I realize this is an old thread, but just in case someone looks at this, as I did, I would like to disagree with Peter's assertion that the # of parameters needed to describe an object is equal to how many points needed to derive a similar object. With respect to the parabola noted here, 4 points in the general case can derive 2 different parabolas. It is simple enough to envision a "kite" arrangement of points (0,0; 0,1; 1,0; 3,3) As this is symmetric, it is easy to see how one parabola constucted could be flipped about the x=y axis and create a second parabola.
One way of seeing how # of points to define vs # of points to derive can be different is to look at a a set of 2 parallel lines, a distance k from an arbitrary line. (Similar to the focus and directrix in a parabola) But it is easy to see how given 3 points, there are 3 different sets of parallel lines that could be constructed through these points.
Another way to look at parabolas is the general conic equation has 6 variables - the requirement that the determinant be = 0 for parabolas essentially reduces this to 5.