Why isn't the answer to TMUA-P1-MM8-006 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P1-MM8-006 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P1-MM8-006?

No — calculators are banned in the TMUA. TMUA-P1-MM8-006 is set for a no-calculator target of about 180 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P1-MM8-006?

TMUA-P1-MM8-006 is Paper P1 · Graphs of functions (specification code MM8), level 2. Drill more of the same topic from this page.

TMUA-P1-MM8-006 — Paper P1 · Graphs of functions · Level 2

P1MM8Level 2~180s target

Spec: MM8 · No calculator · ~180s

A cubic

y = ax^3 + \dots

with

a > 0

has three distinct real roots. Which best describes its graph?

Options

starts bottom-left, crosses the $x$ -axis three times, ends top-right
starts top-left, crosses the $x$ -axis three times, ends bottom-right
touches the $x$ -axis once and stays above it
crosses the $y$ -axis three times
starts bottom-left and ends bottom-right

Letters only — the answer key is hidden. Work it out, then check the solution below.

Hint (check trope)

No written hint yet — check the trope: cubic-shape, leading-coefficient. Name the trap first, work the stem by hand, then open the solution below.

Show solution

A positive cubic goes

-\infty \to +\infty

and three distinct real roots mean three separate

x

-intercepts. Trap: the “top-left to bottom-right” version describes a negative leading coefficient — plausible if you link “three crossings” to any cubic shape.

Keep practising

Same topic · same level
P1MM8Level 2
TMUA-P1-MM8-004
How is the graph of y = f(x) + 2 obtained from the graph of y = f(x)?
Graphs of functions
Open TMUA-P1-MM8-004 →
Same trap: cubic-shape
P1MM8Level 3
TMUA-P1-MM8-022
The cubic y = (x-1)^2(x+2) touches the x -axis at one point. Where?
Graphs of functions
Open TMUA-P1-MM8-022 →
Step up a level
P1MM8Level 3
TMUA-P1-MM8-009
Which statement correctly describes the stationary points of y = x^3 - 3x^2?
Graphs of functions
Open TMUA-P1-MM8-009 →

Browse more

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Options

starts bottom-left, crosses the

x

-axis three times, ends top-right

starts top-left, crosses the

x

-axis three times, ends bottom-right

touches the

x

-axis once and stays above it

crosses the

y

-axis three times

starts bottom-left and ends bottom-right

Letters only — the answer key is hidden. Work it out, then check the solution below.

Keep practising

Same topic · same level

P1MM8Level 2

TMUA-P1-MM8-004

How is the graph of y = f(x) + 2 obtained from the graph of y = f(x)?

Graphs of functions

Open TMUA-P1-MM8-004 →

Same trap: cubic-shape

P1MM8Level 3

TMUA-P1-MM8-022

The cubic y = (x-1)^2(x+2) touches the x -axis at one point. Where?

Graphs of functions

Open TMUA-P1-MM8-022 →

Step up a level

P1MM8Level 3

TMUA-P1-MM8-009

Which statement correctly describes the stationary points of y = x^3 - 3x^2?

Graphs of functions

Open TMUA-P1-MM8-009 →

TMUA-P1-MM8-006 — Paper P1 · Graphs of functions · Level 2

Options

Keep practising

Master Graphs of functions

TMUA-P1-MM8-006 — Paper P1 · Graphs of functions · Level 2

Options

Keep practising

Master Graphs of functions