Interpreting exponential functions
• Solve problems involving exponential functions in many different contexts (F-LE.A.2$^\star$).
• Write exponential expressions in different forms (F-BF.A.1$^\star$).
• Explain what the parameters of an exponential function mean in different contexts (F-LE.B.5$^\star$).
• Use the properties of exponents to write expressions in equivalent forms (A-SSE.B.3c$^\star$).
In Exponential Functions 1, students worked with exponential functions in the form $f(t)=ab^t$ or $f(t)=a(1+r)^t$ and interpreted the parameters $a$, $b$, and $r$ in terms of a context. In this unit they see more complicated forms. In the previous section, students developed an understanding of different compounding intervals and continuous compounding using base $e$. The purpose of this section is to examine some of these different forms and learn to interpret the parameters in terms of a context. Students learn the concept of doubling time and see functions expressed in a form that shows the doubling time; they work algebraically with functions expressed in a form like $f(x) = A(1 + r/n)^{nt}$ that shows the compounding period; and they work with functions written with the base e, $g(x) = Ae^{rt}$ , in many continuous growth contexts. In this section they do not build functions in any of these forms.
Tasks
1
Bacteria Populations
WHAT: Students are given three functions in the form $p(t) = Pe^{rt}$ that model the growth of three bacteria colonies. They are asked to use the structure of exponential expressions (MP.7) to contrast the behavior of the three colonies.
WHY: This task gives students an opportunity to interpret continuous growth functions in a context and set up and solve associated equations. (The solution does show solving a question by using natural log, but this approach is not necessary and would not be expected in this unit. Students could instead take a graphing approach.)
2
Lake Algae
WHAT: Students are given a description of a situation: an algae population on a lake is doubling every day and will completely cover the lake in 30 days. They respond to a series of question to make sense of the situation, and then construct a function to model the percent of the lake covered by algae in terms of days.
WHY: This is a simple example of a situation with a quantity that doubles in preparation for more complicated work with the concept of doubling time. While the context presents a classic example of exponential growth, it approaches it from a non-standard point of view. Instead of giving a starting value and asking for subsequent values, it gives an end value and asks about what happened in the past. Hence, it provides an opportunity to reason abstractly (MP.2).
3
Rising Gas Prices – Compounding and Inflation
WHAT: In the context of price inflation of a single commodity, students are given the price of a gallon of gas in consecutive years, and asked to compute the percent increase. Then they are given the doubling time (in recent history) of the price of a gallon of gasoline, and asked to compute the annual inflation rate, $r$. Finally, they use a table of real data to compare gasoline inflation with overall inflation in specific years.
WHY: The purpose of this task is to give students an opportunity to explore various aspects of exponential models in the context of a real world problem with ties to developing financial literacy skills. This task gives opportunity to work with a question of the form, “here’s the doubling time, find the annual percent increase” in a real-world context.
4
Forms of exponential expressions
WHAT: The Forms of Exponential Expressions task compares the same exponential function in four different forms and asks students to show that they are the same function. Students are also asked when each form might be useful, based on what information is wanted.
WHY: Students are asked to write equivalent expressions, but the question is posed in an engaging way with a challenge to show expressions are equivalent. (Essentially, students will prove expressions are equivalent, but the word “proof” is not used.) Since the different forms represent a context, students make use of structure (MP.7) to interpret different parameters in representations of exponential functions.