Unit 02 · Lesson 07 · Week 6 · Polynomial & Rational Functions
Polynomial and Rational Modelling
Analyze ratios of polynomials by distinguishing excluded values, holes, vertical asymptotes, horizontal behavior, and sign intervals. In this lesson, the central focus is polynomial and rational modelling. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.
Learning intentions
- Interpret the defining quantities and notation used in polynomial and rational modelling.
- Construct or analyze at least two representations related to polynomial and rational modelling.
- 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
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
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
Polynomial and Rational Modelling
Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.
Name
Date
State the domain of f(x)=(x+1)/(x−4).
Find the vertical asymptote of f(x)=3/(x+2).
Find the horizontal asymptote of (2x²+1)/(x²−9).
Simplify (x²−9)/(x−3) and identify the excluded value.
Solve 1/x+1/(x+2)=1.
On which interval is 1/(x−5) positive?