Polynomial End Behavior

End behavior: what happens as $x \to \pm\infty$

The end behavior of a polynomial depends on just two things: the degree (even or odd) and the sign of the leading coefficient (positive or negative).

The four cases

• Even degree, positive leading coefficient (e.g., $x^2$, $x^4$): Both ends go up ↑↑
• Even degree, negative leading coefficient (e.g., $-x^2$): Both ends go down ↓↓
• Odd degree, positive leading coefficient (e.g., $x^3$): Left goes down, right goes up ↓↑
• Odd degree, negative leading coefficient (e.g., $-x^3$): Left goes up, right goes down ↑↓

The quick rule

The right end always follows the sign of the leading coefficient (positive → up, negative → down). The left end is the same for even degree, opposite for odd degree.

Why only the leading term matters

For very large $|x|$, the highest-degree term dominates. $x^4 - 100x^3 + 5000x^2$: at $x = 1000$, the $x^4$ term is $10^{12}$, swamping the other terms. So end behavior = leading term's behavior.

Try it in the visualization

Toggle degree (odd/even) and sign (+/−) to see all 4 end behavior cases. The arrows show which direction each end goes. Add lower-degree terms and see that they don't change the end behavior — only the leading coefficient sign** (+/−). Even degree: both ends go same direction. Odd degree: ends go opposite directions. Positive leading coefficient: right end goes up. Negative: right end goes down.