# How do you write the equation of the parabola in vertex form given points (3,1) the vertex(5,9)?

Jul 23, 2017

$y = - 2 {\left(x - 5\right)}^{2} + 9$

#### Explanation:

Given -

$\left(3 , 1\right)$ [point on the parabola]
Vertex $\left(5 , 9\right)$

Formula for the parabola in vertex form

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

Here $\left(h , k\right)$ coordinates of the vertex.

Let us substitute $\left(5 , 9\right)$ in the formula.

$y = a {\left(x - 5\right)}^{2} + 9$

To find the value of $a$, substitute $\left(3 , 1\right)$

$1 = a {\left(3 - 5\right)}^{2} + 9$
$1 = a 4 + 9$

Solve for $a$

$4 a + 9 = 1$
$4 a = 1 - 9 = - 8$
$a = - \frac{8}{4}$

$a = - 2$

The equation is -

$y = - 2 {\left(x - 5\right)}^{2} + 9$