Investigation

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## Warm-up solution

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 $y=\frac{1}{2}\;f(x).$