Advanced Syllabus

Functions

MA-F1 Working with FunctionsYear 11

F1.1: Algebraic techniques

• use index laws and surds
• solve quadratic equations using the quadratic formula and by completing the square
• manipulate complex algebraic expressions involving algebraic fractions

F1.2: Introduction to functions

• define and use a function and a relation as mappings between sets, and as a rule or a formula that defines one variable quantity in terms of another
  • define a relation as any set of ordered pairs ($x$, $y$) of real numbers
  • understand the formal definition of a function as a set of ordered pairs ($x$, $y$) of real numbers such that no two ordered pairs have the same first component (or $x$-component)
• use function notation, domain and range, independent and dependent variables
  • understand and use interval notation as a way of representing domain and range, eg $[4,\infty)$
• understand the concept of the graph of a function
• identify types of functions and relations on a given domain, using a variety of methods
  • know what is meant by one-to-one, one-to-many, many-to-one and many-to-many
  • use the vertical line test to identify a function
  • determine if a function is one-to-one
• define odd and even functions algebraically and recognise their geometric properties
• define the sum, difference, product and quotient of functions and consider their domains and ranges where possible
• define and use the composite function $f(g(x))$ of functions $f(x)$ and $g(x)$ where appropriate
  • identify the domain and range of a composite function
• recognise that solving the equation $f(x)=0$ corresponds to finding the values of $x$ for which the graph of $y=f(x)$ cuts the $x$-axis (the $x$-intercepts)

F1.3: Linear, quadratic and cubic functions

• model, analyse and solve problems involving linear functions
  • recognise that a direct variation relationship produces a straight-line graph
  • explain the geometrical significance of $m$ and $c$ in the equation $f(x)=mx+c$
  • derive the equation of a straight line passing through a fixed point ($x_1,y_1)$ and having a given gradient $m$ using the formula $y-y_1=m(x-x_1)$
  • derive the equation of a straight line passing through two points ($x_1$, $y_1$) and ($x_2$, $y_2$) by first calculating its gradient $m$ using the formula $m=\frac{y_2-y_1}{x_2-x_1}$
  • understand and use the fact that parallel lines have the same gradient and that two lines with gradient $m_1$ and $m_2$ respectively are perpendicular if and only if $m_1m_2=-1$
  • find the equations of straight lines, including parallel and perpendicular lines, given sufficient information
• model, analyse and solve problems involving quadratic functions
  • recognise features of the graph of a quadratic, including its parabolic nature, turning point, axis of symmetry and intercepts
  • find the vertex and intercepts of a quadratic graph by either factorising, completing the square or solving the quadratic equation as appropriate
  • understand the role of the discriminant in relation to the position of the graph
  • find the equation of a quadratic given sufficient information
• solve practical problems involving a pair of simultaneous linear and/or quadratic functions algebraically and graphically, with or without the aid of technology; including determining and interpreting the break-even point of a simple business problem
  • understand that solving $f(x)=k$ corresponds to finding the values of $x$ for which the graph $y=f(x)$ cuts the line $y=k$
• recognise cubic functions of the form: $f(x)=kx^3$, $f(x)=k(x-b)^3+c$ and $f(x)=k(x-a)(x-b)(x-c)$, where $a$, $b$, $c$ and $k$ are constants, from their equation and/or graph and identify important features of the graph

F1.4: Further functions and relations

• define a real polynomial $P(x)$ as the expressions $a_nx^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+x_1x+a_0$ where $n=0,1,2,\ldots$ and $a_0,a_1,a_2,\ldots,a_n$ are real numbers
• identify the coefficients and the degree of a polynomial
• identify the shape and features of graphs of polynomial functions of any degree in factored form and sketch their graphs
• recognise that functions of the form $f(x)=\frac{k}{x}$ represent inverse variation, identify the hyperbolic shape of their graphs and identify their asymptotes
• define the absolute value $|x|$ of a real number $x$ as the distance of the number from the origin on a number line without regard to its sign
• use and apply the notation $|x|$ for the absolute value of the real number $x$ and the graph of $y=|x|$
  • recognise the shape and features of the graph of $y=|ax+b|$ and hence sketch the graph
• solve simple absolute value equations of the form $|ax+b|=k$ both algebraically and graphically
• given the graph of $y=f(x)$, sketch $y=-f(x)$ and $f(-x)$ and $y=-f(-x)$ using reflections in the $x$ and $y$-axes
• recognise features of the graphs of $x^2+y^2=r^2$ and $(x-a)^2+(y-b)^2=r^2$, including their circular shapes, their centres and their radii
  • derive the equation of a circle, centre the origin, by considering Pythagoras’ theorem and recognise that a circle is not a function
  • transform equations of the form $x^2+y^2+ax+by+c=0$ into the form $(x-a)^2+(y-b)^2=r^2$, by completing the square
  • sketch circles given their equations and find the equation of a circle from its graph
  • recognise that $y=\sqrt{r^2-x^2}$ and $y=-\sqrt{r^2-x^2} are functions, identify the semicircular shape of their graphs and sketch them

