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9.2 – Exponential Functions
Key Terms
 Base – A number in exponential form. It appears beneath the exponent.
 e – The irrational number e 2.7182818284590…
 You can approximate e by substituting large values of n into the expression
 Compound Interest – Interest applied to both the principal and any previously earned interest.
 Exponent – A small raised number that tells you how many times to multiply the base number by itself.
 Exponential Decay – A condition in which a quantity decreases exponentially over time.
 The quantity decreases at a rate that is proportional to the current value of the quantity.
 Exponential decay can be modeled by the equation , where = the amount at time t, = the initial amount, t = time, and k = a constant called the decay constant.
 If a quantity decays exponentially, its halflife is a constant.
 A situation in which a quantity decreases by a common ratio at regular intervals.
 Exponential Function – A function that has the form , where the coefficient a is a constant, the base b is positive but not equal to 1, and the exponent x is any number.
 Exponential Growth – A condition in which a quantity increases exponentially over time.
 The quantity increases at a rate that is proportional to the current value of the quantity.
 Exponential growth can be modeled by the equation , where = the amount at time t, = the initial amount, t = time, and k = a constant called the growth constant.
 If a quantity grows exponentially, its doubling time is a constant.
 A situation in which a quantity increases by a common ratio at regular intervals.
 HalfLife – The time it takes for a quantity to decrease to onehalf of its size or value.
 If a quantity is decaying exponentially, its halflife is a constant.
 Input – A number that is entered into a function. The input variable in a function is the independent variable.
 Output – The result of a function. The output variable in a function is the dependent variable.
 Principal – The amount of money invested or borrowed.
Notes
What if… 
 What if someone gave you 2 cents ($0.02) each day… how long would it take for you to reach $1,000,000?
 1,000,000 divided by 0.02 gives you 50,000,000 days! That’s 137,000 years!
 This is an example of linear growth.
 However, what if someone gives you 2 cents on June 1st with one of two options (below). Which would you rather:
 Would you prefer to double your pennies from the previous day for a month, or
 Would you rather receive 1 million dollars right now?
 The exponential function that represents this problem is .
 x: day of the month
 2: doubling (base)
 You start with zero pennies, but then someone gives you 2 pennies on day 1: .
 So, would you rather have $1,000,000 dollars today, or double your pennies everyday for a month?
 I think I’ll take the pennies!

 Exponential Growth and Decay
Exponential Functions 
 Functions with a variable in the exponent
 Formula for Exponential Functions (General Form)
 Examples of Exponential Functions

Inputs and Outputs with Exponential Functions 
 As the input values increase, the function with the variable exponent grows more quickly than the function with the variable base.

Compounding Interest Periodically 
 Compound interest is an example of exponential growth because as more time passes, the amount of money grows faster and faster.
 Interest is money owed (and paid) for borrowing money.
 When you put your money in a bank account, it’s like loaning the money to the bank. For that service, the bank pays you a percentage of the amount you put in.
 If you borrow money from the bank, you have to pay interest to the bank.
 The rule of 72 is a quick way to estimate the effects of compound interest.
 Rule: The number of years it takes an amount to double is approximately the interest rate divided by 72.
 Years to double ≈ 72 ÷ interest rate
 So at 3%, it would take about 24 years for a $200 investment to double to $400.
 Formula for Compound Interest
 A: Amount of money accumulated in the account. Amount depends on time: A depends on t, so A(t).
t: Time that money remains invested at the given interest rate. Number of periods.
P: Principle is the original amount invested.
r: Interest Rate. Ex. 9% = 0.09. (To write a percent as a decimal, divide the percent number by 100).
n: Number of times that interest is compounded per year.

Examples
 Ex 1. Find F(3), when
 Substitute:
 Answer:

 Ex 2. Find F(3), when
 Substitute:
 Answer:

 Ex 3. Find F(4), when
 Substitute:
 Answer:

 Ex 4. Find F(3), when
 Substitute:
 Answer:

 Ex 5. Find F(4), when
 Substitute:
 Answer:

 Ex 6. Find F(2), when
 Substitute:
 Simplify:
 Expand:
 Answer:

 Ex 7. How much would $200 invested at 6% interest compounded annually be worth after 6 years? Round your answer to the nearest cent.
 Formula:
 Substitute:
 Simplify:
 Simplify More:
 Answer: $283.70

 Ex 8. How much would $500 invested at 6% interest compounded monthly be worth after 5 years? Round your answer to the nearest cent.
 Formula:
 Substitute:
 Simplify:
 Simplify More:
 Answer: $674.43

 Ex 9. How much would $400 invested at 9% interest compounded continuously be worth after 3 years? Round your answer to the nearest cent.
 Formula:
 Substitute:
 Simplify:
 Answer: $523.97

Important!
Practice (Apex Study 9.2)
 Try practice problems on Pgs 10, 20
 Mandatory: write and answer problems on Pgs 11, 21
 2 Quizzes
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