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Graphing Absolute Value Functions

Just as the shape of the graph of a linear function is unique to that family, absolute value functions have distinct graphs. Below, these graphs and their characteristics will be explored.
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.

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Theory

Describing Key Features of Absolute Value Functions

Key features are certain characteristics of the graphs of functions that are noteworthy. For absolute value functions, the graph's intercepts, in which intervals the function is increasing or decreasing, minimum and maximum values, and end behavior are important.
Exercise

The graph shows the function f(x)=x+31.f(x)=|x+3|-1.

Describe the function's key features including intercepts, the intervals for which it increases and decreases, its minimum and maximum values, and its end behavior. Then, show the features on the graph.

Solution

To begin, we'll describe each of the features. A graph's xx- and yy-intercepts are the points where the graph intersects with the xx-axis and yy-axis, respectively. It can be seen that f(x)f(x) intersects the xx-axis at two different points, (-4,0)(\text{-} 4,0) and (-2,0).(\text{-} 2,0). The function intersects the yy-axis at (0,2).(0,2). Therefore, the intercepts are (-4,0),(-2,0),and (0,2). (\text{-} 4,0), (\text{-} 2, 0), \text{and } (0,2). Since ff is an absolute value function, it has a V-shaped graph. This means the function will have both an increasing and a decreasing interval. Looking from left to right on the graph, it can be seen that from the left side of the graph until x=-3,x=\text{-} 3, f(x)f(x) decreases. Additionally, from x=-3x=\text{-} 3 to the right side of the graph, f(x)f(x) increases. Thus, we can express the increasing and decreasing intervals of ff as follows. Decreasing Interval:-<x<-3Increasing Interval:--3<x<+\begin{aligned} \textbf{Decreasing Interval} &: \quad \text{-} \infty < x < \text{-} 3 \\ \textbf{Increasing Interval} &: \quad \phantom{\text{-}} \text{-} 3 < x < + \infty \end{aligned} Looking at the graph, we can see that both the left end and the right end extend upward. Thus, the end behavior of f(x)f(x) can be written as follows. As x-, f(x)+As x+, f(x)+\begin{aligned} \text{As}\ x \rightarrow \text{-} \infty , && \ f(x) \rightarrow +\infty \\ \text{As}\ x \rightarrow +\infty , && \ f(x) \rightarrow +\infty \end{aligned} Since the function is decreasing in one interval and increasing in another the function has a minimum. From the graph we can tell that this is in the point (-3,-1).(\text{-} 3, \text{-} 1). Let us now show these features on the graph.

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