MA-F2 Graphing TechniquesYear 12

• apply transformations to sketch functions of the form $y=kf(a(x+b))+c$ where $f(x)$ is a polynomial, reciprocal, absolute value, exponential or logarithmic function and $a$, $b$, $c$ and $k$ are constants
  • examine translations and the graphs of $y=f(x)+c$ and $y=f(x+b)$ using technology
  • examine dilations and the graphs of $y=kf(x)$ and $y=f(ax)$ using technology
  • recognise that the order in which transformations are applied is important in the construction of the resulting function or graph
• use graphical methods with supporting algebraic working to solve a variety of practical problems involving any of the functions within the scope of this syllabus, in both real-life and abstract contexts
  • select and use an appropriate method to graph a given function, including finding intercepts, considering the sign of $f(x)$ and using symmetry
  • determine asymptotes and discontinuities where appropriate (vertical and horizontal asymptotes only)
  • determine the number of solutions of an equation by considering appropriate graphs
  • solve linear and quadratic inequalities by sketching appropriate graphs

Trigonometric Functions

MA-T1 Trigonometry and Measure of AnglesYear 11

T1.1: Trigonometry

• use the sine, cosine and tangent ratios to solve problems involving right-angled triangles where angles are measured in degrees, or degrees and minutes
• establish and use the sine rule, cosine rule and the area of a triangle formula for solving problems where angles are measured in degrees, or degrees and minutes
• find angles and sides involving the ambiguous case of the sine rule
  • use technology and/or geometric construction to investigate the ambiguous case of the sine rule when finding an angle, and the condition for it to arise
• solve problems involving the use of trigonometry in two and three dimensions
  • interpret information about a two or three-dimensional context given in diagrammatic or written form and construct diagrams where required
• solve practical problems involving Pythagoras’ theorem and the trigonometry of triangles, which may involve the ambiguous case, including finding and using angles of elevation and depression and the use of true bearings and compass bearings in navigation

T1.2: Radians

• understand the unit circle definition of $\sin\theta$, $\cos\theta$ and $\tan\theta$ and periodicity using degrees
  • sketch the trigonometric functions in degrees for $0\degree\leq x\leq360\degree$
• define and use radian measure and understand its relationship with degree measure
  • convert between the two measures, using the fact that $360\degree=2\pi$ radians
  • recognise and use the exact values of $\sin\theta$, $\cos\theta$ and $\tan\theta$ in both degrees and radians for integer multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$
• understand the unit circle definition of $\sin\theta$, $\cos\theta$ and $\tan\theta$ and periodicity using radians
• solve problems involving trigonometric ratios of angles of any magnitude in both degrees and radians
• recognise the graphs of $y=\sin x$, $y=\cos x$ and $y=\tan x$ and sketch on extended domains in degrees and radians
• derive the formula for arc length, $l=r\theta$ and for the area of a sector of a circle, $A=\frac{1}{2}r^2\theta$
• solve problems involving sector areas, arc lengths and combinations of either areas or lengths

MA-T2 Trigonometric Functions and IdentitiesYear 11

• define the reciprocal trigonometric functions, $y=\cosec x$, $y=\sec x$ and $y=\cot x$
  • $\cosec A=\frac{1}{\sin A}, \sin A\neq0$
  • $\sec A=\frac{1}{\cos A}, \cos A\neq0$
  • $\cot A=\frac{\cos A}{\sin A}, \sin A\neq0$
• sketch the graphs of reciprocal trigonometric functions in both radians and degrees
• prove and apply the Pythagorean identities $\cos^2x+\sin^2x=1$, $1+\tan^2x=\sec^2x$ and $1+\cot^2x=\cosec^2x$
  • know the difference between an equation and an identity
• use $\tan x=\frac{\sin x}{\cos x}$ provided that $\cos x\leq0$
• prove trigonometric identities
• evaluate trigonometric expressions using angles of any magnitude and complementary angle results
• simplify trigonometric expressions and solve trigonometric equations, including those that reduce to quadratic equations

