### Functions and graphs: Transformations of graphs and functions

### Combination of transformations

The four transformations discussed earlier, namely vertical translation, horizontal translation, vertical multiplication, and horizontal multiplication, can also be combined. The most common transform is \(g(x)=a\cdot f\bigl(b\cdot x+c\bigr)+d\) for a given function \(f(x)\).

Here, the graph of \(f\) is first transated horizontally by \(c\). A horizontal multiplication by \(1/b\) is then applied to the result. Vertical multiplication by \(a\) is applied to this intermediate result and then a vertical translation by \(d\).*Note:* the order of application of the four transformations matters. For example, if you first multiply the function \(f(x)\) by \(b\) and then shift it by \(c\), you get \(x\mapsto f\bigl(b\cdot(x+c)\bigr)\).

Sketch the graph of the function \(x\mapsto \tfrac{1}{2} \sqrt{2x-3}+2\) based on the graph of the square root function.

We show step by step which transformations we successively apply to a graph; the gray graph is always the red result obtained in the previous step.

*Function rule from \(\sqrt{x}\) to \(\sqrt{x-3}\)*

Start with the graph of the square root function and move it by three to the right.

*Function rule from \(\sqrt{x-3}\) to \(\sqrt{2x-3}\)*

Multiply the gray graph horizontally by the factor \(\frac{1}{2}\).

*Function rule from \(\sqrt{2x-3}\) to \(\tfrac{1}{2}\sqrt{2x-3}\)*

Multiply the gray graph vertically by the factor \(\tfrac{1}{2}\).

*Function rule from \(\tfrac{1}{2}\sqrt{2x-3}\) to \(\tfrac{1}{2}\sqrt{2x-3}+2\)*

Move the gray graph upward with \(2\).