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Course Overview

APPCAP Precalculus

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Polynomial and Rational Functions

Polynomial Functions and End Behavior

Connect polynomial degree, leading coefficient, and zeros to graph behavior.

30 min

Understanding Polynomial Functions

Key Features of Polynomials:

A polynomial function p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ has:

Degree (n): Determines the maximum number of turning points (n-1) and zeros (n)

Leading coefficient (aₙ): Determines end behavior

Zeros/Roots: Where p(x) = 0 — the x-intercepts

End Behavior:

Degree	Leading Coeff	As x → -∞	As x → +∞
Even	Positive	f(x) → +∞	f(x) → +∞
Even	Negative	f(x) → -∞	f(x) → -∞
Odd	Positive	f(x) → -∞	f(x) → +∞
Odd	Negative	f(x) → +∞	f(x) → -∞

Multiplicity of Zeros:

• Odd multiplicity: Graph crosses the x-axis

• Even multiplicity: Graph touches and bounces off the x-axis

• Higher multiplicity → flatter approach at the zero

Factored Form Advantage:

p(x) = a(x - r₁)^m₁ · (x - r₂)^m₂ · ... reveals:

• Each rᵢ is a zero

• Each mᵢ determines crossing/bouncing behavior

• a determines vertical stretch and direction

Exam Format: Section I: 28 no-calc MCQ (80 min) + 12 calc-required MCQ (40 min). Section II: 2 calc-required FRQ (30 min) + 2 no-calc FRQ (30 min).

UC Section: C — Mathematics

Grade Level: 10-12