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Exponential Growth

Site: Gladwin
Course: Michigan Algebra I Sept. 2012
Book: Exponential Growth
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Date: Thursday, November 21, 2024, 9:57 AM

Description

Exponential Growth

Table of contents

Growth & Decay

Exponential functions are functions where the base is a constant number and the exponent is a variable. The general exponential function takes the form: y = a· bxwhere ais the initial amount andb is the factor that the amount gets multiplied by each time x is increased by one.

There are two types of exponential functions, exponential growth and exponential decay. Exponential growth functions start out growing slowly and then grow faster and faster. There will be a consistent fixed period during which the function will increase by a fixed proportion. Exponential decay functions start out decreasing quickly and then decrease slower and slower. There is a consistent fixed period during which the function will decrease by a fixed proportion.

In this book, exponential growth functions will be discussed.


Growth Example 1

A colony of bacteria has a population of three thousand at noon on Sunday. During the next week, the colony's population doubles every day. What is the population of the bacteria colony at noon on Saturday?

Step 1. Determine the initial value.

Since the first day recorded is Sunday, the initial population is 3,000.

Step 2. Determine the growth factor.

Since the population is doubling, it is being multiplied by 2 each day. Therefore, the growth factor is 2.

Step 3. Write an exponential function.

The general form of an exponential function is ExpGrowthEx1-1.

This function is ExpGrowthEx1-2.

Step 4. Use the equation to solve the problem.

Since Saturday is 6 days after the initial day, Sunday, we will use x = 6.

ExpGrowthEx1-3

y = 192,000 bacteria

Growth Example 2

The population of a town is estimated to increase by 15% per year. The population today is 20,000. Make a graph of the population function and find out what the population will be ten years from now.

Step 1. Determine the initial value.

The initial population is 20,000.

Step 2. Determine the growth factor.

Since the population is increasing by 15% per year, it is being multiplied by 115% each day. Therefore, the growth factor is 1.15.

Step 3. Write an exponential function.

The general form of an exponential function is ExpGrowthEx2-1.

This function is ExpGrowthEx2-2.

Step 4. Use the equation to solve the problem.

Continued

Step 5. Graph the function.

ExpGrowthEx2-4

*Note: Negative values of x are graphed on this graph. In this case, x = ? 5 represents what the population was five years ago, so it can be useful information. Also, the scale of the y-axis is in thousands.

Interactive Activity

For a visual understanding of exponential growth, select the following link:

Exponential Growth Interactive

Video Lessons

To learn how to evaluate and solving exponential growth, select the following link:

Solving Exponential Growth

Evaluating Exponential Growth

Exponential Growth

Guided Practice

To solidify your understanding of exponential growth, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Practice

Exponential Growth Worksheet

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Answer Key

Exponential Growth Answer Key

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Sources

Embracing Mathematics, Assessment & Technology in High Schools; A Michigan Mathematics & Science Partnership Grant Project

Gloag, Anne & Andrew. "Exponential Functions." February 24,

2010.http://www.ck12.org/flexr/chapter/4478

Holt, Rinehart, & Winston. "Exponential and Logarithmic Functions."

http://my.hrw.com/math06_07/nsmedia/homework_help/alg2/alg2_ch07_01_homeworkhelp.html (accessed September 9, 2010).

Mathwarehouse.com, "Exponential Growth in Real World ." http://www.mathwarehouse.com/exponential-growth/exponential-models-in-real-world.php (accessed 9/15/2010).