Looking at the equation governing this curve:
If -2 is one root, then (x+2) is one factor; if 2 is another root, then (x-2) is another factor.
The equation is then f(x) = a(x-2)(x+2), or f(x) = a(x^2 - 4).
When x=0, f(x) = 8. In other words, the vertex is at (0,8).
We must find the value of a in f(x)=a(x^2-4): 8 = a(0^2) + 4. Then 8 = 4a, and a= 2.
Thus, the equation of this parabola is f(x) = 2(x^2-4). Graph this and see for yourself whether the graph goes through (0,8), (-2,0) and (2,0).