Describe combined transformations of a function

Combined transformations are interpreted in equations such as $g(x)=a\,f(b(x-h))+k$ , where $h$ and $k$ translate the graph, $a$ produces vertical stretch or compression and possible reflection, and $b$ produces horizontal scale change and possible reflection; the learner connects each parameter to its effect on the graph and recognizes that horizontal changes act oppositely inside the function. The understanding includes combining transformations of familiar parent functions and predicting corresponding graph, equation, and key-point changes, but not arbitrary function composition, multivariable transformations, or more advanced abstract treatments.

Detailed Explanation: Describe combined transformations of a function

A combined transformation can be written as

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

Each parameter changes the graph:

$h$ : moves the graph right $h$ units if $h>0$ or left if $h<0$ .
$k$ : moves the graph up $k$ units if $k>0$ or down if $k<0$ .
$a$ : stretches vertically by a factor of $\vert a \vert$ . If $a<0$ , it also reflects the graph across the $x$ -axis.
$b$ : changes the graph horizontally by a factor of $\frac{1}{ \vert b \vert }$ . If $b<0$ , it also reflects the graph across the $y$ -axis.

The horizontal change acts oppositely because it is inside the function. For example, $f(3x)$ makes the graph narrower by a factor of $3$ , not wider.

Example

Let

f(x)=x^2

and

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

We identify the parameters:

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

Apply the transformations to the graph of $y=x^2$ :

$h=1$ : shift the graph right $1$ unit.
$b=3$ : compress it horizontally by a factor of $\frac13$ .
$a=-2$ : stretch it vertically by a factor of $2$ and reflect it across the $x$ -axis.
$k=4$ : shift it up $4$ units.

The vertex of $f(x)=x^2$ is $(0,0)$ . After the transformations, it becomes

(1,4).

So the new graph is a downward-opening parabola with vertex $(1,4)$ .

To find its equation, substitute $f(x)=x^2$ :

g(x)=-2\left[3(x-1)\right]^2+4.

Simplifying gives

\boxed{g(x)=-18(x-1)^2+4}.

For another key point, the point $(1,1)$ on $y=x^2$ changes horizontally by $\frac13$ and vertically by the factor $-2$ , then shifts. Its new coordinates are

x=1+\frac13=\frac43,\qquad y=4+(-2)(1)=2.

Thus $(1,1)$ becomes $\left(\frac43,2\right)$ . These points confirm the shape and position of the transformed graph.

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Function Transformations (Definition) - Double Definition (Values) to Transformation

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Describe combined transformations of a function

Detailed Explanation: Describe combined transformations of a function

Learn by doing: Describe combined transformations of a function

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Function Transformations (Definition) - Double Definition (Values) to Transformation