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

A1.N.Q.A.1

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

A1.N.Q.A.1.A

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

A1.N.Q.A.1.B

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

A1.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.*

A1.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.*

A1.A.REI.D.5

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A1.A.REI.D.6

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). Find approximate solutions by graphing the functions or making a table of values, using technology when appropriate.*

A1.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.*

A1.F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the context of the function it models. *

A1.F.IF.B.6

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate and interpret the rate of change from a graph.*

A1.F.IF.C.7

Graph functions expressed algebraically and show key features of the graph by hand and using technology.*

A1.F.IF.C.9

Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.*

4.08 Linear absolute value functions

Concept summary

Worked examples

Example 1

Solution

Solution

Outcomes

A1.N.Q.A.1

A1.N.Q.A.1.A

A1.N.Q.A.1.B

A1.A.CED.A.2

A1.A.CED.A.3

A1.A.REI.D.5

A1.A.REI.D.6

A1.F.IF.B.4

A1.F.IF.B.5

A1.F.IF.B.6

A1.F.IF.C.7

A1.F.IF.C.9

A1.MP1

A1.MP3

A1.MP6

A1.MP7

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