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Absolute Value Functions

Module by: First Last. E-mail the author

Summary: In this section you will:

  • Graph an absolute value function.
  • Solve an absolute value equation.

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Figure 1: Distances in deep space can be measured in all directions. As such, it is useful to consider distance in terms of absolute values. (credit: "s58y"/Flickr)
The Milky Way.

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will continue our investigation of absolute value functions.

Understanding Absolute Value

Recall that in its basic form f(x)=| x |, f(x)=| x |,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.

A General Note: Absolute Value Function:

The absolute value function can be defined as a piecewise function

f(x)=| x |={ x if x0 x if x<0 f(x)=| x |={ x if x0 x if x<0

Example 1

Problem 1

Using Absolute Value to Determine Resistance

Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often ±1%,±5%, ±1%,±5%,or ±10%. ±10%.

Suppose we have a resistor rated at 680 ohms, ±5%. ±5%.Use the absolute value function to express the range of possible values of the actual resistance.

Solution

We can find that 5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance R Rin ohms,

| R680 |34 |R680|34

Try It:

Exercise 1

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

Solution

using the variable p pfor passing, | p80 |20 | p80 |20

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 2
Graph of an absolute function

Figure 3 shows the graph of y=2| x3 |+4. y=2| x3 |+4. The graph of y=| x | y=| x | 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 ( 3,4 ) ( 3,4 )for this transformed function.

Figure 3
Graph of the different types of transformations for an absolute function.

Example 2

Problem 1

Writing an Equation for an Absolute Value Function Given a Graph

Write an equation for the function graphed in Figure 4.

Figure 4
Graph of an absolute function.
Solution

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.

Figure 5
Graph of two transformations for an absolute function at (3, -2).

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.

Figure 6
Graph of two transformations for an absolute function at (3, -2) and describes the ratios between the two different transformations.

From this information we can write the equation

f(x) = 2|x3|2, treating the stretch as a vertical stretch,or f(x) = |2(x3)|2, treating the stretch as a horizontal compression. f(x) = 2|x3|2, treating the stretch as a vertical stretch,or f(x) = |2(x3)|2, treating the stretch as a horizontal compression.
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.

Q&A:

If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it?

Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for x xand f(x). f(x).

f(x)=a|x3|2 f(x)=a|x3|2

Now substituting in the point (1, 2)

2 = a|13|2 4 = 2a a = 2 2 = a|13|2 4 = 2a a = 2

Try It:

Exercise 2

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.

Solution

f(x)=| x+2 |+3 f(x)=| x+2 |+3

Q&A:

Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?

Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.

No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see Figure 7).

Figure 7: (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.
Graph of the different types of transformations for an absolute function.

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 8=| 2x6 |, 8=| 2x6 |,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.

2x6 = 8 or 2x6 = −8 2x = 14 2x = −2 x = 7 x = −1 2x6 = 8 or 2x6 = −8 2x = 14 2x = −2 x = 7 x = −1

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,

| x |=4, | 2x1 |=3,or | 5x+2 |4=9 | x |=4, | 2x1 |=3,or | 5x+2 |4=9

A General Note: Solutions to Absolute Value Equations:

For real numbers A A and B B, an equation of the form | A |=B, |A|=B, with B0, B0, will have solutions when A=B A=B or A=B. A=B. If B<0, B<0, the equation | A |=B |A|=B has no solution.

How To:

Given the formula for an absolute value function, find the horizontal intercepts of its graph.

  1. Isolate the absolute value term.
  2. Use | A |=B | A |=Bto write A=B A=Bor −A=B, −A=B, assuming B>0. B>0.
  3. Solve for x. x.

Example 3

Problem 1

Finding the Zeros of an Absolute Value Function

For the function f(x)=|4x+1|7, f(x)=|4x+1|7, find the values of x x such that f(x)=0. f(x)=0.

Solution
0 = |4x+1|7 Substitute 0 for f(x). 7 = |4x+1| Isolate the absolute value on one side of the equation. 7 = 4x+1 or −7 = 4x+1 Break into two separate equations and solve. 6 = 4x −8 = 4x x = 6 4 =1.5 x = 8 4 =2 0 = |4x+1|7 Substitute 0 for f(x). 7 = |4x+1| Isolate the absolute value on one side of the equation. 7 = 4x+1 or −7 = 4x+1 Break into two separate equations and solve. 6 = 4x −8 = 4x x = 6 4 =1.5 x = 8 4 =2

The function outputs 0 when x= 3 2 x= 3 2 or x=2. x=2. See Figure 8.

Figure 8
Graph an absolute function with x-intercepts at -2 and 1.5.

Try It:

Exercise 3

For the function f(x)=| 2x1 |3, f(x)=| 2x1 |3, find the values of x x such that f(x)=0. f(x)=0.

Solution

x=1 x=1or x=2 x=2

Q&A:

Should we always expect two answers when solving | A |=B? | A |=B?

No. We may find one, two, or even no answers. For example, there is no solution to 2+| 3x5 |=1. 2+| 3x5 |=1.

Media:

Access these online resources for additional instruction and practice with absolute value.

Key Concepts

  • Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See Example 1.
  • The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See Example 2.
  • In an absolute value equation, an unknown variable is the input of an absolute value function.
  • If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See Example 3.

Section Exercises

Verbal

Exercise 4

How do you solve an absolute value equation?

Solution

Isolate the absolute value term so that the equation is of the form |A|=B. |A|=B.Form one equation by setting the expression inside the absolute value symbol, A, A,equal to the expression on the other side of the equation, B. B.Form a second equation by setting A Aequal to the opposite of the expression on the other side of the equation, B. B.Solve each equation for the variable.

