On Tue, 11 Nov 1997, Eileen M. Klimick Schoaff wrote:
> > 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?
Yes they do. But not if it isn't. You did miss something!
But if we consider any axes, > then there are an infinite number of parabolas passing through three points.
Yes, this is true. But I spoke of 4 points, not 3.
> The generic equation is ax^2 + bxy +cy^2 + dx + ey + f = 0.
This is the general conic, which is usually an ellipse or hyperbola rather than a hyperbola.
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?
Yes, roughly speaking.
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.
You do indeed need 5 points to determine the general conic.
> 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. > > Eileen Schoaff > Buffalo State College > I think as a math educator you really SHOULD have known of the existence of conics other than parabolae!