# Chapter 6, Explorations 18-23

The graph below shows the function \(x_{n+1} = f_4(x_n) = 4x_n(1 − x_n)\). By adjusting the slider for \(n\), you can perform graphical analysis on \(f^n_4(x)\) for \(n=1,2,\ldots,7\). Use this to answer the following questions and complete explorations 18-23 from the text.

- How many critical points does \(y = f_4^n(x)\) have? Is there a general formula for the number of critical points in terms of \(n\)? What are the critical values (i.e. \(y\)-values of these critical points)? Describe the orbits of these points under iteration by \(f_4\).
- What happens to the number of critical points of \(y = f_4^n(x)\) and the distance between them as \(n\) goes to infinity?
- How many fixed points does \(f_4^n(x)\) have? Is there a general formula for the number of fixed points of \(f_4^n(x)\) in terms of \(n\)? Describe the orbits of these points under iteration by \(f_4.\)
- Explain why \(f_4(x)\) has periodic points of all periods. Does \(f_4(x)\) have prime periodic points of all periods? Why or why not?
- What happens to the distance between fixed points of \(y=f_4^n(x)\) as \(n\) goes to infinity? What does this tell you about the distance between periodic points of \(f_4\)?
- When you set \(n=1\) and do the graphical analysis, can you find any periodic orbits (other than the obvious fixed points)? Can you find any eventually fixed points (other than the obvious ones)?

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