Exercise 57 Page 162

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 $v .$

$x$	$- 2 ∣ 3 x + 1 ∣ + 4$	Simplify	$v (x)$
$- 2$	$- 2 ∣ 3 (- 2) + 1 ∣ + 4$	$- 2 ∣ - 5 ∣ + 4$	$- 6$
$- \frac{1}{3}$	$- 2 ∣ ∣ ∣ ∣ ∣ 3 (- \frac{1}{3}) + 1 ∣ ∣ ∣ ∣ ∣ + 4$	$- 2 ∣ 0 ∣ + 4$	$4$
$0$	$- 2 ∣ 3 (0) + 1 ∣ + 4$	$- 2 ∣ 1 ∣ + 4$	$2$
$1$	$- 2 ∣ 3 (1) + 1 ∣ + 4$	$- 2 ∣ 4 ∣ + 4$	$- 4$

Now it is time for doing the same with function $w .$

$x$	$3 ∣ 3 x + 1 ∣ - 6$	Simplify	$w (x)$
$- 1$	$3 ∣ 3 (- 1) + 1 ∣ - 6$	$3 ∣ - 2 ∣ - 6$	$0$
$- \frac{1}{3}$	$3 ∣ ∣ ∣ ∣ ∣ 3 (- \frac{1}{3}) + 1 ∣ ∣ ∣ ∣ ∣ - 6$	$3 ∣ 0 ∣ - 6$	$- 6$
$0$	$3 ∣ 3 (0) + 1 ∣ - 6$	$3 ∣ 1 ∣ - 6$	$- 3$
$1$	$3 ∣ 3 (1) + 1 ∣ - 6$	$3 ∣ 4 ∣ - 6$	$6$

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

Comparison

As we can see from the graphs, the function $w$ is a vertical stretch by a factor of $\frac{3}{2}$ and a reflection in the $x -$ axis of the graph of function $v .$