Question

which equation represents a parabola with a vertex at the origin and a focus at (0,-1)

Answer 1

We are given parabola with vertex at origin (0,0) and given focus (0,-1).

We know vertex is given by (h,k).

Therefore, h=0 and k=0.

Formula for focus is (h, k + p).

On comapring with given focus

k+p = -1.

Plugging value of k=0 in above equation we get

0 +p =-1.

p = -1.

Parabola equation is  4p (y - k)=(x - h)^2

Plugging values of h, k and p in parabola equation, we get

$4(-1) (y-0) = (x-0)^2$

$-4y = x^2$

Dividing both sides by -4, we get

$y=-\frac{1}{4}x^2$.

Therefore, $y=-\frac{1}{4}x^2$ equation represents a parabola with a vertex at the origin and a focus at (0,-1).