The absolute value of a number is its distance from zero on a number line. An absolute value is indicated by vertical lines on either side. For example, the absolute value of -3 is 3, which is written as \left|-3\right| = 3.

An absolute value function is a function that contains a variable expression inside absolute value bars; a function of the form f\left(x\right) = a\left|x - h\right| + k

x	-2	-1	0	1	2	3	4
f\left(x\right)	3	2	1	0	1	2	3

For example, consider the absolute value function f\left(x\right) = \left|x - 1\right| .

We can complete a table of values for the function.

-4

-3

-2

-1

-4

-3

-2

-1

Since the absolute value of an expression is non-negative, the graph of this absolute value function does not go below the x-axis, as the entire expression is inside the absolute value.

If the function has a negative value for k, such as \\y=\left|x-1\right|-2, it can go below the x-axis, but will still have a minimum value, in this case y=-2.

If the function has a negative value of a, the graph of the function will open downwards, and instead have a maximum value.

Worked examples

Example 1

Consider the function f\left(x\right) = \left|3x - 6 \right|.

x	0	1	2	3	4	5
f\left(x\right)

Complete the table of values for this function.

Solution

x	0	1	2	3	4	5
f\left(x\right)	6	3	0	3	6	9

Sketch a graph of the function.

Solution

-2

-1

-2

-1

Outcomes

MA.912.AR.4.3

Given a table, equation or written description of an absolute value function, graph that function and determine its key features.

MA.912.F.1.2

Given a function represented in function notation, evaluate the function for an input in its domain. For a real-world context, interpret the output.

4.09 Linear absolute value functions

Concept summary

Worked examples

Example 1

Solution

Solution

Outcomes

MA.912.AR.4.3

MA.912.F.1.2

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