# 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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