VCE Maths Methods cheat sheet

A function maps each x in its domain to exactly one y; vertical line test fails if any vertical line meets the graph more than once.

Domain of f(g(x)): x must be in domain of g and g(x) must be in domain of f.

Inverse f^{-1} exists only if f is one-to-one; its graph is the reflection of f in y = x, and domain/range swap.

Transformations: y = a f(b(x-h)) + k gives dilation by factor |a| parallel to y-axis, factor 1/|b| parallel to x-axis, then translation h right, k up.

For y = a(x-h)^2 + k, vertex is (h,k); axis of symmetry x = h; turning point is a maximum if a < 0, minimum if a > 0.

Exponential y = a^x has domain R, range (0, ∞), horizontal asymptote y = 0, and passes through (0,1).

Logarithm y = log_a x (a>1) has domain (0, ∞), range R, vertical asymptote x = 0, and passes through (1,0).

Composite function f∘g means f(g(x)); it is defined only where g(x) lies in domain of f.

Absolute value |x| = x if x ≥ 0, -x if x < 0; graph is V-shaped with vertex at (0,0).

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

Discriminant Δ = b² - 4ac: Δ > 0 two distinct real roots; Δ = 0 one repeated root; Δ < 0 no real roots.

Factorisation: a² - b² = (a-b)(a+b); a³ ± b³ = (a ± b)(a² ∓ ab + b²).

Index laws: a^m a^n = a^{m+n}; a^m / a^n = a^{m-n}; (a^m)^n = a^{mn}; a^0 = 1 (a ≠ 0); a^{-n} = 1/a^n.

Log laws: log_a(mn) = log_a m + log_a n; log_a(m/n) = log_a m - log_a n; log_a(m^p) = p log_a m; log_a a = 1.

Change of base: log_a b = log_c b / log_c a; also log_a b = 1 / log_b a.

Solving a^x = b: take logs to get x = log b / log a (any consistent base).

Simultaneous equations: substitution or elimination; number of solutions matches intersection points of graphs.

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

Power rule: d/dx (x^n) = n x^{n-1}; d/dx (constant) = 0; d/dx (e^{kx}) = k e^{kx}.

Product rule: (uv)' = u'v + uv'; quotient rule: (u/v)' = (u'v - uv')/v².

Chain rule: d/dx f(g(x)) = f'(g(x)) g'(x).

d/dx (ln x) = 1/x for x > 0; d/dx (sin x) = cos x; d/dx (cos x) = -sin x; d/dx (tan x) = sec² x.

Stationary points occur where f'(x) = 0; classify by sign change of f' or by f''(x): f'' > 0 local minimum, f'' < 0 local maximum.

Antiderivative: ∫ x^n dx = x^{n+1}/(n+1) + C for n ≠ -1; ∫ (1/x) dx = ln|x| + C; ∫ e^{kx} dx = (1/k)e^{kx} + C.

Definite integral ∫_a^b f(x) dx gives signed area between curve and x-axis from x=a to x=b; reverse limits changes sign.

Average value of f on [a,b] is (1/(b-a)) ∫_a^b f(x) dx.

Mean of n values: x̄ = (Σ x_i)/n; median is the middle value when ordered (average of two middle values if n even).

Sample standard deviation s = √[Σ(x_i - x̄)²/(n-1)]; population σ uses n in the denominator.

For a Normal distribution X ~ N(μ, σ²), about 68% of values lie within μ ± σ, 95% within μ ± 2σ, 99.7% within μ ± 3σ.

Standardise: z = (x - μ)/σ; then use standard Normal tables or technology to find probabilities.

Probability rules: P(A') = 1 - P(A); P(A ∪ B) = P(A) + P(B) - P(A ∩ B); if independent, P(A ∩ B) = P(A)P(B).

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

Binomial distribution: X ~ Bi(n, p) gives P(X = k) = C(n,k) p^k (1-p)^{n-k}; mean np, variance np(1-p).

Correlation coefficient r lies between -1 and 1; r near ±1 indicates strong linear association, r near 0 weak linear association.

Least-squares regression line: y = a + bx with b = r (s_y/s_x) and a = ȳ - b x̄.