Cubic Functions & Graphs (DP IB Applications & Interpretation (AI): SL): Revision Note

Cubic functions & graphs

What are the key features of cubic graphs?

• A cubic graph is of the form $y=a{x}^3+b{x}^2+cx+d$ where $a\ne 0$.

• The value of $a$ affects the shape of the curve

  • If $a$ is positive the graph goes from bottom left to top right

  • If $a$ is negative the graph goes from top left to bottom right

• The y-intercept is at the point $(0,d)$

• The zeros or roots are the solutions to $a{x}^3+b{x}^2+cx+d=0$

  • These can be found using your GDC

  • These are also called the x-intercepts

  • There can be 1, 2 or 3 x-intercepts

    • There is always at least 1

• There are either 0 or 2 local minimums/maximums

  • If there are 0 then the curve is monotonic

  • If there are 2 then one is a local minimum and one is a local maximum