Section+3.1+for+the+last+time

=Finally We're done talking about Section 3.1=

Today we wrapped up our discussion of Section 3.1. The discussion of this section has been so long because I (Mrs. Tyson) was sick last week and last week was the end of the quarter. When I returned we had to spend a few days on Administrivia to finish out the quarter.

Each time we start a new investigation, there are two somewhat competing things that are going on.
 * Some problems serve as REMINDERS of what we learned about the subject in the past.
 * Some problems serve as TEASERS for the new material that we are going to be learning.

Figuring out which is which can be hard... especially if your own memory isn't as good as I want. Which means these new investigation days will sometimes be frustrating.

REMINDERS about Polynomials
We started today with a 10 minute discussion of what we remembered about polynomials from last year. The class was able to generate a pretty solid list of items which matched Mr. Bild's list of skills and theory from last year. Careful use of vocabulary was an issue.


 * ====The degree (the largest exponent) of the polynomial is the same as the number of zeros of the polynomial when we allow for complex zeros and we count zeros with multiplicity more than once.====
 * ====To find the zeros/roots/x-intercepts of a polynomial function, you factor the polynomial.====
 * ====If r is a zero, then (x-r) is a factor.====
 * ====The degree of a factor is the same as the multiplicity of the root that corresponds to that factor.====
 * ====End behavior of a polynomial is determined by the degree (the largest exponent) and the Leading Coefficient (coefficient on the term with the largest exponent -- often the first term but not necessarily). ***come back and add sample graphs here ***====
 * ====A polynomial with even degree and positive leading coefficient "smiles"====
 * ====A polynomial with even degree and a negative leading coefficient "frowns"====
 * ====A polynomial with odd degree and positive leading coefficient====
 * ====increases as x goes to infinity (math geek talk) or goes up on the right.====
 * ====decreases as x goes to negative infinity (math geek talk) or goes down to the left.====
 * ====A polynomial with odd degree and negative leading coefficient====
 * ====decreases as x goes to infinity or goes down on the right====
 * ====increases as x goes to negative infinity or goes up on the left.====
 * ====A degree n polynomial has at most n-1 turns (max/min) in the graph.====

That's a lot of stuff that we remembered. Mrs. Tyson is going to be assuming this knowledge going forward.

We took a quick look at #7, 5, 1, 10 in that order.

Question #7:


 * Reminded us how to look at a graph of a polynomial with clearly labelled zeros and work backwards to find the equation of the polynomial. We have to pay attention to end behavior to find the sign of the Leading Coefficient. We have to pay attention to the zeros to find the factors. We need one more piece of information to be sure we are correct -- in this case we used the y-intercept.

Question #5:


 * Looked at #7 from the other direction. We were given the equation of a polynomial in factored form and asked to graph the polynomial. Plotting the zeros and getting the end behavior was easy. However, we quickly realized that we didn't know how to figure out where the polynomial turned around -- which seemed like something that I would want to know for my graph. Mrs. Tyson reminded us that we could use the 2nd Calc menus on the TI's, but that really finding the turning points was one of the main purposes of calculus and not a point of emphasis in this class.
 * If we are asked to sketch a polynomial by hand (no calc or CAS) then there is no expectation that we are able to accurately find the turning points. Our goal is to get the zeros, end behavior, and y-intercept done correctly.


 * Demonstrated why the function touches the axis and turns around at a root of multiplicity 2 by looking at the graph of a sequence of polynomials where 2 roots are getting closer and closer together. ***again I might want to put the graphs in here***


 * We also used question 5 to talk about challenging ourselves to extend the questions. If this question shows us what the graph looks like at a root of multiplicity 2, maybe we should think about what the graph looks like at a root of multiplicity 3 or 4...

Question #1 and #10


 * Designed to remind us about the general appearance of a polynomial and to think about things like zeros, turns, end behavior, roots with multiplicity, etc.

TEASERS for learning more about Polynomials
IMPORTANT -- the book assumes we have access to a CAS. Mrs. Tyson demonstrated using Wolframalpha today in class. We can use any CAS that we want, but we also need to know how to do some things without a CAS.

Question #4


 * Gave 5 forms for the equation of the same polynomial. We could expand them using the CAS to see that they were all the SAME polynomial
 * WHY would we want lots of forms of a polynomial
 * Form 1 (Factored Form) made it easy to read off the zeros
 * Form 2 (Taylor Expansion at x = 3) --
 * Form 3 (Standard Form) -- easy to read y-int
 * Form 4 (Taylor Expansion at x = -3) -- book says easy to evaluate at x = -3, but when we asked wolfram alpha to do the Taylor expansion we also saw that it was somehow tied up with the tangent line to the polynomial at x = - 3.
 * Form 5 (????) -- shows the end behavior clearly
 * So the important teasers here
 * There's some new form called the Taylor expansion... and it's got something to do with tangent lines
 * There's some other new form that we didn't get a name for that has something to do with end behavior

Question #6

We didn't really have time to get into this one, and Mrs. Tyson says we'll look at something similar tomorrow. There was some wacky looking notation here, so hopefully it will be explained well tomorrow.