Graphic Convergence

The point (x , y) moves to a new position (x� , y�) by the rule :

eg the point (2, 10) moves to (5 , 6) because x = 2 and y = 10 gives:

The Problems :

A Start with the point (0 , 1) and repeat the rule 4 times

B Start with the point (5 , 6) and repeat the rule 4 times

C Plot the points on graph paper, and comment on your results so far.

D Can you guess a point that will not move when the rule is applied ?

This point is described as an 'invariant' of the process.

A slightly different rule is :

E Start with the point (1 , 1) and see where it goes after 4 repeats of this rule

F Do the same for the point (5 , 4)

G What did you notice about the way the 2 different points behaved ?

Can you find the 'invariant' under this rule?