# What is the equation of the parabola with a focus at (0, 2) and vertex at (0,0)?

Apr 26, 2017

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

#### Explanation:

If the focus is above or below the vertex, then the vertex form of the equation of the parabola is:

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

If the focus is to the left or right the vertex, then the vertex form of the equation of the parabola is:

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

Our case uses equation [1] where we substitute 0 for both h and k:

$y = a {\left(x - 0\right)}^{2} + 0 \text{ [3]}$

The focal distance, f, from the vertex to the focus is:

$f = {y}_{\text{focus"-y_"vertex}}$

$f = 2 - 0$

$f = 2$

Compute the value of "a" using the following equation:

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

$a = \frac{1}{4 \left(2\right)}$

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

Substitute $a = \frac{1}{8}$ into equation [3]:

$y = \frac{1}{8} {\left(x - 0\right)}^{2} + 0$

Simplify:

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