Skill: Write transformed functions using mapping notation

To write a mapping, compare the transformed function with the form

$$g(x)=a f\bigl(b(x-h)\bigr)+k.$$

A point $(x,y)$ on the original graph $y=f(x)$ moves according to

$$(x,y)\mapsto\left(\frac{x}{b}+h,\ ay+k\right).$$

The horizontal factor is divided by $b$, while the vertical factor is multiplied by $a$.

Worked example

Suppose

$$g(x)=-2f\bigl(3(x-4)\bigr)+5.$$

We identify the values:

• $a=-2$: reflect in the $x$-axis and stretch vertically by a factor of $2$
• $b=3$: compress horizontally by a factor of $3$
• $h=4$: shift right $4$
• $k=5$: shift up $5$

Substitute these values into the mapping rule:

$$(x,y)\mapsto\left(\frac{x}{3}+4,\ -2y+5\right).$$

So the mapping notation is

$$\boxed{(x,y)\mapsto\left(\frac{x}{3}+4,\ -2y+5\right)}.$$

For example, if $(1,1)$ is a point on $y=f(x)$, its transformed point is

$$\left(\frac{1}{3}+4,\ -2(1)+5\right) = \left(\frac{13}{3},3\right).$$

Thus, $(1,1)$ moves to $\left(\frac{13}{3},3\right)$ on the graph of $g(x)$.