Mathematics Methods Level 4 cheat sheet

For y = a·f(b(x − h)) + k: horizontal scale by 1/|b| (reflect in y-axis if b < 0), vertical scale by |a| (reflect in x-axis if a < 0), then translate right h and up k.

Domain of a composite f(g(x)) requires x in dom g and g(x) in dom f; solve the second as an inequality.

Inverse exists only if f is one-to-one; f⁻¹ swaps domain and range, and the graph of f⁻¹ is the reflection of y = f(x) in the line y = x.

Algebraically, f⁻¹ is found by solving y = f(x) for x, then swapping x and y; check f(f⁻¹(x)) = x on the restricted domain.

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

Sum of two functions: (f + g)(x) = f(x) + g(x) with domain dom f ∩ dom g; quotient additionally excludes points where g(x) = 0.

A relation is a function only if every x-value maps to exactly one y-value; the vertical line test detects failures.

Factor theorem: (x − a) is a factor of P(x) exactly when P(a) = 0; remainder theorem: dividing P(x) by (x − a) leaves remainder P(a).

For a polynomial of degree n, the graph has at most n − 1 turning points and at most n real roots counted with multiplicity.

Multiplicity rule: odd multiplicity crosses the x-axis, even multiplicity touches it; the sign of the leading coefficient fixes end behaviour.

Rational function R(x) = P(x)/Q(x): vertical asymptotes where Q(x) = 0 and P(x) ≠ 0; a hole where both vanish after cancellation.

Horizontal asymptote: y = 0 if deg P < deg Q; y = (leading coefficient ratio) if deg P = deg Q; none (or a slant asymptote) if deg P > deg Q.

Partial fractions decompose a proper rational expression into terms A/(x − a) for distinct linear factors and (Ax + B)/(x² + bx + c) for irreducible quadratics.

To solve an equation with a square root, isolate the radical, square both sides, then reject extraneous roots by substitution.

Laws of logarithms: log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x; these hold only for x, y > 0.

Change of base: log_b x = ln x / ln b = log_a x / log_a b, valid for positive bases not equal to 1.

Identity pair: b^(log_b x) = x for x > 0, and log_b(b^x) = x for all real x.

Exponential growth/decay model y = A·e^(kt): k > 0 grows, k < 0 decays; half-life t½ satisfies e^(k·t½) = 1/2, so t½ = ln(1/2)/k.

To solve a·e^(kx) + c = d, isolate the exponential, take logarithms of both sides, then divide by k.

To solve log equations, combine into a single logarithm, convert to exponential form, and check each root in the original domain.

Graph shapes: y = a^x passes through (0, 1) with horizontal asymptote y = 0; y = log_a x passes through (1, 0) with vertical asymptote x = 0.

Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ.

Compound angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.

Double angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.

General solutions: sin θ = k gives θ = arcsin k + 2nπ or θ = π − arcsin k + 2nπ; cos θ = k gives θ = ±arccos k + 2nπ; tan θ = k gives θ = arctan k + nπ.

Amplitude of y = a sin(bx + c) + d is |a|; period is 2π/|b| (π/|b| for tan); phase shift is −c/b; vertical shift is d.

Sine rule: a/sin A = b/sin B = c/sin C = 2R. Cosine rule: a² = b² + c² − 2bc cos A.

Area of a triangle = ½ab sin C; use the ambiguous case of the sine rule (two possible angles) when the given angle is opposite the shorter side.

Definition: f′(x) = lim_{h→0} [f(x + h) − f(x)]/h; the derivative is the gradient of the tangent and the instantaneous rate of change.

Power rule: d/dx (axⁿ) = nax^(n−1); extend to negative and fractional powers on the appropriate domain.

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); for e^u and ln u the derivatives are e^u·u′ and u′/u.

Trig derivatives: d/dx sin x = cos x; d/dx cos x = −sin x; d/dx tan x = sec²x (x in radians).

Stationary points solve f′(x) = 0; classify with the sign change of f′ or with f″(x): f″ > 0 gives a local minimum, f″ < 0 a local maximum.

For motion, velocity v = ds/dt and acceleration a = dv/dt = d²s/dt²; the object reverses direction where v changes sign.

Fundamental theorem: ∫_a^b f(x) dx = F(b) − F(a) where F′ = f; the definite integral is signed area.

Standard antiderivatives: ∫xⁿ dx = x^(n+1)/(n+1) + C (n ≠ −1); ∫x⁻¹ dx = ln|x| + C; ∫e^(kx) dx = e^(kx)/k + C.

Trig integrals: ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C.

Area between curves: ∫_a^b [upper(x) − lower(x)] dx; find limits by solving upper(x) = lower(x).

Volume of revolution about the x-axis: V = π∫_a^b [f(x)]² dx, provided f(x) ≥ 0 on the interval.

Substitution reverses the chain rule: let u = g(x), du = g′(x) dx; for definite integrals also convert the limits to u-values.

Total change: if F′(x) = f(x), then ∫_a^b f(x) dx = F(b) − F(a), giving net displacement from a velocity function.

Conditional probability: P(A|B) = P(A ∩ B)/P(B) for P(B) > 0; independence means P(A ∩ B) = P(A)P(B).

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the last term is zero.

Binomial distribution: P(X = k) = C(n, k)p^k(1 − p)^(n−k), mean np, variance np(1 − p), for independent trials with constant p.

Normal distribution: standardise with z = (x − μ)/σ, then use tables or technology for areas; about 68% lies within one σ and 95% within two σ.

Expected value E(X) = Σ x·P(x) for discrete variables; variance Var(X) = E(X²) − [E(X)]².

Correlation coefficient r lies in [−1, 1]; r near ±1 indicates a strong linear association, but correlation does not establish causation.

Least-squares regression line ŷ = a + bx has b = r·(s_y/s_x) and passes through the point (x̄, ȳ).

Model cycle: formulate variables and assumptions, build the function, solve, interpret in context, then validate and refine against data.

Choose the family from behaviour: constant difference suggests linear, constant ratio suggests exponential, constant second difference suggests quadratic.

Domain restrictions come from context (time ≥ 0, integer counts, physical limits) and must be stated with the model.

Piecewise models apply different formulas on different intervals; check continuity at the join points and interpret each segment.

Optimisation procedure: write the target as a function of one variable using the constraint, differentiate, solve f′ = 0, and test with the second derivative or endpoints.

Rates link variables through the chain rule: dV/dt = (dV/dr)(dr/dt); related-rates problems differentiate the constraint with respect to time.

Sensitivity check: compare model predictions with observed data and report residuals or percentage error to judge model adequacy.