In this unit, students build on their understanding of linear functions, properties of exponents, and percent change to explore exponential relationships. Students learn that exponential relationships are characterized by a constant quotient over equal intervals, and compare it to linear relationships, which are characterized by a constant difference over equal intervals. They encounter contexts that change exponentially. These contexts are presented verbally and with tables and graphs. They construct equations and use them to model situations and solve problems. At first, students investigate these exponential relationships without using function notation and language so that they can focus on gaining an appreciation for critical properties and characteristics of exponential relationships.

Later, students view these relationships as functions and employ the notation and terminology of functions. They study graphs of exponential functions both in terms of contexts that they represent and abstract functions that don’t represent a particular context, observing the effect of different values of and on the graph of the function represented by . 

The contexts used early in this unit lead to functions where the domain is the integers. In some situations, students will interpret exponential expressions with fractional values in the exponent using graphing, but the connection to roots or logarithms is left for a later course.

Note on materials: Throughout this unit, students should have access to a calculator with an exponent button. Access to graphing technology is necessary for some activities and encouraged throughout the unit. Examples of graphing technology include a handheld graphing calculator, a computer with a graphing calculator application installed, or an internet-enabled device with access to a site like desmos.com/calculator or geogebra.org/graphing. Interactive applets are embedded throughout, and a graphing calculator tool is accessible in the Math Tools in the digital version.

Graph of 3 functions labeled, p, q, and r on grid, origin O. Horizontal axis, time in hours, from 0 to 10, by 2's. Vertical axis, number of bacteria, from 0 to 40,000 by 20,000's. All three functions start at 0 comma 5,000 and trend upward and to the right. Function p passes through 2 comma 20,000 and trends rapidly upward and right. Function q passes through 4 comma11,250 and trends steadily upward and right. Function r passes through 6 comma 7,200 and trends more slowly upward and right.