# How do you write a quadratic equation with vertex (-2,-3) and goes through the point (-4,25)?

Mar 19, 2017

$y = 7 {\left(x + 2\right)}^{2} - 3$

#### Explanation:

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

With a vertex of $\left(\textcolor{red}{- 2} , \textcolor{b l u e}{- 3}\right)$ this becomes
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} {\left(x - \left(\textcolor{red}{- 2}\right)\right)}^{2} + \left(\textcolor{b l u e}{- 3}\right)$
or
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} {\left(x + 2\right)}^{2} - 3$

Given that $\left(x , y\right) = \left(- 4 , 25\right)$ is a solution for the required quadratic we have:
$\textcolor{w h i t e}{\text{XXX}} 25 = \textcolor{g r e e n}{m} {\left(\left(- 4\right) + 2\right)}^{2} - 3$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow \textcolor{g r e e n}{m} {\left(- 2\right)}^{2} = 28$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow \textcolor{g r e e n}{4} \textcolor{g r e e n}{m} = 28$

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

and the complete quadratic equation becomes:
$\textcolor{w h i t e}{\text{XXX}} y = 7 {\left(x + 2\right)}^{2} - 3$