Apply vertical and horizontal stretches and compressions

A learner interprets $y=af(x)$ as a vertical scaling, with $|a|>1$ stretching outputs and $0<|a|<1$ compressing them, and $y=f(bx)$ as a horizontal scaling, whose factor is $1/|b|$ : $|b|>1$ compresses inputs and $0<|b|<1$ stretches them. They can predict corresponding changes to key points, domain, range, and graph shape, understanding that horizontal factors act on input values rather than output values; this scope uses constant real scale factors and excludes more advanced generalized transformations.

Detailed Explanation: Apply vertical and horizontal stretches and compressions

A transformation of the form

y=af(x)

changes the outputs of $f$ :

If $(|a|>1)$ , the graph is stretched vertically by factor $(|a|)$ .
If $(0<|a|<1)$ , the graph is compressed vertically by factor $(|a|)$ .

A transformation of the form

y=f(bx)

changes the inputs of $f$ :

The horizontal scale factor is $\frac{1}{|b|}$ .
If $(|b|>1)$ , the graph is horizontally compressed.
If $(0<|b|<1)$ , the graph is horizontally stretched.

For a point $((u,v))$ on $(y=f(x))$ , the corresponding point on

y=af(bx)

\left(\frac{u}{b},av\right).

The input is divided by $b$ , while the output is multiplied by $a$ .

Worked example

Let

f(x)=\sqrt{x}

and consider

g(x)=2f(3x)=2\sqrt{3x}.

Describe the transformations and find some corresponding points.

Step 1: Identify the vertical transformation

The factor outside the function is $2$ :

g(x)=2f(3x).

Therefore, the outputs are multiplied by $2$ . This is a vertical stretch by factor $2$ .

Step 2: Identify the horizontal transformation

The factor inside the function is $3$ :

f(3x).

Since $(|3|>1)$ , the graph is horizontally compressed by factor

\frac{1}{3}.

This means the input coordinates become one-third as large. The horizontal factor is not $3$ ; it is $\frac13$ .

Step 3: Transform key points

Some points on $(y=f(x)=\sqrt{x})$ are

(0,0),\qquad (1,1),\qquad (4,2).

Use

(u,v)\longrightarrow \left(\frac{u}{3},2v\right).

Thus,

(0,0)\longrightarrow \left(0,0\right),

(1,1)\longrightarrow \left(\frac13,2\right),

and

(4,2)\longrightarrow \left(\frac43,4\right).

So points on $(g(x)=2\sqrt{3x})$ include

(0,0),\qquad \left(\frac13,2\right),\qquad \left(\frac43,4\right).

Step 4: Check the domain and range

For $(g(x)=2\sqrt{3x})$ , the expression inside the square root must be nonnegative:

3x\ge 0 \quad\Rightarrow\quad x\ge 0.

The domain is therefore

[0,\infty).

Because the square-root outputs are nonnegative and are multiplied by $2$ , the range is

[0,\infty).

The graph is vertically stretched, horizontally compressed, and still begins at $((0,0))$ .

Learn by doing: Apply vertical and horizontal stretches and compressions

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Function Transformations (Definition) - Single Transformation Function to Graph

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