ABSOLUTE VALUE FUNCTION
 

 

Absolute Value:

A real number (rational and irrational numbers) can be represented by a position on the number line.  Absolute values will be the distance from the origin (zero) to the numbers.  Absolute value is greater than or equal to zero.  Absolute value can not be negative, distance can not be negative.  Absolute value is written between two bars.

 

Example 1:

 

x = |3|

Go 3 units to the left and 3 units to the right from the origin in the number line.

x = - 3

x =  3

 

 

  • Close dots in the number line indicate that end points are included.
  • Open dots in the number line indicate that end points are not included.

 

 

Example 2:

 

| x 5 | = 2

 

(x 5) will go 2 units to the left and 2 units to the right from the origin.

 

x 5  = - 2

 

x 5  = 2

x = 3

x = 7

 

  

Example 3:

 

| x 4 | > 7

 

(x 4) will go more than 7 units to the left and more than 7 units to the right from the origin.

 

x - 4  = - 7

 

x 4  = 7

x = - 3

x = 11

 

 

 

PROPERTIES OF ABSOLUTE VALUE:

 

.................................................................

|a| = 3 means a = 3 or a = -3

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Non negativity property:   The absolute value of any numbers is greater than zero.  Absolute value can not be negative.  Distance cannot be negative.

 

Example:  |-4| = 4

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Positive definiteness:  The absolute value of zero is zero:

 

Example:

|x| = 0, then x = 0
|0| = 0
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Symmetry property:  A number and its negative have the same absolute value.

|-a| = |a|

 

Example:

|-5| = |5|

5 = 5
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Identity of indiscernible property:  If the subtraction of any two absolute values is 0 then these two absolute values are equal.

 

|x - y| = 0 then x = y

...................................................................

(a2)1/2 = |a|
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Subtraction / addition property:  Addition of any two absolute values is always less than or equal to the sum of the values.

 

|x + y| <= |x| + |y|

 

Example:

 

|3 + (- 8)| < |3| + |-8|

|3 - 8| < 3 + 8

5 < 11

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Multiplicative property:  The absolute value of a product equals to the product of absolute values.

 

|x . y | = |x| . |y|

 

Example:

|-3 x 4| = |-3| x |4|

|-12 |= 3 x 4

12 = 12

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Division property:  The absolute value of a ratio is the ratio of the absolute values

 

|a / b| = |a| / |b|  ((where b is not equal to zero))

|12 / -3| = |12| / |-3|
|-4| = 12 / 3
4 = 4

...........................................................

 

 

GRAPHING ABSOLUTE VALUE FUNCTIONS:

 

 Example 4:

 

Graph f(x) = |x|,  x R

 

First graph f(x) = x

 

Then reflect the portion of the graph below the x-axis in the x-axis as illustrated below:

 

 

 

 

Example 5:

 

Graph y = |x2 - 3x - 28|

 

First graph y = x2 - 3x - 28

 

Then reflect the portion of the graph below the x-axis in the x-axis as illustrated below:

 

 

 

 

Example 6:

 

 Plot   |2x4 x3 - 47x2 + x + 45|

 

First graph y = 2x4 x3 - 47x2 + x + 45

 

Then reflect the portion of the graph below the x-axis in the x-axis as illustrated below: