Graphs of Cubic Polynomials (Cambridge O Level Additional Maths)

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Graphs of Cubic Polynomials

What is a cubic polynomial?

  • A cubic polynomial is a function of the form a x cubed plus b x squared plus c x plus d
    • a comma space b comma space c and d are constants
    • it is a polynomial of degree 3
      • so b comma space c and/or d could be zero
  • To sketch the graph of a cubic polynomial it will need to be in factorised form
    • e.g.  open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses is the factorised form of 2 x cubed minus 3 x squared minus 11 x plus 6

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!

general shape of positive and negative cubic graphs

  • The exact form a particular cubic polynomial will depend on
    • The number (and value) of roots (x-axis intercepts) of the cubic polynomial
    • The y-axis intercept
    • The sign of the coefficient of the x cubed term (a)
      • If a greater than 0 the graph is a positive cubic ('starts' in the third quadrant, 'ends' in the first)
      • If a less than 0 the graph is a negative cubic ('starts' in the second quadrant, 'ends' in the fourth)
    • Turning points

Key features of a polynomial graph - shape, intercept, turning points

How do I sketch the graph of a cubic polynomial?

STEP 1
Find the y-axis intercept by setting x equals 0

STEP 2
Find the x-axis intercepts (roots) by setting y equals 0
(Any repeated roots will mean the graph touches - rather than crosses - the x-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 x and y axes

Examiner Tip

  • In the exam, a cubic polynomial that requires sketching will be given in factorised form
    • i.e. as the product of three linear factors
    • a factor could be repeated
  • Build your sketch up as you work through each step to gradually build a mental picture of it
    • If you need to redraw the graph for your final answer, that's fine!

Worked example

Sketch the graph of y equals open parentheses 2 x minus 1 close parentheses open parentheses x minus 3 close parentheses squared.

STEP 1 - y-axis intercept

y equals open parentheses negative 1 close parentheses open parentheses negative 3 close parentheses squared equals negative 9

STEP 2 - x-axis intercepts

2 x minus 1 equals 0 comma space x equals 1 half

x minus 3 equals 0 comma space x equals 3 (repeated)

STEP 3 - shape, 'start'/'end'

a greater than 0 space open parentheses a equals 2 close parentheses so it is a positive cubic
x equals 3 is a repeated root so the graph will touch the x-axis at this point

STEP 4 - turning points

One turning point (minimum) will need to be where the curve touches the x-axis
The other (maximum) will need to be betweenx equals 1 half and x equals 3

STEP 5 - smooth curve with labelled intercepts

worked example - final answer sketch of cubic showing intersections

Modulus Cubic Graphs

What is a modulus cubic graph?

  • A (factorised) cubic polynomial is of the from straight f open parentheses x close parentheses equals a open parentheses x minus b close parentheses open parentheses x minus c close parentheses open parentheses x minus d close parentheses
  • The graph of y equals straight f open parentheses x close parentheses must cross the x-axis at least once
    • therefore y must take both positive and negative values
  • The modulus cubic graph, y equals vertical line straight f open parentheses x close parentheses vertical line will mean all values of y are positive
    • Any negative values become their positive equivalents
      • e.g.  If straight f open parentheses x close parentheses equals x cubed then straight f open parentheses negative 1 close parentheses equals open parentheses negative 1 close parentheses cubed equals negative 1, but vertical line straight f open parentheses negative 1 close parentheses vertical line equals vertical line open parentheses negative 1 close parentheses cubed vertical line equals vertical line minus 1 vertical line equals 1
  • A modulus cubic graph will not have any negative values
    • the graph will not cross the x-axis
    • the graph will touch the x-axis (at least once)

How do I sketch a modulus cubic graph?

  • Sketch the graph of the (original) cubic polynomial, y equals straight f open parentheses x close parentheses
  • Any parts of this graph that are below the bold italic x-axis should be reflected in the bold italic x-axis to sketch the graph of y equals vertical line straight f open parentheses x close parentheses vertical line

Cubic graphs and their modulus graphs

  • The points at which a modulus graph touches the x-axis are the same as the points at which the original graph intercepts the x-axis (i.e. the roots of straight f open parentheses x close parentheses)
    • Label these points, and the y-axis intercept, on a sketch
  • Notice that the points at which a modulus graph touches the x-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 x-axis intercepts - the roots
      • for the roots b comma space c and d, write the factors open parentheses x minus b close parentheses open parentheses x minus c close parentheses open parentheses x minus d close parentheses
    • the y-axis intercept - to deduce the expression in the form a open parentheses x minus b close parentheses open parentheses x minus c close parentheses open parentheses x minus d close parentheses
      • lots of cubic functions have the same roots but have different coefficients
      • the y-axis intercept should be the product a cross times negative b cross times negative c cross times negative d equals negative a b c d
        • a may often, but not always, be 1

