WACE Maths Methods flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Function definition

A relation in which every element of the domain maps to exactly one element of the range. The vertical line test checks this on a graph.

Vertical line test

A graph represents a function only if no vertical line crosses it more than once. Two intersections at the same x-value mean one input has two outputs.

Inverse function

The inverse f inverse undoes f, exists only when f is one-to-one, and is found by swapping x and y then solving for y. Its graph reflects in y = x.

Quadratic axis of symmetry

For y = ax^2 + bx + c, the axis of symmetry is the vertical line x = -b/(2a). The vertex lies on this line and is a maximum if a < 0.

Discriminant

The quantity b^2 - 4ac for a quadratic. Positive gives two distinct real roots, zero gives one repeated root, and negative gives no real roots.

Composite function domain

For f(g(x)), x must lie in the domain of g, and the output g(x) must lie in the domain of f. Both conditions must hold simultaneously.

Radians to degrees

Multiply radians by 180/pi to get degrees; multiply degrees by pi/180 to get radians. A full circle is 2 pi radians or 360 degrees.

Arc length formula

For a circle of radius r and central angle theta measured in radians, arc length s = r theta. Sector area is half r squared times theta.

Amplitude and period

For y = a sin(bx), amplitude is |a| and period is 2 pi / |b|. The curve oscillates between a and negative a about its centre line.

Pythagorean identity

sin squared x plus cos squared x equals 1 for all x. Dividing by cos squared gives 1 + tan squared x = sec squared x.

Tangent asymptotes

The graph of y = tan x has vertical asymptotes where cos x = 0, that is x = pi/2 + n pi. Its period is pi and its range is all real numbers.

Double angle for sine

sin 2x = 2 sin x cos x. This identity converts products of sine and cosine into a single sine of double the angle, useful in solving equations.

General sine solution

If sin x = k, then x = arcsin k + 2n pi or x = pi - arcsin k + 2n pi for integer n. Both families are needed for full solutions.

Derivative from first principles

The derivative is the limit as h approaches 0 of [f(x + h) - f(x)] divided by h. It gives the instantaneous gradient of the curve at x.

Product rule

If y = uv then dy/dx = u'v + uv'. Differentiate the first factor times the second, plus the first times the derivative of the second.

Quotient rule

For y = u/v, dy/dx = (u'v - uv')/v squared. The numerator order matters: derivative of top times bottom minus top times derivative of bottom.

Chain rule

For y = f(g(x)), dy/dx = f'(g(x)) times g'(x). Differentiate the outer function, keep the inner unchanged, then multiply by the inner derivative.

Trigonometric derivatives

With x in radians, the derivative of sin x is cos x, of cos x is negative sin x, and of tan x is sec squared x.

Stationary point classification

Stationary points satisfy f'(x) = 0. If f''(x) > 0 the point is a local minimum; if f''(x) < 0 it is a local maximum.

Tangent line equation

At x = a the tangent has gradient f'(a) and passes through (a, f(a)). Its equation is y - f(a) = f'(a)(x - a).

Definite integral as area

The definite integral from a to b of f(x) gives signed area between the curve and the x-axis. Regions below the axis contribute negatively, so split at intercepts.

Logarithm definition

log base a of x equals y exactly when a to the power y equals x. The base a must be positive and not equal to 1, and x must be positive.

Logarithm product law

log(xy) = log x + log y and log(x/y) = log x - log y, for positive x and y. These laws let you combine or split logarithmic expressions.

Power law for logarithms

log(x^n) = n log x for positive x. This moves an exponent to the front, which is the key step in solving exponential equations.

Change of base

log base a of x equals ln x divided by ln a. This converts any logarithm into natural logs so a calculator can evaluate it.

Exponential growth model

y = A e^(kt) with k positive describes growth from initial value A. With k negative it describes decay. Doubling time satisfies e^(kt) = 2.

Solving a^x = b

Take logarithms of both sides to get x log a = log b, so x = log b / log a. This works for any positive base a not equal to 1.

Binomial probability formula

For X ~ Bin(n, p), P(X = k) = C(n, k) p^k (1-p)^(n-k). C(n, k) counts the arrangements of k successes among n trials.

Binomial mean and variance

If X ~ Bin(n, p), then E(X) = np and Var(X) = np(1-p). The standard deviation is the square root of np(1-p).

Standard normal z-score

z = (x - mu)/sigma measures how many standard deviations x lies from the mean. Convert to z, then find area using tables or a calculator.

Empirical rule

For a normal distribution, roughly 68% of values lie within one standard deviation of the mean and about 95% within two standard deviations.

Conditional probability formula

P(A given B) = P(A and B) / P(B), provided P(B) > 0. Events A and B are independent when P(A and B) = P(A) times P(B).