Question

What is the equation of the parabola that has a vertex at `(0, 8)` and passes through point `(5,-4)`?

Answer 1

There are an infinite number of parabolic equations that meet the given requirements.
If we restrict the parabola to having a vertical axis of symmetry, then:
`color(white)("XXX")y=-12/25x^2+8`

Explanation:

For a parabola with a vertical axis of symmetry, the general form of the parabolic equation with vertex at `(a,b)` is:
`color(white)("XXX")y=m(x-a)^2+b`

Substituting the given vertex values `(0,8)` for `(a,b)` gives
`color(white)("XXX")y=m(x-0)^2+8`

and if `(5,-4)` is a solution to this equation, then
`color(white)("XXX")-4=m((-5)^2-0)+8 rArr m=-12/25`

and the parabolic equation is
`color(white)("XXX")color(black)(y=-12/25x^2+8)`
graph{y=-12/25*x^2+8 [-14.21, 14.26, -5.61, 8.63]}

However, (for example) with a horizontal axis of symmetry:
`color(white)("XXX")color(black)(x=5/144(y-8)^2)`
also satisfies the given conditions:
graph{x=5/144(y-8)^2 [-17.96, 39.76, -8.1, 20.78]}

Any other choice for the slope of the axis of symmetry will give you another equation.