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Absolute Value Function

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An integer can be positive or negative depending on its value with respect to zero. In many real life problems we need the negative number such as in debit and credit problems. But there are few situations where we just need the numerical value but not the sign attached to it.

 To get the distance between two points we need the absolute value, but to know the direction in which an entity is moving we can take signs. In a Cartesian system or a number line, to get the distance between any two points we need the absolute value.

Absolute value is used to change the negative number into a positive number. In absolute value both the real number and its negative have the same value. Example: |-4| = 4.

Definition

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Absolute Value Function is taken as the distance of the value given from zero. It is a continuous function. This function does increase as we move along the positive scale in the graph and decreases with the negative scale. The symbol ‘|f ()|’ is used to represent the absolute value function.

The absolute function f(x) = |x| can be defined as,
|x|= $\left\{\begin{matrix}
x & if\ x \geq 0\\
 -x& if\ x <0
\end{matrix}\right.$

For Example: For 2 and -2 the distance from zero to the number on the number line is 2 only. It is just that the direction from zero makes the difference. Hence, using the absolute value function

|2| = 2,
 
|-2| = 2.

Domain and range

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A function consists of a dependent variable and an independent variable. In the absolute value function, y = |x|, x is the independent variable and y is the dependent variable. Domain is the set of possible values that can be used for independent variable, and range is the set of values that can be obtained for the dependent variable.

Domain of a function can only have such values which when applied in the function give output as real numbers. Suppose, we have function such that $y = \sqrt{x}$. If we give negative values for this function only imaginary numbers will come as the output. Hence, the domain is positive numbers.
For the absolute value function the domain is the set of real numbers, $\mathbb{R}$ and the range is the set of all non-negative real numbers.

Graph

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An absolute function will always have a V shape
d graph in a XY plane. The values of x can go negative but the values of y will always remain positive and equal to the value of x. For the equation y = x, the graph will be a straight line but for the function y = |x| for the negative values of x the graph will come in the second quadrant as a straight line.
Absolute Value Function
The above graph represents the function y = |x|.

Transformation

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A change in the position, shape or size of a graph is known as transformation of the graph. There are two types of transformations.
Translation: When each point of the graph is shifted in same direction by same distance, such transformation is called translation. It can be horizontal or vertical.

Reflection: When the axis of the graph is changed, it is known as reflection.
For translations, we have horizontal and vertical translations:

1) Horizontal translation: For the absolute value function, to make the graph shift left or right, a constant value k should be added or subtracted to x respectively.

Left shift
y = |x + k|
 Right shift  y = |x - k|

For Example: y = |x - 1| will shift the graph 1 unit right and y = |x + 3| will shift the graph 3 units left.

2) Vertical translation: For a vertical translation in a absolute value function, a constant value k should be added to the function itself.
 Moving up y = |x| + k
 Moving down  y = |x| - k

For Example: y = |x| + 2 will make the graph go 2 points up and y = |x| - 2 will shift the graph downwards by two points.

For reflexive transformations, a minus sign is introduced accordingly. 
Reflect along X-axis
 x = |y|
Reflect along Y-axis  y = |-x|

Examples

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Example 1: Write the function to shift the graph of absolute value function to right by 2 points. Plot the graph.

Solution: The function will be y = |x - 2|. To plot the graph we will check some values.

For x $\geq $ 0

x 0

3
4
y
 2  0 1 2

For x < 0

 x  -1 -2
 -3 -4
 y  3  4  5 6

The graph will be as given here.

Absolute Value Function Graph

Example 2: Plot the graph for reflecting the absolute value function y = |x| along Y-axis.

Solution: The function will be y = -|x|.

For x > 0, y = -(x)

x
0
1
2
y
 0  -1  -2

and for x < 0, y = -[-(-x)].

x
0
-1
-2
 y  0  -1 -2

The graph will be as given here:

Absolute Value Function Graph Example