Exponential functions are functions that have exponents in them. These are typically used in real world problems because in nature things grow at an exponential rate. In other words, it is very rare that something would grow or follow a pattern of 2,4,6,8..., but one could expect it to grow like 2,4,8,16…..

 

The first kind of exponential functions are those that have a variable, or letter, in the exponent. They will look something like the following.

 

f(x)= 3^x

 

f(x) = 3000(5)^2x

 

As one can clearly see, the variable (x) is in the exponent. Look at the following function.

f(x) = 2^x

 

The main thing to look at is the growth factor (2). With the growth factor being a 2, it is a doubling formula. That means that every time that x is increased by 1 the value doubles. For example when x = 1 the function is 2¹ = 2. When x = 2 the result is 2² = 4. The next values are shown in the following chart.

Value of x	Value of 2x
3	8
4	16
5	32
6	64

 

 

From following this pattern, notice that the value doubles each time that x is increased by 1. All exponential functions can be written in the form f(x) = a b^x. The a stands for the initial amount starting with, and the b is called the growth factor. The x would represent the number of cycles that this function will go through.

 

Example

 

If $10 is invested and it doubles every year. How much is the investment worth after 3 years?

 

$10 is considered the initial amount started with, substitute that in for a. The growth factor in this problem would be 2, and the number of cycles is 3. This is shown in the following equation. Let A = the amount the investment is worth.

A = 10(2³)

2³= 8.

A = 10(8)

Multiply.

A = 80

$80

Answer! The investment would be worth $80.

 

To get a formula for finding an amount if something is tripled every time, use 3 as the growth factor.

 

Half-life formulas are formulas used to determine how much of something is left after a certain amount of time. After every cycle the amount is cut in half. For example, if starting out with 1000 the next step would result in 500, then 250, and so on. Use a half- life formula the same way as the doubling or tripling functions previously discussed, only using instead of 2 or 3 for the growth factor.

 

 

 

 

 

Example

 

Suppose there are 200 grams of a substance that had a half-life of 4 months. How much of the substance would be left at the end of 2 years?

Find all the variables. The initial amount is 200. The growth factor is . The number of cycles in this case is 6 since there are 6, 4 month cycles in 2 years.

 

Do the exponents.

 

A = 200 (.015625)

 

Multiply.

 

A = 3.125 grams

 

Answer!

 

Talking about the growth factor in more detail, anytime that the growth factor is larger than 1 then the function is increasing. If the growth factor is less than 1, then the function is decreasing. When trying to calculate an amount based upon a certain percentage increase or decrease. First, look at functions that increase a certain percent.

 

 

To find the growth rate of a function that increases a certain percent, first change the percent that it is increasing to a decimal and add 1 to it.

 

Example

What is the growth factor of a function that increases 3%?

Change 3% to a decimal.

 

.03

 

Add 1 to this.

 

1.03

 

Answer!

 

 

 

 

 

 

Example

 

What is the growth factor of a function that increases 250%?

Change 250% to a decimal.

 

2.50                                                                                                                                   

 

Add 1 to this.

 

3.50                                                                                                                                   

 

Answer!

 

If the function is decreasing a certain percent, first change the percent to a decimal and subtract this amount from 1. This is called the decay factor instead of the growth factor.

 

 

 

 

 

Example

 

What is the decay factor of a function that is decreasing by 2%?

Change 2% to a decimal.

.02

 

Subtract this amount from 1.

.98

 

Answer!

 

 

 

 

 

 

 

Example

 

What is the decay factor of a function that is decreased by 60%?

Change 60% to a decimal.

.60

 

Subtract this amount from 1.

.40

 

Answer!

 

 

 

 

 

 

 

Example

 

If there are 250 grams of a substance that increases 25% every year for 3 years, how much will there be?

 

Find all the variables. The initial amount is 250. The growth factor is 1.25. And the number of cycles is 3.

 

A = 250 (1.25)³

 

Do the exponent.

 

A = 250 (1.953125)

 

Multiply.

