Exercise 62 Page 162

Let's start by recalling the general forms for horizontal and vertical translations of an absolute value function. When considering a horizontal translation, we subtract the number of units that we are translating a graph. Since $x+p$ can be written as $x-(\text{-}p)$ 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 $y=|x|.$

$x$	$|x|$	$y=|x|$
$\text{-}2$	$|\text{-}2|$	$2$
$\text{-}1$	$|\text{-}1|$	$1$
$0$	$|0|$	$0$
$1$	$|1|$	$1$
$2$	$|2|$	$2$

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

$x$	$|x+2|$	$y=|x+2|$
$\text{-}2$	$|\text{-}2+2|$	$0$
$\text{-}1$	$|\text{-}1+2|$	$1$
$0$	$|0+2|$	$2$
$1$	$|1+2|$	$3$
$2$	$|2+2|$	$4$

Finally let's make a table for $y=|x+p|,$ with $p=\text{-}2.$ The resulting function is $y=|x-2|.$

$x$	$|x-2|$	$y=|x-2|$
$\text{-}2$	$|\text{-}2-2|$	$4$
$\text{-}1$	$|\text{-}1-2|$	$3$
$0$	$|0-2|$	$2$
$1$	$|1-2|$	$1$
$2$	$|2-2|$	$0$

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.