Unit 01 · Lesson 03 · Open-Age

Transformations of Parent Functions

Transformations describe how a parent function is stretched, reflected, and shifted. Horizontal changes act inside the input and therefore work inversely; vertical changes act directly on outputs.

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Learning objectives

Explain the central structure of transformations of parent 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 notation, transformations, composition, and inverse thinking reveal the structure and behavior of functions?

Governing relationship

g(x)=a·f(b(x−h))+k.

Transformations describe how a parent function is stretched, reflected, and shifted. Horizontal changes act inside the input and therefore work inversely; vertical changes act directly on outputs.

Worked examples

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

Describe g(x)=−2(x−3)²+5.Start with y=x²; shift right 3, stretch vertically by 2, reflect across the x-axis, then shift up 5.

Transform y=|x| to y=|2x+4|−1.Factor inside: |2(x+2)|−1. Compress horizontally by factor 1/2, shift left 2, then down 1.

Guided and independent practice

Describe y=3√(x+1)−4.
Write an equation for y=x³ shifted right 5 and reflected across the x-axis.
Compare vertical and horizontal stretches.
Create one original example involving transformations of parent 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 two transformed parent functions that meet given 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: Treating x−h as a shift left h. Inside subtraction shifts the graph right.

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?