Understanding Graph Transformations

Introduction to Graph Transformations

Graph Transformations involve modification of a parent function to produce a new function with unique characteristics.
Transformations play an essential role in making sense of real-world problems in Algebra.

Types of Transformations

The basic types of transformations for functions on a graph include translations, reflections, and stretches.

Translations

A translation involves shifting the entire graph to a new location without changing its shape or orientation.
The transformation y=f(x)+c involves a vertical translation c units upwards for positive c or downwards for negative c.
The transformation y=f(x+c) involves a horizontal translation c units to the left for positive c or to the right for negative c.

Reflections

Reflection involves mirroring an original function about an axis.
The transformation y=-f(x) reflects the graph across the x-axis.
The transformation y=f(-x) reflects the graph across the y-axis.

Stretches

A stretch involves changing the size of a graph either vertically or horizontally, altering its shape but not its orientation.
The transformation y=af(x) (a>1) stretches the graph of y=f(x) vertically by a factor of a.
The transformation y=f(ax) (a>1) compresses the graph of y=f(x) horizontally by a factor of a.
If 0<a<1 in the transformations y=af(x) or y=f(ax), the graph shrinks instead of stretches.

Strategies for Graph Transformations

Order in Combining Transformations

Multiple transformations can be applied to a single function. However, the order can affect the final result.

Observation and Analysis

Always observe the original graph first before applying transformations.
Analyse which transformation has been applied to the function when given a transformed graph.

Practice and Understanding

Understanding transformations rather than memorising them will help in dealing with different types of functions.
The key is constant practice with different types and combinations of transformations.

Finally, always remember: Graph transformations are a visual and creative tool in Algebra that can help simplify complex mathematical problems. Practicing with these transformations can lead to a deeper understanding of functions and their behaviours.