Exploring Absolute Value: Graphs, Formulas, and Functions

Comprehensive Definition, Description, Examples & Rules 

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Introduction to Absolute Value

Absolute numbers play an important role in mathematics. It is also referred to as the modulus. You will get to know about the absolute value concept in depth, its significance in mathematics, its graph, usage, formula, and many other related things on this informative page. You can take a worksheet practice test that is in the ending section of this article. 

Definition and concept of absolute value

This concept is known for checking the distance of real numbers from zero on the number line. It is depicted as “lxl”, which refers to bars on the number line. The absolute value doesn’t symbolize the sign. Whether an absolute number is positive or negative, it would be presented as a real number. The absolute value of any number is depicted by two vertical lines lbl; this is referred to as the modulus of b. For 0, there are no negative values. 

For example, if there is -2 on the number line, then the absolute value will be 2. 

Absolute value is symbolized by ‘ l l, for example, -2 = l 2 l 

  • We can also define absolute value as 

(a if a is greater than 0)

(lal) = (-a if a is smaller than 0) 

Significance of absolute value in mathematics

Absolute numbers play a key role in mathematics. You can easily find distance and magnitude without getting confused with positive and negative signs.

  • It is necessary to solve any equations. You won’t be able to solve equations without using absolute numbers. 
  • Absolute value helps to find out the distance between the numbers on the number line. It is important for solving geometric equations. 
  • It is also used in polynomial equations. 
  • It has important significance in graph representation. In V-shaped graphs, it represents the specific point that breaks the line..
  • Absolute value is not that complex, even though it is essential for understanding the y 3 graph  and diverse concepts.

Applications and real-life examples of absolute value

Absolute value is useful in day-to-day life. Are you wondering? Here below, you can find out how you can learn absolute value through daily life practice. 

  • An absolute value is used for measuring the distance on a map. It is important from a geographical perspective.
  • The absolute value is used for measuring and calculating the temperature.
  • It is used in multiple sectors of the economy, and you can better analyze the status through positive and negative signs.
  • It is also used in science streams for experimental purposes.

The general usefulness of absolute value in real life is measured. Whether it is the magnitude or economic profile, you can easily calculate it through absolute value.

Absolute Value Graph

Absolute value graphs have a V-shaped representation. It is the most basic graph in mathematics for students. This V-shaped cross has an origin with an infinite end.

Characteristics of the absolute value graph

Absolute value has major characteristics in math. It contains symmetry, an axis of symmetry, an absolute vertex, and a point known as the corner point. 

  • Symmetry means the representation of both (a,b) and (-a, -b) representations on a single graph. 
  • From Vertix, the V-shaped graph turns to a different side. Vertix lies over the origin, which is referred to as (0, 0). 
  • There are two axes of any graph that make a proper division of it. (a,b).  
  • A corner point is the turning point from which a V-shaped graph takes a turn in a different direction.

The behavior of the graph for positive, negative, and zero input values 

  • Positive input value

(‘+a’ if a is greater than 0) In this case, you can observe the positive graph. 

  • Negative input value

(lal) = (-a if a is smaller than 0) In this case, the graph will represent a negative linear line. 

  • Zero input value

It is neutral. In which the value intersects or crosses the y-axis at its origin.

Graph transformations and translations of the absolute value function

There are four major functions that help transform graphs. 

  • Translation
  • Stretching
  • Shrinking 
  • Reflection  

A graph is transformed by a horizontal axis, which represents the negative absolute value number. On the other hand, A vertical side reflects the positive side number.

Absolute Value Formula

The absolute formula is very simple.

lxl = { x x > 0 }

       {-x x< 0 }

For example 

  • 4 lx -2 l = 8? What is the value of x?
  • x-2= 8/4 
  • x = 2 + 2
  • x = 4, x = – 4

Deriving the absolute value formula

The formula is derived from modulus. The absolute value of any number can be determined by its distance from the zero in a number line.

lxl = { x x > 0 }

       {-x x< 0 }

Mathematical notation and representation of absolute value

Find the value of x. 

  • x – 2 =0 

Solution :  

  • x = 2

Solving equations and inequalities using the absolute value formula 

Solve the equation. 

  • What is the value of x?

x – 9 = 16 


x = 25 (16+9) 

  • find the value of x? 

