# How do you write the equation of each parabola in vertex form given Vertex (5, 12); point (7,15)?

Dec 13, 2017

There are two vertex forms:
$\text{[1] } y = a {\left(x - h\right)}^{2} + k$
$\text{[2] } x = a {\left(y - k\right)}^{2} + h$
where $\left(h , k\right)$ is the vertex and a is determined by the given point.

#### Explanation:

Given $\left(h , k\right) = \left(5 , 12\right)$:

$\text{[1.1] } y = a {\left(x - 5\right)}^{2} + 12$
$\text{[2.1] } x = a {\left(y - 12\right)}^{2} + 5$

Given $\left(x , y\right) = \left(7 , 15\right)$:

$15 = a {\left(7 - 5\right)}^{2} + 12$
$7 = a {\left(15 - 12\right)}^{2} + 5$

$15 = a {\left(2\right)}^{2} + 12$
$7 = a {\left(3\right)}^{2} + 5$

$3 = a {\left(2\right)}^{2}$
$2 = a {\left(3\right)}^{2}$

$a = \frac{3}{4}$
$a = \frac{2}{9}$

Substitute the above values into its respective equation:

$\text{[1.2] } y = \frac{3}{4} {\left(x - 5\right)}^{2} + 12$
$\text{[2.2] } x = \frac{2}{9} {\left(y - 12\right)}^{2} + 5$

Here is a graph with the vertex and the required point (in black), equation [1.2] (in blue), and equation [2.2] (in green):