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{{ courseTrack.displayTitle }} {{ printedBook.courseTrack.name }} {{ printedBook.name }} # Analyzing Graphs of Absolute Value Functions

Concept

## Absolute Value Function

An absolute value function is any function that contains the absolute value of an expression. In other words, any function that can be described as a transformation of the 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 $|\text{-} 5| = 5 \quad \text{and} \quad |5|=5.$ This means, the points $(\text{-} 5,5)$ 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. The function $f(x)=|x|$ is the parent function to the absolute value functions.
Exercise

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

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, $|\text{-} 2| = 2.$ Let's first calculate the $y$-value that corresponds with $x=\text{-} 3.$
$f(x)=|x-2|-3$
$f({\color{#0000FF}{\text{-} 3}})=|{\color{#0000FF}{\text{-} 3}}-2|-3$
$f(\text{-} 3)=|\text{-} 5|-3$
$f(\text{-} 3)= 5-3$
$f(\text{-} 3)= 2$
We have found that $f(\text{-} 3)= 2.$ Thus, the point $(\text{-} 3,2)$ lies on the graph. We can find other points on the graph in the same way.
$x$ $|x-2|-3$ $f(x)$
${\color{#0000FF}{\text{-} 2}}$ $|{\color{#0000FF}{\text{-} 2}}-2|-3$ $1$
${\color{#0000FF}{\text{-} 1}}$ $|{\color{#0000FF}{\text{-} 1}}-2|-3$ $0$
${\color{#0000FF}{0}}$ $|{\color{#0000FF}{0}}-2|-3$ $\text{-} 1$
${\color{#0000FF}{1}}$ $|{\color{#0000FF}{1}}-2|-3$ $\text{-} 2$
${\color{#0000FF}{2}}$ $|{\color{#0000FF}{2}}-2|-3$ $\text{-} 3$
${\color{#0000FF}{3}}$ $|{\color{#0000FF}{3}}-2|-3$ $\text{-} 2$
${\color{#0000FF}{4}}$ $|{\color{#0000FF}{4}}-2|-3$ $\text{-} 1$
${\color{#0000FF}{5}}$ $|{\color{#0000FF}{5}}-2|-3$ $0$
${\color{#0000FF}{6}}$ $|{\color{#0000FF}{6}}-2|-3$ $1$
${\color{#0000FF}{7}}$ $|{\color{#0000FF}{7}}-2|-3$ $2$

To draw the graph, we can plot these points, then connect them with a V-shaped curved. info Show solution Show solution
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 $\left| 2x^2-8 \right|=5.$

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.
Exercise

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

Solution

When solving an equation graphically, the first step is to rearrange the equation so that the constant term is alone on one side. $|2x-5|-3=0 \quad \Leftrightarrow \quad |2x-5|=3$ The left-hand side of the equation can now be expressed as a function, $f(x)=|2x-5|.$ We'll draw the graph of $f.$ The solutions to the equation $y=|2x-5|=3$ are the $x$-coordinates of the points on the graph that have the $y$-coordinate $3.$ The equation has the solutions $x=1$ and $x=4.$ We can verify the solutions by testing them in the equation. We'll start with $x=1.$
$|2x-5|=3$
$|2\cdot {\color{#0000FF}{1}}-5|\stackrel{?}{=}3$
$|2-5|\stackrel{?}{=}3$
$|\text{-} 3|\stackrel{?}{=}3$
$3=3$
Since $x=1$ makes a true statement, it is a solution to the equation. Next, we'll verify $x=4$ in the same way. $|2({\color{#0000FF}{4}})-5|=3 \quad \Leftrightarrow \quad |3|=3$

Since $x=4$ makes a true statement, it is also a solution. Thus, the equation has the solutions $x=1 \quad \text{and} \quad x=4.$

info Show solution Show solution