2 – x = 12 


x = -10

Inequalities equation  

How will you determine the positive and negative cases in this given equation? 

  • 3- x < 4 (negative because x is (-x))
  • x + 3 > 4 (positive because x is positive)

Absolute Value and Graphing

Relationship between absolute value and graphing

Absolute values and graphs are interlinked with each other. It helps to represent the distance on the graph through the vertical and horizontal axes, which are also referred to as the coordinate plane. Absolute-value graphs primarily depict the input value.

Obtaining the graph of an absolute value function 

It includes various steps to obtain the absolute value function: 

 First step 

You are expected to find out the absolute value in the first step. In the graph, you will find the absolute value. F(a) 

Second Step  

In this second step, you need to divide the section into a graph with vertical and horizontal lines. The vertical line should reflect positive numbers, and the horizontal line should reflect negative numbers. 

Third Step  

A graph that is divided into positive and negative planes is referred to as the coordinate plane.  

Fourth Step  

After drawing the coordinate plane, you need to connect two portions with a vertex at the center, or (0, 0). From this point, your graph will take a turn and will represent a V shape.

Key features: vertex, axis of symmetry, and shape of the graph 

In various Key features, the vertex, axis of symmetry, and shape of the graph are most important. 

  • Vertex: The vertex is the center or origin axis in a graph, which represents (0, 0). 
  • Axis of Symmetry: In an absolute graph, the y-axis represents an axis of symmetry that makes an equal division of portions for the negative and positive coordinate planes. 
  • The shape of graph: the shape of absolute graphs represented with V.

Absolute Value of Zero

Understanding the behavior of absolute values when the input is zero

It is reflected as l1l. Here, 1 is an absolute number. In this context, it represents the distance of +1 on a number line in reference to 0.

Significance of the absolute value of zero in mathematics 

Absolute value plays a crucial role in determining the role of zero in mathematics. 

  • You will be amazed to learn about the absolute value of zero. It is zero itself. 
  • It helps in measuring the distance and calculating the positive and negative impacts due to the number line. 
  • In a coordinate plane, a vertex lies on this zero. Zero is the origin of the axis (0, 0).

Examples and applications of the absolute value of zero 

You can use zero as an absolute value in math equations, geometry, etc. 


  • x -3 = 0 

Answer: x = 3. 

It is also used in numerical representation, algebraic representation, graphic representation, and so on.

Absolute Value and Functions

Role of absolute value in function graph

The absolute value represents the symmetry line, vertices, and origin of the axis, and it helps in the formation of a graph through these characteristics.

Effect of absolute value on the shape and behavior of functions 

  • Function behavior depends upon the positive and negative signals of input. 
  • A shape of a graph formed at a vertex, which is also referred to as a corner point, from which the shape of the graph took a turn.

Examples of functions involving absolute values 

There are various examples of absolute functions: 

  • Absolute value function as f(a) =lal. 

lxl = { x x > 0 }

 {-x x< 0 } 

  • Example of an Equation of Absolute Value 

x-8 = 18 

Solution: 26

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Key Takeaways

  1. Absolute value is known for measuring the distance of real numbers from zero on the number line.
  2. Absolute value also represents a graph, which can be transformed by a horizontal axis that highlights the negative absolute number.

  3. On the other hand, the vertical side depicts the positive side number.

  4. Absolute value has significance in day-to-day life. You can easily measure distance, profit, and growth by using these values.

  5. In mathematics, the absolute value can be expressed through graphs, linear equations, geometry, and so on.

  6. You can better grasp the absolute value by practicing on the worksheet provided on this page.


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Frequently Asked Questions

It can be used with complex numbers. For example, c = a + bi. C represents an absolute number with a distance of 0.

The formula is simple for the absolute value formula for negative numbers. For instance, lxl = -x

On a number line, it helps to calculate the distance from zero of positive and negative absolute values.

There are various properties and rules related to absolute freedom. An absolute number is always represented either as zero or positive. It can’t be negative.

absolute value formula can be applied to expressions with variables with variables, for example, 8 x -2.

It is represented as lxl. And sometimes it is represented with a double bar as well. A graph of absolute value includes vertices, coordinate planes, positive and negative inputs, and so on.

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