Functions, relations and graphs

The definitive exam reference, built from the 2006–2026 archive.

Compiled 15 September 2026. Companion to 01-study-design.md; nothing here contradicts it. Where that document establishes what the study design says, this one establishes what VCAA actually asks, how the state actually performs, and what the reports say went wrong.

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0. Sources, scope and method

0.1 The dataset for this area

• 512 question parts, 744 marks, across the twenty November papers 2006–2025 plus 2024/2025/2026 NHT.
• 498 of those parts carry a published pct (percentage of the state awarded full marks). The 14 without are NHT parts, for which VCAA publishes answers but no statistics.
• Functions, relations and graphs is the largest single area of study in the archive by marks: 744 of the 2 379 November marks that carry a topic, or 31.3%. Calculus is second at 729, statistics 586, algebra 320.

Two readings matter.

1. Exam 1 is where this area kills people. Six questions in ten are separators when the content is functions and graphs and the technology is gone. That is worse than the paper-wide rate and worse than the same content in Section B.
2. Section A is the safe harbour — 59.9% mean, only a third separators — right up until the last four questions, where the abstract parameter items live. Every Section A separator from 2018 onwards sits at Q11 or higher except 2023 Q3 and 2022 Q6.

0.2 Marks by year

The 2011 row is small because the 2011 extraction is degraded (01-study-design.md §0.1); do not read a trend into it. The stable picture from 2016 onwards is 10–18 Exam 1 marks and 5–11 Section A marks every year, with Section B swinging between 12 and 33 depending on whether the paper contains a dedicated "functions" question.

0.3 Method note on counts

Phrase counts below were taken across the 40 November paper files (2006–2025). Two caveats that apply to every count: the 2024 November papers extract as empty files, so 2024 never registers a phrase hit; and 2011 is OCR-damaged, so its hits are unreliable. Percentages are always the published pct from [QJSON] and are never estimated.

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1. What the study design puts in this area — and what the exams take from it

1.1 The content dot points, verbatim

[SD], the complete list for this area of study:

• graphs of polynomial functions and their key features
• graphs of the following functions: power functions, y = xⁿ, n ∈ Q; exponential functions, y = aˣ, a ∈ R, in particular y = eˣ; logarithmic functions, y = logₑ(x) and y = log₁₀(x); and circular functions, y = sin(x), y = cos(x) and y = tan(x) and their key features
• transformation from y = f(x) to y = A f(n(x + b)) + c, where A, n, b and c ∈ R, A, n ≠ 0, and f is one of the functions specified above, and the inverse transformation
• the relation between the graph of an original function and the graph of a corresponding transformed function (including families of transformed functions for a single transformation parameter)
• graphs of sum, difference, product and composite functions involving functions of the types specified above (not including composite functions that result in reciprocal or quotient functions)
• modelling of practical situations using polynomial, power, circular, exponential and logarithmic functions, simple transformation and combinations of these functions, including simple piecewise (hybrid) functions.

The area-of-study overview:

"In this area of study students cover transformations of the plane and the behaviour of some elementary functions of a single real variable, including key features of their graphs such as axis intercepts, stationary points, points of inflection, domain (including maximal, implied or natural domain), co-domain and range, asymptotic behaviour and symmetry."

1.2 The key skills that decide the questions

From [SD] Outcome 1 key skills, the four that this area is built on:

• identify key features and properties of the graph of a function or relation and draw the graphs of specified functions and relations, clearly identifying their key features and properties, including any vertical or horizontal asymptotes
• describe the effect of transformations on the graphs of a function or relation
• find the rule of an inverse function and give its domain and range
• find the rule of a composite function and give its domain and range
• sketch by hand graphs of polynomial functions up to degree 4; simple power functions, y = xⁿ where n ∈ N, y = aˣ, (using key points (−1, 1/a), (0, 1) and (1, a)); logₑ(x); log₁₀(x); and simple transformations of these

Note the recurring clause: "and give its domain and range" appears in both the inverse and the composite key skill. It is not decorative. Across the archive, the domain half of an inverse or composite question is consistently worth a mark of its own and consistently answered worse than the rule.

1.3 What the exams actually test — phrase counts across the November papers

Read against the dot points, the mapping is:

• "graphs … and their key features" is by a distance the most examined phrase. Thirty-two explicit Sketch the graph instructions in seventeen of twenty papers, and a matching State/Find the range in nine years. This dot point is examined every single year in both papers.
• The transformation dot point is examined every year, but its surface form migrated. transformation T — the matrix or mapping-notation form — occurs seventeen times across 2006–2020 and zero times after 2020. sequence of transformations occurs in 2007, 2013, 2015, 2020, 2023, 2025. The 2023 study design deleted "the matrix representation of points and transformations of the plane" and the key skill "apply matrices to transformations"; the papers agree.
• "families of transformed functions for a single transformation parameter" is the phrase that licenses the hardest items in the area. It is new-ish emphasis and it is where the state collapses: see §4, question type T20, which produces more separators than any other type.
• Composite functions appear in ten of twenty years. The [SD] exclusion — "not including composite functions that result in reciprocal or quotient functions" — is respected in every archive instance found.
• Hybrid (piecewise) functions never appear under that word in the question stems; VCAA writes the brace-defined rule and calls it nothing, or calls it "the piecewise function w" (2023 Exam 2 Section B Q2). Searching for "hybrid" returns three files, all reports.
• amplitude occurs only four times in twenty years (2006, 2010, 2014, 2023) while period occurs thirty-three times in sixteen. If you are budgeting revision, period is eight times more likely to be asked by name than amplitude.

1.4 The one piece of key knowledge that has left

01-study-design.md §3.2 records that the 2016–2022 dot point on functional relations (f(x + k) = f(x), f(xy) = f(x)f(y), and so on) was deleted in 2023. That deletion matters here more than anywhere: eight of the 239 separators in this area are functional-equation items (2006 Exam 2 Section A Q17, 2007 Exam 2 Section A Q17, 2007 Exam 2 Section B Q4di and Q4dii, 2012 Exam 2 Section A Q19, 2015 Exam 2 Section A Q18, 2016 Exam 2 Section A Q11, 2017 Exam 2 Section A Q13). They are marked in §4 as out of scope. Do not drill them.

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2. The complete catalogue of question types

Thirty-two types, grouped. For each: VCAA's literal wording, what it tests, the method, archive instances with ref and pct, mark allocation, and the reports' own account of the failures.

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Group A — Sketching

T1. By-hand sketch with labelled key features (Exam 1)

Wording. 2016 Exam 1 Q3a, verbatim: "Sketch the graph of f. Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation." 2023 Exam 1 Q3a: "Sketch the graph of f(x) = 2/(x−1) + 3 on the axes below, labelling all asymptotes with their equations and axial intercepts with their coordinates." 2025 Exam 1 Q3c: "Sketch the graph of y = f(x) for x ∈ [−π/2, 3π/2] on the axes below. Label the endpoints with their coordinates."

What it tests. Shape, position, curvature, asymptotic behaviour, and — decisively — obeying the labelling instruction to the letter. The instruction line is the marking scheme.

Method. (i) Read the label list and write it down as a checklist before drawing. (ii) Compute every listed feature exactly, in coordinate form. (iii) Draw the asymptotes as dashed lines first. (iv) Draw the curve, using the printed gridlines. (v) Stop at the stated domain and mark the endpoints closed or open.

Instances. 2023 E1 Q3a 66% (3 marks) · 2018 E1 Q3b 66% (3) · 2014 E1 Q5b 63% (2) · 2022 E1 Q6a 57% (2) · 2016 E1 Q3a 57% (3) · 2021 E1 Q4a 56% (3) · 2015 E1 Q4b 54% (2) · 2012 E1 Q5a 46% (3) · 2024 E1 Q3a 42% (3) · 2019 E1 Q4b 37% (2) · 2017 E1 Q3b 36% (3) · 2025 E1 Q3c 33% (2) · 2007 E1 Q3a 25% (3) · 2006 E1 Q4b 14% (3).

Marks. 2 or 3. Never more.

Traps, in the reports' words. 2006 E1 Q4b: "It was disappointing to see the number of graphs that were either not smooth or not symmetric, or both… x-intercepts were often incorrect and not written as coordinates, the y-intercept was often ignored and graphs sometimes did not cover the entire domain." 2017 E1 Q3b: "graphs that looked more like an inverted parabola rather than a cubic, due to lack of recognition of a stationary point located at the y-intercept." 2024 E1 Q3a: "Common errors included not labelling the asymptotes… incorrectly determining the coordinates of intercepts, and not indicating the symmetry of the curve. Some students mistakenly sketched a hyperbola." 2025 E1 Q3c: "Students are encouraged to pay attention to the symmetry of the curve and to use the grid lines to assist with accurately positioning the curve… some students drew graphs that looked more like parabolas."

T2. Technology-active sketch with stated accuracy (Exam 2 Section B)

Wording. 2012 Exam 2 Section B Q2a: "Sketch the graph of y = f(x) on the set of axes below. Label the axes intercepts with their coordinates and label each of the asymptotes with its equation." The Exam 2 variant adds a rounding instruction: 2006 Exam 2 Section B Q1c-type wording, "Label any stationary points with their coordinates, correct to two decimal places."

What it tests. Whether you can transfer a calculator display onto printed axes without losing the features. The maths is free; the marks are in the transcription.

Method. Define the function on CAS with the exact domain. Read off every labelled feature to one more decimal place than asked, then round. Plot the named features first as points, then join.

Instances. 2012 E2 B Q2a 43% (3) · 2008 E2 B Q1ci 42% (2) · 2008 E2 B Q4e 26% (2) · 2011 E2 B Q1c 5% (2, 2011 caveat) · non-separator comparators: 2024 E2 B Q3bi 63% (2), 2025 E2 B Q1b 62% (2), 2015 E2 B Q4b 67% (2).

Marks. 2 or 3.

Traps. 2008 E2 B Q1ci: "The local maximum had to be labelled with its coordinates… others rounded incorrectly or gave their answers correct to only one decimal place when the question asked for two. The x-axis must be scaled." 2025 E2 B Q1b: "Other students did not scale their graphs well on the axes. The x-intercept was often closer to 2 than 1. Some students did not draw the stationary point of inflection correctly."

T3. Sketch the inverse on the given axes

Wording. 2023 Exam 1 Q7b, verbatim: "Sketch the graph of the inverse function y = f⁻¹(x) on the axes above. Label any endpoints and axial intercepts with their coordinates."

What it tests. That the inverse is the reflection in y = x, and that the domain of f⁻¹ is the range of f — visible as where the sketch stops.

Method. Reflect the named points of f through y = x by swapping coordinates. Reflect the asymptotes the same way (x = a ↔ y = a). Draw y = x faintly if it is not printed. Stop the curve at the reflected endpoints.

Instances. 2023 E1 Q7b 47% (2) · 2010 E2 B Q1aii 44% (3) · 2008 E1 Q10b 19% (1) · 2007 E2 B Q4cii 15% (2).

Marks. 1–3.

Traps. 2007 E2 B Q4cii: "Many students drew the graph of h⁻¹ : [−∞, 1) → R… Only 15 percent of students restricted the domain correctly." 2008 E1 Q10b (sketch y = f(f⁻¹(x)) over its maximal domain): "Students should be aware that the composition of a function and its inverse results in the function y = x over the appropriate domain. Many graphs were either exponential or logarithmic. Students who drew the line y = x often drew it over R even if they had stated the correct domain in part a."

T4. Deduce and sketch the derivative graph

Wording. 2024 Exam 1 Q7c requires the graph of f′ across a stated domain, with the given local maxima and minima of f used as x-intercepts of f′. 2021 Exam 2 Section A Q8: "The graph of the function f is shown below… The graph corresponding to f′ is".

What it tests. The sign-and-zero correspondence between a graph and its gradient function, including at endpoints.

Method. Mark where f has stationary points → x-intercepts of f′. Mark where f rises/falls → f′ above/below the axis. Endpoints of f give endpoints of f′ at the same x-values, not on the x-axis.

Instances. 2021 E2 A Q8 40% (1) · 2024 E1 Q7c 15% (3). The same task classified under Calculus: 2009 E2 B Q1eiii 17% (3).

Marks. 1 in Section A, 2–3 by hand.

Traps. 2024 E1 Q7c: "Many students calculated the coordinates of the endpoints correctly but then positioned them incorrectly. Some students failed to label the coordinates of the endpoints while others incorrectly had the endpoints on the x-axis."

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Group B — Key features and notation

T5. State the range (or an interval) in correct set notation

Wording. 2023 Exam 1 Q7a: "State the range of f." 2025 Exam 1 Q3a: "State the range of f." 2006 Exam 1 Q7b: "State the range of the function with rule y = |f(x)| and domain [−5, 1]."

What it tests. Reading a range off a restricted domain, and writing it with the right brackets.

Method. Evaluate the function at both endpoints; find any interior stationary value; the range is between the least and greatest of those. Square bracket where the value is attained, round where it is not.

Instances. 2025 E1 Q3a 81% (1) · 2023 E1 Q7a 90% (1) · 2017 E1 Q7a 65% (1) · 2021 E1 Q3a 78% (1) · then the separators: 2007 E2 A Q6 39% · 2006 E1 Q7b 35% · 2018 E2 A Q3 48% · 2020 E2 A Q18 43% · 2024 E1 Q7bii 20% · 2019 E1 Q8c 1% (2).

Marks. 1, occasionally 2 when domain and range are both wanted.

