Leading Coefficient Test

The graph of the polynomial function f( x )= a n x n + a n1 x n1 +...+ a 1 x+ a 0 eventually rises or falls depends on the leading coefficient ( a n ) and the degree of the polynomial function.

Case
End Behavior of graph
When n is odd and a n is positive Graph falls to the left and rises to the right
When n is odd and a n is negative Graph rises to the left and falls to the right
When n is even and a n is positive Graph rises to the left and right
When n is even and a n is negative Graph falls to the left and right

Example:

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function f( x )= x 3 +5x .

Solution:

Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right as shown in the figure.