A = 488.28 grams

 

Answer!

 

 

 

 

 

 

Example

 

If you have $500 and it decreases 3% at the end of every year for 5 years, how much money will you have?

 

Find the variables. The initial amount is $500. The decay factor is .97. The number of cycles is 5.

A = 500 (.97)⁵

 

Do the exponent.

A = 500 (.8587)

 

Multiply.

A= $429.37

 

Answer!

 

Note- In the previous problem, it was stated that at the end of every year the amount would decrease. This is very important because on dealing with money in particular many times interest rates are not figured at the end of every year but rather are calculated semi-annually, which is twice a year, or quarterly, which is four times a year. There are many other different periods in which to compound interest and they will be discussed next.

 

 

Compound interest is the type of interest that a bank or credit card company may use to calculate interest on accounts and determine the amount in an account. Here is the formula used.

 

 

Where A = amount in the account after the interest

P = principal or amount to start with

t = number of years

r = interest rate (as a decimal)

n = number of times a year that interest is compounded.

 

Here is a chart that tells how many times a year interest is compounded.

Annually	1 time a year
Semi-annual	2 times a year
Quarterly	4 times a year
Monthly	12 times a year
Weekly	52 times a year

 

 

Example

 

Invest $5000 for 3 years at 6.5%. How much money would you then have? Calculate if compounded quarterly.

 

First, find out what all the variables are.

P = 5000

t = 3 years

r = .065

n = compounded 4 times a year.

 

Note- Using this formula, do not find the growth rate by adding 1 to the interest rate. Just use the actual decimal value for the interest rate in decimal form.

 

 

 

 

Simplify the exponent.

 

 

Simplify the parentheses.

 

 

 

Do the exponent.

 

 

 

Multiply.

 

 

 

Answer! Rounded off to the nearest cent.

 

 

 

 

 

 

 

Example

 

If you invest $25,000 at 4% compounded annually. How much money will you have at the end of 7 years?

 

Find all the variables.

P = 25,000

t = 7

r = .04

n = 1

 

 

 

Simplify the exponent and the parentheses.

 

 

 

 

Do the exponent.

 

 

 

 

Multiply.

 

 

 

 

Answer!

 

 

 

 

 

 

 

Example

 

Use the previous amounts only compound the interest quarterly.

 

Find all the variables.

P = 25,000

t = 7

r = .04

n = 4

 

 

 

 

Simplify the exponent and the parentheses.

 

 

 

 

Do the exponent.

 

 

 

Multiply.

 

 

 

 

Answer!

 

 

More money was made from the interest when it was compounded quarterly as opposed to being compounded annually. Now use the same example and compound the interest daily.

 

 

 

 

Example

 

Use the previous amounts and compound the interest daily.

 

Find all the variables.

 

P = 25,000

t = 7

r = .04

n = 365

 

 

 

Simplify the exponent and the parentheses.

 

 

 

Do the exponent.

 

 

 

Multiply.

 

 

 

Answer!

 

When continuing to compound the interest over a smaller and smaller amount of time the amount will get larger and larger. This is shown by the three previous examples. Compounding the interest continually, meaning the smallest amount of time, will produce the greatest amount of return and is given by the following formula.

A = P e^rt

 

Where P is the principal, r is the interest rate, and t is the number of years. The other variable, e, is actually not a variable but a number, much like p. Look on a calculator there will be an e button on there somewhere. It may look like e^x. The number e is approximately 2.718281828. Use this number or the button on a calculator to compound something continuously.

 

 

 

 

 

 

Example

 

Invest $25,000 at 4% for 6 years, compounded continuously.

 

Find the variables.

P = 25,000

r = .04

t= 7

 

 

 

Simplify the exponent.

 

 

 

Use the e button to calculate e^.28 as accurately as possible.

 

 

 

Multiply.

 

 

 

 

Answer!

 

The previous 4 examples were all done with the same variables except the amount of times that it was compounded. Notice the highest possible return occurs when one compounds continuously.