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Exponential functions are useful for modeling a wide variety of real-life scenarios. This lesson will introduce the concepts of exponential growth and exponential decay. How they are used to model real-life situations will also be understood.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Carbon Decay and Archaeology

While watching a documentary about ancient civilizations, Ali wondered how scientists could determine the age of an object belonging to such ancient civilizations.

Scientist Explaining How Old a Civilization Is

Ali recalled that his uncle, Mr. Jones, is an archaeologist! Ali called and asked him how are the ages of ancient objects determined.

Mr Jones Explaining Carbon Dating

Carbon is a substance present in organisms that, once they expire, begin to be released from an object's body at a slow rate. To date an object this way consists of measuring the amount of carbon in a sample and comparing it to known values of different ages.

Ali Asking Why Carbon Dating Works
Well, carbon has a half-life of years. That means that after years there will be half as much carbon in the sample. In light of all this information, is it possible to write a function that can be used in carbon dating?
Explore

Exploring Exponential Functions

Consider the following exponential function.
How does its graph change when the value of changes? To visualize its effect, move the slider in the following applet.
Exponential Function Change Base Applet
Think about these two questions.
  • What is the behavior of the function when ?
  • What is the behavior of the function when ?
Discussion

Exponential Growth

For an exponential function where and the values increases as the values increases. Therefore, it can be classified as an increasing function.
Such exponential functions are called exponential growth functions.
When a quantity increases by the same factor over equal intervals of time, it is said that such a quantity is in exponential growth. Exponential growth is modeled using exponential functions where and
In this form, is the initial amount, the base is the growth factor, and usually represents time. Like any other exponential function, also represents the intercept.
Graph of exponential growth functions
As shown, the greater the base the faster the exponential function grows. Since the base is greater than it can be written as the sum of and some positive number This constant can then be interpreted as the rate of growth, in decimal form.
Exponential Growth Equation Parts
For example, means that the quantity increases by over every unit of time.
Example

Studying Bacterial Growth

Magdalena, excited for biology lab, is exploring about bacterial growth.

Magdalena With a Microscope
Bacteria are known to duplicate themselves within a certain amount of time. This means that after some time, there will be twice the amount of bacteria as before. Magdalena and her partners are studying E. coli, a bacteria responsible for many stomach related diseases. E. coli is known to duplicate about every minutes.
Bacteria Replication Animation
Magdalena begins her experiment with only 1 bacteria of E. coli in her petri dish. She watches how it duplicates as time goes by.
a Write an equation that models this exponential growth in terms of the time elapsed since the start of the experiment, in minutes.
b How much bacteria will there be after two hours?

Hint

a The initial amount of bacteria is and they duplicate each minutes.
b Write hours as minutes. Then substitute that into the equation obtained in Part A.

Solution

a Exponential growths are modeled using exponential functions of the following form.
In the above equation, represents the of bacteria. Since the experiment starts with bacteria, equals The bacteria themselves every minutes. That means the should be equal to
Notice that this function does not represent the situation exactly. The reason being that, for example, substituting for doubles the number of the bacteria. That is, the bacteria doubles itself every minute. For the equation to give the number of bacteria duplicating every minutes, must be divided by
Therefore, correctly models the situation for E. coli. Comparing the number of bacteria on the applet and the values found by the equation, it can be checked if the equation is correct or not.
Time Elapsed, Number of Bacteria,
b Since the exponential growth function is a function of time in minutes, hours must expressed in minutes as well. That can be done using a conversion factor.
The converted time can now be substituted into the equation.
It has been found that there will be bacteria after hours. That is amazing, considering Magdalena began with only bacteria of E. coli.
Discussion

Compound Interest

Applications of exponential growth can also be encountered in the world of finance. Some people use the power of compound interest to grow their wealth exponentially.

Compound interest is the interest earned depending on both the initial investment and previously earned interest. To find the balance of an account that earns compound interest, an exponential growth function can be used.
Compound interest formula
In this function, stands for the principal, or the initial amount of money, is the interest rate in decimal form, and is the number of times the interest is compounded per year. For an account with the principal and an annual interest of compounded twice a year, the balance in the account after years is shown in the graph.
Compound interest graph
Notice that the function grows continuously, whereas, in reality, the account balance only increases at the times of compound. When calculating compound interest, the number of compounding periods creates a difference. That is, the higher the number of compounding periods, the greater the amount of compound interest.
Compound interest graph for different n
Example

Investing Into a Savings Account

Kriz, determined and focused, won an online video game competition. The first place prize was

Kriz First Price in Video Games

Kriz decides to not spend the prize money. Instead, their parent suggests placing all of it into a Certificate of Deposit. This is a type of savings account with compound interest. The catch is that the money cannot be taken out for a certain period of time. Ngân Hàng, a local bank, offers a Certificate of Deposit with the interest rate at compounded monthly.

a Write an equation that models the local bank's compound interest.
b How much money will Kriz have years after opening the account? Round the amount to decimal places.

