Calculator Power Plays Cheat Sheet

Calculator Power Plays

TI-84 Reference Guide | NYS Algebra 2 Regents

1. Graphing & Analysis

Find Zeros: [2nd] [TRACE] → [2: zero].

Find Intersects: [2nd] [TRACE] → [5: intersect].

Root Finder: [APPS] → [PolySmlt2] → [1].

2. Math Templates

Fraction: [ALPHA] [Y=] → [1: n/d]

Log with Base: [MATH] → [A: logBASE(]

Summation: [MATH] → [0: summation]

Finance Solver: [APPS] → [Finance] → [1: TVM Solver]

Quick Note: Ans Feature

Press [2nd] [(-)] to use your previous answer in a new calculation. Avoid re-typing long decimals!

3. Regression & Stats

Data Entry: [STAT] → [Edit]. (L1, L2).

Regression: [STAT] → [CALC] → [4: LinReg] or [0: ExpReg].

Correlation (r): Run DiagnosticOn from [2nd] [0] (CATALOG) to see "r" values.

4. Normal Dist

Area/Percent: [2nd] [VARS] → [2: normalcdf]

Syntax: (low, high, μ, σ)

*Use -1E99 for lower bound ( [2nd] [,] for E ).

Table Trick

Press [2nd] [GRAPH] to view a table of values for any equations in your Y= list.

Always check: Is your calculator in RADIAN mode for Trig Modeling?

PAGE 02

Step-by-Step Examples

Exponential Regression Walkthrough

Problem: Determine the growth model \(y = ab^x\) for the population: (0, 50), (1, 82), (2, 135).

1. Input Data:

[STAT] [1: Edit]. Enter x-values in L1 and y-values in L2.

2. Run Regression:

[STAT] [CALC] [0: ExpReg]. Use L1 and L2, then scroll to "Calculate".

Final Answer:

y = 50(1.64)^x

Check: a=50 is the initial value (at x=0).

Normal Probability Walkthrough

Problem: Find the percentage of kitten weights above 7 lbs given \(\mu = 5\) lbs and \(\sigma = 1.2\) lbs.

Menu Path: [2nd] [VARS] → [2: normalcdf]

• Lower: 7
• Upper: 1E99
• Mean (μ): 5
• Std Dev (σ): 1.2

Sketch Normal Curve Below:

Answer: ≈ 0.0478 (4.8%)

Calculator = Secondary Check

While these shortcuts are incredibly powerful, remember that Algebra 2 Regents "Show Work" questions require algebraic evidence. Use the calculator to verify your answers before moving on!

Regents Exam Strategy Tips

Regents Game Plan

Strategic Guide for the Algebra 2 Regents Exam

Duration

3 Hours

1. Scoring & Time Management

Part I

24 MC

60 mins

Part II

8 SA

45 mins

Part III

4 SA

45 mins

Part IV

1 SA

30 mins

Strategy in Action: Maximize Partial Credit

Common 2-Point Question:

"Determine if (x-2) is a factor of f(x) = x³ - 4x + 1."

The "Justify" Response:

f(2) = (2)³ - 4(2) + 1
f(2) = 8 - 8 + 1 = 1
Since f(2) ≠ 0, (x-2) is NOT a factor.

Pro Tip:

Notice how we showed the substitution and the final comparison to zero. This ensures full credit even if you make a calculation error!

2. Never Give Away Points

• The Rounding Trap: Check the prompt for "nearest tenth," "hundredth," or "integer." Never round intermediate steps; only round the final answer.
• The Units Trap: If the question is about "years" or "bacteria," include the units in your answer box unless they are already printed there.
• Explain vs Justify: "Explain" means write a sentence. "Justify" allows math. If it says explain, use words to describe your reasoning.

PAGE 02

Final Checks & Checklist

3. High-Frequency Topic Flash Tips

Focus & Directrix

The vertex is always in the middle. Distance from vertex to focus (p) = distance from vertex to directrix.

Recursive Sequences

Must include a starting term like a1. The formula an = an-1 + 4 means "previous term plus 4."

Inverse Functions

Switch x and y, then solve. Graph of f-1(x) is a reflection over the line y = x.

Causation

Correlation does NOT mean causation. Only a controlled experiment with random assignment shows causation.

Avoid Silly Errors

Parentheses Matter

Negative numbers must be in parentheses when squared!
(-5)² = 25 vs. -5² = -25.

Extraneous Solutions

Always check your answers for radical and rational equations in the original equation.

Calc Safety Checks

Radian Mode: Required for Trig modeling / Sinusoidal graphs.

Diagnostics ON: [2nd][0] → DiagnosticOn for correlation (r).

Fraction Template: [ALPHA][Y=] → [1] for input precision.

Exam Morning Readiness

Calculator Fully Charged

Pencils and a clear plastic ruler

Balanced breakfast for focus

Water bottle and a light snack

Precision and Persistence

Formula Weapons Cheat Sheet

Regents Review Essentials

Formula Weapons Cheat Sheet

NYS Algebra 2 Regents: Ultimate Core Reference Guide (Page 1 of 3)

Must Memorize

Exam Prep Tool For Quick Study

The Vault: Formulas NOT Provided on the Regents Exam

The official NYS Reference Sheet is missing many critical equations. Use this cheat sheet to memorize these key relationships before sitting for the exam.

CONICS

Parabola Focus & Directrix

For a vertical parabola with vertex (h, k) and focal distance p from vertex to focus/directrix:

y = 1 4p (x - h)2 + k

Vertex: (h, k) Focus distance: |p|

FINANCE

Compound Interest (Periodic)

For an investment compounded n times per year at annual rate r over t years:

A(t) = P ( 1 + r n ) nt

Quarterly: n = 4 Monthly: n = 12

FINANCE

Continuous Compounding

For compound interest growing continuously, use the natural base e ("PERT" formula):

A(t) = P • ert

Natural base constant: e ≈ 2.71828

ALGEBRA

Log-Exponential Loop

Convert base b exponent equations directly into logarithmic form to solve for powers:

logb(y) = x  ⇔  bx = y

Remember: "The base of the log is the base of the exponent."

TRIGONOMETRY

Sine & Cosine Curve Mechanics

For any sinusoidal wave of the form y = A sin(B(x-C)) + D or y = A cos(B(x-C)) + D:

Amplitude Amp = |A|

Midline / Vertical y = D

Period (Radians) P = 2π |B|

Frequency (B) B = 2π Period

C represents: Phase Shift (Left/Right) Frequency B represents: Number of cycles in 2π

NYS Next Gen Regents Standards Match Core Formula Cheat Sheet: 1 of 3

Regents Review Essentials

Core Algebraic Structures

NYS Algebra 2 Regents: High-Yield Factoring, Roots, & Powers (Page 2 of 3)

Must Memorize

Exam Prep Tool For Quick Study

POLYNOMIALS

Remainder Theorem

When dividing a polynomial P(x) by (x - a), the remainder is always:

Remainder = P(a)

If P(a) = 0, then (x - a) is a factor.

FACTORING

Sum/Difference of Cubes

Use SOAP (Same, Opposite, Always Positive) ordering:

a3 + b3 = (a + b)(a2 - ab + b2)

a3 - b3 = (a - b)(a2 + ab + b2)

Sign Order: Same, Opposite, Always +

QUADRATICS

The Discriminant

Determines root types from quadratic ax2 + bx + c = 0:

Discriminant = b2 - 4ac

> 0: 2 Real • = 0: 1 Real • < 0: 2 Complex

COMPLEX

Powers of Imaginary Unit i

Pattern of 4. Divide exponents by 4 & use remainder:

i1 = i

i2 = -1

i3 = -i

i4 = 1

Standard Complex Form: a + bi

ALGEBRA

Logarithmic Properties & Laws

Product Rule log(xy) = log(x) + log(y)

Quotient Rule log(x/y) = log(x) - log(y)

Power Rule log(xy) = y • log(x)

Change of Base: loga(b) = log(b) / log(a) Natural Logs: ln(e) = 1 • ln(1) = 0

SWISS ARMY KNIFE

Functions, Rates, & Exponent Mechanics

Average Rate of Change f(b) - f(a) b - a

Rational Exponents xa/b = b√xa

Even Functions f(-x) = f(x) y-axis symmetry

Odd Functions f(-x) = -f(x) origin symmetry

Radian to Degree: Degrees = Radians • (180/π) Degree to Radian: Radians = Degrees • (π/180)

NYS Next Gen Regents Standards Match Core Formula Cheat Sheet: 2 of 3

Regents Review Essentials

Formula Decoding & Statistics

NYS Algebra 2 Regents: Reference Sheet & Data Mastery (Page 3 of 3)

Provided on Exam

Exam Prep Tool For Quick Study

SERIES

Geometric Series Formula

GIVEN

Sn = a1 - a1rn 1 - r

a1 1st Term

r Ratio

n # Terms

TRIG

Pythagorean Identity

GIVEN

cos2(θ) + sin2(θ) = 1

sin2(θ) = 1 - cos2(θ)

cos2(θ) = 1 - sin2(θ)

The Statistics Weapons & Calculator Arsenal

TI-84 normalcdf

Use to find percentages, proportions, or probability under a Normal Curve:

normalcdf(Lower, Upper, Mean, SD)

"Less Than" x Lower = -1E99, Upper = x

"Greater Than" x Lower = x, Upper = 1E99

Z-Score (Standardized Value)

Measures standard deviations away from the mean. NOT on Reference Sheet!

z = x - x̄ σ

TI-84 1-Var Stats Dictionary

Press STAT → CALC → 1:1-Var Stats:

x̄ Mean (Average)

Med Median (Middle)

σx Population SD

Sx Sample SD

n Sample Size

Mode Most Frequent

*Always use σx for standard deviation on the NYS Regents.

