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Algebra 1 View details

6. Exponential Growth and Decay

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Chapter 6

6.

Exponential Growth and Decay

This lesson explores the mathematical concepts of exponential growth and decay, emphasizing their real-world applications, particularly in finance and biology. For example, understanding exponential growth can help you appreciate how compound interest works in a savings account, allowing your money to grow over time. On the flip side, exponential decay is crucial in calculating the depreciation of assets like cars. These concepts are not just theoretical; they are tools that can help you make informed decisions in various aspects of life, from managing your finances to understanding biological phenomena like bacterial growth.

Lesson Settings & Tools

	12 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Exponential Growth and Decay

Slide of 12

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-14 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.

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

Carbon-14 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-14 in a sample and comparing it to known values of different ages.

Well, carbon-14 has a half-life of 5730 years. That means that after 5730 years there will be half as much carbon-14 in the sample. In light of all this information, is it possible to write a function that can be used in carbon-14 dating?

Explore

Exploring Exponential Functions

Consider the following exponential function. y=2 * b^x How does its graph change when the value of b 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 b>1?
What is the behavior of the function when b<1?

Discussion

Exponential Growth

For an exponential function y=a(b)^x, where a>0 and b>1, the y-values increases as the x-values increases. Therefore, it can be classified as an increasing function. y = ab^x ⇓ Increasing when a>0 and b >1 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 a >0 and b>1. y = a(b)^t In this form, a is the initial amount, the base b is the growth factor, and t usually represents time. Like any other exponential function, a also represents the y-intercept.

Graph of exponential growth functions

As shown, the greater the base b, the faster the exponential function grows. Since the base b is greater than 1, it can be written as the sum of 1 and some positive number r. This constant r can then be interpreted as the rate of growth, in decimal form.

Exponential Growth Equation Parts

For example, r=0.06 means that the quantity increases by 6 % over every unit of time.

Example

Studying Bacterial Growth

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

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 20 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 1 and they duplicate each 20 minutes.

b Write 2 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.

y=a* b^t In the above equation, a represents the initial number of bacteria. Since the experiment starts with 1 bacteria, a equals 1. The bacteria duplicate themselves every 20 minutes. That means the base should be equal to 2. y= 1( 2)^t ⇒ y=2^t Notice that this function does not represent the situation exactly. The reason being that, for example, substituting 1 for t 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 20 minutes, t must be divided by 20. y =2^t ⇒ y=2^(t 20) Therefore, y=2^(t20) correctly models the situation for E. coli. Comparing the number of bacteria on the applet and the y-values found by the equation, it can be checked if the equation is correct or not.

Time Elapsed, t	y=2^(t20)	Number of Bacteria, y
0	2^(020)=1	1
20	2^(2020)=2^1	2
40	2^(4020)=2^2	4
60	2^(6020)=2^3	8
80	2^(8020)=2^4	16

b Since the exponential growth function is a function of time in minutes, 2 hours must expressed in minutes as well. That can be done using a conversion factor.

2h* 60min/1h = 120min The converted time can now be substituted into the equation.

y=2^(t20)

t= 120

y=2^(12020)

Calculate quotient

y=2^6

Calculate power

y=64

It has been found that there will be 64 bacteria after 2 hours. That is amazing, considering Magdalena began with only 1 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 A of an account that earns compound interest, an exponential growth function can be used.

Compound interest formula

In this function, P stands for the principal, or the initial amount of money, r is the interest rate in decimal form, and n is the number of times the interest is compounded per year. For an account with the principal $100 and an annual interest of 18 % compounded twice a year, the balance in the account after t 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 n 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 $1000!

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 3 % compounded monthly.

a Write an equation that models the local bank's compound interest.

b How much money will Kriz have 2 years after opening the account? Round the amount to 2 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 2 for t in the equation obtained in Part A.

Solution

a The local bank offers Kriz an interest rate of 3 % that is compounded monthly. Using the exponential growth function that models compound interest, first write the percentage as a decimal.

3 % ⇒ 0.03 It is given that the interest is compounded monthly. Therefore it will compound 12 times a year. Knowing that Kriz will store all of the $ 1000 in prize money, there is now enough information to write the equation that models this compound interest.

A=P(1+r/n)^(nt)

SubstituteValues

Substitute values

A= 1000(1+0.03/12)^(12t)

▼

Simplify right-hand side

Calculate quotient

A=1000(1+0.0025)^(12t)

Add terms

A=1000(1.0025)^(12t)

Using this equation, Kriz can calculate their earning after t years.

b In order to find how much money Kriz would have in their savings account after 2 years, 2 will be substituted for t into the equation solved for in Part A.

