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College Algebra with Corequisite Support

3.6 Absolute Value Functions

College Algebra with Corequisite Support3.6 Absolute Value Functions
  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Chapter Review
      1. Key Terms
      2. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 7.1 Systems of Linear Equations: Two Variables
    3. 7.2 Systems of Linear Equations: Three Variables
    4. 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 7.4 Partial Fractions
    6. 7.5 Matrices and Matrix Operations
    7. 7.6 Solving Systems with Gaussian Elimination
    8. 7.7 Solving Systems with Inverses
    9. 7.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 8.1 The Ellipse
    3. 8.2 The Hyperbola
    4. 8.3 The Parabola
    5. 8.4 Rotation of Axes
    6. 8.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 9.1 Sequences and Their Notations
    3. 9.2 Arithmetic Sequences
    4. 9.3 Geometric Sequences
    5. 9.4 Series and Their Notations
    6. 9.5 Counting Principles
    7. 9.6 Binomial Theorem
    8. 9.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
  12. Index

Learning Objectives

In this section you will:

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

Corequisite Skills

Learning Objectives

  1. Solve absolute value equations (IA 2.7.1)
  2. Identify graphs of absolute value functions (IA 3.6.2)

Objective 1: Solve absolute value equations (IA 2.7.1)

Recall that in its basic form, ๐‘“(๐‘ฅ)=|๐‘ฅ|, the absolute value function is one of our toolkit functions. The absolute value function is often thought of as providing the distance the number is from zero on a number line. Numerically, for whatever the input value is, the output is the magnitude of this value.

The absolute value function can be defined as a piecewise function

f(x)=|x|=-xf(x)=|x|=-x , when x<0x<0 or xx , when xโ‰ฅ0xโ‰ฅ0

Example 1

Solve absolute value equations

โ“ Solve for x: |x|=5

โ“‘ Solve for x: |2xโ€“5|=3

Practice Makes Perfect

Solve absolute value equations.

1.

Solve for x: |x|=0|x|=0

2.

Solve for x: |x|=2|x|=2

3.

Solve for x: |xโ€“2|=6|xโ€“2|=6

4.

Solve for x: |2x+1|=7|2x+1|=7

5.

Solve for t: |3tโ€“1|=โ€“2|3tโ€“1|=โ€“2

6.

Solve for z: |2zโ€“3|โ€“4=1|2zโ€“3|โ€“4=1

7.

Solve for x: โ€“2|xโ€“3|+8=โ€“4โ€“2|xโ€“3|+8=โ€“4

Objective 2: Identify and graph absolute value functions (IA 3.6.2)

Absolute value functions have a โ€œVโ€ shaped graph. If scanning this function from left to right the corner is the point where the graph changes direction.

f(x)=|x|=-xf(x)=|x|=-x , when x<0x<0 or xx , when xโ‰ฅ0xโ‰ฅ0

This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). The point (0, 0) where the line changes slope is called the vertex. Next to the graph are the following: โ€œf of x equalsabsolute value of xโ€, โ€œDomain: (negative infinity, infinity)โ€, and โ€œRange: [0, infinity)โ€.
Practice Makes Perfect

Identify and graph absolute value functions. Graph each of the following functions. Label at least one point on your graph.

8.

y=|x|โ€“2y=|x|โ€“2

blank graph
9.

y=|2x|+3y=|2x|+3

blank graph
10.

y=|xโ€“4|+3y=|xโ€“4|+3

blank graph
11.

y=โ€“|x+6|โ€“4y=โ€“|x+6|โ€“4

blank graph
12.

y=โ€“|xโ€“2|y=โ€“|xโ€“2|

blank graph
13.

y=3|x+1|+4y=3|x+1|+4

blank graph
14.

Find the domain and range of the following absolute value function.

The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from 0 to 12. The vertex is at the point (0, 4). The line goes through the points (negative 2, 6) and (2, 6).
The Milky Way.
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)

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.

Absolute Value Function

The absolute value function can be defined as a piecewise function

f(x)=| x |={ x if xโ‰ฅ0 โˆ’x if x<0 f(x)=| x |={ x if xโ‰ฅ0 โˆ’x if x<0

Example 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 ยฑ1%,ยฑ5%, ยฑ1%,ยฑ5%, or ยฑ10%. ยฑ10%.

Suppose we have a resistor rated at 680 ohms, ยฑ5%. ยฑ5%. Use the absolute value function to express the range of possible values of the actual resistance.

Try It #1

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

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.

Graph of an absolute function
Figure 2

Figure 3 shows the graph of y=2| xโ€“3 |+4. y=2| xโ€“3 |+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.

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

Example 2

Writing an Equation for an Absolute Value Function Given a Graph

Write an equation for the function graphed in Figure 4.

