# 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 $(1+\frac{1}{n})^{n}$
• 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 $A_{t}=A_{0}e^{-kt}$, 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.
• 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:
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 Growth and Decay Exponential Functions • Functions with a variable in the exponent • Formula for Exponential Functions (General Form) • Examples of Exponential Functions • $F(x)=2\bullet 3^{x}$ • $F(x)=-4\bullet 5^{x}$ • $F(x)=(\frac{1}{2})^{x}$ • $F(x)=\frac{2}{3}\bullet 0.1^{x}$ 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. • x: input • F(x): outputs Exponential Growth Exponential Decay • An exponential growth function represents a quantity that has a constant doubling time. • Ex. $-4\bullet 5^{2t}$ • Ex. $2\bullet 3^{6t}$ • An exponential decay function represents a quantity that has a constant halving time (half-life). • The exponent will either be negative (like $3^{-x}$) or the base will be between 0 and 1 (like $0.3^{x}$). • Ex. $2\bullet 0.4^{3t}$ • Ex. $8^{-2t}$ • 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. • $A_0$, b, and k all stand for fixed numbers and t is the only variable. 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. • Example Compounding Interest Continuously • Continuous compounding means that the interest instead gets calculated at every instant in time, not just at periodic intervals. • So, what if you invest$1 at 100% interest for 1 year?
• This is known as continuous compounding.

Answer: When the interest is compounded continuously,
the value of the account is e dollars (about $2.72)! • Formula for Compounding Interest Continuously • $A(t)=P\bullet e^{rt}$ • A(t): amount of money in the account after t years • P: principal – original amount (in dollars) you put into the account • r: interest rate (written as a decimal) • t: number of years the interest is compounded • 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 • Example: You invest$500 in a savings account. The account pays 4% interest compounded continuously. How much money is in the account after 7 years?
• P: $500 • r: 0.04 • t: 7 • Setup: $A(7)=500\bullet e^{7\bullet 0.04}$ • Simplify: $A(7)=500\bullet e^{0.28}$ • Simplify: $A(7)=500\bullet 1.3231298$ • Answer:$661.55

## Examples

 Ex 1. Find F(3), when $F(x)=3^{x}$ Substitute: $F(3)=3^{3}$ Answer: $3\bullet 3\bullet 3=27$ Ex 2. Find F(3), when $F(x)=(\frac{1}{6})^{x}$ Substitute: $F(3)=(\frac{1}{6})^{3}$ Answer: $(\frac{1}{6})\bullet (\frac{1}{6})\bullet (\frac{1}{6})=(\frac{1}{216})$ Ex 3. Find F(4), when $F(x)=(\frac{1}{3})\bullet 4^{x}$ Substitute: $F(4)=(\frac{1}{3})\bullet 4^{4}$ Answer: $(\frac{1}{3})\bullet 4\bullet 4\bullet 4\bullet 4=\frac{256}{3}$ Ex 4. Find F(3), when $F(x)=4\bullet (\frac{1}{3})^{x}$ Substitute: $F(3)=4\bullet (\frac{1}{3})^{3}$ Answer: $4\bullet (\frac{1}{3})\bullet (\frac{1}{3})\bullet (\frac{1}{3})$ Ex 5. Find F(-4), when $F(x)=2^{-4}$ Substitute: $F(-4)=\frac{1}{2^{4}}$ Answer: $(\frac{1}{2\bullet 2\bullet 2\bullet 2})=\frac{1}{16}$ Ex 6. Find F(2), when $F(x)=2\bullet (\frac{1}{2^{3t}})$ Substitute: $F(2)=2\bullet (\frac{1}{2^{3\bullet 2}})$ Simplify: $2\bullet (\frac{1}{2^{6}})$ Expand: $2\bullet (\frac{1}{2\bullet 2\bullet 2\bullet 2\bullet 2\bullet 2})$ Answer: $2\bullet (\frac{1}{64})=\frac{1}{32}$ 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: $A(t)=P(1+\frac{r}{n})^{nt}$ Substitute: $A(6)=200(1+\frac{0.06}{1})^{1\bullet 6}$ Simplify: $A(6)=200(1.06)^{6}$ Simplify More: $A(6)=200(1.06)^{6}$ 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: $A(t)=P(1+\frac{r}{n})^{nt}$ Substitute: $A(5)=500(1+\frac{0.06}{12})^{12\bullet 5}$ Simplify: $A(5)=500(1+0.005)^{60}$ Simplify More: $A(5)=500(1.005)^{72}$ 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: $A(t)=P\bullet e^{rt}$ Substitute: $A(3)=400\bullet e^{0.09\bullet 3}$ Simplify: $A(3)=400\bullet 2.718^{0.27}$ Answer:$523.97