Exercise 32 Page 160

To graph the functions without going through the entire process of transforming the parent function, we can make two tables of values. Then we only need to plot the ordered pairs that we find.

Graph

Let's begin with creating a table of values of function $u .$

$x$	$∣ x - 1 ∣ + 2$	Simplify	$u (x)$
$- 3$	$∣ - 3 - 1 ∣ + 2$	$∣ - 4 ∣ + 2$	$6$
$- 1$	$∣ - 1 - 1 ∣ + 2$	$∣ - 2 ∣ + 2$	$4$
$1$	$∣ 1 - 1 ∣ + 2$	$∣ 0 ∣ + 2$	$2$
$3$	$∣ 3 - 1 ∣ + 2$	$∣ 2 ∣ + 2$	$4$

Now let's do the same with function $v .$

$x$	$∣ ∣ ∣ ∣ ∣ - \frac{1}{2} x - 1 ∣ ∣ ∣ ∣ ∣ + 2$	Simplify	$v (x)$
$- 4$	$∣ ∣ ∣ ∣ ∣ - \frac{1}{2} (- 4) - 1 ∣ ∣ ∣ ∣ ∣ + 2$	$∣ 1 ∣ + 2$	$3$
$- 2$	$∣ ∣ ∣ ∣ ∣ - \frac{1}{2} (- 2) - 1 ∣ ∣ ∣ ∣ ∣ + 2$	$∣ 0 ∣ + 2$	$2$
$2$	$∣ ∣ ∣ ∣ ∣ - \frac{1}{2} (2) - 1 ∣ ∣ ∣ ∣ ∣ + 2$	$∣ - 2 ∣ + 2$	$4$
$4$	$∣ ∣ ∣ ∣ ∣ - \frac{1}{2} (4) - 1 ∣ ∣ ∣ ∣ ∣ + 2$	$∣ - 3 ∣ + 2$	$5$

Finally we can plot these ordered pairs on one coordinate plane.

Comparison

We can rewrite $v (x)$ as $u (- \frac{1}{2} x),$ which is a horizontal stretch by a factor of $2$ followed by a reflection in the $y -$ axis. Note that the $y -$ intercept is the same for both graphs.