A fixed point is unstable if nearby points move
away from the fixed point under graphical iteration. 

Click the animation to stop. 
A fixed point is unstable if at the fixed point the tangent line to the graph
lies outside the region between +45° and 45°. 
That is, its slope is greater than +1 or less than 1. 
The region between these lines is shaded in blue.
If the graph crosses the diagonal outside the
blue region (on both sides), the fixed point is unstable. 
