# 6.2 – Exponential Functions

## Key Terms

• e – The irrational number e 2.7182818284590…
• You can approximate e by substituting large values of n into the expression $(1+\frac{1}{n})^{n}$
• Compound Interest – Interest applied to both the principal and any previously earned interest.
• 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 $A_{t}=A_{0}e^{-kt}$, where $A_{t}$ = the amount at time t, $A_{0}$ = the initial amount, t = time, and k = a constant called the decay constant.
• If a quantity decays exponentially, its half-life 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 $F(x)=a\bullet b^{x}$, 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 $A_{t}=A_{0}e^{kt}$, where $A_{t}$ = the amount at time t, $A_{0}$ = 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.
• Half-Life – The time it takes for a quantity to decrease to one-half of its size or value.
• If a quantity is decaying exponentially, its half-life is a constant.
• Principal – The amount of money invested or borrowed.

## Notes

What if…
• What if someone gives you 2 cents on April 1st, which would you rather:
1. Would you prefer to double your pennies from the previous day for a month, or
2. Would you rather receive 1 million dollars right now?

•  The exponential function that represents this problem is $F(x)=2^{x}$.
• x: day of the month
• 2: doubling (base)
• You start with zero pennies, but then someone gives you 2 pennies on day 1: $F(1)=2^{1}=2$.

• 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 Functions • Functions with a variable in the exponent • Formula for Exponential Functions (General Form) • Exponential Growth and Decay Exponential Growth Exponential Decay • An exponential growth function represents a quantity that has a constant halving time. • An exponential decay function represents a quantity that has a constant halving time. • Either the exponent is negative (like $3^{-x}$) or the base will be between 0 and 1 (like $0.3^{x}$). • Exponential growth and decay functions are written in standard form as $F(t)=A_0\bullet b^{kt}$, where $A_0$ is an initial amount, b is the growth factor, k is the growth rate, and t is time. Compounding Interest Periodically • Compound interest is an example of exponential growth because as more time passes, the amount of money grows faster and faster. • Formula • 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. Use decimal values for calculations. n: Number of times that interest is compounded per year. • Example Compounding Interest Continuously • The number e • $e\approx 2.7182818284590...$ • The number e is an irrational number. • This means that its decimal representation goes on and on forever and does not repeat. • You can round the value of e to 2.718, unless otherwise instructed. • You can approximate e by substituting large values of n into the expression: $(1+\frac{1}{n})^{n}$ • To evaluate e in the Windows calculator, type the exponent first, then “Inv,” then $e^{x}$. • Ex. Evaluate $e^{-2}$. • Type $2, \pm, Inv, e^{x}$ • Answer: 0.135 • If you invest$1 in a savings account at 100% interest. What is the value of the account in 1 year?