CC-5
Content Standards
Analyzing Function
Intervals
F.IF.4 For a function that models a relationship between
two quantities, interpret key features of graphs and tables…
F.IF.6 Calculate and interpret the average rate of change
of a function (presented symbolically or as a table) over
a specified interval. Estimate the rate of change from
a graph.
What was the
driver doing from
time 4.5 to 5.5?
MATHEMATICAL
PRACTICES
Lesson
V
Vocabulary
t
tJODSFBTJOH
function
tEFDSFBTJOH
function
tDPOTUBOUGVODUJPO
tUVSOJOHQPJOU
tBWFSBHFSBUFPG
change
The graph illustrates a 15-minute
drive to a local shopping mall.
During what time intervals was
the driver increasing his speed?
During which time intervals did
the driver slow down? During
what interval of time was the
greatest change in speed? Explain
your reasoning.
Speed (mi/h)
Objectives To find and estimate the average rate of change of a function over a specific
interval
To interpret intervals where a function increases and/or decreases
50
45
40
35
30
25
20
15
10
5
0
Q What are the starting time and speed? What is the
speed after 1 minute? [(0 min, 0 mi/h); 25 mi/h]
Q What do the horizontal lines on the graph
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
represent? [where the speed is constant]
Q What might have been happening from 3.5
minutes to 4.5 minutes? [The car was stopped,
Time (minutes)
perhaps at a traffic light.]
Q How will the greatest change in speed be
represented on the graph? [the steepest line]
ANSWER See Solve It in Answers on next page.
CONNECT THE MATH Have students identify the
Interval Notation
coordinates of each interval endpoint and discuss
the meaning of each coordinate. (0, 0) represents
0 minutes and 0 miles per hour. (1, 25) represents
1 minute and 25 miles per hour. The steepness of
the line, or slope, indicates the change in speed.
The steeper the slope, the greater the change in
speed. Horizontal lines indicate a constant speed.
You can write the intervals where a function is increasing or decreasing in a compact form.
Use a bracket when the endpoint is included in the interval. Use a parenthesis when the
endpoint is not included in the interval.
Interval Notation
ation
x8
(`, 8)
4 x
[4, `)
3 x 6
(3, 6]
PURPOSE To present students with a problem
FACILITATE
Essential Understanding You can describe a function’s behavior on an interval,
including whether the function is increasing or decreasing, whether its values are
positive or negative, and what the average rate of change of the function is.
Inequality
Solve It!
in which they determine where the graph of a
function is increasing and decreasing
PROCESS Students may identify the sections of the
graph where the line is sloping up from left to right
as the time increases and where the line is sloping
down from left to right as the time increases.
Students may make a table of the increasing,
decreasing, and unchanging intervals.
In the Solve It, you had to identify when the graph was increasing and decreasing. You
also had to determine when the greatest change in speed occurred.
Key Concept
1 Interactive Learning
The symbols ` and ` represent positive infinity
and negative infinity, respectively. Each symbol
always appears with a parenthesis. ` means the
interval has no right endpoint, and ` means the
interval has no left endpoint.
2 Guided Instruction
TAKE NOTE Point out to students that the use of
CC-5
CC-5
Preparing to Teach
HSM12_CC_TransitionKit_CC05.indd 1
BIG idea Function
ESSENTIAL UNDERSTANDINGS
• The behavior of a function on an
interval can be described as increase or
decreasing.
• The values of a function on an interval
can be described as positive or negative.
• The average rate of change of a
function on an interval can be found
as a ratio.
Math Background
In this lesson, students learn how to
interpret key features of graphs and
tables, including writing and interpreting
the intervals on which a function
increases, decreases, is positive, and is
negative. Students will also determine and
1
Analyzing Function Intervals
a bracket or a parenthesis for an interval indicates
whether the endpoint is included. For x , 8, the
endpoint 8 is not included, so use a parenthesis:
(2`, 8). Note that parentheses are always used
with infinity symbols. For x # 8, now the value 8 is
included, so use a bracket: (2`, 8].
