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.
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.
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Symmetry of cubic functions ![[0_0]](CubicSymmetry_files/CubicSymmetry_0_0_0.png)
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