## All parabola are similar

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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*) = *a**x*^{2} + *b**x* + *c*), until its vertex coincides with that of the red parabola (*y* = *h*(*x*) = *x*^{2}).

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*) = *x*^{2}).

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.

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Dilation of parabola

**Instructions**: 1. Drag slider *k* to dilate (enlarge/reduce) the purple parabola *y* = *f*(*x*) = *x*^{2} 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* = *ax*^{2} 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.