MA-T3 Trigonometric Functions and GraphsYear 12

• examine and apply transformations to sketch functions of the form $y=kf(a(x+b))+c$, where $a$, $b$, $c$ and $k$ are constants, in a variety of contexts, where $f(x)$ is one of $\sin x$, $\cos x$ or $\tan x$, stating the domain and range when appropriate
  • use technology or otherwise to examine the effect on the graphs of changing the amplitude (where appropriate), $y=kf(x)$, the period, $y=f(ax)$, the phase, $y=f(x+b)$, and the vertical shift, $y=f(x)+c$
  • use $k$, $a$, $b$, $c$ to describe transformational shifts and sketch graphs
• solve trigonometric equations involving functions of the form $kf(a(x+b))+c$, using technology or otherwise, within a specified domain
• use trigonometric functions of the form $kf(a(x+b))+c model and/or solve practical problems involving periodic phenomena

Calculus

MA-C1 Introduction to DifferentiationYear 11

C1.1: Gradients of tangents

• distinguish between continuous and discontinuous functions, identifying key elements which distinguish each type of function
  • sketch graphs of functions that are continuous and compare them with graphs of functions that have discontinuities
  • describe continuity informally, and identify continuous functions from their graphs
• describe the gradient of a secant drawn through two nearby points on the graph of a continuous function as an approximation of the gradient of the tangent to the graph at those points, which improves in accuracy as the distance between the two points decreases
• examine and use the relationship between the angle of inclination of a line or tangent, $\theta$, with the positive $x$-axis, and the gradient, $m$, of that line or tangent, and establish that $\tan\theta=m$

C1.2: Difference quotients

• describe the behaviour of a function and its tangent at a point, using language including increasing, decreasing, constant, stationary, increasing at an increasing rate
• interpret and use the difference quotient $\frac{f(x+h)-f(x)}{h}$ as the average rate of change of $f(x)$ or the gradient of a chord or secant of the graph $y=f(x)$
• interpret the meaning of the gradient of a function in a variety of contexts, for example on distance–time or velocity–time graphs

C1.3: The derivative function and its graph

• examine the behaviour of the difference quotient $\frac{f(x+h)-f(x)}{h}$ as $h\to0$ as an informal introduction to h the concept of a limit
• interpret the derivative as the gradient of the tangent to the graph of $y=f(x)$ at a point $x$
• estimate numerically the value of the derivative at a point, for simple power functions
• define the derivative $f'(x)$ from first principles, as $\displaystyle\lim_{h\to0}$$\frac{(fx+h)-f(x)}{h}$ and use the notation for the derivative: $\frac{dy}{dx}=f'(x)=y'$, where $y=f(x)$
• use first principles to find the derivative of simple polynomials, up to and including degree 3
• understand the concept of the derivative as a function
• sketch the derivative function (or gradient function) for a given graph of a function, without the use of algebraic techniques and in a variety of contexts including motion in a straight line
  • establish that $f'(x)=0$ at a stationary point, $f'(x)>0$ when the function is increasing and $f'(x)<0$ when it is decreasing, to form a framework for sketching the derivative function
  • identify families of curves with the same derivative function
  • use technology to plot functions and their gradient functions
• interpret and use the derivative at a point as the instantaneous rate of change of a function at that point
  • examine examples of variable rates of change of non-linear functions

C1.4: Calculating with derivatives

• use the formula $\frac{d}{dx}\left(x^n\right)=nx^{n-1}$ for all real values of $n$
• differentiate a constant multiple of a function and the sum or difference of two functions
• understand and use the product, quotient and chain rules to differentiate functions of the form $f(x)g(x)$, $\frac{f(x)}{g(x)}$ and $f(g(x))$ where $f(x)$ and $g(x)$ are functions
  • apply the product rule: If $h(x)=f(x)g(x)$ then $h'(x)=f(x)g'(x)+f'(x)g(x)$, or if $u$ and $v$ are both functions of $x$ then $\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$
  • apply the quotient rule: If $h(x)=\frac{f(x)}{g(x)}$ then $h'(x)=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$, or if $u$ and $v$ are both functions of $x$ then $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$
  • apply the chain rule: If $h(x)=f(g(x))$ then $h'(x)=f'(g(x))g'(x)$, or if $y$ is a function of $u$ and $u$ is a function of $x$ then $\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$
• calculate derivatives of power functions to solve problems, including finding an instantaneous rate of change of a function in both real life and abstract situations
• use the derivative in a variety of contexts, including to finding the equation of a tangent or normal to a graph of a power function at a given point
• determine the velocity of a particle given its displacement from a point as a function of time
• determine the acceleration of a particle given its velocity at a point as a function of time