Margin of Error (95% Confidence)

Maximum expected difference between a sample statistic and the true population parameter:

ME ≈ 2 • Standard Deviation (σ)

95% Confidence Interval = Mean ± ME

NYS Next Gen Regents Standards Match Core Formula Cheat Sheet: 3 of 3

Regents Mastery Blueprint Answer Key

Regents Mastery Blueprint

Teacher Answer Key & Facilitation Guide • Algebra 2

Page 1 of 4

UNIT 1: ESSENTIAL REVIEW Algebraic Foundations

Q1: Simplify (x + 2)^2 - (x - 3) • Q2: Solve (1/2)(x - 4) = (1/3)x + 2

Q1 Correct Simplification:

x^2 + 3x + 7

Step-by-step: Expand (x + 2)^2 = x^2 + 4x + 4. Distribute negative: -(x - 3) = -x + 3. Combine like terms: x^2 + (4x - x) + (4 + 3) = x^2 + 3x + 7.

Q2 Correct Solve:

x = 24

Step-by-step: Multiply entire equation by 6 (LCD) to clear fractions: 3(x - 4) = 2x + 12 which simplifies to 3x - 12 = 2x + 12. Subtract 2x and add 12 to find x = 24.

Grading Tip: Watch for distribute negative errors -(x-3) to -x-3, which yields x^2+3x+1. Award 1 of 2 partial credit points.

UNIT 2: FUNCTIONS Concepts & Inverses

If f(x) = 2x - 5 and g(x) = x^2, find g(f(4)) and determine f^-1(x).

g(f(4)) Correct Evaluation:

9

Step-by-step: Evaluate inner function first: f(4) = 2(4) - 5 = 3. Substitute 3 into g(x): g(3) = (3)^2 = 9.

f^-1(x) Correct Inverse:

f^-1(x) = (x + 5) / 2 or 0.5x + 2.5

Step-by-step: Set to y: y = 2x - 5. Switch x and y: x = 2y - 5. Solve for y: x + 5 = 2y, which simplifies to y = (x + 5) / 2.

Calculator Check: Students can test equivalence by entering f(f^-1(x)) in [Y1]. The [2nd][GRAPH] (TABLE) should show y = x.

UNIT 3: LINEAR FUNCTIONS Piecewise Modeling

Taxi: $5 base plus $2/mi for first 10 mi, then $1/mi after. Write & graph piecewise.

Piecewise Equation:

C(x) = 2x + 5 (for 0 ≤ x ≤ 10) AND C(x) = x + 15 (for x > 10)

Derivation: For x ≤ 10, standard rate is 2x+5. At 10 miles, cost is 2(10)+5 = 25. For x > 10, cost is 25 + 1(x-10) which simplifies to x+15.

Key Graph Marks:
• Y-intercept at (0, 5)
• Node change at (10, 25) (continuous)
• Secondary slope of 1 up to (20, 35)

Common Trap: Students write 1x+5 for the second domain, failing to accumulate the first 10 miles' cost. This is a 2-point conceptual loss.

UNIT 4: EXPONENTIALS Modeling with Points

Determine the equation of the exponential function that passes through (0, 5) and (3, 40).

Correct Model Equation:

y = 5(2)^x

Step-by-step Solution:
1. Standard Exponential form is y = a(b)^x.
2. Substitute (0,5) to find a: 5 = a(b)^0, which simplifies to a = 5.
3. Substitute (3,40) to find b: 40 = 5(b)^3, which simplifies to 8 = b^3, so b = 2.
4. Write final equation with parameters: y = 5(2)^x.

TI-84 Stat Play: Students can enter coordinates into list L1 and L2. Then run [STAT] → [CALC] → [0: ExpReg] to instantly find a and b.

Algebra 2 Regents Mastery Suite Confidential Teacher Material

Regents Mastery Blueprint

Teacher Answer Key & Facilitation Guide • Algebra 2

Page 2 of 4

UNIT 5: LOGARITHMS Algebraic Equations

Solve algebraically to the nearest thousandth: 4(1.05)^(2t) = 20

Correct Answer:

t ≈ 16.493

Step-by-step Solution:
1. Divide by 4: (1.05)^(2t) = 5.
2. Apply natural logs on both sides: ln((1.05)^(2t)) = ln(5).
3. Move exponent using power law: 2t * ln(1.05) = ln(5).
4. Isolate t: t = ln(5) / (2 * ln(1.05)) ≈ 1.6094379 / 0.0975803 ≈ 16.4934.

Regents Alert: Rounding to the tenth or hundredth (e.g. 16.5 or 16.49) is a flat -1 point deduction. Emphasize tracking thousands place!

UNIT 6: SEQUENCES & SERIES Sigma Notation Sums

Evaluate: Sum from n=1 to 5 of (3n - 2)

Correct Evaluation Answer:

35

Step-by-step expansion: Evaluate term formula (3n-2) for all index steps n=1, 2, 3, 4, 5:
• n = 1: 3(1) - 2 = 1
• n = 2: 3(2) - 2 = 4
• n = 3: 3(3) - 2 = 7
• n = 4: 3(4) - 2 = 10
• n = 5: 3(5) - 2 = 13
Sum: 1 + 4 + 7 + 10 + 13 = 35.

TI-84 Check: Tell students to press [MATH] → [0: summation], input parameters to instantly double check their arithmetic sum!

UNIT 8: TRANSFORMATIONS Coordinate Mapping

If (4, -10) is on graph of y = f(x), what point must be on g(x) = f(x+2) + 1?

Correct Mapping coordinate:

(2, -9)

Transformation logic:
• f(x+2) shifts graph horizontal left 2 units, subtracting 2 from x: 4 - 2 = 2.
• + 1 shifts graph vertical up 1 unit, adding 1 to y: -10 + 1 = -9.

Grading Trap: Students shift right instead of left, writing (6, -9). This is a conceptual shift error (award 1pt of 2).

UNIT 7: PARABOLAS Focus & Directrix

Write the equation of the parabola with focus (3, 5) and directrix y = 1.

Correct Parabola Equation:

y = (1/8)(x - 3)^2 + 3

Step-by-step:
1. Vertex (h,k) is midpoint of focus (3,5) and directrix y=1, yielding h=3 and k=(5+1)/2=3. Vertex is (3,3).
2. Distance p from vertex to focus is 5 - 3 = 2.
3. Standard Formula: y = (1/(4p))(x-h)^2 + k, which gives y = (1/8)(x-3)^2 + 3.

High-Yield Trap: Students often mistake focus distance p as the distance from focus to directrix (4) instead of vertex to focus (2). Watch closely!

Algebra 2 Regents Mastery Suite Confidential Teacher Material

Regents Mastery Blueprint

Teacher Answer Key & Facilitation Guide • Algebra 2

Page 3 of 4

UNIT 9: COMPLEX NUMBERS Quadratic Formula

Solve for x and express roots in a+bi form: x^2 + 4x + 20 = 0

Correct roots (a+bi form):

x = -2 ± 4i

Step-by-step quadratic formula:
x = (-4 ± root(4^2 - 4(1)(20))) / (2(1))
x = (-4 ± root(16 - 80)) / 2 = (-4 ± root(-64)) / 2
Convert negative root: root(-64) = 8i, so x = (-4 ± 8i) / 2, which gives x = -2 ± 4i.

Calculator Tip: Press [MODE] → [a+bi]. This tells the system to evaluate negative square roots directly instead of throwing an error.

UNIT 10: POLYNOMIALS Long Division

Divide x^3 + 2x^2 - 5x + 6 by (x - 1).

