Video summary

2026 Algebra 2 Regents Review (EVERYTHING YOU NEED TO KNOW!)

Main summary

Key takeaways

Educational

Main ideas, concepts, and lessons (organized by unit/topic)

Exam overview (Algebra 2 Regents structure)

  • 4 major content clusters/units on the exam:

    • Algebra 2 Number and Quantity: 5–12%
    • Functions: 35–44%
    • Algebra 2 Functions (already included above) + Transformations etc. (covered under Functions)
    • Statistics and Probability: 14–21%

  • Number of questions & time:

    • 3 hours
    • 37 questions
    • 4 parts

  • Scoring breakdown (points):

    • Part 1: 24 multiple choice
      • 2 points each → 48 total
    • Part 2: 8 short answer
      • 2 points each → 16 total
    • Part 3: work… multi-part (described as worth 4 points each)
      • 4 questions → 16 total
    • Part 4: 1 final question
      • 6 points (often includes graph drawing)

  • Total points: 86 points (raw score total)

  • Score conversion / passing targets (approximate, may vary by exam):

    • Scaled score is positively scaled (scaled score > raw score unless you get 100)
    • Level 5: about 77% raw = 66/86
    • Level 4: about 47/86
    • Level 3 / pass: about 20/86 = ~30%
    • Rule of thumb: around one-third of points to pass.

Unit 1: Number and Quantity

1) Number system classification

  • Rational numbers

    • Can be written as terminating or repeating decimals
    • Integers
      • Whole numbers
        • Natural/counting numbers (typically 1 and greater in the video’s description)
    • Diagram logic (overlapping sets):
      • Any natural number is also whole → integer → rational
      • Whole numbers are integers and rationals but not necessarily naturals

  • Irrational numbers

    • Decimals that neither terminate nor repeat
    • Examples: π (pi), e

2) Basic number properties

Mentions that you must know properties related to:

  • Radicals
  • Rational exponents
  • Logs
  • Exponents

Instructional note: Teacher advises pausing to write these down if taking notes.

3) Polynomial division (long division for polynomials)

  • Concept: like standard long division, but:

    • divisor/dividend are polynomials
    • output is a quotient polynomial

  • Example described: divide (2x² + 7x + 6) by (x + 2)

    • Steps summarized:
      • Choose multiplier so the leading term cancels
      • Subtract and repeat until fully reduced
    • Outcome in the example:
      • No remainder
      • Uses Factor Theorem:
        • If remainder is 0, then (x + 2) is a factor

  • Remainder Theorem

    • If remainder exists, remainder relates to the polynomial value after substitution into the divisor’s root.

4) Rational functions

  • Definition: fractions where numerator and denominator are polynomials
  • Emphasis: use fraction rules to simplify/manage them.

5) Rationalizing denominators (important)

  • Problem type: expressions with a square root in the denominator
  • Goal: remove the radical from the denominator
  • Method:
    • Multiply numerator and denominator by the conjugate of the denominator.
    • Conjugate rule:
      • conjugate switches the sign between the two terms (e.g., (B-\sqrt{C}) becomes (B+\sqrt{C}), or vice versa).
  • Result: square root ends up in the numerator, none in the denominator.

6) Factoring (multiple methods)

  • Method 1: Greatest common factor (GCF)
    • Factor out the common factor first.
  • Method 2: Difference of squares
    • Pattern:
      • (a^2 - b^2 = (a+b)(a-b))
  • Method 3: Trinomial factoring
    • For quadratics (x^2 + bx + c):
      • find two numbers that:
        • multiply to (c)
        • add to (b)
    • Example given:
      • numbers that multiply to 3 and add to -4 lead to ((x-1)(x-3))
  • Method 4: Factoring by grouping
    • Used when degree is 3 or greater (often)
    • Steps:
      • make two groups
      • factor out each group’s GCF
      • check resulting binomials match, then combine to factor completely

7) Complex numbers

  • Imaginary unit:
    • (i = \sqrt{-1})
    • therefore (i^2 = -1)
  • Complex number form:
    • (a + bi), where (a,b) are real
  • Rationalizing complex expressions
    • same idea as rationalizing radicals:
    • multiply by the conjugate (change sign of the imaginary term)

Unit 2: Functions

Topics included (as listed by the speaker)

  • Domain and range
  • Composition functions
  • Inverse functions
  • One-to-one and on-to
  • End behavior
  • Multiplicity
  • Transformations
  • Applications
  • Logarithms
  • Regression functions

1) Vertical line test (function definition)

  • If a vertical line hits the graph:
    • more than once for the same x-value → not a function
    • exactly once → is a function

2) Domain

  • Definition: set of x-values where the function produces an output / exists
  • Rational function rule: exclude values where denominator = 0
    • Example: if a factor makes the denominator zero at (x=-2), then (-2) is excluded.