MA-C2 Differential CalculusYear 12

C2.1: Differentiation of trigonometric, exponential and logarithmic functions

• establish the formulae $\frac{d}{dx}(\sin x)=\cos x$ and $\frac{d}{dx}(\cos x)=-\sin x$ by numerical estimations of the limits and informal proofs based on geometric constructions
• calculate derivatives of trigonometric functions
• establish and use the formula $\frac{d}{dx}\left(a^x\right)=(\ln a)a^x$
  • using graphing software or otherwise, sketch and explore the gradient function for a given exponential function, recognise it as another exponential function and hence determine the relationship between exponential functions and their derivatives
• calculate the derivative of the natural logarithm function $\frac{d}{dx}(\ln x)=\frac{1}{x}$
• establish and use the formula $\frac{d}{dx}\left(log_ax\right)=\frac{1}{x\ln a}$

C2.2: Rules of differentiation

• apply the product, quotient and chain rules to differentiate functions of the form $f(x)g(x)$, $\frac{f(x)}{g(x)}$ and $f(g(x))$ where $f(x)$ and $g(x)$ are any of the functions covered in the scope of this syllabus, for example #xe^x, $\tan x, $\frac{1}{x^n}$, $x\sin x$, $e^{-x}\sin x$ and $f(ax+b)$
  • use the composite function rule (chain rule) to establish that $\frac{d}{dx}\left\{e^{f(x)}\right\}=f'(x)e^{f(x)}$
  • use the composite function rule (chain rule) to establish that $\frac{d}{dx}\left\{\ln f(x)\right\}=\frac{f'(x)}{f(x)}$
  • use the logarithmic laws to simplify an expression before differentiating
  • use the composite function rule (chain rule) to establish and use the derivatives of $\sin(f(x))$, $\cos(f(x))$ and $\tan(f(x))$

MA-C3 Applications of DifferentiationYear 12

C3.1: The first and second derivatives

• use the first derivative to investigate the shape of the graph of a function
  • deduce from the sign of the first derivative whether a function is increasing, decreasing or stationary at a given point or in a given interval
  • use the first derivative to find intervals over which a function is increasing or decreasing, and where its stationary points are located
  • use the first derivative to investigate a stationary point of a function over a given domain, classifying it as a local maximum, local minimum or neither
  • determine the greatest or least value of a function over a given domain (if the domain is not given, the natural domain of the function is assumed) and distinguish between local and global minima and maxima
• define and interpret the concept of the second derivative as the rate of change of the first derivative function in a variety of contexts, for example recognise acceleration as the second derivative of displacement with respect to time
  • understand the concepts of concavity and points of inflection and their relationship with the second derivative
  • use the second derivative to determine concavity and the nature of stationary points
  • understand that when the second derivative is equal to 0 this does not necessarily represent a point of inflection

C3.2: Applications of the derivative

• use any of the functions covered in the scope of this syllabus and their derivatives to solve practical and abstract problems
• use calculus to determine and verify the nature of stationary points, find local and global maxima and minima and points of inflection (horizontal or otherwise), examine behaviour of a function as $x\to\infty$ and $x\to-\infty$ and hence sketch the graph of the function
• solve optimisation problems for any of the functions covered in the scope of this syllabus, in a wide variety of contexts including displacement, velocity, acceleration, area, volume, business, finance and growth and decay
  • define variables and construct functions to represent the relationships between variables related to contexts involving optimisation, sketching diagrams or completing diagrams if necessary
  • use calculus to establish the location of local and global maxima and minima, including checking endpoints of an interval if required
  • evaluate solutions and their reasonableness given the constraints of the domain and formulate appropriate conclusions to optimisation problems

