Definition: End behavior describes what happens to the values of a function as x grows very large in the positive direction and in the negative direction.
What is end behavior?
It tells you where the graph is heading at the far left and far right. We write it with limits or arrows, such as “as x goes to infinity, f(x) goes to negative infinity.”
How do you read it from a polynomial?
Only the leading term matters for large |x|. Look at its degree and sign.
- Even degree, positive coefficient: both ends rise.
- Even degree, negative coefficient: both ends fall.
- Odd degree, positive coefficient: left falls, right rises.
- Odd degree, negative coefficient: left rises, right falls.
Worked example: f(x) = -4x5 + 9x3 – 7. The leading term is -4x5, odd degree with a negative coefficient. So as x goes to infinity, f goes to negative infinity, and as x goes to negative infinity, f goes to positive infinity.
What goes wrong?
Students look at the constant term or the first term as written instead of the highest power. Mathbot says: rewrite in descending order first.
Where to learn it
- Polynomial Functions: Zeros, Ends, and Graphs – lesson
- Polynomial and Rational Functions: Zeros and Asymptotes – lesson