An **absolute value function** is an important function in algebra that consists of the variable in the absolute value bars. The general form of the absolute value function is f(x) = a |x - h| + k and the most commonly used form of this function is f(x) = |x|, where a = 1 and h = k = 0. The range of this function f(x) = |x| is always non-negative and on expanding the absolute value function f(x) = |x|, we can write it as x, if x ≥ 0 and -x, if x < 0.

In this article, we will explore the definition, various properties, and formulas of the absolute value function. We will learn graphing absolute value functions and determine the horizontal and vertical shifts in their graph. We shall solve various examples based related to the function for a better understanding of the concept.

1. | What is Absolute Value Function? |

2. | Absolute Value Function Definition |

3. | Absolute Value Function Graph |

4. | Absolute Value Equation |

5. | Graphing Absolute Value Functions |

6. | FAQs on Absolute Value Function |

## What is Absolute Value Function?

An absolute value function is a function in algebra where the variable is inside the absolute value bars. This function is also known as the modulus function and the most commonly used form of the absolute value function is f(x) = |x|, where x is a real number. Generally, we can represent the absolute value function as, f(x) = a |x - h| + k, where a represents how far the graph stretches vertically, h represents the horizontal shift and k represents the vertical shift from the graph of f(x) = |x|. If the value of 'a' is negative, the graph opens downwards and if it is positive, the graph opens upwards.

## Absolute Value Function Definition

The absolute value function is defined as an algebraic expression in absolute bar symbols. Such functions are commonly used to find distance between two points. Some of the examples of absolute value functions are:

- f(x) = |x|
- g(x) = |3x - 7|
- f(x) = |-x + 9|

All the above given absolute value functions have non-negative values, that is, their range is all real numbers except negative numbers. All these functions change their nature (increasing or decreasing) after a point. We can find those points by expressing the absolute value function f(x) = a |x - h| + k as,

f(x) = a (x - h) + k, if (x - h) ≥ 0 and

= – a (x - h) + k, if (x - h) < 0

## Absolute Value Function Graph

In this section, we will understand how to plot the graph of the common form of the absolute value function f(x) = |x| whose formula can also be expressed as f(x) = x, if x ≥ 0 and -x, if x < 0. Let us consider different points and determine the value of the function using the formula and plot them on a graph.

x | f(x) = |x| |
---|---|

-5 | 5 |

-4 | 4 |

-3 | 3 |

-2 | 2 |

-1 | 1 |

0 | 0 |

1 | 1 |

2 | 2 |

3 | 3 |

4 | 4 |

5 | 5 |

## Absolute Value Equation

Now that we have understood the meaning of the absolute value function, now we will understand the meaning of the absolute value equation f(x) = a |x - h| + k and how the values of a, h, k affect the value of the function.

- The value of 'a' determines how the graph of f(x) stretches vertically
- The value of 'h' tells the horizontal shift
- The value of 'k' tells the vertical shift

The vertex of the absolute value equation f(x) = a |x - h| + k is given by (h, k). We can also find the vertex of f(x) = a |x - h| + k using the formula (x - h) = 0. On determining the value of x, we substitute the value into the equation to find the value of k.

Let us consider an example and find the vertex of an absolute value equation.

**Example 1:** Consider the modulus function f(x) = |x|. Find its vertex.

**Solution:** Compare the function f(x) = |x| with f(x) = a |x - h| + k. We have a = 1, h = k = 0. So, the vertex of the function is (h, k) = (0, 0).

**Example 2:** Find the vertex f(x) = |x - 7| + 2.

**Solution:** On comparing f(x) = |x - 7| + 2 with f(x) = a |x - h| + k, we have the vertex (h, k) = (7, 2).

We can find it using the formula. So, we have (x - 7) = 0

⇒ x = 7

Now, substitute x = 7 into the equation f(x) = |x - 7| + 2, we have

f(x) = |7 - 7| + 2

= 0 + 2

= 2

So, the vertex of absolute value equation f(x) = |x - 7| + 2 using the formula is (7, 2).

## Graphing Absolute Value Functions

In this section, we will learn graphing absolute value functions of the form f(x) = a |x - h| + k. The graph of an absolute value function is always either 'V-shaped or inverted 'V-shaped depending upon the value of 'a' and the (h, k) gives the vertex of the graph. Let us plot the graph of two absolute value functions below.

f(x) = 2 |x + 2| + 1 and g(x) = -2 |x - 2| + 3

On comparing the two absolute value functions with the general form, a is positive in f(x), so it will open upwards and its vertex is (-2, 1). For g(x), the value of a = -2 which is negative, so the graph will open downwards and its vertex is (2, 3). The image below shows the graph of the absolute value functions f(x) and g(x).