3) Radical function domain rule

  • Rule: radicand must satisfy the real-output condition:
    • described as radicand ≥ 0
  • Example:
    • (\sqrt{x-5}) implies (x-5 \ge 0 \Rightarrow x \ge 5)

4) Range (notes)

  • Speaker says range is generally not required algebraically for the Regents, but it is:
    • the set of all y-values the function can output.

5) One-to-one and on-to

  • One-to-one:
    • No repeating x-values or y-values
    • Uses both:
      • vertical line test
      • and horizontal line test (to prevent repeated y-values)
  • On-to:
    • every x-value has a defined y output (described as “most normal functions”)

6) Composition functions

  • Notation:
    • (f \circ g(x)) means (f(g(x)))
  • Method:
    • replace the input of the outer function with the expression from the inner function

7) Inverse functions

  • Meaning:
    • reflection across the line (y=x) for one-to-one functions
  • How to compute:
    • swap x and y in the equation and solve for y
  • Notation reminder:
    • (f^{-1}(x)) denotes the inverse.

8) End behavior (polynomials)

  • Key idea:
    • Odd degree: ends go in opposite directions
    • Even degree: ends go in the same direction
    • sign of leading coefficient determines up/down direction
  • Graph-based fallback:
    • “Just graph it” to confirm.

9) Multiplicity

  • How many times the graph “touches/bounces” at a root
  • Higher multiplicity → the graph behavior “stretches/touches” more.

Transformations (detailed methodology)

Types to know (6 total)

  • Horizontal transformations (inside the parentheses):

    1. Horizontal translations/shifts
    2. Horizontal dilations (scaling)
    3. Reflection over the y-axis

  • Vertical transformations (outside the parentheses):

    1. Vertical translations
    2. Vertical dilations
    3. Reflection over the x-axis

How to read transformation equations (rules)

Horizontal shifts

  • Use (f(x \pm a))
  • Direction:
    • (x + a) → shifts left
    • (x - a) → shifts right

Horizontal dilations

  • Use (f(kx)) or (f(\frac{1}{k}x))
  • Effects:
    • multiply by k inside → graph becomes skinnier
    • multiply by 1/k inside → graph becomes wider

Horizontal reflection (y-axis)

  • Replace (x) with (-x): (f(-x))

Vertical translations

  • (f(x) + a) shifts up
  • (f(x) - a) shifts down

Vertical dilations

  • (k \cdot f(x)) scales vertically
  • (\frac{1}{k}\cdot f(x)) scales vertically the other way

Vertical reflection (x-axis)

  • (-f(x))

Mnemonics + ordering (important)

  • Mnemonics (style):
    • “H via y” / “V x”: horizontal inside parentheses; vertical outside parentheses
  • Order matters: HDRV
    • Horizontal translations
    • Dilations
    • Reflections
    • Vertical translations
  • Memory trick: “Helicopters do rise vertically” (for HDRV order)

Even/odd functions

  • Even function:
    • symmetric about y-axis
    • (f(-x)=f(x))
  • Odd function:
    • rotational symmetry about origin (180°)
    • (f(-x)=-f(x))

Logarithms, regression, and applications

Regression

  • Use calculator regression to match data points
  • Mentioned: “recognize models” (specific list not provided)
  • Power functions show up less, but other models do.

Logarithm applications

  • Used in real-life word problems, including:
    • exponential growth/decay modeling
    • exponential compounding formulas
  • Exponential growth/decay equation form:
    • (A(T)=P(1 \pm r)^T)
    • meanings:
      • (P) = initial value
      • (1 \pm r) = growth/decay factor
      • (r) is in decimal (e.g., 3% → 0.03)
      • (T) = time

Compound interest

  • Uses (n) = number of compounds per year
  • Common choices listed:
    • monthly, quarterly, daily, yearly

Continuous compounding

  • Uses (e) (~2.72)
  • Based on:
    • limit of ((1 + 1/n)^n) as (n \to \infty)
  • Speaker says: mainly need to use the equation, not derive it.