Correct Rational Result:

x^2 + 3x - 2 + 4/(x - 1)

Synthetic Division setup: Root c = 1 on coefficients:
1 | 1 2 -5 6 → Quotient Row: 1, 3, -2, Remainder: 4 Resulting quotient is x^2 + 3x - 2 with remainder 4. Expressed rationally as: x^2 + 3x - 2 + 4/(x - 1).

Quick Check: Use Remainder Theorem: evaluate f(1) = 1^3 + 2(1)^2 - 5(1) + 6 = 4. Remainder must equal 4.

UNIT 11: TRIGONOMETRY Sine / Cosine Wave

Max height 50, min 10. Cycle repeats every 4 seconds. Write the model.

Correct Equations (Both Accepted):

1. y = 20 * sin((pi/2) * t) + 30 (midline start)

2. y = -20 * cos((pi/2) * t) + 30 (minimum start)

Parameters:
• Midline (D): (max + min) / 2 = (50 + 10) / 2 = 30.
• Amplitude (A): (max - min) / 2 = (50 - 10) / 2 = 20.
• Frequency (B): Period = 2*pi/B, so 4 = 2*pi/B, which yields B = pi/2.

Regents Trap: Students mistakenly write frequency B = 4 or B = 4*pi. Frequency is derived from the period length!

UNIT 12: PROBABILITY Conditional Rule

Survey: 40 like apples, 30 like oranges, 10 like both. Find P(Apple | Orange).

Correct Conditional Probability:

10/30 or 1/3 or 33.3%

Step-by-step: Apply formula P(A|B) = P(A and B) / P(B):
• Intersection (Apples and Oranges) = 10.
• Restrict denominator to condition (Oranges) = 30.
• Calculate: P(Apple|Orange) = 10/30 = 1/3.

Visual Helper: Draw a Venn diagram. Orange-only has 20. Apples-only has 30. Intersection has 10. Fraction is: 10 / (20 + 10) = 10/30.

Algebra 2 Regents Mastery Suite Confidential Teacher Material

Regents Mastery Blueprint

Teacher Answer Key & Facilitation Guide • Algebra 2

Page 4 of 4

UNIT 13: STATISTICS & SIMULATIONS Inferences & Significance

Q: A simulation of 100 trials shows a mean difference of 0.05 and a standard deviation of 0.02. If a result of 0.10 was observed, is it significant? Justify.

CORRECT CONCLUSION:

Yes, the result is statistically significant.

Required justification math:
1. Compute 95% Confidence Interval using standard bounds:
mean ± 2*(standard deviation) = 0.05 ± 2(0.02) = [0.01, 0.09].
2. Observe that 0.10 is greater than 0.09, lying outside the confidence interval boundaries.

Grading Rubric Breakdown:

• 2 Marks: Correct conclusion ("Yes") with interval calculations [0.01, 0.09] and clear comparison statement.
• 1 Mark: Correctly calculates interval but makes incorrect conclusion, OR writes "Yes" with vague explanation and no math.
• 0 Marks: "Yes" with zero mathematical backing.

Z-Score Alternative: Graders must accept z-score argument: Z = (0.10 - 0.05) / 0.02 = 2.5. Since 2.5 is greater than 2.0, the observation lies in the extreme outer tails, validating significance.

NYS Regents Grading Standards & Best Practices

Computational vs. Conceptual

Computational Error (-1 pt): An arithmetic or algebra slip (e.g., 2 * 3 = 5) but all subsequent steps are correct. Multiple slips in one problem still cap at -1.
Conceptual Error (Half Credit Max): Utilizing incorrect mathematical logic (e.g., using linear rules on exponential growth, or using midline as amplitude).

"Explain" vs. "Justify":
• Explain: Requires complete sentences describing math logic (e.g., "The function has a denominator of zero at x = 2").
• Justify: Accepts pure math calculations showing truth values; words are optional if equations are clear.

NYS Penalties & Pitfalls

Rounding (-1): Any rounding error on final numeric answers is automatic -1. Emphasize tracking of "nearest thousandth".

Variables (-1): Using standard x instead of the defined variable (e.g., t in Unit 5) results in a -1 penalty.

Calculator Explanations: Students writing "I put it in my calculator" receive 0pts. They must specify "From ExpReg..." or coordinates.

Mastery Suite Cross-References

Use the "Calculator Power Plays Cheat Sheet" to walk students through TI-84 validation processes for Units 1, 4, 5, 9, 10, and 13.

Formula Sheet Calculator Playbook

Algebra 2 Regents Mastery Suite Confidential Teacher Material

Next Gen Shockwave Handout

Next-Gen Shockwave Briefing

June 2026 Algebra 2 Regents • Transition & Grading Intel

Page 1 of 3

ATTENTION STUDENTS: June 2026 marks the first-ever solo administration of the Next Generation Algebra 2 Regents Exam. The old Common Core syllabus is completely retired. This briefing details the structural curriculum shifts, new problem formats, and strict new grading guidelines.

Core Curricular Shifts: Old vs. Next-Gen

Fading Out (Prior Exams)

• Arcane Derivations: Deriving trig double-angle identities and algebraic fraction rules.
• Manual Stats: Formulaic calculation of standard deviation and basic z-score sheets.
• Generic Growth Problems: Simple linear vs. exponential growth questions with minimal context.
• Rote factoring: Abstract algebraic manipulation without graphic validation.

No longer represents active high-yield exam categories.

Stepping In (June 2026 Next-Gen)

• Trig Modeling: High focus on real-world sinusoidal data modeling and unit circle parameters.
• Statistical Simulations: Dot plots representing computer-generated trials and margin of error.
• Financial Modeling: Geometric series formulas to model retirements, mortgages, and assets.
• Structure & Roots: Higher focus on Remainder Theorem and focus-directrix models.

Primary targets for June 2026 high-credit scores.

The 3 Rigid Next-Gen Grading Pitfalls (Avoid at All Costs)

Pitfall #1

Variable Infidelity

If a problem defines a variable like t (time) or p (population) and you write the final equation using standard x or y, it is a flat -1 point penalty.

Survival: Circle defined variables in the text before beginning.

Pitfall #2

Explain vs. Justify

Next-Gen rubrics strictly distinguish these: "Explain" requires complete, grammatically sound English sentences. "Justify" requires mathematical evidence (calculations, coordinates, or table values).

Survival: Doing the math is not enough for an explanation.

Pitfall #3

Precision Truncation

The Next-Gen standard defaults to higher decimal precision. Rounding to the nearest tenth or hundredth when a question specifies "nearest thousandth" is an immediate -1 point deduction.

Survival: Keep unrounded values in your calculator.

TI-84 Power Plays for the Next-Gen Exam

1. Statistical Simulations: Use mean ± 2σ to quickly establish the 95% confidence interval boundaries on trial simulations.

2. Fractional Exponents: Use [ALPHA] → [Y=] (n/d) to input complex fraction power values like xab = b√xa without losing algebraic accuracy.

3. Trigonometric Mode: ALWAYS set your TI-84 to RADIAN mode for sinusoidal wave graphing to prevent zero-credit wave models.

4. Sequence Mode: Press [MODE] → [SEQ]. This opens recursive inputs like an = an-1 + d for populate-table checks.

ALERT: Acknowledge and adapt. Target June 2026 Mastery.

NYS Next-Gen Standards

Next-Gen Question Spotlights

June 2026 Algebra 2 Regents • Real-World Mock Questions

Page 2 of 3

Spotlight #1: Statistical Simulation Analysis NYS Standard S-IC.B.5

Scenario: A researcher conducts a simulation of 100 trials to analyze the difference in mean recovery times between a treatment group and a control group. The simulation yields a mean difference of 0.35 days and a standard deviation of 0.18 days.

a) Calculate the 95% confidence interval for the mean difference of the simulation. Round your bounds to the nearest thousandth.

b) During a clinical trial, an observed mean difference of 0.75 days was recorded. Is this observed difference statistically significant? Justify your response.

Spotlight #2: Sinusoidal Real-World Modeling NYS Standard F-TF.B.5

Scenario: A passenger boards a Ferris wheel at its lowest point. The wheel has a maximum height of 55 meters, a minimum height of 5 meters, and completes one full rotation every 6 minutes.

a) If the passenger boards at t = 0, write a sinusoidal equation that models the passenger's height, H(t), in meters, after t minutes.

b) Explain how you determined the frequency (or B value) of your sinusoidal equation.

WARNING: Practice under realistic exam conditions. Target June 2026 Mastery.

