Glossary for Lesson 29: Exponential Functions

Exponential Function

An exponential function is a function that can be written in the form f(x)=a^x .  Here, a is called the base of the exponential function, and a must be greater than 0 and can’t equal 1.  An exponential function can also be translated vertically or horizontally, making it in the form f(x) = k + a^{(x-h)}

Exponential Growth

If an exponential function f(x) = a^x has a base a>1, then it shows behavior that is known as exponential growth.  An exponential growth function will be increasing everywhere in its domain, and it will have a graph resembling the one below. 

Exponential Decay

If an exponential function f(x) = a^x has a base a<1, then it shows behavior that is known as *exponential decay*.  Its equation can also be written in the form f(x)=a^{-x} , as long as a>1.  An exponential decay function will be decreasing everywhere in its domain, and it will have a graph resembling the one below. 

Growth Rate

For an exponential function f(x) = a^x, where a>1, the growth rate is the number that the value of the function multiplies by if we increase x by 1. This number is a, 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 f(x)=3^x, the growth rate will be 3, since each time x goes up by 1, f(x) multiplies by 3

Decay Rate

For an exponential function f(x) = a^{-x} , where a>1 , the decay rate is the number that the value of the function is divided by if we increase x by 1. This number is a, 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 f(x)=3^{-x}, the decay rate will be 3, since each time x goes up by 1, f(x) is divided by 3

Interest, Principal, and Balance

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

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 n times per year, then we can expression the account balance A after a certain number of years t as
A=P(1+\frac{r}{n})^{nt}
Here, P is the principal, and r is the yearly interest rate. 

e

The number e is an irrational number that has special significance in calculus, probability, finance, physics, and other areas.  It is approximately equal to 2.71828182845904523536028747135266249775724709369995...

Continuously Compounded Interest

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 A after some number of years t :
A=Pe^{rt}
Here, P is the principal, and r is the yearly interest rate.