Cubic Graphs (Cambridge International IGCSE Maths)

Cubic Graphs

What is a cubic?

• A cubic 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 (order) 3

    • So $b,c$ and/or $d$ could be zero
    • but $a$ cannot be equal to zero

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

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

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

  • E.g. $x^{3} - 4 x = x ( x^{2} - 4 ) = x ( x - 2 ) ( x + 2 )$
  • 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 ($x$-axis intercepts)

  • 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 bottom left, 'ends' in the top right)
    • If $a<0$ 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 $y=x^{3}$ and $y=-x^{3}$:

  • Do not have a maximum or minimum (turning points)
  • Only cross the $x$-axis once, at $x=0$

How do I sketch the graph of a cubic?

• STEP 1 Find the $y$-axis intercept by setting $x=0$

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

  • In factorised form, this can be done by inspection

    • A cubic of the form $y = ( x - p ) ( x - q ) ( x - r )$ has roots at $p,q$ and $r$
    • E.g. $y = ( x - 2 ) ( x - 3 ) ( x + 5 )$ has roots at 2, 3, and -5

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

    • The graph does not cross the $x$-axis
    • E.g. $( x - 2 )^{2} ( x + 1 )$* touches *the $x$-axis at $x=2$, and intersects *the $x$-axis at $x=-1$

• 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 $x$ and $y$ axes

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

STEP 1 Find the $y$-axis intercept by substituting in $x=0$

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

STEP 2 Find the $x$-axis intercepts by solving $y=0$ Either bracket can be equal to zero

$\begin{gathered} ( 2 x - 1 ) & = 0 \\ x & = 1 \frac{1}{2} \end{gathered}$

$\begin{gathered} ( x - 3 )^{2} & = 0 \\ x = 3 \end{gathered}$ (repeated solution, as there are two $( x - 3 )$ brackets

STEP 3 Consider the shape, and the 'start' and 'end' points:

$a > 0 ( a = 2 )$ 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 Consider the 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 the two roots $x=\frac{1}{2}$ and $x=3$

STEP 5 Sketch a smooth curve with labelled intercepts