3.06 Absolute value

Absolute value can be thought of as the distance a number is from $0$0 on a number line. For example, the absolute value of $-3$−3 and $3$3 is $3$3.

Number line

Notation

Absolute value is represented mathematically by two vertical lines on either side of a value. For example $\left|3\right|$|3| means "the absolute value of $3$3," and equals $3$3,  $\left|-9\right|$|−9| means "the absolute value of $-9$−9" and equals $9$9 and the $\left|-x\right|$|−x| means "the absolute value of $-x$−x " and equals $x$x. 

Except for $0$0, the absolute value of any real number is the positive value of that number. The absolute value of an expression is the positive value of the number returned by the expression.

The graph of $f(x)=|x|$f(x)=|x|

Let's consider the basic absolute value function $f(x)=|x|$f(x)=|x|. This looks similar to the linear function $f(x)=x$f(x)=x but with absolute value signs. We know that $f(x)=x$f(x)=x represents a straight line through the origin on the coordinate plane. Introducing the absolute value means that negative function values become positive  as shown in the table and on the diagram below. Notice that this occurs at the $x$x intercept of the graph of $f(x)=x$f(x)=x. 

Table of values:

Graph:

$f(x)=\left|x\right|$f(x)=|x|

The graph of function of the form $f(x)=|ax+b|$f(x)=|ax+b|

Let's consider a more complicated function like $y=|5x-15|$y=|5x−15|. The $x$x intercept of the function $y=5x-15$y=5x−15 is the critical value below which it is negative. This will be the vertex of the absolute value function.

To find the $x$x intercept we make $y=0$y=0:

Hence, as this graph has a positive gradient function values will be negative for $x<3$x<3. For $x>3$x>3, the graph of $y=|5x-15|$y=|5x−15| will be identical the graph of $y=5x-15$y=5x−15. For the other values, the negative $y$y values turn positive.

We can think of absolute value functions as being defined by two functions: one for values of $x$x that would make the original function positive and another for values of $x$x that make the original function negative.

So $y=|5x-15|$y=|5x−15| can be redefined as:

In the diagram below, the lines corresponding to the two parts of a function definition are shown lightly with the 'composite' absolute value function drawn more heavily.

Piecewise definition of $f(x)=|ax+b|$f(x)=|ax+b|

Other functions of the form $|ax+b|$|ax+b| have a shape similar to the one illustrated above. The 'V' shape is typical with the vertex located on the horizontal axis at the point $x$x that makes $ax+b=0$ax+b=0. 

When the coefficient $a$a is large, the 'V' shape is narrow. This is because $a$a controls the gradients. One side has a gradient of $a$a while the other has a gradient of $-a$−a. Because of this, these absolute value functions will always be symmetrical. 

If a negative sign is placed in front of the absolute value symbol, the effect is to invert the 'V' shape of the graph. All the function values are negative when $y=-|ax+b|$y=−|ax+b|.

As you can see, our understanding of the features of straight lines are very helpful in sketching absolute value functions.

Practice Questions

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Transformations of the absolute value function

We will also need to know how to sketch absolute value functions that look like $y=|x|+2$y=|x|+2 or $y=5-|2x|$y=5−|2x|. In these cases, the vertex will not be located on the $x$x axis. It is easiest to think about these graphs like we do about simple parabolas like $y=x^2+3$y=x2+3 whose vertices has been shifted up or down.

Graph of $y=\left|x\right|+2$y=|x|+2 and $y=5-2\left|x\right|$y=5−2|x|

And so, we can see that the domain of any absolute value function is the same as a linear function - all real $x$x values are possible. The range, however, will vary, depending on if there has been a vertical shift or a negative sign in front of the absolute value symbols.

Practice Questions

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question 5