"Can you take three given points on the coordinate plane and describe the set of x, y pairs that could potentially represent the point of symmetry of a cubic that contains those three points?"
My first step was to draw a few cubic functions with the online function grapher. Here's what I made:
As you can see, I marked three points that both cubic functions go through then I marked the point of symmetry of both functions. Since any given three points cannot constrain a cubic function, there will always be multiple (and in fact infinite) possibilities for the point of symmetry.
I agree that the solution set contains an infinite number of curves. Then the question would be what characterizes all of those curves? The point of symmetry will move, of course (as you have shown), but does that mean it can be anywhere? Or are there limitations on where the point of symmetry can be?
ReplyDeleteYou might want to first investigate the quadratic situation that we did in class with The Case of the Missing Vertex. See if you can get a description of the solution set for that problem, where two points are fixed and the vertex is allowed to move. If you got an expression for K in terms of H, you would be able to describe all of the locations where the vertex can lie.