Maths Advanced syllabus, dot point by dot point

Describe the rate of change of a quantity, distinguishing the average rate of change over an interval (the gradient of the chord joining two points on the curve) from the instantaneous rate of change at a point (the gradient of the tangent there), estimate an instantaneous rate of change, read and interpret rates from a graph, and recognise that the gradient of the tangent at each point defines a new function, the derivative

What is the difference between an average rate of change and an instantaneous rate of change, how does each one show up as a gradient on a graph (a chord for the average, a tangent for the instant), how do you estimate an instantaneous rate when you cannot read the tangent exactly, and how does collecting the gradient at every point build a brand new function, the derivative?

Year 11: Exponential and Logarithmic Functions

Define logarithms as indices, convert between index form and logarithmic form, apply the logarithm laws (product, quotient, power), use the logarithms of 1 and of the base, change the base, and work with common logarithms

What is a logarithm really, how do you switch between index form and log form, and how do the log laws let you expand, contract, evaluate and solve - including the cases a calculator only does in base 10?

Define a radian as the angle whose arc length equals the radius, convert between degrees and radians using pi rad = 180 deg, know the exact radian values of the common angles, and use the radian formulas for arc length L = r theta, sector area A = (1/2) r^2 theta and the area of a segment as the sector minus the triangle

What is a radian, why does measuring an angle by arc length over radius make pi rad = 180 deg, what are the exact radian values of the common angles, and how do the clean formulas arc length L = r theta, sector area A = (1/2) r^2 theta and segment area (sector minus triangle) follow once an angle is measured in radians?

Define Euler's number e as the base for which y=e^x has gradient 1 at the y-intercept, work with the natural logarithm ln x = log_e x as the inverse of e^x, sketch y=e^x and y=ln x as reflections in y=x, transform y=e^x, and solve e^x=k and ln x=k by converting between the two forms

What is the special number e (about 2.718), why is it singled out as the base whose graph y=e^x has gradient exactly 1 where it crosses the y-axis, what is the natural logarithm ln x as its inverse, and how do you convert between e^x and ln to solve equations and transform the curve?

Graph exponential and logarithmic functions, identify their asymptotes, use the reflection in the line y=x that makes them inverse functions, apply translations, reflections and dilations, and state the domain and range of each

What do the graphs of y=a^x and y=log_a x actually look like, why are they exact mirror images of each other, where are their asymptotes, and how do translations, reflections and dilations move them while you read off the domain and range?

Apply the index laws to expressions with rational indices: use zero, negative and fractional indices, simplify and evaluate index expressions, solve simple index equations, and write numbers in scientific notation

What do the index laws really mean once the index can be zero, negative or a fraction, and how do you simplify index expressions, solve a simple index equation and use scientific notation without a slip?

Year 11: Functions

Define and use function notation, distinguish a function from the more general relation using the vertical line test, and classify relations as one-to-one, many-to-one, one-to-many or many-to-many using the horizontal line test

What makes a rule a function rather than just a relation, how does $f(x)$ notation let us name and evaluate that rule, and how do the vertical and horizontal line tests sort every graph into one of four types?

Sketch a parabola by finding its intercepts by factoring, complete the square to write a quadratic in vertex form and read off the turning point and axis of symmetry, use the quadratic formula and the discriminant to find and classify the roots, and find the maximum or minimum value of a quadratic

How do you sketch a parabola from its equation, find its turning point and axis of symmetry, decide how many times it crosses the x-axis, and read off its maximum or minimum value?

24 min read →

How do you sketch the graph of a power function such as $y = x^n$ , a cubic or polynomial that has been factored into linear factors, a circle centred at the origin, and the rectangular hyperbola $y = \frac{1}{x}$ ?

