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Mathematics LibreTexts

2.5 Absolute Value Functions

So far in this chapter we have been studying the behavior of linear functions. The Absolute Value Function is a piecewise-defined function made up of two linear functions.  The name, Absolute Value Function, should be familiar to you from Section 1.2.  In its basic formFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.gif it is one of our toolkit functions. 

Definition: Absolute Value Function

The absolute value function can be defined as

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif

 

The absolute value function is commonly used to determine the distance between two numbers on the number line. Given two values a and b, then File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image006.gif will give the distance, a positive quantity, between these values, regardless of which value is larger.

Example 1

Describe all values, x, within a distance of 4 from the number 5.

We want the distance between x and 5 to be less than or equal to 4.  The distance can be represented using the absolute value, giving the expression

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image008.gif

Example 2

A 2010 poll reported 78% of Americans believe that people who are gay should be able to serve in the US military, with a reported margin of error of 3%[1]. The margin of error tells us how far off the actual value could be from the survey value[2].  Express the set of possible values using absolute values.

Since we want the size of the difference between the actual percentage, p, and the reported percentage to be less than 3%,

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image010.gif

Try it Now

1. Students who score within 20 points of  80 will pass the test. Write this as a distance from 80 using the absolute value notation.

Important Features

The most significant feature of the absolute value graph is the corner point where the graph changes direction.  When finding the equation for a transformed absolute value function, this point is very helpful for determining the horizontal and vertical shifts.

 

Example 3

Write an equation for the function graphed below.

Graphs

The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 and down 2 from the basic toolkit function.  We might also notice that the graph appears stretched, since the linear portions have slopes of 2 and -2.  From this information we can write the equation:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image014.gif,  treating the stretch as a vertical stretch

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image016.gif,  treating the stretch as a horizontal compression

 

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

 

If you had not been able to determine the stretch based on the slopes of the lines, you can solve for the stretch factor by putting in a known pair of values for x and f(x)

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image018.gif                Now substituting in the point (1, 2)

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image020.gif

Try it Now

2. Given the description of the transformed absolute value function write the equation.  The absolute value function is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units,

 

The graph of an absolute value function will have a vertical intercept when the input is zero.  The graph may or may not have horizontal intercepts, depending on how the graph has been shifted and reflected.  It is possible for the absolute value function to have zero, one, or two horizontal intercepts.

 

Zero horizontal intercepts                   One                             Two

Graphs  Graphs  Graphs

To find the horizontal intercepts, we will need to solve an equation involving an absolute value.

Notice that the absolute value function is not one-to-one, so typically inverses of absolute value functions are not discussed.

Solving Absolute Value Equations

To solve an equation like File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image028.gif, we can notice that the absolute value will be equal to eight if the quantity inside the absolute value were 8 or -8.  This leads to two different equations we can solve independently:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image030.gif       or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image032.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image034.gif                       File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image036.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image038.gif                          File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image040.gif

 

 

Solutions to Absolute Value Equations

An equation of the form File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image042.gif, with File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image044.gif, will have solutions when

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image046.gif or File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image048.gif

 

Example 4

Find the horizontal intercepts of the graph of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image050.gif

 

The horizontal intercepts will occur when File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image052.gif.  Solving,

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image054.gif          Isolate the absolute value on one side of the equation

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image056.gif                Now we can break this into two separate equations:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image058.gif     or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image060.gif

 

The graph has two horizontal intercepts, at File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image062.gif and x = -2

Example 5

Solve File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image064.gif

Isolating the absolute value on one side the equation,

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image064.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image067.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image069.gif

At this point, we notice that this equation has no solutions – the absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value.

Try it Now

3. Find the horizontal & vertical intercepts for the functionFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image071.gif

Solving Absolute Value Inequalities

When absolute value inequalities are written to describe a set of values, like the inequality File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image008.gif we wrote earlier, it is sometimes desirable to express this set of values without the absolute value, either using inequalities, or using interval notation.

