Chapter 3, Explorations 57 and 58

Minimize the left sidebar so you can see both graphs. The left figure always shows a graph of \(y = f_a(x)\). The right figure shows a graph of \(y = f_a^n(x)\) where \(n\) can also be set using a slider.

Exploration 57 - For each of the \(a\)-values given in your text, the logistic equation has an attracting periodic orbit. Select the appropriate value of \(n\) and determine its period. How many periodic points are there of this period? How many prime periodic orbits are there of this period?

Exploration 58 - For each parameter value use the slider to change \(n\) so that it agrees with the period of the attracting periodic orbit you found in Exploration 57. For example, if you found an attracting period 2 orbit in Exploration 57, make \(n\) equal to 2 using the slider. What can you say about the derivative of the right graph at the attracting periodic points? What about the repelling ones?