NYS Next-Gen Standards

Next-Gen Question Spotlights

June 2026 Algebra 2 Regents • Real-World Mock Questions

Page 3 of 3

Spotlight #3: Financial Modeling (Geometric Series) NYS Standard A-SSE.B.4

Scenario: An educator deposits $200 at the end of every month into an investment annuity account paying 6% annual interest compounded monthly. The multiplier r is defined as 1 + 0.06/12 = 1.005.

a) Write a geometric series expression for the total amount of money in the annuity after 30 years (360 monthly deposits). Then, calculate the total accumulated value to the nearest dollar. Use formula: Sn = a1(1 - rn)1 - r

Spotlight #4: Polynomial Structure & Remainder Theorem NYS Standard A-APR.B.2

Scenario: A cubic polynomial function is defined as P(x) = 2x3 - kx2 + 5x - 6, where k is an unknown real constant.

a) If (x - 2) is a factor of polynomial P(x), determine the value of k. Show all algebraic work.

b) State the theorem you used to solve part (a) and write a single-sentence explanation of how it justifies your setup.

WARNING: Practice under realistic exam conditions. Target June 2026 Mastery.

NYS Next-Gen Standards

Next Gen Shockwave Answer Key

Next-Gen Shockwave Key

Teacher Answer Key & Facilitation Guide • Mock Exam Spotlights

Page 1 of 2

Teacher Facilitation Notes: Use this separate key to grade the student 3-page Next Gen Shockwave Handout. This key details the step-by-step math and highlights standard Next-Gen rubrics for grading.

Spotlight #1 Solution: Statistical Simulations S-IC.B.5 • 2 Points

a) Calculate 95% Confidence Interval • Answer:

[-0.010, 0.710]

Steps: Given mean difference = 0.35 and standard deviation = 0.18.
95% Interval is defined as: mean ± 2*(standard deviation)
Lower Bound = 0.35 - 2(0.18) = 0.35 - 0.36 = -0.010
Upper Bound = 0.35 + 2(0.18) = 0.35 + 0.36 = 0.710
Interval: [-0.010, 0.710] (thousandths place required: three decimal places).

b) Is observed difference significant? • Answer:

Yes, the difference is statistically significant.

Justification Statement: Yes, the observed mean difference of 0.75 days is statistically significant because it lies outside of the 95% confidence interval boundaries of [-0.010, 0.710]. This implies that a difference of 0.75 days has less than a 5% probability of occurring due to random sampling chance alone.

NYS Grading Standard: Award 1 point for the correct confidence interval with work. Award 1 point for the correct "Yes" conclusion with a mathematically valid comparison (e.g., stating 0.75 is greater than upper bound 0.710). No credit for pure "Yes" without explanation.

Spotlight #2 Solution: Sinusoidal Modeling F-TF.B.5 • 4 Points

a) Sinusoidal Model Equation • Answer:

H(t) = -25 cos((π/3) * t) + 30

Steps to determine parameters:
1. Midline (D): (max + min)/2 = (55 + 5)/2 = 30.
2. Amplitude (A): (max - min)/2 = (55 - 5)/2 = 25.
3. Frequency (B): Period is 6 minutes. Period = 2π/B → 6 = 2π/B → B = 2π/6 = π/3.
4. Trig Function choice: Passenger boards at lowest point at t=0, requiring a negative cosine wave:
H(t) = -25 cos((π/3) * t) + 30.

b) Frequency Explanation • Answer:

"The frequency is determined by dividing the standard 2π period of a cosine wave by the given 6-minute period of the Ferris wheel, which reduces to π/3."

Required for Explain credit: Students must write a full verbal description of the division of 2π by the period length 6.

NYS Grading Standard: Award 2 points for the correct equation. Deduct 1 point for variable infidelity if students write standard "y" and "x" instead of H(t) and t. Award 2 points for part (b) explanation (1 for mathematical formula, 1 for full sentence).

Algebra 2 Regents Mastery Suite Confidential Teacher Material

Next-Gen Shockwave Key

Teacher Answer Key & Facilitation Guide • Mock Exam Spotlights

Page 2 of 2

Spotlight #3 Solution: Financial Modeling A-SSE.B.4 • 2 Points

a) Geometric Expression & Valuation • Answers:

Expression: S360 = 200(1 - 1.005360)1 - 1.005

Final Calculated Value: $200,903

Detailed Math Steps:
1. Compounding: 6% annual / 12 months = 0.5% monthly interest → r = 1 + 0.005 = 1.005.
2. Total months n = 30 years * 12 months = 360 months.
3. Plug terms into geometric sum formula: S360 = 200 * (1 - 1.005360) / (1 - 1.005).
4. Evaluate: 1.005360 ≈ 6.022575 → 1 - 6.022575 = -5.022575.
5. Division: -5.022575 / -0.005 = 1004.515 → S360 = 200 * 1004.515 = $200,903 (to nearest dollar).

NYS Grading Standard: Award 1 point for writing a mathematically valid geometric series sum expression. Award 1 point for evaluating the expression correctly to $200,903. Deduct 1 point for rounding errors (e.g. failing to round to the nearest whole dollar).

Spotlight #4 Solution: Polynomial Structure A-APR.B.2 • 4 Points

a) Algebraic value of k • Answer:

k = 5

Steps to solve:
Since (x - 2) is a factor, setting x = 2 as a root must yield P(2) = 0.
P(2) = 2(2)3 - k(2)2 + 5(2) - 6 = 0
2(8) - 4k + 10 - 6 = 0
16 - 4k + 4 = 0 → 20 - 4k = 0 → 4k = 20 → k = 5.

b) Theorem & Justification • Answer:

Theorem: Factor Theorem (or Remainder Theorem).

Justification: "The Factor Theorem justifies setting P(2) equal to zero because it states that a linear expression (x - c) is a factor of a polynomial if and only if evaluating the polynomial at x = c yields a remainder of exactly zero."

NYS Grading Standard: Award 2 points for part (a) showing correct substitution and algebraic flow to k = 5. Award 2 points for part (b) naming the Factor Theorem and explaining its mathematical relationship to factor zero remainders.

Algebra 2 Regents Mastery Suite Confidential Teacher Material

Next Gen Mock Exam

The University of the State of New York

REGENTS HIGH SCHOOL EXAMINATION

NEXT GENERATION ALGEBRA II

Mock Practice Examination for Classroom Assessment

Student Name:

Teacher / Period:

School Name:

DO NOT OPEN THIS EXAMINATION BOOKLET UNTIL THE SIGNAL IS GIVEN.

This exam consists of four parts, with a total of 9 representative questions. You must answer all questions in this examination. Record your answers to the Part I multiple-choice questions on this page or a separate answer sheet. Write your answers to the Part II, III, and IV questions directly in this booklet. All work must be written in ink, except for graphs and drawings, which should be done in pencil.

A graphing calculator (TI-84 plus recommended) and a straightedge must be available for your use during this exam.

PART I: Multiple Choice Answer all questions in this part • 2 Credits Each

Directions: For each statement or question, choose the word or expression that, of those given, best completes the statement or answers the question. Record your choice in the space provided.

1

Which expression is equivalent to \(\left(\frac{x^8}{y^{-4}}\right)^{1/4}\) for all positive values of \(x\) and \(y\)?

(1) \(x^2 y\)

(2) \(\frac{x^2}{y}\)

(3) \(x^2 \sqrt[4]{y}\)

(4) \(x^2 y^4\)

Answer Choice: ______

2

A Ferris wheel rotates such that a passenger's height, \(H(t)\), in feet above the ground, can be modeled by the equation \(H(t) = -50\cos\left(\frac{\pi}{5}t\right) + 60\), where \(t\) represents the time in minutes. What is the minimum height, in feet, that a passenger reaches on the Ferris wheel?

(1) 10

(2) 50

(3) 60

(4) 110

Answer Choice: ______

NYS Algebra II Mock Exam — Page 1 [OVER]

3

A researcher conducts a randomized study comparing a new fertilizer to a standard treatment. The treatment group's mean tomato yield exceeded the control group's yield by 2.5 pounds. A statistical simulation of 100 trials, under the hypothesis of random assignment, resulted in a mean difference of 2.5 or greater in exactly 4 of the trials. Based on this simulation, what is the correct conclusion?

(1) The result is statistically significant because a difference of 2.5 or greater occurred less than 5% of the time.

(2) The result is not statistically significant because the simulation proved that random assignment is likely to blame.

(3) The new fertilizer is guaranteed to produce exactly 2.5 more pounds of tomatoes in all applications.

(4) An increase of 2.5 pounds has a 4% probability of occurring on any individual plant.

Answer Choice: ______

4

A student starts a savings account by depositing $50 in the first week. Each subsequent week, they increase their deposit amount by 5% over the previous week. Which expression represents the total amount deposited, in dollars, after 20 weeks?

(1) \(\sum_{n=1}^{20} 50(1.05)^n\)

(2) \(\sum_{n=1}^{20} 50(1.05)^{n-1}\)

(3) \(\sum_{n=1}^{20} 50(0.05)^{n-1}\)

(4) \(\sum_{n=1}^{20} 50(1.5)^{n-1}\)

Answer Choice: ______

PART II: Short Answer Answer all questions in this part • 2 Credits Each

Directions: Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. All answers should be written in pen, except for graphs and drawings, which should be done in pencil.

