Even degree:

• $a_n > 0$: rises both sides ↗↗
• $a_n < 0$: falls both sides ↘↘

Zeros and Multiplicity

Zero: value where $f(x) = 0$ (x-intercept)

Multiplicity: how many times the factor appears

Even multiplicity: graph touches x-axis and bounces Odd multiplicity: graph crosses x-axis

Example: $f(x) = (x - 2)^2(x + 1)$

• Zero at $x = 2$ (multiplicity 2, bounces)
• Zero at $x = -1$ (multiplicity 1, crosses)

Turning Points

A polynomial of degree $n$ has at most $n - 1$ turning points.

Y-Intercept

Evaluate $f(0)$ to find where graph crosses y-axis.

Key Features

1. Degree determines end behavior
2. Leading coefficient affects direction
3. Zeros show x-intercepts
4. Multiplicity affects crossing behavior

• Zero at $x = 2$, multiplicity 2 (even, bounces)

From $(x + 1)$:

• Zero at $x = -1$, multiplicity 1 (odd, crosses)

Answer:

• $x = 0$ (mult. 3, crosses)
• $x = 2$ (mult. 2, bounces)
• $x = -1$ (mult. 1, crosses)

Answer: Maximum of 5 turning points