Polynomial end behavior can be determined by the features of the term with the highest power. Look at this term carefully to provide this information.
- Determine whether the power is odd or even
If the largest term in the polynomial has an even power, it starts and ends with very large numbers. Consider f(x) = x², f(-100) = 10,000 and f(75) = 5625, so both the negative and positive values of x yield end behavior on the same side of the x-axis. However, if the largest term is odd, one end of the graph starts on one side of the x-axis, the function's graph crosses the x-axis on one or more occasions, and the other end leaves on the other side.
- Use the sign to determine the direction
Note that f(x) = x² yields large positive values regardless of the sign for x. Looking at the same values of x for g(x) = -x², the values for g are the exact opposite of the values for f. The sign of the highest power term determines whether the end behavior is positive on both ends or negative on both ends. For odd-powered largest terms, a positive sign gives large positive values of the function for large positive values of x; thus, the function starts on the negative side of the x-axis and ends on the positive side. The opposite is true if the largest term is odd-powered and negative.
- Do not be misled by larger secondary terms.
Even if there are second terms in the polynomial with large coefficients (for instance, x²-500,000x), this does not change the end behavior. It simply indicates that this behavior is not going to present itself until larger values are plotted.