Analyzing Graphs of Absolute Value Functions

a

To match the graph with its function, we will graph the function by making a table of values. To graph the given function using a table of values, we will substitute some

x -

values into the rule and solve for the corresponding

y -

values. Let's first calculate the

y -

value that corresponds to

x = - 10 .

y = ∣ x + 6 ∣ - 4

Substitute

x = - 10

y = ∣ - 10 + 6 ∣ - 4

AddTermsAdd terms

y = ∣ - 4 ∣ - 4

AbsNeg

∣ - 4 ∣ = 4

y = 4 - 4

SubTermSubtract term

y = 0

We have found that

y = 0

when

x = - 10 .

Therefore, the point

(- 10, 0)

lies on the graph. We can find other points on the graph in the same way.

$x$	$∣ x + 6 ∣ - 4$	$y$
$- 10$	$∣ - 10 + 6 ∣ - 4$	$0$
$- 8$	$∣ - 8 + 6 ∣ - 4$	$- 2$
$- 6$	$∣ - 6 + 6 ∣ - 4$	$- 4$
$- 4$	$∣ - 4 + 6 ∣ - 4$	$- 2$
$- 2$	$∣ - 2 + 6 ∣ - 4$	$0$
$0$	$∣ 0 + 6 ∣ - 4$	$2$
$2$	$∣ 2 + 6 ∣ - 4$	$4$
$4$	$∣ 4 + 6 ∣ - 4$	$6$

To draw the graph, we will plot and connect the obtained points. Do not forget that the graph of an absolute value function has a V-shape!

The graph given in option C is the graph of the function.

b

Starting from

x = - 4,

let's make a table of values for the absolute value function.

$x$	$∣ x - 6 ∣ - 4$	$y$
$- 4$	$∣ - 4 - 6 ∣ - 4$	$6$
$- 2$	$∣ - 2 - 6 ∣ - 4$	$4$
$0$	$∣ 0 - 6 ∣ - 4$	$2$
$2$	$∣ 2 - 6 ∣ - 4$	$0$
$4$	$∣ 4 - 6 ∣ - 4$	$- 2$
$6$	$∣ 6 - 6 ∣ - 4$	$- 4$
$8$	$∣ 8 - 6 ∣ - 4$	$- 2$
$10$	$∣ 10 - 6 ∣ - 4$	$0$

To draw the graph, we will plot and connect the obtained points. Do not forget that the graph of an absolute value function has a V-shape!

As a result, the graph of the function is given in option A.

c

We will proceed in the same way as we did in the previous parts.

$x$	$∣ x - 6 ∣ + 4$	$y$
$0$	$∣ 0 - 6 ∣ + 4$	$10$
$2$	$∣ 2 - 6 ∣ + 4$	$8$
$4$	$∣ 4 - 6 ∣ + 4$	$6$
$6$	$∣ 6 - 6 ∣ + 4$	$4$
$8$	$∣ 8 - 6 ∣ + 4$	$6$
$10$	$∣ 10 - 6 ∣ + 4$	$8$

To draw the graph, we will plot and connect the obtained points. Do not forget that the graph of an absolute value function has a V-shape!

The graph we have above matches the graph given in option B.

Analyzing Graphs of Absolute Value Functions 1.12 - Solution