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. For p(x) = 3x^4 + 2, describe the end behavior.
- The degree 4 is even, so both ends point the same direction.
- The leading coefficient 3 is positive, so the right end rises.
- Same direction and right end up means both ends go up.
Answer: both ends up
Practice problems
1. For p(x) = 3x^4 + 2, describe the end behavior.
Choices: both ends up · both ends down · left end down, right end up · left end up, right end down
Show solution
- The degree 4 is even, so both ends point the same direction.
- The leading coefficient 3 is positive, so the right end rises.
- Same direction and right end up means both ends go up.
Answer: both ends up
2. Which phrase describes the ends of p(x) = -4x^6 + 3?
Choices: both ends down · both ends up · left end down, right end up · left end up, right end down
Show solution
- The degree 6 is even, so both ends point the same direction.
- The leading coefficient -4 is negative, so the right end falls.
- Same direction and right end down means both ends go down.
Answer: both ends down
3. For p(x) = 5x^3 - 4, describe the end behavior.
Choices: left end down, right end up · left end up, right end down · both ends up · both ends down
Show solution
- The degree 3 is odd, so the ends point in opposite directions.
- The leading coefficient 5 is positive, so the right end rises.
- Opposite directions with right end up means the left end goes down.
Answer: left end down, right end up
4. The graph of p(x) = -6x^5 + 1 has which end behavior?
Choices: left end up, right end down · left end down, right end up · both ends up · both ends down
Show solution
- The degree 5 is odd, so the ends point in opposite directions.
- The leading coefficient -6 is negative, so the right end falls.
- Opposite directions with right end down means the left end goes up.
Answer: left end up, right end down
5. Which two features of a polynomial determine its end behavior?
Choices: the degree and the leading coefficient · the constant term and the number of terms · the y-intercept and the middle coefficient · the number of zeros and the constant term
Show solution
- For large |x| the leading term dominates every other term.
- The degree (even or odd) sets whether the ends match or oppose.
- The leading coefficient's sign sets which way the right end points.
Answer: the degree and the leading coefficient
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