Sketch the absolute-value reflection of the cubic.

01 / Standard#Your turn

y=-f(|x|),\quad -2\le x\le2

Official paper · jm02-2026 · 2(a)(v) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Retain the right-hand part of f and reflect it in the y-axis.

Hint 2

The leading minus then reflects the result in the x-axis.

Worked solution

Write an even expression for the transformed curve.
$y=-|x|^3+x^2+|x|$
Use the right half of the old curve and its mirror; there are peaks at ±1 and a corner at the origin.
$(0,0),\ (\pm1,1),\ (\pm2,-2),\quad x\text{-intercepts}:0,\ \pm\frac{1+\sqrt5}2$

The graph is even, with maxima at (±1,1) and endpoints (±2,−2).

Checks and common pitfalls: f(|x|) copies only the original right half, not the original left half.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Write an even expression for the transformed curve.
$y=-|x|^3+x^2+|x|$
Use the right half of the old curve and its mirror; there are peaks at ±1 and a corner at the origin.
$(0,0),\ (\pm1,1),\ (\pm2,-2),\quad x\text{-intercepts}:0,\ \pm\frac{1+\sqrt5}2$

Think first. Reveal a hint when the class is ready.

2026 JM02