Polynomials & Functions
End Behavior of Polynomials
End behavior describes what happens to a polynomial as x → +∞ and x → −∞. It depends on the leading coefficient and the degree (even or odd).
Watch & Learn
Watch a clear, friendly video explanation of End Behavior of Polynomials:
Watch on YouTube →Opens a YouTube search for the best tutorial videos on this topic.
Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
Describe end behavior of y = 3x⁴ − 2x + 1
Step-by-Step Solution
Leading term: 3x⁴. Degree = 4 (even). Leading coefficient = 3 (positive).
Even degree + positive leading coefficient: both ends go UP.
As x → +∞, y → +∞. As x → −∞, y → +∞.
Answer
Both ends rise (up on left, up on right)
Problem
Describe end behavior of y = −2x³ + x
Step-by-Step Solution
Leading term: −2x³. Degree = 3 (odd). Leading coefficient = −2 (negative).
Odd degree + negative: left end goes UP, right end goes DOWN.
As x → −∞, y → +∞. As x → +∞, y → −∞.
Answer
Left end rises, right end falls
Problem
Describe end behavior of y = x³ − 5x
Step-by-Step Solution
Leading term: x³. Degree = 3 (odd). Leading coefficient = 1 (positive).
Odd degree + positive: left end goes DOWN, right end goes UP.
Answer
Left end falls, right end rises
Problem
Describe end behavior of y = −x² + 4
Step-by-Step Solution
Leading term: −x². Degree = 2 (even). Leading coefficient = −1 (negative).
Even degree + negative: both ends go DOWN.
As x → ±∞, y → −∞.
Answer
Both ends fall
Problem
Describe end behavior of y = 5x⁵ − 3x²
Step-by-Step Solution
Leading term: 5x⁵. Degree = 5 (odd). Leading coefficient = 5 (positive).
Odd degree + positive: left end falls, right end rises.
Answer
Left end falls, right end rises
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
End behavior of y = 2x²?
End behavior of y = −x⁴ + 3x?
End behavior of y = x⁵?
End behavior of y = −3x³ + 2x²?
End behavior of y = 4x⁶ − x?
End behavior of y = −x²?
End behavior of y = x³?
End behavior of y = −2x⁵ + x?
What determines end behavior?
End behavior of y = 7x⁴ − 3x² + 1?
End behavior of y = −x³ + 100?
End behavior of y = 6x² + 5x − 2?
Finished all 12? Give yourself a pat on the back — then check your work!
Keep going — you're on a roll!
Every topic you master is another step on your journey.
Explore More Topics