Polynomial End Behavior

Polynomial p(x)

Enter your polynomial

Use ^ for exponents or Unicode superscripts: x^4 or x⁴. Supports + − * and spaces.

Examples — all 4 cases

End Behavior Analysis

Enter a polynomial and press Analyze End Behavior to see how the function behaves at the extremes.

Leading Term

—

Degree (even / odd)

—

Leading coeff. (pos / neg)

As x → +∞:

y → ?

As x → −∞:

y → ?

All 4 Cases — matching case highlighted

Even + Positive

↑ Both ends up
x→±∞: y→+∞

Even + Negative

↓ Both ends down
x→±∞: y→−∞

Odd + Positive

↓ left, ↑ right
−∞:y→−∞, +∞:y→+∞

Odd + Negative

↑ left, ↓ right
−∞:y→+∞, +∞:y→−∞

The Four End Behavior Patterns

Only the leading term (highest-degree term) determines end behavior. All other terms become negligible as x grows large in magnitude.

Degree	Lead coeff.	x → −∞	x → +∞
Even	Positive (a > 0)	y → +∞ ↑	y → +∞ ↑
Even	Negative (a < 0)	y → −∞ ↓	y → −∞ ↓
Odd	Positive (a > 0)	y → −∞ ↓	y → +∞ ↑
Odd	Negative (a < 0)	y → +∞ ↑	y → −∞ ↓

For even degree: both ends go the same direction. For odd degree: ends go opposite directions. The sign of the leading coefficient flips the pattern.

The "Highway" Analogy

Think of end behavior as the "highway" of the function — where does it go at the far left and far right of its domain?

No matter how many twists and turns a polynomial makes in the middle, the ends of its graph always follow a predictable pattern set by just two things: degree and leading coefficient sign.

Imagine driving on a highway that stretches infinitely in both directions. Your starting and ending destinations (up or down on the graph) are determined only by those two facts about the leading term — everything else is just scenery along the way.

Even degree → both ends are "symmetric" (same destination)
Odd degree → ends go in opposite directions
Positive leading coeff. → right end goes up (+∞)
Negative leading coeff. → right end goes down (−∞)

Polynomials still feel unpredictable?