Read: Define and Evaluate Exponential Functions
Learning Objectives
- Define an exponential function and it's domain and range
- Evaluate an exponential function
- Define and evaluate a compound interest formula
Year | Stores, Company A | Description of Growth | Stores, Company B |
Starting with each | |||
They both grow by stores in the first year. | + of | ||
Store A grows by , Store B grows by | of | ||
Store A grows by , Store B grows by | of |

Notice that the domain for both functions is , and the range for both functions is . After year , Company B always has more stores than Company A.
Consider the function representing the number of stores for Company B
In this exponential function, represents the initial number of stores, represents the growth rate, and represents the growth factor. Generalizing further, we can write this function as , where is the initial value, is called the base, and x is called the exponent. This is an exponential function.
Exponential Growth
A function that models exponential growth grows by a rate proportional to the current amount. For any real number x and any positive real numbers a and b such that , an exponential growth function has the form
where
- a is the initial or starting value of the function.
- b is the growth factor or growth multiplier per unit x.
To evaluate an exponential function with the form , we simply substitute x with the given value, and calculate the resulting power. For example:
Let . What is ?
To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:
Let . What is ?
Note that if the order of operations were not followed, the result would be incorrect:
Example
Let . Evaluate without using a calculator.Answer:
Follow the order of operations. Be sure to pay attention to the parentheses.
Example
At the beginning of this section, we learned that the population of India was about billion in the year , with an annual growth rate of about . This situation is represented by the growth function , where t is the number of years since . To the nearest thousandth, what will the population of India be in ?Answer:
To estimate the population in , we evaluate the models for , because is years after . Rounding to the nearest thousandth,
There will be about billion people in India in the year .
The annual percentage rate (APR) of an account, also called the nominal rate, is the yearly interest rate earned by an investment account. The term nominal is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.
We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time t, principal P, APR r, and number of compounding periods in a year n:
The Compound Interest Formula
Compound interest can be calculated using the formula
where
- A(t) is the account value,
- t is measured in years,
- P is the starting amount of the account, often called the principal, or more generally present value,
- r is the annual percentage rate (APR) expressed as a decimal, and
- n is the number of compounding periods in one year.
Example
If we invest $3,000 in an investment account paying interest compounded quarterly, how much will the account be worth in years?Answer:
Because we are starting with $3,000, P = . Our interest rate is , so r = . Because we are compounding quarterly, we are compounding times per year, so n = . We want to know the value of the account in years, so we are looking for A , the value when t = .
The account will be worth about $4,045.05 in years.
Example
A Plan is a college-savings plan that allows relatives to invest money to pay for a child’s future college tuition; the account grows tax-free. Lily wants to set up a account for her new granddaughter and wants the account to grow to $40,000 over years. She believes the account will earn compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now?Answer:
The nominal interest rate is , so r = . Interest is compounded twice a year, so .
We want to find the initial investment, P, needed so that the value of the account will be worth $40,000 in years. Substitute the given values into the compound interest formula, and solve for P.
Lily will need to invest $13,801 to have $40,000 in years.