z-functions and difference functions

In the applet below, you can see the z-function f(z) = zⁿ (green) with initially n = 2, and its difference function (red) Df(z) = f(z+1) – f(z).

1. Describe the relation between the graphs and the tables of f and of Df.
2. Starting with Df(n) we build DDf(n)? Describe the function DDf(n).
3. Describe the types of growth of f and Df. Can you give an equation for the function Df?
4. Now shift the slider to 3. What types of growth do f, D(f) and D(D(f)) have?
5. Experiment with the slider and formulate a conjecture on how the difference functions behave for different exponents of f.

Andreas Bauer, created with GeoGebra