4.7 Transforming Functions
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Learning Objectives
- Understand how functions can be transformed through shifts, reflections, and stretches
- Apply transformations in the correct order
- Use function notation to describe transformations clearly
- Recognise how transformations affect the graph shape and position
- Practice combining multiple transformations
Key Concepts
Transforming Functions: The Big Picture
Function transformations involve changing the graph’s position, shape, or orientation by modifying the equation.
Transformations include:
- **Translations** (shifting the graph)
- **Reflections** (flipping over an axis)
- **Stretches/Compressions** (rescaling in x or y)
- **Combinations** (multiple transformations applied in sequence)
Function Notation Recap
- **y = f(x) + a** → vertical translation **up** by a
- **y = f(x - a)** → horizontal translation **right** by a
- **y = -f(x)** → reflection in **x-axis**
- **y = f(-x)** → reflection in **y-axis**
- **y = af(x)** → vertical stretch by **a**
- **y = f(ax)** → horizontal stretch by **1/a**
Order of Transformations
When applying multiple transformations, the **order matters**.
- Generally: apply **stretches first**, then **reflections**, then **translations**.
This prevents misinterpreting how inputs or outputs are altered by each transformation.
Example Combination
**y = -2f(x - 3) + 1** applies:
- **Horizontal translation right 3** (x - 3)
- **Vertical stretch by 2** (2f)
- **Reflection in x-axis** (-f)
- **Vertical shift up 1** (+1)
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