Question

A.) For this linear function, what happens to the function’s values after many iterations?

Answer 1

It can be helpful to use technology (a spreadsheet program or graphing calculator) to help you with iterated functions. What you are doing is evaluating the function using its output as new input.

(A) Whenever the magnitude of the slope of a function is less than 1, it will iterate toward the point where it intersects the line y = x. Here the magnitude o the slope is 1/2, so the final value in this case will approach
.. x = y = (-1/2)x +3
.. 3/2x = 3 . . . . . . . . add x/2
.. x = 2 . . . . . . . . . . . multiply by 2/3

You can see several iterations in the table in the figure here.


(B) The values get closer to 2 in each case.

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Iteration of this function is like drawing a spiral on the graph. Start with some point on the function curve, such as (0, 3). Draw a horizontal line to the line y=x. this gives you x=3. Now draw a vertical line to the function curve. This will give you the point (3, f(3)) = (3, 1.5). Draw a horizontal line to y = x, repeating as many times as you like. You will see the points get closer and closer to (2, 2) with each loop around the spiral.

If the slope of the function is greater than 1, the "spiral" will diverge instead of converging.