5

If \(x - 2\) is a factor of \(P(x) = 2x^3 - 3x^2 - x + k\), determine the value of \(k\). Justify your answer using algebraic work.

Student Work Area

6

A bacterial culture starts with 500 cells and grows continuously at an hourly rate of 12%. Write a continuous growth function, \(A(t)\), to represent the bacteria population after \(t\) hours. Then, determine the population to the nearest whole number after 8 hours.

Student Work Area

NYS Algebra II Mock Exam — Page 2 [OVER]

PART III: Medium Response Answer all questions in this part • 4 Credits Each

Directions: Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. All answers should be written in pen, except for graphs and drawings, which should be done in pencil.

7

Solve the following equation algebraically for all values of \(x\). State any extraneous solutions that arise during your solving process. \(\frac{2}{x-3} + \frac{x}{x+3} = \frac{18}{x^2-9}\)

Student Work Area

8

A survey of 150 high school students was conducted to analyze course preferences. The results comparing a student's grade level and their advanced math course are shown in the table below:

a) Find the probability that a student takes Calculus, given that they are in the 12th grade.

Work Box

b) Are the events "being in 12th grade" and "taking Calculus" independent? Justify your answer.

Work Box

NYS Algebra II Mock Exam — Page 3 [OVER]

PART IV: Comprehensive 6-Credit Response Answer all questions in this part • 6 Credits Total

Directions: Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For this comprehensive question, a correct numerical answer with no work shown will receive only 1 credit. All answers should be written in pen, except for graphs and drawings, which should be done in pencil.

9

The depth of water, in feet, at a local boat dock is monitored over a 12-hour period starting at midnight (\(t = 0\)). At midnight, the depth is at its maximum of 12 feet. Exactly 6 hours later, at 6:00 AM, the water level reaches its low tide minimum of 4 feet. The depth of the water can be modeled by a sinusoidal function of the form \(D(t) = a\cos(bt) + d\).

a) Determine values for \(a\), \(b\), and \(d\). Write the complete trigonometric equation for \(D(t)\).

Work Box

b) A container ship requires a minimum water depth of exactly 6 feet to dock safely. Algebraically determine, to the nearest tenth of an hour, the interval of hours \(t\) between \(t = 0\) and \(t = 12\) during which the boat can safely dock.

Work Box

c) Graph the function \(y = D(t)\) on the axes below over the interval \(0 \le t \le 12\).

Be sure to label your axes, amplitude peaks, and low tide values.

0 2 4 6 8 10 12 14 Water Depth (feet) 0 2 4 6 8 10 12 Hours after Midnight (t)

NYS Algebra II Mock Exam — Page 4 END OF EXAMINATION

Next Gen Mock Exam Answer Key

FOR TEACHERS ONLY

The University of the State of New York

REGENTS HIGH SCHOOL EXAMINATION

NEXT GENERATION ALGEBRA II

OFFICIAL RATING GUIDE & ANSWER KEY

SCORING MC & CONSTRUCTED RESPONSES:

• Part I (Multiple Choice): Award exactly 2 credits for each correct answer. Do not award partial credit.

• Parts II, III, and IV (Constructed): Follow the specific rubrics provided. Full credit requires complete algebraic justification and correct notations. State model-solution pathways and highlight common point-loss traps below.

PART I: Multiple Choice Answer Key 2 Credits Each • No Partial Credit

1

Which expression is equivalent to \(\left(\frac{x^8}{y^{-4}}\right)^{1/4}\)?

Correct Answer Choice: (1) \(x^2 y\) CCSS.Math.Content.HSN.RN.A.2

Detailed Teacher Walkthrough:

Apply the power of a quotient rule: \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\)
\(\left(\frac{x^8}{y^{-4}}\right)^{1/4} = \frac{(x^8)^{1/4}}{(y^{-4})^{1/4}}\)
Multiply exponents: \(8 \times \frac{1}{4} = 2\) for \(x\), and \(-4 \times \frac{1}{4} = -1\) for \(y\).
This yields: \(\frac{x^2}{y^{-1}}\). Since \(\frac{1}{y^{-1}} = y\), the simplified term is \(x^2 y\).

2

Ferris wheel minimum height modeled by \(H(t) = -50\cos\left(\frac{\pi}{5}t\right) + 60\)...

Correct Answer Choice: (1) \(10\) CCSS.Math.Content.HSF.TF.B.5

Detailed Teacher Walkthrough:

The midline (vertical shift) is \(d = 60\) feet.
The amplitude is \(|a| = |-50| = 50\) feet.
The minimum value of a negative cosine function occurs at its starting point (\(t = 0\)) or at the bottom of the wave:
\(\text{Minimum} = d - |a| = 60 - 50 = 10\) feet.
Verification: \(H(0) = -50\cos(0) + 60 = -50(1) + 60 = 10\).

NYS Algebra II Mock Exam Key — Page 1 [OVER]

3

Statistical simulation significance conclusion...

Correct Choice: (1) statistically significant because probability < 5%

Justification: The simulated probability (p-value) of getting a mean difference of \(2.5\) or greater due to random assignment is \(4/100 = 4\%\). In NYS standards, a threshold of \(5\%\) (\(\alpha = 0.05\)) is standard for statistical significance. Since \(4\% < 5\%\), we reject the null hypothesis of random assignment and conclude the fertilizer had a statistically significant effect.

4

Sigma notation of savings deposit...

Correct Choice: (2) \(\sum_{n=1}^{20} 50(1.05)^{n-1}\)

Justification: Week 1 deposit is \(a_1 = \$50\). Week 2 is \(50(1.05)^1\). The common ratio is \(r = 1.05\). The explicit formula for the \(n\)-th week is \(a_n = 50(1.05)^{n-1}\). The sum over \(20\) weeks is represented by the summation of \(a_n\) from \(n = 1\) to \(20\), which is \(\sum_{n=1}^{20} 50(1.05)^{n-1}\). Choice (1) starts week 1 at \(50(1.05)^1 = 52.50\), which is incorrect.

PART II: Short Answer Key 2 Credits Each • Show All Work

Q5: Find \(k\) such that \(x-2\) is a factor of \(P(x) = 2x^3 - 3x^2 - x + k\).

MODEL SOLUTION

Teacher Simulation of Perfect Student Response: According to the Remainder Theorem, if \((x - 2)\) is a factor, then \(P(2) = 0\).
Substitute \(x = 2\) into the polynomial:
\(P(2) = 2(2)^3 - 3(2)^2 - (2) + k = 0\)
\(2(8) - 3(4) - 2 + k = 0\)
\(16 - 12 - 2 + k = 0\)
\(2 + k = 0 \implies \mathbf{k = -2}\)

[2 Credits] Correct answer \(k = -2\) with full justification.

[1 Credit] Arithmetic error with correct conceptual setup.

Q6: Bacterial continuous growth \(A(t) = P e^{rt}\) for \(500\) cells, \(12\%\) growth. Find after 8 hrs.

MODEL SOLUTION

Teacher Simulation of Perfect Student Response: Continuous growth equation: \(A(t) = P e^{rt}\)
Where \(P = 500\), \(r = 0.12\) (growth rate). Equation: \(A(t) = 500 e^{0.12t}\)
To find the population at \(t = 8\):
\(A(8) = 500 e^{0.12(8)} = 500 e^{0.96}\)
Using calculator: \(e^{0.96} \approx 2.611696\)
\(A(8) = 500 \times 2.611696 \approx 1305.85 \implies \mathbf{1306}\) bacteria.

[2 Credits] Both correct formula & rounded \(1306\) cells.

[1 Credit] Formula correct, but failed to calculate or round.

NYS Algebra II Mock Exam Key — Page 2 [OVER]

PART III: Medium Response Key 4 Credits Each • Comprehensive Walkthroughs

Q7: Solve algebraically: \(\frac{2}{x-3} + \frac{x}{x+3} = \frac{18}{x^2-9}\)

MODEL SOLUTION

Multiply entire equation by the Lowest Common Denominator (LCD): \((x-3)(x+3) = x^2-9\). Note restrictions: \(x \neq 3, x \neq -3\).
\(2(x+3) + x(x-3) = 18\)
\(2x + 6 + x^2 - 3x = 18\)
\(x^2 - x + 6 = 18 \implies x^2 - x - 12 = 0\)
Factor the quadratic: \((x - 4)(x + 3) = 0\)
Set each factor to zero: \(x - 4 = 0 \implies \mathbf{x = 4}\) and \(x + 3 = 0 \implies \mathbf{x = -3}\).
⚠️ Extraneous Solution Check: If \(x = -3\), the term \(\frac{x}{x+3}\) has a zero denominator. Thus, \(\mathbf{x = -3}\) is an extraneous solution.
The only valid solution is \(x = 4\).

