NSWMaths Advanced

Year 11: Functions

9 dot points across 9 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

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
A focused answer to the Year 11 Maths Advanced groundwork: expanding brackets and the special expansions, the four factoring methods (including non-monic quadratics), simplifying and combining algebraic fractions, and solving linear, literal and simultaneous equations, with worked examples and original practice questions.
16 min answer →

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?

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
A Year 11 Maths Advanced answer on domain and range: finding the natural domain by avoiding division by zero and square roots of negatives, reading the range from a sketch, restricted domains, and reading a graph backwards with horizontal lines, with worked parabola and semicircle examples and practice questions.
20 min answer →

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?

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
A Year 11 Maths Advanced answer on functions and notation: the function machine and $f(x)$ notation, evaluating and substituting expressions into a function, the difference between a function and a relation, the vertical line test, and one-to-one, many-to-one, one-to-many and many-to-many classification via the horizontal line test, with worked examples and practice questions.
19 min answer →

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
A Year 11 Maths Advanced answer on power, polynomial and circle graphs: the shapes of even and odd powers of x, sketching a factored cubic with a sign table, circles centred at the origin, and the rectangular hyperbola and its asymptotes, with worked examples and practice questions.
23 min answer →

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
A focused Year 11 Maths Advanced answer on intervals, inequalities and absolute value: open, closed and unbounded intervals on the number line, solving linear inequalities and the one rule that reverses the sign, reading a quadratic inequality off the parabola, and absolute value as distance with the |x| < k and |x| > k templates, with worked examples and practice questions.
18 min answer →

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
A Year 11 Maths Advanced answer on the straight line: gradient as rise over run and as tan of the angle of inclination, the gradient-intercept, point-gradient and general forms, the parallel and perpendicular gradient tests, and the length and midpoint of an interval, with worked examples and practice questions.
22 min answer →

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?

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
A Year 11 Maths Advanced answer on the parabola: sketch by factoring for the intercepts, complete the square for the vertex form and turning point, use the quadratic formula and the discriminant to count and classify the roots, find the axis of symmetry, and read off the maximum or minimum value, with worked examples and practice questions.
24 min answer →

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?

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
A focused answer to the Year 11 Maths Advanced number groundwork: the sets N, Z, Q and R, why a surd is irrational, simplifying surds by taking out square factors, surd arithmetic and binomial expansions, rationalising single-term and binomial-surd denominators with the conjugate, and the index laws for integer indices, with worked examples and original practice questions.
17 min answer →

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?

Translate a known graph vertically and horizontally and reflect it in the $x$ -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
A Year 11 Maths Advanced answer on transformations, symmetry and composite functions: translating and reflecting a known curve, testing even and odd functions algebraically, absolute-value graphs by transformation, and composite functions with their domain, with worked examples and practice questions.
24 min answer →

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

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?

What makes a rule a function rather than just a relation, how does f(x)f(x)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?

How do you sketch the graph of a power function such as y=xny = x^ny=xn, a cubic or polynomial that has been factored into linear factors, a circle centred at the origin, and the rectangular hyperbola y=1xy = \frac{1}{x}y=x1​?

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?

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?

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?

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?

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∣y = |x|y=∣x∣, and form the composite function f(g(x))f(g(x))f(g(x)) and find its domain?

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?

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 $y = \frac{1}{x}$ ?

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?