Graphs of Cubic Polynomials (Cambridge International IGCSE Further Maths)

Graphs of cubic polynomials

What is a cubic polynomial?

• A cubic polynomial is a function of the form $ax^{3}+bx^{2}+cx+d$

  • $a,b,c$ and $d$ are constants

  • it is a polynomial of degree 3

    • so $b,c$ and/or $d$ could be zero

• To sketch the graph of a cubic polynomial it will need to be in factorised form

  • e.g. $( 2 x - 1 ) ( x + 2 ) ( x - 3 )$ is the factorised form of $2x^{3}-3x^{2}-11x+6$

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!

• 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 $x^{3}$ term ($a$)

    • If $a>0$ the graph is a positive cubic ('starts' in the third quadrant, 'ends' in the first)
    • If $a<0$ 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 $y$-axis intercept by setting $x=0$

STEP 2 Find the $x$-axis intercepts (roots) by setting $y=0$ (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

Exam Hint

• 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 $y = ( 2 x - 1 ) ( x - 3 )^{2}$.

STEP 1 - $y$-axis intercept

$y = ( - 1 ) ( - 3 )^{2} = - 9$

STEP 2 - $x$-axis intercepts

$2x-1=0,x=\frac{1}{2}$

$x-3=0,x=3$ (repeated)

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

$a > 0 ( a = 2 )$ so it is a positive cubic $x=3$ 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 between$x=\frac{1}{2}$ and $x=3$

STEP 5 - smooth curve with labelled intercepts

Modulus Cubic Graphs

What is a modulus cubic graph?

• A (factorised) cubic polynomial is of the from $\mathrm{f} ( x ) = a ( x - b ) ( x - c ) ( x - d )$

• The graph of $y = \mathrm{f} ( x )$ must cross the $x$-axis at least once

  • therefore $y$ must take both positive and negative values

• The modulus cubic graph, $y = \vert \mathrm{f} ( x ) \vert$ will mean all values of $y$ are positive

  • Any negative values become their positive equivalents

    • e.g. If $\mathrm{f} ( x ) = x^{3}$ then $\mathrm{f} ( - 1 ) = ( - 1 )^{3} = - 1$, but $\vert \mathrm{f} ( - 1 ) \vert = \vert ( - 1 )^{3} \vert = \vert - 1 \vert = 1$

• 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, $y = \mathrm{f} ( x )$
• Any parts of this graph that are below the $x$-axis should be reflected in the $x$-axis to sketch the graph of $y = \vert \mathrm{f} ( x ) \vert$

• 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 $\mathrm{f} ( x )$)

  • 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 $b,c$ and $d$, write the factors $( x - b ) ( x - c ) ( x - d )$

  • the $y$-axis intercept - to deduce the expression in the form $a ( x - b ) ( x - c ) ( x - d )$

    • lots of cubic functions have the same roots but have different coefficients

    • the $y$-axis intercept should be the product $a\times-b\times-c\times-d=-abcd$

      • $a$ may often, but not always, be 1

Exam Hint

• Sketching the (original) graph of $y = \mathrm{f} ( x )$ 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 $y = \vert \mathrm{f} ( x ) \vert$ where $\mathrm{f} ( x )$ is a positive cubic polynomial.

Deduce an expression for $\mathrm{f} ( x )$ leaving your answer in factorised form, with each factor having integer coefficients.

$\mathrm{f} ( x )$ 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.

The $x$-axis intercepts indicate the roots.

$( x + 4 ) ( x + 1 \frac{1}{2} ) ( x - 3 )$

The expression for $\mathrm{f} ( x )$ is of the form $a ( x + 4 ) ( x + 1 \frac{1}{2} ) ( x - 3 )$. Use the $y$-axis intercept to deduce the value of $a$. From the sketch of $y = \mathrm{f} ( x )$ the $y$-axis intercept is -12 (not 12).

$\begin{gathered} a \times 4 \times 1 \frac{1}{2} \times - 3 & = - 12 \\ - 6 a & = - 12 \\ a = 2 \end{gathered}$

The final answer requires each factor to have integer coefficients so multiply 2 into $( x + 1 \frac{1}{2} )$ and write down $\mathrm{f} ( x )$.

$f(x)=(x+4)(2x+1)(x-3)$

Solving Cubic Inequalities Graphically

What is a cubic inequality?

• A cubic function is of the form $\mathrm{f} ( x ) = a x^{3} + b x^{2} + c x + d$ where $a,b,c$ and $d$ are constants

• A cubic inequality can be any of the following

  • $\mathrm{f} ( x ) > 0$
  • $\mathrm{f} ( x ) \ge 0$
  • $\mathrm{f} ( x ) < 0$
  • $\mathrm{f} ( x ) \le 0$

• 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 $( x - p ) ( x - q ) ( x - r )$ where $p,q$ 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. $( 2 x - 3 ) ( x - 4 ) ( x + 1 ) \le 0$

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.

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. $x\leq-1,\frac{3}{2}\leq x\leq4$

Worked Example a) Sketch the graph of $y = \mathrm{f} ( x )$ where $\mathrm{f} ( x ) = ( 3 x - 1 ) ( x - 3 )^{2}$.

*This is a positive cubic polynomial with roots *$x=\frac{1}{3}$ and $x=3$

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

a) Hence solve the inequality $\mathrm{f} ( x ) > 0$.

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

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

$x>\frac{1}{3},x\neq3$