Math Methods syllabus, dot point by dot point

Algebraic manipulation of polynomial, exponential and logarithmic expressions, including index laws, logarithm laws, factorisation, and the solution of linear, quadratic, polynomial, exponential and logarithmic equations

Average rates of change between two points, the gradient of a chord, the gradient at a point as a limit, and the derivative of polynomial functions using the power rule

Sketch cubic and quartic polynomials, identifying intercepts, end behaviour, turning points and points of inflection, and using factored form to read roots and multiplicities

Linear, quadratic, cubic and quartic polynomial functions, basic exponential functions $y = a^x$, logarithmic functions $y = \log_a(x)$, and the standard transformations (dilation, reflection, translation)

Define inverse and composite functions, identify when an inverse function exists (one-to-one), find inverse functions algebraically, and graph inverse and composite functions

Sketch and analyse linear functions of the form $y = mx + c$, including finding gradient, $x$- and $y$-intercepts, equations of parallel and perpendicular lines, and solving linear equations and inequalities

Apply the factor theorem and the remainder theorem to factorise polynomials and to solve polynomial equations

Counting principles (multiplication principle, permutations and combinations), set notation, simple probability, conditional probability and the addition / multiplication rules

Apply the rules of probability (addition, multiplication, conditional), the counting principles (permutations and combinations), and use these to find probabilities in compound experiments

Sketch and analyse quadratic functions in standard, factored and turning-point form, including finding vertex, axis of symmetry, intercepts and using the discriminant to classify roots

Simplify and operate on surd expressions and apply the laws of indices to rational and negative exponents

Apply translations, dilations and reflections to the graph of a function $y = f(x)$, including the form $y = a f(b(x - h)) + k$ and the effect of each parameter on the graph

Use differentiation to analyse the behaviour of functions, including locating and classifying stationary points, finding tangent and normal equations, and solving optimisation problems

Define and apply the binomial distribution to model the number of successes in $n$ independent Bernoulli trials, including computing probabilities, expected value $np$ and variance $np(1-p)$

Antidifferentiation as the reverse of differentiation, the antiderivative of polynomial functions via the power rule, the constant of integration, and the use of an initial condition to determine a specific antiderivative

Sketch and analyse trigonometric functions $y = a\sin(b(x - h)) + k$ and $y = a\cos(b(x - h)) + k$, identifying amplitude, period, phase and vertical translation, and solve trig equations over a specified interval

Differentiate exponential ($e^x$, $a^x$) and logarithmic ($\ln x$, $\log_b x$) functions, including composite functions via the chain rule

Differentiate sine, cosine and tangent functions and their compositions via the chain rule

Define a discrete random variable and its probability distribution, and compute expected value (mean) and variance for given distributions

Sketch and analyse exponential functions of the form $y = a \cdot b^{x - h} + k$, identifying key features (intercepts, asymptote, domain, range) and applying transformations

Composite functions $f \circ g$ and $g \circ f$, the existence and form of inverse functions $f^{-1}$, the relationship between a function and its inverse (reflection in $y = x$, domain and range swap), and the one-to-one restriction

Define logarithms as the inverse of exponentials, apply the laws of logarithms, sketch logarithmic graphs and solve exponential equations using logs

Bernoulli trials and sequences of Bernoulli trials, sample data analysis (mean, median, mode, range), simulation of random processes, and the relationship between theoretical probability and observed relative frequency

Trigonometric functions $y = \sin(x)$, $y = \cos(x)$ and $y = \tan(x)$, the unit circle, exact values at standard angles, transformations of trig graphs, and solving trigonometric equations

Bernoulli trials, the binomial distribution $X \sim \mathrm{Bi}(n, p)$, its probability function $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$, mean $E(X) = np$, and variance $\mathrm{Var}(X) = np(1-p)$

Graphs of circular functions $f(x) = \sin(x)$, $f(x) = \cos(x)$ and $f(x) = \tan(x)$, their key features (period, amplitude, asymptotes), exact values at standard angles, and graphs of the form $f(x) = a\sin(b(x - h)) + k$

