PhD a.k.a. A Higher Degree

The graph of a polynomial has no breaks, holes, or gaps.

Pretending like i didnt use paint and my lines were actually acurate, we can see what i mean.  The first graph is good cause it has no breaks, the second is not a polynomial because it has breaks.  The third is "theoretically" good cause it has no sharp turns unlike the 4th, which has many sharp turns.

THE LEADING COEFFICIENT

Now, we examine End Behavior.  End behavior is where the right and left ends of a polynomial graph points too.  In the first of the two graphs above, the two ends point in different directions, this lets us know that the leading coefficient is negative.  In the second graph, the two ends are pointed in the same direction, thus, the leading coefficient is positive.

The notation we use to show the first graph's end behavior is:

for the left end and

               Those last 6 lines took me just about 45 minutes

Now we can apply the things we have learned to some equations.

Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right. Like this:

It can be shown that for a polynomial function f of degree n, the following statements are true.

1.  The graph of f has at most n real zeros.

2.  The function f has at most n-1 relative extrema (relative minimums or maximums).