Cubic Graphs (Cambridge (CIE) IGCSE International Maths): Revision Note

Cubic Graphs

What is a cubic?

• A cubic is a function of the form

  • and are constants

  • It is a polynomial of degree (order) 3

    • So  and/or  could be zero

• To sketch the graph of a cubic polynomial needs to be in factorised form

  • E.g. is the factorised form of 

• You do not need to factorise a cubic unless it is a more simple one

  • E.g.

  • You should be able to expand three brackets to find the expanded form of a cubic

What does the graph of a cubic look like?

• In general the graph of a cubic will take one of the four forms

  • All are smooth curves

• The exact form of a particular cubic will depend on:

  • The number (and value) of roots (-axis intercepts)

  • The -axis intercept

  • The sign of the coefficient of the term ()

    • If the graph is a positive cubic ('starts' in the bottom left, 'ends' in the top right)

    • If the graph is a negative cubic ('starts' in the top left, 'ends' in the bottom right)

  • The turning points

• Cubics can have two turning points

  • a maximum point and a minimum point

• However, note that the graphs of and :

  • Do not have a maximum or minimum (turning points)

  • Only cross the -axis once, at

How do I sketch the graph of a cubic?

• STEP 1
  Find the -axis intercept by setting 

• STEP 2
  Find the -axis intercepts (roots) by setting

  • In factorised form, this can be done by inspection

    • A cubic of the form has roots at and

    • E.g. has roots at 2, 3, and -5

  • Any repeated roots will mean the graph touches the -axis

    • The graph does not cross the -axis

    • E.g. touches the -axis at , and intersects the -axis at

• STEP 3
  Consider the shape of the graph

  • Is it a positive cubic or a negative cubic?

  • Where does the graph 'start' and 'end'?

• STEP 4
  Consider where any turning points should go

• STEP 5
  Sketch the graph with a smooth curve
  Label points where the graph intercepts the  and axes