Overview
This lecture gives a comprehensive review of all main topics and problem types for the 2025 Algebra 2 Regents Exam, covering number systems, algebra, functions, trigonometry, statistics, and probability.
Exam Structure & Scoring
- Four main units: Number and Quantity (5-12%), Algebra (35-44%), Functions (30-40%), Statistics and Probability (14-21%).
- 3-hour exam with 37 questions in four parts (multiple choice, short answer, multi-part, graph drawing).
- Raw score out of 86; scale raises your raw score.
- Passing = 20/86 (30%); Level 4 = 47/86; Level 5 = 66/86.
Number Systems and Properties
- Real numbers include rational (terminating/repeating decimals), integers, whole, natural, and irrational (non-terminating/non-repeating decimals).
- Polynomial division uses long division; remainder means divisor is not a factor.
- Rational functions are fractions with polynomials; same fraction rules apply.
- To rationalize denominators with roots or complex numbers, multiply by the conjugate.
Factoring Polynomials
- Methods: GCF, difference of squares, factoring by grouping, PolyRoots calculator.
- Factor quadratics by finding two numbers that add to b and multiply to c.
- Factoring by grouping: group terms, factor out GCF, combine common binomial.
Complex Numbers
- Form: a + bi; i is the imaginary unit (i² = -1).
- Rationalize complex denominators by multiplying top and bottom by the conjugate.
Functions
- Must pass vertical line test (one y for each x).
- Domain: x-values that yield outputs; exclude values making denominator zero or radicand negative.
- Range: set of y-values.
- One-to-one functions pass horizontal line test; onto functions have outputs for all possible y-values.
- Composition: f(g(x)) = f of g of x.
- Inverse: swap x and y and solve for y; reflection over y=x.
Function Transformations
- Horizontal translations/dilations & y-axis reflection: inside parentheses.
- Vertical translations/dilations & x-axis reflection: outside parentheses.
- Order: Horizontal (shift, dilation, reflection), then vertical.
- Even functions: y-axis symmetry; odd: rotational symmetry.
Regression & Logarithms
- Use calculator for regression (linear, quadratic, cubic, exponential, logarithmic).
- Exponential growth/decay: A = P(1 + r)^t; compound interest: A = P(1 + r/n)^(nt); continuous: A = Pe^(rt).
- Logs: log_b(x); common applications in growth/decay problems.
Trigonometric Functions
- Degrees↔radians: multiply/degrees by π/180 or 180/π.
- SOHCAHTOA defines sine, cosine, tangent.
- Special triangles: 30-60-90 (ratios 1:√3:2), 45-45-90 (1:1:√2).
- Unit circle: x = cos(θ), y = sin(θ).
- Trig identities: reciprocal, Pythagorean (sin²θ + cos²θ = 1), tangent/cotangent.
Trigonometric Graphs
- General form: y = A·sin(B(x–C)) + D; amplitude = |A|, period = 2π/B, shifts by C/D.
- Identify sine (starts at 0), cosine (starts at max), tangent (distinct shape).
Linear Equations and Systems
- Slope = change in y / change in x.
- Slope-intercept: y = mx + b; point-slope: y – y₁ = m(x – x₁).
- For 3-variable systems: eliminate variables stepwise, substitute to solve.
Quadratics
- Standard form: y = ax² + bx + c.
- Axis of symmetry: x = –b/(2a).
- Quadratic formula: x = [–b ± √(b²–4ac)]/(2a).
- Discriminant (b²–4ac) tells number and type of real roots.
- Parabola focus/directrix: vertex form and geometric properties.
Sequences and Series
- Arithmetic: add/subtract common difference.
- Geometric: multiply/divide by common ratio.
- Explicit formulas find nth term; recursive finds next term.
- Sigma notation and geometric series formula sum terms.
Statistics & Probability
- Types of studies: sample surveys, observational, controlled experiments.
- Key stats terms: mean (average), median, mode, standard deviation.
- Normal distribution: 68-95-99.7 rule.
- Confidence interval: CI = x̄ ± Z*(σ/√n).
- Probability rules: unions/intersections, complements, independent/dependent events.*
Key Terms & Definitions
- Real Number — any rational or irrational number.
- Rational Function — a function as a ratio of two polynomials.
- Complex Number — number in the form a + bi.
- Domain — all input values for which the function is defined.
- Range — all possible output values of a function.
- Amplitude — half the distance from max to min of a trig function.
- Period — length of one complete cycle in a trig function.
- Standard Deviation — measure of how spread out values are from the mean.
- Confidence Interval — range likely to contain the population mean.
- Discriminant — value b²–4ac in a quadratic; tells type/number of roots.
Action Items / Next Steps
- Practice factoring and polynomial/rational operations.
- Review function transformations and trig identities.
- Memorize special triangle side ratios and unit circle values.
- Use calculator for regression, systems solving, and normal distribution.
- Complete assigned practice Regents problems or review packets.