This Unit continues the discussion of functions by examining exponential functions. Models of exponential growth and decay are numerous such as growth or decay of populations—from bacteria, amoebas, radioactive material and money, to mammals (including people). Doubling, tripling, halving, and so on, are all intuitive situations for students to help them make sense of exponential functions.
The growth pattern in exponential functions is multiplicative. That is, for each additive change in the independent variable, there is a multiplicative change in the dependent variable. For example, in Problem 1.1, students look at the number of ballots created by repeatedly cutting in half a sheet of paper. As the number of cuts increases by one, the number of ballots increases by a factor of two. This factor is called the growth factor.
Investigation 1 continues to look at doubling, tripling, and quadrupling patterns. It ends by contrasting linear and exponential growth factors. Investigation 2 introduces the y‑intercept, or initial value, which it is sometimes called in exponential growth situations. Investigation 3 introduces growth rates that are not whole numbers and leads to growth rates, usually expressed as percents. Investigation 4 introduces growth factors that are less than one, but greater than zero. These are exponential decay situations. In Investigation 5, patterns with exponents are explored.
The Unit ends by looking at the effects of growth factors and y‑intercepts on graphs of exponential functions. Since exponential growth patterns can grow rapidly, students may encounter answers on their calculators expressed in scientific notation. Therefore, scientific notation is introduced in Investigation 1 and used throughout the Unit.Student Textbook