8/2/11 10:47:58 AM
interpret the average rate of change over
a specific interval. Students should be able
to determine coordinates from a graphed
function and understand how to write
and interpret an inequality. They will use
this information to write intervals using
interval notation.
Mathematical Practices
Use appropriate tools strategically.
In determining and interpreting the
intervals for a graphed function, students
will graph functions by hand and with a
graphing calculator. They will then analyze
the graphs to determine intervals, turning
points, and rates of change over specific
intervals.
CC-5
1
Problem 1 Writing an Inequality in Interval Notation
Problem 1
Write each inequality using interval notation.
Q When an endpoint is included in the interval, should
you use a bracket or a parenthesis? [bracket]
Q What does it mean if an inequality gives only one
endpoint? [The other endpoint is ` or 2`.]
Got It?
Q Is the endpoint 9 included in the interval? What
A 5 ' x ' 6
How is this interval
different from the
interval in part (A)?
The left endpoint is
not included in the
interval, and the interval
continues to infinity on
the right.
Since the endpoints are included in the inequality, use brackets. The interval
is F5, 6G .
B x / 15
The endpoints are not included in this inequality, so use parentheses. You want
all values greater than 15. The interval is (15, @).
Got It? 1. Write 9 x using interval notation.
does this mean? [Yes; you use a bracket.]
Q Should you use positive or negative infinity as
If the values of f (x) increase as x increases on an interval [a, b], we say that the function
is increasing on the interval. If the values of f (x) decrease as x increases on [a, b], we say
that the function is decreasing on the interval. If the values of f (x) remain unchanged as
x increases on [a, b], we say that the function is constant on the interval. The points on
a graph where it changes from increasing to decreasing or vice versa are called turning
points. These are the maxima and minima of the graph.
the other endpoint? Explain. [negative infinity,
because the interval is all values less than or
equal to 9]
Problem 2
Q How many increasing and/or decreasing intervals
are shown on the graph? Explain. [three; two
decreasing intervals and one increasing interval]
Q How many turning points are shown on the
graph? Explain. [two, one at 24 where f(x)
changes from decreasing to increasing, and
one at 0 where f(x) changes from increasing to
decreasing]
Problem 2 Identifying Intervals Where f (x) Is Increasing or
How do the relative
maximum and
relative minimum
help you write
increasing and
decreasing intervals?
These points on the
graph are the turning
points of the function.
They are endpoints of the
intervals.
Decreasing
Use the graph of f (x) to identify where the function
f (x) is increasing, where f (x) is decreasing, and the
turning points of the function.
Step 1
shown on the graph? [two increasing intervals
and one decreasing interval]
Q Will you use brackets or parentheses to write the
intervals? Explain. [The endpoints are included in
4
2
O
2
4
6
2
4
6
2
Step 3
The turning points are the ordered pairs at
which the graph changes from increasing to
decreasing or decreasing to increasing. The
turning points are at (4, 3), and (0, 5).
Identify the intervals where g (x) is increasing,
the intervals where g (x) is decreasing, and the
turning points of g (x).
2
x
6
To find where f (x) is decreasing, you need to
determine the x-intervals where the graph of f (x)
falls as you move from left to right. You can see
that the graph is decreasing when 7 x 4 and
0 x 6. Write these intervals as F7, 4G and [0, 6].
Got It? 2. The graph of g (x) is shown at the right.
each interval, so I will use brackets.]
2
Step 2
Got It?
Q How many increasing and decreasing intervals are
To find where f (x) is increasing, you need to
determine the x-intervals where the graph of
f (x) is rising as you move from left to right.
You can see that the graph is increasing when
4 x 0. The interval is F4, 0G .
f(x)
4
4 g(x)
2
x
6 4
O
4
Common Core
Answers
HSM12_CC_TransitionKit_CC05.indd 2
Solve It!
