VCE 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 each element of the domain is paired with exactly one element of the range; passes the vertical line test.

One-to-one function

A function where each output value corresponds to exactly one input value; passes both vertical and horizontal line tests, so an inverse function exists.

Inverse function

For a one-to-one function f, f^{-1} reverses the mapping: f^{-1}(f(x)) = x. Its graph is the reflection of y = f(x) in the line y = x.

Composite function domain

For f(g(x)), the domain consists of all x in the domain of g such that g(x) lies in the domain of f.

Quadratic vertex form

y = a(x-h)^2 + k has vertex (h,k) and axis of symmetry x = h; it opens upward if a > 0 and downward if a < 0.

Exponential function properties

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

Logarithm function properties

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

Quadratic formula

The solutions of ax² + bx + c = 0, a ≠ 0, are x = [-b ± √(b² - 4ac)]/(2a).

Discriminant

For ax² + bx + c = 0, Δ = b² - 4ac. If Δ > 0 there are two distinct real roots; Δ = 0 gives one repeated root; Δ < 0 gives no real roots.

Index laws

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

Logarithm 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; log_a 1 = 0.

Change of base formula

log_a b = log_c b / log_c a for any valid base c; in particular log_a b = 1 / log_b a.

Solving exponential equations

To solve a^x = b, take logarithms of both sides to obtain x = log b / log a, using the same base throughout.

Difference of squares

a² - b² factorises as (a - b)(a + b); this identity simplifies many algebraic expressions and rational functions.

Derivative from first principles

f'(x) = lim_{h→0} [f(x+h) - f(x)]/h, provided the limit exists; it gives the instantaneous rate of change.

Power rule

For any real n, d/dx (x^n) = n x^{n-1}; the derivative of a constant is zero.

Product rule

If y = uv, then dy/dx = u'v + uv', where u' and v' are the derivatives of u and v with respect to x.

Quotient rule

If y = u/v with v ≠ 0, then dy/dx = (u'v - uv')/v².

Chain rule

For a composite function y = f(g(x)), dy/dx = f'(g(x)) g'(x); differentiate the outer function and multiply by the derivative of the inner function.

Stationary points

Points where f'(x) = 0; they are local maxima, local minima, or points of inflection, classified by the sign change of f' or by f''.

Second derivative test

At a stationary point where f'(x) = 0, if f''(x) > 0 the point is a local minimum; if f''(x) < 0 it is a local maximum.

Antiderivative of x^n

For n ≠ -1, ∫ x^n dx = x^{n+1}/(n+1) + C; for n = -1, ∫ (1/x) dx = ln|x| + C.

Definite integral

∫_a^b f(x) dx gives the signed area between the curve and the x-axis from x = a to x = b; areas below the axis count as negative.

Average value of a function

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

Mean and median

The mean is the sum of values divided by the number of values; the median is the middle ordered value, or the average of the two middle values.

Standard deviation

A measure of spread: s = √[Σ(x_i - x̄)²/(n-1)] for a sample; the population standard deviation uses n in the denominator.

Empirical rule

For a Normal distribution, approximately 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.

Standard normal variable

z = (x - μ)/σ converts a Normal variable X with mean μ and standard deviation σ into a standard Normal variable with mean 0 and standard deviation 1.

Addition rule

P(A ∪ B) = P(A) + P(B) - P(A ∩ B); if A and B are mutually exclusive, P(A ∩ B) = 0.

Conditional probability

P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0; it gives the probability of A given that B has occurred.

Binomial distribution

For n independent trials each with success probability p, P(X = k) = C(n,k) p^k (1-p)^{n-k}; mean np, variance np(1-p).

Correlation coefficient

r measures the strength and direction of linear association between two variables; it lies between -1 and 1, with values near ±1 indicating strong linear association.