**Important Notes on Absolute Value Function**

- The general form of the absolute value function is f(x) = a |x - h| + k, where (h, k) is the vertex of the graph.
- An absolute value function is a function in algebra where the variable is inside the absolute value bars.
- The graph of an absolute value function is always either 'V-shaped or inverted 'V-shaped depending upon the value of 'a'.

**☛ Related Articles:**

- Constant Function
- Inverse of a Function
- Graphing Functions

## FAQs on Absolute Value Function

### What is Absolute Value Function?

An **absolute value function** is an important function in algebra that consists of the variable in the absolute value bars. The general form of the absolute value function is f(x) = a |x - h| + k, where (h, k) is the vertex of the function.

### What is an Example of Absolute Value Function?

Some of the examples of absolute value functions are:

- f(x) = |x|
- g(x) = 2 |3x - 5| + 5
- f(x) = |-x - 9|
- f(x) = 3 |x|

### How To Find the Vertex of an Absolute Value Function?

The general form of the absolute value function is f(x) = a |x - h| + k, where (h, k) is the vertex of the function. So, to find the vertex of the function, we compare the two equations and determine the values of h and k.

### What Does the Value of k Do to the Absolute Value Function?

The value of 'k' in f(x) = a |x - h| + k tells us the vertical shift from the graph of f(x) = |x|. The graph moves upwards if k > 0 and moves downwards if k < 0.

### Why is An Absolute Value Function Not Differentiable?

An absolute value function f(x) = a |x - h| + k is not differentiable at the vertex (h, k) because the left-hand limit and the right-hand limit of the function are not equal at the vertex.

### Is an Absolute Value Function Even or Odd?

The absolute value function f(x) = |x| is an even function because f(x) = |x| = |-x| = f(-x) for all values of x.

### How to Write an Absolute Value Function as a Piecewise Function?

We can write the absolute value function f(x) = |x| as a piecewise function as, f(x) = x, if x ≥ 0 and -x, if x < 0.

## FAQs

### What is an example of an absolute value function equation? ›

Some of the examples of absolute value functions are: **f(x) = |x|** **g(x) = |3x - 7|** **f(x) = |-x + 9|**

**What is absolute value examples and answers? ›**

Definitions: The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign. For example, **the absolute value of 5 is 5, and the absolute value of −5 is also 5**. The absolute value of a number may be thought of as its distance from zero along real number line.

**What is the absolute function function equation? ›**

The general form of an absolute value function is **f(x)=a|x-h|+k**. From this form, we can draw graphs.

**What is absolute value as an equation? ›**

To solve an equation containing absolute value, isolate the absolute value on one side of the equation. Then set its contents equal to both the positive and negative value of the number on the other side of the equation and solve both equations. Solve | x | + 2 = 5.

**What is the simple absolute value equation? ›**

An absolute value equation in the form **|ax+b|=c | a x + b | = c** has the following properties: If c<0,|ax+b|=c has no solution. If c=0,|ax+b|=c has one solution. If c>0,|ax+b|=c has two solutions.

**Does absolute value have 2 answers? ›**

**Absolute values will have two solutions when they are equations, functions, in the inequalities that will give a set of results**. For a specific number with an absolute value, it will only have one result which will always be positive.

**What are the 4 steps to solving an absolute value equation? ›**

Step 1: Isolate the absolute value | |3x - 6| - 9 = -3 |3x - 6| = 6 |
---|---|

Step 2: Is the number on the other side of the equation negative? | No, it's a positive number, 6, so continue on to step 3 |

Step 3: Write two equations without absolute value bars | 3x - 6 = 6 |

Step 4: Solve both equations | 3x - 6 = 6 3x = 12 x = 4 |

**What is the absolute value of 10 answer? ›**

In the case of the absolute value of 10, or **|10|**, we have that 10 is positive, so the absolute value of 10 is 10. Geometrically speaking, we define the absolute value of a number x as the distance x is from 0 on the number line.

**What is the absolute value 8th grade? ›**

An absolute value is **the numerical distance from zero** and can be used in equations and graphed as dilations or reflections.

**What's the absolute value of 4? ›**

The absolute value of 4 is **4** and –3 is 3. Subtract the smaller number from the larger and you get 4 – 3 = 1. The larger absolute value in the equation was 4 or a positive number so you give the result a positive result.

