Deep Dive: Cubic Graph Sketching From Factored Form

Provenance (2026-09-29). Read in full for this pass: research/veridian/WMA11-verified-facts.md (§1's verbatim spec table for 1.10/1.11; §2's assessment structure; §3's mark-scheme conventions, specifically the M/A/B definitions and the differentiation method-mark wording; §4.1's two named anchors for this topic; and the complete cubic-graph-sketching-from-factored-form.ts mark-scheme- bullet-coverage audit log entry, 2026-09-14, both anchor transcriptions read in full); the full lesson file content/veridian/curriculum/wma11/cubic-graph-sketching-from-factored-form.ts (all 802 lines, read to confirm exactly what it already teaches — its mechanism block, both marked-solutions, all seven trap-taxonomy entries and its beyond-spec triple-root block — so nothing below restates them); content/veridian/technique-hub/wma11/signal-topic-brief-cubic-graph-sketching-from-factored- form.md (in full, for its frequency count and mark-tariff-pattern finding); content/veridian/ technique-hub/wma11/codex-formula-and-method-bank.md (§1.1.7–1.1.9 and §2.1.5, its factorisation/ graphs/transformations entries and five-step method); and content/veridian/question-bank/wma11.ts (grepped for "cubic" — no entry exists there for this topic, so no overlap to avoid on that front). Every numeric claim below was independently computed and cross-checked with node -e before being written in. Not affiliated with or endorsed by Pearson Edexcel. Every quotation is short (under ~15 words) and attributed to a named document; no full question, mark-scheme extract or examiner- report passage is reproduced at length.

#CODEX Deep-Dive — Cubic Graph Sketching From Factored Form (WMA11, spec 1.10/1.11)

#What this document is, and isn't

The lesson file already teaches this content pedagogically in full: a mechanism block deriving why a repeated root touches rather than crosses, a worked chain and chain-drill sketching a cubic from scratch, two marked-solutions (one VERIDIAN-original, one the real Jan 2023 Q10(a) transcription), seven trap-taxonomy entries, and a beyond-spec block on the triple-root case. The signal brief already states the real frequency count and the five confirmed traps. The formula bank already gives the compact five-step sketching method and a one-line note on reciprocal functions and transformations (both explicitly out of scope here — see below).

None of those three documents states the complete real classification of how a cubic's roots can be structured (there are exactly four shapes, not two — the lesson and the real anchors between them only instantiate two of the four), derives why a cubic can only ever have 0 or 2 turning points and never exactly 1 (the lesson's own reference card states this as a bald fact; this document proves it, using only P1-legal tools already established as prerequisites), states the general rule for an explicit leading constant multiplied onto the whole factorisation (a real feature of the one real anchor that uses a non-unit leading coefficient, which none of the lesson's own worked or quiz content exercises directly), or closes with a consolidated ledger of every real exam angle this specific topic has actually been tested from, built directly from the facts bank's own citations rather than re-derived from scratch.

Scope, stated up front. This document is the cubic slice of spec 1.11 only — reciprocal-function sketching (y = k/x, y = k/x², asymptote vocabulary) and trigonometric-graph sketching are different sub-skills of the same spec line and are not covered here, matching the lesson's own declared scope. Graph transformations (spec 1.12) are a separate topic with its own lesson and are not covered here either. The honest evidentiary picture: exactly two real WMA11 anchors have been found for this topic in the series the facts bank has mined question-by-question (Oct 2021 Q6(a), Jan 2023 Q10(a)) — both already fully transcribed in the lesson. Everything new below that has no real per-question anchor says so plainly, rather than being dressed up with an invented citation.

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#Part 1 — The four facts factored form hands you, and why testing one point per interval is enough

Spec 1.10 covers "factorisation (degree n ≥ 3; notation f(x) may be used)" (verbatim, facts bank §1); spec 1.11 covers "sketching curves defined by simple equations" and "geometrical interpretation of algebraic solution of equations" (verbatim, same source) for the cubic case specifically. Together they hand a factorised cubic's sketch exactly four facts, and no more:

1. Every root and its multiplicity — read directly off each linear factor.
2. The y-intercept — f(0), found by substituting into every factor at once.
3. The end-behaviour direction — the sign of the leading coefficient.
4. The full sign pattern along the whole number line — which regions are positive, which negative.

