Download Georgia Department of Education Accelerated Mathematics III Unit 8

Georgia Department of Education
Accelerated Mathematics III
Frameworks
Student Edition
Unit 8
Investigations of Functions
2nd Edition
April, 2011
Georgia Department of Education
Georgia Department of Education
Accelerated Mathematics III
Unit 8
2nd Edition
Table of Contents
INTRODUCTION: ..................................................................................................3
Combining Functions Learning Task ...................................................................5
Composition of Functions Learning Task ...........................................................9
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Dr. John D. Barge, State School Superintendent
April, 2011  Page 2 of 12
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Georgia Department of Education
Accelerated Mathematics III
Unit 8
2nd Edition
Accelerated Mathematics III – Unit 8
Investigations of Functions
Student Edition
INTRODUCTION:
In this unit, students are given the opportunity to delve more deeply into functional
relationships. The tasks in Unit 8 offer contextual situations in which students will represent and
interpret functions numerically, algebraically, and graphically. Students will apply their prior
knowledge of all the functions they studied in their previous mathematics courses using
technology when appropriate. The tasks will focus on different ways of combining functions to
create a new function and will investigate the characteristics of the new function relative to the
characteristics of the functions that were combined. While each element of this standard could be
addressed independently, we have chosen not to do so in this unit. Emphasis is on recognizing
the transformations of functions by finding the function that models given data, then using these
functions to create new functions that meet a contextual need. There was not an attempt to
address each of the functions mentioned in element a, but rather the use of the ones that most
appropriately fit the data situations.
KEY STANDARDS ADDRESSED:
MA3A4. Students will investigate functions.
a. Compare and contrast properties of functions within and across the following types:
linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric,
and piecewise.
b. Investigate transformations of functions.
c. Investigate characteristics of functions built through sum, difference, product, quotient,
and composition.
RELATED STANDARDS ADDRESSED:
MA3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
MA3P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
April, 2011  Page 3 of 12
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Georgia Department of Education
Accelerated Mathematics III
Unit 8
2nd Edition
MA3P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and
others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
MA3P4. Students will make connections among mathematical ideas and to other
disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a
coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
MA3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
Unit Overview:
During our elementary years, we learned to add, subtract, multiply, and divide whole numbers. In
middle school we expanded those operations to integers, fractions, decimals, and finally real
numbers. In high school mathematics courses we learned to use these same operations for
complex numbers thus expanding our number system. However, these operations are not limited
to just numerical representations. Since variables and functions represent numbers and their
relationships, we can expand these operations to functions and thus expand the relationships that
we can represent by the resulting functions. In the following tasks, we will see how each of these
four operations can be used to represent relationships among and between data sets and make
interpolations and extrapolations using the new functions formed. Explorations of these new
functions will require use of our knowledge of the basic functions as well as the transformations
of these functions. Since this unit is likely the culminating unit of your high school mathematics
experience, much emphasis is placed on recognizing transformations of functions through use of
data sets, writing the functions that model the data, and analyzing the resultant function when
combinations are made.
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Dr. John D. Barge, State School Superintendent
April, 2011  Page 4 of 12
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Georgia Department of Education
Accelerated Mathematics III
2nd Edition
Unit 8
Combining Functions Learning Task:
Part I Per Capital Crime Rate
The number of violent crimes committed in major cities is one statistics that is used to determine
the safety rating of that city. In this task, we will examine data from two cities to not only make
conclusions about those cities, but to examine the relationships of the crime rates to other factors
relative to each city. In table 1, the number of violent crimes committed in each city is given by
year. In table 2, the population of each city is given by year.
TABLE 1:
2000
2001
2002
2003
2004
2005
City A
793
795
807
818
825
831
City B
448
500
525
566
593
652
Year
By just looking at the raw data for the number of crimes, which city would you predict is safer?
Why?
Year
2000
2001
2002
2003
2004
2005
City A
61,000
62,100
63,220
64,350
65,510
66,690
City B
28,000
28,588
29,188
29,801
30,427
31,066
By just looking at the raw data for the population, which city would you predict is safer? Why?
Do you think that these two data sets could be related? How? Why?
If in fact they are then we need to look at another relationship other than number of crimes per
year and number of people per year. We need to look at the relationship between the number of
crimes and the number of people or the per capita crime rate, that is the number of crimes per
person in each city.
To do that, we first have to define functions to represent the data as we have it. Let C(t) be the
function that represents the number of crimes in t year, where t is measured in number of years
since 2000. That means that for city A, C(0) = ?, C(1) = ? C(4) = ?
