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

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-12: Why the 150s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~150s (Blows 90s Clock)

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).

The Fast ShortcutTarget: 40s (Saves 110s)

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).

Where Students Lose Time: 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.

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