Section 2.6 Parameterized families of functions
Rather than explore the behavior of a single function at a time, we can introduce a parameter and explore the range of behavior that arises in a whole family of functions. Two important examples are
The quadratic family: \(f_c(x)=x^2+c\)
The logistic family: \(f_{\lambda}(x)=\lambda x(1-x)\)
The cobweb plots shown back in Figure 2.9 are all chosen from the logistic family with \(\lambda=2.8\text{,}\) \(\lambda=3.2\text{,}\) and \(\lambda=4\text{.}\) Even in those three pictures with graphs that look so very similar, we see three different types of behavior: an attractive fixed point, an attractive orbit of period two, and chaos (which can be given a very technical meaning).
Figure 2.24 shows some cobweb plots for the quadratic family of functions. Note that the behavior we see is very similar to the behavior we see for the logistic family - a fact that will become more understandable once we study conjugacy in section Section 2.7
