
Left and Right Behavior
We will investigate the outer shape of several polynomials and explore the
following rules:

Even 
Odd 
Pos 
UU 
DU 
Neg 
DD 
UD 
Where Even and Odd refers to the degree of the polynomial, Pos and Neg refers
to the leading coefficient, And a U or a D refers to the left and right behavior
of the curve.
Example
3x^{7} + 4x^{4}  1
has degree 7 which is odd and has leading coefficient 3 which is negative.
Hence the left and right behavior is UD, i.e. the curve goes up on
the left and down on the right.

Max and Min
Theorem
If f(x) is a polynomial of degree n then f(x) has at most
n  1 relative extrema. Where relative extrema are lumps of the graph.

Example:
4x^{5} + 2x^{3}  x^{2} + 7x + 12
Has at most 4 relative extrema.

Three Step Procedure For Graphing Polynomials
Step 1: Factor the polynomial into linear factors of the form
ax + b
Step 2 : Determine the left and right behavior of the graph and the
shape of the graph near each x intercept.
Step 3: Connect the dots.
Example:
Graph
y = x^{4}  10x^{2} + 9

We have
y = (x^{2} 9)(x^{2}  1) = (x
 3)(x + 3)(x  1)(x + 1)

The left behavior is up and the right behavior is up.
Near x = 3 the graph is positive on the left and negative on the right.
Near x = 1 the graph is negative on the left and positive on the right
Near x = 1 the graph is positive on the left and negative on the
right.
Near x = 3 the graph is negative on the left and positive on the right
