# What is the vertex form of 5y=-x^2 + 9x +8?

May 9, 2018

$y = - \frac{1}{5} {\left(x - \frac{9}{2}\right)}^{2} + \frac{113}{20}$

#### Explanation:

We need the form of: $y = \text{something}$ so divide all of both sides by 5 giving:

$y = - \frac{1}{5} {x}^{2} + \frac{9}{5} x + \frac{8}{5} \text{ } \ldots \ldots . E q u a t i o n \left(1\right)$

Write as:

$\textcolor{g r e e n}{y = - \frac{1}{5} \left({x}^{2} - \textcolor{red}{9} x\right) + \frac{8}{5}}$

Halve the $\textcolor{red}{9}$ and write as:

$\textcolor{g r e e n}{y = - \frac{1}{5} {\left(x - \frac{\textcolor{red}{9}}{2}\right)}^{2} + k + \frac{8}{5}} \text{ } \ldots . E q u a t i o n \left(2\right)$

The $k$ is a correction factor as by doing the above you have added a value that is not in the original equation.

Set $\textcolor{g r e e n}{- \frac{1}{5} {\left(- \frac{\textcolor{red}{9}}{2}\right)}^{2} + k = 0}$

$\implies k = + \frac{81}{20}$

Substitute for $k$ in $E q u a t i o n \left(2\right)$ giving:

$\textcolor{g r e e n}{y = - \frac{1}{5} {\left(x - \frac{\textcolor{red}{9}}{2}\right)}^{2} + \frac{81}{20} + \frac{8}{5}} \text{ } \ldots . E q u a t i o n \left({2}_{a}\right)$

$y = - \frac{1}{5} {\left(x - \frac{9}{2}\right)}^{2} + \frac{113}{20}$