[4 Credits] Correct value \(x = 4\) with full work, including checking and explicitly rejecting \(x = -3\).

[3 Credits] Solves correctly but does not reject \(x = -3\) (answers \(x = 4\) and \(x = -3\)).

Q8: Course preference statistics table conditional probability & independence.

MODEL SOLUTION

a) Conditional Probability: We need \(P(\text{Calculus} | \text{12th Grade})\).
Look at the "12th Grade" row only:
Total 12th graders = \(90\).
12th graders in Calculus = \(55\).
\(P(\text{Calc} | \text{12th}) = \frac{55}{90} = \frac{11}{18} \approx \mathbf{0.611}\) (or \(61.1\%\)).

b) Independence Justification: Let's compare \(P(\text{Calculus} | \text{12th})\) and \(P(\text{Calculus})\).
Overall probability of taking Calculus:
\(P(\text{Calculus}) = \frac{70}{150} \approx \mathbf{0.467}\) (or \(46.7\%\)).
Since \(P(\text{Calculus} | \text{12th}) \neq P(\text{Calculus})\)
(\(0.611 \neq 0.467\)), the events are not independent.

[4 Credits] Correct probability \(0.611\) and "not independent" with correct mathematical justification comparing probabilities.

[2 Credits] Correct part (a) but incorrect or missing math argument for part (b).

NYS Algebra II Mock Exam Key — Page 3 [OVER]

PART IV: Comprehensive 6-Credit Response Key 6 Credits • Full Algebraic Solving & Graphing Solutions

a) Determine \(a\), \(b\), \(d\) and write equation:

• Midline (d): \(\frac{\text{Max} + \text{Min}}{2} = \frac{12 + 4}{2} = \mathbf{8}\)
• Amplitude (a): \(\frac{\text{Max} - \text{Min}}{2} = \frac{12 - 4}{2} = \mathbf{4}\)
• Period: High-to-low is 6 hours, so complete wave cycle is \(12\) hours.
\(b = \frac{2\pi}{\text{Period}} = \frac{2\pi}{12} = \mathbf{\frac{\pi}{6}}\)
Since depth starts at a maximum (12 feet at \(t=0\)), use positive cosine:
\(D(t) = 4\cos\left(\frac{\pi}{6}t\right) + 8\)

b) Algebraically find interval where depth \(\ge 6\) feet:

Set \(D(t) \ge 6\):
\(4\cos\left(\frac{\pi}{6}t\right) + 8 = 6\)
\(4\cos\left(\frac{\pi}{6}t\right) = -2 \implies \cos\left(\frac{\pi}{6}t\right) = -0.5\)
The reference angle is \(\theta = \arccos(0.5) = \frac{\pi}{3}\).
Since cosine is negative, solutions are in QII and QIII:
\(\frac{\pi}{6}t = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \implies t = \frac{2\pi}{3} \times \frac{6}{\pi} = \mathbf{4.0}\) hours.
\(\frac{\pi}{6}t = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \implies t = \frac{4\pi}{3} \times \frac{6}{\pi} = \mathbf{8.0}\) hours.
Depth is \(\ge 6\) feet at the start of the cycle up to 4 hours, and after 8 hours to 12 hours.
Intervals: \([0, 4.0]\) and \([8.0, 12.0]\) hours

c) Ideal Graph Model Solution:

A perfect cosine curve with critical coordinates clearly marked:
• Maxima: \((0,12)\) and \((12,12)\)
• Minimum: \((6,4)\)
• Midline Intersections: \((3,8)\) and \((9,8)\)

(0,12) (6,4) (12,12) 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12

CONSTRUCTED PART IV RUBRIC CRITERIA [6 Credits Total]

• [6 Credits] Correct equation, fully correct algebraic interval calculation with check, and perfect graph with critical coordinates labeled.

• [4-5 Credits] Conceptual correct pathways but with 1-2 arithmetic mistakes, or correct interval and equation but minor curve drawing defects.

NYS Algebra II Mock Exam Key — Page 4 OFFICIAL SCORING GUIDE END

Algebra 1 Mastery Blueprint Study Guide

Regents Mastery Blueprint

Algebra 1 (Next Generation) | Essential Review Guide

Name: ___________________________

The Algebra 1 Playbook

1. Solving for X

The Golden Rule

Whatever you do to one side, you MUST do to the other. Use inverse operations to isolate the variable.

• Distribute: \(a(b+c) = ab + ac\)
• Combine Like Terms: terms with same variable and exponent.
• Literal Equations: Solving for a variable (e.g., Solve \(V = lwh\) for \(h\)).

The Inequality Flip

When you multiply or divide both sides of an inequality by a negative number, you MUST flip the sign! (\(<\) becomes \(>\)).

Practice: Multistep Solving

Solve for \(x\): \(3(x - 4) + 2x = 8x - 18\)

Systems of Equations

Methods:

• Substitution: Use when one variable is already isolated (e.g., \(y = 2x\)).
• Elimination: Add or subtract equations to "cancel" a variable.
• Graphing: The solution is the point of intersection \((x, y)\).

Practice: Elimination

2x + y = 10
3x - y = 5

Regents Mastery Blueprint

Unit 2: Functions

2. The Function Concept

Function Rules: Each input \(x\) has exactly one output \(y\). Passes the Vertical Line Test.

Domain & Range: Domain is the set of inputs (\(x\)); Range is the set of outputs (\(y\)).

Function Notation: \(f(x)\) is the output for a given input \(x\). If \(f(x) = 2x + 1\), then \(f(3) = 2(3) + 1 = 7\).

Rate of Change: For a function, ROC is the slope: \(\frac{\Delta y}{\Delta x}\).

Calculator: The Table

Type any function into [Y=]. Press [2nd] [GRAPH] to see the table. This is the fastest way to check Domain/Range or evaluate \(f(x)\)!

Key Features

• - Zeros/Roots: where graph crosses \(x\)-axis (\(y=0\)).
• - Y-intercept: where graph crosses \(y\)-axis (\(x=0\)).
• - Relative Max/Min: peaks and valleys.

Practice: Evaluating Functions

Given \(g(x) = x^2 - 4\). Find \(g(-3)\) and determine the range if the domain is \(\{ -2, 0, 2 \}\).

Regents Mastery Blueprint

Unit 3: Linear Functions

3. Linear Change & Modeling

Equation: \(y = mx + b\)

Slope (\(m\)): \(\frac{y_2 - y_1}{x_2 - x_1}\). Represents constant rate of change.

Y-intercept (\(b\)): The "start" or "initial" value of a scenario.

Arithmetic Sequences: Linear patterns. Add/Subtract same amount each time.

Practice: Writing Linear Models

A fitness club charges a $50 joining fee and $30 per month. Write a linear equation \(C(m)\) to model the total cost for \(m\) months. How much will it cost for 1 year?

Pro Tip: If they give you two points, find the slope first, then plug one point back into \(y = mx + b\) to solve for \(b\)!

Regents Mastery Blueprint

Unit 4: Polynomials & Factoring

4. Working with Polynomials

Operations: Add/Subtract (Combine Like Terms). Multiply (Distribute/Box Method).

Standard Form: exponents in decreasing order (e.g., \(x^2 + 5x + 6\)).

Degree: The highest exponent in the polynomial.

Leading Coefficient: The number in front of the highest power.

The Factoring Tree

Step 1: GCF

Always check for Greatest Common Factor first.

Step 2: DOTS

Difference of Two Squares: \(a^2 - b^2 = (a+b)(a-b)\).

Step 3: Trinomials

Find numbers that multiply to \(c\) and add to \(b\).

Practice: Factor Completely

Factor completely: \(2x^2 - 18\)

Regents Mastery Blueprint

Unit 5: Quadratics

5. Solving & Graphing Quadratics

Standard Form: \(y = ax^2 + bx + c\)

Vertex Form: \(y = a(x-h)^2 + k\). Vertex is \((h, k)\).

Quadratic Formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Axis of Symmetry: \(x = -b / 2a\)

Practice: Quadratic Solving

Solve algebraically using any method: \(x^2 + 6x - 16 = 0\)

Calculator: Finding Roots

[Y=]. Type equation. [GRAPH].
[2nd] [TRACE] → [2: zero]. Select left and right bounds to find intercepts.

Discriminant (\(b^2 - 4ac\))

Pos: 2 real roots. Zero: 1 real root. Neg: No real roots.

Regents Mastery Blueprint

Unit 6: Exponential Functions

6. Growth & Decay

Model: \(y = a(b)^x\). \(a\) is initial, \(b\) is multiplier.

