Graph Transformations
Year 10 (IGCSE) 📊 Algebra Translate, reflect and stretch graphs of functions.
🔄 Translations
Adding a constant inside or outside the function shifts the entire graph.
⚡ Translation Rules
$$y = f(x) + a \quad \text{shifts UP by } a$$$$y = f(x + a) \quad \text{shifts LEFT by } a$$ Changes inside the brackets affect the $x$-direction and work in the opposite direction to what you might expect.
Example: $y = (x-3)^2$ shifts $y = x^2$ to the right by 3 units.
↔️ Reflections
A negative sign creates a reflection in one of the axes.
⚡ Reflection Rules
$$y = -f(x) \quad \text{reflection in the } x\text{-axis}$$$$y = f(-x) \quad \text{reflection in the } y\text{-axis}$$Example: $y = -\sin(x)$ reflects $\sin(x)$ in the $x$-axis, flipping all the peaks and troughs.
🔍 Stretches
Multiplying by a constant stretches or compresses the graph.
⚡ Stretch Rules
$$y = af(x) \quad \text{vertical stretch, factor } a$$$$y = f(ax) \quad \text{horizontal stretch, factor } \frac{1}{a}$$Example: $y = 2\sin(x)$ doubles the amplitude. $y = \sin(2x)$ halves the period (compresses horizontally by factor $\frac{1}{2}$).
Ready to test yourself? Click the Quiz tab above to answer questions on this topic!
Interactive Demonstration — Graph Transformations
🟠 y = x² (original) 🔵 transformed
🔄 Graph Transformation Tool
Use the Demo tab to visualise transformations interactively. Here, find image coordinates under a transformation.