Study Notes
The transformation of graphs involves translating, stretching, and reflecting graphs of functions. Translations move the graph without changing its shape, while stretches change the size of the graph. Reflections flip the graph over a specified axis.
- Translation — Moving a graph along the x-axis or y-axis without changing its shape.
Example: The graph of y = f(x) + a is a translation of y = f(x) by a units in the y-direction. - Vertical Stretch — Changing the size of a graph by multiplying the function by a constant outside.
Example: The graph of y = af(x) is a vertical stretch of y = f(x) by a factor of a. - Horizontal Stretch — Changing the size of a graph by multiplying the function by a constant inside.
Example: The graph of y = f(ax) is a horizontal stretch of y = f(x) by a factor of 1/a. - Reflection — Flipping a graph over the x-axis or y-axis.
Example: The graph of y = -f(x) is a reflection of y = f(x) in the x-axis.
Exam Tips
Key Definitions to Remember
- Translation: Moving a graph along the axes without changing its shape.
- Vertical Stretch: Multiplying the function by a constant outside to change its size.
- Horizontal Stretch: Multiplying the function by a constant inside to change its size.
- Reflection: Flipping a graph over an axis.
Common Confusions
- Confusing vertical and horizontal stretches.
- Misinterpreting the direction of translations.
Typical Exam Questions
- What is the effect of y = f(x) + a on the graph? It translates the graph up by a units.
- How does y = af(x) affect the graph? It stretches the graph vertically by a factor of a.
- What happens to the graph of y = f(-x)? It reflects the graph in the y-axis.
What Examiners Usually Test
- Understanding of how translations affect the graph's position.
- Ability to identify and apply stretches and reflections.
- Combining multiple transformations correctly.