Plumpton High School Mathematics Advanced HSC 2026 Examination Mapping

Functions and Graphing Techniques

• MA-F1.3: Linear, Quadratic, and Cubic Functions

  • Find the equations of straight lines, including parallel and perpendicular lines, given sufficient information.

  • Relevant Outcomes: MA11-1, MA11-2.

• MA-F1.4: Further Functions and Relations

  • Identify the shape and features of graphs of polynomial functions of any degree in factored form.

  • Sketch polynomial graphs based on these features.

  • Relevant Outcomes: MA11-1, MA11-2.

• MA-F2: Graphing Techniques

  • Odd and Even Functions:

    • Define odd and even functions algebraically.

    • Recognise their geometric properties.

  • Graphing Methods:

    • Select and use appropriate methods to graph a given function.

    • Find intercepts.

    • Consider the sign of $f(x)$.

    • Utilise symmetry.

  • Asymptotes and Discontinuities:

    • Determine vertical and horizontal asymptotes and discontinuities where appropriate.

  • Transformations:

    • Use the constants $k$, $a$, $b$, and $c$ to describe transformational shifts.

    • Sketch graphs incorporating these transformations.

  • Relevant Outcomes: MA12-1.

Statistical Analysis

• MA-S1.1: Probability and Venn Diagrams

  • Multi-stage Experiments:

    • Use arrays and tree diagrams to determine outcomes and probabilities.

  • Independent Events:

    • Use the multiplication law for independent events $A$ and $B$: $P(A \cap B) = P(A)P(B)$.

    • Recognise the symmetry of independence in simple probability situations.

  • Relevant Outcomes: MA11-7, MA12-8.

• MA-S2.1: Data (Grouped and Ungrouped) and Summary Statistics

  • Organise, interpret, and display data into appropriate tabular and/or graphical representations.

  • Includes but is not limited to: Pareto charts, cumulative frequency distribution tables or graphs, parallel box-plots, and two-way tables.

  • Relevant Outcomes: MA12-8.

• MA-S2.2: Bivariate Data Analysis

  • Pearson’s Correlation Coefficient (r):

    • Calculate and interpret $r$ using technology to quantify the strength of a linear association of a sample.

  • Line of Best Fit:

    • Use the appropriate line of best fit, found both by eye and by applying the equation of the fitted line.

    • Make predictions using interpolation or extrapolation.

  • Relevant Outcomes: MA12-8.

Calculus: Differentiation

• MA-C1.4: Calculating with Derivatives

  • Use the derivative in various contexts.

  • Find the equation of a tangent or a normal to a graph of a power function at a given point.

  • Relevant Outcomes: MA11-1, MA11-5.

• MA-C2.1: Differentiation of Trigonometric, Exponential, and Logarithmic Functions

  • Establish and use the formula: $\frac{d}{dx}(a^x) = (\ln(a))a^x$

  • Relevant Outcomes: MA12-3, MA12-6, MA12-7.

• MA-C2.2: Rules of Differentiation

  • Apply the product, quotient, and chain rules to differentiate functions of the forms:

    • $f(x)g(x)$

    • $\frac{f(x)}{g(x)}$

    • $f(g(x))$

  • Apply these rules to functions within the syllabus scope, including: $xe^x$, $\tan(x)$, $\frac{1}{x^n}$, $x\sin(x)$, $e^{-x}\sin(x)$, and $f(ax + b)$.

  • Relevant Outcomes: MA12-3, MA12-6, MA12-7.

• MA-C3.1: The First and Second Derivatives

  • Use the second derivative to determine concavity and the nature of stationary points.

  • Relevant Outcomes: MA12-3, MA12-6.

• MA-C3.2: Applications of the Derivative

  • Graph Sketching and Analysis:

    • Determine and verify the nature of stationary points.

    • Find local and global maxima and minima.

    • Identify points of inflection (horizontal or otherwise).

    • Examine function behaviour as $x \rightarrow \infty$ and $x \rightarrow -\infty$.

