Leading Coefficient Test
Now that we know how zeros affect a polynomial graph, the next step is to figure out the end behavior.
The first term, \(a_n x^n\), of a polynomial provides end behavior information.
| First Term of the Polynomial | End Behavior of Graph |
|---|---|
| When \(n\) is even and \(a_n\) is positive | The graph appears from the top left of the coordinate plane and leaves at the top right. Graph rises to the left and right. |
| When \(n\) is even and \(a_n\) is negative | Ends of the graph appears from the bottom left of the coordinate plane and leaves at bottom right. Graph falls to the left and right. |
| When \(n\) is odd and \(a_n\) is positive | The graph appears from the bottom left of the coordinate plane and leaves at the top right. Graph falls to the left and rises to the right. |
| When \(n\) is odd and \(a_n\) is negative | The graph appears from the top left of the coordinate plane and leaves at the bottom right. Graph falls to the right and rises to the left. |
Go to Practice (Polynomial Characteristics)