4.1PureCore

Cubic Graphs

A cubic has the form $y = ax^3 + bx^2 + cx + d$ . Its graph has a distinctive double-bend shape. If you can factorise it, the roots and the sign of $a$ are enough to produce a clear sketch.

30 min Graphs and Transformations

Edexcel AS level Maths: 4.1 Cubic GraphsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Recognise the shape of a cubic from the sign of a
Find the roots of a factorised cubic
Understand repeated roots and what they look like
Sketch a cubic showing roots and the y-intercept

Shape and orientation

For a positive leading coefficient the curve rises from bottom-left to top-right; for a negative one it falls from top-left to bottom-right. Cubics always have this overall direction with up to two turning points in between.

a > 0:\ \nearrow \quad\quad a < 0:\ \searrow

The sign of

a

fixes the overall direction of the cubic.

Roots from a factorised cubic

If the cubic is given as a product of brackets, each bracket gives a root where the curve crosses the $x$ -axis. Set each factor to zero.

Example 1Find where

y = (x - 1)(x + 2)(x - 3)

crosses the

x

-axis.

1Set each bracket to zero:

x = 1

x = -2

x = 3

Answerroots at

x = -2, 1, 3

Repeated roots and the y-intercept

A squared factor like $(x - 2)^2$ gives a repeated root where the curve $\textbf{touches}$ the axis rather than crossing it. The $y$ -intercept is found by setting $x = 0$ .

Example 1Describe the graph of

y = (x + 1)(x - 2)^2

1Roots: crosses at

x = -1

, touches at

x = 2

(repeated).

y

-intercept:

x = 0

gives

y = (1)(4) = 4

Answercrosses at

-1

, touches at

2

y

-intercept

4

Tip — A repeated (squared) factor means the curve just touches the x-axis there, like a parabola.

Formula recap

y = ax^{3} + bx^{2} + cx + d

General cubic.

(x - r) \Rightarrow \text{root at } x = r

Each factor gives a crossing point.

(x - r)^{2} \Rightarrow \text{touches at } x = r

Repeated root.

Common mistakes to avoid

Drawing a repeated root as a normal crossing.

A squared factor means the curve touches the axis there, not crosses.

Forgetting the y-intercept on a sketch.

Set x = 0 to find where the curve meets the y-axis.

Key takeaways

Sign of a sets the cubic’s overall direction.
Each linear factor gives a root (crossing point).
A squared factor gives a repeated root where the curve touches the axis.
Find the y-intercept by setting x = 0.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Cubic Graphs. Use the feedback to decide what needs another look.

4 past-paper questions · 10 marks

Practise this topic

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