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5. Exponential & Piecewise Functions
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United States of AmericaTennessee
Integrated Math 2

5.03 Absolute value functions

Lesson
Practice
Lesson

Concept summary

The parent absolute value function f\left(x\right) = \left|x\right| takes an input and gives an output of the absolute value of that number. The equation of absolute value function contains a variable expression inside absolute value bars; a function of the form f\left(x\right) = a\left|x - h\right| + k.

The absolute value function f\left(x\right)=\left|x\right| has two cases to consider:

  • If x \geq 0, then f\left(x\right)=x
  • If x < 0, then f\left(x\right)=-x

As a result, the graph of an absolute value function looks like two rays that meet at a common point, called its vertex.

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The parent function has a minimum vertex, but with a reflection the vertex can be a maximum.

To sketch the graph of an absolute value function we can create a table of values or consider the key features of the graph from its equation.

For an absolute value function of the form y=a\left|x-h\right|+k:

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  • The vertex is \left(h,k\right)
  • If a>0, the function will open upwards, minimum vertex
  • If a<0, the function will open downwards, maximum vertex
  • The rate of change will be -a to the left of the vertex, and a to the right of the vertex

Worked examples

Example 1

Consider the function f\left(x\right) = \left|x - 1\right| + 2

a

State the coordinates of the vertex.

Approach

The vertex of an absolute value function in the form f\left(x\right)=a\left\vert x-h\right\vert +k is always \left(h,k\right).

Solution

We can see that this function has a vertex at \left(1, 2\right) by looking at the equation.

b

Draw a graph of the function.

Approach

Use key features, including the vertex and rate of change, to sketch the graph.

Solution

From part (a) the function has a vertex of \left(1,2\right).

And as a=1 we know:

  • The function will open upwards, since a>0.
  • The rate of change will be -1 to the left of the vertex, and 1 to the right of the vertex.
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A rate of change of 1 means for each one across the graph rises one.

Plot the vertex, then plot a second point one across and one up from the vertex. Join the points to create half the function.

The graph is symmetrical about the vertical line through the vertex. So we can reflect the line created to complete the graph.

This results in the following graph:

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Reflection

Alternatively, to graph the function we could create a table of values by evaluating the function {y=\left|x - 1\right| + 2} for different values of x. Given the vertex is at x=1, we can create a table with one or two points either side of this to generate the graph.

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c

State the domain and range of the function, using interval notation.

Approach

Remember that the domain is the set of all x-values that can be put into the function, while the range is the set of all function values (i.e. y-values) that can be obtained by the function.

Solution

Looking at the graph, we can see that the domain will be "all real values", as any value of x can be used as an input.

The vertex of this function is at \left(1, 2\right) and is a minimum point, so the range will be "all values greater than or equal to 2."

Using interval notation, we have:

  • Domain: \left(-\infty, \infty\right)
  • Range: \left[2, \infty\right)

Outcomes

M2.F.IF.C.6

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

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

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