Absolute Value Graphs

You have explored the graphs of three important functions: - x² (in the quadratic functions unit), sin x and cos x (in the trigonometry unit). Now you will examine a few more of the 'basic' functions that you will need to graph, starting with the graph of the absolute value function.

Absolute Value refers to a number's distance from zero on a numberline. As you can see from the numberline below, both 3 and -3 are equidistant from zero (they are both 3 units away from zero). Therefore, the absolute value of BOTH 3 and -3 is 3.

Absolute Value is written using two straight parenthesis. For example, to say "The absolute value of 3" you would write |3|. To say "The absolute value of -3" you would write |-3|.

Example 1

Evaluate |5|.

Solution

This question is asking for the absolute value of 5. Since the number 5 is 5 units away from zero:

|5| = 5

Example 2

Evaluate |-7|.

Solution

What is the absolute value of -7? Since -7 is seven units away from zero:

|-7| = 7

Graphing f(x) = |x|

Now that you know how absolute value works, it is time to graph the basic absolute value function. When you want to graph a function but are unsure of how that function works, you can always start with a table of values.

Let's graph f(x) = |x| using a table of values:

Let's plot these points on a graph:

These points are going to be connected with straight lines to get the final graph of f(x) = |x|:

This is the basic graph of the absolute value function f(x) = |x|.

Transformation Example

The transformations that you have already seen also apply to the graph of the absolute value function. Specifically, the standard form of the absolute value function is:

y = a|x - h| + k
where
a is vertical stretch
h is horizontal shift
k is vertical shift

Example

Sketch the graph of f(x) = 2|x - 3| -4

Solution

For this graph we can focus on three important points: (0,0), (-1,1) and (1,1). Start by sketching the basic graph of f(x) = |x| with the three important points shown:

This is the graph of y = |x|.

Now start by applying the transformations in order:

1. In this example, a is 2. Multiply all the y - coordinates by 2. The important points ((-1,1), (0,0), (1,1)) become (-1,2), (0,0) and (1,2). The new graph looks as follows:

This is the graph of y = 2|x|.

2. In this example, h is positive 3. (Again, if this does not make sense, refer back to the lesson on graphing quadratic functions for an explanation of how h works.) This means that every point on this graph should be shifted 3 units to the right. This gives you the following graph:

This is the graph of y = 2|x - 3|.

3. The last transformation to be performed is k, which is -4 in this example. This means that we have to move all of the points DOWN 4 units. This will give us our final graph:

This is your final graph of y = 2|x - 3| - 4.

Now that you have explored absolute value graphs, you can head back to the Polynomial and Rational Functions main page.