The driver was increasing his speed for the first
minute and also between 4.5 and 5.5 minutes. He
increased at a faster rate from 4.5 to 5.5 minutes.
The segment from 4.5 to 5.5 minutes is steeper
than the segment from 0 to 1 minute. The driver
had to slow down twice.
Got It?
1. (2`, 9]
2. increasing f27, 25g and [0, 7]; decreasing
f25, 0g; turning points at (25, 2) and (0, 22)
2
Common Core
8/2/11 10:48:02 AM
The average rate of change of a function over an interval [a, b] is the ratio of the change
in the function value to the length of the interval. This is computed as f (b) f (a) .
Problem 3
b a
Problem 3 Finding the Average Rate of Change Over
Q What points do you need to find the average
f(x)
rate of change? [the endpoints of the interval,
an Interval
How can you find
the average rate of
change without the
graph?
Substitute the interval’s
endpoints a and b into
the function rule to
find f(a) and f(b), then
use the average rate of
change formula.
x3
(22, 0) and (0, 4)]
4
x2
The function f (x) 4x 4 is shown by the graph. What is
the average rate of change of the function over the interval F2, 0G ?
Q Why isn’t the turning point at (21, 6) important
2
in solving this problem? [Only the endpoints are
x
From the graph you can see the value of f (2) 0 and the value of
f (0) 4. Substitute these values into the formula for the average rate of
change.
O
3
needed to solve for the average rate of change
over a specific interval.]
3
2
f (b) f (a)
f (0) f (2)
40
4
2
ba
0 (2)
02
2
4
Got It?
6
The average rate of change of the function over the interval F2, 0G is 2.
Q What values of f(x) do you need to find to solve
this problem? [f(21) and f(1)]
Got It? 3. What is the average rate of change of the function described by the graph over
the interval F1, 1G ?
Problem 4
Problem 4 Interpreting the Average Rate of Change Over an Interval
The graph shows the balance in a bank account in which $5000 was invested at 8%,
compounded monthly. Estimate and interpret the average rate of change for this
investment over the interval [15, 20].
Q What does the interval [15, 20] represent? [the
balance of the bank account from 15 to 20 years]
Q What do the values 17,000 and 25,000
represent? [the amount in the bank account at
Use the graph to estimate f (x) for
U
ea x-value, and then substitute
each
th values into the rate of change
these
fo
formula.
TThe values of f (x)
when
w
x 15 and
w
when x 20
The graph of the value of
the account over time
From the graph you can see the value of f (15) 17,000
and the value of f (20) 25,000. Substitute these values
into the formula for the average rate of change.
The average rate of change of the function over the
interval [15, 20] is 1600. This means that from years 15
to 20 the investment is increasing at an average rate of
$1600 per year.
f(x)
$45,000
$35,000
Dollars
f (20) f (15)
25,000 17,000
8000
1600
20 15
20 15
5
15 years and at 20 years]
$25,000
$15,000
$5,000
0
Got It?
0 3
9
15
x
27
21
Q Before solving for the average rate of change, use
the graph to predict whether the average rate of
change will be greater for [15, 20] or [20, 26].
Explain. [the interval [20, 26], because the graph
Years
Got It? 4. a. What is the average rate of change for this
investment over the interval [20, 26]?
b. Interpret this rate of change.
is steeper]
CC-5
3
Analyzing Function Intervals
Additional Problems
HSM12_CC_TransitionKit_CC05.indd 3
8/2/11 10:48:07 AM
ANSWER f22, 6)
2. The graph of f(x) is shown below.
Identify where f(x) is increasing, where
f(x) is decreasing, and the turning
points of f(x).
3. The function f(x) 5 2 2 5x 1 1
is shown by the graph. What is
the average rate of change of the
function over the interval f22, 3g ?
