NY Regents Algebra II cheat sheet

i² = −1; powers of i repeat with period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, so i^n depends on n mod 4.

(a + bi) + (c + di) = (a + c) + (b + d)i; subtract by combining real parts and imaginary parts separately.

(a + bi)(c + di) = (ac − bd) + (ad + bc)i, obtained by distributing and then replacing i² with −1.

The conjugate of a + bi is a − bi, and (a + bi)(a − bi) = a² + b², a real number; divide complex numbers by multiplying numerator and denominator by the conjugate of the denominator.

a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m, and a^(−n) = 1/a^n for a ≠ 0.

√(ab) = √a·√b and √(a/b) = √a/√b; simplify a radical by factoring out perfect-square (or perfect-nth-power) factors.

For ax² + bx + c = 0 with a ≠ 0, the discriminant b² − 4ac gives two complex conjugate roots when negative, one repeated real root when zero, and two distinct real roots when positive.

Solving √(x + k) = m requires isolating the radical, squaring both sides, and substituting each candidate back into the original equation to reject extraneous roots.

Remainder Theorem: dividing polynomial p(x) by (x − a) leaves remainder p(a); Factor Theorem: (x − a) is a factor exactly when p(a) = 0.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a) solves ax² + bx + c = 0 for a ≠ 0, and it applies even when the discriminant is negative.

For ax² + bx + c = 0, the two roots sum to −b/a and multiply to c/a.

Completing the square rewrites ax² + bx + c as a(x + b/(2a))² + (c − b²/(4a)), which reveals the vertex and solves the equation.

The vertex of y = ax² + bx + c occurs at x = −b/(2a); vertex form y = a(x − h)² + k has vertex (h, k) and axis of symmetry x = h.

Logarithm rules: log_b(mn) = log_b m + log_b n; log_b(m/n) = log_b m − log_b n; log_b(m^p) = p·log_b m; log_b b = 1 and log_b 1 = 0.

Change of base: log_b a = (log_c a)/(log_c b), for example log_b a = (ln a)/(ln b).

To solve b^(f(x)) = c, take a logarithm of both sides and use the power rule to bring the exponent down, since log_b(b^y) = y.

A rational expression is undefined wherever its denominator equals zero; clear denominators when solving rational equations, then check each solution against the excluded values.

Transformations of y = f(x): f(x) + k shifts the graph up k units; f(x − h) shifts it right h units; a·f(x) stretches vertically by |a| and reflects across the x-axis when a < 0.

(f ∘ g)(x) = f(g(x)); the domain of the composition excludes values not in the domain of g and values where g(x) is not in the domain of f.

Inverse relation: interchange x and y, then solve for y; the graphs of f and its inverse are mirror images across the line y = x.

f is even when f(−x) = f(x) for all x in its domain (y-axis symmetry) and odd when f(−x) = −f(x) (origin symmetry).

Domain rules: exclude inputs making a denominator zero, require even-index radicands to be nonnegative, and require logarithm arguments to be positive.

Exponential models: A = P(1 + r)^t for growth or decay per period, A = Pe^(rt) for continuous change, and half-life problems use A = P(1/2)^(t/h).

Arithmetic sequence: a_n = a₁ + (n − 1)d with sum S_n = n(a₁ + a_n)/2; geometric sequence: a_n = a₁r^(n−1) with S_n = a₁(1 − r^n)/(1 − r) for r ≠ 1.

Exponential and logarithmic functions are inverses: y = b^x is equivalent to x = log_b y, so log_b(b^x) = x and b^(log_b x) = x for x > 0.

Circle equation: (x − h)² + (y − k)² = r² has center (h, k) and radius r; convert x² + y² + Dx + Ey + F = 0 by completing the square in both variables.

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

Law of sines: a/sin A = b/sin B = c/sin C, applicable when two angles or an angle–opposite-side pair are known.

Law of cosines: c² = a² + b² − 2ab·cos C, applicable to SAS and SSS triangles.

Area of a triangle = (1/2)ab·sin C, where C is the included angle between sides a and b.

Radian–degree conversion: π radians = 180°, so degrees × (π/180) gives radians and radians × (180/π) gives degrees.

For a central angle θ measured in radians and radius r: arc length s = rθ and sector area = (1/2)r²θ.

Pythagorean identity: sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ; on the unit circle the point (cos θ, sin θ) corresponds to angle θ.

Conditional probability: P(A|B) = P(A and B)/P(B), provided P(B) > 0.

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); when A and B are mutually exclusive, P(A or B) = P(A) + P(B).

Events A and B are independent exactly when P(A and B) = P(A)·P(B).

Binomial probability: P(k successes in n trials) = nCk · p^k · (1 − p)^(n−k) for n independent trials with success probability p.

Permutations nPr = n!/(n − r)! count arrangements where order matters; combinations nCr = n!/(r!(n − r)!) count groups where order does not matter.

Expected value: E(X) = Σ x·P(x), the long-run average of a discrete random variable over many trials.

z = (x − μ)/σ; for normal data about 68% of values lie within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.

Residual = observed value − predicted value; the correlation coefficient r satisfies −1 ≤ r ≤ 1 and describes the direction and strength of a linear association.