The lesson's own teach block walks fact 4 by stepping rightward across each root and flipping the sign at every simple root crossed. Why this single-point-per-interval test is actually watertight, not just a convenient shortcut, is worth stating explicitly: a polynomial is continuous everywhere, and once it is given in fully factored form, its zeros are exactly the listed roots — nowhere else. A continuous function can only change sign by passing through zero (it cannot jump from positive to negative without a value in between equalling zero), so between any two consecutive roots the sign is genuinely locked — checking it once anywhere in that interval settles the whole interval, not merely the point checked. This is a fact about this specific, already-factored expression, not a general claim about finding roots of an arbitrary polynomial (that broader machinery — the factor theorem — is explicitly P2 content, excluded at P1; nothing here needs it, because the roots are already given).

One useful corollary, not stated explicitly in the lesson: the identical method that reads f(x) > 0 off a sketch reads f(x) < 0 with zero extra work — it is the other set of regions from the same sign walk, never a separate calculation.

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#Part 2 — The complete real classification: every way a real cubic's roots can be structured

A cubic has degree exactly 3, so its roots' multiplicities (real and complex together) must sum to 3. Enumerating every way that can happen over the reals gives exactly four shapes — not the two the lesson's own worked and real-anchor content instantiate:

(a) Three distinct simple real roots (1+1+1). The real anchor for this shape is Jan 2023 Q10(a) (already fully transcribed in the lesson): the sketch given for f(x) = (3x+20)(x+6)(2x−3) crosses three times. A fresh micro-example, not reusing any lesson number: f(x) = (x+5)(x−2)(x−4), expanding to x³−x²−22x+40. Verified: f(0) = 40, matching (5)(−2)(−4) = 40.

(b) One simple root and one double (repeated) root (1+2). The real anchor is Oct 2021 Q6(a): f(x) = 2(x+1)(x−3)² — one crossing, one touch. A fresh micro-example: f(x) = (x−3)(x+1)², expanding to x³−x²−5x−3. Verified: f(0) = −3, matching (−3)(1)² = −3.

(c) One triple real root (3). The lesson's own beyond-spec block already derives this case abstractly (f(x) = (x−p)³·h(x)) with no concrete number attached; a genuinely new, fully-worked instance: f(x) = (x−2)³, expanding to x³−6x²+12x−8. Verified: f(0) = −8, matching (−2)³ = −8. No real per-question WMA11 anchor exists for this shape — exactly as the lesson's own beyond-spec block already discloses, and repeated here rather than silently dropped.

(d) One real root and one irreducible quadratic factor (1 + a non-real conjugate pair). This is the one shape neither the lesson nor the two real anchors instantiate at all — both real anchors, and every teaching cubic the lesson itself uses, factorise completely into real linear brackets. A cubic does not have to: f(x) = (x−2)(x²+x+1) is a genuine, fully real-valued cubic with exactly one real root. Verified: the quadratic factor's discriminant is 1²−4(1)(1) = −3 < 0 (spec 1.4, already an assumed prerequisite for this lesson), so x²+x+1 has no real roots at all — x = 2 is the only real root, however many turning points the curve has (Part 3 below). Expanding gives x³−x²−x−2; verified: f(0) = −2, matching (−2)(1) = −2 directly from the factored form. No real per-question WMA11 anchor exists for this shape either. Spec 1.10's own worked example (x³+4x²+3x = x(x+1)(x+3), fully linear) and both real anchors for this exact topic all factorise completely into real linear brackets — whether WMA11 would ever hand a P1 student a cubic with a genuinely irreducible quadratic factor to sketch is not confirmed either way by anything reviewed for this document. Case (d) is included purely to complete the real mathematical classification of what a cubic can be, not because any evidence suggests it is examined at this level.

