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Visual, dynamic proof that all cubic polynomials are affine equivalent

Instructions: 1. Drag p and q to move the blue cubic, which is the translation of the general purple one, until its point of symmetry coincides at the origin with the point of symmetry of the red cubic (y = x^3)

2. Click on black 'Show Function Plot', and then drag k, to shear the translated cubic in the y-direction until it has a horizontal point of inflection.

3. Click on green 'Show Function Plot', and then drag m, to stretch the translated, sheared cubic in the y-direction until it coincides with the red cubic.

4. Change the parameters of the general purple cubic by dragging a, b, c and d, and repeat steps 1, 2 and 3 above.

Also read my article at The affine equivalence of cubic polynomials

In addition, have a look at my article at All cubic polynomials are point symmetric

For further useful background look at the JavaSketchpad sketch (refresh page if necessary) at An area preserving transformation: shearing