We will explore two approaches to solving absolute value inequalities:

  1. Using the graph
  2. Using test values

 

Example 6

Solve File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image008.gif

 

With both approaches, we will need to know first where the corresponding equality is true.  In this case we first will find where File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image074.gif.  We do this because the absolute value is a nice friendly function with no breaks, so the only way the function values can switch from being less than 4 to being greater than 4 is by passing through where the values equal 4.  Solve File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image074.gif,

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image076.gif       or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image078.gif

 

To use a graph, we can sketch the function File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image080.gif.  To help us see where the outputs are 4, the line File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image082.gif could also be sketched.

Graphs

 

On the graph, we can see that indeed the output values of the absolute value are equal to 4 at x = 1 and x = 9.  Based on the shape of the graph, we can determine the absolute value is less than or equal to 4 between these two points, when File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image086.gif.  In interval notation, this would be the interval [1,9].

 

As an alternative to graphing, after determining that the absolute value is equal to 4 at x = 1 and x = 9, we know the graph can only change from being less than 4 to greater than 4 at these values.  This divides the number line up into three intervals:  x<1, 1<x<9, and x>9.  To determine when the function is less than 4, we could pick a value in each interval and see if the output is less than or greater than 4.

Interval         Test x               f(x)                  <4 or >4?

x<1                0                      File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image088.gif        greater

1<x<9           6                      File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image090.gif         less

x>9                11                    File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image092.gif       greater

 

Since File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image086.gif is the only interval in which the output at the test value is less than 4, we can conclude the solution to File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image008.gif is File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image086.gif.

Example 7

Given the function File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image095.gif, determine for what x values the function values are negative.

We are trying to determine where f(x) < 0, which is when File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image097.gif.  We begin by isolating the absolute value:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image099.gif       when we multiply both sides by -2, it reverses the inequality

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image101.gif

Next we solve for the equality File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image103.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image105.gif     or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image107.gif

We can now either pick test values or sketch a graph of the function to determine on which intervals the original function value are negative.  Notice that it is not even really important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image109.gif and File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image111.gif, and that the graph has been reflected vertically. 

Graphs

From the graph of the function, we can see the function values are negative to the left of the first horizontal intercept at File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image109.gif, and negative to the right of the second intercept at File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image111.gif.  This gives us the solution to the inequality:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image115.gif

In interval notation, this would be File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image117.gif

 

Try it Now

4.  SolveFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image119.gif

Important Topics of this Section

  • The properties of the absolute value function
  • Solving absolute value equations
  • Finding intercepts
  • Solving absolute value inequalities

 

Try it Now Answers

1.  Using the variable p, for passing,  File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image121.gif

2. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image071.gif

3. f(0) = 1, so the vertical intercept is at (0,1).  f(x)= 0 when x = -5 and x = 1 so the horizontal intercepts are at (-5,0) & (1,0)

4. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image124.gifor File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image126.gif; in interval notation this would be File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image128.gif

Section 2.5 Exercises

 

Write an equation for each transformation of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image130.gif

 

1.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image132.jpg                             2.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image134.jpg

 

3.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image136.jpg                             4.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image138.jpg

 


Sketch a graph of each function

5. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image140.gif                                       6. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image142.gif

7. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image144.gif                                       8. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image146.gif

9. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image148.gif                                      10. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image150.gif

 

Solve each the equation

11. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image152.gif                                              12. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image154.gif 
13. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image156.gif                                              14. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image158.gif

15. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image160.gif                                        16. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image162.gif 
 


Find the horizontal and vertical intercepts of each function

17. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image164.gif                                  18. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image166.gif

19. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image168.gif                                  20. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image170.gif

Solve each inequality

21. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image172.gif                                                  22. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image174.gif

23. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image176.gif                                                  24. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image178.gif

25. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image180.gif                                                26. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image182.gif
 

 


[1] http://www.pollingreport.com/civil.htm, retrieved August 4, 2010

[2] Technically, margin of error usually means that the surveyors are 95% confident that actual value falls within this range.

Contributors

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)