MA-C4 Integral CalculusYear 12

C4.1: The anti-derivative

• define anti-differentiation as the reverse of differentiation and use the notation $\int f(x)\,dx$ for anti- derivatives or indefinite integrals
• recognise that any two anti-derivatices of $f(x)$ differ by a constant
• establish and use the formula $\int x^n\,dx=\frac{1}{n+1}x^{n+1}+c$, for $n\neq-1$
• establish and use the formula $\int f'(x)[f(x)]^n\,dx=\frac{1}{n+1}[f(x)]^{n+1}+c$ where $n\neq-1$ (the reverse chain rule)
• establish and use the formulae for the anti-derivatives of $\sin(ax+b)$, $\cos(ax+b)$ and $\sec^2(ax+b)$
• establish and use the formulae $\int e^x\,dx=e^x+c$ and $\int e^{ax+b}$\, dx=\frac{1}{a}e^{ax+b}+c$
• establish and use the formulae $\int\frac{1}{x}\,dx=\ln|x|+c$ and $\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+c$ for $x\neq0$, $f(x)\neq0$, respectively
• establish and use the formulae $\int a^x\,dx=\frac{a^x}{\ln a}+c$
• recognise and use linearity of anti-differentiation
  • examine families of anti-derivatives of a given function graphically
• determine indefinite integrals of the form $\int f(ax+b)\,dx$
• determine $f(x)$, given $f'(x)$ and an initial condition $f(a)=b$ in a range of practical and abstract applications including coordinate geometry, business and science

C4.2: Areas and the definite integral

• know that 'the area under a curve' refers to the area between a function and the $x$-axis, bounded by two values of the independent variable and interpret the area under a curve in a variety of contexts
• determine the approximate area under a curve using a variety of shapes including squares, rectangles (inner and outer rectangles), triangles or trapezia
  • consider functions which cannot be integrated in the scope of this syllabus, for example $f(x)=\ln x$, and explore the effect of increasing the number of shapes used
• use the notation of the definite integral $\int_a^b f(x)\,dx$ for the area under the curve $y=f(x)$ from $x=a$ to $x=b$ if $f(x)\geq0$
• use the Trapezoidal rule to estimate areas under curves
  • use geometric arguments (rather than substitution into a given formula) to approximate a definite integral of the form $\int_a^b f(x)\,dx$, where $f(x)\geq0$, on the interval $a\leq x\leq b$, by dividing the area into a given number of trapezia with equal widths
  • demonstrate understanding of the formula:
    $\int_a^bf(x)\,dx\approx\frac{b-a}{2n}\left[f(a)+f(b)+2\left\{f\left(x_1\right)+\dots+f\left(x_{n-1}\right)\right\}\right]$ where $a=x_0$ and $b=x_n$, and the values of $x_0$, $x_1$, $x_2$, $\ldots$, $x_n$ are found by dividing the interval $a\leq x\leq b$ into $n$ sub-intervals
• use geometric ideas to find the definite integral $\int_a^bf(x)\,dx$ where $f(x)$ is positive throughout an interval $a\leq x\leq b$ and the shape of $f(x)$ allows such calculations, for example when $f(x)$ is a straight line in the interval or $f(x)$ is a semicircle in the interval
• understand the relationship of position to signed areas, namely that the signed area above the horizontal axis is positive and the signed area below the horizontal axis is negative
• using technology or otherwise, investigate the link between the anti-derivative and the area under a curve
  • interpret $\int_a^bf(x)\,dx$ as a sum of signed areas
  • understand the concept of the signed area function $F(x)=\int_a^xf(t)\,dt$
• use the formula $\int_a^bf(x)\,dx=F(b)-F(a)$, where $F(x)$ is the anti-derivative of $f(x)$, to calculate definite integrals
  • understand and use the Fundamental Theorem of Calculus, $F'(x)=\frac{d}{dx}\left[\int_a^xf(t)\,dt\right]=f(x)$ and illustrate its proof geometrically
  • use symmetry properties of even and odd functions to simplify calculations of area
  • recognise and use the additivity and linearity of definite integrals
  • calculate total change by integrating instantaneous rate of change
• calculate the area under a curve
• calculate areas between curves determined by any functions within the scope of this syllabus
• integrate functions and find indefinite or definite integrals and apply this technique to solving practical problems

Exponential and Logarithmic Functions

MA-E1 Logarithms and ExponentialsYear 11

E1.1: Introducing logarithms

• define logarithms as indices: $y=a^x$ is equivalent to $x=\log_ay$, and explain why this definition only makes sense when $a>0$, $a\neq1$
• recognise and sketch the graphs of $y=ka^x$, $y=ka^{-x}$ where $k$ is a constant, and $y=log_ax$
• recognise and use the inverse relationship between logarithms and exponentials
  • understand and use the fact that $log_aa^x=x$ for all real $x$ and $a^{\log_ax}=x$ for all $x>0$

