Topic Brief: Cubic Graph Sketching From Factored Form

Provenance: compiled 2026-09-29. Source is research/veridian/WMA11-verified-facts.md §4.1 and its "Lesson-audit log" entry for cubic-graph-sketching-from-factored-form.ts (both read in full for this document) — no fresh PDF extraction was done here; the facts bank's own re-extraction (pdftotext -layout/-raw cross-checked, every numeric claim independently recomputed with node -e) is the primary source being drawn on. This is a narrow expansion of the single-line entry ("Polynomial factorisation / cubic sketching | 1.10/1.11 | 2 of 5 | Jan 2023 Q10, Oct 2021 Q6(a)") already in signal-frequency-and-boundaries.md §1 of this same folder, and it does not repeat that document's whole-paper tables or codex-formula-and-method-bank.md §2.1.5's step-by-step method — both are cross-referenced, not restated. Not affiliated with or endorsed by Pearson Edexcel.

#SIGNAL Topic Master Brief — Cubic Graph Sketching From Factored Form (spec 1.10/1.11)

#Frequency: seen in 2 of 5 series checked

The facts bank fully mined five WMA11 examiner reports question-by-question: Jan 2019, Oct 2020, Oct 2021, Jan 2023, Jan 2025. A factorised-cubic sketching or sketch-reading question is confirmed in 2 of those 5: Oct 2021 Q6(a) and Jan 2023 Q10(a).

Four further series have been touched by this research pass but not on equal terms: Oct 2022 and Jan 2024 were confirmed as genuine mark schemes but never mined topic-by-topic, and June 2022 / June 2023 were each read for one specific, different question only (Q5 and Q1–Q2 respectively — neither this topic). So "2 of 5" is the honest count where a non-appearance would actually mean something; across those other four series, whether this topic came up is simply unknown, not confirmed absent.

#The two confirmed appearances

Oct 2021 Q6(a) (3 marks) — f(x) = 2(x+1)(x−3)², "sketch a graph of C, showing the coordinates of the points where C cuts or meets the coordinate axes." One simple root, one repeated (squared) root. Real mark scheme: M1 for the correct positive-cubic shape (the scheme explicitly tolerates near-vertical ends and, for this question specifically, a "cusp like appearance for the turning points"); A1 for the curve reaching the x-axis at −1 and having its minimum at 3 without crossing there; B1 for "y intercept at 18."

Jan 2023 Q10(a) (2 marks) — sketch already given, of f(x) = (3x+20)(x+6)(2x−3), three distinct simple roots; "state the values of x for which f(x) > 0." M1 for one correct region on its own; A1 for both regions stated together, correctly bounded and ordered.

#Mark-tariff pattern: a low-tariff opening sub-part of a longer question

Neither confirmed appearance is a stand-alone question. Both sit as part (a) of a larger multi-part item that immediately pivots to a different spec point: Oct 2021 Q6 continues into (b) expanding the same cubic and (c) a tangent-at-a-stationary-point differentiation question (its own real, verified 4-mark tariff — B1ft/M1/A1/A1); Jan 2023 Q10 continues into (b) expanding the brackets and (c) a further tangent/intersection question, with the whole of Q10 worth 10 marks against this topic's 2. On the only two confirmed instances, the sketching-or-reading sub-skill itself is consistently a small, early payoff (2–3 marks) that then hands off to expansion and differentiation for the rest of the question's marks — worth knowing so a student doesn't mistake "I got the sketch right" for "I've banked most of this question's marks."

#The recurring angle: factored, positive, linear factors only — but two different jobs

Both real cubics are handed over already fully factored into linear brackets, with a positive leading coefficient once multiplied out (2×1×1 for Oct 2021; 3×1×2 for Jan 2023) — on this evidence, WMA11's factored-cubic content sticks to positive cubics built from linear factors alone, not a mix that includes an irreducible quadratic factor. Beyond that shared shape, the two anchors ask for opposite directions of work: Oct 2021 Q6(a) asks the student to generate the sketch from the factors; Jan 2023 Q10(a) hands the sketch over already drawn and asks the student to read a solution set off it. They also don't cross-confirm each other's root pattern — Oct 2021's cubic has one simple root plus one repeated root (the case that produces a touch-and-turn), while Jan 2023's has three distinct simple roots (three genuine crossings, no touching point at all). Each root-multiplicity flavour is therefore confirmed exactly once, not twice.

#Confirmed trap set (both series, verified against the real mark scheme and examiner report)

• Cusp at the turning point — noted by the Oct 2021 examiner report ("a few candidates were inclined to feature a cusp at the minimum point"), but the real mark scheme for that specific question explicitly condones it; it did not, in fact, cost marks on this series.
• Y-intercept calculated but never labelled on the sketch — a separate, genuinely mark-costing omission on the same Oct 2021 question ("failed to label the y intercept"), independent of the cusp habit above.
• Critical values given instead of inequalities — on Jan 2023 Q10(a), the report attributes losing both marks, for a large share of candidates, to writing the roots alone without converting them into a solution-set inequality.
• One-sided region — the single most common one-mark answer on the same question paired a correct x > 3/2 with an incorrect, unbounded x > −6 for the other region, silently absorbing the negative stretch between the two roots.
• Boundary order reversed — a smaller, distinct error on the same question: the two correct boundary values for a compound inequality written with the more negative value on the wrong side, producing an inequality with no solutions at all.

Only about a third of Jan 2023 candidates scored both marks on that sub-part, per the examiner report's own figure — cited here as a real, reported statistic on that one question, not a general likelihood claim for the topic.

#If this topic appears: what to expect

Expect a small (2–3 mark) opening sub-part rather than a whole question, built on a positive cubic already given as a product of linear factors. Expect one of the two confirmed jobs — either sketch it and label every axis crossing (including the y-intercept, on the drawing itself, not just in your working), or read a sign-based solution set off a sketch you're given. If it's the sketching job, get the crossing-vs-touching shape right and don't worry disproportionately about a slightly cusp-like turning point if your overall shape and labelled points are correct — but still draw it smoothly where you can, since the leniency seen here is confirmed for only one series, not a general rule. If it's the solution-set job, finish the job: state inequalities, not bare values, bound every region on both sides, and put the more negative boundary on the left.