4.1PureCore

Cubic Graphs

A cubic has the form $y = a x^{3} + b x^{2} + c x + d$ . 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 Video by Zeeshan Zamurred Graphs and Transformations

Edexcel AS level Maths: 4.1 Cubic GraphsWatch the full walkthrough 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 : ↗ a < 0 : ↘

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.

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 $touches$ the axis rather than crossing it. The $y$ -intercept is found by setting $x = 0$ .

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 = a x^{3} + b x^{2} + c x + d

General cubic.

(x - r) \Rightarrow root at x = r

Each factor gives a crossing point.

(x - r)^{2} \Rightarrow 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.

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