# Chapter 6, Application 9

Use the graph below of the function $$x_{n+1} = f_a(x_n) = ax_n(1 − x_n)$$ for values of $$a \in (1,3)$$. Use this graph to repeat applications 2-8 in Chapter 6 of the text. The questions are repeated here.

• What are the fixed points and what are their stabilities?
• Describe the orbits of all initial conditions that are less than the smallest of the two fixed points.
• Describe the orbits of all initial conditions that are between the two fixed points.
• On your graph, locate the two points that are eventually fixed points. There is one that lands on each of the two fixed points.
• Describe the orbits of all initial conditions that are between the fixed point $$x=0$$ and the eventually fixed point $$x=1.$$
• Describe the orbits of all initial conditions that are between the eventually fixed point $$x=1$$ and the other eventually fixed point.
• Describe the orbits of all initial conditions that are greater than the largest of the two eventually fixed points.

a: