To determine the end behavior of a polynomial, the key features to identify are the Degree of the polynomial and the sign of the leading coefficient. The leading coefficient is the coefficient in front of the input variable with the largest exponent, typically located at the front of the equation, but not always. If the polynomial is factored, the leading coefficient is usually the value outside the factors.
| Positive Leading Coefficient | Negative Leading Coefficient | |
|---|---|---|
| Even Degree | ||
| Odd Degree |
Each of the pictures above is valid only for the end behavior! Anything can happen near the vertical axis and the functions shown are simple representations of polynomial functions as placeholders for the more general even and odd polynomials.
For each of the images above, there is a corresponding algebraic way we represent the end behavior:
| Positive Leading Coefficient | Negative Leading Coefficient | |
|---|---|---|
| Even Degree | \[ \solve{ y\rightarrow \infty &\text{{as}}& x\rightarrow -\infty\\ y\rightarrow \infty &\text{{as}}& x\rightarrow \infty } \] | \[ \solve{ y\rightarrow -\infty &\text{{as}}& x\rightarrow -\infty\\ y\rightarrow -\infty &\text{{as}}& x\rightarrow \infty } \] |
| Odd Degree | \[ \solve{ y\rightarrow -\infty &\text{{as}}& x\rightarrow -\infty\\ y\rightarrow \infty &\text{{as}}& x\rightarrow \infty } \] | \[ \solve{ y\rightarrow \infty &\text{{as}}& x\rightarrow -\infty\\ y\rightarrow -\infty &\text{{as}}& x\rightarrow \infty } \] |
And a corresponding verbal description:
| Positive Leading Coefficient | Negative Leading Coefficient | |
|---|---|---|
| Even Degree | The function goes up to the left and up to the right. | The function goes down to the left and down to the right. |
| Odd Degree | The function goes down to the left and up to the right. | The function goes up to the left and down to the right. |