4.6 Stretching Graphs
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Learning Objectives
- Understand vertical and horizontal stretches of graphs
- Use function notation to describe stretches
- Recognise the difference between multiplying inside and outside the function
- Apply stretches to standard graphs like quadratics and exponentials
- Sketch transformed graphs from their equations
Key Concepts
What Is a Stretch?
A **stretch** changes the shape of a graph by making it **taller or wider** — it alters the scale but keeps the origin fixed (unless combined with a translation).
Stretches either pull the graph **vertically** (up/down) or **horizontally** (left/right).
Vertical Stretches: y = a·f(x)
Multiplying **outside** the function affects the **y-values**, causing a vertical stretch or compression.
- If a > 1 → the graph stretches vertically (gets taller)
- If 0 < a < 1 → the graph compresses vertically (gets shorter)
Example: y = 3·x² is y = x² stretched **vertically** by a factor of 3
Horizontal Stretches: y = f(ax)
Multiplying **inside** the function affects the **x-values**, causing a horizontal stretch or compression.
- If a > 1 → the graph compresses horizontally (gets narrower)
- If 0 < a < 1 → the graph stretches horizontally (gets wider)
This feels 'reversed' — because the **input** is scaled, not the output.
Example: y = (2x)² is y = x² compressed **horizontally** by a factor of ½
Key Reminder
- Outside multiplication affects **y** → vertical stretch by **a**
- Inside multiplication affects **x** → horizontal stretch by **1/a**
Always check if multiplication happens inside or outside the brackets!
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