Analyzing Graphs of Absolute Value Functions

To graph the given absolute value function, we will make a table of values. We can substitute some $x\text{-}$values into the rule and solve for the corresponding $y\text{-}$values. Recall that the absolute value of a number is always positive. Let's first calculate the $y\text{-}$value for $x=\text{-}4.$ We have found that $f(\text{-}4)=19.$ Therefore, the point $(\text{-}4,19)$ lies on the graph. We can find other points on the graph in the same way.

$x$	$\frac{4}{3}|(x-5)|+7$	$f(x)$
$\text{-}4$	$\frac{4}{3}|(\text{-}4-5)|+7$	$19$
$\text{-}1$	$\frac{4}{3}|(\text{-}1-5)|+7$	$15$
$2$	$\frac{4}{3}|(2-5)|+7$	$11$
$5$	$\frac{4}{3}|(5-5)|+7$	$7$
$8$	$\frac{4}{3}|(8-5)|+7$	$11$
$11$	$\frac{4}{3}|(11-5)|+7$	$15$
$14$	$\frac{4}{3}|(14-5)|+7$	$19$

To draw the graph, we will plot and connect these points. Recall that the graph of an absolute value function has a V-shape!