Sketch the graphs of power functions

y = x^n

and contrast even and odd powers, sketch a cubic or higher polynomial that is factored into linear factors using a sign table, sketch circles

x^2 + y^2 = r^2

centred at the origin and recognise shifted circles, and sketch the rectangular hyperbola

y = \frac{1}{x}

with its asymptotes

23 min read →

How do we expand, factor and simplify algebraic expressions, and solve linear and simultaneous equations fluently enough to support the whole Advanced course?

Use algebraic techniques to expand, factor and simplify expressions and algebraic fractions, and to solve linear and simultaneous equations

Find the natural domain of a function (avoiding division by zero and square roots of negatives), determine the range from a sketch, work with restricted domains, and read the range from a graph using horizontal lines

What inputs is a function allowed to take, what outputs does it actually reach, and how do you read both straight off a graph, including a restricted piece and a graph read backwards with horizontal lines?

Work with the real number system and surds: simplify surds, add, multiply and expand surdic expressions, rationalise single- and binomial-surd denominators, and apply the index laws to expressions with integer indices

What are the real numbers, and how do we simplify surds, rationalise denominators and apply the index laws without leaving a surd in the wrong place or a slip in the arithmetic?

17 min read →

How do we describe a piece of the number line, solve linear and quadratic inequalities without losing the direction of the sign, and read absolute value as distance?

Use interval notation and number-line graphs, solve linear and quadratic inequalities, and work with the absolute value definition to solve equations and inequalities of the form |x| < k and |x| > k

Translate a known graph vertically and horizontally and reflect it in the

How do you transform a known curve by translating and reflecting it, test a function for even or odd symmetry, sketch an absolute-value graph from $y = |x|$ , and form the composite function $f(g(x))$ and find its domain?

x

-axis and the

y

-axis, recognise and test even functions (symmetric about the

y

-axis,

f(-x) = f(x)

) and odd functions (symmetric about the origin,

f(-x) = -f(x)

), sketch absolute-value graphs as transformations of

y = |x|

, and form composite functions

f(g(x))

and determine their domain

24 min read →

How do you measure the steepness of a line, write its equation in the form a question wants, decide when two lines are parallel or perpendicular, and find the length and midpoint of the interval joining two points?

Work with the gradient of a line as rise over run and as the tangent of its angle of inclination, write the equation of a line in gradient-intercept, point-gradient and general form, use the parallel and perpendicular gradient conditions, and find the length and midpoint of an interval

22 min read →

Year 11: Trigonometry

What do the graphs of $y = \sin x$ , $y = \cos x$ and $y = \tan x$ look like in degrees, what do amplitude and period mean and how do you read them, and how do you solve an equation like $2\cos x = 1$ or $\sin x = -\tfrac12$ for every solution in a stated range such as $0\degree$ to $360\degree$ ?

Sketch the graphs of

y = \sin x

y = \cos x

in degrees over one or more periods, identify their amplitude (where it exists) and period and the key maximum, minimum, zero and intercept points, and solve trigonometric equations of the form

y = \tan x

\sin x = k

\cos x = k

for all solutions in a given domain using the graph together with the related acute angle and the ASTC rule

\tan x = k

Solve three-dimensional problems involving right-angled triangles, including finding the relevant right-angled triangle inside a solid such as a rectangular prism or pyramid, the angle between a line and a plane, and problems that combine right-triangle results across different planes

How do you solve a problem set in a three-dimensional solid by finding the one right-angled triangle inside it that carries the answer, and how do you measure the angle a line makes with a plane?

Establish and apply the sine rule

How do you find an unknown side or angle in a triangle that has no right angle, when do you reach for the sine rule and when for the cosine rule, why can the sine rule give two answers (the ambiguous case), and how do you find a triangle's area from two sides and the angle between them?

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

(including the ambiguous case when finding an angle), the cosine rule

a^2 = b^2 + c^2 - 2bc\cos A

to find a side and its rearrangement to find an angle, and the area rule

\text{Area} = \dfrac12 ab\sin C

, to solve problems involving non-right-angled triangles

Use the trigonometric ratios sine, cosine and tangent to find unknown sides and angles in right-angled triangles, including the exact ratios of 30, 45 and 60 degrees, and apply them to angles of elevation and depression and to compass and true bearings

How do the sine, cosine and tangent ratios fix the unknown sides and angles of a right-angled triangle, and how do they solve angles of elevation and depression and compass bearings?

