Select students to share their responses and explanations for each description. If not brought up by a student, point out that all four examples of exponential equations show the same initial values and exponents but different bases and, as such, different growth factors. The numbers 0.5, 1.25, 0.75, and 2 were used. Tell students that, in this lesson, we will encounter another number that is used as a base in many contexts.

If time allows, invite students to write a situation to match , and then select 2–3 students to share.

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the representations. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement. 

The function presented in the table is not, strictly speaking, exponential, as the successive quotients have slightly different values. If students notice or wonder about this, tell them that the values have been rounded to the nearest whole number.  

Then tell students that is an important constant in mathematics. Its value is about 2.718. It is irrational, so it can’t be represented with a fraction, and its decimal representation never repeats or terminates. (Other examples of irrational numbers are and .) It is sometimes called Euler’s number, and is named after the 18th century mathematician Leonhard Euler.

Explain that is used by scientists, engineers, economists, and others because it has useful properties in other mathematical areas such as calculus. Tell students that they will learn much more about the many uses of , its properties, and where it comes from in future courses. For now, it is enough to know that represents a number that is approximately 2.718.

This activity serves as a brief and light introduction to how is visible in the behaviors of certain functions. The standards addressed in this course don’t require a deep understanding of the meaning of , so the observations from this activity should not be assessed.

has a special connection to the expression . Jumping to this expression would likely be an opaque process for most students, so they first investigate the range of behavior of its component pieces, for very small and very large positive values of .

Entering increasingly larger or smaller values of in the functions and making observations about them is a chance to look for and express regularity through repeated reasoning (MP8).

Students may substitute 0 for in order to see what happens for small positive values of . If these students are not sure how to think about the functions , , or , where 0 is not in the domain, consider asking:

• “Can you explain how you found the output when is small and positive.”

• “How could you use a value very close to 0 to figure out the value of the function?”

Invite students to share their observations for the first question and the supporting work (a table, a spreadsheet, or graphs) that led to these observations, if available.

Direct students’ attention to the reference created using Collect and Display. Ask students to share what they noticed about the two functions and . Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

If not already observed by students, highlight that:

• The value of function gets closer and closer to as gets larger and larger. Display a graph representing , and trace it out for large values of . The curve essentially flattens out as increases.

  

  Here is a graph of along with a graph of the horizontal line given by .

  

  As the input grows, the two graphs are difficult to distinguish. The function has a horizontal asymptote of . (The exact value of the horizontal asymptote for is one way to define the number .)

• When is a very large value, is very small (as in function ). So in the expression , we are expressing as . (It is critical that the "huge" number is the multiplicative inverse of the "small" number.)
• The values of and are very close for very small, positive values of . In other words, raising to an increasingly small positive exponent gives essentially the same values as adding 1 to . Display a graph such as the following, or use graphing technology and zoom in toward smaller and smaller values of .

  

Tell students that these behaviors turn out to be useful in mathematical and scientific applications and that they will learn a lot more about them in future math courses.