Sketch the cubic on the stated closed interval.

01 / Standard#Your turn

y=x^3-x^2-x,\quad -2\le x\le2

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Include zeros, extrema, inflection and endpoints.

Hint 2

Use the sign chart to join the points smoothly.

Worked solution

Factor to find all three intercepts.
$f(x)=x(x^2-x-1),\quad x=0,\ \frac{1-\sqrt5}2,\ \frac{1+\sqrt5}2$
Plot the closed endpoints and the previously found critical points.
$f(-2)=-10,\ f(2)=2,\quad\left(-\frac13,\frac5{27}\right),\ (1,-1),\ \left(\frac13,-\frac{11}{27}\right)$
Increase to x=−1/3, decrease to x=1, then increase; change concavity at x=1/3.

The original generated plot shows the complete graph on [−2,2].

Checks and common pitfalls: Do not extend the answer beyond the requested interval.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Factor to find all three intercepts.
$f(x)=x(x^2-x-1),\quad x=0,\ \frac{1-\sqrt5}2,\ \frac{1+\sqrt5}2$
Plot the closed endpoints and the previously found critical points.
$f(-2)=-10,\ f(2)=2,\quad\left(-\frac13,\frac5{27}\right),\ (1,-1),\ \left(\frac13,-\frac{11}{27}\right)$
Increase to x=−1/3, decrease to x=1, then increase; change concavity at x=1/3.

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

2026 JM02