Lesson_Presentation_Graphing_Special_Functions.pptx

Lesson Menu
Five-Minute Check (over Lesson 2–4)
Mathematical Practices
Then/Now
New Vocabulary
Example 1: Piecewise-Defined Function
Example 2: Write a Piecewise-Defined Function
Example 3: Real-World Example: Use a Step Function
Key Concept: Parent Functions of Absolute Value Functions
Example 4: Absolute Value Functions

Over Lesson 2–4
You are asked to sketch the graph of a linear
function that is decreasing for all values of x.
Which end behavior should your graph show?
A. As x → ∞, f(x) → ∞ and as x → −∞, f(x) → −∞.
B. As x → ∞, f(x) → ∞ and as x → −∞, f(x) → ∞.
C. As x → ∞, f(x) → −∞ and as x → −∞, f(x) → −∞.
D. As x → ∞, f(x) → −∞ and as x → −∞, f(x) → ∞.

Over Lesson 2–4
Choose the correct nonlinear graph for the given key
features. The function is continuous and symmetric
about the line x = 2. The function has a maximum at
(2, –1). As x → ∞, f(x) → −∞ and as x → −∞, f(x) → −∞.
A. B.
C. D.

Over Lesson 2–4
A. The function is nonlinear.
B. The function has a minimum at (0, 5).
C. The function is increasing for x < 0 and
decreasing for x > 0.
D. As x → ∞, f(x) → ∞ and as x → −∞, f(x) → ∞.
Patrick correctly sketched a graph based on some
given key features. His graph was an upward
opening parabola symmetric about the y-axis. Which
could not have been a key feature he was given?

Over Lesson 2–4
A. The function is decreasing.
B. The y-intercept is (0, –3).
C. The function is positive for x > 0.
D. The function has a maximum at (5, 2).
The function f(x) is linear. Based on the table of
values, which key feature will the graph of f(x) show?
x f(x)
1 –2
3 0
5 2

• piecewise-defined function
• piecewise-linear function
• step function
• greatest integer function
• absolute value function

Piecewise-Defined Function
Step 1 Graph the linear
function f(x) = x – 1
for x ≤ 3. Since 3
satisfies this
inequality, begin
with a closed circle
at (3, 2).

Step 2 Graph the constant
function f(x) = –1 for
x > 3. Since x does
not satisfy this
inequality, begin with
an open circle at
(3, –1) and draw a
horizontal ray to the
right.

Answer: The function is
defined for all values of x,
so the domain is all real
numbers. The values that are
y-coordinates of points on the
graph are all real numbers
less than or equal to 2, so the
range is {f(x) | f(x) ≤ 2}.

Identify the domain and range.

A. domain: all real numbers
range: all real numbers
B. domain: all real numbers
range: {y|y > –1}
C. domain: all real numbers
range: {y|y > –1 or y = –3}
D. domain: {x|x > –1 or x = –3}
range: all real numbers

Write a Piecewise-Defined Function
Write the piecewise-defined
function shown in the graph.
Examine and write a function for
each portion of the graph.
The left portion of the graph is a graph of f(x) = x – 4.
There is a circle at (2, –2), so the linear function is
defined for {x | x < 2}.
The right portion of the graph is the constant function
f(x) = 1. There is a dot at (2, 1), so the constant function
is defined for {x | x ≥ 2}.

Write a Piecewise-Defined Function
Write the piecewise-defined function.
Answer:

Identify the piecewise-defined
function shown in the graph.
A.
B.
C.
D.

Use a Step Function
PSYCHOLOGY One psychologist charges for
counseling sessions at the rate of $85 per hour or
any fraction thereof. Draw a graph that represents
this situation.
Understand The total charge must be a multiple of
$85, so the graph will be the graph of a
step function.
Plan If the session is greater than 0 hours, but
less than or equal to 1 hour, the cost is
$85. If the time is greater than 1 hour,
but less than or equal to 2 hours, then
the cost is $170, and so on.

Use a Step Function
Solve Use the pattern of times and costs to make a
table, where x is the number of hours of the
session and C(x) is the total cost. Then draw
the graph.

Use a Step Function
Answer:
Check Since the psychologist rounds any fraction of
an hour up to the next whole number, each
segment on the graph has a circle at the left
endpoint and a dot at the right endpoint.

SALES The Daily Grind charges $1.25 per pound of
meat or any fraction thereof. Draw a graph that
represents this situation.
A. B.
C. D.

Absolute Value Functions
Graph y = |x| + 1. Identify the domain and range.
Create a table of values.
x |x| + 1
–3 4
–2 3
–1 2
0 1
1 2
2 3
3 4

Absolute Value Functions
Graph the points and
connect them.
Answer:
The domain is all real
numbers. The range is
{y | y ≥ 1}.

A. y = |x| – 1
B. y = |x – 1| – 1
C. y = |x – 1|
D. y = |x + 1| – 1
Identify the function shown by
the graph.

Lesson_Presentation_Graphing_Special_Functions.pptx

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