2
x
1
⫺2
O
⫺2
x
⫺4
O
2
4
⫺2
3
⫺4
ANSWER increasing f25, 21g and
[3, 5]; decreasing f21, 3g ; turning
points (21, 3) and (3, 22)
$100 Invested for 45 Years
at 5 Percent Interest
$800
$600
$400
$200
$0
1
9
17
25
33
41
Years of Saving
⫺4
⫺6
the average rate of change for this investment
over the interval [15, 29].
f(x)
4
f(x)
⫺5
x2
Balance of Savings
1. Write the inequality 22 # x , 6
using interval notation.
x3
ANSWER 1
4. The graph shows the balance in a
bank account in which $100 was
invested at 5% interest, compounded
annually. Estimate and interpret
ANSWER The average rate of change is
approximately 14.29. The investment is
increasing at an average rate of $14.29 per
year from years 15 to 29.
CC-5
3
A function is positive on an interval if f (x) 0, for every x-value in the interval. A function
is negative on an interval if f (x) 0 for every x-value in the interval.
Problem 5
Q Are x or y-values used to write intervals? [x-values]
Q How do you determine which intervals are positive
Problem 5 Identifying Intervals Where a Function is Positive or
Negative
and which are negative? [Intervals for which f(x)
The graph of f (x) x 3 2x 2 is shown at the right. Find the
intervals where f (x) is positive and the intervals where f (x) is negative.
is above the x-axis are positive and intervals
below the x-axis are negative.]
Got It?
Step 1
Why do you use
parentheses for the
endpoints 0 and 2 of
the intervals?
Since f(0) 0 and
f(2) 0, the function
is neither positive nor
negative at the endpoints
0 and 2. So the endpoints
are not included in these
intervals.
ERROR PREVENTION
Students may confuse decreasing intervals with
negative intervals and may confuse increasing
intervals with positive intervals. Point out that
an interval may be decreasing and positive
or increasing and negative. The two sets of
properties—positive/negative and increasing/
decreasing—are independent.
6 f(x)
4
Identify the x-intercepts of the graph.
The x-intercepts are 0 and 2. These two intercepts divide the
x-axis into three intervals.
Step 2
Identify the intervals for which the graph is above the x-axis.
The function is positive on the intervals (@, 0) and (0, 2).
Step 3
Identify the intervals for which the graph is below the x-axis.
The function is negative on the interval (2, @).
2
x
O
3
3
2
4
6
Got It? 5. The graph of g (x) x 4 5x 2 4 is shown to the right.
Find the intervals where g(x) is positive and the intervals
where g(x) is negative.
g(x)
2
x
O
3
3
4
Problem 6
Problem 6 Using a Calculator to Find Key Features of a Graph
Stress that when using a graphing calculator, the
endpoints are often estimates.
A Determine the intervals where the graph of f (x) x 3 3x 1 is increasing or decreasing
and find the turning points. Round all decimals to two decimal places.
Use the
Q How might the table feature save you time when
trace
key to identify the turning points of the graph. The x-values of
these turning points are the endpoints of your intervals.
finding the average rate of change for a function
on a graphing calculator? [For integer endpoints,
Y1=X3–3X+1
exact values for f(x) are listed.]
Y1=X3–3X+1
Q Describe the difference between an increasing
interval and a positive interval. [An increasing
interval rises from left to right on the graph of
the function. A positive interval is an interval
over which the function is above the x-axis.]
X=–1.06383
4
x
⫺4
⫺2
O
2
4
⫺2
⫺4
ANSWER Positive (21, 2) (2, `)
Negative (2`, 21)
4
Common Core
Y=–.9875172
Common Core
Additional Problems
f(x)
X=1.0638298
The turning points are approximately (1.06, 2.99) and (1.06, 0.99). The function is
increasing on the interval (@, 1.06], decreasing on the interval F1.06, 1.06G , and
increasing on the interval [1.06, @).
4
5. The graph of f(x) 5 x3 2 3x2 1 4 is
shown below. Find the intervals where
f(x) is positive and the intervals where
f(x) is negative.
