Are you looking for a quadratic that goes through those 3 points, or are controlled by those 3 points?
If controlled, a point p on a curve c can be found using t, where 0 < t < 1 from start to finish.
v0 = (1-t)*x + t *y
v1 = (1-t)*y + t * z
p = (1-t)*v0 + t*v1
if you are looking for a curve that fits those 3 points, i'm not 100% sure, but I think that more than 1 curve could fit.
EDIT: I stand corrected. Essentially you create the quadratic for each point and then with those 3 equations, Solve for a,b, and c.
EDIT2: I think i feel vindicated, multivariable quadratics would allow for an infinite number of quadratics that fit those points