Here is a graph of a function, \(f(x)\).

For each of the following transformations, sketch the transformed graph and write its equation in terms of \(f\).

Translation by \(\begin{pmatrix}-2\\0\end{pmatrix}\)

This is a translation to the left by \(2\) units.

Algebraically, when translating \(2\) units left we replace \(x\) with \(x+2\), so we can write \[y=f(x+2).\]

Translation by \(\begin{pmatrix}0\\-4\end{pmatrix}\)

The graph is translated \(4\) units downwards.

We can write this as \[y=f(x)-4.\]You can think of this as replacing \(y\) with \(y+4\), which is consistent with how we treated \(x\) in the first example.

Stretch by factor \(3\) parallel to \(x\)

The graph is made \(3\) times as wide.

We replace the \(x\) with \(\frac{x}{3}\), so we can write this as \[y=f\left(\frac{x}{3}\right).\]

Stretch by factor \(\frac{1}{2}\) parallel to \(y\)

The graph is made half as tall.

We replace \(y\) with \(2y\) and write this as \[y=\frac{1}{2}\;f(x).\]