Invite students to share some issues they came across when evaluating the expressions with a calculator and how to resolve them. Make sure that students recognize that grouping symbols may be needed when entering an expression as an exponent, and that calculation results may vary because of the level of precision of the calculator.

Select groups of students to share their observations and graphs (or consider displaying graphs of the pairs of models for the four colonies for all to see). Here are some questions for discussion:

• “Which pair of graphs seem most alike?” (Colony 4's graphs are almost the same until you got to large values of . At the end of year 2, and are both about 11.3 thousand.)
• “Which pair of graphs seem least alike?” (Colony 1's graphs look different after just a few months. After 2 years, , while .)
• “For Colony 2, what is the growth factor for 1 month predicted by each model?” ( predicts a growth factor of 1.03, while predicts a growth factor of , which is a bit more than 1.03.)
• “If the first student models the population of another colony of insects as , what is a model that the second student might write that is an exponential function with a base of ?” ( )

Tell students that exponential functions that involve small but ongoing growth (such as population growth or inflation) can be modeled in different ways. A factor written in the form of highlights that the percent growth rate per unit time (for example, represents 3% per unit time) is applied continuously. The model predicts a change by a factor of for each unit of time. While these sound similar, they are not equivalent.

Students will learn more about in future courses. For now, it is sufficient simply to know that scientists and mathematicians often find it helpful to use to model exponential growth and decay in cases where the growth or decay is assumed to happen continuously.

This activity further familiarizes students with the way continuously compounded models are typically written. It makes explicit the format of an equation used to represent such a model, which students will encounter more of in future lessons. Students also interpret the parameters of such equations in context and use the given structure to complete partially built equations. In this course, though, students are not assessed on generating a model to represent a situation characterized by continuous growth.

Only examples of growth situations are included in the activity. Decay situations are intentionally excluded so students don’t have to make sense of a negative factor in the exponent of an expression while also deciphering a new form for writing exponential functions. In synthesizing the activity, however, the teacher could choose to mention that in decay situations the takes on a negative value to represent a percent decrease.

If students do not yet correctly relate the given exponential expression to the given situation, consider asking:

• “Can you explain how you completed the missing parts of the function.”

• “How could you use the example to help you identify the parts of each expression?”

Select students to share how they completed and interpreted the missing information, where possible drawing attention to the structure of the exponential model given. Invite students to discuss how this form is like and unlike the equations students have seen prior to this point ( ). In particular, highlight that the growth factor, , is expressed in terms of both and . 

Tell students that the value for is intertwined with the meaning of when working with an expression of the form . If is measured in years, then represents the growth rate per year. If is measured in days, then represents the growth rate per day.

Use this activity to allow students to practice analyzing graphs of exponential functions involving and setting an appropriate graphing window on their graphing technology in order to produce useful graphs.

Invite selected students to share their strategies for finding the output from a graph.

Students have graphed exponential functions and analyzed the graphs to solve problems prior to this point. Though the functions here use base , the analytical work is fundamentally the same as that when the base is another number. Students will also have opportunities to graph exponential functions with base later in the unit.

Focus the discussion on how students set the graphing window and how they used the graphs to answer the questions.

Depending on the graphing technology used, students may use the following strategies:

• Visually estimate the value of the output given a certain input, or the input given an output.
• Use a tracing tool to trace the graph, and identify the coordinates when the input or output has a certain value.
• To find an unknown input: Graph a horizontal line at a given -value (for example, , a doubled bacteria population), find the intersection of that line and the graph of the exponential function, and identify the -value at that intersection.
• To find an unknown output: Use a table, or type something like after was defined.

Some students may also notice that they could calculate the output value given an exponential equation and an input value, but had to rely on the graph to find an unknown input value because they were unsure how to solve for when the equation has as a base. Tell students that in a future lesson they will learn how to use logarithms to solve this type of equation algebraically, similarly to how they have used logarithms previously.