Traps. 2006 E1 Q7b: "Common incorrect responses were R, (0, 20), [20, 0], R⁺, and [0, 320]." Note [20, 0] — reversed. 2023 E1 Q7a: "The most common errors involved stating the correct range values with incorrect brackets or swapping the interval values… Students are reminded that mathematical notation is a precise language." 2025 E1 Q3a: "It is important to note that incorrectly stating the range of values in terms of y (as in −1 ≤ y ≤ 3) is not acceptable notation" — a range is a set, not an inequality in the dependent variable.

T6. Maximal domain

Wording. 2022 Exam 1 Q5b: "Find the maximal domain of f, where f(x) = logₑ(x² − 2x − 3)." 2024 NHT Exam 1 Q3a: "State the maximal domain for f(x)." Section A form, 2022 Exam 2 Q13: "The function f(x) = logₑ((x − a)/(x + a)), where a is a positive real constant, has the maximal domain…"

What it tests. Knowing which operation constrains the domain (log argument > 0; even root ≥ 0; denominator ≠ 0), then solving a quadratic inequality and writing the answer as a union.

Method. Write the constraint. Factorise. Sketch the parabola or use a sign diagram. Read off the intervals. Join with ∪, never ∩.

Instances. 2022 E1 Q5b 27% (3) · 2021 E2 B Q3a 36% (2) · 2022 E2 A Q13 39% · 2021 E2 B Q1fi 33% (1) · 2021 E2 B Q1b 42% (1) · 2019 E1 Q8b 9% (1) · 2011 E1 Q4b–4b 1% (2, 2011 caveat).

Marks. 1–3.

Traps. 2022 E1 Q5b: "A common error was writing the interval as an intersection not a union." 2011 E1 Q4b: "Common incorrect responses included [−3, 3] (the domain of f(x)); x ≥ 2 (as the 'intersection' of x ≤ −8 with x ≥ 2); or x ≤ −8 (as the 'union' of x ≤ −8 with x ≥ 2)." 2019 E1 Q8b: the maximal domain of a difference of logs requires the intersection of the two separate log constraints and the excluded zero — answer (−1, 0) ∪ (0, 1).

T7. Period, amplitude, asymptotes and symmetry of circular functions

Wording. 2006 Exam 1 Q4a: "write down the amplitude and period of the function." 2021 Exam 1 Q3b: "State the period of g." Section A form, 2019 Exam 2 Q1: "The period and range of f are respectively…" Parameter form, 2018 Exam 2 Q11: "The graph of y = tan(ax), where a ∈ R⁺, has a vertical asymptote x = 3π and has exactly one x-intercept in the region (0, 3π). The value of a is…"

What it tests. period = 2π/n for sin and cos, π/n for tan — and the ability to run it backwards from a printed graph.

Method. Identify n from the rule. Period = 2π/|n| (sin, cos) or π/|n| (tan). Amplitude = |A|. Range = [c − |A|, c + |A|]. For tan, asymptotes at n(x + b) = π/2 + kπ.

Instances. 2021 E1 Q3b 89% (1) · 2013 E2 B Q1b 88% (1) · 2019 E2 B Q3a 76% (1) · 2006 E1 Q4a 79% (2) · 2010 E1 Q4a 64% (2) · separators: 2018 E2 A Q11 26% · 2020 E1 Q3 23% (3, find a and b in y = tan(ax + b) from a graph) · 2023 E2 A Q18 29% (number of local minima of sin(ax) on [−a, a]) · 2021 E2 B Q5c 21% (smallest h with f(h − x) = f(x)) · 2025 E2 B Q4c 15% (find k and the largest a with f(x + k) = −f(x)).

Marks. 1–3.

Traps. 2006 E1 Q4a: "The most common error was stating the period as 2π." 2010 E1 Q4a: "it was disappointing to see the number of students who had a period of 6π." The symmetry variants (2021 Q5c, 2025 Q4c) are separators because they require reasoning about a shift that maps the graph to itself or its negative, which is a half-period statement, not a formula.

T8. Stationary points, turning points, inflection, and intervals of monotonicity

Wording. 2023 Exam 2 Section B Q1b: "Find the coordinates of the stationary points of f." 2009 Exam 2 Section B Q1a: "State the interval for which the graph of f is strictly decreasing." 2023 Exam 2 Section B Q3e: "Find the largest interval of x values for which h is strictly decreasing. Give your answer correct to two decimal places."

What it tests. That an interval of monotonicity is closed at a stationary point, and that "coordinates" means an ordered pair.

Method. Solve f′(x) = 0. Substitute back for the y-coordinate. For monotonicity, the answer is the closed interval between consecutive stationary x-values, intersected with the domain.

Instances. 2009 E2 B Q1a 72% (2) · 2018 E2 B Q1a 95% (1) · 2023 E2 B Q1b 68% (2) · separators: 2013 E2 A Q19 50% · 2012 E2 A Q8 49% · 2007 E2 B Q2c 48% (2) · 2008 E2 B Q3f 43% (2) · 2023 E2 B Q3e 35% (1) · 2013 E1 Q9a 50% (1).

Marks. 1–2.

Traps. 2023 E2 B Q3e: "Round brackets were often seen; these were incorrect as the largest interval of x values was required, which included the interval endpoints… These students have incorrectly interpreted the question requirements as asking for intervals where the function is strictly increasing." 2008 E2 B Q3f: "Many students gave the correct coordinates for A but then gave the coordinates for B, not C. Some gave their answers as 16 and 24" — that is, as numbers rather than coordinates.

T9. Solve an inequality by reading your own sketch

Wording. 2023 Exam 1 Q3b: "Find the values of x for which f(x) ≥ 1." 2021 Exam 1 Q4b: "Find the values of x for which 1/(x − 2) + 2 ≥ 3." 2025 Exam 1 Q7dii: "Write down the values of x for which f(x)g(x) ≤ 0."

What it tests. Whether you use the graph you were just made to draw, rather than restarting algebraically.

Method. Solve the corresponding equation for the boundary values. Then read the graph — already drawn in the previous part — to decide which side of each boundary satisfies the inequality, remembering that an asymptote breaks the interval.

Instances. 2023 E1 Q3b 38% (1) · 2021 E1 Q4b 32% (1) · 2025 E1 Q7dii 27% (1).

Marks. 1. Always.

Traps. 2023 E1 Q3b: "many students did not use their graph from part 3a. to assist them to correctly identify the interval required." 2021 E1 Q4b: "Most students attempted to solve algebraically instead of using the graph, and only obtained the lower bound of inequality." This type is a 1-mark separator three times out of three. It is the cheapest mark in the area and the state loses it every time.

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Group C — Transformations

T10. Describe a sequence of transformations in words

Wording. 2015 Exam 2 Section B Q4c: "State a sequence of two transformations that maps the graph of y = f(x) to the graph of y = h(x)." 2017 Exam 2 Section B Q4gi: "Describe the transformation that maps the graph of g₁ onto the graph of g_k." 2023 Exam 2 Section B Q5a: "Complete a possible sequence of transformations to map f to g." 2025 Exam 2 Section B Q1e: "Write down a possible sequence of three transformations to map from g to h." 2026 NHT Exam 2: "State a sequence of two transformations, a dilation followed by a translation, that will map the graph of y = h(t) to the graph of y = p(t)" — followed immediately by the same question with "a translation followed by a dilation".

What it tests. The vocabulary, the direction, and the order.

Method. Write g(x) in the form A f(n(x + b)) + c. Then, reading from the inside out: dilation factor 1/|n| from the y-axis; reflection in the y-axis if n < 0; translation −b in the x-direction; dilation factor |A| from the x-axis; reflection in the x-axis if A < 0; translation c in the y-direction. A vertical translation may be moved anywhere in the sequence after the vertical dilation; a horizontal translation may not be moved across a horizontal dilation without changing its magnitude.

Instances. 2020 E2 B Q1d 76% (1) · 2018 E2 B Q3b 78% (1) · separators: 2017 E2 B Q4a 48% (2) · 2016 E2 B Q1e 10% (3, matrix form) · 2015 E2 A Q11 24% · 2015 E2 B Q4c 41% (2) · 2014 E2 B Q5b 37% (1) · 2007 E2 B Q3di 13% (2) · 2017 E2 B Q4gi 31% (1) · 2017 E2 B Q4gii 30% (1) · 2023 E2 B Q5a 35% (2) · 2024 E1 Q5b 2% (2) · 2024 E2 B Q1di 37% (1) · 2024 E2 B Q1dii 5% (2) · 2025 E2 A Q20 36%.

Marks. 1–3.

Traps. The 2024 Exam 1 report, on the 2%-scoring Q5b: "Many students were able to list one transformation, usually the dilation; however, frequently the incorrect axis or direction was specified. Students are urged to use the correct language when referring to transformations." And in general comments: "Students need to use correct mathematical language when describing transformations of graphs and students are encouraged to follow the study design for the correct expression of these descriptions. For dilations, students should be familiar with both 'parallel to an axis' and 'from an axis' descriptions." 2024 E2 B Q1dii: "The vertical translation could be completed at any stage in the sequence. The other transformations had to be in the correct order." 2023 E2 B Q5a: "The order of the transformations needed to be correct, as well as the wording. A common incorrect answer was 'reflect in the y-axis and then translate 2 units to the left'." 2015 E2 B Q4c: "Many students did not give the correct wording when describing the dilations." 2017 E2 B Q4gi: "for example, 'dilation of a factor 1/k in the y-axis'" — in is wrong; the words are from or parallel to.

T11. Apply a given sequence and find the image rule, domain, or feature

Wording. 2013 Exam 1 Q9c: "The following sequence of transformations is applied to the graph of the function g: [−2, 2] → R, g(x) = |f(x)| + 1. • a translation of one unit in the negative direction of the x-axis • a translation of one unit in the negative direction of the y-axis • a dilation from the x-axis of factor 1/3. Find i. the rule of the image of g after the sequence of transformations has been applied ii. the domain of the image of g after the sequence of transformations has been applied." 2021 Exam 1 Q5b: "Let the graph of h be a transformation of the graph of f where the transformations have been applied in the following order: • dilation by a factor of 1/2 from the vertical axis (parallel to the horizontal axis) • translation by two units to the right… State the rule of h and the coordinates of the horizontal axis intercepts of the graph of h."

What it tests. Executing transformations in the stated order — including the composition rule that a dilation applied after a translation changes the translation's effect.

Method. Use x′ and y′ mapping equations, one line per transformation, in order. Solve for x and y in terms of x′, y′. Substitute into the original rule. Then rename. Track the domain by applying only the horizontal parts of the map to the endpoints.

Instances. 2013 E1 Q9ci 16% (2) · 2013 E1 Q9cii 39% (1) · 2021 E1 Q5b 18% (2) · 2012 E1 Q5bii 26% (2) · 2007 E1 Q8b 20% (2) · 2022 E1 Q6cii 46% (1) · 2022 E1 Q6ciii 12% (1) · 2016 E2 A Q20 17% · 2017 E2 A Q10 47% · 2012 E2 A Q22 32% · 2009 E2 A Q9 49%.

Marks. 1–3.

Traps. 2013 E1 Q9ci: "The order of transformations may have been different from what many students had experienced and proved to be the stumbling block for most." 2013 E1 Q9cii: "The frequency of [−2, 2] as a preferred solution raises the concern that students believe that the domain does not change under transformations. They may have confused this with rules applied to either composite functions or the addition, subtraction and multiplication of functions." 2021 E1 Q5b: "Some could not properly express the horizontal dilation, and, with the translation, many did not use brackets around the term." 2012 E1 Q5bii: "It is important that the original function and image functions have distinct names" — the 2013 report repeats this: "Note the use of a different function name for the image."

T12. Matrix / mapping-notation transformation: find the parameters

Wording. 2010 Exam 1 Q6: "The transformation T : R² → R² is defined by T([x, y]) = [3 0; 0 2][x, y] + [−1, 4]. The image of the curve y = 2x² + 1 under the transformation T has equation y = ax² + bx + c. Find the values of a, b and c." 2017 Exam 2 Section B Q4a: "The transformation T : R² → R², T([x, y]) = [x + c, y + d] maps the graph of y = 2ˣ onto the graph of f. State the values of c and d." 2012 Exam 2 Section B Q2e: "A transformation T: R² → R² that maps the graph of f to the graph of the function g… has rule T([x, y]) = [a 0; 0 1][x, y] + [c, d], where a, c and d are non-zero real numbers. Find the values of a, c and d."

What it tests. Reading a matrix as a pair of mapping equations, and inverting them.

Method. Write x′ = ax + c, y′ = by + d. Invert: x = (x′ − c)/a, y = (y′ − d)/b. Substitute into the original equation. Rearrange to the image form and equate coefficients.

Status. This type has not appeared since 2020. 01-study-design.md §6.1 records the deletion of the matrix key knowledge and key skill in 2023; the corpus confirms zero transformation T hits in 2021, 2022, 2023, 2025 papers. It remains in the archive as good transformation practice but is no longer an examinable form.

Instances. 2006 E2 A Q13 33% · 2010 E1 Q6 23% (3) · 2013 E2 A Q20 25% · 2012 E2 B Q2e 8% (2) · 2016 E2 B Q1e 10% (3) · 2017 E2 B Q4a 48% (2) · 2018 E1 Q9c 34% (1) · 2019 E2 B Q3e 10% (2) · 2020 E2 A Q13 26% · 2020 E2 B Q5h 2% (1).

Traps. 2010 E1 Q6: "Students who attempted to define the transformations… often misinterpreted both the dilations and the horizontal translation, and then did not rearrange to construct the new rule. Students who chose to determine the inverse matrix often made errors in their inverse and/or the multiplication of the matrices." 2012 E2 B Q2e: "Some students gave the transformations that map the graph of g to f" — i.e. the inverse direction.