Average and instantaneous rates of change, the definition of the derivative as a limit $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, and the use of this definition to differentiate from first principles

The product, quotient and chain rules of differentiation, and the derivatives of standard functions $x^n$ for $n \in Q$, $e^x$, $\ln(x)$, $\sin(x)$, $\cos(x)$ and $\tan(x)$

Discrete random variables, their probability distributions, the expected value (mean) $E(X) = \sum x P(X = x)$, the variance $\mathrm{Var}(X) = E(X^2) - [E(X)]^2$ and the standard deviation $\mathrm{sd}(X) = \sqrt{\mathrm{Var}(X)}$

Graphs of exponential functions $f(x) = a^x$ (in particular $f(x) = e^x$) and logarithmic functions $f(x) = \log_a(x)$ (in particular $f(x) = \ln(x)$), including their key features and the inverse relationship

The factor theorem and the remainder theorem for polynomial functions, the method of equating coefficients, and the factorisation of cubic and quartic polynomials over the rationals

Applications of differentiation to optimisation problems (maximising or minimising a quantity subject to constraints) and to rates of change in modelled real-world contexts

Graphs of polynomial functions and key features including stationary points and points of inflection, intercepts, asymptotes, end behaviour, and the graphs of power functions $f(x) = x^n$ for $n \in Q$ and the modulus function $f(x) = |x|$

Random experiments, sample spaces, events and probabilities, including the addition rule, conditional probability $P(A|B) = \frac{P(A \cap B)}{P(B)}$, the multiplication rule, and the concept of independence

Solution of polynomial equations of low degree with real coefficients, exponential and logarithmic equations using properties such as $a^x = e^{x\ln a}$, and circular equations using exact unit-circle values

Equations of tangents and normals to graphs of functions, stationary points and points of inflection, use of the first and second derivatives to classify stationary points, and curve sketching

Transformations from $y = f(x)$ to $y = A f(n(x - b)) + c$ (dilation, reflection, translation), composite functions $(f \circ g)(x) = f(g(x))$ and the conditions for their existence, and inverse functions $f^{-1}$ with the link to one-to-one functions

Antidifferentiation as the reverse of differentiation, including the antiderivatives of $x^n$ for $n \in Q$ and $n \neq -1$, $e^{kx}$, $\frac{1}{x}$, $\sin(kx)$ and $\cos(kx)$, and the use of the constant of integration

The use of definite integrals to find the area between a curve and the $x$-axis, and the area between two curves on a closed interval, including handling sign changes of the integrand

Applications of integration including the average value of a function on a closed interval, total change from a rate of change function, and kinematics (displacement and distance from velocity)

Approximate confidence intervals for a population proportion $p$ based on the sample proportion $\hat{p}$, including the standard 90, 95 and 99 percent intervals and their interpretation

Continuous random variables, their probability density functions, cumulative distribution functions, expected value (mean), variance and standard deviation, and computation of probabilities as definite integrals

The definite integral, the fundamental theorem of calculus linking definite integration to antidifferentiation, and the properties of the definite integral over intervals

Hybrid (piecewise-defined) functions, their continuity and differentiability conditions, inverse functions $f^{-1}$ where defined, and the reflection of $y = f(x)$ in the line $y = x$

The use of substitution to evaluate integrals of the form $\int f(g(x)) g'(x) \, dx$, recognising the reverse of the chain rule

The normal distribution with mean $\mu$ and standard deviation $\sigma$, the standard normal $Z$, the use of the empirical 68/95/99.7 rule, and computation of normal probabilities and inverse probabilities using technology or standard tables

The application of differentiation, including the chain rule, to related rates of change problems involving two or more time-dependent quantities

The sample proportion $\hat{p}$ as a random variable, the sampling distribution of $\hat{p}$ for repeated samples of size $n$ from a population with true proportion $p$, and the normal approximation for large $n$