End Behavior
A free Precalculus lesson from the “Polynomial Functions” unit, with a worked example and practice problems including step-by-step solutions.
Even-degree polynomials have ends in the same direction; odd-degree polynomials have ends in opposite directions. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Predict end behavior from degree and leading coefficient
- Use end behavior in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. An even-degree polynomial with a positive leading coefficient has end behavior:
- Even degree means the ends go the same direction.
- Positive leading coefficient makes the right end rise.
- Both ends rise.
Answer: both ends up
Practice problems
1. An even-degree polynomial with a positive leading coefficient has end behavior:
Choices: both ends up · both ends down · left down and right up · left up and right down
Show solution
- Even degree means the ends go the same direction.
- Positive leading coefficient makes the right end rise.
- Both ends rise.
Answer: both ends up
2. An even-degree polynomial with a positive leading coefficient has end behavior: (variation 2)
Choices: both ends up · both ends down · left down and right up · left up and right down
Show solution
- Even degree means the ends go the same direction.
- Positive leading coefficient makes the right end rise.
- Both ends rise.
Answer: both ends up
3. An odd-degree polynomial with a negative leading coefficient has end behavior:
Choices: left up and right down · left down and right up · both ends up · both ends down
Show solution
- Odd degree means opposite directions.
- Negative leading coefficient makes the right end fall.
- So the left end rises and the right end falls.
Answer: left up and right down
4. An odd-degree polynomial with a negative leading coefficient has end behavior: (variation 2)
Choices: left up and right down · left down and right up · both ends up · both ends down
Show solution
- Odd degree means opposite directions.
- Negative leading coefficient makes the right end fall.
- So the left end rises and the right end falls.
Answer: left up and right down
5. Which feature controls polynomial end behavior most directly?
Choices: degree and leading coefficient · constant term only · middle coefficient only · number of terms only
Show solution
- The leading term dominates for large |x|.
- Its degree controls same/opposite directions.
- Its coefficient controls up/down orientation.
Answer: degree and leading coefficient
Practice this interactively with instant feedback and an AI tutor.
Practice End Behavior Take the free placement check