T13. Fit transformation parameters so the image equals a named target

Wording. 2019 Exam 1 Q2c: "Let g be the function obtained by applying the transformation T to the function f, where T([x, y]) = [x + c, y + d] and c, d ∈ R. Find the values of c and d given that g = f⁻¹." 2023 Exam 2 Section B Q1d: "Let h : R → R, h(x) = (x − a)(x − b)², where h(x) = f(x) + k and a, b, k ∈ R. Find the possible values of a and b."

What it tests. Working backwards from a property of the image (it equals the inverse; it has a repeated root) to the parameters.

Method. Write the transformed rule with the unknown parameters. Write the target rule. Equate coefficients — the reports repeatedly prefer this to technology here.

Instances. 2019 E1 Q2c 24% (1) · 2023 E2 B Q1d 13% (4) · 2013 E2 B Q1fi 10% (2) · 2013 E2 B Q1fii 12% (1) · 2007 E2 B Q3dii 13% (2).

Marks. 1–4. The 4-mark instance (2023) is the single highest-value separator in the modern era of this area.

Traps. 2019 E1 Q2c: "Some students had the incorrect sign for c and d. Other students attempted dilations rather than translations as specified by the question." 2023 E2 B Q1d: "Those who used method 1 [equating coefficients] were generally successful. Those who used method 2 [using transformations] often had sign errors in their expressions for k… Some students only gave one set of values for a and b."

T30. Image of a single point under a transformation

Wording. 2018 Exam 2 Q4: "The point A(3, 2) lies on the graph of the function f. A transformation maps the graph of f to the graph of g, where g(x) = ½f(x − 1). The same transformation maps the point A to the point P. The coordinates of the point P are…" 2025 Exam 2 Q15: "The graph of y = g(x) passes through the point (1, 3). The graph of y = 1 − g(2x + 3) must pass through the point…"

What it tests. That you can transform a point without ever writing the function's rule.

Method. Set the arguments equal: if the image is g(n(x + b)), then the new x satisfies n(x + b) = old x. Apply the vertical operations to the old y in order.

Instances. 2018 E2 A Q4 48% · 2025 E2 A Q15 44% · 2024 E2 A Q12 47% · 2024 E2 B Q3aii 26% (2).

Marks. 1–2.

Traps. 2024 E2 A Q12 report: "The graph of h has been dilated by a factor of ½ from the y-axis and translated a unit left. The local minimum of h is at approximately (−0.61, −0.37). This would become (−0.31, −0.37) and then (−1.31, −0.37) after each of the transformations." 2024 E2 B Q3aii: "Many students did not realise they only needed to translate the point… Others translated the local minimum."

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Group D — Inverses

T14. Find the rule and domain of the inverse

Wording. 2018 Exam 1 Q5, verbatim and complete: "Let f : (2, ∞) → R, where f(x) = 1/(x − 2)². State the rule and domain of f⁻¹." 2023 Exam 1 Q7c: "Determine the equation and the domain for the inverse function f⁻¹." 2017 Exam 2 Section B Q4b: "Find the rule and domain for f⁻¹, the inverse function of f."

What it tests. The swap-and-solve routine, the choice of branch when the algebra produces ±, and — the mark most often lost — the domain.

Method. (1) Write y = f(x). (2) Write a new line that says "for the inverse, swap x and y": x = f(y). (3) Solve for y. (4) Choose the sign using the range of the inverse, which is the domain of the original. (5) State dom f⁻¹ = ran f. (6) Present as f⁻¹ : [domain] → R, f⁻¹(x) = ….

Instances. 2006 E1 Q2a 63% (2) · 2019 E1 Q2a 57% (2) · 2012 E1 Q3 56% (2) · 2017 E2 B Q4b 64% (2) · 2016 E2 B Q4bi 56% (2) · 2010 E2 B Q1ai 57% (3) · 2022 E2 B Q4d 51% (3) · separators: 2018 E1 Q5 46% (3) · 2006 E1 Q2b 45% (1, domain only) · 2008 E1 Q10a 45% (2) · 2009 E1 Q3 38% (3) · 2016 E1 Q5bii 24% (2) · 2023 E1 Q7c 21% (2) · 2016 E1 Q5bi 16% (2) · 2015 E2 A Q2 50% · 2022 E2 A Q6 47% · 2023 E2 B Q5b 39% (2).

Marks. 2 or 3. When the question says "rule and domain", one of the marks is the domain.

Traps. 2009 E1 Q3: "Few students realised that the inverse function, f⁻¹, required the rule and the domain to be specified. The definition of f should have provided students with a clue." The 2007 Exam 2 report states this as standing policy: "When the inverse function is asked for, the domain must be given. Students will be penalised in the future if the domain is left out. If only the rule for the inverse function is asked for, the domain does not have to be given." 2012 E1 Q3: "It is important that students do not proceed directly from y = 2x³ + 1 to x = 2y³ + 1. This is not correct working. Students need to indicate that new working is starting." 2016 E1 Q5bi: "Students appeared quite adept at the mechanics… However, few students took care to determine the range of the inverse function and select for the negative root." 2023 E1 Q7c: "The most common error was writing the function as the positive arm of the inverse."

T15. Conditions for an inverse to exist; the largest domain restriction

Wording. 2025 Exam 1 Q5b: "The function g(x) has exactly one stationary point, a local minimum. Find the largest value of a such that when g is restricted to the domain (−∞, a] it has an inverse function." Section A form, 2013 Exam 2 Q7: "The function g : [−a, a] → R, g(x) = sin(2x − π/6) has an inverse function. The maximum possible value of a is…" 2022 Exam 2 Q5: "The largest value of a such that the function f : (−∞, a] → R, f(x) = x² + 3x − 10, where f is one-to-one, is…" 2014 Exam 2 Q6: "The function f : D → R with rule f(x) = 2x³ − 9x² − 168x will have an inverse function for…"

What it tests. That an inverse function exists exactly when the function is one-to-one, and that the boundary of the largest such interval is a stationary point (or, for the circular functions, the nearest turning point).

Method. Differentiate, solve f′(x) = 0, take the relevant stationary x-value as the endpoint. Include the endpoint — a one-to-one restriction may close at a turning point.

Instances. 2014 E2 A Q13 42% · 2013 E2 A Q7 37% · 2011 E2 B Q3bii 20% (1, 2011 caveat) · non-separator comparators: 2025 E2 A Q5 76%, 2022 E2 A Q5.

Marks. 1–2.

Traps. 2011 report: "many students did not know that a cubic function could have no stationary points if it is to be a one-to-one function even though earlier parts of the question were helpful in this regard. Some students thought the number of stationary points was required." The "explain" version, 2021 Exam 2 Section B Q3c ("Explain why p is not a one-to-one function", 66%): "Some students wrote that there exists two x-values for every y-value, which is not the case, or p fails the vertical line test." The accepted answers were "Fails the horizontal line test", "many-to-one function", or "there exist two x-values for some y-values".

T16. Where f meets f⁻¹

Wording. 2010 Exam 2 Section B Q1aiii: "Solve g⁻¹(x) = g(x)." 2017 NHT Exam 1 Q8b: "Solve f(x) = f⁻¹(x)." 2020 Exam 1 Q8dii: "Find all values of k for which the graphs of g and g⁻¹ do not intersect."

What it tests. That for an increasing function the intersections of f and f⁻¹ lie on y = x — and awareness that this is a shortcut, not a theorem for decreasing functions.

Method. Solve f(x) = x. For "do not intersect", the boundary case is tangency to y = x: solve f(x) = x and f′(x) = 1 simultaneously.

Instances. 2010 E2 B Q1aiii 66% (2) · 2017 E2 B Q4ii 3% (2) · 2020 E1 Q8dii 3% (2) · 2011 E2 B Q3cii 6% (2, 2011 caveat).

Marks. 1–2.

Traps. 2020 E1 Q8dii: "Some students tried to algebraically find the point of intersection of the graphs of function and its inverse function, with limited progress. This question could also be solved by consideration of the point where the gradient of g(x) was equal to the gradient of y = x." 2010 E2 B Q1aiii: "A small number of students solved g(x) = g(x)" — a null equation — "while others did not give their answers correct to three decimal places."

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Group E — Composition and combination

T17. Find the rule of a composite function (or recognise its graph)

Wording. 2016 Exam 1 Q5ai: "Find the rule for h, where h(x) = f(g(x))." 2019 Exam 1 Q9a: "State the rule of g(f(x))." Section A graph form, 2025 Exam 2 Q13: "The graphs of y = f(x) and y = g(x) are sketched on the same set of axes below. Which of the following could be the graph of y = (g ∘ f)(x)?"

What it tests. Substitution and bracket discipline. The rule is the easy half.

Method. Replace every x in the outer rule by the entire inner rule, in brackets. Do not expand unless asked.

Instances. 2016 E1 Q5ai 95% (1) · 2019 E1 Q9a 94% (1) · 2019 E1 Q9c 86% (1) · 2006 E1 Q1 83% (1) · separators: 2025 E2 A Q13 45% · 2007 E2 A Q22 37% · 2015 E2 A Q22 35% (graph of g(−f(x))) · 2008 E1 Q10c 20% (2) · 2009 E2 B Q1eii 13% (2).

Marks. 1 for the rule; 2 when simplification into a stated form is required.

Traps. 2016 E1 Q5ai: "The few errors tended to be the result of poor notation, for example logₑ x² + 1, rather than a lack of understanding." 2006 E1 Q1: "some students failed to gain the mark because they incorrectly expanded the correct expression. Students should be advised against proceeding beyond what is explicitly asked for."

T18. Domain, range, and existence of a composite

Wording. 2016 Exam 1 Q5aii: "State the domain and range of h." 2017 Exam 1 Q7b: "Let g : (−∞, c] → R, g(x) = x² + 4x + 3, where c < 0. i. Find the largest possible value of c such that the range of g is a subset of the domain of f. ii. For the value of c found in part b.i., state the range of f(g(x))." [SAMP] Exam 1 Q2c: "Determine the maximal domain, D, such that g ∘ h exists." 2023 Exam 2 Q20: "The largest interval of x values for which (f ∘ g)(x) and (g ∘ f)(x) both exist is…"

What it tests. The condition ran g ⊆ dom f, and the fact that ran(f ∘ g) = f(ran g), not ran f.

Method. (1) dom(f ∘ g) = dom g, possibly restricted. (2) Compute ran g over that domain. (3) Check ran g ⊆ dom f; if not, shrink dom g until it is — that is what "largest possible value of c" means. (4) The range is the image of ran g under f; sketch f over ran g and read it off.

Instances. 2017 E1 Q7c 30% (1) · 2017 E1 Q7bi 29% (2) · 2023 E2 A Q20 30% · 2017 E1 Q7bii 20% (1) · 2020 E2 A Q20 18% · 2016 E1 Q5aii 15% (2).

Marks. 1–2. Every archive instance of this type is a separator.

Traps. 2016 E1 Q5aii: "A small proportion of students gained full marks… students appeared to experience difficulty in determining the range of the composite function, h. A quick sketch over the given domain would have been helpful." 2017 E1 Q7bii, the report spelling the whole idea out: "Since (−∞, −3] is the domain of g, the range of g is the same as the domain of f. Hence, in this case, the range of f(g(x)) is the same as the range of f." 2017 E1 Q7c: "Most students could identify the composite function but struggled with determining its range."

T19. Sum, difference and product functions

Wording. 2023 Exam 2 Q3: "Two functions, p and q, are continuous over their domains, which are [−2, 3) and (−1, 5], respectively. The domain of the sum function p + q is…" 2018 Exam 2 Section B Q2di requires the graph of b(t) + b(t − 6). 2025 Exam 1 Q7di: "State the coordinates of the stationary point of inflection for the graph of y = f(x)g(x)."

What it tests. That dom(f + g) = dom f ∩ dom g, and that a product's key features come out of the factorised form.

Method. For domain: intersect, keeping bracket types. For graphs: addition of ordinates, or define the sum on CAS. For products: factorise and read multiplicities — a triple factor is a stationary point of inflection.

Instances. 2023 E2 A Q3 47% · 2025 E1 Q7di 44% (1) · 2018 E2 B Q2di 35% (2) · 2021 E2 B Q5f 13% (1, bounding sin + cos) · non-separator: 2013 E2 A Q5 75%, 2021 E2 A Q10 70% (maximal domain of f + g).

Marks. 1–2.

Traps. 2025 E1 Q7di: "Some students gave the equation of the product curve as their answer, rather than the coordinate of the stationary point of inflection. Some students chose to use the expanded form… This approach often made the question unnecessarily difficult." 2018 E2 B Q2di: "Students could use addition of ordinates or define the function b₂(t) = b(t) + b(t − 6) and use technology to sketch the graph." 2021 E2 B Q5f: "Some students considered the maximum value only and not the minimum value."

T29. Hybrid (piecewise) functions

Wording. 2023 Exam 2 Section B Q2diii: "Sketch the graph of the piecewise function w on the axes below, showing the coordinates of the endpoints." 2023 NHT Exam 2 Q1e: "Show that the function C₂ is not continuous for t > 0." 2021 Exam 2 Q19: "Which one of the following functions is differentiable for all real values of x?"

What it tests. Continuity (equal values at the join), smoothness (equal derivatives at the join), and endpoint conventions.

Method. For continuity, set the two branch rules equal at the join. For differentiability, also set the two derivatives equal. For sketching, compute every endpoint of every branch and mark it open or closed.

