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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. f(x)=|x|. Since the absolute value of an expression is never negative, the graph of the function f(x)=xf(x)=|x| will always lie on or above the xx-axis. Note that -5=5and5=5. |\text{-} 5| = 5 \quad \text{and} \quad |5|=5. This means, the points (-5,5)(\text{-} 5,5) and (5,5)(5,5) both lie on f(x)=x.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)=xf(x)=|x| is the parent function to the absolute value functions.
Exercise

Graph the absolute value function f(x)=x23f(x)=|x-2|-3 using a table of values.

Solution
To graph the given function using a table of values, we can substitute various xx-values into the rule and solve for the corresponding yy-values. The absolute value of a number is always the positive value of that number. For instance, -2=2.|\text{-} 2| = 2. Let's first calculate the yy-value that corresponds with x=-3.x=\text{-} 3.
f(x)=x23f(x)=|x-2|-3
f(-3)=-323f({\color{#0000FF}{\text{-} 3}})=|{\color{#0000FF}{\text{-} 3}}-2|-3
f(-3)=-53f(\text{-} 3)=|\text{-} 5|-3
f(-3)=53f(\text{-} 3)= 5-3
f(-3)=2f(\text{-} 3)= 2
We have found that f(-3)=2.f(\text{-} 3)= 2. Thus, the point (-3,2)(\text{-} 3,2) lies on the graph. We can find other points on the graph in the same way.
xx x23|x-2|-3 f(x)f(x)
-2{\color{#0000FF}{\text{-} 2}} -223|{\color{#0000FF}{\text{-} 2}}-2|-3 11
-1{\color{#0000FF}{\text{-} 1}} -123|{\color{#0000FF}{\text{-} 1}}-2|-3 00
0{\color{#0000FF}{0}} 023|{\color{#0000FF}{0}}-2|-3 -1\text{-} 1
1{\color{#0000FF}{1}} 123|{\color{#0000FF}{1}}-2|-3 -2\text{-} 2
2{\color{#0000FF}{2}} 223|{\color{#0000FF}{2}}-2|-3 -3\text{-} 3
3{\color{#0000FF}{3}} 323|{\color{#0000FF}{3}}-2|-3 -2\text{-} 2
4{\color{#0000FF}{4}} 423|{\color{#0000FF}{4}}-2|-3 -1\text{-} 1
5{\color{#0000FF}{5}} 523|{\color{#0000FF}{5}}-2|-3 00
6{\color{#0000FF}{6}} 623|{\color{#0000FF}{6}}-2|-3 11
7{\color{#0000FF}{7}} 723|{\color{#0000FF}{7}}-2|-3 22

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 2x28=5. \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 yy-coordinate that equals the constant.
Exercise

Solve the equation graphically. 2x53=0 |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. 2x53=02x5=3 |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)=2x5.f(x)=|2x-5|. We'll draw the graph of f.f.

The solutions to the equation y=2x5=3 y=|2x-5|=3 are the xx-coordinates of the points on the graph that have the yy-coordinate 3.3.

The equation has the solutions x=1x=1 and x=4.x=4. We can verify the solutions by testing them in the equation. We'll start with x=1.x=1.
2x5=3|2x-5|=3
215=?3|2\cdot {\color{#0000FF}{1}}-5|\stackrel{?}{=}3
25=?3|2-5|\stackrel{?}{=}3
-3=?3|\text{-} 3|\stackrel{?}{=}3
3=33=3
Since x=1x=1 makes a true statement, it is a solution to the equation. Next, we'll verify x=4x=4 in the same way. 2(4)5=33=3 |2({\color{#0000FF}{4}})-5|=3 \quad \Leftrightarrow \quad |3|=3

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

info Show solution Show solution
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