Chapter 2.2

Identifying Polynomial Graphs:

-First things first- the graph of a polynomial function is continuous, meaning it has no breaks, holes, or gaps. Also, it only has soft, round turns- no sharp, pointed turns. By sticking to this defintion, you should be able to distinguish a polynomial graph!

-Polynomials can have up to many degrees, but here are some of the basics so you can get a feel for them:







0 degree



1 degree



2 degree





3 degree



4 degree



5 degree

-As you may notice, every odd polynomial has at least one x-intercept, and the degree number is the maximum number of x-intercepts that polynomial can have.
-The maximum number of Relative Extrema for each polynomial is the degree number minus one (n-1)

-Now we will discuss a little bit about end behavior. When you have an even function, the ends of the graph will be doing the same thing, and when you have an odd function, the ends of a graph will be doing opposite things. But what are the ends of the graph defind as?
        -Left end: x approaches negative infinty
        -Right end: x approaches postive infinity


Now we have gone over the basic things of polynomial functions, let's move on to:

LIMIT NOTATION:

  This is read: "The limit of f(x) as x approaches (negative/positive) infinity"
The infinity symbols on the picture can be replaced with either negative or positive infinity depending on your graph.

So, how do we use this?
Well, let's take a polynomial F(x)= -73x^7+2x^3+5
 You can take a few routes:
       -Plug in a number as close to infity as you can, and see what happens when that number is   negative and positive to see what happens as x approaches negative or postivie infinity.

       -When the leading coefficient (the one with the highest degree) is so much bigger than the other numbers that the other numbers don't take much of an effect, you can focus on the leading coefficient.

                   -If the leading coefficient is negative, you know that: as the limit of f(x) as x approaches infinity, the graph decreases, or f(x) goes to negative infinity. Knowing then that the graph is odd, the left end (as x approaches negative infinity), f(x) will be increasing towards infinity. If the graph was even, the left and right end would be doing the same thing.

So that's basically it- identifying polynomials and being able to use limit notation. If more help is needed, feel free to visit the video:

http://www.youtube.com/watch?v=ar24CNFZfOo

 And one things the book mentions:
Intermediate Value Theorm:
Let a and b be real numbers such that a<b If f is a polynomial function such that f(a) does not equal f(b), then in the interval [a,b], f takes on every value between f(a) and f(b).
This is a way to help locate real zeros of a polynomial function!



-Maggie