Graphs and transformations (Higher Tier)

Understanding Graphs and Transformations

A graph is a visual representation of a mathematical function or relation, showing how the values of the function change as the input values change.
A transformation involves changing the appearance of a graph in some way, such as shifting it, stretching it, reflecting it, or rotating it.
Transformations apply to the graph of any function, not just to lines or parabolas. They do not change the basic shape of the graph, only its size and position.
The four basic types of transformations are translations (shifting left or right, up or down), reflections (in the x-axis, y-axis, or origin), rotations (around a point, usually the origin), and dilations (stretching or shrinking).

Recognising Different Types of Transformations

A shift to the right or left is a horizontal translation, represented by the equation (y = f(x – h)), where h is the amount of the shift.
A shift up or down is a vertical translation, represented by the equation (y = f(x) + k), where k is the amount of the shift.
A vertical reflection is produced by the equation (y = -f(x)), which flips the graph over the x-axis.
A horizontal reflection is produced by the equation (y = f(-x)), which flips the graph over the y-axis.
A vertical stretch or shrinkage is represented by the formula (y = af(x)), where a is the stretch factor; a value greater than 1 stretches the graph, and a value between 0 and 1 shrinks it.
A horizontal stretch or shrinkage is represented by the formula (y = f(bx)), where b is the stretch factor; a value greater than 1 shrinks the graph, and a value between 0 and 1 stretches it.

Applying Transformations to Functions

To apply a transformation to a function, you modify the function’s equation according to the type and amount of the transformation.
You can combine multiple transformations by applying them one at a time, in a specific order. Usually, you apply translations last.
When applying transformations, be aware of invariant points, which are points that do not change their position during the transformation.

Interpreting the Effects of Transformations

The effect of a transformation on the graph of a function can often give you important clues about the function’s behaviour.
For example, if the graph of a function has been reflected in the x-axis, it means that the function’s values have been negated.
Similarly, a shift of the graph to the right signifies that the function’s inputs are reduced by a certain amount, indicating a delay or lag in the function’s response.