Growth: \(b = 1 + r\). Decay: \(b = 1 - r\).

Geometric Sequences: Exponential patterns. Multiplied by same amount (\(r\)) each time.

Practice: Exponential Modeling

A population of 500 rabbits triples every year. Write an equation \(P(t)\) to model the population after \(t\) years. How many rabbits will there be in 4 years?

Regents Mastery Blueprint

Unit 7: Statistics

7. Data & Analytics

Central Tendency: Mean (average), Median (middle), Mode (most common).

Spread: Range (\(\max-\min\)), Interquartile Range (\(Q3 - Q1\)), Standard Deviation.

Outliers: Data points far from the rest. Affect mean more than median.

Correlation vs Causation: Just because two things change together doesn't mean one causes the other!

Visualizing Data

• Box Plot: Shows 5-number summary (Min, Q1, Med, Q3, Max).
• Histogram: Frequency of data in ranges (intervals).
• Dot Plot: Dots representing each data point.

Sketch a box plot here

Calculator: 1-Var Stats

[STAT] → [Edit]. Enter data in L1.
[STAT] → [CALC] → [1: 1-Var Stats].
\(\bar{x}\) = Mean | \(\sigma x\) = Std Dev | Med = Median.

Practice: Correlation

Describe a real-world scenario with a strong negative correlation.

Regents Game Plan

Algebra 1 Strategy

1. Mastering the Format

• Multiple Choice: 24 questions (48 points). Use "Plug and Chug" strategy! Check your answers by plugging them back into the equation.
• Free Response: 13 questions (38 points). SHOW ALL WORK. Graders give partial credit for correct steps even if the final answer is wrong.

2. Never Give Points Away

Rounding Rule: Check for "tenth," "hundredth," or "integer." Rounding to the tenth when it asks for the hundredth is -1 point automatically.

The Inequality Rule: Did you flip the sign when dividing by a negative? Check this every time!

State vs Explain: "State" means just the answer. "Explain" means write a complete sentence using math words.

Fatal Mistakes

• Standard Form: Always write your final quadratic and polynomial answers in standard form (highest to lowest exponent).
• Units: If the question is about "money" or "feet," include the symbol/label.
• Graphing: Use a ruler! For parabolas, make sure you show the vertex and at least 5 clear points.

Last Minute Checks

• DiagnosticOn: Make sure you run this if doing any regression.
• Zoom 6: The standard graphing window. Use this to reset your view if your graph is "missing."
• Verify Zeroes: Plug your calculated zeroes back into the original equation to ensure they result in 0.

Algebra 1 Mastery Blueprint | Complete

Algebra 1 Calculator Power Plays Cheat Sheet

Calculator Power Plays

Algebra 1 Regents Reference | Next Generation

1. Templates & Reset

Reset RAM (Pre-Exam):
[2nd] [MEM] → [7] → [1] → [2].

Pretty Fractions:
[ALPHA] [Y=] → [1].

Decimal to Frac:
[MATH] → [1].

2. Graphing Essentials

Find Zeros:
[2nd] [TRACE] → [2]. Select bounds.

Find Intersect:
[2nd] [TRACE] → [5]. Hit [ENTER] 3x.

Find Vertex (Max/Min):
[2nd] [TRACE] → [3:min] or [4:max].

Reset View: [ZOOM] → [6].

3. Regressions (Fit)

Enter Lists: [STAT] → [1]. (L1, L2).

Linear (ax+b): [STAT] → [CALC] → [4].

Exponential: [STAT] → [CALC] → [0:ExpReg].

Missing "r"? [2nd] [0] → scroll to DiagnosticOn → [ENTER] [ENTER].

4. 1-Var Stats

Results: [STAT] → [CALC] → [1].

• - \(\bar{x}\): Mean
• - \(\sigma x\): Standard Deviation
• - Med: Median
• - \(Q_1\)/\(Q_3\): Quartiles

5. Equation Solver

Numeric Solver: [MATH] → [B:Solver]. Type eq as \(0 = \text{expression}\) to find \(x\). Best for checking complex equations!

6. MC Testing Trick

The "TEST" Menu: Use this to check if a choice is correct without solving.

1. Store value for X: 5 → [STO] → [X].
2. Type equation: \(2x+5\)
3. Select symbol: [2nd] [MATH] → [1: =].
4. Finish eq: \(15\). Hit [ENTER].

1 = TRUE | 0 = FALSE

The table [2nd][GRAPH] is your best friend for factoring. If your factored expression doesn't have the same Y-values as the original, you made a mistake!

Algebra 1 Regents Exam Strategy Tips

Algebra 1 Game Plan

Strategic Success for the Next Generation Regents Exam

Format

NYS Regents

1. Points on the Board

Part I

24

Multiple Choice
(48 pts Total)

Part II

8

2-Pt Short Answer
(16 pts Total)

Part III

4

4-Pt Multi-Step
(16 pts Total)

Part IV

1

6-Pt Deep Dive
(6 pts Total)

2. Time Management

0-75 mins:

Part I (MC)

Aim for 3 minutes per question. Use the calculator to verify every bubble.

75-120 mins:

Parts II & III

Tackle your strongest topics first. Show all work to secure partial credit.

120-150 mins:

Part IV (Big One)

Usually 3 parts (a, b, c). Even if you can't do part 'a', try to finish 'b' and 'c'.

Final 30:

CHECK ROUNDING!

The Partial Credit Secret

Algebra 1 graders award points for correct mathematical procedures even if you made a simple calculation mistake earlier in the problem.

Example: If you get the wrong linear equation in part 'a' but you graph THAT wrong equation perfectly in part 'b', you can still earn full points for part 'b'!

Regents Survival

Avoid Common Trap Doors

The "Point Killers"

Standard Form Check

When a question asks for standard form, exponents must decrease: \(x^2 + 5x - 6\).

The Negative Flip

If you divide an inequality by a negative, FLIP the sign! Failure to flip is a common point loss.

Rounding Precision

Check for tenth, hundredth, or integer. -1 point if you rounding to the wrong place value.

High-Yield Habits

The 5-Point Rule

For any parabola graph, plot at least 5 points: the vertex plus two points on each side.

Explain with Logic

If it says "Explain," write a full sentence. Using only numbers is often zero credit for that part.

Plug and Check

For systems and equations, plug your answers back into the original. If the equation isn't true, fix it!

Regents Morning Prep

Calculator Fully Charged

Pencils and a Ruler

Balanced Meal for Focus

Check Exam Room & Time

Geometry Mastery Blueprint Study Guide

Regents Mastery Blueprint

Geometry (NYS) | The Ultimate Review Guide

Name: ___________________________

Spatial Sense & Logic

1. Rigid Motions & Transformations

The "Rigid" Trio

Rigid motions preserve distance (length) and angle measure. The preimage and image are congruent.

• Reflection: Flip over a line. \(r_{x-axis}(x,y) = (x, -y)\).
• Rotation: Turn around a point. Counter-clockwise is positive!
• Translation: Slide. \(T_{a,b}(x,y) = (x+a, y+b)\).

Dilations (Non-Rigid)

Preserve angle measure but NOT distance. Preimage and image are similar.

Scale factor \(k\): \((x,y) \rightarrow (kx, ky)\).
Center of dilation is usually origin \((0,0)\).

Practice: Point Mapping

Reflect \((3, -4)\) over the line \(y = x\), then rotate the result \(90^\circ\) CCW about the origin.

Coordinate Tools

Slope (\(m\))

\(\frac{y_2 - y_1}{x_2 - x_1}\)

Parallel: same slope. Perpendicular: neg reciprocal.

Midpoint (\(M\))

\((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\)

The average of the coordinates.

Distance (\(d\))

\(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)

Pythagorean theorem in disguise!

Regents Mastery Blueprint

Unit 2: Congruence & Similarity

Triangle Congruence (\(\cong\))

Valid Shortcuts:

SSS, SAS, ASA, AAS, HL

NEVER Use:

AAA or SSA (Ass!)

CPCTC

"Corresponding Parts of Congruent Triangles are Congruent."
Use this AFTER you prove triangles are \(\cong\).

Similarity (\(\sim\))

Triangles are similar if their angles are \(\cong\) and sides are proportional.

• - AA\(\sim\): Two \(\cong\) angles.
• - SAS\(\sim\): 2 prop. sides, 1 \(\cong\) inc. angle.
• - SSS\(\sim\): 3 proportional sides.

Right Triangle Similarity (Altitudes)

When an altitude is drawn to the hypotenuse of a right triangle, it creates 3 similar triangles.

