Write transformed functions using mapping notation

Mapping notation describes how a point $(x,y)$ on $y=f(x)$ moves to a point on its transformed graph: for $g(x)=a f\bigl(b(x-h)\bigr)+k$ , the mapping is $(x,y)\mapsto\left(\frac{x}{b}+h, ay+k\right)$ , where signs indicate reflections and magnitudes indicate stretches or compressions. This understanding connects graphical transformations with function equations, especially the reciprocal effect of horizontal scale factors; it is limited to standard real-valued transformations of familiar parent functions rather than abstract or multidimensional generalizations.

Detailed Explanation: 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 $g(x)$ .

Learn by doing: Write transformed functions using mapping notation

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Function Transformations - Mapping Notation - Action to Double Transformation

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