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#Part 3 — Why a cubic has exactly 0 or 2 turning points, and never exactly 1

The lesson's own reference card states "a cubic has at most 2 turning points, never 1" as a bald fact. Here is why, using nothing beyond tools this same paper already requires: differentiate once (spec 4.2, the power rule already used as an optional check in the lesson's own worked chain) and apply the discriminant (spec 1.4) to the result.

For f(x) = ax³+bx²+cx+d (a≠0), f′(x) = 3ax²+2bx+c — a genuine quadratic, since 3a≠0. Its own discriminant is (2b)²−4(3a)(c) = 4(b²−3ac). Exactly the same three-way discriminant split spec 1.4 already teaches for any quadratic applies here, one level up:

• Discriminant of f′ positive → f′ has two distinct real roots, each a genuine sign change in f′ (the discriminant-positive case always crosses, never touches — spec 1.4's own result, applied to f′ itself) → f has exactly two genuine turning points.
• Discriminant of f′ zero → f′ has one repeated real root, which touches zero rather than crossing it (the discriminant-zero case, again spec 1.4's own result) → f′ never actually changes sign there, so f's gradient momentarily reaches zero and immediately resumes its previous direction: a stationary point of inflection, not a turning point → f has zero turning points (this is exactly case (c) above: verified, f′(x) = 3(x−2)² for f(x) = (x−2)³, discriminant 0, and f′ never goes negative).
• Discriminant of f′ negative → f′ has no real roots at all, never changes sign, so f is strictly monotonic everywhere → zero stationary points of any kind.

So the count is 2 (discriminant of f′ positive) or 0 (discriminant of f′ ≤ 0) — never 1, because a quadratic's own discriminant has no case that produces "one crossing and nothing else."

This directly resolves a trap the turning-point count alone would set for an unwary reader: two turning points do not guarantee three real roots. Case (d)'s own micro-example proves it — verified, f(x) = (x−2)(x²+x+1) has discriminant of f′ equal to 16 (positive, so genuinely two turning points, at x = −1/3 and x = 1) yet both turning-point values come out negative (f(−1/3) ≈ −1.81, f(1) = −3): the curve rises from below the axis, reaches a local maximum that is still below the axis, falls to a local minimum, then rises once and only once through x = 2. The root count is fixed the moment the factorisation is written down (one linear factor, one irreducible quadratic factor, so exactly one real root) — the two turning points change the curve's wiggle, not how many times it can possibly cross. Counting turning points and counting roots are two different questions, answered by two different tools (calculus for the first, the factorisation itself for the second) — a distinction the lesson's sketch-from-factors method never needs to blur, precisely because it never needs calculus at all to get the roots right.

The same discriminant-of-f′ machinery is exactly what spec 1.11's own clause — "use of intersection points of graphs to solve equations" — needs for a horizontal line y = k rather than just y = 0: the number of real solutions to f(x) = k depends on where k sits relative to the two turning-point y-values (call them the local-max value M and local-min value m, with M > m for a positive cubic). For k above M or below m, only one branch can reach it: one real solution. For m < k < M, the line passes through all three "sections" of the curve: three real solutions. At k = M or k = m exactly, the line is tangent to a turning point: two distinct real solutions, one of them a repeated root. This is the same real, verified mechanism the lesson's own teach block gestures at with a large-|k| heuristic ("only the branch heading off to infinity can meet it") — stated here as the complete rule it is a special case of, rather than restated as the heuristic alone. Finding the exact values of M and m needs the differentiation this sketch-only skill doesn't itself require — which is exactly why the real Oct 2021 Q6 anchor's own later part (c), on the same cubic, is a tangent-at-a-stationary-point differentiation question (already logged in the facts bank, not re-derived here): the real exam structure already pairs "sketch it" with "now differentiate it," for precisely this reason.

