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Graphing Absolute Value Functions
To see what the graph of y = |x| looks like, let’s create a table of values.

xy
-33
-22
-11
00
11
22
33


To graph these values, simply plot the points and see what happens.



Whenever you have an absolute value graph, the general shape will look like a “v” (or in some cases, an upside down “v” as we will see later).


Let's Practice:
  1. Graph y = |x+2|

  2. We know what the general shape should look like, but let’s create a table of values to see exactly how this graph will look.

    xy
    -3|-3 + 2| = |-1| = 1
    -2|-2 + 2| = |0| = 0
    -1|-1 + 2| = |1| = 1
    0|0 + 2| = |2| = 2
    1|1 + 2| = |3| = 3
    2|2 + 2| = |4| = 4
    3|3 + 2| = |5| = 5


    So our graph of y = |x + 2| looks like



    Notice that the graph in this example looks almost identical to the graph of y = |x| except that it was shifted to the left 2 units. This will be important as we try to make generalizations later in the lesson.

  3. Graph y = |x| - 4

  4. The table of values looks like this:

    xy
    -55 - 4 = 1
    -44 - 4 = 0
    -33 - 4 = -1
    -22 - 4 = -2
    -11 - 4 = -3
    00 - 4 = -4
    11 - 4 = -3


    Which makes the graph look like this:



    Notice that the graph in this example is the same shape as except that it has been moved down 4 units.

  5. Graph y = -|x|

  6. In creating the table of values, be careful of your order of operations. You should find the absolute value of x first and then change the sign of that answer.

    x|x|y
    -33-3
    -22-2
    -11-1
    000
    11-1
    22-2
    33-3


    So the graph of looks like:



    In this example, we have the exact same shape as the graph of y = |x| only the “v” shape is upside down now.
Based on the examples we’ve seen so far, there appears to be a pattern when it comes to graphing absolute value functions.
  • When you have a function in the form y = |x + h| the graph will move h units to the left.
    When you have a function in the form y = |x - h| the graph will move h units to the right.
     
  • When you have a function in the form y = |x| + k the graph will move up k units.
    When you have a function in the form y = |x| - k the graph will move down k units.
     
  • If you have a negative sign in front of the absolute value, the graph will be reflected, or flipped, over the x-axis.
Keep in mind that you can also have combinations that change the absolute value graph more than once. You can practice these transformations with this EXCEL Modeling worksheet.



Examples
Graph each of the following functions. You should try to use the rules shown above, but if you want to check yourself, make a table of values to make sure you are on the right track.
Example y = |x - 3|
What is your answer?
 
Example y = |x + 1| - 3
What is your answer?
 
Example y = -|x| + 2
What is your answer?
 
Example y = |x - 2| + 1
What is your answer?
 
Example y = -|x + 4|
What is your answer?
 
Example y = |x - 1| + 3
What is your answer?
 



S Taylor

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