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.
Point Symmetry
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. A useful application of this point symmetric property of cubic polynomials is that one can use it to easily find not only their turning points, but also their local maxima and minima using only transformations as shown in Taylor & Hansen (2008). It's a interesting property of cubic polynomials which not only provides an intrinsically beautiful result, but provides an easily accessible investigation with dynamic geometry for learners and students. It is surprising though that this property unfortunately does not seem to be well-known at high school as evidenced by a recent paper by Lanz (2023).
Moreover, as can be seen from the above, there is in general a close relationship between the symmetry of the graph of a particular function and the graph of its derivative function, and this relationship has been explored further in De Villiers (1991).
References
De Villiers, M. (1991). Vertical line and point symmetries of differentiable functions. International Journal of Mathematical Education in Science and Technology, 22(4), pp. 621-644.
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.
Lanz, J. (2023). 'Symmetry' of Cubic Functions. Learning & Teaching Mathematics, No. 35, Dec, pp. 23-25.
Taylor, R.D. (Jr.) & Hansen, R. (2008). Optimization of Cubic Polynomials without Calculus. Mathematics Teacher, Vol. 101, no. 6, Feb, pp. 408-411.
Some Related Links
An example: the point symmetry of a cubic polynomial (with reference to my 2004 paper)
The affine invariance of the conics
Vertical Line and Point Symmetries of Differentiable Functions
Three Circle Geometry Theorem Proofs by Transformations
An area preserving transformation: shearing
A Rectangle Angle Trisection Result
All parabola are similar - i.e. have the same shape
Mystery Transformation
Some Transformations of Graphs
Miscellaneous Dynamic Transformations (of Geometric Figures & Graphs)
International Mathematical Talent Search (IMTS) Problem Generalized
Some External Related Links
Technologically Embodied Geometric Functions
YouTube: Point Symmetry in Graphs
Symmetry Challenges
Cubic function at Wikipedia.
Symmetry in Geometry
YouTube: Point Symmetry
Graphs and Symmetry
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; 30 Nov 2024; 4 Dec 2024.