# Functions

## Absolute Value Functions


### 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, 𝑓(𝑥)=|𝑥|, 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
  , when  or  , when  

### Practice Makes Perfect


Solve absolute value equations.


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


Absolute value functions have a “V” shaped graph. If scanning this function from left to right the corner is the point where the graph changes direction.

  , when  or  , when  

### Practice Makes Perfect

Identify and graph absolute value functions. Graph each of the following functions. Label at least one point on your graph.
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 
    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 .
 shows the graph of 
  The graph of 
  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 
    for this transformed 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 
    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,

### Key Concepts

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

### Section Exercises


### Verbal


### Algebraic

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

### Graphical

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

### Technology

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

### Extensions

For the following exercises, solve the inequality.

### Real-World Applications