Instances. 2007 E2 B Q2a 44% (1, find m so the model is continuous) · 2023 E2 B Q2di 40% (1) · 2007 E2 B Q2b 37% (3, sketch) · 2021 E2 A Q19 35% · 2023 E2 B Q2diii 24% (3) · 2022 E1 Q7c 16% (2).

Marks. 1–3.

Traps. 2007 E2 B Q2a: "Many students did not know to substitute t = 8 or t = 16 into the equation… It was stated in the introduction that the concentration of insects in the gorge was a continuous function of time." 2007 E2 B Q2b: "The curve had to be continuous and should have been smooth at t = 8 and t = 16… Others drew cusps instead of turning points." 2022 E1 Q7c: "It was clear that some students interpreted the word 'endpoints' as only the right-hand end of each tile… A percentage of students tried to prove the derivatives of f and g were equal at the endpoints. This was not the intention of the question."

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Group F — Families, parameters and counting solutions

T20. For which values of the parameter does … ? (families of transformed functions)

The single most productive separator type in the area: 19 of the 239.

Wording. 2014 Exam 2 Section B Q5c: "Find the values of d such that the graph of y = f(x + d) has i. one positive x-axis intercept ii. two positive x-axis intercepts." 2012 Exam 2 Q16: "The graph of a cubic function f has a local maximum at (a, −3) and a local minimum at (b, −8). The values of c, such that the equation f(x) + c = 0 has exactly one solution, are…" 2021 Exam 2 Q18: "The maximum number of solutions for the equation f(x − k) = g(x), where k ∈ R, is…" 2019 Exam 2 Section B Q1biii: "Find the values of d ∈ R for which f(x) + d is always negative." 2018 Exam 2 Section B Q1hi: "Find the values of a for which p has only one stationary point."

What it tests. Holding a family of curves in mind and counting intersections as a parameter slides. The study-design phrase is "families of transformed functions for a single transformation parameter".

Method. (1) Identify the fixed graph and the moving graph. (2) Find the critical positions — where the moving graph is tangent to the fixed one, or passes through a turning point or endpoint, or where a root becomes repeated. (3) The answer is an interval or union of intervals bounded by those critical values. (4) Decide each endpoint's bracket by testing the critical case itself. CAS sliders are explicitly endorsed here: the 2024 NHT report says "This value can also be found using the slider functionality on CAS."

Instances. 2020 E2 B Q5b 46% (1) · 2021 E2 A Q18 39% · 2015 E2 A Q21 37% · 2019 E2 B Q1biii 35% (1) · 2012 E2 A Q16 34% · 2012 E2 B Q5bi 34% (2) · 2025 E1 Q9bi 22% (2) · 2014 E2 B Q5cii 19% (1) · 2018 E2 B Q1hi 18% (1) · 2018 E2 A Q18 14% · 2006 E2 B Q3e 13% (4) · 2023 E2 B Q5d 13% (1) · 2012 E2 B Q5bii 11% (1) · 2024 E2 B Q5d 9% (2) · 2014 E2 B Q5ci 7% (1) · 2022 E2 B Q4ei 6% (1) · 2021 E1 Q9bii 4% (1) · 2020 E2 B Q5g 3% (1) · 2021 E2 B Q5g 2% (1).

Marks. Usually 1. Occasionally 2 or 4.

Traps. 2014 E2 B Q5ci: "(1, 3) and 1 < d < 3 were common incorrect answers" — the correct answer was [1, 3). 2012 E2 B Q5bii: "Some students included extra solutions, ignoring the requirement that a had to be a positive real number." 2019 E2 B Q1biii: "Common incorrect answers were d = −1/e, d ≤ −1/e, d > 1/e…" — the answer was d < −1/e. 2018 E2 B Q1hi: "Common incorrect answers were a ≤ 1, a < 0 or a > 0." The pattern is unmistakable: students find the critical value and then get the inequality direction or the bracket wrong.

T31. Recover the rule from a printed graph

Wording. 2019 Exam 1 Q8a: "Find the rule of f" (a degree-4 polynomial, with a labelled graph). 2015 Exam 2 Q3: "The rule for a function with the graph above could be…" 2011 Exam 2 Q15: "The graph shown could have equation…"

What it tests. Reading multiplicity from shape (touch = even, cross = odd, flatten-and-cross = triple) and then fixing the dilation factor from one extra labelled point.

Method. Write y = A(x − r₁)^{m₁}(x − r₂)^{m₂}… from the intercepts and their behaviour. Substitute the one labelled non-intercept point to find A.

Instances. 2015 E2 A Q3 20% · 2019 E1 Q8a 14% (1) · 2011 E2 A Q15 6% (2011 caveat).

Marks. 1.

Traps. 2019 E1 Q8a: "This question was well attempted but not done well, with many students overlooking the dilation factor." The graph showed turning points at (±½, 1) and intercepts at (±1, 0) and the origin; the answer f(x) = −4x²(x² − 1) requires a turning point to pin A = −4.

T28. Functional equations and abstract function properties — removed from the course in 2023

Wording. 2015 Exam 2 Q18: "For which one of the following functions is the equation f(x + y) − f(x − y) = 4 f(x) f(y) true for all x ∈ R and y ∈ R?" 2016 Exam 2 Q11: "The function f has the property f(x) − f(y) = (y − x) f(xy) for all non-zero real numbers x and y." 2007 Exam 2 Section B Q4d: "Show that f(u) f(v) = f(u) + f(v) − f(u + v)."

Status. The 2016–2022 dot point on "simple functional relations such as f(x + k) = f(x), f(xⁿ) = n f(x)…" was deleted in 2023 along with the matching key-knowledge point (01-study-design.md §3.2). No instance appears after 2017. Do not drill these.

Instances (archival only). 2015 E2 A Q18 48% · 2007 E2 A Q17 47% · 2016 E2 A Q11 47% · 2017 E2 A Q13 46% · 2012 E2 A Q19 45% · 2006 E2 A Q17 37% · 2007 E2 B Q4dii 29% (1) · 2007 E2 B Q4di 26% (2).

Why they are still worth reading once. The 2007 report's comment on Q4di is a general rule that still applies: "As it was a 'show that' question, full working needed to be shown. Some students assumed that the left hand side equalled the right hand side at the start. This will be penalised in the future. Some students let u and v equal an integer value, which is not acceptable as it only establishes (or not) a result for that particular set of values."

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Group G — Modelling

T21. Fit model parameters, usually as a "show that"

Wording. 2023 Exam 2 Section B Q2a: "Consider the function h(t) = −60cos(bt) + c for some b, c ∈ R… Show that b = π/15 and c = 75." 2025 Exam 2 Section B Q2a: "Write down two simultaneous equations in terms of A and k. Solve them, using algebra, to show that A = 18/27 and k = (3/14)logₑ(2)." 2010 Exam 2 Section B Q3b: "Show that the total surface area (including the base), S m², of the pyramid WABCD is given by S = 400(cos²(x) + cos(x)sin(x))."

What it tests. Translating a physical description into equations, and then displaying the algebra. In a "show that", the answer is printed; the marks are entirely for the route.

Method. Extract one equation per given fact — a point on the graph, a maximum, a period, a derivative that is zero. Solve simultaneously. Write each step; never jump from the equations to the printed answer.

Instances. 2020 E2 B Q1a 83% (1) · 2022 E2 B Q2b 68% (2) · 2023 E2 B Q2a 66% (2) · separators: 2018 E2 B Q1f 49% (1) · 2006 E1 Q11 46% (5) · 2010 E2 B Q3b 42% (2) · 2006 E2 B Q4bi 41% (2) · 2025 E2 B Q2fi 39% (2) · 2009 E2 B Q2c 35% (1) · 2009 E2 B Q4ei 34% (1) · 2006 E2 B Q4bii 34% (1) · 2009 E2 B Q2aii 33% (2) · 2006 E2 B Q4e 31% (3) · 2009 E2 B Q2d 30% (2) · 2014 E2 B Q3d 29% (3) · 2025 E2 B Q2fii 28% (2) · 2006 E2 B Q4biii 27% (1) · 2006 E2 B Q4biv 26% (2) · 2007 E2 B Q4a 26% (2) · 2010 E2 B Q3d 22% (1) · 2011 E1 Q10–10a 6% (2, 2011 caveat).

Marks. 1–5.

Traps. 2009 E2 B Q2aii: "For 'show that' questions, students must ensure they provide sufficient relevant working using the mathematical notation." 2010 E2 B Q3b: "This was a 'show that' question and adequate working was required. There was poor use of brackets and cos x² was often seen." 2006 E2 B Q4biv: "Many students simply substituted the values for p and q into the equation and showed that the equation held with these values. Unfortunately there are infinitely many such pairs of values." 2006 E1 Q11 makes the same point: "Sometimes the values for m and n appeared from simply looking at the diagram… This was not an appropriate solution as there could be an infinite set of solutions." 2020 E2 B Q1a: "Some students assumed the result in their proof, rather than showing it."

T22. Interpret a circular or exponential model in context

Wording. 2017 Exam 2 Section B Q2a: "State the minimum and maximum heights of P above the ground." Q2b: "For how much time is Sammy in the capsule?" 2007 Exam 2 Section B Q2e asks for the total time above a threshold. 2020 Exam 2 Q12 gives a clock face and asks which of five rules gives the height of the minute-hand tip.

What it tests. Whether mathematical features (range, period, solutions of an equation) get translated back into the language of the context, with units and to the stated accuracy.

Method. Range → min and max. Period → duration of one cycle. "For how long is X above k" → solve f(t) = k, subtract the t-values within one period. Re-read what was wanted: a time, a duration, or a value.

Instances. 2017 E2 B Q2a 89% (1) · 2017 E2 B Q2b 84% (1) · 2019 E2 B Q3c 73% (1) · separators: 2007 E2 B Q2fii 49% (1) · 2020 E2 A Q12 45% · 2007 E2 B Q2d 41% (1) · 2007 E2 B Q2e 35% (2) · 2009 E2 B Q2e 23% (2) · 2007 E2 B Q2g 14% (3).

Marks. 1–3.

Traps. 2017 E2 B Q2b: "A common incorrect answer was 15 minutes" — half a period. 2007 E2 B Q2d: "Many students did not give an exact answer… Students must remember that an exact answer is required unless a numerical answer is asked for." 2009 E2 B Q2e: "Students often gave the final answer as 6 km — the distance from P. Students should reread questions before moving on." 2007 E2 B Q2g: "Many students rounded too soon."

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Group H — Where this area meets calculus and coordinate geometry

These types are classified under Functions, relations and graphs in [QJSON] because the object of the question is a graph and its features, even though the tool is calculus. They account for a large share of the Section B separators.

T23. Tangents, normals and the coordinate geometry of a graph

Wording. 2016 Exam 2 Section B Q1d: "Find the equations of the tangents to the graph of f that have a gradient of 1." 2020 Exam 2 Section B Q5a: "Let g_a be the function representing the tangent to the graph of f at x = a… Let (b, 0) be the x-intercept of the graph of g_a. Show that b = 2a³/(3a² − 1)." 2025 Exam 2 Section B Q4fi: "Show that t(x) = cos(p)(x − p) + sin(p) + 1." 2012 Exam 2 Section B Q2d: "Find the coordinates of the points on the graph of y = f(x) such that the tangents to the graph at these points intersect at (−1/2, 7/4)."

What it tests. y − f(a) = f′(a)(x − a) handled with a parameter rather than a number, and then used.

Method. Write the general tangent at x = a in point-gradient form and keep it symbolic. Impose the extra condition (a given gradient, a point it passes through, perpendicularity) and solve for a.

Instances. 2013 E2 B Q4ai 50% (2) · 2020 E2 B Q5a 49% (3) · 2025 E2 B Q4fi 49% (2) · 2016 E2 B Q1d 47% (2) · 2020 E2 B Q4ei 44% (1) · 2017 E2 B Q1dii 42% (2) · 2008 E2 B Q4aiii 39% (3) · 2020 E1 Q7biv 29% (1) · 2020 E2 B Q4eii 28% (1) · 2020 E2 B Q4dii 26% (3) · 2025 E2 B Q4fii 23% (2) · 2020 E2 B Q5c 23% (1) · 2006 E2 B Q1c 17% (2) · 2017 E1 Q9c 17% (2) · 2012 E2 B Q2d 14% (4) · 2021 E1 Q9a 13% (2) · 2012 E2 B Q5c 11% (2) · 2014 E2 B Q5fii 11% (3) · 2017 E1 Q9d 9% (4) · 2020 E2 B Q5e 7% (3).

Marks. 1–4.

Traps. 2020 E2 B Q5a: "many students left out brackets when multiplying by the gradient, writing y − f(a) = f′(a)x − a." 2014 E2 B Q5fii: "Many students did not read the question carefully and tried to find the equation of the tangent to g(x) at x = 3/2. (3/2, −12) was not a point on g(x)." 2020 E2 B Q4dii: "Many students successfully found that the point of intersection of the two tangents occurred at x = 0.80 but then substituted this into f(x)… instead of substituting it into one of the two tangent equations. Others rounded too early." 2025 E2 B Q4fii: "Some students gave their answers in coordinate form without stating the minimum and maximum values of the y-intercept."

T24. Area bounded by graphs

Wording. 2023 Exam 2 Section B Q1cii: "Write down an expression using definite integrals that gives the area of the regions bound by f and g." 2018 Exam 2 Section B Q1e: "Find the total area of the regions bounded by the tangent l and y = f(x). Express your answer in the form a√b/c, where a, b and c are positive integers." 2017 Exam 2 Q17 prints an even function with four x-intercepts and asks which integral expression gives the bounded area.

