# Modeling with Exponential Functions

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## Objective

SWBAT build an exponential function that can model population growth.

#### Big Idea

This lesson lets students use real world data to build an exponential growth model.

## Launch

15 minutes

Students should read the first slide of modeling_exponential_funtions_launch with a partner and make sense of what the table is displaying.  After about 2-3 minutes, I will put up slide #2 and have students begin to think about how they will make a model that will represent this data.

Students should be thinking about whether the data is linear or exponential.  They can then try to determine the parameters. I do not plan to give students any hints as to which type of function would be used to model the data.  As students are working, I will circulate around the room to make note of which students are looking for a common difference or a common ratio.

In this task, the data is exponential so the students should be trying to find the common ratio and starting "a" value.  In the 10-15 minutes of this segment of the lesson, I let students take the problem as far as they can.  If they are able to find the function, ask them to plot the points and graph the function using a calculator or computer graphing tool (MP5).

## Investigation

20 minutes

This segment of the lesson will allow students to investigate the problem in greater depth.  The questions on this investigation formalize the thought process that we want all students to go through when making a determination about linear versus exponential models.  Questions 1-5 ask questions to make determinations and justifications about the type of function chosen.  Questions #6 and #7 look at the domain and range of the function in the context of the given question (MP4).

Question #8 requires students to think about how to solve an exponential equation.  Recall from a previous lesson that this can be done by making the exponential equation into two functions. Students can then graph both functions and determine the point of intersection.

Question #9 asks students to compare two exponential models with the same growth rate.  This will revisit the concept of transformations of functions.  Students will see that since both functions have the same growth rate, the function with the larger y-intercept will always have a greater value for a given x-value.

## Closure

5 minutes

This Closing Activuty asks students to think about negative input values for this function.  Since the common ratio between consecutive y-values is approximately 1.2 students can divide by 1.2 to determine the population in 1999.  This quantitative reasoning is imperative to understanding the concept behind exponential models (MP2).  Often students realize that they are always multiplying by the same value to get consecutive y-values.  Students sometimes forget that they can also do the inverse operation (divide) to work backwards in this model.

I plan to have students work individually on this problem and collect the results.  I want to assess the level of conceptual understanding that students have obtained on the content covered in this unit.

## Resources

Lesson is adapted from Houghton Mifflin Harcourt Publishing Company, Oncore Mathematics Algebra 1, copyright 2012.