Cubic Graphs (Edexcel IGCSE Maths B): Revision Note

Cubic Graphs

What is a cubic?

• A cubic is a function of the form $a{x}^3+b{x}^2+cx+d$

  • $a,b,c$ 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 zero

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

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

• Some cubics are simple to factorise

  • E.g. $x^3-4x=x\left(x^2-4\right)=x\left(x-2\right)\left(x+2\right)$

• For more complicated cubics you may be able to factorise them using the factor theorem and algebraic division

• You should also 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 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 (no 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 $y=\left(x-p\right)\left(x-q\right)\left(x-r\right)$ has roots at $p,q$ and $r$

    • E.g. $y=\left(x-2\right)\left(x-3\right)\left(x+5\right)$ 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. ${\left(x-2\right)}^2\left(x+1\right)$ 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