### How do you use absolute functions? ›

**Returns the absolute value of a number**. The absolute value of a number is the number without its sign.

**How do you use absolute value? ›**

The most common way to represent the absolute value of a number or expression is to **surround it with the absolute value symbol: two vertical straight lines**. |6| = 6 means “the absolute value of 6 is 6.” |–6| = 6 means “the absolute value of –6 is 6.”

**What is the absolute value of 3? ›**

For example, the absolute value of 3 is **3**, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

**What is the absolute value of 9? ›**

The absolute value of 9 is 9. (9 is **9 places from 0**.) The absolute value of -4 is 4.

**What is the absolute value of 2? ›**

For example,**|2|** represents the absolute value of 2.

**What is the absolute value of 12? ›**

It's easy to understand that the absolute value of 12 and -12 are identical: they're both 12, since **each number is 12 units away from zero**. Absolute value is always positive — to get the absolute value of a negative number, just take away the minus sign.

**How many answers should you get for an absolute value problem? ›**

An absolute value equation may have **one solution, two solutions, or no solutions**.

**Do you do absolute value first? ›**

Summary. Evaluating these expressions isn't much harder than regular expressions. Absolute value bars act like parentheses; **you should do the operations inside them first**.

**How do you solve square root equations? ›**

**Solve a radical equation.**

- Isolate the radical on one side of the equation.
- Square both sides of the equation.
- Solve the new equation.
- Check the answer.

**What is 100 in absolute value? ›**

The absolute value of the number 100 is 100, in mathematical notation **|100| = 100**. The absolute value of zero is zero, |0| = 0, the absolute value of −6.3 is 6.3, | − 6.3| = 6.3.

### What is the absolute value of 11 answer? ›

The absolute value of any positive number is **the number itself**, so 11 has 11 as an absolute value.

**What is the absolute value of 16 answer? ›**

Hence, the answer is **16**.

**What is absolute value 6th grade? ›**

The absolute value of a number is **its distance from zero on the number line**. Since absolute value is a distance, it is always greater than or equal to zero. Example: The absolute value of a number is its distance from zero on the number line.

**What is the absolute value of 22? ›**

22 is **22 units from zero on the number line**. This means that the absolute value of 22 is 22. Notice that our sign didn't change. The absolute value of a number will always be positive.

**What is the absolute value of 23? ›**

The absolute value of a number is the distance of that number from zero. The absolute value of 23 is 23 because it is **23 units from zero**.

**What is a real life example of absolute value equation? ›**

A geophysicist uses absolute value to look at the total amount of energy used. In an energy wave, there are both negative and positive directions of movement. Another example is **when scuba divers discuss their location in regards to sea level**. “50 feet below sea level” doesn't have to be represented as -50 feet.

**What are the types of absolute value function? ›**

**There are 4 main types of absolute value equations regarding whether there are;**

- absolute value and a static value.
- absolute value and an expression involving unknown pronumerals.
- two absolute values in both sides.
- two absolute values and a value.

**What is the equation for an absolute value function graph? ›**

We can graph any absolute value equation of the form **y=k|x-a|+h** by thinking about function transformations (horizontal shifts, vertical shifts, reflections, and scalings).

**What function is used for absolute value? ›**

Returns the absolute value of a number. The absolute value of a number is the number without its sign.

**What is a real life example of an equation? ›**

Suppose we rent a car with a charge of $200 plus $25 for every hour. Here you don't know how many hours you will travel so by using "t" to represent the number of hours to your destination and "x" to represent the cost of that taxi ride, this can be framed in an equation as **x = 25 × t + 200**.

### What makes an absolute value equation all real numbers? ›

**If the absolute value is greater than or greater than or equal to a negative number**, the solution is all real numbers. The absolute value of something will always be greater than a negative number.

**What are the 4 properties of absolute value? ›**

Absolute value has the following fundamental properties: **Non-negativity |a| ≥ 0**. **Positive-definiteness |a| = 0a = 0**. **Multiplicativity |ab| = |a| |b|**

**What is the absolute value of 4? ›**

The absolute value of 4 is **4** and –3 is 3. Subtract the smaller number from the larger and you get 4 – 3 = 1. The larger absolute value in the equation was 4 or a positive number so you give the result a positive result.

**What are all the absolute values of 4? ›**

Answer and Explanation: **The absolute value of 4 is 4**. The formal definition of the absolute value of a number x, denoted |x|, is the distance that x is from 0 on a number line.