Algebra 2 Regents Flashcards

Sequences and Series

• A sequence is an ordered list of numbers, defined formally as a function with positive integers as its domain.
• The first term in a sequence is denoted as f(1) or a₁.
• Recursive formulas derive terms by performing operations on previous terms.
• Explicit formulas derive terms using the term's index or position in the sequence.
• The common difference (d) is the difference between any two consecutive terms in an arithmetic sequence.
• The common ratio (r) is the ratio between any two consecutive terms in a geometric sequence.

Arithmetic and Geometric Sequences

• An arithmetic sequence is characterized by constant addition to obtain the next term.
• A geometric sequence utilizes a constant multiplication pattern to find the next term.
• Recursive formulas for arithmetic and geometric sequences are used but need to be specified.
• Explicit formulas for both types of sequences must be defined for full understanding.

Summations and Series

• Summation (Sigma) notation represents the sum of a sequence's terms.
• A series is defined as the sum of the terms of a sequence.
• Arithmetic series and geometric series formulas provide methods to calculate these sums.

Exponential Functions

• Basic exponential function form includes a as the y-intercept and b as the base (multiplier).
• Decreasing exponential functions occur when 0 < b < 1.
• Increasing exponential functions occur when b > 1.
• Exponential regression fits data to an exponential function.

Exponent Laws

• Exponent Law 1: (x²)(x³) = x⁵ demonstrates the product of powers.
• Exponent Law 2: x⁸÷x⁵ = x³ illustrates the quotient of powers.
• Negative exponents transform into reciprocal forms: 4⁻² = 1/16.
• Exponent Law 4 states 5⁰ = 1.
• Exponent Law 5: (x³)⁵ = x¹⁵ shows power of a power rule.
• Exponent Law 6: (2xy²)³ expands to 8x³y⁶ through multiplication.
• Exponent Law 7 simplifies fractions with exponents: (3/x⁴)³ = 27/x¹².
• Square roots represented as exponents: 25 ^ ½ = 5.
• Exponent Law 9: 4^ 3/2 simplifies to 8 by rewriting as (√4)³.

Asymptotes

• A horizontal asymptote is a horizontal line that a graph approaches at infinity.
• A vertical asymptote is a vertical line that a graph approaches where the function is undefined.

Logarithms and Inverse Functions

• Inverse functions switch x and y, reflecting symmetrically across the line y = x.
• Common logs have base 10 (y = log(x)), while natural logs use base e (y = ln(x)).

Interest and Growth Models

• Formulas for compound interest and continuous compound interest provide methods for financial calculations.
• The half-life formula is used in decay models.

Additional Concepts

• The equation of the unit circle is x² + y² = 1, centered at (0, 0) with a radius of 1.
• In the unit circle, sin(θ) corresponds to the y-coordinate, while cos(θ) corresponds to the x-coordinate.