Unit 04 · Lesson 06 · Open-Age

Transformations and Features of Rational Functions

Transformed reciprocal functions use h and k to locate asymptotes and a/b to control orientation and scale. Feature analysis should be tied to the original equation.

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Learning objectives

Explain the central structure of transformations and features of rational functions using precise mathematical language.
Apply the governing relationship and verify results with an alternate representation or substitution.
State domain, unit, precision, and context restrictions whenever they affect the result.
Analyze a misconception and communicate a correction with evidence.

Essential question: How do excluded values, asymptotes, holes, and rates of change shape rational expressions and models?

Governing relationship

y=a/[b(x−h)]+k.

Transformed reciprocal functions use h and k to locate asymptotes and a/b to control orientation and scale. Feature analysis should be tied to the original equation.

Worked examples

Read each example, close it, and reconstruct the reasoning independently. Then verify by substitution, graph, table, or alternate method.

Describe y=−3/(x+2)+4.Vertical asymptote x=−2; horizontal asymptote y=4; reflected and vertically stretched.

Find an equation with asymptotes x=5, y=−1 through (6,2).Use y=a/(x−5)−1. Substitute: 2=a−1, so a=3.

Guided and independent practice

Graph a translated reciprocal.
Find a parameter from a point.
Compare two rational transformations.
Create one original example involving transformations and features of rational functions and solve it completely.
Use the Khaemenes Scientific Calculator to verify one result, then document the setup and reasonableness check.
Write a one-paragraph explanation connecting symbolic, graphical, numerical, and contextual representations.

Differentiated pathways

Foundation

Review prerequisite vocabulary, annotate each worked step, complete the first four prompts, and use a table or graph to support symbolic work.

Core

Complete every prompt, explain domain and verification, correct errors, and submit the applied task with clear mathematical communication.

Extended

Generalize the relationship, compare methods, use technology strategically, analyze sensitivity, and create a proof, counterexample, or advanced model.

Application laboratory

Design a rational response curve satisfying location and scale constraints.

Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.

Misconception and correction

Common misconception: Using h instead of −h when reading x−h.

Write the first invalid step, the governing rule that corrects it, and one counterexample or verification that demonstrates the correction.

Mastery evidence

Accurate symbolic work with preserved equivalence.
Correct domain, interval, unit, and precision statements.
At least one independent verification.
A correction explaining why an incorrect approach fails.
A clear application conclusion that distinguishes mathematical result from recommendation.

Reflection

What structure did you recognize? Which representation was most useful? What assumption or restriction mattered most? What will you do differently on the next problem?