Types of Graphs (Leaving Cert Mathematics): Revision Notes

Cubic

Introduction

A cubic function is a polynomial function of degree 3. It is written in the general form:

where $a,b,c,d$ are constants and $a \ne 0$.

The graph of a cubic function is more complex than linear or quadratic graphs. It can have one or two turning points and can cross the $x$-axis up to three times.

Characteristics

• The shape of the graph depends on the sing and magnitude of $a$.
  • If $a>0$ : The graph starts from $- \infty$ (left) and rises to $+ \infty$ (right)
  • If $a<0$ : The graph starts from $+ \infty$ (left) and falls to $+ \infty$ (left)
• The roots are the points where the graph crosses the $x$-axis, $f(x)=0$. A cubic can have :
  • One real root (and two complex roots)
  • Three real roots.