# How do you write a quadratic equation with vertex; ( -2,1 ); point: ( 1,-8 )?

Nov 5, 2017

$y = \frac{7}{9} {\left(x + 2\right)}^{2} + 1$

#### Explanation:

The general vertex form for a quadratic with vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$ is
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$
(where $\textcolor{g r e e n}{m}$ can be thought of as a "spread" factor).

Given the vertex $\left(\textcolor{red}{- 2} , \textcolor{b l u e}{1}\right)$
this becomes
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} {\left(x - \left(\textcolor{red}{- 2}\right)\right)}^{2} + \textcolor{b l u e}{1} = \textcolor{g r e e n}{m} {\left(x + 2\right)}^{2} + 1$

If $\left(x , y\right) = \left(1 , 8\right)$ is a solution to this equation,
then
$\textcolor{w h i t e}{\text{XXX}} 8 = \textcolor{g r e e n}{m} {\left(1 + 2\right)}^{2} + 1$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow 7 = \textcolor{g r e e n}{m} \times 9$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow \textcolor{g r e e n}{m} = \textcolor{g r e e n}{\frac{7}{9}}$

$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{\frac{7}{9}} {\left(x + 2\right)}^{2} + 1$