Algebra 2: 6.1 - Review Notes

Exponential Growth and Decay

• Definition of Exponential Functions: An exponential function can represent either growth or decay based on its form.   - Exponential Growth: The function has a positive base greater than 1.   - Exponential Decay: The function has a base between 0 and 1.

Examples

1. Function Analysis: Determine if functions represent growth or decay.    - Example: $y = 5 imes e^{0.7x}$ \      - Analysis: This represents exponential growth since the base (e^{0.7}) > 1.    - Example: $y = 3 imes (0.5)^x$ \      - Analysis: This represents exponential decay since the base (0.5) < 1.

Compound Interest Calculation

• When money is deposited, it earns interest, which may be compounded at different frequencies.

• Formula for Compound Interest: $A = P imes \bigg(1 + rac{r}{n}\bigg)^{nt}$   - Where:     - $A$ = the amount of money accumulated after n years, including interest.     - $P$ = principal amount (the initial amount of money).     - $r$ = annual interest rate (decimal).     - $n$ = number of times that interest is compounded per year.     - $t$ = number of years the money is invested or borrowed.

Application Example

• You deposit $6,000 into an account at an interest rate of 3.25%.

a. Compounding Monthly

• Parameters: $P = 6000, r = 0.0325, n = 12, t = 8$

• Formula: $A = 6000 imes \bigg(1 + rac{0.0325}{12}\bigg)^{12 imes 8}$

b. Compounding Quarterly

• Parameters: $n = 4$

• Formula: $A = 6000 imes \bigg(1 + rac{0.0325}{4}\bigg)^{4 imes 8}$

c. Compounding Daily

• Parameters: $n = 365$

• Formula: $A = 6000 imes \bigg(1 + rac{0.0325}{365}\bigg)^{365 imes 8}$

d. Compounding Continuously

• Formula: $A = Pe^{rt}$

• Substituting Values: $A = 6000 imes e^{0.0325 imes 8}$

Depreciation of Asset Value

• Depreciation Formula: $A = P imes (1 - r)^t$   - Where:     - $A$ = value after t years.     - $P$ = initial value.     - $r$ = rate of depreciation (as a decimal).     - $t$ = number of years.

Application Example

• You bought a boat for $120,000 with a depreciation rate of 8.5%.

• Formula: $A = 120000 imes (1 - 0.085)^5$

Value of a Rare Coin

• The value of the rare coin is described by the function: $Y = 0.25(1.06)^t$   - Where:     - $Y$ = value of the coin in dollars.     - $t$ = number of years since minted.

Analysis

1. Growth or Decay: The model represents exponential growth since the base (1.06) > 1.

2. Annual Percent Increase: The annual percent increase is 6% (derived from (1.06 - 1) imes 100).

3. Original Value: The original value of the coin is $0.25$ dollars (when t = 0).

4. Estimate Future Value: To find when the coin's value will be $0.60$, set up the equation:    - $0.60 = 0.25(1.06)^t$    - Solve for $t$.

Simplifying Expressions with Exponential Components

• Examples of simplifying exponential expressions include:   1. $e^4 imes e^{-1} ightarrow e^{4-1} = e^3$   2. $(2e^{3x})^5 ightarrow 2^5 imes e^{15x} = 32e^{15x}$   3. $e^{-3} imes e^{8} ightarrow e^{5}$   4. $12e^5 imes 4e^{2} ightarrow (12 imes 4)e^{5+2} = 48e^{7}$    

• Operations involving multiple exponentials also involve adding exponents with like bases and applying multiplication rules.