# Find the equation of a parabola with vertex (-4,2) and directrix y=5?

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Mar 7, 2018

$y = - \frac{1}{12} {\left(x - \left(- 4\right)\right)}^{2} + 2$

#### Explanation:

The vertex form for the equation of a parabola with a horizontal directrix is:

$y = \frac{1}{4 f} {\left(x - h\right)}^{2} + k \text{ [1]}$

where $\left(h , k\right)$ is the vertex and the equation of the directrix is, $y = k - f \text{ [2]}$

Substitute the vertex $\left(- 4 , 2\right)$ into equation [1]:

$y = \frac{1}{4 f} {\left(x - \left(- 4\right)\right)}^{2} + 2 \text{ [1.1]}$

Substitute $k = 2$ and $y = 5$ into equation [2]:

$5 = 2 - f$

$f = - 3$

Substitute $f = - 3$ into equation [1.1]:

$y = - \frac{1}{12} {\left(x - \left(- 4\right)\right)}^{2} + 2 \text{ [1.2]}$

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