Algebra 2 Benchmark #3 - Equations Flashcards

Logarithmic and Exponential Formulas

• Change of Base Formula: Transforms logarithmic expressions from one base to another for easier calculations.
• Compound Interest Formula: Used to calculate accumulated interest on an investment; expressed as ( A = P(1 + \frac{r}{n})^{nt} ), where ( A ) is the amount, ( P ) the principal, ( r ) the interest rate, ( n ) the number of times interest is compounded per year, and ( t ) the number of years.
• Half-Life Formula: Determines the remaining quantity after a certain time based on its half-life; given by ( y = C (1/2)^{t/h} ), where ( C ) is the initial amount, ( t ) is time, and ( h ) is the half-life duration.
• Doubling Formula: Describes exponential growth, indicating that quantity doubles over time; modeled as ( y = C (2)^{t/T} ), where ( C ) is initial amount, ( t ) is time, and ( T ) is doubling time.
• Continuous Growth Formula: Represents growth compounding continuously, formulated as ( A = Pe^{rt} ) with ( P ) as the initial investment, ( r ) the growth rate, and ( t ) time.
• Exponential Decay Formula: Used in contexts of decline; represented by ( y = C (1 - r)^{t/T} ) where terms follow as in previous formulas.
• Exponential Growth Formula: Indicates growth based on fixed intervals, given as ( y = C (1 + r)^{t/T} ) following similar variables.

Circle and Trigonometric Concepts

• Equation of a Circle: Standard form is ( (x-h)^2 + (y-k)^2 = r^2 ); defines a circle with center ( (h, k) ) and radius ( r ).
• Trigonometric Functions:
  • Sine: Defined as ( \sin θ = \frac{y}{r} ).
  • Cosine: Defined as ( \cos θ = \frac{x}{r} ).
  • Tangent: Expressed as ( \tan θ = \frac{y}{x} ).
  • Cosecant: Inversely related to sine; ( \csc θ = \frac{r}{y} ).
  • Secant: Inversely related to cosine; ( \sec θ = \frac{r}{x} ).
  • Cotangent: Inversely related to tangent; ( \cot θ = \frac{x}{y} ).

Trigonometric Identities

• Tangent Identity: ( \tan θ = \frac{\sin θ}{\cos θ} ).
• Cotangent Identity: ( \cot θ = \frac{\cos θ}{\sin θ} ).

Additional Concepts

• Average Rate of Change: Calculated as the ratio of the change in ( y ) to the change in ( x ).
• Period Equation: Represents the duration of one complete cycle in periodic functions, given by ( \text{Period} = \frac{2π}{\text{frequency}} ).
• Reference Angle: The acute angle formed between the terminal side of an angle in standard position and the x-axis.
• Amplitude: Maximum height of a wave's crest from its midline.
• Midline: A horizontal axis serving as a reference for the oscillation of periodic function graphs.
• Negative Angle: Refers to angles measured in a clockwise direction.
• Positive Angle: Refers to angles measured in a counterclockwise direction.
• Phase Shift: A horizontal displacement in the graph of a periodic function.
• Frequency: The count of complete waves passing a specific point in a designated time frame.