Question

What is the equation of the parabola that has a vertex at `(-2, -4)` and passes through point `(-3,-15)`?

Answer 1

`y=-11(x+2)^2-4`

Explanation:

The general form of a parabolic equation with vertex `(a,b)` is
`color(white)("XXX")y=m(x-a)^2+b` for some constant `m`

Since the required parabola has a vertex at `(-2,-4)` this becomes:
`color(white)("XXX")y=m(x+2)^2-4`

and since `(x,y)=(-3,-15)` is a solution to this equation:
`color(white)("XXX")-15=m(-3+2)^2-4`

#color(white)("XXX")-11 = m

So the equation of the parabola can be written as
`color(white)("XXX")y=(-11)(x+2)^2-4`

graph{-11(x+2)^2-4 [-12.24, 13.06, -16.24, -3.59]}