# What is the equation of the parabola that has a vertex at  (2, -5)  and passes through point  (3,-105) ?

Feb 22, 2016

$y = - 100 {\left(x - 2\right)}^{2} - 5$

#### Explanation:

Note: The standard form of a parabola is $y = a {\left(x - h\right)}^{2} + k$ , in which the $\left(h , k\right)$ is the vertex.

This problem given the vertext $\left(2 , - 5\right)$ , which mean $h = 2 , k = - 5$

Passes through the point $\left(3 , - 105\right)$ , which mean that $x = 3 , y = - 10$

We can find $a$ by substitute all the information above into the standard form like this

$y = a {\left(x - h\right)}^{2} + k$
$y = a {\left(x - \textcolor{red}{2}\right)}^{2} \textcolor{red}{- 5}$

$\textcolor{b l u e}{- 105} = a {\left(\textcolor{b l u e}{3 - \textcolor{red}{2}}\right)}^{2} \textcolor{red}{- 5}$

$- 105 = a {\left(1\right)}^{2} - 5$

$- 105 = a - 5$

$- 105 + 5 = a$

$a = - 100$

The standard equation for the parabola with the given condition is

$y = - 100 {\left(x - 2\right)}^{2} - 5$