# 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 form it is one of our toolkit functions.

Definition: Absolute Value Function |
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The absolute value function can be defined as
The absolute value function is commonly used to determine the distance between two numbers on the number line. Given two values |

Example 1 |
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Describe all values, We want the distance between |

Example 2 |
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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, |

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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 |
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Write an equation for the function graphed below.
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: , treating the stretch as a vertical stretch , 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 Now substituting in the point (1, 2) |

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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

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 , 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:

or

Solutions to Absolute Value Equations

An equation of the form , with , will have solutions when

or

Example 4 |
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Find the horizontal intercepts of the graph of
The horizontal intercepts will occur when . Solving, Isolate the absolute value on one side of the equation Now we can break this into two separate equations: or
The graph has two horizontal intercepts, at and |

Example 5 |
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Solve Isolating the absolute value on one side the equation, 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. |

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3. Find the horizontal & vertical intercepts for the function

### Solving Absolute Value Inequalities

When absolute value inequalities are written to describe a set of values, like the inequality 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:

- Using the graph
- Using test values

Example 6 |
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Solve
With both approaches, we will need to know first where the corresponding or
To use a graph, we can sketch the function . To help us see where the outputs are 4, the line could also be sketched.
On the graph, we can see that indeed the output values of the absolute value are equal to 4 at
As an alternative to graphing, after determining that the absolute value is equal to 4 at
1<
Since is the only interval in which the output at the test value is less than 4, we can conclude the solution to is . |

Example 7 |
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Given the function , determine for what We are trying to determine where when we multiply both sides by -2, it reverses the inequality Next we solve for the equality or 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 and , and that the graph has been reflected vertically.
From the graph of the function, we can see the function values are negative to the left of the first horizontal intercept at , and negative to the right of the second intercept at . This gives us the solution to the inequality: In interval notation, this would be |

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4. Solve

### 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,

2.

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. or ; in interval notation this would be

### Section 2.5 Exercises

Write an equation for each transformation of

1. 2.

3. 4.

Sketch a graph of each function

5. 6.

7. 8.

9. 10.

Solve each the equation

11. 12.

13. 14.

15. 16.

Find the horizontal and vertical intercepts of each function

17. 18.

19. 20.

Solve each inequality

21. 22.

23. 24.

25. 26.

[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)