Assume that P ( x ) is a polynomial  along with degree n.  Thus we know that the polynomial have to look like,

                                                    P ( x ) =axⁿ..............

We don't know if there are any other terms in the polynomial; however we know that the first term will need to be the one listed as it has degree n. Now we have the following facts regarding the graph of P ( x ) at the ends of the graph.

1.   If a> 0 and the value of n is even then the graph of P ( x )will increase without restricts positively at both endpoints.  A good instance of this is the graph of x².

2.   If a= 0 and the value of n is odd then the graph of P ( x ) will increase without bound positively at the right end and decrease without bound at the left end.  A good instance of this is the graph of x³.

3.   If a= 0 and the value of n is even then the graph of P (x ) will decrease without  any bound positively at both of the endpoints.  A good instance of this is the graph of -x².

4.   If a= 0 and the value of n is odd then the graph of P ( x ) will decrease without  any bound positively at the right end and increase without  any bound at the left end.  A good instance of this is the graph of -x³.

Okay, now that we've obtained all that out of the way finally we can give a procedure for getting a rough sketch of the graph of a polynomial.