11.2 Exponential Functions

General Form

Let a be a constant and let x be a variable. Then the general form of an exponential function is:

€

f x( ) = ax

What does it mean?

We talk about exponentials in 2 ways:

Exponential Growth

Exponential Decay

Growth

In the equation below, when a > 1, we have exponential growth.

Exponential growth is when something is growing without bound.

€

f x( ) = ax

Decay

In the equation below, when 0<a<1, we have exponential decay.

This happens when something is decaying toward extinction.

€

f x( ) = ax

Compounding InterestCompounding interest occurs when you

earn interest on top of interest.An example would be a savings account.The following formula assumes that you are

putting a lump sum into an account and not touching it for a number of years.

Compounding InterestLet A represent the amount of money in an account.Let P represent the principal investment.Let r represent the interest rate (APR) as a decimal.Let n represent the number of compounds per year.Let t represent the time in year.Then the formula for compounding interest is:

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A = P 1 +r

n ⎛ ⎝

⎞ ⎠

n⋅t

Compounding Interest

Determine the future value of an account that has an initial investment of $3000 and is compounded quarterly at a rate of 6.5% for 20 years.

€

A = P 1 +r

n ⎛ ⎝

⎞ ⎠

n⋅t

Compounding Interest

€

P = 3000

n = 4

r = 0.065

t = 20

€

A = P 1+r

n ⎛ ⎝

⎞ ⎠

n⋅t

€

A = 3000 1 +0.065

4 ⎛ ⎝

⎞ ⎠

4⋅20

This is a number... use the calculator

1 2 4 4 4 3 4 4 4 = $10,893.50

AnnuityA series of payments made at equal

intervals is known as an annuity.An example would be a retirement fund

where a certain dollar amount is taken from each paycheck and put into an account.

Notice this differs from the previous situation in that there are scheduled payments/contributions.

Present ValueThe present value of an annuity is the amount

of money you currently have in the account.The following formula could be used to

determine what the payment would be, for example, on a loan.

Formula:

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APresent Value

{ = PNumber of dollars paid

per payment period

{ ⋅1− 1 + r n( )

−n⋅t

rn( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

r is again the APR

n is the number of payment intervals per year

ExampleYou take out a loan for a new car to the tune

of $18,000. You land a cool 6.8% for 48 months. What will your monthly payment be?

€

$18,000 = P ⋅1− 1+ 0.068

12( )−48

0.06812( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

P = $429.36 per month

Future ValueThe future value of an annuity is how much you

can expect to have in the future assuming you make the scheduled contributions of a predetermined amount.

You would use this formula to calculate, for example, the future value of a retirement fund.

Formula:

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F = P ⋅1+ r n( )

n⋅t−1

rn( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

ExampleYou pledge to invest $180 per month into your

child’s college savings plan. If the plan pays 4.6% APR, how much will it be worth in 18 years when they start college?

€

F = 180 ⋅1+ 0.046

12( )12⋅18

−1

0.04612( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

= $60,344.60

Pg. 612 # 17-25 [3], 41, 42

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