There are couple ways to go about this. If the polynomial is odd the graph with be symmetric to the origin, and if the polynomial even the graph with be symmetric to the y-axis. An odd degree polynomial end points with go in opposite directions, even degree polynomial end points will go in the same direction. A positive even degree poly will open up(smile), a negative even degree poly will open down (frowny face). I can go on if needed, I hope this helps :)
