NY Regents Algebra I cheat sheet

Rational numbers can be written as a fraction a/b with integers a and b, b ≠ 0; their decimal expansions terminate or repeat.

Irrational numbers have non-terminating, non-repeating decimals; examples include √2, √3, π, and e.

For any real number a, √(a²) = |a|, so √(x²) = x only when x ≥ 0.

Product rule for radicals: √(ab) = √a · √b for a ≥ 0, b ≥ 0; quotient rule: √(a/b) = √a / √b for a ≥ 0, b > 0.

Rationalizing a denominator multiplies numerator and denominator by the conjugate or the radical needed to clear it, e.g. 1/√2 = √2/2.

Rules of exponents: a^m · a^n = a^(m+n), a^m / a^n = a^(m−n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^0 = 1 for a ≠ 0.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiplying shifts the exponent by adding, dividing by subtracting.

When adding or subtracting radicals, combine only like radicals (same index and radicand), e.g. 3√5 + 2√5 = 5√5; √2 + √3 cannot be simplified.

Unit conversion uses dimensional analysis: multiply by conversion factors written as fractions equal to 1 so unwanted units cancel.

Slope between two points: m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have slopes whose product is −1.

Slope-intercept form y = mx + b; point-slope form y − y₁ = m(x − x₁); standard form Ax + By = C.

Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a) solves ax² + bx + c = 0 with a ≠ 0.

Discriminant b² − 4ac: positive gives two real solutions, zero gives one real solution, negative gives no real solutions (two complex).

Factoring patterns: difference of squares a² − b² = (a − b)(a + b); perfect square trinomials a² ± 2ab + b² = (a ± b)².

Zero product property: if ab = 0 then a = 0 or b = 0, so solve factored quadratics by setting each factor to zero.

Vertex form of a quadratic: y = a(x − h)² + k has vertex (h, k); axis of symmetry x = h = −b/(2a) in standard form.

Systems of two linear equations: substitution isolates one variable; elimination adds or subtracts multiples of equations to cancel a variable.

Exponential growth/decay: y = a(1 + r)^t for growth and y = a(1 − r)^t for decay, where r is the rate per period as a decimal.

A relation is a function if every input (x-value) has exactly one output; the vertical line test confirms this on a graph.

Function notation: f(x) means the output of f at input x; evaluate by substituting the value for every x in the rule.

Domain is the set of allowed inputs; range is the set of resulting outputs; for real-valued functions, exclude values making a denominator zero or a radicand negative.

Linear functions have constant rate of change (slope); exponential functions have constant ratio between successive outputs.

Transformations: y = f(x) + k shifts up k; y = f(x − h) shifts right h; y = −f(x) reflects over the x-axis; y = af(x) stretches vertically by |a|.

The inverse of a function swaps inputs and outputs; on a graph it reflects over the line y = x; f⁻¹ exists only if f is one-to-one.

Average rate of change on [a, b] is (f(b) − f(a))/(b − a), the slope of the secant line between the two points.

Sequences: arithmetic sequences add a common difference d with aₙ = a₁ + (n − 1)d; geometric sequences multiply by a common ratio r with aₙ = a₁r^(n−1).

Absolute value function y = |x| has a V-shaped graph with vertex at the origin; y = |x − h| + k shifts the vertex to (h, k).

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope criteria: segments are parallel when slopes are equal and perpendicular when slopes are negative reciprocals.

Pythagorean theorem a² + b² = c² applies to right triangles, with c the hypotenuse; converse tests whether a triangle is right.

Special right triangles: 45-45-90 has side ratios 1 : 1 : √2; 30-60-90 has ratios 1 : √3 : 2.

Circle: circumference C = 2πr, area A = πr²; equation (x − h)² + (y − k)² = r² has center (h, k) and radius r.

Volume formulas: cylinder V = πr²h, cone V = (1/3)πr²h, sphere V = (4/3)πr³, prism V = Bh.

Triangle angle sum is 180°; exterior angle equals the sum of the two remote interior angles.

Similar figures have proportional corresponding sides and equal corresponding angles; areas scale by the square of the scale factor and volumes by its cube.

Coordinate geometry proof strategy: compute slopes, lengths, and midpoints to verify parallel sides, right angles, or congruent diagonals.

Mean = sum of values ÷ count; median is the middle value of ordered data; mode is the most frequent value.

Interquartile range IQR = Q3 − Q1; outliers are commonly values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures spread about the mean; adding a constant to every value leaves standard deviation unchanged, while multiplying by a constant multiplies it by |constant|.

A residual is actual value minus predicted value; a good linear fit has residuals randomly scattered around zero with no pattern.

Correlation coefficient r ranges from −1 to 1; r near ±1 indicates a strong linear association, and correlation does not imply causation.

Probability of an event = favorable outcomes ÷ total equally likely outcomes; P(not A) = 1 − P(A).

For independent events, P(A and B) = P(A)·P(B); for mutually exclusive events, P(A or B) = P(A) + P(B).

Conditional probability P(A|B) = P(A and B)/P(B); two-way tables give joint, marginal, and conditional frequencies.

A random sample supports generalization to the population; a biased or self-selected sample does not.