An archived instance of discourse for discussion in a Fractal Introductory Colloquium.

A graphical iteration problem


Let's try to analyze the behavior of the function
$$f(x) = -0.9x(x+1)(x-2).$$
First, let's start with just a graph of $f$ together with a graph of $y=x$. Have a look at that on either Desmos or on Easy Plot or even WolframAlpha (if you can figure out how to plot them both together).

  • How many fixed points do you see?
  • How many are attractive and how many repulsive?

I suppose you could try our cobweb tool to verify your conjectures.


When the functions above are graphed they look like the following:

-There is 1 fixed point

-Also, I don't understand the difference between attractive and repulsive


That looks great! To understand the difference between attractive and repulsive - place your pen down on the line $y=x$ near the fixed point zero but not exactly at zero. Perform graphical analysis by moving vertically to the graph and then horizontally back to the line. Does the process lead you toward zero or away? That's the difference between attractive and repulsive.

Also, I think there are two more fixed points - a total of three.


Here's the graph of the function along with the line y=x:

There would be three fixed points, where the graph of the function crosses the y=x line: at about 1.55, 0, and -0.57

I believe that at 1.55 and 0 the graph would be repulsive, while at the other point it'd be attractive, but I could be wrong so I'll edit this later if I can get the cobweb plot tool to work.