What it tests. ∫(upper − lower) with the right terminals, and the treatment of regions below the axis.

Method. Find the intersections. Write one integral of upper − lower per region between consecutive intersections. Do not negate; do not split unnecessarily; keep the dx.

Instances. 2018 E2 B Q1e 49% (2) · 2017 E1 Q9a 47% (2) · 2017 E2 B Q1dii 42% (2) · 2025 E2 A Q17 38% · 2019 E2 B Q3f 36% (2) · 2023 E1 Q5b 35% (3) · 2022 E1 Q7b 32% (3) · 2013 E2 B Q4aii 26% (3) · 2008 E2 B Q2cii 26% (2) · 2017 E2 A Q17 21% · 2018 E1 Q9aii 19% (1) · 2020 E2 B Q4eiii 11% (3) · 2013 E2 B Q4di 3% (2) · 2017 E2 B Q4iii 2% (1).

Marks. 1–3.

Traps. 2017 E2 B Q1dii: "∫x − x³ − kx dx was a common error, leaving out the brackets… The easiest approach was to use 'upper function subtract lower function'." 2018 E2 B Q1e: "Some students split the integral, which was unnecessary. Others put a negative sign in front of the integral for the bounded area below the x-axis." 2015 E2 B Q1d: "The 'dx' was often missing and brackets were used poorly." 2020 E2 B Q4eiii: "Others used areas of triangles… The area from x to y is a trapezium, not a triangle."

T25. Minimum distance and optimisation on a curve

Wording. 2016 Exam 2 Section B Q4c: "The point P(c, d) is on the graph of f. Find the exact values of c and d such that the distance of this point to the origin is a minimum, and find this minimum distance." 2020 Exam 1 Q7c: "Find the value, k, that gives the shortest possible distance between the graph of the function y = f(x − k) and point P." 2022 Exam 2 Q9: "The shortest distance, d, from the origin to the point (x, y) on the graph of f is given by…"

What it tests. Setting up a distance function of one variable and minimising it — and knowing when a geometric argument beats calculus.

Method. Write D² = (x − x₀)² + (f(x) − y₀)². Minimise D², not D. Check the endpoints of the domain as well as the stationary points, because [SD] explicitly names "identification of interval endpoint maximum and minimum values".

Instances. 2022 E2 A Q9 50% · 2018 E2 A Q17 45% · 2006 E1 Q9a 37% (1) · 2022 E2 B Q1e 18% (4) · 2016 E2 B Q4c 14% (3) · 2020 E1 Q7c 10% (2) · 2019 E2 B Q5f 4% (2) · 2007 E2 B Q1d 3% (2) · 2013 E2 B Q4dii 3% (2) · 2013 E2 B Q4diii 2% (2) · 2021 E1 Q9ci 1% (2).

Marks. 1–4.

Traps. 2020 E1 Q7c: "Many students used the distance formula and then attempted to differentiate and equate to zero (often with limited success due to error in differentiation or algebra). Students who used a geometric approach tended to score more highly." 2007 E2 B Q1d: "Many students found the minimum surface area instead of the maximum. Some students chose the wrong endpoint." 2013 E2 B Q4dii: the maximum of A(k) occurred at the endpoint k = 8; Q4diii's minimum occurred at the turning point. 2016 E2 B Q4c: "Some students did not consider the domain and gave two sets of values for c and d… Others had the correct answers for c and d but did not work out the minimum distance." 2021 E1 Q9ci: "Students are reminded of the need to consider domains when defining functions. Many were able to write the rule, but very few stated the domain."

T26. Angles: between a tangent and an axis, or between two lines

Wording. 2020 Exam 2 Section B Q4b: "Find the obtuse angle that the tangent to f at x = 1 makes with the positive direction of the horizontal axis. Give your answer correct to the nearest degree." 2017 Exam 2 Section B Q4h: "The lines L₁ and L₂ are the tangents at the origin to the graphs of g_k and g_k⁻¹ respectively. Find the value(s) of k for which the angle between L₁ and L₂ is 30°."

What it tests. gradient = tan(θ), plus calculator-mode discipline.

Method. θ = tan⁻¹(m); if an obtuse angle is wanted, add 180°. For the angle between two lines, use the tangent-difference identity. Set the calculator to degrees.

Instances. 2018 E2 B Q3d 61% (1) · 2020 E2 B Q4b 37% (1) · 2017 E2 B Q2d 36% (1) · 2017 E2 B Q2g 7% (1) · 2017 E2 B Q2h 2% (2) · 2017 E2 B Q4h 2% (2).

Marks. 1–2.

Traps. 2017 E2 B Q2d: "Many students knew to get tan⁻¹(65/500) but they did not specify 'degree' for their technology." 2017 E2 B Q2g: "Some students used radians instead of degrees." 2017 E2 B Q2h: "Some students wrote 7 minutes without showing any working. As indicated in the instructions on the examination, for questions worth more than one mark, appropriate working must be shown."

T27. Solve a trigonometric equation over an interval, or give the general solution

Wording. 2021 Exam 1 Q3c: "Solve 2sin(2x) = −√3 for x ∈ R." 2022 Exam 1 Q6b: "Find all values of k such that f(k) = 0 and k ∈ [0, 2π]." 2025 Exam 1 Q3b: "Solve f(x) = 0 for x." 2006 Exam 2 Section B Q1d asks for the general solution of |sin(x)| = 0.5.

What it tests. The by-hand guarantee from [SD]: "solve by hand equations of the form sin(ax + b) = c, cos(ax + b) = c and tan(ax + b) = c with exact value solutions over a given interval."

Method. Set u = ax + b and transform the interval for x into an interval for u. Find the reference angle from the exact-value table. Place solutions in the correct quadrants and add period multiples until the u-interval is exhausted. Convert back. For a general solution, use x = 2nπ ± α (cos), x = nπ + (−1)ⁿα (sin), x = nπ + α (tan), and state n ∈ Z.

Instances. 2012 E1 Q6a 65% (2) · 2025 E1 Q3b 51% (3) · separators: 2022 E1 Q6b 46% (3) · 2008 E2 B Q4c 37% (2) · 2021 E1 Q3c 17% (3) · 2006 E2 B Q1d 15% (2).

Marks. 2–3.

Traps. 2021 E1 Q3c: "The construction of a general solution, while attempted, was not done well… or lacked a correct number categorisation of n." 2022 E1 Q6b: "It is expected that students will have a way of remembering the exact values of sin, cos and tan for values between 0 and π/2… Errors included not finding the third and fourth angle correctly." 2025 E1 Q3b: "Some students only gave two of the solutions, not taking into account the period of the function. Some students gave a general solution to the equation without indicating the particular solutions." 2006 E2 B Q1d: "some used R… or J instead of Z. Some students interpreted the period as 2π rather than π, suggesting that they had ignored the absolute value part."

T32. Approximate an area from a graph (trapezium rule, rectangles)

Wording. 2023 Exam 1 Q4: "Use two trapeziums of equal width to approximate the area between the curve, the x-axis and the lines x = 1 and x = 3." 2021 Exam 2 Section B Q2d: "Approximate ∫f(x)dx using four rectangles of equal width and the right endpoint of each rectangle." 2018 Exam 2 Q16: "Jamie's approximation as a fraction of the exact area is…"

What it tests. That "approximate" forbids integration, and that the trapezium formula on the sheet must be applied with the right h.

Method. Compute the strip width h. Evaluate the function at every strip boundary. Apply A ≈ (h/2)[f(x₀) + 2f(x₁) + … + f(xₙ)], or sum the individual trapezium areas.

Instances. 2018 E2 A Q16 49% · 2023 E1 Q4 45% (2) · 2021 E2 B Q2d 16% (1).

Marks. 1–2.

Traps. 2023 E1 Q4: "any attempt to calculate this area using integral calculus was not acceptable. Some students gave the formula as stated on the formula sheet; however, many did not proceed to identify and substitute the correct values… Common errors involved incorrect values of h… Arithmetic manipulation errors (frequently) arose from dealing with the different denominators of the fractions."

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3. The standard wordings — a phrasebook

VCAA reuses sentences. Each one carries a specific obligation, and the reports penalise students who answer a neighbouring question instead. These are quoted from the papers, with the years in which the exact form appears.

3.1 Sketching

3.2 Key features

3.3 Transformations

3.4 Inverses and composites

3.5 Parameters, families and proof

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4. The separators

239 of the 498 parts in this area with published statistics (48.0%) are separators — pct ≤ 50, meaning at most half the state earned full marks. Every one is listed below, grouped by question type, in the format ref — pct% — what it asked.

Two data caveats carried from 01-study-design.md §0.1: 2011 is OCR-damaged and its percentages should not be quoted; the 2024 November papers extract as empty files, so 2024 descriptions are reconstructed from the reports and one 2024 item could not be recovered at all.

4.1 Group A — Sketching (18 separators)

T1 · By-hand sketch with labelled key features — 8

• 2006 Exam 1 Q4b — 14% — sketch y = 5cos(2x) + 3 on [−π, π]; label axis intercepts and endpoints as coordinates
• 2007 Exam 1 Q3a — 25% — sketch a graph with a cusp at x = 0 and behaviour either side of x = 3, using open/closed endpoints correctly
• 2025 Exam 1 Q3c — 33% — sketch y = 2cos(2x) − 1 on [−π/2, 3π/2]; label the endpoints
• 2017 Exam 1 Q3b — 36% — sketch (x + 2)²(x − 1) on [−3, 0]; label intercepts and stationary points
• 2019 Exam 1 Q4b — 37% — sketch g(x) = 1 − f(x) over a printed cosine graph; label intersections and endpoints
• 2024 Exam 1 Q3a — 42% — sketch a reciprocal-square graph; label asymptotes, intercepts, and show the symmetry
• 2012 Exam 1 Q5a — 46% — sketch f : [0, 5] → R, f(x) = −|x − 3| + 2; label intercepts and endpoints
• 2013 Exam 1 Q9b — 46% — sketch g(x) = |f(x)| + 1 on [−2, 2]; label the endpoints

T2 · Technology-active sketch with stated accuracy — 4

• 2011 Exam 2 Section B Q1c — 5% — sketch y = h(x), labelling all intercepts and turning points (2011 caveat)
• 2008 Exam 2 Section B Q4e — 26% — sketch h: exact stationary point, both asymptote equations, exact y-intercept
• 2008 Exam 2 Section B Q1ci — 42% — sketch the density graph and label the local maximum to two decimal places
• 2012 Exam 2 Section B Q2a — 43% — sketch y = 1/(2x − 4) + 3; label intercepts and both asymptotes with equations

T3 · Sketch the inverse — 4

• 2007 Exam 2 Section B Q4cii — 15% — sketch h⁻¹ on the correctly restricted domain
• 2008 Exam 1 Q10b — 19% — sketch y = f(f⁻¹(x)) over its maximal domain (the line y = x, restricted)
• 2010 Exam 2 Section B Q1aii — 44% — sketch g⁻¹ with exact intercepts and the asymptote y = −4
• 2023 Exam 1 Q7b — 47% — sketch y = f⁻¹(x) on the printed axes; label endpoints and axial intercepts

T4 · Derivative graph — 2

• 2024 Exam 1 Q7c — 15% — sketch f′ across the stated domain, with endpoints positioned off the x-axis
• 2021 Exam 2 Section A Q8 — 40% — choose the graph of f′ given the graph of f

4.2 Group B — Key features and notation (32 separators)

T5 · State the range or an interval — 8

• 2019 Exam 1 Q8c — 1% — range of h = logₑ(g(x)) − logₑ(x³ + x²) over its maximal domain
• 2024 Exam 1 Q7bii — 20% — state an interval from substituted endpoint values, in correct bracket notation
• 2024 Exam 2 Section B Q5biv — 34% — state an exact interval; round brackets were the common error
• 2006 Exam 1 Q7b — 35% — range of y = |f(x)| on the domain [−5, 1]
• 2007 Exam 2 Section A Q6 — 39% — range of f(x) = 3|sin(2x) − 1| + 2 on [0, π/3]
• 2020 Exam 2 Section A Q18 — 43% — range of h(x) = a/x + b on [−a, 0) ∪ (0, a]
• 2007 Exam 1 Q3b — 45% — state the range (R \ {0, 3}) of the graph drawn in part a
• 2018 Exam 2 Section A Q3 — 48% — range of f(x) = 1/x on [a, b) for positive a, b

T6 · Maximal domain — 8

• 2011 Exam 1 Q4b–4b — 1% — maximal domain of a square-root composite, requiring a union (2011 caveat)
• 2011 Exam 1 Q4b–4b — 1% — duplicate record of the same part; the damaged 2011 extraction carries this label twice
• 2019 Exam 1 Q8b — 9% — maximal domain of a difference of logarithms: (−1, 0) ∪ (0, 1)
• 2022 Exam 1 Q5b — 27% — maximal domain of logₑ(x² − 2x − 3), as a union
• 2021 Exam 2 Section B Q1fi — 33% — state the domain of the general-width box-volume function
• 2021 Exam 2 Section B Q3a — 36% — maximal domain and range of q(x) = logₑ(x² − 1) − logₑ(1 − x)
• 2022 Exam 2 Section A Q13 — 39% — maximal domain of logₑ((x − a)/(x + a))
• 2021 Exam 2 Section B Q1b — 42% — state the domain of V_box for the folded-cardboard model

