Sketch y=f(x) for −1≤x≤3.

01 / Standard#Your turn

f(x)=2x^3-9x^2+12x-5

Official paper · jm02-2021 · 2(a)(iv) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Mark the endpoints, intercepts, stationary points and inflection.

Hint 2

Combine derivative signs with the concavity change.

Worked solution

Factor f to locate both x-intercepts, including the double root.
$f(x)=(x-1)^2(2x-5)$
The double zero touches the axis; the simple zero crosses it.
$(-1,-28),\ (0,-5),\ (1,0),\ (3/2,-1/2),\ (2,-1),\ (5/2,0),\ (3,4)$
Increase to x=1, decrease to x=2, then increase; concavity changes at 3/2. Keep the two domain endpoints.

The plotted cubic is restricted to [−1,3], with the listed key points.

Checks and common pitfalls: The double root x=1 touches the axis rather than crossing it.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Factor f to locate both x-intercepts, including the double root.
$f(x)=(x-1)^2(2x-5)$
The double zero touches the axis; the simple zero crosses it.
$(-1,-28),\ (0,-5),\ (1,0),\ (3/2,-1/2),\ (2,-1),\ (5/2,0),\ (3,4)$
Increase to x=1, decrease to x=2, then increase; concavity changes at 3/2. Keep the two domain endpoints.

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

2021 JM02