# Suppose a parabola has vertex (4,7) and also passes through the point (-3,8). What is the equation of the parabola in vertex form?

May 16, 2017

Actually, there are two parabolas (of vertex form) that meet your specifications:

$y = \frac{1}{49} {\left(x - 4\right)}^{2} + 7$ and $x = - 7 {\left(y - 7\right)}^{2} + 4$

#### Explanation:

There are two vertex forms:

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

where $\left(h , k\right)$ is the vertex and the value of "a" can be found by using one other point.

We are given no reason to exclude one of the forms, therefore we substitute the given vertex into both:

$y = a {\left(x - 4\right)}^{2} + 7$ and $x = a {\left(y - 7\right)}^{2} + 4$

Solve for both values of a using the point $\left(- 3 , 8\right)$:

$8 = {a}_{1} {\left(- 3 - 4\right)}^{2} + 7$ and $- 3 = {a}_{2} {\left(8 - 7\right)}^{2} + 4$

$1 = {a}_{1} {\left(- 7\right)}^{2}$ and $- 7 = {a}_{2} {\left(1\right)}^{2}$

${a}_{1} = \frac{1}{49}$ and ${a}_{2} = - 7$

Here are the two equations:

$y = \frac{1}{49} {\left(x - 4\right)}^{2} + 7$ and $x = - 7 {\left(y - 7\right)}^{2} + 4$

Here is an image containing both parabolas and the two points:

Please observe that both have the vertex $\left(4 , 7\right)$ and both pass through the point $\left(- 3 , 8\right)$