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Leading Coefficient Test

The graph of the polynomial function $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + ... + a_{1} x + a_{0}$ 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 $f (x) = - x^{3} + 5 x$ .

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.

Math diagram

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