Introduction
Absolute value functions can be used to model situations
in which there is a peak in a phenomenon. One such
example is a movie orchestra instructed to play at a
higher sound level for a number of measures, then to play
at a lower level, only to return back to the higher sound
level. This represents a peak in the sound level. Sound
engineers might graph the absolute value function for this
situation in order to observe the sound levels at each
measure and make adjustments quickly without having to
perform cumbersome calculations. In this lesson, you will
create and use various absolute value functions and their
graphs to solve a variety of real-world problems.
1
6.2.3: Absolute Value Functions
Key Concepts
• An absolute value function is a function of the form
f (x) = a x - h + k where x is the independent variable
and a, h, and k are real numbers. An absolute value
function is a special type of piecewise function
because the function values on either side of its
maximum or minimum point are equal; for example, if
f (x) = x - 6 , then f(3) = f(9), since f (3) = 3 - 6 = -3 = 3
and f (9) = 9 - 6 = 3 = 3 .
• In general terms, the maximum or minimum function
value occurs at the point (h, k).
2
6.2.3: Absolute Value Functions
Key Concepts, continued
• For an absolute value function of the form
f (x) = a x - h + k, there is one extreme value for the
range at y = k.
• The graph of the basic absolute value function, f (x) = x ,
is comprised of lines that “open upward” with a right
angle at the vertex, which is at the origin of (0, 0); the
function also has symmetry about the y-axis.
3
6.2.3: Absolute Value Functions
Key Concepts, continued
• The graph of the basic negative absolute value
function, f (x) = - x , is comprised of lines that “open
downward” with a right angle at the vertex (the origin),
and also has symmetry about the y-axis.
4
6.2.3: Absolute Value Functions
Key Concepts, continued
5
6.2.3: Absolute Value Functions
Key Concepts, continued
• The domain of an absolute value function is all real
numbers, while the range is the set of non-negative
real numbers: [0, ∞).
• Note that in the positive absolute value function,
f (x) = x , the graph increases from y = 0 as the
x-values approach negative infinity. That is, as the
x-values decrease, the y-values increase.
• Additionally, f (x) = x increases from y = 0 as x
approaches positive infinity—as the x-values
increase, the y-values also increase.
6
6.2.3: Absolute Value Functions
Key Concepts, continued
• However, in the negative absolute value function,
f (x) = - x , the graph decreases from y = 0 as the
x-values approach negative infinity; as the x-values
decrease, the y-values also decrease.
• Additionally, f (x) = - x decreases from y = 0 as x
approaches positive infinity—as the x-values
increase, the y-values decrease.
• If a in the general form f (x) = a x - h + k is positive,
the extreme value is the minimum of the range, and
there is no maximum of the range.
7
6.2.3: Absolute Value Functions
Key Concepts, continued
• If a in the general form f (x) = a x - h + k is negative,
the extreme value is the maximum of the range, and
there is no minimum of the range.
• Graphing calculators can be an efficient way to graph
absolute value functions. Follow the directions specific
to your model.
8
6.2.3: Absolute Value Functions
Key Concepts, continued
On a TI-83/84:
Step 1: Press [Y=]. At [Y1], press [2ND][0] to bring up
the Catalog menu. Select “abs(”. Enter the
expression contained within the absolute value
bars, using [X, T, θ, n] for x. Enter the
remaining part of the expression by moving the
cursor to the left and/or right of the absolute
value bars. Enter additional absolute value
functions on separate lines.
Step 2: Press [GRAPH]. Adjust the viewing window
as needed.
9
6.2.3: Absolute Value Functions
Key Concepts, continued
On a TI-Nspire:
Step 1: Press the [home] key. Arrow down and over to
the graphing icon and press [enter].
Step 2: Enter the expression at the bottom of the
application, using x as the variable. Type
“abs(x)” to enter an absolute value; for
example, typing “abs(x+4)–5” is equivalent to
the expression f (x) = x . Press [enter]. Adjust
the viewing window as needed.
