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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 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 nonnegative 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.
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 nonnegative 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 'Vshaped or inverted 'Vshaped 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 'Vshaped or inverted 'Vshaped depending upon the value of 'a'.
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Absolute Value Function Examples

Example 1: Find the vertex of the absolute value function f(x) = 2 x  1  3
Solution: On comparing the function f(x) = 2 x  1  3 with f(x) = a x  h + k, the vertex of the function is (h, k) = (1, 3)
Answer: Vertex = (1, 3)

Example 2: Plot the graph of absolute value function f(x) =  x + 2  3
Solution: As we can see, the value of a is 1 in f(x) =  x + 2  3 which is negative. SO, the graph opens downwards and hence will be inverted Vshaped. The vertex of the graph is (h, k) = (2, 3).
So, the graph of the given absolute value function f(x) =  x + 2  3 is given by,

Example 3: Find the derivative of the absolute value function f(x) = x.
Solution: We can write f(x) = x as,
f(x) = x, if x ≥ 0 and
f(x) = x, if x < 0
Now, we know that the derivative of x is 1 and the derivative of x is 1. So, the derivative of f(x) = x is given by,
d(x)/dx = 1, if x > 0 and
= 1, if x < 0
Answer: d(x)/dx = 1, if x > 0 and 1, if x < 0.
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 lefthand limit and the righthand 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.
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