First, we will take a look at the given graph.

As we can see, each $y -$ value of $f$ is shifted $1$ unit to the right and $3$ units down. To write the function rule $g (x)$ let's look at a table of transformations.

Transformations of $f (x)$
Horizontal Translations	$Translation right h units, h > 0 y = f (x - h)$
Horizontal Translations	$Translation left h units, h > 0 y = f (x + h)$
Vertical Translations	$Translation up k units, k > 0 y = f (x) + k$
Vertical Translations	$Translation down k units, k > 0 y = f (x) - k$

Using the table, let's apply the first transformation, a horizontal translation

1

unit to the right, by substituting

(x - 1)

for each

x -

term.

f (x) = ∣ x ∣

Substitute

$x = x - 1$

f (x - 1) = ∣ x - 1 ∣

Next, we will apply the second transformation, a translation

3

units down, by subtracting

3

from both sides of the equation.

f (x - 1) = ∣ x - 1 ∣

SubEqn

$LHS - 3 = RHS - 3$

f (x - 1) - 3 = ∣ x - 1 ∣ - 3

Finally, we can replace

f (x - 1) - 3

with

g (x)

and complete the function rule.

g (x) = ∣ x - 1 ∣ - 3

Exercise