17 min read →

Why is $\sin^2\theta + \cos^2\theta = 1$ true for every angle, how does it link to $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ , and how do you use the two together to prove a trigonometric identity and to simplify a trigonometric expression?

Prove and apply the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

and its rearrangements, and the ratio identity

\tan\theta = \dfrac{\sin\theta}{\cos\theta}

, to prove further trigonometric identities by transforming one side to the other and to simplify trigonometric expressions

17 min read →

How do sine, cosine and tangent make sense for an angle that is not acute, and how do you find the exact value of something like $\cos 150\degree$ or $\tan 210\degree$ using the unit circle, the ASTC sign rule and the related acute angle?

Extend the definitions of sine, cosine and tangent to any angle using the unit circle and the four quadrants, use the ASTC rule for the signs of the ratios, find the related acute angle, determine exact values of the trigonometric functions at angles around the circle, and find one ratio given another together with the quadrant, all in degrees

Year 12: Calculus

Use the trapezoidal rule to estimate areas and definite integrals, and determine whether the estimate is an over- or under-estimate

How can we estimate the area under a curve, or a definite integral, when we only have a table of values or an integral we cannot evaluate exactly?

Apply calculus to motion in a straight line, with displacement, velocity and acceleration as derivatives and integrals with respect to time

How do we use calculus to analyse the motion of a particle moving in a straight line?

Solve problems involving related and general rates of change, including integrating a given rate dQ/dt to recover a quantity with an initial condition, and interpreting rate graphs

How do we use calculus to model the rate at which a quantity changes, recover the quantity from its rate, and read a rate graph?

Apply the product, quotient and chain rules, and differentiate exponential, logarithmic and trigonometric functions

How do we differentiate functions built from products, quotients and compositions of standard functions?

Use the rules of differentiation to find the derivatives of

How do the chain and quotient rules give the derivatives of the reciprocal trigonometric functions, and how are those derivatives used in tangent, normal and stationary point problems?

\operatorname{cosec} x

\sec x

\cot x

Find antiderivatives of standard functions, apply integration by substitution and evaluate definite integrals using the Fundamental Theorem of Calculus

How do we find antiderivatives and use the Fundamental Theorem of Calculus to evaluate definite integrals?

Use definite integrals to solve problems involving the areas of regions bounded by the graph of the continuous function

How do we find the area between a curve and the y-axis, and how does the reflection of exponential and logarithmic graphs in y = x help with areas?

y = f(x)

, the

y

-axis and the lines

y = a

changes sign in the interval

y = b

, in cases where

x \geq 0

throughout

a \leq y \leq b

x \leq 0

throughout

a \leq y \leq b

or where

x

a \leq y \leq b

with or without the graph provided

Calculate the area under a curve, the area between two curves, and the volume of a solid of revolution about the

How do we use definite integrals to compute areas between curves and volumes of solids of revolution?

x

y

axis

Establish and solve differential equations of the form

How do we model continuous growth, decay and cooling using differential equations?

\frac{dN}{dt} = k N

and apply them to growth, decay and Newton's law of cooling

\frac{dT}{dt} = k(T - T_a)

Use the first and second derivatives to find stationary points, points of inflection, and to solve optimisation and related rates problems

How can the derivative be used to analyse curves and solve optimisation and rate problems?

Find derivatives and integrals of

How do we differentiate and integrate exponential and logarithmic functions, and how do they appear in modelling?

e^x

, including composed forms, and apply them to modelling problems

\ln x

Find derivatives and integrals of

How do we differentiate and integrate trigonometric functions and use them to model periodic phenomena?

