Transformations of Functions: Reflections and Stretches

Active learning works well for transformations because students need to see, touch, and manipulate the effects of stretches and reflections. Moving beyond static images lets students internalize how equations change graphs through direct experience, which builds lasting intuition about function behavior.

• Year 10
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Compare the graphical effects of vertical stretches (k f(x)) versus horizontal stretches (f(x/k)) on a given function.
2. Explain how reflections across the x-axis (-f(x)) and y-axis (f(-x)) alter the graph of a function.
3. Analyze a sequence of transformations to accurately map the graph of y = f(x) onto the graph of y = a f(b(x-c)) + d.
4. Synthesize understanding of reflections and stretches to predict the final graph of a composite transformation.
5. Create a new function's equation given a series of transformations applied to a parent function's graph.

Stations Rotation

⏱ 35 min·Pairs

Desmos Sliders: Stretch and Reflect Quadratics

Pairs access Desmos and graph y = x². Add sliders for vertical stretch factor k and horizontal factor m in k(x/m)², plus checkboxes for reflections -f(x) and f(-x). Students predict changes, adjust sliders, and note equation-graph links. Conclude with sketching a transformed cubic.

Prepare & details

Differentiate between a vertical and horizontal stretch/compression.

Facilitation Tip: During Desmos Sliders, circulate and ask pairs to predict the slider’s effect before they move it, so they connect the algebra to the visual change.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 25 min·Small Groups

Card Sort: Match Graph to Transformation

Prepare cards showing original graphs, transformed versions, and equations. Small groups sort matches for stretches and reflections on quadratics and exponentials. Groups justify choices, then share one mismatch with the class for discussion.

Prepare & details

Explain how reflections across axes affect the equation of a function.

Facilitation Tip: For the Card Sort, listen for students using precise terms like ‘compress’ or ‘flip’ as they match graphs to equations, reinforcing vocabulary through discussion.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 40 min·Small Groups

Relay Race: Build Transformation Sequences

Divide class into teams. Display start graph; first student draws one transformation, next adds another to match target graph. Teams race while explaining steps aloud. Debrief on order effects and equation updates.

Prepare & details

Construct a sequence of transformations to map one function onto another.

Facilitation Tip: In the Relay Race, step in immediately if teams skip writing intermediate steps, since these reveal whether they understand order or just mimic patterns.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 30 min·Whole Class

Human Graph: Physical Transformations

Select student volunteers to form a line graph of y = x. Class directs stretches by spacing adjustments and reflections by mirroring positions. Record before-after photos; whole class analyses equation changes.

Prepare & details

Differentiate between a vertical and horizontal stretch/compression.

Facilitation Tip: When running Human Graph, assign each student a point to track through all transformations so they see unchanged intercepts and scaled distances firsthand.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Teach this topic by starting concrete with graphs students know, like quadratics or absolute value, before moving to abstract equations. Use contrasting examples side-by-side to highlight differences between vertical and horizontal changes. Avoid rushing to rules; instead, build the rules from observed patterns. Research shows that students grasp transformations best when they physically manipulate or visualize changes before formalizing them.

What to Expect

Successful learning looks like students confidently predicting where key points move, describing transformations in precise language, and sequencing multiple steps without confusion. They should connect equations to visual shifts and justify their reasoning with clear examples.

Watch Out for These Misconceptions

Common Misconception

During Desmos Sliders, watch for students thinking vertical stretches move x-intercepts.

What to Teach Instead

Have students plot three points on y = x², then slide the vertical stretch factor and observe that only y-values change while x-intercepts remain at zero. Ask them to explain why this happens in pairs before moving to the next slider.

Common Misconception

During Desmos Sliders, watch for students believing horizontal stretches widen the graph when k > 1.

What to Teach Instead

Ask students to compare the graphs of y = x² versus y = (0.5x)² and y = (2x)² on the same axes. Have them measure the width at y = 1 and record how k > 1 actually compresses the graph toward the y-axis.

Common Misconception

During Relay Race, watch for students assuming the order of transformations never matters.

What to Teach Instead

Assign teams two sequences with the same transformations in different orders and ask them to sketch both results. Then have them present why the sequences produce different graphs, using specific points to justify their reasoning.

Assessment Ideas

Quick Check

After Desmos Sliders, ask students to sketch y = x², y = 3x², and y = (1/2)x² on the same axes and label each. Then have them write the equation for the reflection of y = x² across the x-axis without using software.

Exit Ticket

After Card Sort, provide f(x) = |x| and ask students to write g(x) that is horizontally stretched by a factor of 2 and reflected across the y-axis. Collect sketches and equations to check if they used f(x/2) and f(-x) correctly.

Discussion Prompt

During Human Graph, present two transformed graphs and ask pairs to describe the sequence of transformations mapping the original to the new graph. Have them justify their answers by identifying how key points like intercepts or vertices moved.

Extensions & Scaffolding

• Challenge students finishing early to create a function family of their own and write a set of transformations that maps one member to another, then swap with a partner to solve.
• For students struggling, provide pre-labeled axes with key points plotted and ask them to apply single transformations one at a time, tracing each change step by step.
• Allow extra time for students to explore how combinations of reflections and stretches affect symmetry or special points like vertices, using graphing software to test hypotheses.

Key Vocabulary

Vertical Stretch	A transformation that scales the y-values of a function by a constant factor k, resulting in the equation y = k f(x). If k > 1, the graph stretches away from the x-axis; if 0 < k < 1, it compresses towards the x-axis.
Horizontal Stretch	A transformation that scales the x-values of a function by a constant factor of 1/k, resulting in the equation y = f(x/k). If k > 1, the graph stretches away from the y-axis; if 0 < k < 1, it compresses towards the y-axis.
Reflection across x-axis	A transformation that flips the graph of a function over the x-axis, changing the sign of the output. The equation changes from y = f(x) to y = -f(x).
Reflection across y-axis	A transformation that flips the graph of a function over the y-axis, changing the sign of the input. The equation changes from y = f(x) to y = f(-x).
Transformation Sequence	A series of geometric transformations applied to a function's graph, such as stretches, compressions, and reflections, performed in a specific order to achieve a desired final graph.