Leg Rule: \(\frac{\text{hyp}}{\text{leg}} = \frac{\text{leg}}{\text{proj}}\)

Altitude Rule: \(\frac{\text{seg 1}}{\text{alt}} = \frac{\text{alt}}{\text{seg 2}}\)

Sketch altitude diagram here

Regents Mastery Blueprint

Unit 3: Trig & Right Triangles

SOH CAH TOA

SINE

\(\sin \theta = \frac{\text{Opp}}{\text{Hyp}}\)

COSINE

\(\cos \theta = \frac{\text{Adj}}{\text{Hyp}}\)

TANGENT

\(\tan \theta = \frac{\text{Opp}}{\text{Adj}}\)

Cofunctions (High Yield)

In a right triangle, the sine of one acute angle equals the cosine of the other acute angle.

\(\sin A = \cos B \iff A + B = 90^\circ\)

Calculator: Trig Mode

Press [MODE]. Make sure DEGREE is highlighted! If you use Radians for Geometry trig, you will get every answer wrong.

Practice: Finding Angles

A 20-ft ladder leans against a wall, reaching a height of 16 ft. Find the angle of elevation to the nearest degree.

Regents Mastery Blueprint

Unit 4: Circles

Circle Properties

Central Angle: Angle \(=\) Arc.

Inscribed Angle: Angle \(=\) \(\frac{1}{2}\) Arc.

Thales' Theorem: Angle inscribed in a semicircle is a right angle.

Equation of a Circle:
\((x - h)^2 + (y - k)^2 = r^2\). Center: \((h, k)\), Radius: \(r\).

Completing the Square: Often required to get the general circle equation into standard form.

Angles Outside & Inside

Inside (Vertex inside):
\(\frac{\text{Arc 1} + \text{Arc 2}}{2}\)

Outside (Vertex outside):
\(\frac{\text{Big Arc} - \text{Small Arc}}{2}\)

Practice: Angle Solving

Two secants intersect outside a circle. If the intercepted arcs are \(120^\circ\) and \(40^\circ\), find the measure of the angle.

Radius & Tangent

A tangent line is always perpendicular to the radius at the point of tangency.
Pythagorean Theorem often follows!

Arc Length / Sector

Arc: \(\frac{n}{360} \cdot 2\pi r\)
Area: \(\frac{n}{360} \cdot \pi r^2\)

Regents Mastery Blueprint

Unit 5: Quadrilaterals & Proofs

The Parallelogram Hierarchy

Parallelogram: Opp sides \(\parallel\) & \(\cong\). Opp angles \(\cong\). Diagonals bisect each other.

Rectangle: All props of para + right angles and \(\cong\) diagonals.

Rhombus: All props of para + 4 \(\cong\) sides. Diagonals are \(\perp\) and bisect angles.

Square: The King! Props of Para, Rect, and Rhombus combined.

Proof Toolbox

To prove a Quad is a Rectangle using coordinate geometry:

• Distance: Show both pairs of opp. sides \(\cong\) AND diagonals \(\cong\).
• Slope: Show adjacent sides are perpendicular (neg. reciprocal slopes).

Strategy

For 2-column proofs, always start with given info and look for shared sides (Reflexive) or vertical angles.

Coordinate Proof Space

Prove triangle ABC with A(0,0), B(6,0), and C(3, 4) is isosceles.

Regents Mastery Blueprint

Unit 6: Volume & Spatial

Volume Formulas

Prism/Cyl

\(V = B h\)

Pyramid/Cone

\(V = \frac{1}{3} B h\)

Sphere

\(V = \frac{4}{3} \pi r^3\)

Cavalieri's Principle

If two solids have the same height and same cross-sectional area at every level, then they have the same volume.

Mass, Density, Volume

\[D = \frac{m}{V}\]

Regents modeling questions often ask you to find the mass of a sculpture or the cost of material based on volume and density.

Practice: Volume Modeling

A wooden cone has a radius of 5cm and height 12cm. If the density of wood is 0.7 g/cm³, find the total mass to the nearest tenth.

Geometry Game Plan

Regents Strategy

1. Mastering the Format

• Multiple Choice: 24 questions (48 points). Always sketch a diagram if one isn't provided! Many answers can be eliminated by spatial logic alone.
• Constructions: Expect one construction (Part II). Keep your arcs clear! Common ones: Perpendicular Bisector, Angle Bisector, Hexagon in a Circle.

2. Never Give Points Away

Rounding Rule: Check for "tenth," "hundredth," or "in terms of \(\pi\)." Leaving \(\pi\) in the answer is a "exact" answer.

The Degree Rule: Always check your calculator mode. One wrong mode kills every trig problem.

Show Work: In Part II-IV, even if the question is "State the volume," showing the substitution into the formula saves you points if you type it wrong into the calculator.

Fatal Mistakes

• Dilations not at Origin: If a line is dilated from a point ON the line, the equation doesn't change! If dilated from origin, multiply the y-int by \(k\).
• Arc vs Angle: Central angles = Arc; Inscribed = 1/2 Arc. Don't flip them.
• Area vs Volume: Area is 2D (squared units); Volume is 3D (cubed units).

Last Minute Checks

• Ruler Usage: Use a straightedge for all transformations and graphs. Hand-drawn lines lose points.
• Proofs: Ensure every "Statement" has a matching "Reason." Never leave a reason blank.
• Density: Remember density is weight per unit of volume. Always find volume first.

Geometry Mastery Blueprint | Complete

Geometry Calculator Power Plays Cheat Sheet

Calculator Power Plays

Geometry Regents Reference Guide | NYS Edition

1. Trig Setup

VITAL

Degree Mode:
[MODE] → Highlight DEGREE.

KEY

Finding Angles:
[2nd] [SIN/COS/TAN]. Result is in degrees.

2. Circles

Circle Eq: \((x-h)^2 + (y-k)^2 = r^2\)

Find Radius: If given Area, type \(\sqrt{\text{Area} / \pi}\) to find \(r\).

Completing the Square: Use calc to divide \(x\)-coeff by 2 and square it quickly.

3. Precisions

The Pi Button: Use [2nd] [^]. Avoid 3.14 unless requested!

Fractions: [ALPHA] [Y=] → [1] for scale factor ratios.

4. Coordinate Geom

Distance Formula: Use \(\sqrt{(X2-X1)^2 + (Y2-Y1)^2}\).

Slopes: \((Y2-Y1)/(X2-X1)\).

Area of Polygon: Use the "Box Method" (Draw box around shape, subtract triangles) - use calc for easy subtractions!

5. Advanced Solver

Find Intersection: [Y1]=line, [Y2]=parabola. [2nd][TRACE][5]. Hit [ENTER] 3x.

Numeric Solver: [MATH] → [Solver]. Solve for missing lengths in proportions!

6. Stats (Density)

Density Calc: Type \(\text{Mass} / \text{Volume}\).
Total Cost: Type \(\text{Volume} \times \text{Density} \times \text{Price}\).

TRICK: Check Scale Factor!
When dilating a line NOT through the origin, the equation of the line remains the SAME if the point is on the line. Use the calculator table to verify points on the new line!

Always write down the formula you are using. If volume is 1000, ensure it fits the diagram logically!

Geometry Regents Exam Strategy Tips

Geometry Game Plan

Strategic Success for the NYS Geometry Regents Exam

Format

Standard Regents

1. Points on the Table

Part I

24

Multiple Choice
(48 pts Total)

Part II

7

2-Pt Short Answer
(14 pts Total)

Part III

3

4-Pt Multi-Step
(12 pts Total)

Part IV

1

6-Pt Big Response
(6 pts Total)

Mastering the 180 Minutes

• 0-60 mins: Part I (MC). Focus on speed and precision. Flag hard ones.
• 60-90 mins: Part II (Short Answer). Includes your construction!
• 90-130 mins: Parts III & IV. Coordinate and Triangle proofs.
• Final 50: The "π" Check. Exact vs Rounding. Check Trig Mode!

Pro Strategy: For the 6-point proof, always write the Givens and Statements/Reasons header. Graders give 1 point for just showing you know the format!

Geometric Essentials

Final Exam Readiness

The "Point Killers"

Compass Arcs

NEVER erase your construction arcs. If they can't see how you did it, you get zero credit.

Trig Mode Error

CHECK YOUR MODE. Degrees for triangles; Radians ONLY for modeling/calc (rare in Geom).

Precision Fail

"In terms of π" means leave the symbol. Do NOT multiply it out to 3.14!

High-Yield Habits

Negative Reciprocals

Perpendicular lines: flip fraction and change sign. Example: 2/3 becomes -3/2.

Mark Your Diagram

When you read a Given, mark it on the picture immediately. It helps you see SAS or ASA.

Reference Sheet

Density, Volume, and Area formulas are all there. Don't memorize; just know where to look!

Regents Morning Prep

Compass & Straightedge

Pencils and a Ruler

Calculator in Degree Mode

Water and a Light Snack