Mathematics IV Unit 8 1st Edition
Mathematics IV Frameworks
Student Edition
Unit 8
Investigations of Functions
1st Edition
January, 2011
Georgia Department of Education
Table of Contents
INTRODUCTION:….. ..3
Unit Overview:...... 4
Combining Functions Learning Task: Part I- Per Capita Crime Rate...... 5
Combining Functions Learning Task: Part II -Per Capita food supply...... 7
Combining Functions learning task: Part III -Interpreting Tables...... 9
Composition of Functions Learning Task: Part I -Is Your Heart Rate Normal for that Medication?...... 10
composition of functions learning task: Part II- Interpreting tables…………………………………………………………………………………………..12
Mathematics IV – 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 Mathematics I, II, and III 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:
MM4A4. 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:
MM4P1. 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.
MM4P2. 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.
MM4P3. 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.
MM4P4. 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.
MM4P5. 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.
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:
Year / 2000 / 2001 / 2002 / 2003 / 2004 / 2005City A / 793 / 795 / 807 / 818 / 825 / 831
City B / 448 / 500 / 525 / 566 / 593 / 652
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 / 2005City 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) = ?
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) represent 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 RA(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 / 2005RA(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?
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?
Write the function rule for this function, calling it S(t).
Is this function exponential, linear, both, or neither? Why?
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:
A. n(x) = f(x) + g(x) What kind of function is n(x)? Why?
B. p(x) = 2f(x)g(x) - f(x)
C. q(x) = g(x)/f(x)
adapted from Functions Modeling Change, A Preparation for Calculus, Connally, Hughes-Hallett, Gleason, et al. John Wiley & Sons, 1998.
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 levelin 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.