Transformations of Graphs

Translation of Graphs

A translation moves a graph without changing its shape or size.

Vertical translation

\[y=\mathrm{f}(x)+d\]

translates the graph of \(y=\mathrm{f}(x)\) vertically by \(d\) units.

• If \(d>0\), the graph moves upwards.
• If \(d<0\), the graph moves downwards.

This translation is represented by the vector

\[\begin{pmatrix}0\\d\end{pmatrix}\]

Horizontal translation

\[y=\mathrm{f}(x-c)\]

translates the graph of \(y=\mathrm{f}(x)\) horizontally by \(c\) units.

• If \(c>0\), the graph moves to the right.
• If \(c<0\), the graph moves to the left.

This translation is represented by the vector

\[\begin{pmatrix}c\\0\end{pmatrix}\]

Combined translation

The graph \(y=\mathrm{f}(x-c)+d\) is obtained by translating \(y=\mathrm{f}(x)\) by the vector

\[\begin{pmatrix}c\\d\end{pmatrix}\]

Equivalently, every point moves according to \((x,y)\mapsto(x+c,y+d)\).