End Behavior Calculator
Find polynomial end behavior instantly from a full expression or just degree and leading coefficient. Get clear left-end and right-end results with step-by-step logic.
Supported format: x, x^n, constants, plus/minus. Example: 4x^6 - x + 9
Quick examples
Enter your polynomial and click Calculate End Behavior.
End Behavior Rule Table
Polynomial end behavior is based on the leading term ax^n. You only need parity of n and sign of a.
Even degree, positive leading coefficient
x -inf -> f(x) +inf | x +inf -> f(x) +inf
Even degree, negative leading coefficient
x -inf -> f(x) -inf | x +inf -> f(x) -inf
Odd degree, positive leading coefficient
x -inf -> f(x) -inf | x +inf -> f(x) +inf
Odd degree, negative leading coefficient
x -inf -> f(x) +inf | x +inf -> f(x) -inf
Worked Examples
f(x) = 2x^4 - x + 3
Leading term 2x^4: even degree + positive coefficient, so both ends go up.
f(x) = -3x^5 + 2x^2 - 1
Leading term -3x^5: odd degree + negative coefficient, so left end up and right end down.
f(x) = x^7 - 10
Leading term x^7: odd degree + positive coefficient, so left down and right up.
f(x) = -4x^2 + 8x - 5
Leading term -4x^2: even degree + negative coefficient, so both ends go down.
Frequently Asked Questions
How do you find end behavior of a polynomial?
Use the leading term (the term with highest exponent). End behavior is determined by two things: degree parity (even or odd) and leading coefficient sign (positive or negative). Even + positive means both ends go up; even + negative means both ends go down; odd + positive means left down right up; odd + negative means left up right down.
What if the polynomial has many terms?
Only the leading term controls end behavior as x approaches positive or negative infinity. Lower-degree terms become negligible for very large absolute x values.
Can I use degree and leading coefficient only?
Yes. If you already know degree n and leading coefficient a, you can get end behavior directly without entering the full polynomial.
What is an example of end behavior?
For f(x) = -3x^5 + 2x^2 - 1, the degree is odd (5) and the leading coefficient is negative (-3), so as x approaches negative infinity f(x) approaches positive infinity, and as x approaches positive infinity f(x) approaches negative infinity.
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