Understanding Absolute Value

Recall that in its basic form [latex]\,f\left(x\right)=|x|,\,[/latex] the absolute value function is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign. Knowing this, we can use absolute value functions to solve some kinds of real-world problems.

Graphing an Absolute Value Function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in Figure 2.

Figure 3 shows the graph of [latex]\,y=2|x–3|+4.\,[/latex] The graph of [latex]\,y=|x|\,[/latex] has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at [latex]\,\left(3,4\right)\,[/latex] for this transformed function.

Writing an Equation for an Absolute Value Function Given a Graph

Write an equation for the function graphed in Figure 4.

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The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. See Figure 5.

We also notice that the graph appears vertically stretched because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance, as shown in Figure 6.

From this information, we can write the equation

[latex]\begin{array}{cccc}\hfill f\left(x\right)& =& 2|x-3|-2,\hfill & \phantom{\rule{0 em}{0ex}}\text{treating the stretch as }a\text{ vert. stretch,or}\hfill \\ \hfill f\left(x\right)& =& |2\left(x-3\right)|-2,\hfill & \phantom{\rule{-0.7 em}{0ex}}\text{treating the stretch as }a\text{ hori. compression}.\hfill \end{array}[/latex]

Analysis

Note that these equations are algebraically equivalent—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.

Solving an Absolute Value Equation

In Other Type of Equations, we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as [latex]\,8=|2x-6|,\,[/latex] we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.

Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,

Finding the Zeros of an Absolute Value Function

For the function[latex]\,f\left(x\right)=|4x+1|-7,[/latex] find the values of[latex]\,x\,[/latex]such that[latex]\,f\left(x\right)=0.[/latex]

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[latex]\begin{array}{cccccccc}\hfill 0& =& |4x+1|-7\hfill & & & & & \text{Substitute 0 for }f\left(x\right).\hfill \\ \hfill 7& =& |4x+1|\hfill & & & & &\text{Isolate the absolute value}.\hfill \\ & & & & & & & \\ & & & & & & & \\ & & & & & & & \\ \hfill 7& =& 4x+1\hfill & \text{or}& \hfill \phantom{\rule{0em}{0ex}}-7& =& 4x+1\hfill & \text{Break into two equations}.\hfill \\ \hfill 6& =& 4x\hfill & & \hfill -8& =& 4x\hfill & \\ & & & & & & & \\ \hfill x& =& \frac{6}{4}=1.5\hfill & & \hfill x& =& \frac{-8}{4}=-2\hfill & \end{array}[/latex]

The function outputs 0 when[latex]\,x=\frac{3}{2}\,[/latex]or[latex]\,x=-2.[/latex] See Figure 8.