Y=2.9875172
Answers
HSM12_CC_TransitionKit_CC05.indd 4
6. Consider the function
f(x) 5 x4 2 6x2 1 2 .
a. Determine the intervals over
which the function is increasing or
decreasing.
b. Determine the average rate of
change over the interval f21, 2g .
c. Identify the intervals where the
function is positive and where it is
negative.
ANSWER
a. increasing over f21.75, 0g
and f1.75, `) ; decreasing over
(2`, 21.75) and [0, 1.75]
b. 21
c. positive over (2`, 22.4) ,
(20.6, 0.6) , and (2.4, `) ; negative
over (22.4, 20.6) and (0.6, 2.4)
8/2/11 10:48:10 AM
Got It? (continued)
3. 23
4. a. 2500
b. The investment is increasing at an average
rate of $2,500 per year from years 20 to 26.
5. negative on the intervals (2`, 22), (21, 1)
and (2, `), positive on the intervals (22, 21)
and (1, 2)
6. increasing (2`, 0g and f1.35, `), decreasing
[0, 1.35], negative (2`, 0) and (0, 2), positive
(2, `); the average rate of change over
f23, 1g is 11
B Determine the average rate of change over the interval [1, 3].
Use the table feature to identify the values of f (x) when
x 1 and x 3.
Substitute the values into the rate of change formula.
f (3) f (1)
19 (1)
20
10
31
31
2
C Identify the intervals where the function is positive and the
intervals where the function is negative.
X
Y1
0
1
2
3
4
5
6
1
–1
3
19
53
111
199
X50
Press 2nd trace and then select 2:zero to identify the x-intercepts of the graph.
Use these values to write the intervals.
Got It?
Zero
X=–1.879385
Zero
X=.34729636
Y=0
Y=0
Zero
X=1.5320889
Y=0
Q Will finding the intervals where the graph
increases, decreases, is positive, and is negative
help you determine the average rate of change
over the interval f23, 1g ? Explain. [No; the
The function is negative on the interval (@, 1.88), positive on the interval (1.88, 0.35),
negative on the interval (0.35, 1.53) and positive on the interval (1.53, @).
Got It? 6. Identify the intervals where the function f (x) x 3 2x 2 is increasing,
average rate of change only depends on the
x- and y-values of the function for the two
endpoints.]
decreasing, positive, and negative. Then find the rate of change over the
interval F3, 1G .
3 Lesson Check
Lesson Check
Do you know HOW?
Do you UNDERSTAND?
For each graph, determine the intervals where f (x) is
increasing and where f (x) is decreasing.
1.
2
2 f(x) x
O
2
2.
f(x)
6
O
• For Exercise 1, remind students that the graph
of a function increases or decreases from left to
right.
• For Exercise 3, encourage students to graph the
function to find the endpoints for the interval and
then determine the rate of change.
5. Error Analysis Your friend has concluded that
whenever a function is increasing on an interval, the
function must be positive, and when the function
is decreasing, the function must be negative. How
would you correct your friend’s misunderstanding?
x
4
Do you know HOW?
4. Writing If the values of f (x) do not change as x
increases on an interval, what can you conclude
about the behavior of the function? Explain.
4
4
MATHEMATICAL
PRACTICES
4
Do you UNDERSTAND?
4
• For Exercise 4, it may help students to make a
sample graph.
8
Close
1
3. What is the average rate of change of f (x) xx 2
over the interval F1, 2G ?
Q Describe the difference between a decreasing
CC-5
Answers
HSM12_CC_TransitionKit_CC05.indd 5
Analyzing Function Intervals
5
interval and a negative interval. [A decreasing
interval slopes down from left to right on the
graph of the function. A negative interval is an
interval over which the graph of the function is
below the x-axis. A decreasing interval may be
above, below, or intersect the x-axis.]
