Assume that P ( x ) is a polynomial along with degree n. Thus we know that the polynomial have to look like,
P ( x ) =ax^{n}..............
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 x^{2}.
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 x^{3}.
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 -x^{2}.
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 -x^{3}.
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.