Cubic Models (DP IB Applications & Interpretation (AI)): Revision Note

Cubic models

What are the parameters of a cubic model?

• A cubic model is of the form 

• The value of represents the value of the function when

  • This is the value of the function when the independent variable is not present

  • This is usually referred to as the initial value

• The value of has the biggest impact on the rate of change of the function

  • If has a large absolute value then the rate of change varies rapidly

  • If has a small absolute value then the rate of change varies slowly

What can be modelled as a cubic model?

• If the graph of the data has exactly one maximum and one minimum within an interval

• If the graph is monotonic with no maximum or minimum

  • D(t) is the vertical distance below a starting point of a bungee jumper t seconds after jumping

  • V(x) is the volume of a cuboid of length x cm that can be made with a 200 cm² of cardboard

What are possible limitations of a cubic model?

• Cubic models have no global maximum or minimum

  • This means the function is unbounded

  • In real-life this might not be the case

    • The function might have a maximum or minimum

    • To overcome this you can decide on an appropriate domain so that the outputs are within a range

• The values of a cubic model could be negative

  • In some contexts negative values might not be appropriate

    • You can adjust the domain to ensure the model is never negative