Prior to this lesson, students studied situations characterized by exponential change using descriptions, tables, graphs, and equations. In this lesson, they start to view these relationships as exponential functions. This means choosing an independent and dependent variable and expressing the relationships using function language and, in some cases, function notation. For an exponential relationship, either variable can be the independent variable but one choice gives an exponential function while the other gives a logarithmic function (which is outside the scope of this course). The contexts here are chosen and presented so that it is more natural to choose an independent variable that leads to an exponential function.
Students use variables to represent one real-world quantity as a function of another, which is an example of decontextualizing and reasoning abstractly (MP2).
- Determine and explain (orally and in writing) whether relationships—in descriptions, tables, equations, or graphs—are functions.
- Use function notation to write equations that represent exponential relationships.
Let’s explore exponential functions.
Devices that can run Desmos (recommended) or other graphing technology should be available as an option for students to select during the lesson. Graphing technology is required for the optional activity "Deciding on Graphing Window". It is ideal if each student has their own device. (Desmos is available under Math Tools.)
- I can use function notation to write equations that represent exponential relationships.
- When I see relationships in descriptions, tables, equations, or graphs, I can determine whether the relationships are functions.
An exponential function is a function that has a constant growth factor. Another way to say this is that it grows by equal factors over equal intervals. For example, \(f(x)=2 \boldcdot 3^x\) defines an exponential function. Any time \(x\) increases by 1, \(f(x)\) increases by a factor of 3.