I am learning graph skectching.
Its well known that quadratic polynomials over reals are symmetric about their minima/maxima.
But today I discovered an interesting result that Cubic polynomials are centrally symmetric about their only point of inflection. But my proof is using mundane algebra and i am unable to find an intuitive explanation for this. So, does any one know of any elegant way to prove it or at least make it seem plausible.
Also my main question, is are there generalisation of the above results for n degree polynomials? It doesn't generalise in a straightforward way since obviously in general it is not true that even degree polynomials are symmetric about the line $x=t$ where $t$ is the only root of the $n-1$ th derivative of the polynomial of degree $n$ where n is even.
But are there generalisations in other directions or any analoggous results or applications of symmetry of graphs of polynomials?