Transformations of Functions

Year Level

Year 7-10 (adjusted activities for learners working between Year 2 to Year 7 levels)

Duration

45 minutes

Unit Context

Unit 3: Algebra in Everyday Life Lesson 17 of 30: Transformations of Functions

NSW Curriculum Links

Stage 4 Mathematics (Years 7-8)

ACMNA214 — Use algebraic expressions to represent situations and describe relationships.
ACMNA217 — Create and interpret graphs from given information, including identifying key features such as intercepts and gradients.
ACMNA228 — Investigate the effect of changing parameters in functions, such as ( y = f(x) ), including transformations like translations and stretches.

Stage 5 Mathematics (Years 9-10) (adjust stretch activities)

ACMNA260 — Recognise, sketch and interpret graphs of linear and non-linear functions including transformations.

General Capabilities

Literacy: Interpretation of graphs and algebraic expressions - supports dyslexia-friendly strategies.
Numeracy: Link between numeric calculations and graphical representation.
Personal and Social Capability: Collaborative learning, respect and empathy.
ICT Capability: Use of graphing technology or apps.

Learning Objectives

By the end of the lesson, students will be able to:

Understand and describe how transformations affect the graph of a function (translations, stretches, reflections).
Identify horizontal/vertical shifts and stretches using function notation ( y = f(x) ), ( y = f(x - a) ), ( y = kf(x) ), etc.
Apply transformations to simple linear and quadratic functions and visualise their impact on graphs.
Connect transformations to real-world contexts (e.g., movements, scaling objects).
Build foundational algebraic understanding and graph interpretation skills through repetition, visual aids, and real-world examples.

Resources

Whiteboard or interactive board
Graph paper and pencils
Printed function rule cards (e.g., ( y = x^2 ), ( y = 2x ), ( y = (x - 3)^2 ), etc.)
Graphing calculators or tablets with graphing apps (optional)
Visual aids showing transformations (slides or posters)
Dyslexia-friendly printed handouts with clear fonts and spacing
Real-world images connected to transformations (e.g., shadows moving on the wall, resizing photos)

Lesson Structure

1. Introduction and Recap (5 minutes)

Begin with a brief refresher on what functions and their graphs are — highlight the relationship between the algebraic expression and its shape.
Use simple terms with visual examples; e.g., graph ( y = x^2 ) on graph paper.
Introduce the concept of "Transformation" as a change in the position or size of a graph.
Objective: Link to students’ prior knowledge, activate memories, and prepare them for new material.

Differentiation: Use large print visuals, simple language, and physical gestures to reinforce understanding.

2. Explicit Teaching: Types of Transformations (10 minutes)

Introduce key transformation types with simple examples:
Translations (Shifts):
Horizontal: ( y = f(x - a) ) shifts the graph right by ( a ) units.
Vertical: ( y = f(x) + b ) shifts the graph upwards by ( b ) units.
Stretches/Compressions:
Vertical stretch: ( y = kf(x) ) makes the graph taller for ( k > 1 ).
Vertical compression for ( 0 < k < 1 ).
Reflections:
( y = -f(x) ) reflects the graph over the x-axis.
Use graphs side-by-side (original vs transformed) to emphasise the effect.
Discuss the difference between changes "inside" and "outside" the function (domain vs range) in a simple way.

Dyslexia-Friendly: Use bullet points, colour coding for shifts (e.g., blue for right, red for left), and use diagrams with clear spacing.

3. Guided Practice: Matching Activity (10 minutes)

Distribute function cards and corresponding graph cards (some showing base functions, some transformed).
In pairs, students match the function expression with the correct transformed graph.
Encourage the use of vocabulary: shift, stretch, reflection, vertical, horizontal.
Teacher circulates to support, especially for students with learning difficulties, breaking down steps as needed.

Differentiation:

For learners with more difficulty: Provide one transformation at a time (e.g., only translations first).
For advanced students: Challenge them to predict the algebraic expression of a given transformed graph.

4. Real-World Connection and Discussion (8 minutes)

Present real-life scenarios where transformations happen:
E.g., a shadow moving and stretching as the sun moves (vertical stretch);
A picture being moved across a screen (translation);
Reflection of objects in a mirror (reflection).
Ask students to suggest functions that might model these transformations.
Briefly discuss why understanding transformations is useful in practical contexts like engineering, design, or animation (careers connection).

5. Independent Activity: Plot and Transform (7 minutes)

Students complete a worksheet with a simple graph ( y = x^2 ) and draw its transformations based on different algebraic rules given (translated, stretched, reflected).
Students label the transformed graphs with sentences explaining the transformation (e.g., "This graph is shifted 3 units to the right").

Differentiation:

Provide graph templates for those needing extra support.
Extension: Ask advanced students to write a function rule for their transformed graphs.

6. Summary and Reflective Discussion (5 minutes)

Recap key ideas: ask students what types of transformations they discovered.
Invite a few students to explain an example in their own words.
Connect learning to real-world useful skills: understanding changes visually and algebraically.
Brief preview of next lesson: combining transformations and exploring composite transformations.

Assessment

Formative assessment:
Observation of student participation during matching and class discussions.
Review of independent activity to check accuracy and understanding.
Use questioning to identify misconceptions about transformation types and their notation.
Differentiation: Provide oral feedback, use visual supports, rephrase questions and allow additional time if needed.

Differentiation Strategies

Use multi-sensory learning methods: visual aids, manipulative cards, oral explanations.
Chunk information into smaller parts with frequent checks for understanding.
Use consistent, simple language and reinforce vocabulary with examples and gestures.
Allow peer support and collaborative learning, respecting social and emotional needs.
Provide print materials in dyslexia-friendly fonts (e.g., OpenDyslexic) and with clear spacing.
Scaffold worksheets with worked examples and use colour coding to highlight changes.
Use technology if available (graphing calculators or apps) for interactive learning and instant visual feedback.

Extension Activities for Advanced Learners

Explore composite transformations: applying two or more transformations in sequence.
Investigate other functions, such as cubic or absolute value functions, and predict transformation outcomes.
Use graphing software to experiment with sliders that modify parameters interactively.
Challenge to create real-world problems involving transformations, including workplace or design scenarios.

This lesson and its approaches are carefully designed to meet the NSW Curriculum standards for Mathematics Stage 4 and Stage 5, tailored for a diverse class including learners with autism and learning difficulties, promoting engagement through repetition, visualisation, and authentic connections. It also supports a caring learning environment aligned with your role in career advising and social development.

Thank you for the opportunity to create this plan to support your important work with your students.

Teaching Instructions