[alogo] Symmetry of cubic functions

The graph of the general cubic function y = ax3+bx2+cx+d is symmetric with respect to the inflection point A, which is the point, where the second derivative y'' = 6ax+2b vanishes.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]

To prove it calculate f(k), where k = -b/(3a), and consider point K = (k,f(k)). Then translate the origin at K and show that the curve takes the form y = ux3+vx, which is symmetric about the origin.
Note that the graphs of all cubic functions are affine equivalent.

See Also



Michael de Villiers All cubic polynomials are point symmetric Learning & Teaching Mathematics, No. 1, April 2004, pp. 12-15.

Return to Gallery

Produced with EucliDraw©