Introduction to Mathematical Modelling
Mathematical modelling is the process of using functions to represent real-world situations. A good model captures the essential behaviour of a phenomenon so we can analyse it and make predictions.
In IB Math AI, you are expected to recognise which type of function best fits a given situation or data set, find the parameters of that function, and interpret the results in context.
The main model types you need to know at SL are:
- Linear — constant rate of change
- Quadratic — parabolic shape, single turning point
- Cubic — up to two turning points and one inflection point
- Exponential — constant percentage growth or decay
- Direct/Inverse variation — power functions
- Sinusoidal — periodic, wave-like behaviour
- Logistic — growth that levels off at a maximum value
Before doing any algebra, always plot the data. A quick sketch or scatter plot will often tell you which model family is appropriate.
Linear Models
A function of the form , where is the slope (rate of change) and is the y-intercept (initial value).
Linear models are used whenever a quantity changes at a constant rate. Key features:
- Straight-line graph
- Constant difference between consecutive y-values for equal x-intervals
- : increasing; : decreasing; : constant (horizontal line)
Profit model
A company has fixed costs of $2000 and earns $500 profit for each unit sold. Model the total profit as a function of units sold .
- : each additional unit adds $500 to profit
- : before any sales, the company is $2000 in debt
- Break-even point: units
To check if data is linear, calculate the first differences (subtract consecutive y-values). If they are constant, the relationship is linear.