To see what the graph
of y = |x| looks like, let’s create a table
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:
- 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.
|-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.
- Graph y = |x| - 4 The table of values looks like this:
|-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.
- 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.
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
- 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