##
* *Reflection: Developing a Conceptual Understanding
Effect of Changing b in f(x) = (b)^x - Section 2: Investigation

The rate of exponential growth/decay is not immediately intuitive to students through this investigation. When students did a similar quadratics investigation they noticed that the closer the "a" value was to zero the wider the graph and the larger the "a" value the narrower the graph. I will say that the students did very well using the Desmos.com calculator to notice what values led to faster/slower growth or decay. However, they were really not sure why. Before teaching this lesson, I was trying to think of a way to make this clear. The students were really into the exponential growth/decay formula so I decided to link the idea to that concept. I showed students the following:

If you had the function f(x)=(0.1)^x and the function g(x)=(0.9)^x which one would decay more slowly? (students would say g(x))

So if we think about the exponential decay function g(x)=(1-___)^x, what would have to go in the blank to get the function g(x) above? (students say .1)

And what growth rate does this represent (students say 10%)

Now try the same thing for f(x). I let students work and they saw that the value must be 90% which is a much faster growth rate.

We could use the same logic for values greater than 1. This really clicked with the students and they could undersand why values close to 0 actually had a faster growth rate!

*Developing a Conceptual Understanding: Understanding the Base*

# Effect of Changing b in f(x) = (b)^x

Lesson 7 of 13

## Objective: SWBAT determine which values of the base will lead to exponential growth and which will lead to exponential decay.

## Big Idea: This investigation allows students to discover how the value of the base of an exponential function determines the appearance of that function.

*40 minutes*

#### Launch

*10 min*

Students will work in pairs on this matching activity. Explain to students that they are trying to group the 8 cards into two categories. Give students about 4-5 minutes to group their cards. Then go card by card and have a member from each group share why they put the card into the category that they did (MP3). Rather quickly, the titles exponential growth and exponential decay should emerge. Question students to find out how much they know about exponential change. For example, when discussing the table representation you could ask, "How do you know this change is exponential and not linear?" Help guide students towards discovering the common ratio between terms and showing that there is no common difference (MP2). For the 4 functions, students may have to graph them on their calculator to see the appearance of the graph. This lesson is all about discovering why values of b less than 1 lead to exponential decay and those greater than 1 lead to exponential growth. Explain to students that at the conclusion of the lesson they will be able to tell much more quickly which functions represent growth or decay.

#### Resources

*expand content*

#### Investigation

*25 min*

Take a few minutes at the beginning of the investigation to help students set up their Desmos screen in the correct way. (**NOTE: **You can change the values of the slider by double clicking on the min and max value).

The first three questions in this investigation will let students experiment with values of b. The students will be able to see how the graph's appearance changes based on these values. Students should note that the closer the value of b is to zero the less quickly the function decays. Likewise, the closer the value of b is to 1 (if it is greater than 1) the less quickly the function grows. In contrast to this, values slightly less than 1 lead to rapid exponential decay and values much greater than 1 (3, 4, 5, etc.) lead to rapid exponential growth (MP1). Stop students after approximately 10 minutes and have them discuss their observations on these three questions. Students most likely investigated many different values. Because of this, after a student shares their findings look for others who agree and can add to the idea based on their own work. Also, if necessary, find students who disagree with an opinion and can give a counterexample based on their own work (MP3).

The next portion of the lesson will allow students to investigate the values of b through a more iterative process (MP8). By calculating all 18 values in the table, students will begin to notice that if a number less than 1 is raised to a power the value decreases. Also, if a number greater than 1 is raised to a power the value increases. Students will also see this reflected in the graphs of the two functions. Once students have had 5-10 minutes to complete the investigation have several students share out their generalization. Make note of unifying themes on the board (phrases, vocabulary words, etc.) Then, as a class try to craft an all-encompassing generalization.

#### Resources

*expand content*

#### Closure

*5 min*

**Question: Use the Desmos calculator to let the value of b=0. What do you see? Why does the line only extend to the right and not to the left?**

This question can be answered either as a ticket out or as a class discussion. Many students will be able to explain that 0 raised to any power is zero. Students may or may not know about undefined terms. Based on the work students have done with exponents, question them to determine how they could rewrite 0^-1, 0^-2, etc. They can then try to determine why these terms do not exist (MP2).

**NOTE: **One of the best explanations for not being able to divide by 0 is as follows. Multiplication is the inverse of division. That said, if we said that a number divided by 0 was equal to another number we would have a/0=b. This would mean that 0(b)=a. We know that 0 times any number is always 0 so the original statement is not true.

Many students think that 1/0=0 by the same reasoning above that would mean that 0 times 0 is equal to 1 which is certainly not true.

*expand content*

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- LESSON 1: Getting Started: Investigating Exponents
- LESSON 2: Geometric Sequences
- LESSON 3: Geometric Sequences and Exponential Functions
- LESSON 4: Zombies: Exploring Exponential Growth
- LESSON 5: More With Exponential Growth
- LESSON 6: Graphing Exponential Decay Functions
- LESSON 7: Effect of Changing b in f(x) = (b)^x
- LESSON 8: Transforming Exponential Functions
- LESSON 9: Comparing Geometric and Arithmetic Sequences
- LESSON 10: Solving Equations Involving Exponents
- LESSON 11: Comparing Linear and Exponential Functions Day 1
- LESSON 12: Comparing Exponential and Linear Functions Day 2
- LESSON 13: Modeling with Exponential Functions