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#Part 4 — The leading constant multiplier: a real exam feature none of the lesson's own worked content exercises

Every teaching cubic in the lesson's mechanism block, worked chain, chain-drill, diagram, marked- solutions and MCQs carries a leading coefficient of exactly +1 or −1 once expanded — the only place a different leading constant appears anywhere in the lesson is inside the real Oct 2021 anchor itself, f(x) = 2(x+1)(x−3)², used there for its cusp/y-intercept discussion, not as a dedicated "compute this" step.

The general rule, a direct extension of the same substitute-and-multiply logic the lesson already teaches: for f(x) = k·(x−p₁)(x−p₂)⋯, k≠0, neither the roots nor their multiplicities change (k never equals zero, so it can never itself be what makes the product zero) — only the leading coefficient (k times the product of each factor's own x-coefficient) and the y-intercept (k times the product of each factor's constant term) are scaled. A negative k additionally flips the end-behaviour direction, exactly as an overall minus sign does in the lesson's own g(x) = −(x+3)(x−2)² example.

Original micro-example, distinct from the real anchor's own coefficient: f(x) = 3(x+2)(x−1)², expanding to 3x³−9x+6. Roots at x = −2 (simple, crosses) and x = 1 (double, touches) — identical classification to the k = 1 case, unaffected by the constant. Leading coefficient: 3×1×1² = 3 (positive cubic). Y-intercept: verified, f(0) = 3(2)(1)² = 6, matching 3×2×1 = 6 directly from the factors. Verified against the expanded form: f(0) = 6. The one thing worth flagging explicitly, since it is a real place a learner might reach for it wrongly: the constant scales heights, not positions — it changes the y-intercept and the size of any turning-point value, but the x-coordinate of every root, and of every turning point, is completely unaffected by k.

No real per-question WMA11 anchor exists for the leading-constant-multiplier step in isolation. The one real exam sighting of a non-unit leading coefficient on this exact topic (Oct 2021 Q6(a)'s k = 2) folds it into the same B1 shape-and-y-intercept mark as everything else in that question, rather than crediting it as its own separate step — so this document states the general rule as genuine, correct spec 1.10/1.11 content that closes a real gap between what the lesson's own examples use (k = ±1 throughout) and what the one real anchor actually contains (k = 2), without inventing a citation for a mark-scheme line that does not exist on its own.

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#Part 5 — Every real exam angle this topic has actually been tested from

Read every row as "seen in the series named," never as a probability or a frequency claim beyond what is stated — no likelihood percentage and no grade boundary is invented anywhere in this document.

Frequency, stated exactly as the signal brief already establishes it: seen in 2 of the 5 WMA11 series mined question-by-question in this research pass (Jan 2019, Oct 2020, Oct 2021, Jan 2023, Jan 2025) — not restated in further detail here, since the signal brief's own frequency analysis already covers it and this document does not duplicate it.

Mark-tariff pattern, confirmed on both real anchors: neither is a stand-alone question — both sit as a small (2–3 mark) opening sub-part of a larger multi-part item that immediately pivots to a different spec point (expansion, then a tangent/differentiation question, in both cases). A student who nails the sketch or the sign-read has banked a small fraction of that question's total marks, not most of them.

The two confirmed real anchors test opposite directions of work on this topic, and, between them, two of this document's four classified root shapes: Oct 2021 Q6(a) asks the student to generate a shape-(b) sketch from its factors; Jan 2023 Q10(a) hands over an already-drawn shape-(a) sketch and asks the student to read a solution set off it. Shapes (c) and (d) — this document's own Part 2 — have no real per-question anchor found anywhere in the series reviewed, and neither does a dedicated f(x) = k (k≠0) intersection-count question on a factorised cubic, or a standalone leading-constant- multiplier mark (Part 4) — each already disclosed at the point it was introduced above, not repeated here as new information.

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Reciprocal-function sketching (y = k/x, y = k/x², asymptote vocabulary) and graph transformations (spec 1.12) are real, separate spec content and are explicitly not covered by this document, matching the lesson file's own declared scope — see its header note for the reasoning.