E1.2: Logarithmic laws and applications

• derive the logarithmic laws from the index laws and use the algebraic properties of logarithms to simplify and evaluate logarithmic expressions
  $\log_am+\log_an=\log_a(mn)$, $\log_am-\log_an=\log_a\left(\frac{m}{n}\right)$, $\log_a\left(m^n\right)=n\log_am$
  $\log_aa=1$, $\log_a1=0$, $\log_a\frac{1}{x}-\log_ax$
• consider different number bases and prove and use the change of base law $\log_ax=\frac{\log_bx}{\log_ba}$
• interpret and use logarithmic scales, for example decibels in acoustics, different seismic scales for earthquake magnitude, octaves in music or pH in chemistry
• solve algebraic, graphical and numerical problems involving logarithms in a variety of practical and abstract contexts, including applications from financial, scientific, medical and industrial contexts

E1.3: The exponential function and natural logarithms

• establish and use the formula $\frac{d\left(e^x\right)}{dx}=e^x$
  • using technology, sketch and explore the gradient function of exponential functions and determine that there is a unique number $e\approx2.71828182845$, for which $\frac{d\left(e^x\right)}{dx}=e^x$ where $e$ is called Euler's number
• apply the differentiation rules to functions involving the exponential function, $f(x)=ke^{ax}$, where $k$ and $a$ are constants
• work with natural logarithms in a variety of practical and abstract contexts
  • define the natural logarithm $\ln x=\log_ex$ for the exponential function $f(x)=e^x$
  • recognise and use the inverse relationship of functions $y=e^x$ and $y=\ln x$
  • use the natural logarithm and the relationships $e^{\ln x}=x$ where $x>0$, and $\ln\left(e^x\right)=x$ for all real $x$ in both algebraic and practical contexts
  • use the logarithmic laws to simplify and evaluate natural logarithmic expressions and solve equations

E1.4: Graphs and applications of exponential and logarithmic functions

• solve equations involving indices using logarithms
• graph an exponential function of the form $y=a^x$ for $a>0$ and its transformations $y=ka^x+c$ and $y=ka^{x+b}$ where $k$, $b$ and $c$ are constants
  • interpret the meaning of the intercepts of an exponential graph and explain the circumstances in which these do not exist
• establish and use the algebraic properties of exponential functions to simplify and solve problems
• solve problems involving exponential functions in a variety of practical and abstract contexts, using technology, and algebraically in simple cases
• graph a logarithmic function $y=\log_ax$ for $a>0$ and its transformations $y=k\log_ax+c$, using technology or otherwise, where $k$ and $c$ are constants
  • recognise that the graphs of $y=a^x$ and $y=\log_ax$ are reflections in the line $y=x$
• model situations and solve simple equations involving logarithmic or exponential functions algebraically and graphically
• identify contexts suitable for modelling by exponential and logarithmic functions and use these functions to solve practical problems

Statistical Analysis

MA-S1 Probability and Discrete Probability DistributionsYear 11

S1.1: Probability and Venn diagrams

• understand and use the concepts and language associated with theoretical probability, relative frequency and the probability scale
• solve problems involving simulations or trials of experiments in a variety of contexts
  • identify factors that could complicate the simulation of real-world events
  • use relative frequencies obtained from data as point estimates of probabilities
• use arrays and tree diagrams to determine the outcomes and probabilities for multi-stage experiments
• use Venn diagrams, set language and notation for events, including $\bar{A}$ (or $A'$ or $A^c$) for the complement of an event $A$, $A\cap B$ for '$A$ and $B$', the intersection of events $A$ and $B$, and $A\cup B$ for '$A$ or $B$', the union of events $A$ and $B$, and recognise mutually exclusive events
  • use everyday occurrences to illustrate set descriptions and representations of events and set operations
• establish and use the rules: $P(\bar{A})=1-P(A)$ and $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
• understand the notion of conditional probability and recognise and use language that indicates conditionality
• use the notation $P(A|B)$ and the formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$, $P(B)\neq0$ for conditional probability
• understand the notion of independence of an event $A$ from an event $B$ is defined by $P(A|B)=P(A)$
• use the multiplication law $P(A\cap B)=P(A)P(B)$ for independent events $A$ and $B$ and recognise the symmetry of independence in simple probability situations

