# How do you find the equation of a parabola with vertex at the origin and directrix x=-2?

Jan 24, 2017

${y}^{2} = 8 x$

#### Explanation:

The standard form of the parabola is ${y}^{2} = 4 a x$, giving a parabola with its axis parallel to the $x$-axis, vertex at the origin, focus $\left(a , 0\right)$ and directrix $x = - a$. So in your case $a = 2$, giving ${y}^{2} = 4 a x$.

Alternatively, you can work from a definition of a parabola, which is the set of all points $\left(x , y\right)$ such that the distance from the point to the directrix $x = - 2$ is the same as the distance to the focus (2,0)#.
(The vertex is half-way between the focus and the directrix.)

${\left(x - \left(- 2\right)\right)}^{2} = {\left(x - a\right)}^{2} + {y}^{2}$
$\cancel{{x}^{2}} + 4 a x + \cancel{4} = \cancel{{x}^{2}} - 4 a x + \cancel{4} + {y}^{2}$
${y}^{2} = 4 a x + 4 a x = 8 a x$