Visual, dynamic proof that all parabola are similar (for a > 0, since for a < 0, a reflection in x-axis is required)
Instructions: 1. Drag p and q to move the blue parabola, which is the translation of the general purple one, until its vertex coincides with that of the red parabola (y = x^2)
2. Click on Show Function Plot, and then drag k, to enlarge/reduce the translated parabola until it coincides with the red one.
3. Change the parameters of the general purple parabola by dragging a (keep a > 0), b and c, and repeat steps 1 and 2 above.
Also read my article at All parabola similar? Never!