IB Math AA SL cheat sheet

Arithmetic sequences: uₙ = u₁ + (n − 1)d and Sₙ = n/2 (2u₁ + (n − 1)d) = n/2 (u₁ + uₙ), where d = u₂ − u₁.

Geometric sequences: uₙ = u₁r^(n−1); Sₙ = u₁(rⁿ − 1)/(r − 1) for r ≠ 1; S∞ = u₁/(1 − r) exists only when |r| < 1.

Logarithm laws: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a xⁿ = n log_a x; log_a 1 = 0; log_a a = 1.

Change of base: log_a x = log_b x / log_b a, so log_a x = ln x / ln a; also a^(log_a x) = x.

Exponent laws: a^m a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/aⁿ; a^(1/n) = ⁿ√a; a⁰ = 1.

Binomial theorem: (a + b)ⁿ = Σ_{r=0}^{n} C(n,r) a^(n−r) b^r, with C(n,r) = n!/(r!(n − r)!); the term in b^r is C(n,r) a^(n−r) b^r.

Solving aˣ = b: write both sides with the same base, or take logs to get x = ln b / ln a (equivalently log b / log a).

Quadratic ax² + bx + c = 0: Δ = b² − 4ac; Δ > 0 gives two distinct real roots, Δ = 0 one repeated root, Δ < 0 no real roots (a conjugate complex pair in ℂ).

Complex numbers: i² = −1; z = a + bi has |z| = √(a² + b²), conjugate z* = a − bi, z z* = |z|²; polar form z = r(cos θ + i sin θ), where products multiply moduli and add arguments.

Translations: y = f(x) + d shifts the graph up by d; y = f(x − c) shifts it right by c (so f(x + c) shifts left by c).

Stretches: y = a f(x) is a vertical stretch of factor a; y = f(bx) is a horizontal stretch of factor 1/b.

Completing the square gives f(x) = a(x − h)² + k, so the vertex is (h, k) and the axis of symmetry is x = h; a > 0 opens upward.

Inverse functions: f⁻¹ exists only when f is one-to-one; its graph is the reflection of y = f(x) in the line y = x, and domain and range swap.

Composites: (f ∘ g)(x) = f(g(x)) - g acts first; the composite is defined only where g(x) lies in the domain of f.

Domains: ln x and log_a x require x > 0; √x requires x ≥ 0; a rational function is undefined where its denominator is zero, producing a vertical asymptote.

Rational asymptotes: numerator degree less than denominator gives y = 0; equal degrees give y = (ratio of leading coefficients); greater degree gives no horizontal asymptote.

Factor and remainder theorems: (x − a) is a factor of f(x) exactly when f(a) = 0; the remainder on dividing f(x) by (x − a) is f(a).

Symmetry: f is even when f(−x) = f(x) (symmetric about the y-axis) and odd when f(−x) = −f(x) (rotational symmetry about the origin).

Sine rule: a/sin A = b/sin B = c/sin C; use it with two angles and a side, or the ambiguous SSA case where two triangles may satisfy the data.

Cosine rule: a² = b² + c² − 2bc cos A; use it for SAS or SSS, rearranged as cos A = (b² + c² − a²)/(2bc) when finding an angle.

Area of any triangle = ½ab sin C, where C is the angle included between sides a and b.

Radians: 180° = π; arc length l = rθ and sector area A = ½r²θ, with θ measured in radians.

Identities: sin²θ + cos²θ = 1; tan θ = sin θ/cos θ; sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

Trigonometric equations: take the principal solution from the inverse function, then add the period (2π for sin and cos, π for tan) and select solutions inside the required interval.

Sinusoidal graphs: y = a sin(b(x − c)) + d has amplitude |a|, period 2π/b, horizontal shift c and vertical shift d.

Vectors: a·b = |a||b| cos θ = a₁b₁ + a₂b₂; a and b are perpendicular exactly when a·b = 0; θ = arccos(a·b/(|a||b|)); the unit vector in the direction of a is a/|a|.

Lines: gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines have m₁ = m₂, perpendicular lines have m₁m₂ = −1; distance = √((Δx)² + (Δy)²) and the midpoint averages the coordinates.

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events P(A ∩ B) = 0 so the probabilities simply add.

Conditional probability: P(A | B) = P(A ∩ B)/P(B) with P(B) > 0; A and B are independent exactly when P(A ∩ B) = P(A)P(B), i.e. P(A | B) = P(A).

Discrete random variables: E(X) = Σ x P(X = x); Var(X) = E(X²) − [E(X)]²; E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X).

Binomial distribution: X ~ B(n, p) needs a fixed number n of independent trials, two outcomes and constant p; P(X = r) = C(n,r) p^r (1 − p)^(n−r), E(X) = np, Var(X) = np(1 − p).

Normal distribution: X ~ N(μ, σ²) is standardised by z = (x − μ)/σ; probabilities come from the standard normal, and the inverse normal finds x from a given probability.

Summary statistics: mean, median and mode locate the centre; standard deviation measures spread about the mean; quartiles give IQR = Q3 − Q1, and outliers lie below Q1 − 1.5 IQR or above Q3 + 1.5 IQR.

Linear transformations of data: adding a constant shifts mean and median by that constant but leaves standard deviation and IQR unchanged; multiplying by a changes SD and IQR by |a|.

Bivariate data: Pearson's r lies between −1 and 1 and measures the strength and direction of linear association; the line of best fit by eye passes through (x̄, ȳ); correlation does not imply causation.

Histograms with unequal class widths: frequency density = frequency / class width, so the area of each bar is proportional to the frequency.

Derivative from first principles: f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.

Standard derivatives: xⁿ → nx^(n−1); sin x → cos x; cos x → −sin x; eˣ → eˣ; ln x → 1/x for x > 0.

Rules: product (uv)′ = u′v + uv′; quotient (u/v)′ = (u′v − uv′)/v²; chain dy/dx = dy/du · du/dx.

Tangent and normal at x = a: tangent gradient m = f′(a) with equation y − f(a) = m(x − a); the normal gradient is −1/f′(a).

Stationary points: solve f′(x) = 0; f″(x) > 0 gives a local minimum, f″(x) < 0 a local maximum, f″(x) = 0 is inconclusive; f′(x) > 0 on an interval means f is increasing there.

Standard integrals: ∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫x⁻¹ dx = ln|x| + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫eˣ dx = eˣ + C.

Definite integrals: ∫_a^b f(x) dx = F(b) − F(a) gives the signed area between the curve and the x-axis; regions below the axis count as negative, so split the interval or use |f(x)|.

Area between curves: ∫_a^b [f(x) − g(x)] dx with f the upper curve, where a and b are the x-coordinates of the intersection points found from f(x) = g(x).

Kinematics: v = ds/dt and a = dv/dt = d²s/dt²; displacement over an interval is the definite integral of velocity, and distance travelled requires |v|.