WACE Maths Methods cheat sheet

A function is a relation where each x-value maps to exactly one y-value; the vertical line test detects violations.

Domain restrictions arise from denominators (exclude where denominator is zero), even roots (radicand must be non-negative), and logarithms (argument strictly positive).

For y = a f(b(x - h)) + k, a is vertical dilation (reflection in x-axis if a < 0), b is horizontal dilation by 1/|b|, h shifts right, k shifts up.

Inverse functions exist only if f is one-to-one; find f inverse by swapping x and y then solving, and the graph is a reflection in y = x.

For a quadratic y = ax^2 + bx + c, the axis of symmetry is x = -b/(2a) and the vertex is at that x with y found by substitution.

The discriminant b^2 - 4ac determines root count: positive gives two real roots, zero gives one repeated root, negative gives no real roots.

Composite f(g(x)) is defined only where x is in the domain of g and g(x) is in the domain of f.

Transformations of y = 1/x and y = x^2 should be sketched with asymptotes or intercepts labelled; asymptotes occur where the function is undefined or tends to a limit.

In a right triangle: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent; SOH-CAH-TOA.

Radians: pi radians = 180 degrees; arc length s = r theta and sector area A = (1/2) r^2 theta with theta in radians.

For y = a sin(b(x - c)) + d, amplitude is |a|, period is 2 pi / |b| (or pi/|b| for tan), phase shift is c, vertical shift is d.

Pythagorean identity sin^2 x + cos^2 x = 1; also tan x = sin x / cos x, and 1 + tan^2 x = sec^2 x.

Exact values: sin 30 = 1/2, cos 30 = sqrt(3)/2, sin 45 = cos 45 = 1/sqrt(2), sin 60 = sqrt(3)/2, cos 60 = 1/2.

General solutions: if sin x = k then x = arcsin k + 2n pi or x = pi - arcsin k + 2n pi; if cos x = k then x = +/- arccos k + 2n pi.

The sine and cosine graphs have period 2 pi and range [-1, 1]; tangent has period pi and vertical asymptotes where cos x = 0.

Double angle formulas: sin 2x = 2 sin x cos x and cos 2x = cos^2 x - sin^2 x = 1 - 2 sin^2 x = 2 cos^2 x - 1.

Derivative from first principles: f'(x) = limit as h tends to 0 of [f(x + h) - f(x)] / h.

Power rule: d/dx (x^n) = n x^(n-1); derivative of a constant is 0; derivative of a sum is the sum of derivatives.

Product rule: (uv)' = u'v + uv'; quotient rule: (u/v)' = (u'v - uv')/v^2; chain rule: d/dx f(g(x)) = f'(g(x)) g'(x).

d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec^2 x, all with x in radians.

d/dx (e^x) = e^x and d/dx (ln x) = 1/x for x > 0; d/dx (a^x) = a^x ln a.

The derivative gives the gradient of the tangent; the equation of the tangent at x = a is y - f(a) = f'(a)(x - a).

Stationary points occur where f'(x) = 0; use the second derivative or a sign table of f' to classify maximum, minimum, or inflection.

Integration reverses differentiation: integral of x^n dx = x^(n+1)/(n+1) + C for n not equal to -1; definite integrals give signed area, so split where the curve crosses the axis.

For y = a^x with a > 1, the graph is increasing with y-intercept 1, horizontal asymptote y = 0, and range (0, infinity).

Logarithm definition: log_a x = y is equivalent to a^y = x; log_a 1 = 0 and log_a a = 1.

Log laws: log(xy) = log x + log y, log(x/y) = log x - log y, log(x^n) = n log x, and log_a x = ln x / ln a.

e is the base where the gradient of y = e^x equals its value; ln is the inverse of e^x, so e^(ln x) = x for x > 0 and ln(e^x) = x for all x.

To solve a^x = b, take logarithms of both sides: x = log b / log a (or ln b / ln a).

To solve equations with logs, combine using log laws then convert to exponential form, and check solutions satisfy the original domain.

Exponential growth and decay: y = A e^(kt) with k > 0 for growth and k < 0 for decay; doubling time satisfies e^(kt) = 2.

The graph of y = ln x has vertical asymptote x = 0, passes through (1, 0), and has domain x > 0.

For a discrete random variable, sum of all probabilities equals 1 and E(X) = sum of x P(X = x); Var(X) = E(X^2) - [E(X)]^2.

For a binomial distribution X ~ Bin(n, p): P(X = k) = C(n, k) p^k (1-p)^(n-k), with E(X) = np and Var(X) = np(1-p).

A binomial setting requires a fixed number of independent trials, two outcomes per trial, and constant probability p.

The normal distribution N(mu, sigma^2) is symmetric about mu; approximately 68% of values lie within one standard deviation and 95% within two.

Standardise with z = (x - mu)/sigma, then use standard normal tables or a calculator to find probabilities; P(X < a) is the area to the left.

For a continuous random variable, the total area under the density curve is 1 and P(a < X < b) is the area between a and b.

Conditional probability: P(A|B) = P(A and B)/P(B); events are independent when P(A and B) = P(A) P(B).

Sample mean of n independent observations has standard deviation sigma / sqrt(n); the mean of the sample means equals the population mean.