The end behavior of a function describes how the function behaves as the input variable xxx approaches very large positive or negative values (i.e., as x→+∞x\to +\infty x→+∞ or x→−∞x\to -\infty x→−∞). It shows the trend of the graph at the far left and far right ends along the xxx-axis
. Specifically, end behavior tells us whether the function values f(x)f(x)f(x) increase or decrease without bound, or approach some finite value, as xxx moves toward infinity or negative infinity. For example, a function might grow larger and larger (approach +∞+\infty +∞) or decrease without bound (approach −∞-\infty −∞) as x→+∞x\to +\infty x→+∞
. In polynomial functions, the end behavior is determined primarily by the leading term—the term with the highest exponent—and its coefficient. The degree (even or odd) and the sign of the leading coefficient dictate the direction of the graph’s ends:
- If the degree is even and the leading coefficient is positive, both ends of the graph rise to +∞+\infty +∞.
- If the degree is even and the leading coefficient is negative, both ends fall to −∞-\infty −∞.
- If the degree is odd and the leading coefficient is positive, the left end falls to −∞-\infty −∞ and the right end rises to +∞+\infty +∞.
- If the degree is odd and the leading coefficient is negative, the left end rises to +∞+\infty +∞ and the right end falls to −∞-\infty −∞
Thus, end behavior provides a concise way to understand the "long-term" trend of a function’s graph as you look far to the left or right along the xxx-axis