T7 · Circular key features and symmetry — 5

• 2025 Exam 2 Section B Q4c — 15% — find k and the largest a with f(x + k) = −f(x) on [0, a]
• 2021 Exam 2 Section B Q5c — 21% — smallest positive h for which f(h − x) = f(x)
• 2020 Exam 1 Q3 — 23% — find a and b in y = tan(ax + b) from two labelled points on one period
• 2018 Exam 2 Section A Q11 — 26% — a for y = tan(ax) with asymptote x = 3π and one intercept in (0, 3π)
• 2023 Exam 2 Section A Q18 — 29% — number of local minima of f(x) = sin(ax) on [−a, a]

T8 · Stationary points, key points, monotonicity — 8

• 2023 Exam 2 Section B Q3e — 35% — largest interval on which h is strictly decreasing, to two decimal places
• 2009 Exam 2 Section B Q2bii — 41% — exact distance between two x-intercepts of the model
• 2008 Exam 2 Section B Q3f — 43% — coordinates of two named points on the graph
• 2007 Exam 2 Section B Q2c — 48% — the minimum concentration and the t-values at which it occurs
• 2012 Exam 2 Section A Q8 — 49% — where the gradient of a cubic with four named ordinates is negative
• 2013 Exam 1 Q9a — 50% — value of a, the x-intercept of (x − 1)² − 2
• 2013 Exam 2 Section A Q19 — 50% — x-coordinate of the third turning point of e^(x/3)sin(x)
• 2009 Exam 2 Section B Q2biii — 50% — exact vertical offset of the model, converted to metres

T9 · Solve an inequality from your own graph — 3

• 2025 Exam 1 Q7dii — 27% — values of x with f(x)g(x) ≤ 0
• 2021 Exam 1 Q4b — 32% — values of x with 1/(x − 2) + 2 ≥ 3
• 2023 Exam 1 Q3b — 38% — values of x with f(x) ≥ 1, read from the hyperbola sketched in part a

T31 · Recover the rule from a printed graph — 3

• 2011 Exam 2 Section A Q15 — 6% — equation of a printed circular graph (2011 caveat)
• 2019 Exam 1 Q8a — 14% — rule of a quartic that touches at the origin, including the dilation factor
• 2015 Exam 2 Section A Q3 — 20% — rule of a graph with a repeated root and a negative leading coefficient

T32 · Approximate an area from a graph — 3

• 2021 Exam 2 Section B Q2d — 16% — approximate ∫f using four right-endpoint rectangles
• 2023 Exam 1 Q4 — 45% — two trapeziums of equal width, by hand, no calculus permitted
• 2018 Exam 2 Section A Q16 — 49% — a three-rectangle approximation as a fraction of the exact area

T29 · Hybrid (piecewise) functions — 6

• 2022 Exam 1 Q7c — 16% — determine all four endpoints of two tile-edge functions and confirm they match
• 2023 Exam 2 Section B Q2diii — 24% — sketch the piecewise function w showing endpoint coordinates
• 2021 Exam 2 Section A Q19 — 35% — which hybrid function is differentiable for all real x
• 2007 Exam 2 Section B Q2b — 37% — sketch a three-branch model, smooth at both joins
• 2023 Exam 2 Section B Q2di — 40% — find k and m so the piecewise wheel model is continuous
• 2007 Exam 2 Section B Q2a — 44% — find m making the insect-concentration model continuous at t = 8 and t = 16

4.3 Group C — Transformations (41 separators)

T10 · Describe a sequence of transformations — 11

• 2024 Exam 1 Q5b — 2% — describe the dilation and translation that make two population models agree
• 2024 Exam 2 Section B Q1dii — 5% — a dilation and two translations, in a valid order
• 2007 Exam 2 Section B Q3di — 13% — state a sequence of two transformations taking f to h
• 2015 Exam 2 Section A Q11 — 24% — the single dilation mapping y = 8x³ + 1 onto y = x³ + 1
• 2017 Exam 2 Section B Q4gii — 30% — the transformation mapping g₁⁻¹ onto g_k⁻¹
• 2017 Exam 2 Section B Q4gi — 31% — the transformation mapping g₁ onto g_k
• 2023 Exam 2 Section B Q5a — 35% — complete a sequence of transformations mapping f to ½f(2x)
• 2025 Exam 2 Section A Q20 — 36% — which sequence does not map f(x) = aˣ onto g(x) = a²ˣ⁺²
• 2014 Exam 2 Section B Q5b — 37% — describe the translation mapping y = f(x) onto y = g(x)
• 2024 Exam 2 Section B Q1di — 37% — translate 1 unit right and a stated amount up
• 2015 Exam 2 Section B Q4c — 41% — a sequence of two dilations mapping 2sin(x) to ⅓sin(3x)

T11 · Apply a given sequence and find the image — 11

• 2022 Exam 1 Q6ciii — 12% — hence state the domain D of h after the translations
• 2013 Exam 1 Q9ci — 16% — rule of the image of g after three stated transformations
• 2016 Exam 2 Section A Q20 — 17% — value of a definite integral after a reflection-and-dilation transformation
• 2021 Exam 1 Q5b — 18% — rule of h and its x-intercepts after a dilation then a translation
• 2007 Exam 1 Q8b — 20% — x-value at which 3f(x − 1) + 2 first attains its maximum
• 2012 Exam 1 Q5bii — 26% — rule of the image after a reflection and a translation
• 2012 Exam 2 Section A Q22 — 32% — local extrema of y = −|f(x − 2)|
• 2013 Exam 1 Q9cii — 39% — domain of the image after the same three transformations
• 2022 Exam 1 Q6cii — 46% — smallest positive horizontal translation mapping h onto g
• 2017 Exam 2 Section A Q10 — 47% — which graph results from applying a dilation matrix to 3sin(2x + π/4)
• 2009 Exam 2 Section A Q9 — 49% — equation of the tangent at (3, 8) to y = f(x − 2) + 3

T12 · Matrix / mapping-notation parameters — 10 (type retired after 2020)

• 2020 Exam 2 Section B Q5h — 2% — restrictions on m, n, h, k so T preserves the parallel-tangent property
• 2012 Exam 2 Section B Q2e — 8% — find a, c, d in T mapping f to g(x) = 1/x
• 2016 Exam 2 Section B Q1e — 10% — find a and b in T mapping f to f′
• 2019 Exam 2 Section B Q3e — 10% — find a, b, c, d so the transformed integrals reproduce a given area
• 2010 Exam 1 Q6 — 23% — image of y = 2x² + 1 under T; find a, b, c
• 2013 Exam 2 Section A Q20 — 25% — T maps f to y = x²; recover the rule of f
• 2020 Exam 2 Section A Q13 — 26% — which T maps y = cos(x) onto y = cos(2x + 4)
• 2006 Exam 2 Section A Q13 — 33% — which T maps y = logₑ(x) to y = logₑ(2x − 4) + 3
• 2018 Exam 1 Q9c — 34% — value of a in the translation mapping x·sin(x) onto (3 − x)sin(x)
• 2017 Exam 2 Section B Q4a — 48% — state c and d in the translation mapping y = 2ˣ onto f

T13 · Fit transformation parameters to a named target — 5

• 2013 Exam 2 Section B Q1fi — 10% — find k and m from two given transformed values
• 2013 Exam 2 Section B Q1fii — 12% — hence the coordinates of the image point
• 2007 Exam 2 Section B Q3dii — 13% — find a cubic with the required roots using the part-i transformations
• 2023 Exam 2 Section B Q1d — 13% — find all a and b with (x − a)(x − b)² = f(x) + k
• 2019 Exam 1 Q2c — 24% — find c and d so the translated f equals f⁻¹

T30 · Image of a single point — 4

• 2024 Exam 2 Section B Q3aii — 26% — translate a single named point to a stated image point
• 2025 Exam 2 Section A Q15 — 44% — point through which y = 1 − g(2x + 3) must pass, given (1, 3) on g
• 2024 Exam 2 Section A Q12 — 47% — image of a local minimum under a dilation then a translation
• 2018 Exam 2 Section A Q4 — 48% — image of A(3, 2) under the transformation giving g(x) = ½f(x − 1)

4.4 Group D — Inverses (16 separators)

T14 · Rule and domain of the inverse — 10

• 2016 Exam 1 Q5bi — 16% — rule for k⁻¹, choosing the negative branch
• 2023 Exam 1 Q7c — 21% — equation and domain of f⁻¹ for f : (−∞, 1] → R, f(x) = x² − 2x
• 2016 Exam 1 Q5bii — 24% — domain and range of k⁻¹
• 2009 Exam 1 Q3 — 38% — rule and domain of f⁻¹, both required by the definition given
• 2023 Exam 2 Section B Q5b — 39% — domain and range of the inverse of the strictly increasing branch g₁
• 2006 Exam 1 Q2b — 45% — state dom f⁻¹ explicitly as a set
• 2008 Exam 1 Q10a — 45% — rule and domain of f⁻¹ for f(x) = e^(2x) − 1
• 2018 Exam 1 Q5 — 46% — rule and domain of f⁻¹ for f : (2, ∞) → R, f(x) = 1/(x − 2)²
• 2022 Exam 2 Section A Q6 — 47% — which listed pair of functions are not inverses
• 2015 Exam 2 Section A Q2 — 50% — the inverse of f : (−2, ∞) → R, f(x) = 1/√(x + 2), with its domain

T15 · Existence of an inverse; largest restriction — 3

• 2011 Exam 2 Section B Q3bii — 20% — how many stationary points a cubic may have and still be one-to-one (2011 caveat)
• 2013 Exam 2 Section A Q7 — 37% — maximum a for which sin(2x − π/6) on [−a, a] has an inverse
• 2014 Exam 2 Section A Q13 — 42% — which domain makes cos(logₐ(x)) one-to-one

T16 · Where f meets f⁻¹ — 3

• 2017 Exam 2 Section B Q4ii — 3% — the value of k for which g_k(x) = g_k⁻¹(x) has exactly one solution
• 2020 Exam 1 Q8dii — 3% — all k for which the graphs of g and g⁻¹ do not intersect
• 2011 Exam 2 Section B Q3cii — 6% — coordinates of the intersection of q and q⁻¹ (2011 caveat)

4.5 Group E — Composition and combination (15 separators)

T17 · Composite rule or composite graph — 5

• 2009 Exam 2 Section B Q1eii — 13% — derivative of f(g(x)) where f carries a modulus
• 2008 Exam 1 Q10c — 20% — express f(−f⁻¹(2x)) in the form ax/(bx + c)
• 2015 Exam 2 Section A Q22 — 35% — which graph is g(−f(x))
• 2007 Exam 2 Section A Q22 — 37% — which graph is f(g(x))
• 2025 Exam 2 Section A Q13 — 45% — which graph could be (g ∘ f)(x)

T18 · Domain, range and existence of a composite — 6 (every archive instance is a separator)

• 2016 Exam 1 Q5aii — 15% — domain and range of h(x) = logₑ(x² + 1)
• 2020 Exam 2 Section A Q20 — 18% — a possible interval D for g = log₂(f(x)) with range [−1, 0]
• 2017 Exam 1 Q7bii — 20% — range of f(g(x)) for the c found in the previous part
• 2017 Exam 1 Q7bi — 29% — largest c such that ran g is a subset of dom f
• 2017 Exam 1 Q7c — 30% — range of f(h(x)) where h(x) = x² + 3
• 2023 Exam 2 Section A Q20 — 30% — largest interval on which both f∘g and g∘f exist

T19 · Sum, difference and product functions — 4

• 2021 Exam 2 Section B Q5f — 13% — explain why sin + cos cannot exceed 2 nor fall below −2
• 2018 Exam 2 Section B Q2di — 35% — graph of the sum function b(t) + b(t − 6)
• 2025 Exam 1 Q7di — 44% — coordinates of the stationary point of inflection of f(x)g(x)
• 2023 Exam 2 Section A Q3 — 47% — domain of the sum function p + q from two half-open domains

4.6 Group F — Families, parameters, and retired abstract items (27 separators)

T20 · For which values of the parameter … ? — 19

• 2021 Exam 2 Section B Q5g — 2% — greatest possible minimum value of g_a over the family
• 2020 Exam 2 Section B Q5g — 3% — values of w for which the stated tangent property holds
• 2021 Exam 1 Q9bii — 4% — values of q for which both intersection coordinates are positive
• 2022 Exam 2 Section B Q4ei — 6% — range of k for which the enclosed area A(k) is positive
• 2014 Exam 2 Section B Q5ci — 7% — values of d giving y = f(x + d) exactly one positive x-intercept
• 2024 Exam 2 Section B Q5d — 9% — the intervals satisfying a stated condition on the family
• 2012 Exam 2 Section B Q5bii — 11% — set of a for which two graphs have two distinct intersections
• 2006 Exam 2 Section B Q3e — 13% — values of k for which 3 − ke^x − e^(−x) = 0 has real solutions
• 2023 Exam 2 Section B Q5d — 13% — n such that the turning point of h always lies on y = 2xⁿ
• 2018 Exam 2 Section A Q18 — 14% — which statement about two competing power functions must be false
• 2018 Exam 2 Section B Q1hi — 18% — values of a for which p has only one stationary point
• 2014 Exam 2 Section B Q5cii — 19% — values of d giving two positive x-intercepts
• 2025 Exam 1 Q9bi — 22% — coordinates, in terms of w, of the minimum of y = (x − 1)(x − w)
• 2012 Exam 2 Section A Q16 — 34% — values of c for which f(x) + c = 0 has exactly one solution
• 2012 Exam 2 Section B Q5bi — 34% — solve g(x) = k(x) for x in terms of the parameter a
• 2019 Exam 2 Section B Q1biii — 35% — values of d for which f(x) + d is always negative
• 2015 Exam 2 Section A Q21 — 37% — condition on m, c, a for y = mx + c and y = ax² to have no intersection
• 2021 Exam 2 Section A Q18 — 39% — maximum number of solutions of f(x − k) = g(x)
• 2020 Exam 2 Section B Q5b — 46% — values of a for which the tangent's x-intercept b does not exist

