Deep Dive: Modulus and Graph Transformations

Provenance (2026-09-29): read/checked in full for this pass — research/veridian/WMA13-verified-facts.md (grepped and read in full for "modulus", "transformation", "1.3", "1.4", "Q8", "Jan 2024", plus the spec table in §1, the mark-scheme-convention material in §4, and the modulus-and-graph-transformations.ts audit-log entry in §9); content/veridian/curriculum/wma13/modulus-and-graph-transformations.ts (the full 1046-line lesson, read start to end — both prequestions, both teach blocks, the two mechanism blocks, the concept ladder, both diagrams, the worked chain, the chain-drill, the marked solution, all six trap-taxonomy entries, all five MCQs, the beyond-spec evenness note, and all nineteen flashcards — to identify what already exists and avoid duplicating it); content/veridian/technique-hub/wma13/codex-formula-and-method-bank.md (§2.3–2.4), forge-exam-technique-guide.md (§1, §2.2, §5.1) and signal-frequency-and-boundaries.md (§1) — all grepped for 1.3/1.4 coverage, all three giving compressed skeletons or bare frequency counts, no theory or exam-angle synthesis. content/veridian/question-bank/wma13.ts was grepped for modulus/f(|x|)/transform — no dedicated 1.3/1.4 items beyond the lesson's own five MCQs, so no duplication risk. Five real primary Pearson questions were opened fresh for this pass and are not cited anywhere in the four files above: October 2022 Q7, January 2024 Q1(c) and Q8 in full, June 2022 Q5 (all three parts, yielding two distinct exam angles below), and June 2024 Q1 — June 2022 and June 2024 re-extracted directly this session with pdftotext -layout (June 2024 also cross-checked with -raw) against the real PDFs at ~/Documents/SOL/Papers/IAL_Paper3_Complete/IAL_Materials/Mathematics/<series>/; October 2022 and January 2024 read from that same directory's existing per-series pdftotext -layout conversions (spot-checked against a second, independently-compiled consolidated conversion at ~/Documents/SOL/Papers/IAL_Paper3_Complete/Master_Consolidated_Docs/Mathematics_Paper3_{Mark_Schemes,Examiner_Reports,Past_Papers}.md, which agreed character-for-character on every line quoted below). Every coordinate, critical value and parameter this document states as a real answer was independently recomputed with sympy before being written in, not copied from a mark scheme and trusted — the working is shown at each point it matters.

#CODEX Deep-Dive: Modulus and Graph Transformations (WMA13, spec 1.3–1.4)

#What this document is for

modulus-and-graph-transformations.ts already derives both modulus constructions and both single/combined transformation mechanisms from first principles, drills them through a worked chain, a chain-drill and a twelve-mark marked solution, and carries six real, evidence-grounded traps anchored on Jan 2022 Q7(b), Jun 2023 Q6(d), Jan 2024 Q1, and Summer 2026 Q3(c). codex-formula-and-method-bank.md compresses two of those same anchors into mark-scheme skeletons; signal-frequency-and-boundaries.md states spec 1.3 has been located in 5 of 13 series checked and spec 1.4 in 2 of 13, with no further detail. None of the three lays out, in one place, the single mechanism that connects the two spec points, and none had yet opened the six real questions this pass located that are not cited anywhere else in this course's WMA13 material. This document does three things: (1) states the one-sentence unifying idea underneath most of this topic's real traps — identifying which candidate case, arm or point actually applies is the graded skill, not the algebra that follows it; (2) documents six real exam angles mined fresh this pass, each named with its real anchor, five with a short original micro-example; (3) synthesises every real exam angle this specific topic has actually been tested from, with an updated, honestly-sourced frequency count.

Spec 1.3's own guidance (verified-facts §1, transcribed from the spec PDF) asks for the graphs of $y=|ax+b|$, $y=|f(x)|$ and $y=f(|x|)$, given $y=f(x)$. Spec 1.4 states directly that "the graph of y = f(ax + b) will not be required."

