# Cobweb plots

A cobweb plot is a simple and efficient tool to visualize functional iteration. The basic idea is simple: Suppose we graph the function $$f$$ together with the line $$y=x$$; something like so:

In this example, the two graphs intersect; that point of intersection is a fixed point. Now, suppose we're on the line at the point $$(x_i,x_i)$$. If we move vertically to the graph of the function, we preserve the $$x$$ coordinate but change the $$y$$ coordinate to $$f(x_i)$$. Thus, we arrive at the point $$(x_i,f(x_i)) = (x_i,x_{i+1})$$. If we then move horizontally back to the line $$y=x$$ we now preserve the $$y$$ coordinate but change the $$x$$ coordinate so that the two are the same. Thus, we arrive at the point $$(x_{i+1},x_{i+1})$$.

In summary: The process of moving vertically from a point on the line $$y=x$$ to the graph of $$f$$ and back to the line is a geometric representation of one application of the function $$f$$.

Repeated application of this process represents repeated application of $$f$$, i.e. iteration. Here's what that process looks like for a logistic function.

It turns out that the process is quite sensitive to the slope of the function at the point of intersection. Here's a slightly steeper logistic function.

You can explore this sensitivity with our cobweb tool.