Let's start by recalling the general forms for horizontal and vertical translations of an absolute value function.

Vertical translation : Horizontal translation : g (x) = ∣ x ∣ + k g (x) = ∣ x - h ∣

When considering a horizontal translation, we subtract the number of units that we are translating a graph. Since

x + p

can be written as

x - (- p)

and we know that

p

is a positive number, the graph of

y = ∣ x + p ∣

represents a horizontal translation of the graph of

y = ∣ x ∣

in the negative direction. Let's make a table of values for

y = ∣ x ∣ .

$x$	$∣ x ∣$	$y = ∣ x ∣$
$- 2$	$∣ - 2 ∣$	$2$
$- 1$	$∣ - 1 ∣$	$1$
$0$	$∣ 0 ∣$	$0$
$1$	$∣ 1 ∣$	$1$
$2$	$∣ 2 ∣$	$2$

Let's now make a table of values for $y = ∣ x + p ∣ .$ We will arbitrarily let $p = 2,$ resulting in the function $y = ∣ x + 2 ∣ .$

$x$	$∣ x + 2 ∣$	$y = ∣ x + 2 ∣$
$- 2$	$∣ - 2 + 2 ∣$	$0$
$- 1$	$∣ - 1 + 2 ∣$	$1$
$0$	$∣ 0 + 2 ∣$	$2$
$1$	$∣ 1 + 2 ∣$	$3$
$2$	$∣ 2 + 2 ∣$	$4$

Finally let's make a table for $y = ∣ x + p ∣,$ with $p = - 2 .$ The resulting function is $y = ∣ x - 2 ∣ .$

$x$	$∣ x - 2 ∣$	$y = ∣ x - 2 ∣$
$- 2$	$∣ - 2 - 2 ∣$	$4$
$- 1$	$∣ - 1 - 2 ∣$	$3$
$0$	$∣ 0 - 2 ∣$	$2$
$1$	$∣ 1 - 2 ∣$	$1$
$2$	$∣ 2 - 2 ∣$	$0$

We can plot and connect the obtained points in the same coordinate plane. Keep in mind that the graph of an absolute value function has a V shape.

Though it may seem counterintuitive, the graph of $y = ∣ x + 2 ∣$ is a negative horizontal translation of the graph of $y = ∣ x ∣$ by $2$ units. Conversely, the graph of $y = ∣ x - 2 ∣$ is a positive horizontal translation of the graph of $y = ∣ x ∣$ by $2$ units.

Exercise