Pure Mathematics 3
Topic Brief: Modulus and Graph Transformations
Provenance: written 2026-09-29. Source: research/veridian/WMA13-verified-facts.md, grepped directly for "modulus" and "1.4"/"transformation" (§1 spec table; §5, "1.3 The modulus function" and "1.4 Transformations"; §9 lesson-audit log entry for modulus-and-graph-transformations.ts), cross-checked against this folder's own signal-frequency-and-boundaries.md (§1 frequency table, rows 1.3 and 1.4) for the series counts. Also checked before writing: content/veridian/curriculum/wma13/modulus-and-graph-transformations.ts (the lesson file itself, including its header-comment sourcing notes, so this brief does not re-run its six worked traps or its diagram) and content/veridian/question-bank/wma13.ts (grepped for "modulus" and "transformation" — zero hits, so no question-bank entries exist yet for this topic to avoid duplicating). No codex-deep-dive document exists yet for this topic in this folder, and no prior signal-topic-brief for it either — this is the first technique-hub document for modulus-and-graph-transformations.ts. No new primary-source PDFs were opened for this pass; every citation below traces back to the verified-facts file.
SIGNAL Topic Master Brief: Modulus And Graph Transformations (WMA13, spec 1.3–1.4)
Scope
Matches the lesson modulus-and-graph-transformations.ts: the modulus function — sketching y = |ax+b|, y = |f(x)|, y = f(|x|), and solving |ax+b| = or > a linear expression (1.3) — plus combined transformations y = af(x), y = f(x)+a, y = f(x+a), y = f(ax) (1.4). These are two separate spec sub-items taught in one lesson; this brief keeps their frequency counts separate below because the real papers treat them separately too (see "What wasn't found").
Every real series that has tested this topic
13 WMA13 series have been checked in the underlying research archive (8 read start-to-finish with full mark schemes and examiner reports; 5 more checked for specific topics or citation-currency only). Between 1.3 and 1.4, this topic has a located, citable question in 6 of the 13: Oct 2020, Jan 2022, Jun 2023, Oct 2023, Jan 2024, Jun 2026. Split by sub-topic: 1.3 (the modulus function itself) is seen in 5 of 13 — Oct 2020, Jan 2022, Jun 2023, Oct 2023, Jan 2024. 1.4 (combined transformations) is seen in only 2 of 13 — Jan 2024 and Jun 2026.
| Series | Q | Sub-topic | What it actually tested |
|---|---|---|---|
| Oct 2020 | Q4(c) | 1.3 | Modulus equation/inequality; one-branch-only error confirmed |
| Jan 2022 | Q7(b) | 1.3 | |linear| > linear inequality; full M1 A1 dM1 A1 breakdown verified from primary source |
| Jun 2023 | Q6(c) | 1.3 | Modulus inequality; same one-branch/wrong-region error family |
| Jun 2023 | Q6(d) | 1.3 | y = f(|x|) graph read via reflective symmetry, not algebra |
| Oct 2023 | Q9(c) | 1.3 | Modulus of a logarithm, |2−4ln(x+1)|=3, region-selection is doubly dependent (ddM1) |
| Jan 2024 | Q1 | 1.4 | Coordinate transformation of a curve; partial-application errors |
| Jan 2024 | Q8 | 1.3 | Parametrised |x|−1 = a−|2x−b| simultaneous equation |
| Jun 2026 | Q3(c) | 1.4 | Point mapped under y=f(2x)+3; B1 dB1; wrong-point trap |
Mark-tariff pattern
Only two of the eight anchors above have a full, verified mark breakdown in the research file; the rest are confirmed real but only as examiner-report prose, not transcribed mark-by-mark, so no tariff is invented for them.
- A plain "|linear| > linear" inequality (Jan 2022 Q7(b)): 4 marks, M1 A1 dM1 A1. One M1 for solving either branch's equation, one A1 covering both critical values together (Pearson does not award a second independent method mark for the second branch), one dM1 for selecting the correct outside region, one A1 for the final two-part answer. This matches the spec's own 1.3 worked example almost exactly (
|2x−1| > x+5). - When modulus is combined with a second technique (Oct 2023 Q9(c), modulus of a log), the region-selection mark deepens to ddM1 — dependent on two prior method marks (forming the equation, then solving it) rather than one. The dependency depth tracks how many real method marks precede that step in that specific question, not a fixed property of "modulus" as a topic.
- The smallest confirmed 1.4 slice is 2 marks (Jun 2026 Q3(c), B1 dB1) for mapping a single named point under a combined transformation — embedded inside a larger 8-mark question whose first two parts (differentiation, quotient rule) are outside this lesson's own scope.
- 1.4 more broadly reads as a small tail-part tariff, not a standalone one. Jan 2024 Q1's transformation errors are documented only in examiner-report prose in the research file, with no mark-by-mark breakdown transcribed — a real gap, left as a gap here rather than guessed at.
Recurring context/angle
Three shapes repeat:
- Solve |linear| = or > linear by case-split, then select the correct combined region. This is the single most repeated 1.3 shape — confirmed as the source of a "select the correct region" error across Jan 2022 Q7, Oct 2020 Q4(c), and Jun 2023 Q6(c), independent of whether the critical values themselves were found correctly.
- A graph-reading argument for y = f(|x|), not pure algebra. Jun 2023 Q6(d) is verified as testing whether candidates understand that the negative-x branch of f(|x|) is a mirror of the positive-x branch — a reasoning mark, not a computation.
- A named point is transformed instead of one computed earlier in the same question. Jun 2026 Q3(c)'s point-mapping trap and Jan 2024 Q1's partial-application errors are both 1.4, but they are distinct mechanisms: Jan 2024 Q1 is about executing a stated transformation incompletely; Jun 2026 Q3(c) is about applying a correct transformation to the wrong point — the point just calculated in an earlier part, rather than the point the question stem separately names.
If this topic appears, here's what to expect
- 1.3 is the more frequently-anchored half by a clear margin (5 of 13 vs 2 of 13) — a modulus equation/inequality is more likely to appear than a standalone combined-transformation question.
- Expect a "|linear| = / > linear" part worth around 4 marks, scored as one method mark per branch's equation-solving, one shared accuracy mark for both critical values, one dependent mark for the region, one final accuracy mark — do not expect a second independent method mark for the second branch.
- If graphs are involved, expect at least one mark to require a stated reflective-symmetry argument for f(|x|), not just a re-sketch.
- If 1.4 appears, expect it as a small 2–3 mark tail part bolted onto a longer question built on different content (a differentiated rational function, a general coordinate-transformation exercise), not as its own multi-part question.
- If a point must be transformed, transform the point the question stem actually names in that part — not a point computed in an earlier part of the same question — even when the transformation arithmetic itself would be flawless.
- Watch for modulus paired with a second technique (logs, in the one confirmed case): the region-selection mark's dependency depth is decided by how many method marks genuinely precede it in that exact question, not by the word "modulus."
What wasn't found
No series in the archive was located testing 1.3 and 1.4 within the same question — every anchor above is one or the other, despite the two being taught together in a single lesson. Jan 2024 Q1's 1.4 errors have no transcribed mark scheme in the research file, only examiner-report prose; that gap is stated here, not filled. No likelihood or percentage claim is made anywhere above — every count is "seen in N of the 13 series checked."
Sources
research/veridian/WMA13-verified-facts.md; signal-frequency-and-boundaries.md (this folder, §1 rows 1.3/1.4); content/veridian/curriculum/wma13/modulus-and-graph-transformations.ts (checked to avoid duplication, not cited for facts).