8/2/11 9:49:35 PM
Lesson Check
1. decreasing (2`, `)
2. increasing (2`, 22g and f2, `),
decreasing f22, 2g
3. 0.25 or 14
4. If x is increasing and f(x) remains the
same for all x-values in the interval,
then the graph does not increase
or decrease. It is constant or is a
horizontal line.
5. A function may be increasing when
the values of f(x) are negative, and a
function can be decreasing when the
values of f(x) are positive.
CC-5
5
MATHEMATICAL
Practice and Problem-Solving Exercises
4 Practice
A
Practice
See Problem 1.
Write each inequality in interval notation.
ASSIGNMENT GUIDE
6. x 20
Basic: 6–18 odd, 19–20, 25–37
PRACTICES
7. 10 x 2
8. 6 x
See Problem 2.
For each graph, determine the following:
a. The intervals where f (x) is increasing
b. The intervals where f (x) is decreasing
Average: 6–18 odd, 19–37
Advanced: 6–18 even, 21–37
9.
10.
24 f(x)
Standardized Test Prep: 25–28
11.
f(x)
8
Mixed Review: 29–37
4
4
MP 1 Make Sense of Problems Ex. 21
MP 1 Make Sense of Problems Ex. 22
MP 3 Construct Viable Arguments Ex. 4
MP 3 Critique the Reasoning of Others Ex. 5
MP 4 Model with Mathematics Ex. 23
O
x
x
2
2
x
O
4
O
2
4
Mathematical Practices are supported by
exercises with red headings. Here are the Practices
supported in this lesson:
f(x)
4
4
4
See Problem 3.
Refer to the graph below. Find the average rate of change
over each interval.
y
2
6 4
HOMEWORK QUICK CHECK
O
2
x
4
6
2
To check students’ understanding of key skills and
concepts, go over Exercises 7, 9, 14, 15, and 17.
12. F2, 0G
13. F1, 1G
14. [1, 4]
15. A model rocket is launched from the ground. The graph of f (t) represents the
height (in feet) of the rocket t seconds after it is launched.
See Problem 4.
140 f(t)
100
60
20
t
2
4
6
Estimate the rocket’s average rate of speed from the time it starts falling to the time
it hits the ground.
Find the intervals where each function is positive and the intervals
where each function is negative.
3x 2
16. f (x) x 3 1
17. f (x) 2
x 1
6
Answers
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
6
(20, `)
(210, 2g
(2`, 26g
a. increasing (2`, 24g and f2, `)
b. decreasing f24, 2g
a. increasing (2`, `)
b. f(x) does not decrease
a. increasing (2`, 0.5g
b. decreasing f0.5, `)
232
1
21
It is falling at an average rate of approximately
53 ft/sec.
positive (2`, 1), negative (1, `)
positive (2`, 0), (0, `)
Common Core
18. f (x) x 7 2x 3
Common Core
18. positive (21.19, 0) and (1.19, `),
negative (2`, 21.19) and (0, 1.19)
HSM12_CC_TransitionKit_CC05.indd 6
Practice and Problem-Solving Exercises
See Problems 5 and 6.
8/2/11 10:48:17 AM
B
Apply
19. The graph of a function has a maximum at x 1 and a minimum at x 5. There
are no other turning points on the graph. Identify the intervals where the function
is increasing and the intervals where it is decreasing.
time. For a majority of the 12-month period,
the stock had a higher value, and the rate of
increase is higher than the bank account.
24. a. As x increases, f(x) increases, except when
x 5 2.
b. When x 5 22, the graph is undefined.
Using the table function will show an error
message.
c. This graph does not contain any decreasing
intervals. The function is increasing for
(2`, 2g and f2, `)
y
20. Use the graph to the right to answer the following questions.
a. Over which interval(s) is the function increasing?
b. Over which interval(s) is the function constant?
c. For which interval(s) is the function positive?
d. What is the average rate of change over the interval
F5, 3G ?
4
2
x
2
4
21. Think About a Plan Use the following clues to draw a rough
graph of the described function.
t The function has turning points at (2, 2) and (0, 3).
t The function is negative on the interval (4, 1).
t One of the x-intercepts is 2.
