A parabola, on the other hand, is determined by any 4 of its points. John Conway>
Am I missing something here? Don't 3 points determine a parabola if the axis of symmetry is either vertical or horizontal? But if we consider any axes, then there are an infinite number of parabolas passing through three points. The generic equation is ax^2 + bxy +cy^2 + dx + ey + f = 0. If Jon Roberts is considering parabolas of the form y = ax^2 + bx + c, then knowing 3 points gives you three equations with three unknowns which can easily be solved -- unless there is no solution.
In the April 1997 issue of the Mathematics Teacher, a colleague of mine, Dr. Ellie Johnson, wrote an article "A Look at Parabolas with a Graphing Calculator". In this article she using the calculator to generate many solutions to the generic equation. Of course this just shows that given three points and restricting yourself to a parabola of the form y = ax^2 + bx + c, you can derive the equation. That does not, of course, construct it.
Does the fourth point determine whether the axis of symmetry is vertical, horizontal, or rotated? In ax^2 + bxy +cy^2 + dx + ey + f = 0, it looks like you need more than 4 points to determine a, b, c, d, e, f.
Then again, I am only a math education person and do not have a PhD in math so I am probably far in the dark.