Unit 1 · Polynomial and Rational Functions
The Complete AP Precalculus Guide · AP PrecalculusUnit 1 · Polynomial and Rational Functions
- The Complete AP Precalculus Guide
- AP Precalculus
- 3 sections
Unit 1: Polynomial and Rational Functions accounts for 30–40% of AP Precalculus multiple-choice content. Section I has 42 multiple-choice questions in 105 minutes and contributes 62.5% of the score. For Section II's 4 free-response questions in 70 minutes (37.5%), be ready to carry the same unit skills and representations into a complete solution. Covariation describes how outputs change as inputs change. Average rate compares two endpoint outputs over a named interval, while local graph behavior adds intercepts, extrema, and direction of change.
- How AP Precalculus assesses this 30–40% of the multiple-choice section · Section I: 42 MCQs in 105 min, 62.5% · Section II: 4 FRQs in 70 min, 37.5% · show the model with factored symbolic form, graph with intercepts and asymptotes, table of average rates over nested intervals
- Key skills Analyze polynomial zeros and end behavior, Analyze rational discontinuities and asymptotes, Construct and transform function models
- How to study for Unit 1 This page turns factored symbolic form, graph with intercepts and asymptotes, table of average rates over nested intervals into one route: factor while retaining the original domain and label every structural feature.
- The organizing decision choose a polynomial or rational model from covariation and preserve zeros multiplicity holes and asymptotes
What AP Precalculus Unit 1 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Analyze polynomial zeros and end behavior
factored symbolic form; Repeated zeros control whether a graph crosses or touchesAPPRECALC-U1-S2Analyze rational discontinuities and asymptotes
graph with intercepts and asymptotes; A canceled factor creates a hole rather than erasing a domain restrictionAPPRECALC-U1-S3Construct and transform function models
table of average rates over nested intervals; End behavior follows degree and leading-coefficient structureUnit 1: Polynomial and Rational Functions accounts for 30–40% of AP Precalculus multiple-choice content.
Official unit name and weighting: College Board course and exam description.
Connect algebraic form to visible function behavior
Connect the published share to the unit model
Covariation describes how outputs change as inputs change. Average rate compares two endpoint outputs over a named interval, while local graph behavior adds intercepts, extrema, and direction of change.
For a polynomial, the leading term controls both ends and factor multiplicity controls crossing or touching at a zero. For a rational function, the original denominator records excluded inputs before any cancellation occurs.
Equivalent-looking formulas are not fully interchangeable when their domains differ. A justification names the feature, points to the factor or leading term that causes it, and checks the conclusion against the representation.
The decision that organizes this unit
Define the system and choose the route before calculating
choose a polynomial or rational model from covariation and preserve zeros multiplicity holes and asymptotes
factor while retaining the original domain and label every structural feature
Mechanism route and repair branches
Relationships to preserve
- Repeated zeros control whether a graph crosses or touches
- A canceled factor creates a hole rather than erasing a domain restriction
- End behavior follows degree and leading-coefficient structure
Representations to read
- factored symbolic form
- graph with intercepts and asymptotes
- table of average rates over nested intervals
Branches to reject
- treating a hole as a vertical asymptote
- ignoring multiplicity at a zero
- extrapolating a polynomial model beyond a defensible domain
| Key concept | Why it's hard | What scores |
|---|---|---|
| Covariation and rate | A quotient is meaningless without interval and units | Shows both endpoint values, the interval, and output units per input unit |
| Zeros and discontinuities | Cancellation can hide an excluded input | Preserves the original domain and distinguishes a zero, hole, and asymptote |
| Equivalent forms | Matching symbols can still describe different domains | States where the forms agree and verifies the relevant features |
How AP Precalculus assesses Polynomial and Rational Functions
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Analyze polynomial zeros and end behavior | factored symbolic form; Repeated zeros control whether a graph crosses or touches | treating a hole as a vertical asymptote |
| Analyze rational discontinuities and asymptotes | graph with intercepts and asymptotes; A canceled factor creates a hole rather than erasing a domain restriction | ignoring multiplicity at a zero |
| Construct and transform function models | table of average rates over nested intervals; End behavior follows degree and leading-coefficient structure | extrapolating a polynomial model beyond a defensible domain |
Resolve the Polynomial and Rational Functions evidence conflict
Carry the model from prompt to check
- Step 1Factor the original numerator and denominator while preserving the original restriction x not equal to 2.
- Step 2Cancel the common x-2 factor only in the simplified rule, not in the domain.
- Step 3Evaluate the simplified rule at x=2 to locate the hole when the remaining denominator is nonzero.
- Step 4Use remaining numerator zeros for intercepts and remaining denominator zeros for vertical asymptotes.
Key terms for Unit 1: Polynomial and Rational Functions
Models, uses, and boundaries
- Analyze Polynomial Zeros and Multiplicity
- A factored polynomial records each zero r sub i with multiplicity m sub i Choose this formula when the prompt asks you to analyze polynomial zeros and multiplicity and the function, domain, interval, or representation matches the symbols shown. A Analyze Polynomial Zeros and Multiplicity solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Analyze Rational Holes and Asymptotes
- A rational function retains the original domain restriction x not equal to c even after cancellation Choose this formula when the prompt asks you to analyze rational holes and asymptotes and the function, domain, interval, or representation matches the symbols shown. A Analyze Rational Holes and Asymptotes solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Predict Polynomial End Behavior
- Polynomial end behavior follows the leading term Choose this formula when the prompt asks you to predict polynomial end behavior and the function, domain, interval, or representation matches the symbols shown. A Predict Polynomial End Behavior solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Apply the Remainder Theorem
- The remainder theorem writes the remainder on division by x minus c as P of c Choose this formula when the prompt asks you to apply the remainder theorem and the function, domain, interval, or representation matches the symbols shown. A Apply the Remainder Theorem solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
AP Precalculus Unit 1 FAQ
How much of AP Precalculus does Unit 1 carry?
Unit 1: Polynomial and Rational Functions accounts for 30–40% of AP Precalculus multiple-choice content.
What is the first move on a Polynomial and Rational Functions problem?
factor while retaining the original domain and label every structural feature
Which relationships should I preserve?
Repeated zeros control whether a graph crosses or touches A canceled factor creates a hole rather than erasing a domain restriction End behavior follows degree and leading-coefficient structure
Which representations should I practice?
Practice moving among factored symbolic form, graph with intercepts and asymptotes, table of average rates over nested intervals.
What error should I check before submitting an answer?
Check for treating a hole as a vertical asymptote; ignoring multiplicity at a zero; extrapolating a polynomial model beyond a defensible domain.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Interpret an interval rate with its sign and units
- Use multiplicity and the leading term to justify a graph
- Record exclusions before canceling a common factor
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related AP Precalculus unit guides
How to study AP Precalculus Unit 1
Start with the organizing decision
Before solving, restate the decision in operational terms: choose a polynomial or rational model from covariation and preserve zeros multiplicity holes and asymptotes. Your first written move should be to factor while retaining the original domain and label every structural feature.
Practice the same idea in several representations
Rotate through factored symbolic form, graph with intercepts and asymptotes, table of average rates over nested intervals. Use each representation to practice Analyze polynomial zeros and end behavior, Analyze rational discontinuities and asymptotes, Construct and transform function models, and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for treating a hole as a vertical asymptote; ignoring multiplicity at a zero; extrapolating a polynomial model beyond a defensible domain. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official College Board course and exam description for the May 2027 administration.
