11–12Mathematics Advanced 11–12 Syllabus

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Overview Rationale Aim Outcomes Content Assessment Glossary Teaching and learning support

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Year 11

Teaching advice

Examples

Working with functions

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
MAV-11-01
applies algebraic techniques and the laws of indices and surds to manipulate expressions and solve problems
MAV-11-02
uses functions and relations to model, analyse and solve problems

Working with functions

Algebraic techniques

Use index laws to simplify expressions and solve problems involving positive, negative, zero or fractional indices
Expand, factorise and simplify algebraic expressions
Simplify expressions involving algebraic fractions
Expand and simplify expressions involving surds
Identify the conjugate of $\sqrt{a} \pm \sqrt{b}$ , and rationalise the denominators of expressions of the form $\frac{a}{\sqrt{b} \pm \sqrt{c}}$ and $\frac{\sqrt{a} \pm \sqrt{d}}{\sqrt{b} \pm \sqrt{c}}$ , where $a$ , $b$ , $c$ and $d$ are positive rational numbers
Solve quadratic equations $a x^{2} + bx + c = 0$ by factorisation, completing the square and using the quadratic formula $x = \frac{- b \pm \sqrt{b^{2} - 4 ac}}{2 a}$ where $a$ , $b$ and $c$ are real numbers and $a \neq 0$
Define the discriminant as $Δ = b^{2} - 4 ac$ and use it to solve problems involving the number of real roots of a quadratic equation and determine the conditions for roots to be equal, distinct, real or rational

Introduction to functions and relations

Describe a relation between two sets as an association between the elements of one set and the elements of the other set
Define a real function $f$ of a real variable $x$ as a relation where each element $x$ of a given set is associated with exactly one element $y$ of the second set
Recognise that a relation is a rule which can be represented by an algebraic formula, a table of values, a set of ordered pairs $(x, y)$ or a graph
Use the notation $f (x)$ to identify the unique value of $y$ associated with $x$ when working with functions, and refer to it as the value of $f$ at $x$
Refer to $x$ in a function $y = f (x)$ as the independent variable of the function, and refer to $y$ as the dependent variable
Substitute numeric and algebraic expressions into the formulas of functions
Apply the vertical line test on the graph of a relation to determine whether it represents a function
Define the domain of a function $f$ as the set of real numbers on which $f$ is defined
Define the range of a function $f$ as the set of values of $f (x)$ obtained as $x$ varies over the domain of $f$
Refer to a value in the domain of a function at which the function is 0 as a zero of the function
Recognise that the $x$ -intercepts of the graph of $y = f (x)$ are its zeroes, the solutions of $f (x) = 0$ , and that the $y$ -intercept is $f (0)$

Linear functions

Determine the equations of straight lines in gradient–intercept form $y = m x + c$ with gradient $m$ and $y$ -intercept $c$
Determine the equations of straight lines in general form $a x + b y + c = 0$ , where $a$ , $b$ and $c$ are constants
Determine the equation of a straight line passing through a point $(x_{1}, y_{1})$ with gradient $m$ using the point–gradient formula $y - y_{1} = m (x - x_{1})$
Determine the equation of a straight line passing through two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ by calculating its gradient $m$ using the formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
Determine the $x$ -intercept and $y$ -intercept of a straight line given its equation
Choose and apply appropriate techniques to graph a straight line given its equation
Find the equation of a line that is parallel or perpendicular to a given line
Solve linear inequalities and graph the solution on a number line

Quadratic and cubic functions

Identify the $x$ -intercepts of a parabola whose quadratic function is expressed in factored form
Use the discriminant to determine the number of $x$ -intercepts on a parabola and justify its position in relation to the $x$ -axis
Show by completing the square on the general quadratic $y = a x^{2} + bx + c$ that the axis of symmetry is $x = - \frac{b}{2 a}$ where $a$ , $b$ and $c$ are constants
Identify the axis of symmetry and vertex of a parabola by completing the square on its quadratic function
Choose and apply appropriate techniques to graph a parabola of the form $y = {ax}^{2} + bx + c$ by identifying its $x$ -intercepts if they exist, its $y$ -intercept, its axis of symmetry using $x = - \frac{b}{2 a}$ and its vertex
Find the equation of a parabola given sufficient graphical features
Use the fact that two quadratic functions are equal for all values of $x$ if and only if the corresponding coefficients are equal to solve related problems
Recognise that solving $f (x) = k$ for some constant $k$ corresponds to finding the $x$ -coordinate(s) of the intersection of the graphs $y = f (x)$ and $y = k$
Solve problems by finding the solution to simultaneous equations involving a linear and a quadratic function, or two quadratic functions, both algebraically and graphically
Solve quadratic inequalities
Recognise and graph cubic functions of the form $f (x) = k x^{3}$ and $f (x) = k (x - a) (x - b) (x - c)$ , where $a$ , $b$ , $c$ and $k$ are constants and $k \neq 0$

