Functions and graphs > Transformations
12345Transformations

## Theory

Consider the basic function $y = f \left(x\right)$.

• The graph of $y = f \left(x\right) + 2$ can be constructed by shifting the graph of $f$ upwards, in the $y$-direction, by $2$ units.

• The graph of $y = f \left(x + 2\right)$ can be constructed by shifting the graph of $f$ in the $x$-direction by –2 units.

These are two examples of a transformation of a graph.
Adding a number in the function rule shifts the graph.
Instead of shifting you often use the term translation.
The characteristics of the transformed functions can be derived from those of the basic function.

Start with the basic function $y = f \left(x\right)$.

• The graph of $y = 2 \cdot f \left(x\right)$ can be constructed by multiplying the graph of $f$ by $2$ in the $y$-direction.

• The graph of $y = f \left(2 \cdot x\right)$ can be constructed by multiplying the graph of $f$ by $\frac{1}{2}$ in the $x$-direction.

These are also transformations of a graph.
By multiplying with a number in the function rule, the graph gets multiplied in the direction of one of the axes. This is called a stretch or decompression w.r.t. one of the axes (vertical or horizontal).
The characteristics of the transformed function can be derived from those of the basic function.