22. Use the following clues to draw a graph of the described function.
t The function decreases on the interval (@, @).
t The x-intercept is 1.
t The y-intercept is 2.
4
6
2
23. Investments The graph shows the value of a stock portfolio over a 12-month period.
Value (dollars)
5000 y
4000
3000
2000
1000
x
0
2
4
6
8
10
Time (months)
The function f (x) 1500(1.0025)x represents the value of a savings account after
x months. Compare the average rate of change for both investments for months
three through six then for months seven through ten. If you were to continue
investing for another year, which would be the better option—the stock portfolio or
the savings account? Explain.
C
Challenge
1
24. Consider the function f (x) xx 2.
a. As the value of x increases, what happens to the value of f (x)?
b. Is there a value of x for which the graph is undefined?
c. What does this tell you about increasing and decreasing intervals for this
function?
CC-5
19. increasing f25, 1g, decreasing
(2`, 25g and f1, `)
HSM12_CC_TransitionKit_CC05.indd 7
20. a. f25, 23g, [0, 3]
b. [3, 5]
c. (24.2, 22), (2, 7)
d. 52
21. Answers may vary. Sample:
4
y
O
⫺2
⫺4
22. Answers may vary. Sample:
4
y
8/2/11 10:48:21 AM
x
⫺4
⫺2
O
2
4
⫺2
⫺4
x
⫺2
7
Analyzing Function Intervals
2
4
23. Answers may vary. Sample: The
stock would be the better option
for another 12 months. For months
three through six, the stock increased
about $233 per month, and the bank
account increased about $150 per
month. For months seven through
ten, the stock increased by about
$250 per month, and the bank
account increased by about $163 per
month. For both intervals, the stock
had a higher rate of increase than the
bank account, even though the value
of the stock dropped some of the
CC-5
7
Standardized Test Prep
Answers
25. For the graph shown at the right, what is the interval of the graph
SAT/ACT
C
I
B
G
7, 25, 28
219, 217, 216.5
71, 11, 24
f(x) 5 (x 1 1.5)2 2 1.25 or
F1, 1G
F2, 2G
F1, @)
26. For the graph shown at the right, what is the average
x
6 4
4
4
3
43
4
O
6
5
28. The table shown below represents a quadratic function.
33. f(x) 5 (x 2 1)2 1 3
34. f(x) 5 2(x 1 1.25)2 2 3.125
2
x
8 6 4
What is the interval where the function is increasing?
25
f(x) 5 2 Q x 1 4 R 2 8
35. up and down
36. up and up
37. up and down
x
2 1
y
11
1
0
1
2
5 7 5
2
O
3
4
1
11
6
F2, 1G
F2, 4G
[0, 4]
2
4
2
4
[1, 4]
y
4
5
12
12
5
12
6
6
for the average rate of change for the interval F8, 4G ?
12
4
4
27. For the graph shown at the right, which is the best estimate
5
2
2
rate of change over the interval F2, 1G ?
3 2 5
f(x) 5 Q x 1 2 R 2 4
5 2
(@, 1G
y
4
where the function is decreasing?
Practice and Problem-Solving
Exercises (continued)
25.
26.
27.
28.
29.
30.
31.
32.
6
Mixed Review
For each function, evaluate f (2), f (0), and f + 12 , .
29. f (b) (6b 5)
30. f (m) (m 17)
31. f (g) (30g 11)
33. f (x) x 2 2x 4
34. f (x) 2x 2 5x
Write each function in vertex form.
32. f (x) x 2 3x 1
Determine the end behavior of the graph of each polynomial function.
35. f (x) 5x 3 4x 2 x
8
Common Core
37. f (x) 9 3x 4 6x 9
Common Core
HSM12_CC_TransitionKit_CC05.indd 8
8
36. f (x) 8x 3x 2 7
8/4/11 1:18:54 AM