\sin

\cos

(with linear inside arguments) and apply them to model simple harmonic and periodic motion

\tan

Year 12: Financial Mathematics

Use simple and compound interest formulas to find future values, present values, interest rates and time periods

How do simple and compound interest accumulate value over time, and how do we move money between present and future?

Derive and use the future value formula for an annuity to find the accumulated value of a series of equal regular contributions

How do equal regular contributions to an investment grow over time, and what is the future value formula for an annuity?

Use the formulas for the nth term and the sum of n terms of a geometric sequence, and the limiting sum, in financial contexts

How do geometric sequences and series model repeated payments and recurring growth, and when does an infinite series converge?

Use the formulas for the nth term and the sum of n terms of an arithmetic sequence in financial and modelling contexts

How do arithmetic sequences and series model quantities that grow by a fixed amount each period, such as simple interest, salary rises and straight-line depreciation?

Use recurrence relations and the present value of an annuity to find loan repayments, outstanding balances and total interest paid

How are reducing-balance loan repayments calculated, and how much of each payment goes to interest versus principal?

Year 12: Functions

Sketch and interpret graphs of exponential and logarithmic functions, including transformations, and use the inverse relationship between them

What are the key features of exponential and logarithmic graphs, and how do transformations and the inverse relationship link them?

Sketch graphs of sums, differences, products, quotients, squares and reciprocals of two known functions

How do we sketch graphs built from sums, differences, products, quotients and reciprocals of standard functions?

Form composite functions, determine when a function has an inverse, find and graph the inverse, and use restriction of domain to invert non-one-to-one functions

When does a function have an inverse, and how do we form, evaluate and graph composite and inverse functions?

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

How do translations, reflections and dilations transform the graph of a function in a predictable way?

Year 12: Statistical Analysis

Define a discrete random variable by its probability distribution, and calculate the expected value, variance and standard deviation

How do we describe a discrete random variable and summarise its distribution with mean and variance?

Construct scatter plots, calculate and interpret Pearson's correlation coefficient, and fit and use the least-squares regression line

How do we describe and model the relationship between two numerical variables?

Use the normal distribution, z-scores, the empirical rule and the standard normal table to find probabilities and percentiles

How do we use the normal distribution and z-scores to compute probabilities and compare observations?

Display and summarise univariate data using frequency and cumulative-frequency tables, histograms, ogives, the mean and standard deviation, the five-number summary, box plots and outliers

How do we display a single set of data and summarise its centre, spread and shape?

Apply tree diagrams, conditional probability, independence and complementary events to solve multi-step probability problems

How do we combine probabilities across several stages, and update a probability once we know something has happened?

Use probability density functions and cumulative distribution functions to find probabilities, medians, modes, means and variances of continuous random variables

How do probability density functions describe continuous random variables, and how do we extract probabilities and summary statistics from them?

Year 12: Trigonometric Functions

Sketch and interpret graphs of

How do amplitude, period, phase shift and vertical shift transform the graphs of sine, cosine and tangent?

y = a \sin(b x + c) + d

y = a \cos(b x + c) + d

, identifying amplitude, period, phase shift and vertical shift

y = a \tan(b x + c) + d

Use radian measure to find arc length, the area of a sector, and the area of a segment of a circle

How are radians defined, and how do we use them to find arc length and sector area?

Solve trigonometric equations over a given interval using exact values, the unit circle, and identities to reduce to a single trig function

How do we find all solutions of a trigonometric equation in a given interval, including equations involving multiple angles and identities?

Use Pythagorean, ratio, double angle and complementary identities to simplify expressions and prove equalities

Which trigonometric identities are essential for simplifying expressions and proving equivalences in HSC Maths Advanced?

Solve problems using the sine rule, cosine rule and area rule, including angles of elevation and depression, bearings and three-dimensional applications

How do the sine, cosine and area rules solve non-right-angled triangles, and how are they applied to elevation, depression, bearings and three-dimensional problems?