Choose a quadratic model when the situation has a curved, parabolic pattern with one turning point. Represent it by , use given data to determine the parameters, and then interpret the model within a realistic domain.
The Modelling Process
This is SL 2.5, Modelling functions in IB Mathematics: Applications and Interpretation.
| Step | What to do |
|---|---|
| Identify the variables | Define the independent variable and dependent variable , including units. |
| Check the shape | A quadratic is appropriate when values increase and then decrease, or decrease and then increase, with a changing rate. |
| Form the model | Write . Three independent conditions are generally needed to determine , , and . |
For example, suppose a ball's height , in metres, is measured at times , , and , where time is in seconds. Let
Substitution gives , , and . Solving these equations gives
The coefficient confirms that the parabola opens downward. Here, is the initial height, while the positive value of represents the model's initial upward rate of change. These interpretations help test whether the fitted equation matches the context.
The vertex occurs at , giving a maximum height of metres. Solving gives , so a sensible domain is .
A common misconception is that a good algebraic fit is automatically realistic. A quadratic may predict negative heights outside the relevant time interval, so its domain must be restricted.
Exam Technique
On a GDC-active Paper 1 or Paper 2 question, show the model and equations before giving calculator results. Examiners may also expect interpretation of the vertex, intercepts, units, and domain in context rather than only the quadratic equation.
