An *exponential function* is a function that can be written in the form . Here, is called the *base* of the exponential function, and must be greater than and can’t equal . An exponential function can also be translated vertically or horizontally, making it in the form .

If an exponential function *exponential growth*. An exponential growth function will be increasing everywhere in its domain, and it will have a graph resembling the one below.

If an exponential function

has a base , then it shows behavior that is known as *exponential decay*. Its equation can also be written in the form , as long as . An exponential decay function will be decreasing everywhere in its domain, and it will have a graph resembling the one below.For an exponential function

, where , the growth rate is the number that the value of the function multiplies by if we increase by . This number is , the base of the exponential function. We can also think of the growth rate as the rate at which the slope of the graph increases. For example, if we consider the function , the growth rate will be , since each time goes up by , multiplies by .For an exponential function

, where , the decay rate is the number that the value of the function is divided by if we increase by . This number is , the base of the exponential function. We can also think of the decay rate as the rate at which the slope of the graph decreases. For example, if we consider the function , the decay rate will be , since each time goes up by , is divided by .When a person decides to invest money in a bank account, the amount of money he or she initially invests is called the *principal*. Once this money is invested, a certain percent of that money will be added into the account at designated times. The money that is added is known as *interest*, and the total amount of money in the account at any given time is called the *balance*. The percent of the account balance that is earned in interest is called the *interest rate*.

*Compound interest* is interest that is added to an account a certain number of times each year, and each time it is added, the amount of money that earns interest is recalculated. Each time interest is added and the balance is recalculated is called a *compounding*. If interest is compounded times per year, then we can expression the account balance after a certain number of years as

Here, is the principal, and is the yearly interest rate.

The number

is an irrational number that has special significance in calculus, probability, finance, physics, and other areas. It is approximately equal to .*Continuously compounded interest* is interest that is compounded at every instant, making the compounding periods infinitesimally short. Through calculus, this yields a formula for the balance after some number of years :

Here, is the principal, and is the yearly interest rate.