75-Archetype SyllabusMathematics 2 • ARCH-MM-12 • Spec MM8

Full 6-Card Lesson 15-Question Timed Drill

UAT-UK SPEC CODE: MM8 (Graphs of functions)ASSUMED MATHS 1 PREREQUISITE: M4.14FREQUENCY: HIGH

Multi-Step Graph Transformations (af(bx+c)+d), Modulus Reflections & Root Counting

Deconstructed from 22 past UAT-UK questions (5.37% of all historical Mathematics 2 papers). Students using standard A-Level classroom methods spend an average of 150s on this archetype—blowing past the 90s ESAT budget. Using the 2-line Sub-90 heuristic below compresses solve time to 40s (saving 110s per question).

Historical Frequency

22 Past Qs

5.37% of Mathematics 2

Classroom Trap Time

150s

+60s over 90s budget

Sub-90 Target Time

40s

Banks +110s on the clock

Dominant UI Format

5-Option A–E

Maths 1 Skill: M4.14

Card 1 & Card 2 Preview: The Classroom Time Trap vs. Sub-90 2-Line Rule

Excerpted from our interactive 6-Step Topic Lesson for ARCH-MM-12.

Launch All 6 Active-Recall Cards

CARD 1 OF 6 • THE CLASSROOM TIME TRAP (150s)

Why Students Bleed Time:

Writing

y = 2f\!\left(2\left(x + \frac{-2}{2}\right)\right) - 1

, the horizontal stretch by

\frac{1}{2}

followed by translation by

-\frac{-2}{2}

maps

x_0 = 4

p = \frac{4 + 2}{2} = 3

. Vertically,

y_0 = 3

maps to

q = 2(3) - 1 = 5

, so

p + q = 8

(~135s).

In 22 past UAT-UK questions (5.37% of Mathematics 2), candidates lose 110s by applying inside-bracket x-transformations in the forward rather than inverse direction. Instead, apply MM8 with Maths 1 skill M4.14.

CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<40s)

Line 1 (Equate Input Argument): Set

2p - 2 = 4 \implies p = 3

immediately.

Line 2 (Evaluate Output):

q = 2(3) - 1 = 5 \implies p + q = 3 + (5) = 8

(20s).

Sanity & Limit Check: Verify units/signs against sign_or_direction_flip (Translating y = f(x + c) by +c (right) instead of -c (left), or stretching f(bx) by factor b instead of 1/b) and fence_post_or_domain_trap (Translating by -c/b vs -c in wrong order for f(bx + c) = f(b(x + c/b))).

Examiner Distractor Blueprint for ARCH-MM-12

FOIL #1sign_or_direction_flip

Translating y = f(x + c) by +c (right) instead of -c (left), or stretching f(bx) by factor b instead of 1/b

Applying inside-bracket x-transformations in the forward rather than inverse direction.

Open Sign / Direction Flip Clinic

FOIL #2fence_post_or_domain_trap

Translating by -c/b vs -c in wrong order for f(bx + c) = f(b(x + c/b))

Failing to factor out b before reading off the horizontal translation -c/b.

Open Fence-Post / Domain Trap Clinic

FOIL #3sign_or_direction_flip

Confusing y = |f(x)| (reflect negative y-parts above x-axis) with y = f(|x|) (mirror x >= 0 across y-axis)

Swapping vertical modulus reflection with horizontal modulus symmetry.

Open Sign / Direction Flip Clinic

FOIL #4multi_statement_linchpin_foil

Missing tangent contact points when counting intersections between y = |f(x)| and y = mx + c

Counting only transverse crossings and missing where the line touches the reflected vertex.

Open Multi-Statement Linchpin Foil Clinic

Try 1 Live Sample Question Under the 90-Second Stopwatch

Press W to open the 2-line dry-wipe scratchpad, select A–E, and lock in to reveal the Dual-View solution.

Full 6-Card Lesson Launch All 15 Questions (ARCH-MM-12)

ARCH-MM-12: Multi-Step Graph Transformations (af(bx+c)+d), Modulus Reflections & Root Counting (Sample)Mathematics 2

00:00.0

SUB-90 ZONE (<40S)

0.0s40s Sub-9090s ESAT Budget150s Classroom

Question 1 of 1ARCH-MM-12 (6-Card Lesson)Format: Standard 5-Option

Right-click any option to cross out • Keys A–E + Enter

Question 1 of 1 • 90s Target

Ready to Start Question 1?

The question stem, diagram, and answer options are hidden behind this pre-start gate. When you click below, the question will reveal and your 90-second split stopwatch will immediately begin ticking.

Mathematics 2•ARCH-MM-12•Standard 5-Option

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