2.1 Polynomial End Behavior

Notes

A polynomial is a sum of terms $a_n x^n + \dots + a_1 x + a_0$. Its degree is the highest power. For large $|x|$ only the leading term matters.

• Even degree, positive leading coefficient: both ends point up.
• Even degree, negative leading coefficient: both ends point down.
• Odd degree, positive leading coefficient: down on the left, up on the right.
• Odd degree, negative leading coefficient: up on the left, down on the right.
• A polynomial of degree $n$ has at most $n - 1$ turning points.
• Example: $f(x) = -2x^3 + 5x$ behaves like $-2x^3$, so $f(x) \to -\infty$ as $x \to \infty$ and $f(x) \to \infty$ as $x \to -\infty$.