Analyzing Graphs of Absolute Value Functions

Concept

Absolute Value Function

An absolute value function is any function that contains the absolute value of a variable expression. These functions can also be described as any function that is a transformation of the absolute value parent function, $f (x) = ∣ x ∣ .$ Since the absolute value of an expression is never negative, the graph of the function $f (x) = ∣ x ∣$ will always lie on or above the $x$ -axis. Note that $∣ - 5 ∣ = 5 and ∣ 5 ∣ = 5 .$ This means that the points $(- 5, 5)$ and $(5, 5)$ both lie on $f (x) = ∣ x ∣ .$ As it turns out, these points lie directly across from each other. In fact, this symmetry exists for all inverse input values. Thus, absolute value graphs have a distinct V-shape.

Any function belonging to the absolute value function family can be written using the equation $y = a ∣ x - h ∣ + k$

where

a,

h,

and

k

are real numbers and

a \neq = 0 .

Graph the absolute value function

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Exercise

Graph the absolute value function $f (x) = ∣ x - 2 ∣ - 3$ using a table of values.

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Solution

To graph the given function using a table of values, we can substitute various

x

-values into the rule and solve for the corresponding

y

-values. The absolute value of a number is always the positive value of that number. For instance,

∣ - 2 ∣ = 2 .

Let's first calculate the

y

-value that corresponds with

x = - 3 .

f (x) = ∣ x - 2 ∣ - 3

Substitute

$x = - 3$

f (- 3) = ∣ - 3 - 2 ∣ - 3

SubTerm

Subtract term

f (- 3) = ∣ - 5 ∣ - 3

AbsNeg

$∣ - 5 ∣ = 5$

f (- 3) = 5 - 3

SubTerm

Subtract term

f (- 3) = 2

We have found that

f (- 3) = 2 .

Thus, the point

(- 3, 2)

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

$x$	$∣ x - 2 ∣ - 3$	$f (x)$
$- 2$	$∣ - 2 - 2 ∣ - 3$	$1$
$- 1$	$∣ - 1 - 2 ∣ - 3$	$0$
$0$	$∣ 0 - 2 ∣ - 3$	$- 1$
$1$	$∣ 1 - 2 ∣ - 3$	$- 2$
$2$	$∣ 2 - 2 ∣ - 3$	$- 3$
$3$	$∣ 3 - 2 ∣ - 3$	$- 2$
$4$	$∣ 4 - 2 ∣ - 3$	$- 1$
$5$	$∣ 5 - 2 ∣ - 3$	$0$
$6$	$∣ 6 - 2 ∣ - 3$	$1$
$7$	$∣ 7 - 2 ∣ - 3$	$2$

To draw the graph, we can plot these points, then connect them with a V-shaped curve.

Method

Solving Absolute Value Equations Graphically

An absolute value equation is an equation that contains the absolute value of a variable expression. An example of this kind of equation is $∣ ∣ ∣ 2 x^{2} - 8 ∣ ∣ ∣ = 5 .$

As is the case with most equations, these can be solved graphically. This is done by moving all terms except the constant term to one side. The function, which is the non-constant side of the equation, is then graphed and the solution(s) to the equation are found as the point(s) on the graph having the

y

-coordinate that equals the constant.

Solve the absolute value equation graphically

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Exercise

Solve the equation graphically. $∣ 2 x - 5 ∣ - 3 = 0$

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Solution

When solving an equation graphically, the first step is to rearrange the equation so that the constant term is alone on one side. $∣ 2 x - 5 ∣ - 3 = 0 \Leftrightarrow ∣ 2 x - 5 ∣ = 3$ The left-hand side of the equation can now be expressed as a function, $f (x) = ∣ 2 x - 5 ∣ .$ We'll draw the graph of $f .$