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

Unit 7: Exponential Functions
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.

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