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BetterGrades Precalculus · Unit 6 · Unit map

Rational Functions

A rational function P/Q excludes every real input where the original denominator Q is zero.

Precalculus · Unit 6

The lesson path

A rational function P/Q excludes every real input where the original denominator Q is zero.

Core sequence

Rational Functions

Move in order when the material is new, or enter at the exact skill you need.

  1. 01Rational functions and their domainsIdentify rational functions and determine all excluded inputs from the original denominator before simplifying.
  2. 02Equivalent formulas, different functions, and holesIdentify removable discontinuities caused by common factors and distinguish formula equivalence from function equality.
  3. 03Vertical asymptotes and local behaviorDetermine vertical asymptotes and describe one-sided behavior from factor signs and multiplicity.
  4. 04Horizontal asymptotes and end behaviorDetermine horizontal asymptotes from degree comparison and interpret them as long-run output behavior.
  5. 05Slant and polynomial asymptotesUse polynomial division to determine nonhorizontal asymptotes and describe the shrinking remainder term.
  6. 06Intercepts and sign chartsFind rational intercepts and determine positive and negative intervals using factored sign structure.
  7. 07Complete rational graph constructionConstruct a complete rational graph using domain, discontinuities, asymptotes, intercepts, signs, and selected values.
  8. 08Rational equations and function intersectionsSolve rational equations and intersections exactly or numerically while rejecting excluded candidates.
  9. 09Rational inequalitiesSolve rational inequalities using critical values, sign intervals, and correct endpoint inclusion.
  10. 10Variation and rational modelsBuild and interpret direct, inverse, joint, and combined variation models with meaningful parameters and domains.
  11. 11Partial-fraction structureDecompose proper rational functions with factorable denominators into simpler component fractions and verify recombination.
Source record

References used for this unit

  • Lippman and Rasmussen, Precalculus Volume 1
  • Utah College Algebra
  • Stitz and Zeager, Precalculus

The learner copy is original BetterGrades material informed by these rights-separated references; no long source passage is reproduced.