Chapter 6, Exploration 48

The graph below is for the function \(x_{n+1} = f_{4.5}(x) = 4.5x(1-x)\). You can select the value of \(n\) to display \(f^n_{4.5}(x)\). Notice that, as \(n\) increases, certain intervals along the \(x\)-axis are colored red.

• How do the intervals relate to properties of the graphs of \(y=f^n_{4.5}(x)\) for each \(n\)?
• How many intervals of each color are there for each \(n\)? Is there a general formula in terms of \(n\)? Why?
• Are there points in the interval [0, 1] whose orbits remain in that interval for all \(n\)? If so, describe them. If not, explain why.
• Does the process demonstrated here remind you of anything else that you have encountered in mathematics? If so, what?