Graphs of Cubic Polynomials (Cambridge O Level Additional Maths)

What is a cubic polynomial?

• A cubic polynomial is a function of the form
  • and are constants
  • it is a polynomial of degree 3
    • so  and/or  could be zero
• To sketch the graph of a cubic polynomial it will need to be in factorised form
  • e.g.  is the factorised form of 

What does the graph of a cubic polynomial look like?

• In general the graph of a cubic polynomial will take one of the four forms
  • All are smooth curves that take some practice to sketch!

• The exact form a particular cubic polynomial will depend on
  • The number (and value) of roots (-axis intercepts) of the cubic polynomial
  • The -axis intercept
  • The sign of the coefficient of the term ()
    • If the graph is a positive cubic ('starts' in the third quadrant, 'ends' in the first)
    • If the graph is a negative cubic ('starts' in the second quadrant, 'ends' in the fourth)
  • Turning points

How do I sketch the graph of a cubic polynomial?

STEP 1
Find the -axis intercept by setting 

STEP 2
Find the -axis intercepts (roots) by setting
(Any repeated roots will mean the graph touches - rather than crosses - the -axis)

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, labelling points where the graph intercepts the  and axes

What is a modulus cubic graph?

• A (factorised) cubic polynomial is of the from
• The graph of  must cross the -axis at least once
  • therefore must take both positive and negative values
• The modulus cubic graph,  will mean all values of are positive
  • Any negative values become their positive equivalents
    • e.g.  If then , but 
• A modulus cubic graph will not have any negative values
  • the graph will not cross the -axis
  • the graph will touch the -axis (at least once)

How do I sketch a modulus cubic graph?

• Sketch the graph of the (original) cubic polynomial, 
• Any parts of this graph that are below the -axis should be reflected in the -axis to sketch the graph of 

• The points at which a modulus graph touches the -axis are the same as the points at which the original graph intercepts the -axis (i.e. the roots of )
  • Label these points, and the -axis intercept, on a sketch
• Notice that the points at which a modulus graph touches the -axis are not smooth
  
  • they are 'pointy' (V-shaped)
  • turning points are smooth (U-shaped)

How do I find a cubic function from its modulus graph?

• To deduce a cubic expression from its modulus graph consider
  • whether the (original) expression could be a positive or negative cubic
    • a positive cubic 'starts' in the third quadrant and 'ends' in the first
    • a negative cubic 'starts' in the second quadrant and 'ends' in the fourth
      • a negative cubic can have a "-" at the start of its expression
  • the -axis intercepts - the roots
    • for the roots  and , write the factors 
  • the -axis intercept - to deduce the expression in the form
    • lots of cubic functions have the same roots but have different coefficients
    • the -axis intercept should be the product
      • may often, but not always, be 1

What is a cubic inequality?

• A cubic function is of the form  where  and  are constants
• A cubic inequality can be any of the following
  • 
  • 
  • 
  • 
• An inequality may need rearranging into one of these forms first before solving
  • Furthermore, solving cubic equations graphically is easiest when the expression has been factorised
    • e.g.  for a cubic with three (real) roots this would be where  and  are the roots

How do I solve a cubic inequality graphically?

STEP 1
If need be, rearrange the inequality so that one side of the inequality is zero
This should leave a cubic polynomial on the other side
Factorise the cubic polynomial if required
e.g.  

STEP 2
Sketch the graph of the cubic polynomial
The -axis intercepts (roots) are crucial to finding the solution but the -axis intercept is not needed
e.g.  

STEP 3
Identify the part(s) of the graph that satisfy the inequality
Highlighting this on the sketch will help
Consider whether you need to include (≤, ≥) or exclude (<, >) the roots
e.g.  

STEP 4
Write down the solutions to the inequality
e.g.