#1. Theory the lesson already derives — stated briefly, not reproduced

$|t|=t$ for $t\geq0$ and $|t|=-t$ for $t<0$; neither modulus construction nor either single transformation is printed in the formula booklet, so both must be reasoned from this definition every time, not recalled as a boxed line. $y=|f(x)|$ reflects in the x-axis only the stretches of $y=f(x)$ where the output was negative; $y=f(|x|)$ discards the whole $x<0$ half and rebuilds it as a mirror of the $x\geq0$ half, which is why $f(|x|)$ is always an even function. Tracking a point $(p,q)$ through each of the four named transformations gives $y=af(x)\to(p,aq)$, $y=f(x)+a\to(p,q+a)$, $y=f(x+a)\to(p-a,q)$, $y=f(ax)\to(p/a,q)$ — an output change moves $q$ in the direction the formula reads, an input change moves $p$ in the opposite direction, and an input change and an output change commute because they touch different coordinates, which is also the real reason $y=f(ax+b)$ (two input changes sharing one coordinate) is excluded. All of this is derived, diagrammed and drilled in full inside the lesson file; nothing above is new, and none of it is re-derived here.

#2. The unifying frame this document adds: identifying the right case is the graded step, the algebra almost never is

Six real anchors already in this course's material and six mined fresh below all reduce to the same underlying skill, stated once here because no existing WMA13 document states it as a single principle: on this topic, marks are lost by applying correct algebra to the wrong candidate — the wrong branch, the wrong arm, the wrong point, or the wrong side of the modulus bars — not by making an arithmetic slip inside the right one. Solving $|A(x)|=B(x)$ loses marks for trying only one branch, not for solving a branch wrong. Transforming a named point loses marks for transforming a different, more recently computed point, not for misapplying the transformation. Sketching $y=f(|x|)$ loses marks for treating the left half as the original curve's own left half, not for drawing the right half wrong. Every sub-case in §3 below is a fresh instance of exactly this one failure shape — the document's real contribution is showing that a single sentence covers all of them, so a candidate checking their own work has one question to ask, not six unrelated ones: "before I compute anything, have I named which specific case/arm/point this step actually needs?"

#3. Six real exam angles mined fresh this pass, not yet in any WMA13 document

#3.1 $y=|f(x)|$ applied to a named point can be confused with reflecting through the origin

Real anchor: January 2024 Q1(c). $P(-4,-3)$ lies on $y=f(x)$; part (c) asks for $P$'s image under $y=|f(x)|$. The lesson's own trap taxonomy already covers parts (a) and (b) of this exact question (combined-transformation-only-partially-applied) but not part (c). Since $|f(x)|$ only ever touches the y-coordinate, $|{-4}|$ is never taken — the correct image is $(-4,3)$ ($x$ untouched, $y=|{-3}|=3$), confirmed by the real mark scheme (B1: "$(-4,3)$"). The examiner report names a distinct, specific error, most commonly a candidate who "change[s] the sign of both the x and y coordinates" — giving $(4,3)$ instead. This is not the input/output confusion the lesson's mechanism block already anticipates ($y=f(-x)$ vs $y=-f(x)$); it is mistaking an output-only modulus reflection for a full point reflection through the origin, which touches both coordinates and belongs to neither $y=|f(x)|$ nor $y=f(|x|)$ at all.

Micro-example. $Q(3,-7)$ lies on $y=g(x)$. Under $y=|g(x)|$: $x$ stays at $3$; $y=|{-7}|=7$. Image: $(3,7)$ — not $(-3,7)$, which would require negating $x$ as well, an operation this transformation never performs.

#3.2 A whole branch of a modulus inequality can be empty — not only a modulus equation

The lesson's own teach block already covers a branch of $|f(x)|=k$ having no real solution (the constant $-k$ falling outside $f$'s range). Real anchor: June 2024 Q1(b), $f(x)=2|x-5|+10$, solve $2|x-5|+10>6x$ algebraically (calculator-technology rubric present, re-confirmed on a modulus question in a fifth series alongside Jan 2023 Q5, Oct 2022 Q7(d) below, and Jan 2024 Q8(c)). Case $x\geq5$: $2(x-5)+10>6x\Rightarrow2x>6x\Rightarrow x<0$ — which contradicts $x\geq5$ entirely, so this branch contributes nothing to the solution set, not merely a rejected root. Case $x<5$: $2(5-x)+10>6x\Rightarrow20>8x\Rightarrow x<2.5$, which does overlap its own region. Final answer, confirmed against the real mark scheme and independently by substitution (e.g. $x=2$: $2|{-3}|+10=16>12$ ✓; $x=6$: $2(1)+10=12>36$ ✗): $x<\frac{5}{2}$. This is a different failure shape from the equation case's empty branch: there, a branch's equation has no root because a discriminant is negative; here, a branch's region is non-empty on its own but is entirely outside the case restriction that produced it, so the whole case is discarded rather than one root within it.

