The "humps" where the graph varies direction from increasing to decreasing or decreasing to increasing is frequently called turning points.

 If we know that the polynomial contains degree n then we will know that there will be at most n -1 turning points in the graph.

Whereas this won't help much with the actual graphing procedure it will be a nice check.  If we contain a fourth degree polynomial with five turning points then we will know that we've done something incorrect as a fourth degree polynomial will contain no more than 3 turning points.

Next, we have to explore the relationship among the x-intercepts of a graph of a polynomial and the zeroes of the polynomial.  Remember again that to determine the x-intercepts of a function we have to solve the equation

Also, remember again that x = r is a zero of the polynomial, P ( x ) , provided P ( r ) = 0 .  However this means that x = r is also a solution to P ( x ) = 0 .

In other terms, the zeroes of polynomial are also the x-intercepts of the graph. Also, remember again that x-intercepts can either cross the x-axis or they can only touch the x-axis without in fact crossing the axis.

Notice as well through the graphs above that the x-intercepts can either flatten as they cross the x-axis or they can go by the x-axis at an angle.