Polynomial Functions and End Behavior

Polynomial Functions

$f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

Rates of Change

Average Rate of Change (AROC)

$\text{AROC} = \frac{f(b) - f(a)}{b - a}$

Concavity from AROC

• If AROC is increasing → concave up
• If AROC is decreasing → concave down

Zeros and Multiplicity

If $(x - c)^k$ is a factor of $f(x)$:

• Odd $k$: graph crosses x-axis at $x = c$
• Even $k$: graph bounces at $x = c$
• Higher $k$ → flatter near the zero

End Behavior (Limit Notation)

$\lim_{x \to \infty} f(x) \quad \text{and} \quad \lim_{x \to -\infty} f(x)$

Determined by the leading term $a_nx^n$:

| $n$ | $a_n > 0$ | $a_n < 0$ | |-----|-----------|-----------| | Even | $+\infty, +\infty$ | $-\infty, -\infty$ | | Odd | $-\infty, +\infty$ | $+\infty, -\infty$ |

Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a, b]$ and $d$ is between $f(a)$ and $f(b)$, then there exists $c \in (a, b)$ with $f(c) = d$.

Factoring Techniques

• Rational Root Theorem: Possible rational roots are $\pm\frac{p}{q}$
• Descartes' Rule of Signs: Count sign changes for positive/negative real zeros
• Polynomial Long Division and Synthetic Division

AP Precalculus Tip: The College Board emphasizes limit notation for end behavior. Always write: $\lim_{x \to \infty} f(x) = +\infty$