Let P(t) be the function that represents the population in t year, where t is measured in number of
years since 2000. That means that for city A, P(0) = ?, P(1) = ?, P(4) = ?
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April, 2011  Page 5 of 12
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Georgia Department of Education
Accelerated Mathematics III
2nd Edition
Unit 8
We have just identified another notational issue. How will we know to which city we are
referring? We can use subscripts to denote the city.
So, CA(0) represents the number of crimes committed in city A during 2000 and PA(2) represents
the population of city A in 2002.
Since the independent variable in our data is time, notice that each function written is dependent
upon time. That means for us to find the per capita crime rate for each city, that is to compare the
number of crimes to the number of people we need the ratio of these two functions. Let R A(t) be
the per capita crime rate in city A and RB(t) be the per capita crime rate in city B. Using C(t) and
P(t) for the appropriate cities, write the functional rule for R(t).
Now that you have the two functions defined, complete the table below showing the per capita
violent crime rate in both cities by year using the data from Table 1 and 2. Write each of the
function values as percents.
t (years)
2000
2001
2002
2003
2004
2005
RA(t)
RB(t)
Now, using this data, which city is safer? Why?
Make any conclusions about the trends you see in the data. What did you base your conclusion
on?
Write a function rule for CA(t), CB(t), PA(t), PB(t), and then using these function rules, write an
explicit function rule for RA(t) and RB(t). Verify that each function gives the correct value that
you calculated from the data in the table above. Using the functions, can you make predictions
about crime rates in the future if the trends in the given data continue?
Since the quotient of the functions gave us per capita crime rate, would the sum, difference, or
product of the two function C(t) and P(t) have any real world meaning in this situation? Why or
why not?
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Dr. John D. Barge, State School Superintendent
April, 2011  Page 6 of 12
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Georgia Department of Education
Accelerated Mathematics III
Unit 8
2nd Edition
Part II Per Capita Food Supply
In 1798, a 32 year-old British economist anonymously published a lengthy pamphlet criticizing
the views of the Utopians who believed that life could and would definitely improve for humans
on earth. The hastily written text, An Essay on the Principle of Population as it Affects the Future
Improvement of Society, with Remarks on the Speculations of Mr. Godwin, M. Condorcet, and
Other Writers, was published by Thomas Robert Malthus. Thomas Malthus argued that because
of the natural human urge to reproduce human population increases geometrically (1, 2, 4, 16, 32,
64, 128, 256, etc.). However, food supply, at most, can only increase arithmetically (1, 2, 3, 4, 5,
6, 7, 8, etc.). Therefore, since food is an essential component to human life, population growth in
any area or on the planet, if unchecked, would lead to starvation. As mathematicians, we know
that if something grows geometrically it can be modeled by an exponential function and if that
something grows arithmetically it can be modeled by a linear function.
Write a function P(t) that gives the population in year t of a country if the initial population is 4
million with the population growing at 5% per year.
Write a function N(t) that gives the number of people a country can supply food for in t year is the
initial food supply can feed 10 million and the number of people for which food is available
increases by 0.75 million per year.
Graph these functions. Is there a place and time where the number of people will exceed the food
supply?
When? How do you know?
If t = 0 denotes the year 2000, what year will the number of people equal the amount of food
available?
In the previous task, we found the quotient of the two functions as the operation needed to
represent the data of interest. What operation would help us be able to easier interpret the food
supply situation?
The difference of the functions would compare when there is a positive, zero, and negative
amounts indicating when there is more than enough, exactly enough, and not enough food for the
population.
Write the function rule for this function, calling it S(t).
Is this function exponential, linear, both, or neither? Why?
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Dr. John D. Barge, State School Superintendent
April, 2011  Page 7 of 12
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Georgia Department of Education
Accelerated Mathematics III
2nd Edition
Unit 8
Graph this resulting function. Is there a maximum value? If so, what is it? What does it mean in
the context of the problem situation?
Can you make a long term prediction regarding the food supply based on this graph? How do you
know?
What function would you write if we wanted to know the per capita food supply? What does this
graph tell us?
Part III Interpreting a Table
Given the functions f and g as defined in the table below.
x
f(x)
g(x)
1
3
2
2
4
1
3
1
4
4
2
3
Complete the tables for the following functions:
n(x) = f(x) + g(x) What kind of function is n(x)? Why?
p(x) = 2f(x)g(x) - f(x)
q(x) = g(x)/f(x)
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April, 2011  Page 8 of 12
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Georgia Department of Education
Accelerated Mathematics III
2nd Edition
Unit 8
Composition of Functions Learning Task
Part I Is Your Heart Rate Normal for that Medication?
