Describe transformations using function notation

Function notation expresses a transformed function in relation to an original function, such as $g(x)=a\,f(b(x-h))+k$ , where $h$ and $k$ translate the graph, $a$ controls vertical stretch or reflection, and $b$ controls horizontal scale or reflection. The learner interprets these parameters consistently across equations, graphs, tables, and verbal descriptions, recognizing that changes inside $f$ affect inputs and therefore act in the opposite horizontal direction from their appearance; the scope is limited to standard translations, reflections, and stretches, not more advanced generalized coordinate transformations.

Detailed Explanation: Describe transformations using function notation

Function notation describes how a new function is changed from an original function. The standard form is

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

$h$ translates the graph horizontally: right if $h>0$ , left if $h<0$ .
$k$ translates the graph vertically: up if $k>0$ , down if $k<0$ .
$a$ $a$ controls vertical changes:
- $\vert a \vert >1$ : vertical stretch
- $0< \vert a \vert <1$ : vertical compression
- $a<0$ : reflection across the $x$ -axis
$b$ $b$ controls horizontal changes:
- $\vert b \vert >1$ : horizontal compression by a factor of $\frac{1}{ \vert b \vert }$
- $0< \vert b \vert <1$ : horizontal stretch by a factor of $\frac{1}{ \vert b \vert }$
- $b<0$ : reflection across the $y$ -axis

The horizontal change can seem backward because it occurs inside the function. For example, $(x-1)$ moves the graph right by $1$ , not left.

Worked example

Suppose

g(x)=-2f\bigl(3(x-1)\bigr)+4.

Describe the transformations from $f(x)$ to $g(x)$ .

First, compare the equation with

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

The parameters are

a=-2,\qquad b=3,\qquad h=1,\qquad k=4.

Now interpret each parameter:

Since $h=1$ , translate the graph right 1 unit.
Since $k=4$ , translate the graph up 4 units.
Since $a=-2$ , reflect the graph across the $x$ -axis and stretch it vertically by a factor of $2$ .
Since $b=3$ , compress the graph horizontally by a factor of $\frac{1}{3}$ .

Therefore, $g(x)$ is obtained from $f(x)$ by a horizontal compression by $\frac13$ , a reflection across the $x$ -axis, a vertical stretch by $2$ , a translation right by $1$ , and a translation up by $4$ .

A point description confirms the horizontal factor. If $(x,y)$ is on $f$ , then its corresponding point on $g$ is

\left(\frac{x}{3}+1,\,-2y+4\right).

The input coordinate is divided by $3$ and then shifted right by $1$ , while the output coordinate is multiplied by $-2$ and shifted up by $4.

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Function Transformations - Mapping Notation - Action to Double Transformation

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