Graph of an absolute function.
Figure 4

Analysis

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

Q&A

If we couldnโ€™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 x and f(x). f(x).

f(x)=a|xโˆ’3|โˆ’2 f(x)=a|xโˆ’3|โˆ’2

Now substituting in the point (1, 2)

2 = a|1โˆ’3|โˆ’2 4 = 2a a = 2 2 = a|1โˆ’3|โˆ’2 4 = 2a a = 2
Try It #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.

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).

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

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=| 2xโˆ’6 |, 8=| 2xโˆ’6 |, 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.

2xโˆ’6 = 8 or 2xโˆ’6 = โˆ’8 2x = 14 2x = โˆ’2 x = 7 x = โˆ’1 2xโˆ’6 = 8 or 2xโˆ’6 = โˆ’8 2x = 14 2x = โˆ’2 x = 7 x = โˆ’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, | 2xโˆ’1 |=3,or | 5x+2 |โˆ’4=9 | x |=4, | 2xโˆ’1 |=3,or | 5x+2 |โˆ’4=9

Solutions to Absolute Value Equations

For real numbers A A and B B , an equation of the form | A |=B, |A|=B, with Bโ‰ฅ0, Bโ‰ฅ0, will have solutions when A=B A=B or A=โˆ’B. A=โˆ’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 |=B to write A=B A=B or โˆ’A=B, โˆ’A=B, assuming B>0. B>0.
  3. Solve for x. x.

Example 3

Finding the Zeros of an Absolute Value Function

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

Try It #3

For the function f(x)=| 2xโˆ’1 |โˆ’3, f(x)=| 2xโˆ’1 |โˆ’3, find the values of x x such that f(x)=0. f(x)=0.

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+| 3xโˆ’5 |=1. 2+| 3xโˆ’5 |=1.

Media

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

3.6 Section Exercises

Verbal

1.

How do you solve an absolute value equation?

2.

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

3.

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?

4.

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

Algebraic

5.

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

6.

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

7.

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

8.

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.

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

9.

f(x)=4| xโˆ’3 |+4 f(x)=4| xโˆ’3 |+4

10.

f(x)=โˆ’3| xโˆ’2 |โˆ’1 f(x)=โˆ’3| xโˆ’2 |โˆ’1

11.

f(x)=โˆ’2| x+1 |+6 f(x)=โˆ’2| x+1 |+6

12.

f(x)=โˆ’5|x+2|+15 f(x)=โˆ’5|x+2|+15

13.

f(x)=2|xโˆ’1|โˆ’6 f(x)=2|xโˆ’1|โˆ’6

14.

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

15.

f(x)=โˆ’|xโˆ’9|+16 f(x)=โˆ’|xโˆ’9|+16

Graphical

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

16.

y=|xโˆ’1| y=|xโˆ’1|

17.

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

18.

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

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

19.

y=| x |โˆ’2 y=| x |โˆ’2

20.

y=โˆ’| x | y=โˆ’| x |

21.

y=โˆ’| x |โˆ’2 y=โˆ’| x |โˆ’2

22.

y=โˆ’| xโˆ’3 |โˆ’2 y=โˆ’| xโˆ’3 |โˆ’2

23.

f(x)=โˆ’|xโˆ’1|โˆ’2 f(x)=โˆ’|xโˆ’1|โˆ’2

24.

f(x)=โˆ’|x+3|+4 f(x)=โˆ’|x+3|+4

25.

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

26.

f(x)=3| xโˆ’2 |+3 f(x)=3| xโˆ’2 |+3

27.

f(x)=| 2xโˆ’4 |โˆ’3 f(x)=| 2xโˆ’4 |โˆ’3

28.

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

29.

f(x)=โˆ’| xโˆ’1 |โˆ’3 f(x)=โˆ’| xโˆ’1 |โˆ’3

30.

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

31.

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

Technology

32.

Use a graphing utility to graph f(x)=10|xโˆ’2| f(x)=10|xโˆ’2| on the viewing window [ 0,4 ]. [ 0,4 ]. Identify the corresponding range. Show the graph.

33.

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

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

34.

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

35.

f(x)=4ร— 10 9 | xโˆ’(5ร— 10 9 ) |+2ร— 10 9 f(x)=4ร— 10 9 | xโˆ’(5ร— 10 9 ) |+2ร— 10 9

Extensions

For the following exercises, solve the inequality.

36.

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.

37.

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

Real-World Applications

38.

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 x represents the distance from city B to city A, express this using absolute value notation.

39.

The true proportion p p of 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.

40.

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 x for the score.

41.

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

42.

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 x inches, express the tolerance using absolute value notation.

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