NYS Math cheat sheet

Rational exponents: a^(m/n) = (n-th root of a)^m = n-th root of a^m, valid for a > 0; for odd n it also works for negative a.

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

Radicals: √(ab) = √a·√b and √(a/b) = √a/√b for a, b ≥ 0; rationalize denominators by multiplying by the conjugate, e.g. 1/(√a+√b) = (√a−√b)/(a−b).

Complex numbers: i = √(−1), i² = −1; for z = a + bi, |z| = √(a²+b²) and the conjugate is a − bi; (a+bi)(a−bi) = a² + b².

Powers of i cycle with period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1; reduce the exponent mod 4 to evaluate i^n.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiply by adding exponents and divide by subtracting them, then adjust the coefficient back into range.

Units and rates: convert by multiplying by conversion factors equal to 1 (e.g. 1 km/1000 m) so unwanted units cancel; dimensional analysis catches setup errors.

Precision: absolute error is |measured − actual|; percent error is (|measured − actual| / |actual|) × 100%, and relative error is the error divided by the actual value.

Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a) solves ax² + bx + c = 0 with a ≠ 0; the discriminant b² − 4ac gives two, one, or no real roots when positive, zero, or negative.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the vertex y-value is found by substitution, and the axis of symmetry is that same vertical line.

Factoring patterns: a² − b² = (a−b)(a+b); a² ± 2ab + b² = (a ± b)²; a³ − b³ = (a−b)(a² + ab + b²); a³ + b³ = (a+b)(a² − ab + b²).

Exponential growth/decay: y = a·b^x where a is the initial value; growth when b > 1, decay when 0 < b < 1, and b = 1 + r for rate r expressed as a decimal.

Compound interest: A = P(1 + r/n)^(nt) for n compounds per year; continuous compounding is A = Pe^(rt).

Systems: substitution isolates one variable; elimination adds or subtracts scaled equations to cancel a variable; a system has no solution when the equations describe parallel lines (same slope, different intercept).

Absolute value equations |x| = k with k ≥ 0 give x = k or x = −k; inequalities |x| < k give −k < x < k while |x| > k gives x < −k or x > k.

Logarithms invert exponentials: log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x − log_b y, log_b(x^p) = p·log_b x, and log_b x = ln x / ln b (change of base).

Function notation: f(g(x)) means substitute g(x) into f; the domain of a composition excludes x-values where g(x) falls outside f's domain.

Inverse functions: f⁻¹ exists only if f is one-to-one; swap x and y and solve, and the graphs of f and f⁻¹ are mirror images across the line y = x.

Transformations of y = f(x): f(x) + k shifts up k; f(x + h) shifts left h; a·f(x) stretches vertically by |a| and reflects across the x-axis when a < 0; f(bx) compresses horizontally by 1/|b|.

Domain restrictions: a rational function excludes zeros of its denominator; a square-root function requires its radicand ≥ 0; a log function requires its argument > 0.

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

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis; odd functions satisfy f(−x) = −f(x) and are symmetric about the origin.

Key parent functions: y = x², y = x³, y = √x, y = |x|, y = 1/x, y = e^x, y = ln x; know each shape, domain, and range.

Sequences are functions on the integers: arithmetic a_n = a₁ + (n−1)d; geometric a_n = a₁·r^(n−1); sum of a finite geometric series is a₁(1 − r^n)/(1 − r) for r ≠ 1.

Distance and midpoint in the plane: d = √((x₂−x₁)² + (y₂−y₁)²); midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

Slope criteria: lines are parallel when slopes are equal; perpendicular when slopes are negative reciprocals (m₁·m₂ = −1).

Circle equation: (x − h)² + (y − k)² = r² has center (h, k) and radius r; complete the square to convert from general form.

Right-triangle relations: Pythagorean theorem a² + b² = c²; 30-60-90 sides are in ratio 1 : √3 : 2; 45-45-90 sides are in ratio 1 : 1 : √2.

Trigonometric ratios in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; identity sin²θ + cos²θ = 1.

Volume formulas: cylinder V = πr²h; cone V = (1/3)πr²h; sphere V = (4/3)πr³; prism/pyramid follow V = Bh and V = (1/3)Bh.

Similar figures: corresponding angles are equal, side lengths scale by factor k, perimeters scale by k, and areas scale by k².

Coordinate transformations: reflection across the x-axis maps (x, y) to (x, −y); across the y-axis to (−x, y); across y = x to (y, x); a 90° counterclockwise rotation about the origin maps (x, y) to (−y, x).

Mean = sum/n; median is the middle value of ordered data (average the two middle values when n is even); mode is the most frequent value.

Standard deviation measures typical distance from the mean; adding a constant to every data value leaves the standard deviation unchanged, while multiplying every value by k multiplies it by |k|.

Normal distribution: about 68% of data lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3; z-score = (x − μ)/σ.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B); P(not A) = 1 − P(A).

Independent events satisfy P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B), and A and B are independent exactly when P(A|B) = P(A).

Permutations count ordered arrangements: nPr = n!/(n−r)!; combinations count unordered selections: nCr = n!/(r!(n−r)!); use combinations when order does not matter.

Line of best fit: the correlation coefficient r lies between −1 and 1; |r| near 1 indicates a strong linear fit, and r² is the fraction of variation explained by the model.

Sampling and study design: random sampling supports generalization to the population; random assignment to treatment groups supports causal conclusions; a sample statistic estimates an unknown population parameter.