Inside This Unit: The Full Breakdown
Polynomial and rational functions describe how quantities change. This unit analyzes rates of change and concavity, the end behavior and zeros of polynomials, and the asymptotes, holes, and behavior of rational functions.
Why it matters
Units 1 carries the heaviest weight on the AP Precalculus exam (30%–40%). The reasoning about rates of change, concavity, and end behavior introduced here recurs in every later unit and is foundational for calculus.
Key concepts
- Average rate of change is the slope of the secant line; how it changes determines concavity.
- A polynomial of degree n has at most n real zeros; even-multiplicity zeros touch the x-axis, odd-multiplicity zeros cross it.
- Non-real zeros of polynomials with real coefficients occur in complex conjugate pairs.
- Rational functions have vertical asymptotes at uncancelled denominator zeros, and horizontal or slant asymptotes determined by comparing degrees.
Rates of Change and Concavity
The average rate of change over [a, b] is [f(b) - f(a)]/(b - a), the slope of the secant line. A function is increasing where output rises and decreasing where it falls. Concavity describes how the rate of change itself behaves: if successive average rates of change increase, the graph is concave up; if they decrease, it is concave down. Using second differences in a table is a quick way to judge concavity and to distinguish polynomial behavior from exponential behavior.
Polynomial Zeros, Multiplicity, and End Behavior
The degree and leading coefficient control end behavior: even degree sends both ends the same direction, odd degree sends them opposite directions, and the leading-coefficient sign sets which way. At a zero, the multiplicity determines local behavior — even multiplicity makes the graph touch and turn, odd multiplicity makes it cross (with a flatter crossing for multiplicity 3 or higher). Complex zeros always come in conjugate pairs for real-coefficient polynomials.
Rational Functions: Asymptotes and Holes
Factor numerator and denominator first. A factor that cancels produces a hole; an uncancelled denominator zero produces a vertical asymptote. End behavior gives the horizontal asymptote: equal degrees yield the ratio of leading coefficients, a smaller numerator degree yields y = 0, and a numerator degree exactly one larger yields a slant asymptote found by long division.
AP exam tip
When describing a rational function, always factor completely first — this single step distinguishes holes from vertical asymptotes and reveals the zeros, which graders expect you to justify explicitly.
Connections to other units
- Unit 2: Comparing polynomial growth to exponential growth motivates the next unit.
- Unit 3: Transformations of parent functions introduced here apply directly to sinusoids.
- Unit 4: Rational and polynomial reasoning extends to implicitly defined conic sections.
