KKhaemenes PrecalculusOpen-Age · Printable Analytical Lesson
Unit 02Course HomeTeacher Key

Unit 02 · Lesson 05 · Week 5 · Polynomial & Rational Functions

Rational Functions: Domain, Holes & Asymptotes

Analyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. In this lesson, the central focus is rational functions: domain, holes & asymptotes. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intentions

  • Interpret the defining quantities and notation used in rational functions: domain, holes & asymptotes.
  • Construct or analyze at least two representations related to rational functions: domain, holes & asymptotes.
  • Show symbolic work with correct domains, units, intervals, or angle measures.
  • Explain why the selected method is valid and check the result against another representation.

Essential question

How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?

Multiple-representation model

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy

Analyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals.

f(x)=\frac{p(x)}{q(x)}q(x)\ne0y=\frac{a_n}{b_n}\text{ when degrees match}

Worked reasoning

Structure to graphFactor first to identify zeros and multiplicities, then combine end behavior with selected points.
Domain auditRecord restrictions from the original denominator before simplifying so holes and asymptotes are classified correctly.

Misconception check: A canceled factor creates a hole, not a vertical asymptote, because the original domain restriction remains.

Communication and reasoning

Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.

Khaemenes Academy · Open-Age Precalculus

Rational Functions: Domain, Holes & Asymptotes

Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.

U02 · L05
Name
Date
1

State the domain of f(x)=(x+1)/(x−4).

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
2

Find the vertical asymptote of f(x)=3/(x+2).

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
3

Find the horizontal asymptote of (2x²+1)/(x²−9).

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
4

Simplify (x²−9)/(x−3) and identify the excluded value.

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
5

Solve 1/x+1/(x+2)=1.

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
6

On which interval is 1/(x−5) positive?

Rational visualAnalyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. xy
β Beta Program