Math Methods syllabus, dot point by dot point

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

How are cubic and quartic polynomials analysed?

5 min read →

How are linear functions analysed?

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

6 min read →

How are quadratic functions analysed?

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

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

What algebraic skills does VCE Math Methods Unit 1 introduce, including index laws, logarithm laws, and the solution of 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

How is the concept of a rate of change introduced in VCE Math Methods Unit 1, leading to the derivative?

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

How are probabilities computed using counting and combinations?

Linear, quadratic, cubic and quartic polynomial functions, basic exponential functions

What functions and relations are introduced in VCE Math Methods Unit 1, and how are they graphed and transformed?

y = a^x

, logarithmic functions

y = \log_a(x)

, and the standard transformations (dilation, reflection, translation)

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

How are polynomial factors found?

5 min read →

How are surds and rational exponents manipulated?

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

6 min read →

What probability and counting principles does VCE Math Methods Unit 1 introduce?

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

Apply translations, dilations and reflections to the graph of a function

How do transformations affect 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

5 min read →

Unit 2: Functions, calculus and probability

Unit overview →

How are discrete random variables analysed?

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

Define and apply the binomial distribution to model the number of successes in

How is the binomial distribution used to model repeated trials?

n

independent Bernoulli trials, including computing probabilities, expected value

np

and variance

np(1-p)

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

How are logarithmic functions used to solve exponential equations?

Differentiate exponential (

How are exponential and logarithmic functions differentiated?

e^x

a^x

) and logarithmic (

\ln x

\log_b x

) functions, including composite functions via the chain rule

Sketch and analyse exponential functions of the form

How are exponential functions analysed?

y = a \cdot b^{x - h} + k

, identifying key features (intercepts, asymptote, domain, range) and applying transformations

Sketch and analyse trigonometric functions

How are circular functions extended to model periodic phenomena?

y = a\sin(b(x - h)) + k

, identifying amplitude, period, phase and vertical translation, and solve trig equations over a specified interval

y = a\cos(b(x - h)) + k

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

How are trigonometric functions differentiated?

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

How are antidifferentiation and the integral introduced in VCE Math Methods Unit 2?

How are trigonometric functions defined and graphed in VCE Math Methods Unit 2?

Trigonometric functions

y = \sin(x)

y = \cos(x)

, the unit circle, exact values at standard angles, transformations of trig graphs, and solving trigonometric equations

y = \tan(x)

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

How is differentiation used to analyse and optimise functions?

6 min read →

How are inverse and composite functions defined and used in VCE Math Methods Unit 2?

Composite functions

f \circ g

, the existence and form of inverse functions

g \circ f

f^{-1}

, the relationship between a function and its inverse (reflection in

y = x

, domain and range swap), and the one-to-one restriction

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

How are Bernoulli trials, sample data and simulation introduced in VCE Math Methods Unit 2?

Discrete random variables, their probability distributions, the expected value (mean)

Unit 3

Unit overview →

How are the distribution, expected value and variance of a discrete random variable defined and computed?

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)}

(dilation, reflection, translation), composite functions

How do transformations, composites and inverses build new functions from old, and what conditions guarantee they exist?

Transformations from

y = f(x)

y = A f(n(x - b)) + c

(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

10 min read →

How do the factor and remainder theorems let us factorise and analyse polynomials by hand?

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

Graphs of circular functions

What are the graphs of the sine, cosine and tangent functions and what features do they have under transformation?

f(x) = \sin(x)

f(x) = \cos(x)

, their key features (period, amplitude, asymptotes), exact values at standard angles, and graphs of the form

f(x) = \tan(x)

f(x) = a\sin(b(x - h)) + k

Graphs of exponential functions

What are the key features of exponential and logarithmic graphs, and how are they related?

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

Graphs of polynomial functions and key features including stationary points and points of inflection, intercepts, asymptotes, end behaviour, and the graphs of power functions

How are polynomial, power and modulus functions defined, and what are the key features of their graphs?

f(x) = x^n

for

n \in Q

and the modulus function

f(x) = |x|

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

How are the first and second derivatives used to find tangent lines, classify stationary points and sketch curves?

10 min read →

What are Bernoulli trials, when does the binomial distribution apply, and how are its probabilities, mean and variance computed?

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)

Solution of polynomial equations of low degree with real coefficients, exponential and logarithmic equations using properties such as

How are polynomial, exponential, logarithmic and circular equations solved exactly, especially without a calculator?

a^x = e^{x\ln a}

, and circular equations using exact unit-circle values

10 min read →

How is differentiation applied to optimisation problems and to interpreting rates of change?

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