# What is the equation of the parabola that has a vertex at  (-4, 2)  and passes through point  (-8,-34) ?

Jun 21, 2018

$y = - \frac{9}{4} {x}^{2} - 18 x - 34$

#### Explanation:

$\text{the equation of a parabola in "color(blue)"vertex form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where "(h,k)" are the coordinates of the vertex and a}$
$\text{is a multiplier}$

$\text{here } \left(h , k\right) = \left(- 4 , 2\right)$

$y = a {\left(x + 4\right)}^{2} + 2$

$\text{to find a substitute "(-8,-34)" into the equation}$

$- 34 = 16 a + 2$

$16 a = - 36 \Rightarrow a = \frac{- 36}{16} = - \frac{9}{4}$

$y = - \frac{9}{4} {\left(x + 4\right)}^{2} + 2 \leftarrow \textcolor{red}{\text{in vertex form}}$

$\text{expanding and rearranging gives}$

$y = - \frac{9}{4} \left({x}^{2} + 8 x + 16\right) + 2$

$y = - \frac{9}{4} {x}^{2} - 18 x - 34 \leftarrow \textcolor{red}{\text{in standard form}}$