Skill: 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

$$y=af(bx)$$

is

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

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 $((0,0))$.