# Parabola focus & directrix review

CCSS Math: HSG.GPE.A.2
Review your knowledge of the focus and directrix of parabolas.

## What are the focus and directrix of a parabola?

Parabolas are commonly known as the graphs of quadratic functions. They can also be viewed as the set of all points whose distance from a certain point (the focus) is equal to their distance from a certain line (the directrix).

## Parabola equation from focus and directrix

Given the focus and the directrix of a parabola, we can find the parabola's equation. Consider, for example, the parabola whose focus is at $(-2,5)$ and directrix is $y=3$. We start by assuming a general point on the parabola $(x,y)$.
Using the distance formula, we find that the distance between $(x,y)$ and the focus $(-2,5)$ is $\sqrt{(x+2)^2+(y-5)^2}$, and the distance between $(x,y)$ and the directrix $y=3$ is $\sqrt{(y-3)^2}$. On the parabola, these distances are equal:
\begin{aligned} \sqrt{(y-3)^2} &= \sqrt{(x+2)^2+(y-5)^2} \\\\ (y-3)^2 &= (x+2)^2+(y-5)^2 \\\\ \blueD{y^2}-6y\goldD{+9} &= (x+2)^2\blueD{+y^2}\maroonD{-10y}+25 \\\\ -6y\maroonC{+10y}&=(x+2)^2+25\goldD{-9} \\\\ 4y&=(x+2)^2+16 \\\\ y&=\dfrac{(x+2)^2}{4}+4\end{aligned}
Write the equation for a parabola with a focus at $(6,-4)$ and a directrix at $y=-7$.
$y=$