Graph functions using combined transformations

A transformed graph is represented by $g(x)=a f(b(x-h))+k$ , where $h$ and $k$ translate the graph, $|a|$ and $|b|$ produce vertical and horizontal stretches or compressions, and negative $a$ or $b$ reflect it across the $x$ - or $y$ -axis. The learner maps points and interprets changes to domain, range, intercepts, and key features, recognizing that the horizontal scale factor is $1/|b|$ and that transformations inside the function act before those outside it. This scope focuses on standard real-valued functions and finite combinations of these transformations, not abstract or more generalized compositions.

Detailed Explanation: Graph functions using combined transformations

To graph a function of the form

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

start with points $(u,f(u))$ on the graph of $f$ . Their transformed coordinates are

\boxed{\left(h+\frac{u}{b},\; af(u)+k\right)}.

This rule shows that:

$h$ shifts the graph horizontally.
$k$ shifts it vertically.
$\vert a \vert$ vertically stretches or compresses the graph.
$\vert b \vert$ changes the horizontal scale by the factor $\frac{1}{ \vert b \vert }$ .
A negative $a$ reflects across the $x$ -axis.
A negative $b$ reflects across the $y$ -direction relative to $x=h$ .

The changes inside $f$ happen before the changes outside it, so be especially careful with $b(x-h)$ .

Example

Let

f(x)=\sqrt{x}

and graph

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

Here,

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

Choose some easy points on $f(x)=\sqrt{x}$ :

(0,0),\quad (1,1),\quad (4,2),\quad (9,3).

For each point $(u,f(u))$ , use

\left(1+\frac{u}{-3},\;-2f(u)+4\right).

Now transform the points:

\begin{array}{c|c} \text{Original point} & \text{Transformed point} \\ \hline (0,0) & \left(1,4\right)\\ (1,1) & \left(\frac23,2\right)\\ (4,2) & \left(-\frac13,0\right)\\ (9,3) & \left(-2,-2\right) \end{array}

Plot these transformed points and connect them with the shape of a square-root graph. Since $a=-2$ , the graph is reflected across the $x$ -axis, vertically stretched by a factor of $2$ , and then shifted up $4$ units. Since $b=-3$ , the graph is reflected horizontally and compressed by a factor of $\frac13$ .

You can also determine the domain and range:

The original function $f(x)=\sqrt{x}$ has domain $[0,\infty)$ .
Its input must satisfy

-3(x-1)\ge 0,

so $x\le 1$ . Therefore,

\boxed{\text{Domain: }(-\infty,1]}.

Since $\sqrt{x}\ge 0$ ,

-2\sqrt{x}+4\le 4.

Therefore,

\boxed{\text{Range: }(-\infty,4]}.

The graph has an $x$ -intercept at

\left(-\frac13,0\right),

and its endpoint is $(1,4)$ .

Learn by doing: Graph functions using combined transformations

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Function Transformations - Mapping Notation - Double Transformation to Action

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