Graphing Absolute Value Functions

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



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|
   
   
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.



x	y
-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.


2. Graph y = |x| - 4
   
   
The table of values looks like this:



x	y
-5	5 - 4 = 1
-4	4 - 4 = 0
-3	3 - 4 = -1
-2	2 - 4 = -2
-1	1 - 4 = -3
0	0 - 4 = -4
1	1 - 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.


3. Graph y = -|x|
   
   
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
-3	3	-3
-2	2	-2
-1	1	-1
0	0	0
1	1	-1
2	2	-2
3	3	-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.