In previous modules, we have examined specific transformations separately.

Translations

We saw that for some function, \(y=f(x)\), \(g(x)=f(x-h)+k\) is a translation of \(y=f(x)\)
horizontally \(h\) units (to the right if \(h \gt 0\) or to the left if \(h \lt 0\))
and vertically \(k\) units (up if \(k \gt 0\) and down if \(k \lt 0\)).

On the graph, \(g(x)=f(x-5)-2\) is a translation of \(y=f(x)\). It has been translated \(5\) units to the right and
\(2\) units down.

Reflections

We then saw that for some function, \(y=f(x)\), \(g(x) = -f(x)\) is a reflection of \(y=f(x)\) about the \(x\)-axis.

And we saw that for some function, \(y=f(x)\), \(g(x) = f(-x)\) is a reflection of \(y=f(x)\) about the \(y\)-axis.

This generalizes so that for some function, \(y=f(x)\), \(g(x) = af(bx)\) is at least a reflection of \(y=f(x)\) about the \(x\)-axis if \(a \lt 0\) and about the \(y\)-axis if \(b \lt 0\). (Horizontal and/or vertical stretches may also be involved.)

Vertical Stretches

Next, we saw that for some function, \(y=f(x)\), \(g(x) = af(x)\), where \(a \gt 0\), is a vertical stretch of \(y=f(x)\) about the \(x\)-axis by a factor of \(a\).

We know that there is a reflection if \(a \lt 0\), so we generalize.

For some function, \(y=f(x)\), \(g(x)=af(x)\) is a vertical stretch of \(y=f(x)\) about the \(x\)-axis by a factor of \(\lvert a \rvert\).

If \(\lvert a \rvert \gt 1\), \(g(x)\) stretches away from the \(x\)-axis.

If \(\lvert a \rvert \lt 1\), \(g(x)\) compresses towards the \(x\)-axis.

On the graph, \(g(x)=-2f(x)\) is a reflection of \(y=f(x)\) about the \(x\)-axis and a vertical stretch of \(y=-f(x)\) about the \(x\)-axis by a factor of \(2\).

Recall that \(A\) is called an invariant point since it is on the line of reflection.

Horizontal Stretches

Finally, we saw that for some function, \(y=f(x)\), \(g(x)=f(bx)\), where \(b \gt 0\), is a horizontal stretch of \(y=f(x)\) about the \(y\)-axis by a factor of \(\dfrac{1}{b}\).

We know that there is a reflection if \(b \lt 0 \), so we generalize.

For some function \(y=f(x)\), \(g(x)=f(bx)\) is a horizontal stretch of \(y=f(x)\) about the \(y\)-axis by a factor of \(\dfrac{1}{\lvert b \rvert}\).

If \(\lvert b \rvert \gt 1\), \(g(x)\) compresses towards the \(y\)-axis.

If \(\lvert b \rvert \lt 1\), \(g(x)\) stretches away from the \(y\)-axis.

On the graph, \(g(x)=f(-2x)\) is a reflection of \(y=f(x)\) about the \(y\)-axis and a horizontal stretch of \(y=f(-x)\) about the \(y\)-axis by a factor of \(\dfrac{1}{2}\).