Review — AP Precalculus | The Complete Equation

AP PRECALCULUS — UNIT 1B · POLYNOMIAL & RATIONAL FUNCTIONS

Unit 1B — Comprehensive Review

Review Notes — Rational Functions, Transformations, and Modeling

💡 Learning Objectives

By the end of this review you will be able to:

Identify zeros, vertical asymptotes, horizontal / slant asymptotes, and holes of a rational function.
Describe the end behavior of a rational function using limit notation.
Perform polynomial long division and apply the Binomial Theorem.
Apply translations and dilations to functions.
Select, construct, and interpret polynomial / rational models for real-world scenarios.

1. Rational Function Feature Map

Every feature of a rational function comes from inspecting N(x) and D(x). Factor both, cancel common factors, and apply the rules above.

2. The Three Cases for End Behavior (1.7)

Case 1 (deg N < deg D): horizontal asymptote y = 0.
Case 2 (deg N = deg D): horizontal asymptote y = aₙ / bₘ (ratio of leading coefficients).
Case 3 (deg N > deg D): no horizontal asymptote. If deg N = deg D + 1, slant asymptote found by long division.
Always express end behavior with limit notation: lim r(x) as x → +∞ and x → −∞.

3. Zeros, VAs, and Holes (1.8, 1.9, 1.10)

Zeros of r: numerator = 0, x in domain.
Vertical asymptote at x = a: factor (x − a) in denominator survives cancellation.
Hole at (a, r̃(a)): factor (x − a) appears in both N and D, cancels; r̃(a) is the y-coordinate.
A hole is a removable discontinuity; a VA is not.

4. Binomial Theorem and Long Division (1.11A, 1.11B)

Binomial expansion: (a + b)ⁿ = Σ C(n, k) aⁿ⁻ᵏ bᵏ for k from 0 to n.
Pascal's Triangle gives the coefficients; general term formula extracts any individual term.
Long division writes r(x) = N/D as q(x) + R(x)/D(x).
Quotient q(x) is the slant asymptote when deg N = deg D + 1.

5. Transformations (1.12A, 1.12B)

Vertical translation: g(x) = f(x) + k. Up if k > 0.
Horizontal translation: g(x) = f(x − h). Right if h > 0 (sign is OPPOSITE to intuition).
Vertical dilation: g(x) = a · f(x). |a| > 1 stretch, |a| < 1 compress, a < 0 reflect across x-axis.
Horizontal dilation: g(x) = f(bx). |b| > 1 COMPRESS (inverse), |b| < 1 stretch, b < 0 reflect across y-axis.
General form: g(x) = a · f(b(x − h)) + k.

6. Modeling (1.13, 1.14)

Select: match shape to function family.
Construct: identify variables, relate them, restrict the domain, apply the model.
Articulate assumptions and check against a known point.
Every interpretation must include units and a complete sentence.

🎯 AP Tip

Unit 1B heavy AP-scored skills: (1) limit notation for end behavior and one-sided limits; (2) fully factoring N and D to distinguish zeros, holes, and VAs; (3) describing transformations in words AND formulas; (4) domain restrictions in modeling. Every single one of these appears frequently on the AP Exam.

Need personalised help?