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.

Polynomial End Behavior Inputs
Enter a polynomial expression or use degree and leading coefficient directly.

Supported format: x, x^n, constants, plus/minus. Example: 4x^6 - x + 9

Quick examples

Result
End behavior depends on parity of degree and sign of leading coefficient.

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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