Curve Fitter 2

In Cartesian coordinates, the squared distance $D^2$ beween two points $(a, b)$ and $(c, d)$ is given by
$$
D^2 = (a-c)^2+(b-d)^2
$$

Imagine that the function $y=f(x)$ has a squared distance $D^2$ between two of its turning points. Which of the following functions also definitely have a squared distance $D^2$ between two of their turning points? ($a$ is a constant)
$$y=f(x)-a\quad\quad y=af(x)\quad\quad y = f(ax)\quad\quad y=f(x-a)$$
Geometrically, what do these transformed equations correspond to?

Use these ideas to help you to find a cubic equation for which the squared distance $D^2$ between the turning points is a whole number.

Very hard extension: Can you find a cubic equation for which the distance between the turning points is a whole number (Warning: we can't find a simple solution to this, so be prepared for an exploration!)