX}} y = m {\left(x - a\right)}^{2} + b$ with vertex at $\left(a , b\right)$

Given
$\textcolor{w h i t e}{\text{XXX}} y = - 2 {x}^{2} - 8 x + 9$

Extract the $m$ factor
$\textcolor{w h i t e}{\text{XXX}} y = \left(- 2\right) \left({x}^{2} + 4 x\right) + 9$

Complete the square:
$\textcolor{w h i t e}{\text{XXX}} y = \left(\textcolor{b l u e}{- 2}\right) \left({x}^{2} + 4 x \textcolor{b l u e}{+ 4}\right) + 9 \textcolor{red}{+ 8}$

Re-write the $x$ expression as a binomial square
$\textcolor{w h i t e}{\text{XXX}} y = \left(- 2\right) {\left(x + 2\right)}^{2} + 17$

Convert the squared binomial into form $\left(x - a\right)$
$\textcolor{w h i t e}{\text{XXX}} y = \left(- 2\right) \left(x - \left(- 2\right)\right) + 17$
which is the vertex form with vertex at $\left(- 2 , 17\right)$
graph{-2x^2-8x+9 [-16.13, 15.93, 6, 22.01]}