Micro-example. Solve $|x+1|-2<3x$. Case $x\geq-1$: $x+1-2<3x\Rightarrow-1<2x\Rightarrow x>-\frac12$. Case $x<-1$: $-(x+1)-2<3x\Rightarrow-x-3<3x\Rightarrow-3<4x\Rightarrow x>-\frac34$ — but this needs $x<-1$ too, and $x>-\frac34$ and $x<-1$ cannot both hold, so this branch is empty. Final answer: $x>-\frac12$ only, confirmed at $x=0$ ($1-2=-1<0$ ✓) and $x=-2$ ($1-2=-1<-6$? ✗, correctly excluded).

#3.3 Distinguishing $y=|x|-1$ from $y=|x-1|$ by which side of the bars the constant sits

Real anchor: January 2024 Q8(b), sketching $y=|x|-1$ (vertex at $(0,-1)$) on the same diagram as a given graph $y=a-|2x-b|$. The examiner report records a specific, nameable slip: a small number of candidates drew "a V shaped graph with a minimum on the x-axis at (1,0)" — i.e. sketching $y=|x-1|$ (vertex $(1,0)$, an input translation) where $y=|x|-1$ (vertex $(0,-1)$, an output translation) was asked for. This is the lesson's own general input-vs-output rule from §1 above, but the real trap is specifically about notational confusion — a constant written outside the closing bar is an output change; a constant written inside the bars, attached to $x$, is an input change — which the lesson's existing worked material never states as its own named risk, because none of its own examples pair the two forms this closely.

Micro-example. $y=|x|+3$ has vertex $(0,3)$ (output change, translate the whole V up 3). $y=|x+3|$ has vertex $(-3,0)$ (input change, translate the whole V left 3 — using the lesson's own rule that an input change moves the opposite way to the sign written). Confirmed: at $x=0$, $|0|+3=3$ and $|0+3|=3$ — the two curves happen to agree at $x=0$ despite having different vertices, which is exactly the kind of single-point coincidence that can make a wrongly-sketched vertex look locally plausible.

#3.4 A modulus expression with an unknown parameter: the same mechanics, evaluated at a given numeric $x$

Real anchors: June 2022 Q5(a)-(b), $f(x)=|kx-9|-2$ for a positive constant $k$; January 2024 Q8(c), $y=a-|2x-b|$ ($a,b$ positive, $a>b$) meeting $y=|x|-1$ at the stated values $x=-3$ and $x=5$. Neither question is genuinely new mechanics — both are the lesson's own two-branch method — but both require tracking a case split through an unresolved letter rather than a number, which the lesson's own worked material never does.

June 2022 Q5, verified: the y-intercept is $A=(0,7)$ ($f(0)=|-9|-2=7$) and the minimum is $B=(9/k,-2)$ ($kx-9=0\Rightarrow x=9/k$). Solving $|kx-9|-2<0$ (i.e. $|kx-9|<2$) gives $7<kx<11$, so $\frac{7}{k}<x<\frac{11}{k}$ (division by $k$ preserves direction since $k>0$) — the real mark scheme's own critical values, confirmed independently with sympy. This is the lesson's own worked-chain method (|3x-2|>2x+1 in the lesson) with every number replaced by an expression in $k$, and nothing about the method changes.

