Starting out with the following two observations that:
1) All straight lines y = mx + c are congruent (that is, any two straight lines can be mapped exactly onto each other by the isometric transformations, that is reflections, rotations and translations)
2) All parabola y = ax2 + bx + c are similar (that is, any two parabola can be mapped exactly onto each other by a combination of the similar transformations, i.e. enlargements or reductions in combination with the isometries.
it is natural to conjecture the following:
3) All graphs of cubic polynomial functions of the form y = ax3 + bx2 + cx + d are affine equivalent (that is, the graphs of any two cubic polynomials can be mapped exactly onto each other by a combination of the affine transformations, that is shears, stretches or the general linear transformations, including the similarity transformations (enlargements/reductions)).
Visual Dynamic Proof
The dynamic sketch below provides a visual proof of no. 3 above. Starting with the graph of an arbitrary cubic polynomial y = ax3 + bx2 + cx + d it is shown how one can translate, shear & stretch it to map exactly on to the graph of a standard cubic polynomial y = x3.
Visual, dynamic proof that all cubic polynomials are affine equivalent
Instructions
4. Drag sliders p and q to move the thin blue cubic g(x) = f(x - q) + p, which is the translation of the general purple one f(x) = ax3 + bx2 + cx + d, until its point of symmetry coincides at the origin with the point of symmetry of the red cubic h(x) = x3.
5. Click on the black 'Show Function Plot' button, and then drag k, to shear the translated cubic q(x) = g(x) + kx in the y-direction until it has a horizontal point of inflection.
6. Click on the green 'Show Function Plot', and then drag m, to stretch the translated, sheared cubic r(x) = m.q(x) in the y-direction until it coincides with the red cubic.
7. Change the parameters of the general purple cubic by dragging a, b, c and d, and repeat steps 4, 5 and 6 above.
Note: as shown in de Villiers (2004), the same basic affine transformations as above can be carried out on any cubic polynomial function y = ax3 + bx2 + cx + d to show that it is point symmetric.
References
De Villiers, M. (1993). Transformations: A golden thread in school mathematics. Spectrum, 31(4), Oct, pp. 11-18.
De Villiers, M. (2003). The Affine Equivalence of Cubic Polynomials. KZN Mathematics Journal, Nov, pp. 5-10.
De Villiers, M. (2004). All cubic polynomials are point symmetric. Learning & Teaching Mathematics, No. 1, April 2004, pp. 12-15.
Related Links
Three Circle Geometry Theorem Proofs by Transformations
The affine invariance of the conics
An area preserving transformation: shearing
All parabola are similar - i.e. have the same shape
Some Transformations of Graphs
Miscellaneous Dynamic Transformations (of Geometric Figures & Graphs)
International Mathematical Talent Search (IMTS) Problem Generalized
External Links
Cubic function at Wikipedia.
Affine transformation at Wikipedia.
Affine Transformation by Robert Fisher, Simon Perkins, Ashley Walker, Erik Wolfart.
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Created by Michael de Villiers, 30 Nov 2008; updated to WebSketchpad 29 Jan 2024.