# What is the equation in standard form of the parabola with a focus at (2,-4) and a directrix of y= 6?

Nov 7, 2016

$y = - \frac{1}{20} {x}^{2} + \frac{1}{5} x + \frac{4}{5}$

#### Explanation:

The general form for the equation of a horizontal directrix is $y = k - f$

The focus of a parabola with a horizontal directrix is of the general form $\left(h , k + f\right)$

Therefore, we can write these 3 equations that will help us:

h = 2

-4 = k + f

6 = k - f

solve the last two equation for k and f:

$k = 1$
$f = - 5$

The vertex form of the equation of this type of parabola is:

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

Because we are not given the value of a, substitute $\frac{1}{4 f}$ for $a$:

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

Substitute our known values into the above equation:

$y = \frac{1}{4 \left(- 5\right)} {\left(x - 2\right)}^{2} + 1$

Expand the square:

$y = - \frac{1}{20} \left({x}^{2} - 4 x + 4\right) + 1$

Distribute $- \frac{1}{20}$:

$y = - \frac{1}{20} {x}^{2} + \frac{1}{5} x - \frac{1}{5} + 1$

Combine the constants:

$y = - \frac{1}{20} {x}^{2} + \frac{1}{5} x + \frac{4}{5}$