Skip to content
VirtusAcademy

Cubic Graphs

FoundationHigherAQAEdexcelOCR

Get to grips with cubic graphs using these Foundation and Higher GCSE Maths practice questions. The worksheet focuses on plotting cubic graphs, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. Cubic graphs have an S-shape with up to two turning points.

Free downloads

These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A cubic graph comes from an equation whose highest power is \(x^3\). The curve has a distinctive shape with up to two turning points and a characteristic S-like bend.

The sign of the \(x^3\) coefficient sets the overall direction. A positive coefficient means the curve rises from bottom left to top right; a negative one reverses it, falling from top left to bottom right.

A cubic can cross the horizontal axis up to three times, so it can have one, two or three roots. Plotting requires care with negative values, since cubing a negative keeps it negative, unlike squaring.

Revision notes

The shape

A positive \(x^3\) coefficient gives a curve rising overall from bottom left to top right, usually with a bend in the middle.

A negative coefficient flips it. Some cubics have two turning points, and some have none at all, rising steadily throughout.

Cubing negatives

Unlike squaring, cubing preserves the sign, so \((-2)^3 = -8\).

This is the most common source of table errors. For \(y = x^3\) at \(x = -3\), the value is \(-27\), not \(27\).

Roots

A cubic crosses the horizontal axis between one and three times.

If the equation factorises, each factor gives a root. For \(y = x(x-2)(x+1)\), the roots are \(0\), \(2\) and \(-1\).

Key points

  • A cubic has \(x^3\) as its highest power.
  • A positive coefficient rises from bottom left to top right.
  • A negative coefficient falls from top left to bottom right.
  • Cubing a negative gives a negative.
  • A cubic can have up to two turning points.
  • It crosses the x-axis between one and three times.

Worked examples

Example 1

Find \(y\) when \(x = -2\) for \(y = x^3\).

Working

\[(-2)^3 = -2 \times -2 \times -2\]cube the negative value
\[= -8\]two negatives give a positive, then the third makes it negative

Example 2

Find \(y\) when \(x = 3\) for \(y = x^3 - 4x\).

Working

\[3^3 = 27\]cube the x value
\[4 \times 3 = 12\]work out the second term
\[27 - 12 = 15\]subtract to find y

Example 3

State the roots of \(y = x(x - 3)(x + 2)\).

Working

\[\text{Set each factor to zero}\]the roots are where y = 0
\[x = 0, 3, -2\]state all three roots

Common mistakes

  • Cubing a negative and getting a positive.

    (−2)³ is −8. The sign is preserved when cubing, unlike squaring.

  • Drawing the curve with straight sections.

    A cubic is smooth throughout, including through its turning points.

  • Assuming there are always three roots.

    A cubic may cross the axis once, twice or three times.

  • Getting the overall direction wrong.

    Check the sign of the x³ term before drawing.

Exam tips

  • Take particular care cubing negative values in the table.
  • Draw the curve smoothly with a freehand line.
  • Plot enough points to reveal both turning points.
  • Check the direction against the sign of the x³ coefficient.

Key terms

Cubic
An expression whose highest power is \(x^3\).
Turning point
Where the curve changes direction.
Root
Where the curve crosses the x-axis.
Coefficient
The number multiplying a term.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.