# What is the vertex of  y= -2x^2+8x-(x-1)^2?

Nov 22, 2017

Vertex at $\left(x - v , {y}_{v}\right) = \left(1 \frac{2}{3} , 7 \frac{1}{3}\right)$

#### Explanation:

Convert the given equation $y = - 2 {x}^{2} + 8 x - {\left(x - 1\right)}^{2}$
into vertex form:
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$ with vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$

$y = - 2 {x}^{2} + 8 x - {\left(x - 1\right)}^{2}$

$\textcolor{w h i t e}{\text{XXX}} = - 2 {x}^{2} + 8 x - {x}^{2} + 2 x - 1$

$\textcolor{w h i t e}{\text{XXX}} = - 3 {x}^{2} + 10 x - 1$

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{g r e e n}{- 3} \left({x}^{2} - \frac{10}{3} x\right) - 1$

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{g r e e n}{- 3} \left({x}^{2} - \frac{10}{3} x + {\left(\frac{{\cancel{10}}^{5}}{{\cancel{6}}_{3}}\right)}^{2}\right) - 1 - \left(\textcolor{g r e e n}{- 3}\right) \cdot {\left(\frac{5}{3}\right)}^{2}$

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{g r e e n}{- 3} {\left(x - \textcolor{red}{\frac{5}{3}}\right)}^{2} - 1 + \frac{25}{3}$

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{g r e e n}{- 3} {\left(x - \textcolor{red}{\frac{5}{3}}\right)}^{2} + \textcolor{b l u e}{\frac{22}{3}}$

which is the vertex form with the vertex at
$\textcolor{w h i t e}{\text{XXX}} \left(\textcolor{red}{\frac{5}{3}} , \textcolor{b l u e}{\frac{22}{3}}\right) = \left(\textcolor{red}{1 \frac{2}{3}} , \textcolor{b l u e}{7 \frac{1}{3}}\right)$