Chapter 5, Explorations 5-8

The graph of \(y = f_a(x) = ax(1-x)\) is shown, and the value of \(a\) is adjusted with the slider. As stated in the text, these quesetions should be able to be answered using methods from calculus or algebra, but you may find this tool helpful. Answer the following questions:

• How does changing the parameter \(a\) change the graph of \(y = f_a(x)\)?
• What is the critical point as a function of the parameter \(a\)? Is it a max or a min? What is the critical value (i.e. the \(y\)-value of the critical point)?
• What are the \(x\) and \(y\) intercepts of \(f_a(x)\) in terms of \(a\)? For what values of \(a\) are there \(x\)-intercepts? What happens to the \(x\)-intercepts as \(a\) increases?
• We know that the intersections between the graph and the line \(y=x\) correspond to fixed points of the dynamical system. For what values of \(a\) are there fixed points? What are formulas for them in terms of \(a\)? Describe what happens to them as \(a\) increases.