T28 · Functional equations and abstract properties — 8 (removed from the study design in 2023; do not drill)

• 2006 Exam 2 Section A Q17 — 37% — which rule satisfies f((x + y)/2) = (f(x) + f(y))/2
• 2012 Exam 2 Section A Q19 — 45% — which rule satisfies two stated symmetry properties
• 2017 Exam 2 Section A Q13 — 46% — which statement about h(x) = 1/(x − 1) is not true
• 2007 Exam 2 Section A Q17 — 47% — which rule satisfies f(f(x)) = x on its maximal domain
• 2016 Exam 2 Section A Q11 — 47% — which rule satisfies f(x) − f(y) = (y − x)f(xy)
• 2015 Exam 2 Section A Q18 — 48% — which rule satisfies f(x + y) − f(x − y) = 4f(x)f(y)
• 2007 Exam 2 Section B Q4di — 26% — show that f(u)f(v) = f(u) + f(v) − f(u + v)
• 2007 Exam 2 Section B Q4dii — 29% — hence show the corresponding result for v = −u

4.7 Group G — Modelling (24 separators)

T21 · Fit model parameters, usually "show that" — 18

• 2011 Exam 1 Q10–10a — 6% — express two triangle sides in terms of a and an angle (2011 caveat)
• 2010 Exam 2 Section B Q3d — 22% — show the pyramid volume equals (4000/3)(cos⁴x − 2cos⁶x)
• 2006 Exam 2 Section B Q4biv — 26% — find p and q, showing they are the only such pair
• 2007 Exam 2 Section B Q4a — 26% — explain why a = 1 (asymptote) and b = −1 (passes through the origin)
• 2006 Exam 2 Section B Q4biii — 27% — show the derivative factorises as (12px − 17)(x − 1)
• 2025 Exam 2 Section B Q2fii — 28% — determine the anti-derivative constant consistent with the stated points
• 2014 Exam 2 Section B Q3d — 29% — find A to the nearest integer from n(0.5) = 0.74 and n′(0.5) = 0
• 2009 Exam 2 Section B Q2d — 30% — solve the fitted logarithmic model for w
• 2006 Exam 2 Section B Q4e — 31% — find a and b from g(0) = 7 and g′(0) = 4.25
• 2009 Exam 2 Section B Q2aii — 33% — show a = 1, b = −6, c = 16 from three simultaneous equations
• 2006 Exam 2 Section B Q4bii — 34% — find f(1) in terms of p
• 2009 Exam 2 Section B Q4ei — 34% — express the statue's height above ground as 14 − t
• 2009 Exam 2 Section B Q2c — 35% — express k in terms of w for the logarithmic model
• 2025 Exam 2 Section B Q2fi — 39% — show that the anti-derivative cannot pass through both given points
• 2006 Exam 2 Section B Q4bi — 41% — express q in terms of p from f′(1) = 0
• 2010 Exam 2 Section B Q3b — 42% — show S = 400(cos²x + cos x sin x) for the pyramid's surface area
• 2006 Exam 1 Q11 — 46% — find a from a definite-integral condition, then the two x-intercepts m and n
• 2018 Exam 2 Section B Q1f — 49% — state the value of a for which p(x) = f(x) for all x

T22 · Interpret a circular or exponential model in context — 6

• 2007 Exam 2 Section B Q2g — 14% — combine two safe-period calculations into a single elapsed time
• 2009 Exam 2 Section B Q2e — 23% — distance from Q, not from P, at which the model reaches zero
• 2007 Exam 2 Section B Q2e — 35% — total time, exactly, for which the concentration is above the safety level
• 2007 Exam 2 Section B Q2d — 41% — exact time at which the safety level is first reached
• 2020 Exam 2 Section A Q12 — 45% — which rule gives the height of a minute-hand tip above the clock base
• 2007 Exam 2 Section B Q2fii — 49% — evaluate the fitted geometric model at n = 4 and convert to minutes and seconds

4.8 Group H — Calculus and coordinate geometry on graphs (52 separators)

T23 · Tangents, normals and coordinate geometry — 19

• 2020 Exam 2 Section B Q5e — 7% — values of a for which the tangents at x = a and x = b are parallel
• 2017 Exam 1 Q9d — 9% — coordinates of C, the intersection of two tangents, at θ = 45°
• 2012 Exam 2 Section B Q5c — 11% — midpoint of the two intersection points, in terms of a
• 2014 Exam 2 Section B Q5fii — 11% — equations of the tangents to g passing through a given external point
• 2021 Exam 1 Q9a — 13% — show that the line through A(2, 0) and P has equation y = −x/(3√3) + 2/(3√3)
• 2012 Exam 2 Section B Q2d — 14% — coordinates of the two points whose tangents meet at (−½, 7/4)
• 2006 Exam 2 Section B Q1c — 17% — value of m, from the x-intercepts of tangents one period apart
• 2017 Exam 1 Q9c — 17% — equation of the line through B and C for θ = 45°
• 2020 Exam 2 Section B Q5c — 23% — state the nature of the tangent line g_a when b does not exist
• 2025 Exam 2 Section B Q4fii — 23% — minimum and maximum values of the tangent's y-intercept over p
• 2020 Exam 2 Section B Q4dii — 26% — coordinates of the intersection of two perpendicular tangents
• 2020 Exam 2 Section B Q4eii — 28% — rule of the second line segment joining Q to (3, f(3))
• 2020 Exam 1 Q7biv — 29% — give the equation of one of the tangents through the external point P
• 2008 Exam 2 Section B Q4aiii — 39% — sketch the normal and give its exact axis intercepts
• 2020 Exam 2 Section B Q4ei — 44% — rule of the first line segment joining (0, f(0)) to Q(n, f(n))
• 2016 Exam 2 Section B Q1d — 47% — equations of both tangents to f with gradient 1
• 2020 Exam 2 Section B Q5a — 49% — show that the tangent's x-intercept is b = 2a³/(3a² − 1)
• 2025 Exam 2 Section B Q4fi — 49% — show that t(x) = cos(p)(x − p) + sin(p) + 1
• 2013 Exam 2 Section B Q4ai — 50% — equation of the line through a point with a stated gradient relation

T24 · Area bounded by graphs — 14

• 2017 Exam 2 Section B Q4iii — 2% — limiting value of the bounded area as k increases
• 2013 Exam 2 Section B Q4di — 3% — express A(k), the bounded area, in terms of k
• 2020 Exam 2 Section B Q4eiii — 11% — value of n making two regions equal, one of which is a trapezium
• 2018 Exam 1 Q9aii — 19% — evaluate ∫x sin(x) over [nπ, (n+1)π] for odd n
• 2017 Exam 2 Section A Q17 — 21% — which integral expression gives the area bounded by an even function
• 2008 Exam 2 Section B Q2cii — 26% — solve the area equation 7(a² − 1)/a² = 7 for a
• 2013 Exam 2 Section B Q4aii — 26% — exact area between the line and the curve
• 2022 Exam 1 Q7b — 32% — show by integration that the Type B tile is half-coloured
• 2023 Exam 1 Q5b — 35% — hence find all k with ∫₀^(π/3) sin = ∫_k^(π/2) cos, −3 < k < 2
• 2019 Exam 2 Section B Q3f — 36% — k making a rectangle equal in area to one period of the signal
• 2025 Exam 2 Section A Q17 — 38% — which graph satisfies ∫₁² f > ∫₁³ f
• 2017 Exam 2 Section B Q1dii — 42% — area between y = x and the cubic, as a single integral
• 2017 Exam 1 Q9a — 47% — exact area between y = √x(1 − x) and the x-axis on [0, 1]
• 2018 Exam 2 Section B Q1e — 49% — total area between the tangent l and the quartic, in surd form

T25 · Minimum distance and optimisation on a curve — 11

• 2021 Exam 1 Q9ci — 1% — define the area function g, including its domain
• 2013 Exam 2 Section B Q4diii — 2% — minimum of A(k), at the turning point
• 2007 Exam 2 Section B Q1d — 3% — maximum surface area, at the correct endpoint
• 2013 Exam 2 Section B Q4dii — 3% — maximum of A(k), at the endpoint k = 8
• 2019 Exam 2 Section B Q5f — 4% — b minimising the total shaded area for the inverse
• 2020 Exam 1 Q7c — 10% — k giving the shortest distance from P to the translated graph
• 2016 Exam 2 Section B Q4c — 14% — exact c, d minimising the distance from the origin, and that distance
• 2022 Exam 2 Section B Q1e — 18% — b in terms of a minimising the shaded area
• 2006 Exam 1 Q9a — 37% — express the rectangle's area as A = 18a − 6a³
• 2018 Exam 2 Section A Q17 — 45% — b for which the turning point of y = x² − 2bx + 1 is closest to the origin
• 2022 Exam 2 Section A Q9 — 50% — correct expression for the shortest distance from the origin to a point on f

T26 · Angles — 5

• 2017 Exam 2 Section B Q2h — 2% — minutes, to the nearest minute, from an angle difference
• 2017 Exam 2 Section B Q4h — 2% — values of k for which the angle between two tangents is 30°
• 2017 Exam 2 Section B Q2g — 7% — second viewing angle, to two decimal places, in degrees
• 2017 Exam 2 Section B Q2d — 36% — angle θ, to two decimal places, in degrees
• 2020 Exam 2 Section B Q4b — 37% — obtuse angle between the tangent and the positive horizontal direction

T27 · Trigonometric equations over an interval, and general solutions — 4

• 2006 Exam 2 Section B Q1d — 15% — general solution of |sin(x)| = 0.5, with n ∈ Z
• 2021 Exam 1 Q3c — 17% — solve 2sin(2x) = −√3 for x ∈ R (general solution)
• 2008 Exam 2 Section B Q4c — 37% — solve a tangent equation for a
• 2022 Exam 1 Q6b — 46% — all four values of k in [0, 2π] with f(k) = 0

4.9 Not recoverable

• 2024 Exam 2 Section A Q6 — 50% — published answer B; the 2024 November paper extracts as an empty file and the report carries no commentary for this item, so the wording could not be recovered from this corpus.

4.10 Which types produce separators most often

Removing the two retired types (T12 and T28, 18 separators between them) from the live syllabus leaves 221 separators in content that is still examinable. Of those, the transformation cluster (T10 + T11 + T13 + T30) accounts for 31 and the inverse/composite cluster (T14 + T15 + T16 + T17 + T18) accounts for 27.

4.11 What the reports say went wrong

Across 239 separators the reports return to six failures, and they are remarkably stable across twenty years.

1. Notation, not mathematics. The single most-repeated complaint. 2023 Exam 1 Q7a: "Students are reminded that mathematical notation is a precise language." 2007 Exam 1 Q3b: "union and intersection were sometimes confused as was the use of round, square or curly brackets." 2022 Exam 1 Q5b: "A common error was writing the interval as an intersection not a union. Students need to practise using correct mathematical notation." 2016 Exam 1 Q5bii: "Again, poor notation was evident."

2. The domain is forgotten. 2009 Exam 1 Q3: "Few students realised that the inverse function, f⁻¹, required the rule and the domain to be specified." 2021 Exam 1 Q9ci: "Many were able to write the rule, but very few stated the domain." 2013 Exam 1 Q9cii: "The frequency of [−2, 2] as a preferred solution raises the concern that students believe that the domain does not change under transformations."

3. Transformation language is wrong, or the order is wrong. 2024 Exam 1 general comments: "Students need to use correct mathematical language when describing transformations of graphs… For dilations, students should be familiar with both 'parallel to an axis' and 'from an axis' descriptions." 2023 Exam 2 Q5a: "The order of the transformations needed to be correct, as well as the wording." 2012 Exam 2 Q2e: "Some students gave the transformations that map the graph of g to f" — the wrong direction entirely.

4. Exact values are given as decimals, or decimals to the wrong precision. 2007 Exam 2 Q2d: "Students must remember that an exact answer is required unless a numerical answer is asked for in the question." 2009 Exam 2 Q2bii: "An exact answer was required." 2008 Exam 2 Q1ci: "others rounded incorrectly or gave their answers correct to only one decimal place when the question asked for two." 2014 Exam 2 Q3bii: "Students should always work to suitable accuracy in intermediate calculations to support rounding the answer to the required accuracy."

5. The answer is the right number in the wrong form. 2008 Exam 2 Q3f: "Some gave their answers as 16 and 24" when coordinates were required. 2018 Exam 2 Q1hii: "The minimum value needed to be stated, not just the coordinates of the turning point." 2025 Exam 2 Q2dii: "Others gave the coordinates of the turning point without stating the maximum value." 2022 Exam 2 Q1a: "An equation was required. The most common errors were x, 0, (0, 0), y-axis."

6. Students restart instead of using the previous part. 2023 Exam 1 Q3b: "many students did not use their graph from part 3a." 2021 Exam 1 Q4b: "Most students attempted to solve algebraically instead of using the graph." 2018 Exam 1 Q9aii: "Students generally did not relate this question to the previous question." 2014 Exam 2 Q5b: "Some students did not make the connection between Question 5a. and Question 5b." 2012 Exam 2 Q2d: "Others did not realise that they could have used the result from part c."

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5. What makes a hard one hard

The archive lets us be specific. Five mechanisms account for nearly every very-low-percentage item in this area.