A=1000(1.0025)^(12t)

t= 2

A=1000(1.0025)^(12( 2))

▼

Simplify right-hand side

Multiply

A=1000(1.0025)^(24)

Use a calculator

A=1061.757044 ...

Round to 2 decimal place(s)

A=1061.76

Kriz will have $1061.76 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 0 and less than 1, 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 a>0 and a base b that is between 0 and 1. y = a(b)^t In this form, a is the initial amount, the base b is the decay factor, and t usually represents time. Like any other exponential function, a also represents the y-intercept.

Exponential Decay Applet

As seen, the closer the base gets to 0, the faster the exponential function decays. Since the base b is less than 1, it can be written as 1 minus a positive number r between 0 and 1. This constant r can be interpreted as the rate of decay, in decimal form.

Exponential Decay Equation Parts

A value of 0.12, for instance, would mean that the quantity decreases by 12 % 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 bought his car five years ago at the same dealer for $20 000. Mr. Peterson states that the car depreciates at a rate of 15 % annually.

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 $ 20 000, which is the initial amount. y=a(1-r)^t ⇓ y= 20 000(1-r)^t The car depreciates at a rate of 15 % annually. That value needs to be written as a decimal. 15 % ⇒ 0.15 There is now enough information to write an equation for the value of the old car.

y=20 000(1-r)^t

r= 0.15

y=20 000(1- 0.15)^t

Subtract term

y=20 000(0.85)^t

It has been five years since Diego bought his car. Therefore, to find the current value of the car, 5 should be substituted for t into the function.

y=20 000(0.85)^t

t= 5

y=20 000(0.85)^5

Use a calculator

y=8874.10625

Round to 2 decimal place(s)

y=8874.11

Considering the depreciation rate of 15 %, Mr.Peterson offers $8874.11 for Diego's 5-year-old car. If Diego accepts the offer, he will have a capital of $8874.11 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 y=ab^t can model exponential decay as well as exponential growth. Identify the rate of decay or growth r for the indicated function. Write the corresponding rate in decimal form.

Exponential Growth or Decay Equation

Closure

Modeling Carbon-14 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.

Since the half-life of carbon-14 is 5730 years, an initial amount A of carbon-14 will decay by half that amount in 5730 years. Let y be the final amount of carbon-14 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 b that is between 0 and 1. y=a(b)^t Since the given information is about the half-life of carbon-14, the base is 12, or 0.5. It is also given that the initial amount of carbon is represented by A. y=a(b)^t ⇓ y= A( 0.5)^t The half-life of carbon-14 is 5730 years, which means that after 5730 years, only half of the initial amount will remain. Therefore, t should be divided by 5730 so that for every 5730 years half the previous amount remains. y=A(0.5)^(t 5730) Wow! That is a great amount of time that carbon-14 takes in order to decay. That fact makes for a good reason to use it when determining the age of a fossil.

Exponential Growth and Decay

Exercises

Identify the rate of growth of the following exponential functions. Write your answer as a percentage.

y=5(1+0.25)^t

y=2.3(1.3)^t

y=1.8^t

Let's start by recalling the general form of exponential growth functions.

Exponential Growth |- y=a(1+r)^t

In this case a>0 is the initial amount and r>0 is the rate of growth written in decimal form. y=5(1+ 0.25)^t Our function represents exponential growth with a rate of growth of r= 0.25. To rewrite the rate of growth as a percentage, we move the decimal point 2 places to the right. r=0.25 ⇔ r=25 % Therefore, the rate of growth is 25 %.

Let's begin by rewriting the given function to match the exponential growth formula.

This time our function represents exponential growth with a rate of growth of r= 0.3. To rewrite the rate of growth as a percentage, we move the decimal point 2 places to the right. r=0.3 ⇔ r=30 % Therefore, the rate of growth is 30 %.

Let's begin by rewriting the given function to match the exponential growth formula.

This exponential growth function has a rate of growth of r= 0.8. To rewrite the rate of growth as a percentage, we move the decimal point 2 places to the right. r=0.8 ⇔ r=80 % Therefore, the rate of growth is 80 %.

Write a function that represents the following situation. A population of500 000 increases by8 % each year.

Let's start by recalling the general form of exponential growth functions.