S1.2: Discrete probability distributions

• define and categorise random variables
  • know that a random variable describes some aspect in a population from which samples can be drawn
  • know the difference between a discrete random variable and a continuous random variable
• use discrete random variables and associated probabilities to solve practical problems
  • use relative frequencies obtained from data to obtain point estimates of probabilities associated with a discrete random variable
  • recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes
  • examine simple examples of non-uniform discrete random variables, and recognise that for any random variable, $X$, the sum of the probabilities is 1
  • recognise the mean or expected value, $E(X)=\mu$, of a discrete random variable $X$ as a measure of centre, and evaluate it in simple cases
  • recognise the variance, $\text{Var}(X)$ and standard deviation ($\sigma$) of a discrete random variable as measures of spread, and evaluate them in simple cases
  • use $\text{Var}(X)=E((X-\mu)^2)=E(X^2)-\mu^2$ for a random variable and $\text{Var}(x)=\sigma^2$ for a dataset
• understand that a sample mean, $\bar{x}$, is an estimate of the associated population mean $\mu$, and that the sample standard deviation, $s$, is an estimate of the associated population standard deviation, $\sigma$, and that these estimates get better as the sample size increases and when we have independent observations

MA-S2 Descriptive Statistics and Bivariate Data AnalysisYear 12

S2.1: Data (grouped and ungrouped) and summary statistics

• classify data relating to a single random variable
• organise, interpret and display data into appropriate tabular and/or graphical representations including Pareto charts, cumulative frequency distribution tables or graphs, parallel box-plots and two-way tables
  • compare the suitability of different methods of data presentation in real-world contexts
• summarise and interpret grouped and ungrouped data through appropriate graphs and summary statistics
• calculate measures of central tendency and spread and investigate their suitability in real-world contexts and use to compare large datasets
  • investigate real-world examples from the media illustrating appropriate and inappropriate uses or misuses of measures of central tendency and spread
• identify outliers and investigate and describe the effect of outliers on summary statistics
  • use different approaches for identifying outliers, for example consideration of the distance from the mean or median, or the use of below $Q_1-1.5\times IQR$ and above $Q_3+1.5\times IQR$ as criteria, recognising and justifying when each approach is appropriate
  • investigate and recognise the effect of outliers on the mean, median and standard deviation
• describe, compare and interpret the distributions of graphical displays and/or numerical datasets and report findings in a systematic and concise manner

S2.2: Bivariate data analysis

• construct a bivariate scatterplot to identify patterns in the data that suggest the presence of an association
• use bivariate scatterplots (constructing them where needed), to describe the patterns, features and associations of bivariate datasets, justifying any conclusions
  • describe bivariate datasets in terms of form (linear/non-linear) and in the case of linear, also the direction (positive/negative) and strength of association (strong/moderate/weak)
  • identify the dependent and independent variables within bivariate datasets where appropriate
  • describe and interpret a variety of bivariate datasets involving two numerical variables using real-world examples in the media or those freely available from government or business datasets
• calculate and interpret Pearson’s correlation coefficient ($r$) using technology to quantify the strength of a linear association of a sample
• model a linear relationship by fitting an appropriate line of best fit to a scatterplot and using it to describe and quantify associations
  • fit a line of best fit to the data by eye and using technology
  • fit a least-squares regression line to the data using technology
  • interpret the intercept and gradient of the fitted line
• use the appropriate line of best fit, both found by eye and by applying the equation of the fitted line, to make predictions by either interpolation or extrapolation
  • distinguish between interpolation and extrapolation, recognising the limitations of using the fitted line to make predictions, and interpolate from plotted data to make predictions where appropriate
• solve problems that involve identifying, analysing and describing associations between two numeric variables
• construct, interpret and analyse scatterplots for bivariate numerical data in practical contexts
  • demonstrate an awareness of issues of privacy and bias, ethics, and responsiveness to diverse groups and cultures when collecting and using data

MA-S3 Random VariablesYear 12

S3.1: Continuous random variables

• use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable
• understand and use the concepts of a probability density function of a continuous random variable
  • know the two properties of a probability density function: $f(x)\geq0$ for all real $x$ and $\int_{-\infty}^\infty f(x)\,dx=1$
  • define the probability as the area under the graph of the probability density function using the notation $P(X\leq r)=\int_a^rf(x)\,dx$, where $f(x)$ is the probability density function defined on $[a, b]$
  • examine simple types of continuous random variables and use them in appropriate contexts
  • explore properties of a continuous random variable that is uniformly distributed
  • find the mode from a given probability density function
• obtain and analyse a cumulative distribution function with respect to a given probability density function
  • understand the meaning of a cumulative distribution function with respect to a given probability density function
  • use a cumulative distribution function to calculate the median and other percentiles