Reciprocal functions

Graph functions of the form $f (x) = \frac{k}{x}$ , where $k$ is a constant and $k \neq 0$ , and identify their hyperbolic shape and their asymptotes
Describe the behaviour of $f (x) = \frac{k}{x}$ as $x \to \infty$ and $x \to - \infty$

Constructing and using functions

Construct and use linear functions to model and solve problems in real-world situations, identifying the independent and dependent variables and any restrictions on these variables, and justify conclusions in the context of the problem
Use linear inequalities to model and solve problems in real-world situations, and justify conclusions in the context of the problem
Solve practical problems involving a pair of simultaneous linear equations both algebraically and graphically, with and without graphing applications, and justify conclusions in the context of the problem
Construct and use simultaneous equations to model and solve a problem where cost and revenue are represented by linear equations, identify and analyse the break-even point, and justify conclusions in the context of the problem
Model and solve practical problems involving quadratic functions and justify conclusions in the context of the problem

Direct and inverse variation

Develop models of the form $y = k x^{n}$ , where $k$ is a non-zero constant, from descriptions of situations in which one quantity varies directly with another
Develop the model $y = \frac{k}{x^{n}}$ , where $k$ is a non-zero constant, from descriptions of situations in which one quantity varies inversely with another
Evaluate $k$ in the equations $y = k x^{n}$ and $y = \frac{k}{x^{n}}$ , given one pair of values for the variables, and use the resulting formula to find other values of the variables
Analyse and solve problems involving direct and inverse variation

Circles and semicircles

Derive the equation of a circle of radius $r$ with centre at the origin by considering Pythagoras’ theorem
Graph circles of the form $x^{2} + y^{2} = r^{2}$ from their equations
Determine the equation of a circle of the form $x^{2} + y^{2} = r^{2}$ given its graph
Identify and graph the semicircles $y = \sqrt{r^{2} - x^{2}}$ , $y = - \sqrt{r^{2} - x^{2}}$ , $x = \sqrt{r^{2} - y^{2}}$ and $x = - \sqrt{r^{2} - y^{2}}$

Properties of functions, relations and graphs

Extend the definitions of domain and range to relations
Recognise domains and ranges of functions and relations given in interval notation, as inequalities and as worded descriptions
Determine and describe the domain and range of functions and relations, using interval notation, inequalities or worded descriptions
Define a function to be even if its graph is unchanged under reflection in the $y$ -axis, and odd if its graph is unchanged under rotation of $180 °$ about the origin
Develop and use the tests that a function $f (x)$ is odd if $f (- x) = - f (x)$ and a function $f (x)$ is even if $f (- x) = f (x)$
Solve problems involving even and odd functions
Use the composite function $f (g (x))$ , where the output of $g (x)$ becomes the input of $f (x)$
Determine the equations of composite functions

Piecewise-defined functions

Interpret piecewise-defined functions, where the function is defined differently in different parts of the domain
Graph piecewise-defined functions involving functions covered in the scope of the Mathematics Advanced course, test if they are even or odd, and determine the domain and range
Define informally that a function is continuous at a point if the curve can be drawn through the point without lifting the pen off the paper
Identify points where piecewise-defined functions and other functions are not continuous
Define a discontinuity of a function informally as a point where the function is not continuous

Absolute value functions

Define the absolute value $|x|$ of a number $x$ , also known as the magnitude of $x$ , to be the distance from the origin to $x$ on the number line
Establish and use the piecewise definition $|x| = \{\begin{matrix} x, for x \geq 0, \\ - x, for x < 0 \end{matrix}$
Show using numerical substitutions that $\sqrt{x^{2}} = |x|$ and use the result
Graph the function $y = |x|$ , describe its symmetry, and identify its domain and range
Graph $y = |ax + b|$ with and without graphing applications, and identify its symmetry, domain and range
Solve absolute value equations of the form $|ax + b| = k$ algebraically and graphically

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11–12Mathematics Advanced 11–12 Syllabus

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