January 2024 Q8, verified in full: $y=a-|2x-b|$ has maximum $(b/2,a)$, y-intercept $(0,a-b)$, and x-intercepts at $x=\frac{a+b}{2}$ and $x=\frac{b-a}{2}$ — each confirmed by solving $|2x-b|=a$. At the two named intersection points, the sign of each modulus's own argument must be read off for that specific $x$-value, not solved as a general case split: at $x=-3$, $|x|=-x=3$ and (since $2(-3)-b=-6-b<0$ for the eventual $b=3$) $|2x-b|=b-2x$; at $x=5$, $|x|=x=5$ and $|2x-b|=2x-b$. Substituting gives two linear equations, $a-b=8$ and $a+b=14$, solving to $a=11,\ b=3$ — independently confirmed: $|{-3}|-1=2$ and $11-|2(-3)-3|=11-9=2$ ✓; $|5|-1=4$ and $11-|2(5)-3|=11-7=4$ ✓. The real examiner report names the specific error this produces: candidates made "a sign error when substituting $x=-3$" into the wrong case, and many arrived at "incorrect pairings of equations scoring no marks." The technique this genuinely adds beyond the lesson's own method: when a modulus's argument contains an unknown parameter, its sign at a stated numeric $x$ is read from the sketch (which arm of the V that $x$-value visibly sits on), not re-derived algebraically from scratch each time.

Micro-example. $g(x)=m-|3x-c|$ for positive constants $m>c$. Given $g(x)=|x|+1$ meets this graph at $x=-1$ and $x=4$: at $x=-1$, $|x|+1=2$ and (assuming, as the sketch would show, $3(-1)-c<0$) $m-|3(-1)-c|=m-(c+3)$, giving $m-c=5$; at $x=4$, $|x|+1=5$ and (assuming $3(4)-c>0$) $m-(12-c)=5$, giving $m+c=17$. Solving: $m=11,\ c=6$ — checked, $m>c$ as required, and both equations independently satisfied by direct substitution.

#3.5 A V-shaped modulus graph meeting another curve: one equation per arm, and one intersection sometimes needs no algebra at all

Real anchor: October 2022 Q7, $C_1$: $y=5-|3x-22|$ (vertex $(22/3,5)$, y-intercept $(0,-17)$, x-intercepts $x=9$ and $x=17/3$) sketched against $C_2$: $y=\frac{1}{9}x^2-9$ (a U-shape, vertex $(0,-9)$, x-intercepts $\pm9$), then asked for their points of intersection (calculator-technology rubric present, part (d)). $C_1$'s two arms are $y=3x-17$ (valid for $x<22/3$) and $y=27-3x$ (valid for $x\geq22/3$), each its own separate linear equation against $C_2$. Solving the left arm against $C_2$ gives $x=3$ or $x=24$ (verified with sympy); only $x=3$ satisfies that arm's own $x<22/3$ restriction, giving $(3,-8)$. Solving the right arm gives $x=9$ or $x=-36$; only $x=9$ satisfies $x\geq22/3$, giving $(9,0)$ — but $(9,0)$ was already established in part (b) as $C_1$'s own x-intercept, and it is also visibly $C_2$'s x-intercept from the sketch in part (c), so this point can be stated directly without solving an equation a second time. The real examiner report confirms this is exactly where marks were lost: "did not realise that the solution $(9,0)$ could be stated from the graph," and separately, of those who solved the left arm's own equation from scratch, "some did not reject $x=24$" — while others who instead solved the right arm's equation kept $(-36,\text{their }y\text{-value})$ as a second, unrejected answer alongside the genuine $(9,0)$. This is the lesson's own extraneous-root check (§ worked-chain, stage 3) applied twice over — once per arm — plus a genuinely new mechanic: a modulus graph's own x-intercept, once found in an earlier part, can be an intersection point with a second curve without any further algebra, if the second curve happens to share it.

Micro-example. $C_1$: $y=|x-4|-2$ (arms $y=x-6$ for $x\geq4$, $y=2-x$ for $x<4$; x-intercepts $x=2,6$). $C_2$: $y=x^2-36$ (x-intercepts $\pm6$). $C_1$ and $C_2$ share the point $(6,0)$ directly, readable from each curve's own intercepts with no equation needed. Solving the other arm, $2-x=x^2-36\Rightarrow x^2+x-38=0\Rightarrow x=\frac{-1\pm\sqrt{153}}{2}$; only the root satisfying $x<4$ is a genuine second intersection (confirmed: $\frac{-1+\sqrt{153}}{2}\approx5.68$ fails $x<4$ and is rejected; $\frac{-1-\sqrt{153}}{2}\approx-6.68$ satisfies $x<4$ and is the genuine second intersection).

