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Algebra 2 Regents Key Concepts

Jun 23, 2025

Overview

This lecture reviews key concepts and test-taking strategies for the Algebra 2 Regents Exam, covering all major units: number systems, algebra, functions, trigonometry, statistics, and probability.

Exam Format & Scoring

  • Four content units: Number & Quantity (5-12%), Algebra (35-44%), Functions (30-40%), Statistics & Probability (14-21%).
  • 3-hour exam, 37 questions split into 4 parts (24 multiple choice, 8 short answer, 4 extended response, 1 graphing question).
  • Scored out of 86 points; raw scores are positively scaled.
  • Passing requires about 30% (20/86); Level 4 at 47/86, Level 5 at 66/86.

Number Systems & Operations

  • Real numbers include rational (fractions, terminating/repeating decimals) and irrational numbers (non-terminating, non-repeating).
  • Rational numbers contain integers, whole numbers (≥ 0), and natural numbers (≥ 1).
  • Irrational numbers: decimals that do not repeat/terminate (e.g., Ï€, e).
  • Properties of radicals, exponents, and logarithms must be memorized.
  • Polynomial division operates like long division; if no remainder, divisor is a factor (factor theorem).
  • Remainder theorem: remainder = f(a) after division by (x-a).
  • Synthetic division is optional for Regents.
  • Rational functions act like fractions but with polynomials; follow fraction rules.
  • Rationalizing denominators removes square roots from denominators by multiplying by the conjugate.
  • Factoring methods: GCF, difference of squares, grouping, and trinomial factoring.

Complex Numbers

  • Imaginary unit ( i = \sqrt{-1} ); ( i^2 = -1 ).
  • Complex numbers: ( a + bi ), where a, b are real.
  • Rationalize denominators in complex fractions by multiplying by the conjugate.

Functions & Transformations

  • A function passes the vertical line test (one output per input).
  • Domain: allowable input values; range: possible output values.
  • Domains for rational functions exclude values where denominator = 0.
  • Domains for radical functions: radicand ≥ 0.
  • One-to-one functions have unique outputs for each input; onto functions cover all possible outputs.
  • Function composition: plugging one function into another.
  • Inverses: swap x and y, solve for y; graphically, reflect over y = x.
  • End behavior depends on degree and leading coefficient (even/odd).
  • Multiplicity: number of times a root occurs (affects graph "bounces").
  • Transformations:
    • Horizontal (inside parentheses): shift (add/subtract), dilate/stretch (multiply), reflect.
    • Vertical (outside parentheses): shift (add/subtract), dilate/stretch (multiply), reflect.
    • Remember "HIYA" (Horizontal Inside, Y-Axis) and "VXVO" (Vertical, X-Axis, Vertical Outside) for transformations.
    • Transformation order: Horizontal shift, dilation, reflection, vertical shift (HDRV).
  • Even functions: symmetric about y-axis; odd: symmetric about origin.

Advanced Functions: Logarithms & Regression

  • Logarithms: know properties and uses, especially for exponential growth/decay and compound interest.
  • Regression: use calculators to model data; recognize linear, quadratic, exponential, logarithmic, logistic models.

Trigonometric Functions

  • Know radians ↔ degrees conversions: degrees × Ï€/180 = radians, radians × 180/Ï€ = degrees.
  • Main trig functions: sine ( (\sin) ), cosine ( (\cos) ), tangent ( (\tan) ); and their reciprocals.
  • Use SOHCAHTOA for right triangle definitions; memorize values for 30°, 45°, 60° triangles.
  • Trig identities:
    • ( \sin^2\theta + \cos^2\theta = 1 )
    • ( \tan\theta = \frac{\sin\theta}{\cos\theta} )
  • Unit circle: trig values correspond to (x, y) coordinates for angle θ.
  • Inverse trig functions: used to find angles from given trig values.
  • Trig graphs: amplitude (A), frequency (B), phase shift (C), vertical shift (D); period = ( 2\pi/B ).

Algebra: Linear, Quadratic, Sequences

  • Linear equations: degree 1, form ( y = mx + b ), slope = rise/run.
  • Point-slope form: ( y - y_1 = m(x - x_1) ).
  • Three-variable linear systems: eliminate variables step by step.
  • Quadratics: degree 2, standard form ( ax^2 + bx + c ).
  • Axis of symmetry: ( x = -b/2a ).
  • Roots: quadratic formula ( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} ); discriminant indicates root type.
  • Focus/directrix: key features for parabolas; know vertex and standard/vertex forms.
  • Sequences: arithmetic (add/subtract, linear-like) and geometric (multiply/divide, exponential-like).
  • Explicit formulas find any term; recursive formulas find next term.
  • Sigma notation: used for summing sequences; geometric series formula on reference table.

Statistics & Probability

  • Types of studies: sample survey (generalize), observational (causation for group), controlled experiment (inference/generalization possible).
  • Single- and double-blind experiments reduce bias.
  • Mean (μ, xÌ„), standard deviation (σ, s): population vs. sample.
  • Normal distribution: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD.
  • Confidence interval: ( \text{mean} \pm z \cdot \frac{\sigma}{\sqrt{n}} ); memorize common z-values (1.645 for 90%, 1.96 for 95%, 2.575 for 99%).
  • Z-score: ( z = \frac{x - \mu}{\sigma} ).
  • Probability:
    • Union (A or B): ( P(A \cup B) )
    • Intersection (A and B): ( P(A \cap B) )
    • Complement: ( P(A') = 1 - P(A) )
    • Mutually exclusive: events can't both happen.
    • Independent: one event does not affect another; conditional probability for dependence.

Key Terms & Definitions

  • Rational Number — Number with terminating/repeating decimal (includes integers, fractions).
  • Irrational Number — Decimal neither terminates nor repeats (e.g., Ï€).
  • Domain — All input (x) values for which a function is defined.
  • Range — All output (y) values a function can produce.
  • Multiplicity — Number of times a root appears in a polynomial factorization.
  • Amplitude — Height from midline to peak in a trig function.
  • Frequency — Number of cycles in ( 2\pi ) units for trig graphs.
  • Confidence Interval — Estimate range for population parameter.
  • Z-score — Number of standard deviations from mean.

Action Items / Next Steps

  • Memorize formulas for transformations, trig identities, and confidence intervals.
  • Practice solving three-variable linear systems and quadratic equations.
  • Review calculator instructions for regression and normal distribution functions (normCDF).
  • Study special triangles and unit circle values for trig.
  • Prepare for focus/directrix and sequence/series questions.

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