Find the equation of the parabola if the vertex is $\left(0,0\right)$ and focus is $\left(0,-4\right)$.
From the given data, the parabola is open downwards.
Here, vertex $= V = \left(h,k\right) = \left(0,0\right)$
Focus $= F = \left(0, -4\right)$
Distance between vertex and focus $= VF = a$
i.e. $VF = a = \sqrt{\left(0 - 0\right)^2 + \left(0 + 4\right)^2} = 4$
The required equation of the parabola is of the form $\;\;$ $\left(x - h\right)^2 = - 4a \left(y - k\right)$
$\therefore$ $\;$ The required equation is
$\left(x - 0\right)^2 = - 4 \times 4 \times \left(y - 0\right)$
i.e. $x^2 = - 16 y$