A real-world situation can be modelled with a cubic function when the data changes direction or curvature in a way that a linear or quadratic model cannot represent. Write the model as , determine its parameters from data using technology, and then test and interpret it within a reasonable domain.
This is part of SL 2.5: Modelling functions. A cubic graph can have up to two turning points, so it is useful for situations involving alternating growth and decline, such as water levels, profit, or population changes.
| Step | What to do |
|---|---|
| Identify variables | Define the independent variable and dependent variable , including units. |
| Choose the model | Check whether the data's shape supports a cubic rather than a linear, quadratic, or exponential model. |
| Find parameters | Substitute four data points into and solve the resulting system using a GDC. |
| Test and use | Compare the model with the data, set a reasonable domain, and make predictions. |
For example, suppose the water depth , in metres, is recorded hours after midnight:
Substituting these points and solving the four simultaneous equations using technology gives
The predicted depth after five hours is
Therefore, the model predicts a water depth of after five hours. Its sensible domain is , because predictions outside the measured interval are extrapolations and may be unreliable.
A common misconception is that a cubic is appropriate simply because four points can determine its coefficients. The graph's shape and context must also justify the model.
In an IB response, show the cubic equation or GDC set-up before giving numerical results. Define variables, give units, interpret predictions in context, and comment on the model's domain or limitations rather than reporting only a calculator value.
