Skill: Identify vertical translations of functions

A vertical translation changes the output of a function by adding or subtracting a constant:

$$g(x)=f(x)+k$$

• If $k>0$, the graph moves up $k$ units.
• If $k<0$, the graph moves down $\vert k \vert$ units.
• The $x$-coordinates stay the same, so the domain and shape do not change.
• The range moves up or down by the same amount.

Worked example

Let

$$f(x)=\sqrt{x}$$

and

$$g(x)=\sqrt{x}-2.$$

Step 1: Compare $g(x)$ with $f(x)$.

Since $f(x)=\sqrt{x}$, we can write

$$g(x)=f(x)-2.$$

This has the form $f(x)+k$ with $k=-2$.

Step 2: Identify the direction and amount.

Because $k=-2$, every output decreases by $2$. Therefore, the graph of $f$ moves down 2 units.

Step 3: Check how points move.

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

$$(0,0) \quad \text{and} \quad (4,2).$$

Subtracting $2$ from each output gives

$$(0,0)\to(0,-2)$$

and

$$(4,2)\to(4,0).$$

The $x$-coordinates remain unchanged.

Step 4: Describe the domain and range.

The domain of $f(x)=\sqrt{x}$ is

$$x\ge 0.$$

The vertical translation does not change the domain, so $g$ also has domain $x\ge 0$.

The range of $f$ is $y\ge 0$. Moving the graph down $2$ units changes the range to

$$y\ge -2.$$

Thus, $g(x)=\sqrt{x}-2$ is the graph of $f(x)=\sqrt{x}$ shifted down 2 units. Be careful: a change inside the function, such as $f(x+2)$, would be a horizontal translation instead.