In this lesson, students analyze the graph of an exponential function \(f\) given by \(f(x) = a \cdot b^x\). In particular, they study the effect of \(b\) on the shape of the graph. To observe how \(b\) impacts the shape of the graph when \(b> 1\) and when \(0<b<1\), they simultaneously examine several functions of this form, all of which have the same value of \(a\).
Narrow contexts are used in these lessons to encourage greater attention to the growth factors of the different functions, rather than on interpreting the quantities in context. Since students are focusing mainly on understanding how varying \(b\) impacts the graph, they are looking at the structure of an exponential function and its graph (MP7).
One of the activities in this lesson works best when each student has access to devices that can run a Desmos applet because students will benefit from seeing the relationship in a dynamic way.
Lesson overview
 12.1 Warmup: Spending Gift Money (5 minutes)

12.2 Activity: Equations and Their Graphs (15 minutes)
 Includes "Are you Ready for More?" extension problem
 12.3 Activity: Graphs Representing Exponential Decay (15 minutes)
 Lesson Synthesis
 12.4 Cooldown: A Possible Equation (5 minutes)
Learning goals:
 Describe (orally and in writing) the effect of changing \(a\) and \(b\) on a graph that represents \(f(x)=a \cdot b^x\).
 Use equations and graphs to compare exponential functions.
Learning goals (student facing):
 Let’s study and compare equations and graphs of exponential functions.
Learning targets (student facing):
 I can describe the effect of changing \(a\) and \(b\) on a graph that represents \(f(x)=a \cdot b^x\).
 I can use equations and graphs to compare exponential functions.
Required materials:
 Graphing technology
Required preparation:
 Acquire devices that can run Desmos (recommended) or other graphing technology.
 It is ideal if each student has their own device. (Desmos is available under Math Tools.)
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