Many medications have side effects that doctors must consider when prescribing the medication.
For example, many cold medications have the effect of raising the takers blood pressure.
Therefore, even when recommending over the counter cold medications, doctors and pharmacist
must consider whether the patient is already on medication for high blood pressure. If the patient
is on such a medication, then the cold medicine recommend will have to be one that does not
affect blood pressure.
In particular, some medications can increase the heart rate, or beats per minute, of a patient of
normal health in a predictable manner depending on how much medication is in the patient’s
system. Based on research and study of one medication, we shall call only D, one such set of data
is given in the table 1.
D, drug level
in mg
0
50
100
150
200
250
r, heart rate in beats
per minute
60
70
80
90
100
110
Write a function r(D) that describes the relationship between the amount of drug administered and
the resulting heart rate.
What does this function tell us about the relationship between the amount of drug administered
and the resulting heart rate? What general conclusions can we draw?
The data in table 1 represents what happens with the initial introduction of the medication. As
time passes, the medication is processed by the body and the effect will lessen. Table 2 gives the
amount of drug D in the patient’s body after t, hours from injection.
t, time in hours
since injection
0
1
2
3
4
5
6
7
8
D, drug level in
mg
250
200
160
128
102
82
66
52
42
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Dr. John D. Barge, State School Superintendent
April, 2011  Page 9 of 12
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Georgia Department of Education
Accelerated Mathematics III
2nd Edition
Unit 8
Write a function D(t) that describes the relationship between the amount of drug in the patient’s
system and the elapsed time since injection.
What does this function tell us about the relationship between the amount of drug in the patient’s
system and the elapsed time since injection? What general conclusions can we draw?
However, to monitor a patient to be sure that the effect of the drug is as expected, the medical
professionals take the patient’s pulse or heart rate. Therefore, we need to find a table and a
function that will predict the patient’s heart rate as time since injection. Using the data given in
Tables 1 and 2, complete the table below:
t, time in hours
0
1
2
3
4
5
6
7
8
r, heart rate in beats
per minute
Now, find a function that would model this data? How could such a function be formed? In other
words, we need r(t). How do we find it?
What kind of function is r(t)? How do you describe its graph? How does its graph relate to the
graphs of the r(D) and D(t) functions? Graph all three on the same window. What relationship
exists?
Use r(t) to find the predicted heart rate of a patient who was given the medication D 3 hours ago.
How would you use this information to determine if the patient is having an abnormal reaction to
the drug?
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Dr. John D. Barge, State School Superintendent
April, 2011  Page 10 of 12
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Georgia Department of Education
Accelerated Mathematics III
2nd Edition
Unit 8
Part II Interpreting a Table
As the cost of a post-secondary education rises, many students are working jobs to help pay for
college. However, these jobs often affect the number of hours they have to study, so may affect
the number of credit hours they pursue during a semester. Table 1 gives such an example of the
effect of the numbers worked on the number of credits attempted. Table 2 gives an example of
the number of hours needed to study relative to the number of hours attempted.
Table 1
Table 2
Number of
hours worked
per week
Number of
credits
attempted
Number of
credits
attempted
Number of hours
needed to study
per week
0  h 4
18
12
12
4  h 8
17
13
14
8  h  12
16
14
16
12  h  16
15
15
20
16  h  20
14
16
25
20  h  24
13
17
31
24  h  28
12
18
39
Graph Table 1 by hand on grid paper. What type of function does it appear to be? What
transformations need to be made to the basic function to model the data given? Write a function
rule for this table. Call this function C(w).
Graph Table 2 on a graphing utility or on grid paper. What type of function does it appear to be?
What transformations need to be made to the basic function to model the data given? Write a
function rule for this table. Call this function S(C).
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Dr. John D. Barge, State School Superintendent
April, 2011  Page 11 of 12
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Georgia Department of Education
Accelerated Mathematics III
Unit 8
2nd Edition
Complete the table below:
Number of
hours worked
per week
Number of hours
needed to study
per week
0  h 4
4  h 8
8  h  12
12  h  16
16  h  20
20  h  24
24  h  28
As the number of hours worked increases, what happens to the number of hours spent studying?
Graph this data on grid paper. What type of function does it appear to be? How does this
function relate to the ones from Tables 1 and 2? Would the use of the functions you wrote for
Tables 1 and 2 help to find a function rule for this data? Write the function that relates number of
hours spent studying to the number of hours worked per week. Call this function S(w).
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Dr. John D. Barge, State School Superintendent
April, 2011  Page 12 of 12
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