Graphs of Cubic Polynomials
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
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 .
STEP 1 - -axis intercept
STEP 2 - -axis intercepts
(repeated)
STEP 3 - shape, 'start'/'end'
so it is a positive cubic
is a repeated root so the graph will touch the -axis at this point
STEP 4 - turning points
One turning point (minimum) will need to be where the curve touches the -axis
The other (maximum) will need to be between and
STEP 5 - smooth curve with labelled intercepts