Encourage students to use different initial values for a, b, c from their neighbors. They may find a formula for the locus of the vertex by studying the applet, and comparing different configurations. That does not constitute a proof, but it is still satisfying.

Here is an algebraic derivation of the equation in each case, assuming that the coordinates of the vertex are `(h,v)`.

Since `h=(-b)/(2a)`, and `v=ah^2+bh+c`, it follows that `v=-b^2/(4a)+c`

If b and c are fixed, and a varies

`v=b/2*(-b)/(2a)+c=b/2h+c`, so the vertex stays on the line `y=b/2x+c`

If a and c are fixed, and b varies

`v=-a(b^2/(4a^2))+c=-ah^2+c`, so the vertex stays on the parabola `y=-ax^2+c`

If a and b are fixed, and c varies

`h` is constant, so the vertex stays on the vertical line `x=(-b)/(2a)`