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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.

Alg1B 6.2 Pennies2


  • 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)

Alg1B 6.2 Exp Fx


  • Exponential Growth and Decay

Alg1B 6.2 Exp Growth 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

Alg1B 6.2 Compound Interest 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.

Alg1B 6.2 Compound Times

  • Example

 

Alg1B 6.2 Principal Rate Time

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?
    • Answer: When the interest is compounded continuously, the value of the account is e dollars (about $2.72)!

Alg1B 6.2 Compound Continuously

Alg1B 6.2 Compound Interest

  • Example (see Ex 9 below)

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: $523.99

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