Michigan Algebra I Sept. 2012

There are patterns in the graphs of the previous examples. If the degree is odd, the polynomial will have at least one real root and up to as many as the degree of the polynomial. For example, if the degree of the polynomial is five, there can be as few as one or as many as five real roots or x-intercepts.

The leading coefficient determines whether the graph is going from lower left to upper right or from upper left to lower right on the graph. When the polynomials have an even degree, say of degree n, the graph has anywhere from 0 to n real roots. Here the leading coefficient determines whether the graph points up (if it is positive) or down (if it is negative). So by plotting the real roots and using these two "rules", we have an idea of what the graph of the polynomial looks like.

Graphs of polynomials have no holes or breaks in the graph and no sharp corners, are always nice smooth curves. The "humps" where the graphs change direction from increasing to decreasing or decreasing to increasing are often called turning points. If the polynomial has degree n then there will be at most n - 1 turning points in the graph. Therefore, a fourth degree polynomial will have no more than 3 turning points.