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!

]]>**You must complete**Characteristics of quadratic functions (from the Algebra 1 tab) to a score of at least an 80.**You must complete ANY TWO OF**- Identify the direction a parabola opens (to 90+)
- Find the vertex of a parabola (to 90+)
- Find the axis of symmetry of a parabola (to 90+)

Use your graphing calculator or Desmos.com/calculator to make graphs. If you exceed the minimum requirements of this assignment, your work will be recorded as extra credit.

]]>On Monday, we will have our **Unit 3** **Test**. We will spend Friday reviewing, and in preparation for that you should take a look at the review problems for Unit 3: pages 331-336, exercises 1, 3, 7, 9, 11, 15, 19, 20, 23, 29, 32, 40, 41. You’ll get answers and supplemental review tomorrow.

See yesterday’s post for the link to the debrief from the Polygraph activity

Today was Day 3 of the Desmos-based activities we’ve been using in class to re-introduce ourselves to parabolas. In class yesterday, many of you had some difficulty in properly describing parabolas or in identifying features of parabolas to ask about in order to guess which one your partner had chosen. This shows us that there is a **need for a shared vocabulary** when describing these entities. Today’s lesson reviewed that vocabulary.

- Notes on Parabola Vocabulary
- HW 3.3 – Parabola Vocabulary Practice
- Lesson Video (reviewing most of the key vocabulary discussed in the notes above)

- Characteristics of quadratic functions (from the Algebra 1 section)
- Identify the direction a parabola opens
- Find the vertex of a parabola
- Find the axis of symmetry of a parabola

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Your next Personal Progress Check has been assigned. Log into AP Classroom and complete the Unit 3 MCQ Part B PPC by **Monday, November 25**

On Monday, we will have our **Unit 3** **Test**. We will spend Friday reviewing, and in preparation for that you should take a look at the review problems for Unit 3: pages 331-336, exercises 1, 3, 7, 9, 11, 15, 19, 20, 23, 29, 32, 40, 41. You’ll get answers and supplemental review on Friday.

Tomorrow, **Thursday the 21st**, you’ll have an **in class investigative task** for Chapter 12, where you will be asked to design an experiment for a scenario. Unlike other tasks, this will be **closed notes/closed book**, and your hand-written response will be due at the end of the period.

Then we pushed the function a little further, from x^2 – 1 to x^2 – 2. Still with two repellers, this time the system of iterations dissolved into complete **chaos**, careening randomly within the interval -2 to 2, never settling into one spot. Changing the value of the seed by only one ten-thousandth produced a completely different pattern of iterations, demonstrating the *sensitivity to initial conditions* that are characteristic of chaos.

We will see plenty more examples of weird behavior in our iterations in the days to come. For tomorrow, be sure to read the article posted yesterday.

]]>We continued our re-introduction to Parabolas and Quadratic Equations today by playing a game similar to the classic game Guess Who, where you had to ask a series of questions to guess which of 25 parabolas your randomly matched partner selected. This taught us the value of having clearly defined vocabulary to describe the key features of such shapes, as several questions that you asked each other led to confusion!

For tonight’s homework, please complete the Parabola Polygraph Debrief reflection. You will also need to bring in your Chromebook once more tomorrow for the third and final introductory activity to this unit.

]]>For some additional reading about the ongoing Replication Crisis in science, see this article from The Atlantic, or this Crash Course video. Check out this New York Times article for more about Brain Wansink, the Cornell food science researcher who resigned amid this scandal.

]]>- If |A| < 1, the fixed point is
*attracting* - If |A| > 1, the fixed point is
*repelling* - If A > 0, the pattern is
*direct* - If A < 0, the pattern is
*alternating* - If A = 1, there is no fixed point
- If A = -1, the fixed point is
*neutral* - If A = 0, the fixed point is “super-attracting”

We also observed that as |A| gets nearer to 1, the attracting/repelling behavior becomes slower.

We looked at another hypothetical non-linear system, then set to work on exploring more of #4 from the IF sheet, using the Iterated Functions Supplement as a guide. Your homework is to finish up through #4h (by tomorrow), and read Chaos and Fractals in Human Physiology, from the February, 1990 edition of Scientific American

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