Transcript Glencoe Algebra 2

LESSON 2–6
Special Functions
Five-Minute Check (over Lesson 2–5)
TEKS
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 2–5
Which scatter plot represents the data shown in
the table?
A.
B.
C.
D.
Over Lesson 2–5
Which prediction equation represents the data
shown in the table?
A. y = 2x + 94
B. y = 2x + 64
C. y = –2x + 94
D. y = –2x + 64
Over Lesson 2–5
Use your prediction equation to predict the
missing value.
A. $62
B. $72
C. $82
D. $92
Over Lesson 2–5
The scatter plot shows the
number of summer workouts the
players on a basketball team
attended and the number of wins
during the following season.
Predict the number of wins the
team would have if they attended
24 summer workouts.
A. 6
B. 12
C. 24
D. 48
You modeled data using lines of regression.
• Write and graph piecewise-defined
functions.
• Write and graph step and absolute value
functions.
• 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).
Piecewise-Defined Function
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.
Piecewise-Defined Function
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}.
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.
Analyze
The total charge must be a multiple of
$85, so the graph will be the graph of a
step function.
Formulate
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
Determine 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:
Justify
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.
Use a Step Function
Evaluate Though a step function can be
represented by a single expression, the
table allows us to break the expression into
pieces based on the time intervals.
The table resembles a piecewise-defined
function and can be graphed in the
same way. Our graph accurately represents
the described situation.
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}.
Identify the function shown by
the graph.
A. y = |x| – 1
B. y = |x – 1| – 1
C. y = |x – 1|
D. y = |x + 1| – 1
LESSON 2–6
Special Functions