10
6.2.3: Absolute Value Functions
Common Errors/Misconceptions
• misinterpreting variables from a word problem
• not realizing that there are two equations for an
absolute value function
• treating absolute value symbols like parentheses when
solving for zeros
11
6.2.3: Absolute Value Functions
Guided Practice
Example 2
Write the absolute value function represented in the graph.
12
6.2.3: Absolute Value Functions
Guided Practice: Example 2, continued
1. Determine the values of h and k from the
graph.
The standard form of an absolute value function is
f (x) = a x - h + k, where (h, k) is the vertex.
Since (2.5, 3.5) is where the two parts of the graph
meet, we know that this is the vertex; therefore,
h = 2.5 and k = 3.5.
13
6.2.3: Absolute Value Functions
Guided Practice: Example 2, continued
2. Determine the value of a.
Choose one of the given points to substitute into the
general absolute value function, along with the
values of h and k, to find a. A convenient point to use
is (0, 0).
Standard form for an
f (x) = a x - h + k
absolute value function
Substitute 2.5 for h and
f (x) = a x - (2.5) + (3.5)
3.5 for k.
Substitute (0, 0) into
(0) = a (0) - 2.5 + 3.5
the equation.
14
6.2.3: Absolute Value Functions
Guided Practice: Example 2, continued
0 = a -2.5 + 3.5
Simplify.
0 = 2.5a + 3.5
Apply the absolute
value.
–3.5 = 2.5a
a = –1.4
Subtract 3.5 from both
sides.
Divide and apply the
Symmetric Property
of Equality.
The value of a is –1.4. A negative value for a means
the graph opens downward, which corresponds with
the provided graph.
6.2.3: Absolute Value Functions
15
Guided Practice: Example 2, continued
3. Use the values for a, h, and k to write the
equation of the absolute value function.
The standard form of an absolute value function is
f (x) = a x - h + k. Substitute the values a = –1.4,
h = 2.5, and k = 3.5 into this form.
The standard form of the absolute value function
shown in the graph is f (x) = -1.4 x - 2.5 + 3.5 .
✔
16
6.2.3: Absolute Value Functions
Guided Practice: Example 2, continued
17
6.2.3: Absolute Value Functions
Guided Practice
Example 4
Graph f (x) = x , g(x) = x + 5, and h(x) = 5 x using a
graphing calculator. Compare the domains, intercepts,
vertices, and symmetry.
18
6.2.3: Absolute Value Functions
Guided Practice: Example 4, continued
1. Use a graphing calculator to graph each
function.
Follow the directions appropriate to your model.
The graphed functions should resemble the graph on
the following slide.
19
6.2.3: Absolute Value Functions
Guided Practice: Example 4, continued
20
6.2.3: Absolute Value Functions
Guided Practice: Example 4, continued
2. Compare the domains of the functions.
Look at the values of x for which the functions are
defined. For each of the graphs, the lines are
continuous as the values of x go from negative infinity
to positive infinity.
Therefore, the domain for all three graphs is all real
numbers.
21
6.2.3: Absolute Value Functions
Guided Practice: Example 4, continued
3. Compare the x- and y-intercepts of each
function.
f (x) = x and h(x) = 5 x both have x- and y-intercepts
at (0, 0). g(x) = x + 5 has a y-intercept at (0, 5), but no
x-intercept.
22
6.2.3: Absolute Value Functions
Guided Practice: Example 4, continued
4. Compare the vertices of each function.
For these graphs, the vertex will be the lowest point
on the absolute value function.
f (x) = x and h(x) = 5 x each have a vertex at (0, 0);
g(x) = x + 5 has a vertex at (0, 5).
23
6.2.3: Absolute Value Functions
Guided Practice: Example 4, continued
5. Compare the lines of symmetry of each
function.
A line that goes through the vertex point of the
absolute value graph is a line of symmetry.
The line x = 0 (the y-axis) is the line of symmetry for
all three functions.
✔
24
6.2.3: Absolute Value Functions
Guided Practice: Example 4, continued
25
6.2.3: Absolute Value Functions