Visual, dynamic proof that all parabola are similar (for a > 0, since for a < 0, only a reflection in x-axis is required)
Instructions: 1. Drag sliders p and q to translate the blue parabola with the transformation g(x) = f(x-q) + p, which is the translation of the general purple one (f(x) = ax2 + bx + c), until its vertex coincides with that of the red parabola (y = h(x) = x2).
2. Click on Show Function Plot, and then drag slider k, to enlarge/reduce the translated parabola with the dilation transformation q(x) = k.g(x/k) centred at the origin, until it coincides with the red one (y = h(x) = x2).
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.
Read my article at All parabola similar? Never! for more background.
Dilation of parabola
Instructions: 1. Drag slider k to dilate (enlarge/reduce) the purple parabola y = f(x) = x2 with the shown rectangle ABCD by the dilation transformation g(x) = k.f(x/k) centred at the origin to map onto the green one and A'B'C'D'.
2. Note therefore that the value of a in y = ax2 does NOT determine the shape of the parabola. Since they are all similar, they have the same shape, and the parameter a determines the dilation (enlargement or reduction). Note that the scale factor k is the reciprocal of a; i.e. k = 1/a. Hence, for a > 0, a decrease in a lets one 'zoom in' (enlarge) while an increase in a, lets one 'zoom out' (reduce).
Return to other Explorations for Students or to more Dynamic Geometry Sketches
Michael de Villiers, Updated 14 May 2011.