how can I calculate an equation of a parabola in factored form?

Question

The vertex of the parabola is (0,-18) and the x-intercepts are (3,0) and (-3,0)

Answer 1

y = a(x-0)^2 -18 = ax^2 - 18 in vertex form

y = a(x-3)(x+3) = a(x^2 -9) in x intercept form

a>0 given the vertex is below the x axis and there are x intercepts

plug in either x intercept (3,0)

y= ax^2 - 18

0 =a(3^2) - 18

9a = 18

a = 18/9 = 2

y = ax^2 - 18

y = 2x^2 -18

or

y =a(x^2 -9)

y =2(x^2 -9) = 2x^2 - 18

the parabola is

f(x) = 2x^2 - 18, upward opening, with axis of symmetry x=0

or in factored form

f(x) = 2(x-3)(x+3)