Fractals and Chaos Recap for 11/26

We discussed the results of 4g, 4h, and 4i from the Iterated Functions sheet and observed how the discussed the ancient divide and average method of approximating irrational square roots relates to 4i.

We then revisited an observation we made before break: the graph of a function “near” a fixed point will look linear if you magnify it enough, even if the function itself is not linear. As a result, we can use the rules we derived for classifying fixed points based on the slope of a linear function to help us classify fixed points now. For example, if the “nearby” slope around a fixed point is positive and greater than that of y=x (i.e., is greater than 1), then the fixed point will be a “direct attractor.” This raised the question: What if the nearby slope is exactly equal to 1? This occurs when the graph of f(x) is tangent to the graph of y = x at the location of the fixed point.

It turns out that the fixed point has no unique classification, as behavior of seeds to the left of the fixed point differs from the behavior of seeds to the right. A fixed point could be directly attracting on one side, but directly repelling on the other!

Your homework tonight is to finish the Iterated Functions sheet by exploring the equations of part 5. All five pairs of functions have a fixed point at 0, all of which having a nearby slope of 1. Use the Geogebra apps on MathInsight.org to make a graph and test the behavior of seeds on both sides of the fixed point and write down your observations on the Iterated Functions Supplement. All five examples of two equations each: the positive and negative version. Make sure you look at both!

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