Unit: Trigonometric functions

Topics covered

  • Radians/degrees
  • Trig functions and reciprocals
  • Unit circle & special triangles
  • Trig identities
  • Inverses and restricted domains/ranges
  • Trig equation transformations/graphing

1) Degree ↔ radian conversion

  • Degrees → radians:
    • multiply by (\pi/180)
  • Radians → degrees:
    • multiply by (180/\pi)

2) Trig functions definitions (using right triangles)

  • (\sin(\theta)=\frac{y}{r})
  • (\cos(\theta)=\frac{x}{r})
  • (\tan(\theta)=\frac{y}{x})

Reciprocals:

  • (\csc(\theta)=\frac{r}{y})
  • (\sec(\theta)=\frac{r}{x})
  • (\cot(\theta)=\frac{x}{y})

Mnemonic:

  • SOH CAH TOA (opposite/hypotenuse/adjacent)

3) Special triangle values emphasized

  • Angles: 30-60-90 and 45-45-90
  • Core lesson:
    • 30°, 45°, 60° must be exact (calculator may give decimals)
  • Example concepts:
    • (\sin 30^\circ = 1/2)
    • (\sin 45^\circ) requires radical handling (conceptually includes rationalization steps)

4) Identities

  • Reciprocal relationships:
    • trig value is reciprocal of its reciprocal trig function
  • Quotient relationships:
    • (\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)})
    • (\cot(\theta)=\frac{\cos(\theta)}{\sin(\theta)})
  • Pythagorean identities:
    • convert between forms, e.g., express (\cos^2\theta) using (\sin^2\theta)

5) Unit circle (x, y, r meaning)

  • Unit circle:
    • radius (r=1)
  • Point ((x,y)) on unit circle corresponds to:
    • (x = \cos(\theta))
    • (y = \sin(\theta))

Inverse trig:

  • arc-sine/arc-cosine/arc-tangent notation described
  • calculator output depends on degrees vs radians

6) Restricted domains and quadrants

  • Inverse trig functions require restricting domains
  • May yield multiple equivalent angles; quadrant rules fix the correct one
  • Determine reference angle and adjust (add/subtract from 90° or 180° as needed)

7) Graphing trig functions (amplitude/frequency shifts)

  • General form (conceptual):
    • (y = A\sin(B(x-C)) + D)
  • Parameters:
    • (A) = amplitude
    • (B) = frequency factor (affects period)
    • (C) = horizontal shift (positive → right, negative → left)
    • (D) = vertical shift (positive → up, negative → down)
  • Process:
    • determine sine vs cosine from graph behavior
    • identify amplitude, midline, frequency, shifts
    • write final equation using parameters

Unit: Linear equations, quadratics, sequences/series

A) Linear functions (degree 1)

  • Slope:
    • rise/run (change in y over change in x)
    • can use two points
  • Forms:
    • slope-intercept: (y=mx+b)
    • point-slope: (y-y_1=m(x-x_1))

3-variable linear systems

  • Recommendation:
    • use calculator “LinSolve”
  • For short answer:
    • show work; elimination strategy is used
3-variable systems: elimination workflow (6-step plan)
  1. Group the system into equation pairs (first+second, second+third)
  2. Eliminate one variable by multiplying equations so coefficients cancel
  3. Consolidate into a new 2-variable system
  4. Eliminate another variable to reduce to one variable
  5. Solve for that variable
  6. Substitute back to find remaining variables
  • Example result mentioned:
    • (x=0), (y=2), (z=-1)

B) Quadratics (degree 2)

  • Standard form:
    • (y=ax^2+bx+c)
  • Axis of symmetry:
    • (x=\frac{-b}{2a})

  • Root relationships:

    • sum/product linked to coefficients (implied)

  • Quadratic formula and discriminant

    • discriminant: (b^2-4ac)
    • interpretation:
      • negative → no real x-intercepts (complex roots)
      • zero → one real root (touches x-axis)
      • positive → two real roots
    • exact vs irrational:
      • perfect square discriminant → rational roots
      • not perfect square → irrational roots

Focus/Directrix (Parabola “torture focus” topic)

  • Speaker note:
    • on the exam but being removed after 2025–2026
  • Definitions:
    • Focus: a point on the axis of symmetry
    • Directrix: a line perpendicular to the axis of symmetry
    • Any point on the parabola is equidistant from focus and directrix
  • Distances labeled:
    • relate to parameter p
  • Vertex form concepts:
    • uses (h,k) (vertex (H,K))
    • vertical vs horizontal:
      • speaker emphasizes vertical equations use y-structure in the “form” logic
      • horizontal handled by swapping x/y roles
  • Example problem:
    • directrix given as (y=4)

Original video