How do you find the equation of the parabola with vertex (0,0) and focus (0,-3)?

Jun 30, 2018

The equation is $y = - \frac{1}{12} {x}^{2}$

Explanation:

If the vertex is at $\left(0 , 0\right)$ and the focus is at $\left(0 , - 3\right)$

The focus is the mid-point from the vertex to the directrix,

Therefore,

The directrix is $y = 3$

Any point $\left(x , y\right)$ on the parabola is equidistant from the focus and from the directrix.

Therefore,

$\left(y - 3\right) = \sqrt{{\left(x - 0\right)}^{2} + {\left(y + 3\right)}^{2}}$

Squaring both sides

${\left(y - 3\right)}^{2} = {\left(x\right)}^{2} + {\left(y + 3\right)}^{2}$

${y}^{2} - 6 y + 9 = {x}^{2} + {y}^{2} + 6 y + 9$

$- 12 y = {x}^{2}$

$y = - \frac{1}{12} {x}^{2}$

graph{-1/12x^2 [-10, 10, -5, 5]}