For these things, I would suggest just finding the stationary points (even approximately) and intercepts, then draw the general shape.
If the curve is a polynomial, all that's needed is to add an intercept at x = 0 .
Prove every odd function is zero at x = 0:
f(-x) = -f(x) for x = 0, so f(0) = -f(0).
2f(0) = 0, so f(0) = 0, proven as required.
Someone else can do the rest for now.