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

M1.N.Q.A.1

Use units as a way to understand real-world problems.*

M1.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.*

M1.N.Q.A.1.B

Use appropriate quantities in formulas, converting units as necessary.*

M1.A.CED.A.2

Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.*

M1.A.CED.A.3

Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.*

M1.F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.*

5.07 Linear absolute value functions

Concept summary

Worked examples

Example 1

Solution

Solution

Outcomes

M1.N.Q.A.1

M1.N.Q.A.1.A

M1.N.Q.A.1.B

M1.A.CED.A.2

M1.A.CED.A.3

M1.F.IF.B.4

M1.MP1

M1.MP3

M1.MP6

M1.MP7

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