The absolute value of a number 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:

$x$`x`	$-2$−2	$-1$−1	$0$0	$1$1	$2$2
$f(x)=\left\|x\right\|$`f`(`x`)=\|`x`\|	$2$2	$1$1	$0$0	$1$1	$2$2

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 absolute value 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:

$5x-15$5`x`−15	$=$=	$0$0
$5x$5`x`	$=$=	$15$15
$x$`x`	$=$=	$3$3

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.

Note: In this course, you may use technology to graph functions, find intercepts and points of intersection.

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.

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

A linear absolute value function is really a composite of two functions. It is defined by a rule like the following, where the coefficient $a$a can be assumed to be a positive real number.

Particular functions are obtained by substituting appropriate values for the coefficients $a$a and $b$b.

Practice questions

question 1

Consider the function $f\left(x\right)=\left|x\right|$f(x)=|x| that has been graphed. Notice that it opens upwards.

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What is the gradient of the function for $x>0$x>0?
What is the gradient of the function for $x<0$x<0?
The graph below shows the graph of $y$y that results from reflecting $f\left(x\right)$f(x) across the $x$x-axis. State the equation of $y$y.

Loading Graph...
Select all the correct statements.
A downward absolute value function goes from decreasing to increasing.
A
An upward absolute value function goes from decreasing to increasing.
B
An upward absolute value function goes from increasing to decreasing.
C
A downward absolute value function goes from increasing to decreasing.
D

question 2

Consider the graph of function $f\left(x\right)$f(x).

Loading Graph...

State the coordinate of the vertex.
State the equation of the line of symmetry.
What is the gradient of the function for $x>5$x>5?
What is the gradient of the function for $x<5$x<5?
Hence, which of the following statements is true?
The graph of $f\left(x\right)$f(x) is steeper than the graph of $y=\left|x\right|$y=|x|.
A
The graph of $f\left(x\right)$f(x) is not as steep as the graph of $y=\left|x\right|$y=|x|.
B
The graph of $f\left(x\right)$f(x) has the same steepness as the graph of $y=\left|x\right|$y=|x|.
C

question 3

Consider the function $y=\left|x+2\right|$y=|x+2|.

Determine the coordinates of the $y$y-intercept.

Intercept $=$= $\left(\editable{},\editable{}\right)$(,)
State the coordinate of the vertex.

Vertex $=$= $\left(\editable{},\editable{}\right)$(,)
Draw the graph of the function.

Loading Graph...

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

question 4

Consider the function $y=\left|x\right|-5$y=|x|−5.

Does the graph of the function open upwards or downwards?
$Downwards$ .
A
$Upwards$ .
B
State the coordinate of the vertex.
State the equation of the line of symmetry.
Which of the following functions would have narrower graphs than $y=\left|x\right|-5$y=|x|−5?
$y=\frac{\left|x\right|}{2}-5$y=|x|2−5
A
$y=\left|x-5\right|$y=|x−5|
B
$y=\left|x\right|-5$y=|x|−5
C
$y=-4\left|x\right|-5$y=−4|x|−5
D

question 5

An absolute value function $f\left(x\right)$f(x) has its vertex at $\left(-4,0\right)$(−4,0), and goes through the point $A$A$\left(1,-5\right)$(1,−5).

Find the gradient between the vertex and point $A$A.
Hence state the equation of the function, expressing $f\left(x\right)$f(x) in terms of $x$x.
What is the domain and range of $f\left(x\right)$f(x)? Give your answers as intervals.

Domain: $\editable{}$

Range: $\editable{}$
Which of the following functions would have a vertex that is to the $left$ of the vertex of $f\left(x\right)=-\left|x+4\right|$f(x)=−|x+4|?
$y=-\left|x-5\right|$y=−|x−5|
A
$y=-\left|x+5\right|$y=−|x+5|
B
$y=-\left|5x+20\right|$y=−|5x+20|
C

Outcomes

0607C3.6

Use of a graphic display calculator to: sketch the graph of a function, produce a table of values, find zeros, local maxima or minima, find the intersection of the graphs of functions.

0607E3.2F

Recognition of absolute value, f(x) = | ax + b |, function types from the shape of their graphs.

0607E3.3

Determination of the value of at most two of a, b, c or d in simple linear, quadratic, cubic, reciprocal, exponential, absolute value and trignometric functions.

0607E3.6

Use of a graphic display calculator to: sketch the graph of a function produce a table of values, find zeros, local maxima or minima, find the intersection of the graphs of functions.