Sketch the cubic using its zeros, extrema and inflection.

01 / Standard#Your turn

f(x)=x^3-3x^2+2

Official paper · jm02-2022 · 2(a)(v) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Mark the three real zeros and the two extrema.

Hint 2

Use the cubic end behaviour and the concavity change at 1.

Worked solution

The curve rises to (0,2), falls to (2,−2), then rises again.
$x\text{-intercepts}:1-\sqrt3,1,1+\sqrt3$
It is concave down for x<1 and up for x>1, with inflection (1,0).
$x\to\pm\infty\implies f(x)\to\pm\infty$

The plot contains all requested features and continues beyond the window.

Checks and common pitfalls: The central zero is also the inflection, not an extremum.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The curve rises to (0,2), falls to (2,−2), then rises again.
$x\text{-intercepts}:1-\sqrt3,1,1+\sqrt3$
It is concave down for x<1 and up for x>1, with inflection (1,0).
$x\to\pm\infty\implies f(x)\to\pm\infty$

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

2022 JM02