# What is the equation of a parabola with a focus at (-2, 6) and a vertex at (-2, 9)?

Sep 5, 2017

$y = - {x}^{2} / 12 - \frac{x}{3} + \frac{26}{3}$

#### Explanation:

Given -

Vertex $\left(- 2 , 9\right)$
Focus $\left(- 2 , 6\right)$

From the information, we can understand the parabola is in the second quadrant. Since focus lies below the vertex, The parabola is facing down.

The vertex is at $\left(h , k\right)$

Then the general form of the formula is -

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

$a$ is the distance between focus and vertex. It is $3$

Now substitute the values

${\left(x - \left(- 2\right)\right)}^{2} = - 4 \times 3 \times \left(y - 9\right)$
${\left(x + 2\right)}^{2} = - 12 \left(y - 9\right)$

${x}^{2} + 4 x + 4 = - 12 y + 108$

By transpose we get -

$- 12 y + 108 = {x}^{2} + 4 x + 4$

$- 12 y = {x}^{2} + 4 x + 4 - 108$
$- 12 y = {x}^{2} + 4 x - 104$

$y = - {x}^{2} / 12 - \frac{x}{3} + \frac{26}{3}$