Introduction to exponential functions
• Distinguish between the growth laws of linear and exponential functions (F-LE.A.1$^\star$).
• Construct simple exponential models (F-LE.A.2$^\star$).
• Create tables and graphs of exponential functions and understand their behavior in terms of the fundamental growth law(F-IF.C.7e$^\star$).
• Understand the form of the expression $f(x)=ab^x$ for an exponential function in terms of the fundamental growth law (F-LE.B.5$^\star$).
In this section students construct and interpret exponential functions expressed in the form f(x)=ab^x to model various context. They work with contexts where the initial value a and the growth factor b are either given or are directly inferable from the context, or where they must interpret those values in terms of the context. They make tables and graphs of exponential functions and begin to acquire a quantitative sense of exponential growth both numerically and graphically.
Tasks
1
Identifying Exponential Functions
WHAT: The first part asks students to use technology to experiment with the two parameters defining an exponential function, a and b in the function $f(x) = ab^x$, with little guidance. Since it is important for the second part, teachers should encourage students to try a wide range of values, and in particular, values of b both less than and greater than 1. The task includes a Desmos app, in which students can make use of sliders to more viscerally see the effect of changing a and b separately. The second part gives four specific graphs and asks students to find a function to match them. A method is not prescribed, so a teacher may opt to allow students to continue experimenting in the dynamic environment or begin teaching them methods for determining an equation given points on the graph.
WHY: With this task, students will start to get a graphical sense of exponential change. The dynamic environment gives students an opportunity to learn about the capabilities of a dynamic graphing tool (MP5) and get a feel for how the parameters affect the graph without worrying about algebra or computation.
2
Exponential Parameters
WHAT: Students build a function $P(t) = 100(2)^t$ to model the growth of a colony of bacteria given a description of the situation. Then, they are given the function $P(t) = 50(3)^t$ which models a different bacterial colony and are asked to interpret the meaning of the 50 and the 3 in the context.
WHY: The task provides a reasonably straight-forward introduction to the basic parameters of an exponential function in terms of a modeling context. In general, an exponential function $f(t) = ab^t$ has two parameters. The parameter a is interpreted as the starting value (when t represents time), and b represents the growth rate -- the amount the quantity is multiplied by each time the value of t is incremented by 1. The task has students both generate an exponential expression from a contextual description of a repeating process (MP8), and in reverse, make use of structure to interpret parameters in a context from an algebraic expression (MP7).
3
U.S. Population 1790-1860
WHAT: Students compare successive differences and successive quotients for population data, and construct an exponential function of the form $f(x)=ab^x$ to model the data (MP4).
WHY: The purpose of this task is to help students learn that exponential functions are characterized by equal growth factors over equal intervals, and that the growth factor over a unit interval is the base b when the exponential function is expressed in the form $f(x)=ab^x.$
4
Basketball Rebounds
WHAT: Students construct a decaying exponential in the form $f(x)=ab^x$ to model the height of a basketball on successive bounces. The decay factor b comes from rules set by the International Basketball Federation.
WHY: The purpose of this task is to introduce students to an exponential decay function in the form $f(x)=ab^x$, with a concrete interpretation of the decay factor b. The context is designed to make clear the meaning of b, with each successive value marked by a clear physical phenomenon, the bounce, which leads to a reduction in height by a factor of b. Notice that the natural domain of the function is not all real numbers, but rather the set of positive integers. This can lead to a useful discussion about domains and modeling.
The solution to the third part of this task uses logarithms. It is recommended that either the third part be omitted as students have not yet seen logarithms or that the teacher introduce technology to solve the equation without logarithms (MP5).
5
Equal Differences over Equal Intervals 1
WHAT: In the first task (Equal Differences Over Equal Intervals I) students look at successive differences in a table of values of a linear function and formulate the principle that the successive differences are constant. In the second task (Equal Factors Over Equal Intervals) they do the same thing with successive quotients in a table of values of an exponential function.
WHY: It is recommended to give these tasks one after the other so that students can notice that both functions have underlying regular growth laws, but of different natures. Throughout this section, students have been making observations about the differences between linear and exponential functions. Now they formalize those observations and give an algebraic derivation of the growth laws (MP8).
6
Equal Factors over Equal Intervals
WHAT: In the first task (Equal Differences Over Equal Intervals I) students look at successive differences in a table of values of a linear function and formulate the principle that the successive differences are constant. In the second task (Equal Factors Over Equal Intervals) they do the same thing with successive quotients in a table of values of an exponential function.
WHY: It is recommended to give these tasks one after the other so that students can notice that both functions have underlying regular growth laws, but of different natures. Throughout this section, students have been making observations about the differences between linear and exponential functions. Now they formalize those observations and give an algebraic derivation of the growth laws (MP8).