Challenge Level

Prove that the graph of the polynomial

$f(x) = x^3 - 6x^2 +9x +1$

has rotational symmetry, find the centre of rotation and re-write the equation of the graph in terms of new co-ordinates $(u,v)$ with the origin of the new co-ordinate system at the centre of rotation.

Do the same for the graph of the function $g(x) = 2x^3 + 3x^2 +5x +4$ .

Prove that the graphs of all cubic polynomials have rotational symmetry.