# What is the standard form equation of the parabola with a vertex at (0,0) and directrix at x= -2?

May 30, 2017

$x = \frac{1}{8} {y}^{2}$

#### Explanation:

Please observe that the directrix is a vertical line, therefore, the vertex form is of the equation is:

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

where $\left(h , k\right)$ is the vertex and the equation of the directrix is $x = k - \frac{1}{4 a} \text{ [2]}$.

Substitute the vertex, $\left(0 , 0\right)$, into equation [1]:

$x = a {\left(y - 0\right)}^{2} + 0$

Simplify:

$x = a {y}^{2} \text{ [3]}$

Solve equation [2] for "a" given that $k = 0$ and $x = - 2$:

$- 2 = 0 - \frac{1}{4 a}$

$4 a = \frac{1}{2}$

$a = \frac{1}{8}$

Substitute for "a" into equation [3]:

$x = \frac{1}{8} {y}^{2} \leftarrow$ answer

Here is a graph of the parabola with the vertex and the directrix: