# What is the equation of the parabola that has a vertex at  (-2, -4)  and passes through point  (1,5) ?

Jan 5, 2017

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

#### Explanation:

The equation of a parabola in $\textcolor{b l u e}{\text{vertex form}}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where (h ,k) are the coordinates of the vertex and a is a constant.

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

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

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

To find a, substitute the point (1 ,5) into the equation. That is x = 1 and y = 5

$\Rightarrow 5 = a {\left(1 + 2\right)}^{2} - 4$

$\Rightarrow 9 a = 9 \Rightarrow a = 1$

$\text{Thus " y=(x+2)^2-4color(red)" is equation in vertex form}$

Expanding the bracket and simplifying gives.

$y = {x}^{2} + 4 x + 4 - 4$

$\Rightarrow y = {x}^{2} + 4 x \textcolor{red}{\text{ equation in standard form}}$