Exercise 5

How can you tell whether an absolute value function has two x-intercepts without graphing the function?

Exercise 6

When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

Solution

The graph of the absolute value function does not cross the x x-axis, so the graph is either completely above or completely below the x x-axis.

Exercise 7

How can you use the graph of an absolute value function to determine the x-values for which the function values are negative?

Algebraic

Exercise 8

Describe all numbers x xthat are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.

Exercise 9

Describe all numbers x xthat are at a distance of 1 2 1 2 from the number −4. Express this set of numbers using absolute value notation.

Solution

| x+4 |= 1 2 | x+4 |= 1 2

Exercise 10

Describe the situation in which the distance that point x xis from 10 is at least 15 units. Express this set of numbers using absolute value notation.

Exercise 11

Find all function values f(x) f(x)such that the distance from f(x) f(x)to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.

Solution

|f(x)8|<0.03 |f(x)8|<0.03

For the following exercises, find the x- and y-intercepts of the graphs of each function.

Exercise 12

f(x)=4| x3 |+4 f(x)=4| x3 |+4

Exercise 13

f(x)=3| x2 |1 f(x)=3| x2 |1

Solution

( 0,7 ); ( 0,7 );no x x-intercepts

Exercise 14

f(x)=2| x+1 |+6 f(x)=2| x+1 |+6

Exercise 15

f(x)=5|x+2|+15 f(x)=5|x+2|+15

Solution

(0,5),(1,0),(5,0) (0,5),(1,0),(5,0)

Exercise 16

f(x)=2|x1|6 f(x)=2|x1|6

Solution

(0,4),(4,0),(2,0) (0,4),(4,0),(2,0)

Exercise 17

f(x)=|2x+1|13 f(x)=|2x+1|13

Solution

(0,12),(6,0),(7,0) (0,12),(6,0),(7,0)

Exercise 18

f(x)=|x9|+16 f(x)=|x9|+16

Solution

(0,7),(25,0),(7,0) (0,7),(25,0),(7,0)

Graphical

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

Exercise 19

y=|x1| y=|x1|

Solution
Graph of an absolute function with points at (-1, 2), (0, 1), (1, 0), (2, 1), and (3, 2).

Exercise 20

y=|x+1| y=|x+1|

Exercise 21

y=|x|+1 y=|x|+1

Solution
Graph of an absolute function with points at (-2, 3), (-1, 2), (0, 1), (1, 2), and (2, 3).

For the following exercises, graph the given functions by hand.

Exercise 22

y=| x |2 y=| x |2

Exercise 23

y=| x | y=| x |

Solution
Graph of an absolute function.

Exercise 24

y=| x |2 y=| x |2

Exercise 25

y=| x3 |2 y=| x3 |2

Solution
Graph of an absolute function.

Exercise 26

f(x)=|x1|2 f(x)=|x1|2

Exercise 27

f(x)=|x+3|+4 f(x)=|x+3|+4

Solution
Graph of an absolute function.

Exercise 28

f(x)=2|x+3|+1 f(x)=2|x+3|+1

Exercise 29

f(x)=3| x2 |+3 f(x)=3| x2 |+3

Solution
Graph of an absolute function.

Exercise 30

f(x)=| 2x4 |3 f(x)=| 2x4 |3

Exercise 31

f( x )=| 3x+9 |+2 f( x )=| 3x+9 |+2

Solution
Graph of an absolute function.

Exercise 32

f(x)=| x1 |3 f(x)=| x1 |3

Exercise 33

f(x)=| x+4 |3 f(x)=| x+4 |3

Solution
Graph of an absolute function.

Exercise 34

f(x)= 1 2 | x+4 |3 f(x)= 1 2 | x+4 |3

Technology

Exercise 35

Use a graphing utility to graph f(x)=10|x2| f(x)=10|x2| on the viewing window [ 0,4 ]. [ 0,4 ]. Identify the corresponding range. Show the graph.

Solution

range: [ 0,20 ] [ 0,20 ]

Graph of an absolute function.

Exercise 36

Use a graphing utility to graph f(x)=100|x|+100 f(x)=100|x|+100on the viewing window [ 5,5 ]. [ 5,5 ].Identify the corresponding range. Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

Exercise 37

f(x)=0.1| 0.1(0.2x) |+0.3 f(x)=0.1| 0.1(0.2x) |+0.3

Solution

x- x- intercepts:

Graph of an absolute function.

Exercise 38

f(x)=4× 10 9 | x(5× 10 9 ) |+2× 10 9 f(x)=4× 10 9 | x(5× 10 9 ) |+2× 10 9

Extensions

For the following exercises, solve the inequality.

Exercise 39

If possible, find all values of a a such that there are no x- x- intercepts for f(x)=2| x+1 |+a. f(x)=2| x+1 |+a.

Exercise 40

If possible, find all values of a asuch that there are no y y-intercepts for f(x)=2| x+1 |+a. f(x)=2| x+1 |+a.

Solution

There is no solution for a athat will keep the function from having a y y-intercept. The absolute value function always crosses the y y-intercept when x=0. x=0.

Real-World Applications

Exercise 41

Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and x xrepresents the distance from city B to city A, express this using absolute value notation.

Exercise 42

The true proportion p pof people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

Solution

| p0.08 |0.015 | p0.08 |0.015

Exercise 43

Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable x xfor the score.

Exercise 44

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using x xas the diameter of the bearing, write this statement using absolute value notation.

Solution

| x5.0 |0.01 | x5.0 |0.01

Exercise 45

The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is x xinches, express the tolerance using absolute value notation.

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