S3.2: The normal distribution

• identify the numerical and graphical properties of data that is normally distributed
• calculate probabilities and quantiles associated with a given normal distribution using technology and otherwise, and use these to solve practical problems
  • identify contexts that are suitable for modelling by normal random variables, eg the height of a group of students
  • recognise features of the graph of the probability density function of the normal distribution with mean $\mu$ and standard deviation $\sigma$, and the use of the standard normal distribution
  • visually represent probabilities by shading areas under the normal curve, eg identifying the value above which the top 10% of data lies
• understand and calculate the 𝑧-score (standardised score) corresponding to a particular value in a dataset
  • use the formula $z=\frac{x-\mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation
  • describe the $z$-score as the number of standard deviations a value lies above or below the mean
• use $z$-scores to compare scores from different datasheets, for example comparing students' subject examination scores
• use collected data to illustrate the empirical rules for normally distributed random variables
  • apply the empirical rule to a variety of problems
  • sketch the graphs of $f(x)=e^{-x^2}$ and the probability density function for the normal distribution $f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ using technology
  • verify, using the Trapezoidal rule, the results concerning the areas under the normal curve
• use $z$-scores to identify probabilities of events less or more extreme than a given event
  • use statistical tables to determine probabilities
  • use technology to determine probabilities
• use $z$-scores to make judgements related to outcomes of a given event or sets of data

Financial Mathematics

MA-M1 Modelling Financial SituationsYear 12

M1.1: Modelling investments and loans

• solve compound interest problems involving financial decisions, including a home loan, a savings account, a car loan or superannuation
  • identify an annuity (present or future value) as an investment account with regular, equal contributions and interest compounding at the end of each period, or a single-sum investment from which regular, equal withdrawals are made
  • use technology to model an annuity as a recurrence relation and investigate (numerically or graphically) the effect of varying the interest rate or the amount and frequency of each contribution or a withdrawal on the duration and/or future or present value of the annuity
  • use a table of interest factors to perform annuity calculations, eg calculating the present or future value of an annuity, the contribution amount required to achieve a given future value or the single sum that would produce the same future value as a given annuity

M1.2: Arithmetic sequences and series

• know the difference between a sequence and a series
• recognise and use the recursive definition of an arithmetic sequence: $T_n=T_{n-1}+d$, $T_1=a$
• establish and use the formula for the $n^\text{th}$ term (where $n$ is a positive integer) of an arithmetic sequence: $T_n=a+(n-1)d$, where $a$ is the first term and $d$ is the common difference, and recognise its linear nature
• establish and use the formulae for the sum of the first 𝑛 terms of an arithmetic sequence: $S_n\frac{n}{2}(a+l)$ where $l$ is the last term in the sequence and $S_n=\frac{n}{2}\left\{2a+(n-1)d\right\}$
• identify and use arithmetic sequences and arithmetic series in contexts involving discrete linear growth or decay such as simple interest

M1.3: Geometric sequences and series

• recognise and use the recursive definition of a geometric sequence: $T_n=rT_{n-1}$, $T_1=a$
• establish and use the formula for the $n$th term of a geometric sequence: $T_n=ar^{n-1}$, where $a$ is the first term, $r$ is the common ratio and $n$ is a positive integer, and recognise its exponential nature
• establish and use the formula for the sum of the first 𝑛 terms of a geometric sequence: $S_n=\frac{a(1-r^n)}{(1-r)}=\frac{a(r^n-1)}{r-1}$
• derive and use the formula for the limiting sum of a geometric series with $|r|<1:S=\frac{a}{1-r}$
  • understand the limiting behaviour as 𝑛 → ∞ and its application to a geometric series as a limiting sum
  • use the notation $\displaystyle\lim_{n\to\infty}r^n=0$ for $|r|<1$

M1.4: Financial applications of sequences and series

• use geometric sequences to model and analyse practical problems involving exponential growth and decay
  • calculate the effective annual rate of interest and use results to compare investment returns and cost of loans when interest is paid or charged daily, monthly, quarterly or six-monthly
  • solve problems involving compound interest loans or investments, eg determining the future value of an investment or loan, the number of compounding periods for an investment to exceed a given value and/or the interest rate needed for an investment to exceed a given value
  • recognise a reducing balance loan as a compound interest loan with periodic repayments, and solve problems including the amount owing on a reducing balance loan after each payment is made
• solve problems involving financial decisions, including a home loan, a savings account, a car loan or superannuation
  • calculate the future value or present value of an annuity by developing an expression for the sum of the calculated compounded values of each contribution and using the formula for the sum of the first $n$ terms of a geometric sequence
  • verify entries in tables of future values or annuities by using geometric series