Exponential Growth |- y=a(1+r)^t

In this case a>0 is the initial amount and r>0 is the rate of growth written in decimal form. To write the function we first need to define the variables. Let y be the population, and let t be the number of years after the initial value. In this case the initial value is 500 000. a= 500 000 We are told that the population increases 8 % each year. This is the rate of growth. Let's write it as a decimal! r=8 % ⇔ r= 0.08 We now have enough information to write the requested function. Let's substitute our values into the general formula.

We can represent the given situation with the exponential function y=500 000(1.08)^t.

Identify the rate of decay of the following exponential functions. Write the rate of decay as a percentage.

y=8.5(1-0.15)^t

y=4(0.97)^t

y=0.7^t

Let's start by recalling the general form of exponential decay functions.

Exponential Decay |- y=a(1-r)^t

In this case a>0 is the initial amount and r>0 is the rate of decay written in decimal form. y=8.5(1- 0.15)^t Our function represents exponential decay with a rate of decay of r= 0.15. To rewrite the rate of decay as a percentage, we move the decimal point 2 places to the right. r=0.15 ⇔ r=15 % Therefore, the rate of decay is 15 %.

Let's begin by rewriting the given function to match the exponential decay formula.

This time our function represents exponential decay with a rate of decay of r= 0.03. To rewrite the rate of decay as a percentage, we move the decimal point 2 places to the right. r=0.03 ⇔ r=3 % Therefore, the rate of decay is 3 %.

Let's begin by rewriting the given function to match the exponential decay formula.

This exponential decay function has a rate of decay of r= 0.3. To rewrite the rate of decay as a percentage, we move the decimal point 2 places to the right. r=0.3 ⇔ r=30 % Therefore, the rate of decay is 30 %.

Write a function that represents the following situation. A stock valued at$200decreases in value by5 % each year.

Let's start by recalling the general form of exponential decay functions.

Exponential Decay |- y=a(1-r)^t

In this case a>0 is the initial amount and r>0 is the rate of decay written in decimal form. To write the function, we first need to define the variables. Let y be the stock value, and let t be the number of years after the initial value. In this case the initial value is 200. a= 200 We are told that the stock decreases in value by 5 % each year. This is the rate of decay. Let's write it as a decimal! r=5 % ⇔ r= 0.05 We now have enough information to write the requested function. Let's substitute our values into the general formula.

We can represent the given situation with the function y=200(0.95)^t.

Ali, founder of Daytas sunglasses, plans to pay the salespeople a starting annual salary of $30 000. The salary increases by 5 % each year worked at the company. He wants keep track of their salaries over the years.

Write a function that represents the salary of the employee t years after they started working for the company.

What will be the salary of the employee after 5 years of working? Round the salary to two decimal places.

Let's start by recalling the general form of exponential growth functions.

Exponential Growth |- y=a(1+r)^t

In this case a>0 is the initial amount and r>0 is the rate of growth written in decimal form. To write the function, we first need to define the variables. Let y be the salary, and let t be the number of years that the employee has been working in the company. In this case the initial value is 30 000. a= 30 000 We are told that their salary increases 5 % each year. This is the rate of growth. Let's write it as a decimal! r=5 % ⇔ r= 0.05 We now have enough information to write the requested function.

We can find the employee's salary after 5 years of working in the company by evaluating the function we obtained in Part A at t=5. Let's do it!

The employee's annual salary will be about $38 288.45 after working in the company for 5 years. For helping, Ali has presented this pair of sunglasses. Great work!

The purchase price of this car is $20 000. The value of the car depreciates by 8 % for each year beginning from the date of purchase.

Write a function that represents the price of the car t years after it was bought.

What will be the price of the car after 10 years from the time of purchase? Round the price to two decimal places.

Let's start by recalling the general form of exponential decay functions.

Exponential Decay |- y=a(1-r)^t

In this case, a>0 is the initial amount and r>0 is the rate of decay written in decimal form. To write the function we first need to define the variables. Let y be the price of the car, and let t be the number of years that have passed since its purchase. In this case the initial value is 20 000. a= 20 000 We are told that the car depreciates 8 % each year. This is the rate of decay. Let's write it as a decimal! r=8 % ⇔ r= 0.08 We now have enough information to write the requested function.

We can find the price of the car 10 after its purchase by evaluating the function we obtained in Part A at t=10. Let's do it!

We have found that the price of the car will be about 8687.77 dollars 10 years after its purchase.

Exponential Growth and Decay

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