All parabola are similar

Please install Java (version 1.4 or later) to use JavaSketchpad applets.

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.

Please install Java (version 1.4 or later) to use JavaSketchpad applets.

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).

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Michael de Villiers, Updated 14 May 2011.