5.1 Compound transformations, where order changes the answer

A single transformation is a 70–90% question. 2018 Exam 2 Section B Q3b — "Describe the transformation that maps the graph of y = h₂(x) to y = h₃(x)" — scored 78%. 2020 Exam 2 Section B Q1d — reflect and translate, two transformations, order flexible because the translation is vertical — scored 76%.

Now add a horizontal dilation to a horizontal translation. 2024 Exam 2 Section B Q1di asked for a translation only: 37%. Q1dii, the same graph, now requiring "Dilate by a factor of ½ from the vertical axis, translate ½ unit to the right, translate 2 units up OR translate 1 unit to the right, dilate by a factor of ½ from the vertical axis, translate 2 units up": 5%. The report: "The vertical translation could be completed at any stage in the sequence. The other transformations had to be in the correct order." Note that the magnitude of the horizontal translation differs between the two valid orders — ½ versus 1. That is the whole difficulty.

The 2026 NHT paper turns this into an explicit two-part question — the same map asked once as "a dilation followed by a translation" and once as "a translation followed by a dilation" — which is the clearest statement VCAA has made that it knows exactly where the difficulty is.

The reliable model. Write g(x) = A f(n(x + b)) + c. Reading outward from x:

1. horizontal dilation, factor 1/|n|, from the y-axis
2. reflection in the y-axis if n < 0
3. horizontal translation, −b
4. vertical dilation, factor |A|, from the x-axis
5. reflection in the x-axis if A < 0
6. vertical translation, c

Horizontal operations must be applied in the order 1→2→3 as written, or the translation amount changes. Vertical operations must be applied 4→5→6, or the translation amount changes. The two blocks can be interleaved freely.

5.2 Endpoints and domain care

Nearly every 1-mark separator in Groups B and F is a question where the mathematics was done and the interval was written wrongly.

• Closed or open. 2014 Exam 2 Q5ci, 7%: the answer was [1, 3); "(1, 3) and 1 < d < 3 were common incorrect answers." 2023 Exam 2 Q3e, 35%: "Round brackets were often seen; these were incorrect as the largest interval of x values was required, which included the interval endpoints."
• Union or intersection. 2022 Exam 1 Q5b, 27%; 2011 Exam 1 Q4b, 1%. Both answers are (−∞, a) ∪ (b, ∞) shapes and both were widely written as a single interval or as an intersection.
• Order of the endpoints. 2006 Exam 1 Q7b: [20, 0] was a common answer. 2024 Exam 1 Q7bii: "Some students, incorrectly, reversed the order of the interval."
• The domain travels with the function. 2013 Exam 1 Q9cii, 39%: students gave [−2, 2], the original domain, after a horizontal translation. The report calls this out as a belief, not a slip.
• A restricted branch closes at a stationary point. 2025 Exam 1 Q5b and the whole T15 family. The largest a with an inverse on (−∞, a] is the x-coordinate of the stationary point, included.

5.3 Notation as a scoring object

Notation is not presentation in this subject; it is content. The reports treat it as such.

• A range is a set: [−1, 3], not −1 ≤ y ≤ 3 (2025 Exam 1 Q3a).
• An intercept is a coordinate pair: (3, 0), not 3 (2006 Exam 1 Q4b; 2008 Exam 2 Q3f; 2016 Exam 2 Q2bi).
• An asymptote is an equation: x = 2, not 2, and not y = 2 when it is vertical (2008 Exam 2 Q4e: "Some students did not give the equations of the asymptotes or they wrote 'y =' instead of 'x ='.").
• An axis of symmetry is an equation (2022 Exam 2 Q1a, 69% on what the report calls a question that "only required students to find the axis of symmetry of the graph of a quadratic function").
• A coordinate uses round brackets only, even for an included endpoint (2012 Exam 1 Q5a).
• An inverse is written f⁻¹(x) = … with its domain, never f⁻¹ = x = … (2009 report).
• A general solution states the quantifier: n ∈ Z, not n ∈ R and not n ∈ J (2006 Exam 2 Q1d; 2021 Exam 1 Q3c).
• A definite integral carries its dx and brackets the whole integrand (2015 Exam 2 Q1d; 2017 Exam 2 Q1dii).
• CAS output is not mathematics: @n1, ^, E notation and tanh all appear in the reports as things students copied across unedited (2006 Exam 2 Q1d; 2010 Exam 2 Q1ai; 2022 Exam 2 Q4d).

5.4 "State" versus "find" versus "show that" versus "explain"

• "State" and "find" both require the answer, and neither exempts you from working when the part carries more than one mark. 2017 Exam 2 Q2h, 2%: "Some students wrote 7 minutes without showing any working. As indicated in the instructions on the examination, for questions worth more than one mark, appropriate working must be shown."
• "Show that" prints the answer. The marks are for the derivation, and there are three specific ways to lose them all: assuming the result (2020 Exam 2 Q1a, "Some students assumed the result in their proof"); starting from LHS = RHS (2007 Exam 2 Q4di, "This will be penalised in the future"); and verifying with particular values when a general result was asked (2006 Exam 2 Q4biv, "there are infinitely many such pairs of values"; 2007 Exam 2 Q4di, "Some students let u and v equal an integer value, which is not acceptable").
• "Explain why" wants a reason tied to the object in the question. 2021 Exam 2 Q3c: "Others gave the meaning of a one-to-one function without relating it to the question."
• "Hence, or otherwise" is a costed hint. 2023 Exam 1 Q5b: "although not essential, most students did well at using their answer from part 5a."
• "Describe" wants transformation vocabulary; the 2007 report records students who "did not seem to know the meaning of the word 'transformation', as they gave instructions to differentiate."

5.5 The calculus interface

More than a fifth of the separators in this area (T23 + T24 + T25 + T26 = 49) are graph questions answered with calculus. Three things make them hard.

Parameters instead of numbers. A tangent at x = 1 is routine; a tangent at x = a is a separator. 2020 Exam 2 Section B Q5 runs the entire question with a parametric tangent: Q5a (show b = 2a³/(3a² − 1)) 49%, Q5b 46%, Q5c 23%, Q5e 7%, Q5g 3%, Q5h 2%. The percentages fall monotonically as the parameter is used more abstractly.

Endpoint maxima. [SD] names "identification of interval endpoint maximum and minimum values" and VCAA uses it. 2013 Exam 2 Q4dii, 3%: "The maximum occurs at the endpoint, k = 8." 2007 Exam 2 Q1d, 3%: "Many students found the minimum surface area instead of the maximum. Some students chose the wrong endpoint." A stationary point is not automatically the answer.

Geometry beats calculus, sometimes. 2020 Exam 1 Q7c, 10%: "Many students used the distance formula and then attempted to differentiate and equate to zero (often with limited success due to error in differentiation or algebra). Students who used a geometric approach tended to score more highly." On a technology-free paper, recognising that the shortest distance from a point to a parabola lies along the normal, or that a translation can be chosen so the point sits directly below the vertex, is faster and safer than a quartic derivative.

5.6 The structural signature of a hard part

Reading across the 239, the low-percentage parts share a shape:

1. they are the last part of a multi-part question (2017 Exam 2 Q4h/Q4ii/Q4iii at 2%, 3%, 2%; 2020 Exam 2 Q5g/Q5h at 3%, 2%; 2013 Exam 2 Q4di/dii/diii at 3%, 3%, 2%);
2. they carry 1 or 2 marks, not 4 — across the whole area, 323 of the 512 parts are worth 1 mark, 140 are worth 2, 42 are worth 3, and only 7 are worth 4 or 5;
3. they introduce a new letter — k, a, d, p, w — into an otherwise-familiar object;
4. they ask for a set, not a number.

If a part has all four properties, treat it as a 3-minute question and leave it until the paper is otherwise finished.

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6. Worked method sheet

Ten routines, compressed to what fits in your head under time pressure. Each is chosen by yield: the types in §4.10 that produce the most separators, plus the two cheapest recoveries.

M1 — Describe a sequence of transformations (T10)

1. Force g into A f(n(x + b)) + c. Factor n out of the bracket first: f(2x + 6) = f(2(x + 3)), so the translation is 3, not 6.
2. Write the six-step list (§5.1), skipping any with n = 1, A = 1, b = 0, c = 0.
3. Say each one in VCAA's words: - "a dilation by a factor of k from the x-axis" (vertical) — or "parallel to the y-axis". - "a dilation by a factor of k from the y-axis" (horizontal) — or "parallel to the x-axis". - "a reflection in the x-axis" / "in the y-axis". - "a translation of k units in the positive direction of the x-axis" — or "k units to the right".
4. Never write "in the y-axis" for a dilation. Never write a dilation factor as a negative number — that is a reflection plus a dilation.
5. Check by transforming one point.

M2 — Apply a given sequence (T11)

Use mapping equations, one line per transformation, in the printed order.

start      x' = x            y' = y
dilate 1/2 from y-axis   x' = x/2
translate +3 in x        x' = x/2 + 3
translate -1 in y                     y' = y - 1

Then invert: x = 2(x′ − 3), y = y′ + 1. Substitute into y = f(x) to get y′ + 1 = f(2(x′ − 3)), and rename. Transform the domain endpoints with the x-line only. Give the image a new name (g, h), never f.

M3 — Inverse: rule and domain (T14)

1. ran f — get it now, from the graph or the endpoints. This becomes dom f⁻¹.
2. Write y = f(x).
3. New line: "for the inverse, swap x and y" → x = f(y).
4. Solve for y. If a square root appears, choose the sign so that y lies in dom f (that is ran f⁻¹).
5. Write f⁻¹ : ran f → R, f⁻¹(x) = ….

Two-second check: f⁻¹(f(a)) = a for one convenient a.

M4 — Composite: rule, domain, range, existence (T17, T18)

1. Exists? ran g ⊆ dom f. If not, restrict dom g.
2. Rule: substitute the whole of g(x) into f, in brackets.
3. Domain: dom(f ∘ g) = dom g (as restricted).
4. Range: compute ran g first, then sketch f over just that piece and read its range. ran(f ∘ g) = f(ran g) — it equals ran f only when ran g covers all of dom f.

For "find the largest c such that ran g ⊆ dom f": set the boundary of ran g equal to the boundary of dom f and solve for c, then check which root lies in the stated region.

M5 — Maximal domain (T6)

Solve the inequality by factorising and sketching the parabola. Write the answer with ∪. If two constraints apply, intersect the solution sets — and remember that a difference of logs has two constraints, not one.

M6 — By-hand sketch (T1)

1. Copy the label instruction as a checklist.
2. Compute, exactly: y-intercept (x = 0), x-intercepts (y = 0), asymptotes, stationary points if asked, endpoint coordinates.
3. Draw asymptotes dashed and label them with equations.
4. Plot the computed points.
5. Draw a smooth curve with correct curvature, using the printed grid.
6. Stop at the domain. Closed dot for an included endpoint, open circle for excluded.
7. Re-read the checklist.

For y = |f(x)|: trace the parts above the axis, reflect the parts below, and mark the cusps. For y = f(x) + k over a printed graph: move every labelled point vertically by k and trace.

M7 — Families: "for which values of the parameter …" (T20)

1. Name the fixed object and the moving object.
2. Find the critical positions. There are only four kinds: - the moving graph is tangent to the fixed one → solve f = g and f′ = g′ simultaneously; - the moving graph passes through a turning point or endpoint of the fixed one; - a root becomes repeated → discriminant = 0; - a feature reaches an asymptote or a domain boundary.
3. Test one value of the parameter on each side of every critical value, and test the critical value itself.
4. Write the answer as an interval or union, with bracket types decided by step 3.
5. If technology is available, put the parameter on a slider and watch the count change. VCAA endorses this.

M8 — Tangent with a parameter (T23)

y − f(a) = f′(a)(x − a). Keep a symbolic. Then:

Bracket the whole (x − a). When asked for the intersection of two tangents, substitute the intersection x into a tangent equation, not into f.

M9 — Area between graphs (T24)

1. Solve f(x) = g(x) for the terminals.
2. Between consecutive terminals, decide which curve is upper by testing one point.
3. Write ∫ᵃᵇ (upper − lower) dx, one integral per region, with the dx.
4. Never insert a minus sign for a region below the x-axis when you are already subtracting lower — that double-counts.
5. Exact unless told otherwise.

For an approximation question: h is the strip width; the trapezium rule is on the formula sheet; integration is not an acceptable method when the question says "approximate using trapeziums/rectangles".

M10 — Recover the two cheapest marks

The inequality after your own sketch (T9). Three archive instances, all separators, all 1 mark. Solve f(x) = k for the boundary; then look at the graph you just drew and read which side is wanted. Watch for an asymptote splitting the answer into a union.

The range or interval (T5). Evaluate at both endpoints and at any interior stationary point; take the least and greatest; choose brackets by whether each value is attained. Write it as a set.

Both take under ninety seconds and the state loses them at rates of 1%–48%.

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7. One-page summary

• Functions, relations and graphs is 31.3% of the November marks in the archive — the largest area of study.
• 48.0% of its graded parts are separators; on Exam 1 that rises to 60.0%.
• The three highest-yield separator types are families/parameters (T20, 19), tangents and coordinate geometry (T23, 19) and "show that" model fitting (T21, 18).
• Every archive instance of "domain/range/existence of a composite" (T18) is a separator.
• Two types in the archive are no longer examinable: matrix transformations (T12, last seen 2020) and functional equations (T28, last seen 2017). Eighteen of the 239 separators sit in those two types.
• The reports name the same six failures for twenty years: notation, missing domains, transformation language and order, exact values, answer form, and not using the previous part.
• The transformation form y = A f(n(x + b)) + c and the sentence "state the rule and domain" are the two most reliable annual appearances in the subject, and the two most reliable places to lose marks.