Examiner Tip

  • Sketching the (original) graph of y equals straight f open parentheses x close parentheses first is often helpful rather than trying to sketch the modulus graph from scratch
  • If asked to find a cubic function from a given graph
    • there may be more than one possibility
      • consider which parts of the graph may have been reflected in the x-axis
      • could it be either a positive or negative cubic polynomial?
    • the factorised form is usually sufficient (unless a question says otherwise)

Worked example

The diagram below shows the graph of y equals vertical line straight f open parentheses x close parentheses vertical line where straight f open parentheses x close parentheses is a positive cubic polynomial.

cie-adma25-2023-cubic-mod-1-we-question

Deduce an expression for straight f open parentheses x close parentheses leaving your answer in factorised form, with each factor having integer coefficients.

straight f open parentheses x close parentheses is a positive cubic polynomial so the graph should 'start' in the third quadrant and 'end' in the first.
A quick sketch (without labels) can help.

cie-adma25-2023-cubic-mod-2-we-solution

The x-axis intercepts indicate the roots.

open parentheses x plus 4 close parentheses open parentheses x plus 1 half close parentheses open parentheses x minus 3 close parentheses

The expression for straight f open parentheses x close parentheses is of the form a open parentheses x plus 4 close parentheses open parentheses x plus 1 half close parentheses open parentheses x minus 3 close parentheses.
Use the y-axis intercept to deduce the value of a.
From the sketch of y equals straight f open parentheses x close parentheses the y-axis intercept is -12 (not 12).

table row cell a cross times 4 cross times 1 half cross times negative 3 end cell equals cell negative 12 end cell row cell negative 6 a end cell equals cell negative 12 end cell row a equals 2 end table

The final answer requires each factor to have integer coefficients so multiply 2 into open parentheses x plus 1 half close parentheses and write down straight f open parentheses x close parentheses.

Solving Cubic Inequalities Graphically

What is a cubic inequality?

  • A cubic function is of the form straight f open parentheses x close parentheses equals a x cubed plus b x squared plus c x plus d where a comma space b comma space c and d are constants
  • A cubic inequality can be any of the following
    • straight f open parentheses x close parentheses greater than 0
    • straight f open parentheses x close parentheses greater or equal than 0
    • straight f open parentheses x close parentheses less than 0
    • straight f open parentheses x close parentheses less or equal than 0
  • 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 open parentheses x minus p close parentheses open parentheses x minus q close parentheses open parentheses x minus r close parentheses where p comma space q and r 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.  open parentheses 2 x minus 3 close parentheses open parentheses x minus 4 close parentheses open parentheses x plus 1 close parentheses less or equal than 0

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

sketch of a cubic using roots

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.  sketch of a cubic using roots, with highlighting of region less than zero

STEP 4
Write down the solutions to the inequality
e.g.  x less or equal than negative 1 comma fraction numerator space 3 over denominator 2 end fraction less or equal than x less or equal than 4

Worked example

a)
Sketch the graph of y equals straight f open parentheses x close parentheses where straight f open parentheses x close parentheses equals open parentheses 3 x minus 1 close parentheses open parentheses x minus 3 close parentheses squared.

This is a positive cubic polynomial with roots x equals 1 third and x equals 3

x equals 3 is a repeated root so the graph will touch the x-axis at this point

cie-adma25-2023-cubic-ineq-1-we-solution-dark-green

a)
Hence solve the inequality  straight f open parentheses x close parentheses greater than 0.

STEP 1 - the cubic polynomial is already in factorised form

STEP 2 - the graph has been sketched in part (a)

STEP 3 - Highlight the graph to show the parts of the graph which lie above zero

dQsDLIdG_cie-adma25-2023-cubic-ineq-2-we-solution-diff-colours

STEP 4 - Write the solution mathematically. Remember that in this case it is a strict inequality, so it cannot be equal to 3 or 1 third

bold italic x bold greater than bold 1 over bold 3 bold comma bold space bold italic x bold not equal to bold 3

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.