#3.6 The hardest, rarest real angle located: a parameter range for a line to meet a modulus graph exactly twice

Real anchor: June 2022 Q5(c), continuing the same $f(x)=|kx-9|-2$ from §3.4: given that $y=3-2x$ meets $y=f(x)$ at exactly two distinct points, find the range of $k$. The real mark scheme's method: substitute $B=(9/k,-2)$ into $y=3-2x$ to find the boundary value of $k$ at which the line passes exactly through the graph's own minimum point — $-2=3-2(9/k)\Rightarrow k=3.6$ — then states the answer as $k>3.6$ (confirmed independently: computing genuine in-region intersections of $y=3-2x$ against both arms of $f$ for $k=1,2,3$ gives at most one valid intersection each, for $k=3.6$ gives exactly one (the line grazes the vertex), and for $k=4,5,10$ gives exactly two each — matching $k>3.6$ exactly). This is a genuinely distinct sub-case from every other real anchor in this document: it is not solving a modulus equation or inequality at all, but reasoning about how many times a line can cross a V-shape as a parameter changes — worth naming as a real, located exam angle, but rare enough (one located instance across 13 series checked) that it does not merit a full micro-example of its own here; §3.4's parametrised-modulus mechanics are the prerequisite skill it builds on.

#4. Synthesis: every real exam angle this topic has actually been tested from

Updated frequency count, this pass's own finding: spec 1.3 is confirmed tested in 8 of the 13 series this course's WMA13 research has checked (Oct 2020, Jan 2022, Jun 2022, Oct 2022, Jun 2023, Oct 2023, Jan 2024, Jun 2024) — up from the 5 currently recorded in signal-frequency-and-boundaries.md (Oct 2020, Jan 2022, Jun 2023, Jan 2024, Oct 2023), which had not yet located Jun 2022, Oct 2022 or Jun 2024. Spec 1.4 is confirmed in 3 of 13 (Jan 2024, Jun 2024, Jun 2026) — up from the 2 currently recorded there. signal-frequency-and-boundaries.md itself is not edited by this document; its maintainer should treat this table as the update to apply.

#5. Flagged / unverified

• January 2023 Q6 (a further real |3x-5a|-2a question, parametrised in $a$) was located in the consolidated mark-scheme conversion during this pass, but the extracted text's fraction denominators for part (b)'s final answer were internally inconsistent with this document's own hand/sympy re-derivation of the stated equation, and the underlying PDF was not re-opened directly to resolve the discrepancy in the time available. Per this course's standing rule against presenting an unresolved number as fact, it is named here as a real, located anchor and not built into this document at all.
• June 2022 Q5(c)'s exact three-mark boundary (why the critical case is specifically "the line through the minimum," rather than a tangency to one of the two arms) was confirmed by direct substitution across seven values of $k$ (§3.6), which is strong but not exhaustive evidence for every $k$; a full algebraic proof of the boundary was not attempted this pass.
• October 2022 Q7 and January 2024 Q1/Q8 were read from this project's existing per-series pdftotext -layout conversions (already present in the shared research directory) rather than freshly re-downloaded and re-converted by this pass; June 2022 and June 2024 were freshly pdftotext'd directly against the primary PDFs this session. All were spot-checked against a second, independently-compiled conversion where one existed, and every numeric answer was independently recomputed with sympy, but a second -raw extraction pass (this course's usual practice for ambiguous typesetting) was only performed for June 2024.

#What this document deliberately leaves to the existing files

The full narrative derivation of both modulus constructions and all four transformations, the concept ladder, both diagrams, the worked chain, the chain-drill, the twelve-mark marked solution, all six trap-taxonomy entries, the beyond-spec evenness proof, and all nineteen flashcards live in content/veridian/curriculum/wma13/modulus-and-graph-transformations.ts and are not reproduced here. The mark-scheme skeletons for the plain linear inequality and the named-point transformation live in codex-formula-and-method-bank.md §2.3–2.4. The bare frequency count (pending the update in §4 above) lives in signal-frequency-and-boundaries.md. This document's job was the one-sentence unifying frame in §2, the six real exam angles mined fresh in §3, and an honest, updated synthesis of what this topic has actually been tested from.