The following two intuitively obvious theorems proven in De Villiers (1991) do not seem to be well-known:
Theorem 1
A differentiable function y = f(x) is reflective symmetric around a vertical line x = a if and only if its derivative dy/dx = f'(x) is point symmetric around the point (a; 0).
Theorem 2
A differentiable function y = f(x) is point symmetric around a point (a; b) if and only if its derivative dy/dx = f'(x) is reflective symmetric around the vertical line x = a.
(Note: 'Even functions' are defined as functions that are reflective symmetric around the y-axis, and 'odd functions' are defined as ones that are point symmetric with respect to the origin. So the above two theorems merely show a link between these two concepts, and generalizes the relationship slightly.)
Illustration
The dynamic sketch below shows two examples illustrating the two theorems, the blue graphs showing the original symmetric functions, and the red graphs their respective derivatives.
Click on the 'Show Transformed Graph' button in each case, and drag the red buttons (points) to dynamically explore a few more cases.
Hint: Click on the 'Link to' buttons on the bottom to navigate from one sketch to the other.
Vertical Line and Point Symmetries of Differentiable Functions
Challenge
Can you explain why (prove that) these two theorems are true?
What corollaries, and other theorems, follow from these theorems? Explore more.
Published Paper
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.
Some Related Papers
De Villiers, M. (1993). Transformations: A golden thread in school mathematics. Spectrum, 31(4), Oct, pp. 11-18.
De Villiers, M. (1993). The affine invariance and line symmetries of the conics. Australian Senior Mathematics Journal, 7(2), pp. 32-50.
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
All cubic polynomial functions are affine equivalent
An example: the point symmetry of a cubic polynomial (with reference to my 2004 paper)
The affine invariance of the conics
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
Even and odd functions (Wikipedia)
Odd and Even Functions (Newcastle University)
Odd and Even Functions (Australian Mathematical Sciences Institute)
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 with WebSketchpad by Michael de Villiers, 4 Dec 2024.