Functions of graphs can be transformed to show shifts and reflections. Graphic designers and 3D modellers use transformations of graphs to design objects and images.

A translation is a movement of the graph either horizontally parallel to the \(x\)-axis or vertically parallel to the \(y\)-axis.

Functions

The graph of \(y = f(x)\) where \(f(x) = x^2\) is the same as the graph of \(y = x^2\).
Writing equations as functions in the form \(f(x)\) is useful when applying translations and reflections to graphs.

Translations parallel to the y-axis

If \(f(x) = x^2\), then \(f(x) + a = x^2 + a\). Here we are adding \(a\) to the whole function.

The addition of the value \(a\) represents a vertical translation in the graph. If \(a\) is positive, the graph translates upwards. If \(a\) is negative, the graph translates downwards.

Example 1

\[f (x) = x^2\]

Draw the graphs of \(y = f(x)\) and \(y = f(x) + 3\).

This is a translation of \(y = f(x)\) by 3 units in the positive\(y\) direction.

Example 2

\[f(x) = x^2\]

Draw the graphs of \(y = f(x)\) and \(y = f(x) − 2\).

This is a translation of \(y = f(x)\) by 2 units in the negative\(y\) direction.

\(f(x) + a\) represents a translation of the graph of \(f(x)\) by the vector \(\begin{pmatrix} 0 \\ a \end{pmatrix}\).

Translations parallel to the x-axis

If \(f(x) = x^2\) then \(f(x + a) = (x + a)^2\)

Here we add \(a\) to \(x\), not to the whole function. This time we will get a horizontal translation. If \(a\) is positive then the graph will translate to the left. If the value of \(a\) is negative, then the graph will translate to the right.

Example 1

\[f(x) = x^2\]

Draw the graphs of \(y = f(x)\) and \(y = f(x + 3)\).

Example 2

\[f(x) = x^2\]

Draw the graphs of \(y = f(x)\) and \(y = f(x – 2)\).

\(f(x + a)\) represents a translation of the graph of \(f(x)\) by the vector \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\).