Exponential Functions

Applications of Exponential Functions

Exponential Growth and Decay

Exponential growth and decay can be modeled using exponential functions. Exponential growth has an increasing exponential function, while exponential decay has a decreasing exponential function.

Many real-world scenarios can be modeled by exponential functions. Exponential growth is an increase in a quantity by a constant percent rate per unit of time. Scenarios that exhibit exponential growth, such as bacteria growth, interest accrual, and population growth, are modeled by increasing exponential functions.

An exponential growth function, where a>0a>0, has the form:
f(t)=a(1+r)tf(t)=a(1+r)^t
The future amount or quantity is represented by f(t)f(t). The initial amount is represented by aa. The rate of growth as a decimal is represented by rr. Time is represented by tt.
Step-By-Step Example
Using Exponential Growth Functions
The number of members in a subscription shipment club is growing by 6% per year. The number of members in 2018 was 350. Write an exponential growth function to model the situation. Then estimate the number of members in the club in 2023.
Step 1
Substitute known values in the general form of the function:
f(t)=a(1+r)tf(t)=a(1+r)^t
The initial number of members was 350, so a=350a=350. The annual rate of growth is 6%, so r=0.06r=0.06.
f(t)=350(1+0.06)tf(t)=350(1.06)t\begin{aligned}f(t)&=350(1+0.06)^t\\f(t)&=350(1.06)^t\end{aligned}
Step 2

Determine the number of members expected in 2023.

The number of years between 2018 and 2023 is 5, so t=5t=5.
f(5)=350(1.06)5f(5)468.38\begin{aligned}f(5)&=350(1.06)^5\\f(5)&\approx 468.38\end{aligned}
Solution
The club will have about 468 members in 2023 based on the function:
f(t)=350(1.06)tf(t)=350(1.06)^t

Compound interest is a common application of exponential growth. When an account accrues compound interest, the interest is calculated on both the principal (the initial amount of money invested or lent) and on any previously earned interest.

The compound interest function has the form:
A=P(1+rn)ntA=P\left(1+\frac{r}{n}\right)^{nt}
The future value of the account is represented by AA. The principal is represented by PP. The annual interest rate as a decimal is represented by rr. The number of times that interest is compounded each year is represented by nn. The time, in years, is represented by tt. Interest can be compounded at different time intervals, such as annually (one time per year), quarterly (four times per year), or monthly (12 times per year). The formula for compounded interest is:
A=PertA=Pe^{rt}
Step-By-Step Example
Calculate Compound Interest
D'Asia deposited $1,500 in a savings account. The account earns an annual interest rate of 3.5%, compounded monthly. Write a compound interest function to model the situation. Then find the amount in the account after eight years.
Step 1
Substitute known values in the general form of the function:
A=P(1+rn)ntA=P\left(1+\frac{r}{n}\right)^{nt}
The principal is the amount D'Asia deposits, so P=1,500P=1\rm{,}500.

The annual rate of growth is 3.5%, so r=0.035r=0.035.

Compounded monthly means 12 times per year, so n=12n=12.
A=1,500(1+0.03512)12tA=1\rm{,}500\left(1+\frac{0.035}{12}\right)^{12t}
Step 2

Determine the value of the account after eight years.

The time in years is eight, so t=8t=8.
A=1,500(1+0.03512)12(8)A1,983.89\begin{aligned}A&=1\rm{,}500\left(1+\frac{0.035}{12}\right)^{12(8)}\\A&\approx 1\rm{,}983.89\end{aligned}
Solution
The value of the account will be $1,983.89 in eight years based on the function:
A=1,500(1+0.03512)12tA=1\rm{,}500\left(1+\frac{0.035}{12}\right)^{12t}

Exponential decay is a decrease in a quantity by a constant percent rate per unit of time. Scenarios that exhibit exponential decay, such as radioactive decay and depreciation, are modeled by decreasing exponential functions.

An exponential decay function, where a>0a>0, has the form:
f(t)=a(1r)tf(t)=a(1-r)^t
The future amount or quantity is represented by f(t)f(t). The initial amount is represented by aa. The rate of decay as a decimal is represented by rr. Time, in years, is represented by tt.
Step-By-Step Example
Using Exponential Decay Functions
Henry buys a new car for $27,000. The value of the car decreases by 15% each year. Write an exponential decay function to model the situation. Then find the value of the car after four years.
Step 1
Substitute known values in the general form of the function:
f(t)=a(1r)tf(t)=a(1-r)^t
The initial value of the car is $27,000, so a=27,000a=27\rm{,}000. The annual rate of decay is 15%, so r=0.15r=0.15.
f(t)=27,000(10.15)tf(t)=27,000(0.85)t\begin{aligned}f(t)&=27\rm{,}000(1-0.15)^t\\f(t)&=27\rm{,}000(0.85)^t\end{aligned}
Step 2
Determine the value of the car after four years.
f(4)=27,000(0.85)4f(4)14,094.17\begin{aligned}f(4)&=27\rm{,}000(0.85)^4\\f(4)&\approx 14\rm{,}094.17\end{aligned}
Solution
The value of the car will be about $14,094.17 in four years based on the function:
f(t)=27,000(0.85)tf(t)=27\rm{,}000(0.85)^t

Exponential Modeling

Technology can be used to generate an exponential function that best fits the exponential relationship between two variables.

A trend observed in a set of real-world data may not be linear. An exponential relationship is a nonlinear relationship for which an exponential function is an appropriate model. If the pattern formed by a set of data points approximates an exponential growth curve or an exponential decay curve, an exponential model is appropriate.

Exponential regression is the mathematical process for finding the equation of an exponential function that best models a data set. Exponential regression is often performed using technology. A model generated by exponential regression can be used to make predictions.

Step-By-Step Example
Fitting an Exponential Model to Data

The data set in the table represents the number of bacteria in a scientist's sample over time. Use technology to determine an exponential model for the data set. Then use the model to identify the expected number of bacteria after eight hours.

Time in hours (xx) 1 2 3 4 6
Number of Bacteria (f(x)f(x)) 346 1,218 2,254 4,179 14,850
Step 1

Enter the data into a graphing calculator, spreadsheet, or computer program, and identify the equation of a function to model the data by using exponential regression.

The utility may give values for aa and bb, as well as values for r2r^2 and rr. Use the values of aa and bb to write the function that models the data. The variable rr represents the correlation coefficient, which indicates the strength of the correlation. The value of rr is not used in the exponential function.
f(x)=226.741776(2.051403774)xf(x)=226.741776\cdot(2.051403774)^x
Step 2
Use the function to predict the number of bacteria after eight hours.
f(8)=226.741776(2.051403774)8f(8)71,112\begin{aligned}f(8)&=226.741776\cdot(2.051403774)^8\\f(8)&\approx71\rm{,}112\end{aligned}
Solution
There will be about 71,112 bacteria after eight hours based on the function:
f(x)=226.741776(2.051403774)xf(x)=226.741776\cdot(2.051403774)^x