Absolute value function

                          Absolute value function is one of the topics in which we are going to see about solving for x in absolute value and graphing the absolute-value-function.


Absolute value: 
The absolute value of a number is its distance from 0 on a number line. For example, the number "9" is 9 units away from zero. So its absolute value is 9. Negative numbers are more interesting compared to positive numbers,because the number -4 is still 4 units away from 0. The absolute value of the number -4 is therefore positive 4 .

                   Examples of absolute value:

  1. The absolute value of -9 is 9

  2. The absolute value of 5 is 5

  3. The absolute value of 0 is 0

  4. The absolute value of -256 is 256

  5. More examples of absolute value:

    | -5 + 8 | = 3

    | -8 x 2 | = | -16 | = 16

    | -13 + 5 | = | -8 | = 8

    Example problem of solving x in absolute values

    | x + 9 | = 5 

    Solution:

    x + 9 = 5 (or) x + 9 = -5 

    x = 5 – 9 (or) x = -5 -9

    x = -4 (or) x =-14

    The solution is -4 and -14 

    | x - 8 | = -9 

    Solution:

    x – 8 = -9 (or) x - 8 = 9 

    x = -9 + 8 (or) x = 9 + 8

    x = -1 (or) x = 17 

    The solution is -1 and 17



Graphing absolute value function

    | x | > 2 

    Before graphing we need to write the given function as the following 

    x > 2 (or) x < -2

    Therefore the interval notation is (-infinity,-2) and (2, + infinity)

| x | = 4

x = 4 or x =-4

    More examples for graphing 

    An absolute value equation graphs like a "V".

  1. General form for Absolute function y = a |x – h| + k or y – k =a|x-h|. Here (h, k) is vertex.

  2. If “a” is positive the graph will be open upward

  3. If “a” is negative the graph will be open downward.


  4. y = | x -1| 

    Here we need to consider the given question as y-k = a |x-h| Comparing with general form now we haveY-0 = |x -1| So the vertex is (1, 0) and instead of a we have +1.so the graph will be open upward.To find the x-intercept we have to put y=0 so that we will get x =1.There fore the x-intercept is 1.

    y = |x-1|-2 

    y + 2 = |x-1|

    Here vertex is (1,-2) and the graph will be open upward because the value of a is positive.

    x-intercept:

    put y=0

    x – 1 = 2 or x – 1 = -2

    x = 2 + 1 or x = -2 + 1

    x = 3 or x = -1

    By using the above problems you can understand the topic absolute value function better.

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