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
Georgia Department of Education Dr. John D. Barge, State School Superintendent April, 2011 Page 2 of 12 All Rights Reserved
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 All Rights Reserved
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.
Georgia Department of Education Dr. John D. Barge, State School Superintendent April, 2011 Page 4 of 12 All Rights Reserved
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) = ? Georgia Department of Education Dr. John D. Barge, State School Superintendent April, 2011 Page 5 of 12 All Rights Reserved
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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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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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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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
Georgia Department of Education Dr. John D. Barge, State School Superintendent April, 2011 Page 9 of 12 All Rights Reserved
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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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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Georgia Department of Education Accelerated Mathematics III
Unit 8
2nd Edition
Complete the table below: Number of hours worked per week 0
h 4
4
h 8
8
h 12
12
h 16
16
h 20
20
h 24
24
h 28
Number of hours needed to study per week
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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