Hint

a The meaning of compounded monthly is that the interest is compounded each month of the year — meaning twelve times per year.
b Substitute for in the equation obtained in Part A.

Solution

a The local bank offers Kriz an interest rate of that is compounded Using the exponential growth function that models compound interest, first write the percentage as a decimal.
It is given that the interest is compounded monthly. Therefore it will compound times a year. Knowing that Kriz will store all of the in prize money, there is now enough information to write the equation that models this compound interest.
Simplify right-hand side
Using this equation, Kriz can calculate their earning after years.
b In order to find how much money Kriz would have in their savings account after years, will be substituted for into the equation solved for in Part A.
Simplify right-hand side
Kriz will have in their savings account after two years. What a good start to their gaming career!
Discussion

Exponential Decay

When the base of an exponential function is a number greater than and less than the function is said to be decreasing. In such cases, the function represents what is known as exponential decay.

When a quantity decreases by the same factor over equal intervals of time it is said that such quantity is in exponential decay. Exponential decay is modeled using exponential functions with and a base that is between and
In this form, is the initial amount, the base is the decay factor, and usually represents time. Like any other exponential function, also represents the -intercept.
Exponential Decay Applet
As seen, the closer the base gets to the faster the exponential function decays. Since the base is less than it can be written as minus a positive number between and This constant can be interpreted as the rate of decay, in decimal form.
Exponential Decay Equation Parts
A value of for instance, would mean that the quantity decreases by over every unit of time.
Example

Calculating the Devaluation of a Car

Diego has saved for the past few years dreaming of buying a car with a drop top so he can cruise the streets looking fly. Diego runs to the nearest car dealer and is met by Mr. Peterson, a car salesmen. They come to an agreement where Diego trades in his old car to help pay for the new car.

Diego and Mr Peterson Conversation

Diego bought his car five years ago at the same dealer for Mr. Peterson states that the car depreciates at a rate of annually.

Sad Diego Listening to Mr Peterson's Offer
How much is Mr. Peterson going to give Diego for his old car? Round Mr. Peterson's offer to two decimal places.

Hint

Since the value of the car depreciates, the situation can be modeled using an exponential decay function.

Solution

Previously, Diego bought a car with a value that has depreciated since the time of purchase. Therefore, this scenario can be modeled using an exponential decay function. The car was bought for which is the initial amount.
The car depreciates at a rate of annually. That value needs to be written as a decimal.
There is now enough information to write an equation for the value of the old car.
It has been years since Diego bought his car. Therefore, to find the current value of the car, should be substituted for into the function.
Considering the depreciation rate of Mr.Peterson offers for Diego's year-old car. If Diego accepts the offer, he will have a capital of to put toward the purchase of a new car from the dealership.
Pop Quiz

Identifying Exponential Growth or Decay From a Table

Select the option that best describes the table of values given below.

Table of Values representing exponential growth, decay, or linear expression
Pop Quiz

Identifying the Rate of Growth or Decay

Exponential functions can model exponential decay as well as exponential growth. Identify the rate of decay or growth for the indicated function. Write the corresponding rate in decimal form.

Exponential Growth or Decay Equation
Closure

Modeling Carbon Decay

This lesson introduced the interesting concepts of compound interest, exponential growth, and exponential decay. Using the knowledge gained from this lesson, the introductory challenge can be modeled using an exponential decay function. Recall what the archaeologist had to say.

Mr Jones Stating the Half-Life of Carbon-14
Since the half-life of carbon is years, an initial amount of carbon will decay by half that amount in years. Let be the final amount of carbon and write an equation that models this exponential decay.

Hint

Find the decay factor of this situation.

Solution

Exponential decay is modeled using exponential functions with a base that is between and
Since the given information is about the life of carbon the base is or It is also given that the of carbon is represented by
The half-life of carbon is years, which means that after years, only half of the initial amount will remain. Therefore, should be divided by so that for every years half the previous amount remains.
Wow! That is a great amount of time that carbon takes in order to decay. That fact makes for a good reason to use it when determining the age of a fossil.