Polynomial End Behavior

Lesson

End behavior describes what a function does at the far left (as $x \to -\infty$) and the far right (as $x \to \infty$) of its graph. For a polynomial, only one term matters at the extremes: the leading term — the term with the highest power of $x$. Everything else is rounding error compared to it once $x$ gets large.

Two things about the leading term decide everything:

• The degree (the exponent on $x$) — even or odd?
• The leading coefficient — positive or negative?

The four cases:

• Even degree, positive coefficient (like $x^2$): both ends go up.
• Even degree, negative coefficient (like $-x^2$): both ends go down.
• Odd degree, positive coefficient (like $x^3$): down on the left, up on the right.
• Odd degree, negative coefficient (like $-x^3$): up on the left, down on the right.