    • Sketch the graph of the function based on these findings.

  • Optimisation:

    • Define variables and construct functions to represent relationships in optimization contexts.

    • Sketch and complete diagrams as necessary.

    • Solve optimization problems for functions within the syllabus scope.

  • Relevant Outcomes: MA12-3, MA12-6.

Calculus: Integration

• MA-C4.1: The Anti-derivative

  • Establish and use the following formulae:

    • $\int \frac{1}{x}\,dx = \ln|x| + c$ for $x \neq 0$

    • $\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + c$ for $f(x) \neq 0$

  • Relevant Outcomes: MA12-3, MA12-7.

• MA-C4.2: Areas and the Definite Integral

  • General Integration:

    • Integrate functions and find indefinite or definite integrals.

    • Apply integration techniques to solve practical problems.

  • Kinematics:

    • Determine the acceleration of a particle given its velocity at a point as a function of time.

  • Area Under a Curve:

    • 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 $f(x) \geq 0$.

  • Estimation:

    • Use the Trapezoidal rule to estimate areas under curves.

  • Area Between Curves:

    • Calculate areas between curves determined by any functions within the syllabus scope.

  • Relevant Outcomes: MA12-3, MA12-7.

Trigonometric Functions

• MA-T1.1: Trigonometry

  • Solve practical problems involving Pythagoras’ theorem and the trigonometry of triangles.

  • Address the ambiguous case in triangles.

  • Applications include finding and using angles of elevation and depression, and using true bearings and compass bearings in navigation.

  • Relevant Outcomes: MA11-1, MA11-3, MA11-4.

• MA-T1.2: Radians

  • Derive and use the formula for arc length: $l = r\theta$

  • Derive and use the formula for the area of a sector of a circle: $A = \frac{1}{2}r^2\theta$

  • Relevant Outcomes: MA11-1, MA11-3, MA11-4.

• MA-T2: Trigonometric Functions and Identities

  • Prove trigonometric identities.

  • Simplify trigonometric expressions.

  • Solve trigonometric equations, including those that reduce to quadratic equations.

  • Relevant Outcomes: MA11-1, MA11-4.

• MA-T3: Trigonometric Functions and Graphs

  • Transformations and Sketching:

    • Examine and apply transformations to sketch functions of the form $y = kf(a(x + b)) + c$.

    • Constants: $a$, $b$, $c$, and $k$.

    • Function types: $f(x) = \sin(x)$, $f(x) = \cos(x)$, or $f(x) = \tan(x)$.

    • State the domain and range when appropriate.

  • Solving Equations:

    • Solve trigonometric equations involving functions of the form $kf(a(x + b)) + c$ using technology or other methods within a specified domain.

  • Relevant Outcomes: MA12-1, MA12-5.

Financial Mathematics

• MA-M1.1: Modelling Investments and Loans

  • Use a table of future value interest factors to perform annuity calculations.

  • Calculations include: finding the future value of an annuity, determining the contribution amount required for a goal, or finding the single sum equivalent to a given annuity.

  • Relevant Outcomes: MA12-2, MA12-4.

• MA-M1.2: Arithmetic Sequences and Series

  • 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.

    • $S_n = \frac{n}{2}\{2a + (n - 1)d\}$

  • Relevant Outcomes: MA12-2, MA12-4.

• MA-M1.3: Geometric Sequences and Series

  • Sum of n Terms:

    • Establish and use the formula: $S_n = \frac{a(1 - r^n)}{1 - r} = \frac{a(r^n - 1)}{r - 1}$

  • Limiting Sum:

    • Derive and use the formula for the limiting sum of a geometric series when $|r| < 1$: $S = \frac{a}{1 - r}$

  • Relevant Outcomes: MA12-2, MA12-4.

• MA-M1.4: Financial Applications of Sequences and Series

  • Solve problems involving financial decisions.

  • Contexts include, but are not limited to: home loans, savings accounts, car loans, or superannuation.

  • Relevant Outcomes: MA12-2, MA12-4.