We had our parabolas/polynomials quiz today. You got HW 3.8 – Post-quiz HW and Extra Credit. Do one side, your choice, for a homework credit tomorrow. Do both sides for extra credit on the quiz.
Note that this assignment will be checked for a credit, as opposed to the usual homework quiz procedure.
With Mr. Drix, you observed some patterns that emerge in cobweb diagrams for certain types of fixed points. For an attractor, the pattern of steps in the cobweb diagram will be drawn towards the fixed point; for a repeller they move away. Furthermore, if the fixed point is direct, the pattern of steps will look like actual steps as they move towards or away from the fixed point, trapped between the two lines defined by the function we are iterating and y = x. If the fixed point is alternating, on the other hand, the pattern will spiral around the fixed point with each step, producing a picture that most clearly gives the cobweb diagram its name.
From there, we took our first look at non-linear functions, realizing quickly that in addition to having more than one fixed point, such functions can have different types of fixed points. One could be a direct repeller, while another an alternating attractor. The cobweb diagram remains our best way to observe these differences, and we used the Geogebra-based applications found here to create them.
We looked at 4a (y = x^2) together, classifying the fixed point at 1 as a direct repeller and the fixed point at 0 as a direct attractor. We furthermore observed that -1 is a “pre-image” to the fixed point at 1, and therefore we have different behavior for different seeds. Your homework tonight is to continue and look at 4b-4f, using the cobweb diagram app linked above and the Iterated Functions Supplement to keep track of your results.
With Mr. Drix, you continued your work with the Iterated Functions sheet, making some observations about how we might predict the classification of fixed point we get based on the parameters of the function we are iterating (see a friend in class for the exact notes on this if you missed them!)
We also introduced a new way of visualizing the behavior of functions: the cobweb diagram. In this style of graph, we draw the function we are iterating on the same plot as the line y=x. We pick a seed and move vertically to the graph of y=f(x), then horizontally to the line y=x, resulting in the output we just got becoming the input for the next iteration. We then move vertically again to y=f(x) and horizontally again to y=x, continuing until we get a view of the behavior of the function.
Your homework tonight then is to finish question 1 (parts f and g) and to do parts a, b, and c of question 3 of the Iterated Functions sheet. Draw both the time diagram and cobweb diagram for each of these functions, and try to make some observations about how the patterns of the two representations align with each other.
We’ve started down the road towards understanding the mathematics of chaos by iterating linear functions. Imagine a recursively-defined sequence, where each term of the sequence is defined based on the previous value. That’s what we’re doing here: we start with a seed, then plug that seed into a formula. Each output becomes the next step’s input, and we seek to understand the long-term behavior of formulas.
We are primarily working with this classwork sheet: Iterated Functions. We’ve done parts a, b, and c for part 1, looking at the behavior of seeds both numerically and graphically, creating a plot of steps vs. value called a Time Diagram. In all three examples, we found a fixed point: a value of a seed that is constant through the formula (plugging that value in gets that same value out). Some of the fixed points were attractors, and some were repellers. We discussed how iterating the function repeatedly is a fine way to find an attracting fixed point, but obviously won’t work to find a repeller. The only option there is to solve the equation f(x) = x.
Tonight for homework, please do parts d and e of question 1, then all of question 2. You should work on identifying and classifying the fixed point (as an attractor or repeller).