# 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?

n: