Introduction to Mathematical Modelling

In your study of functions, you have learned about many different "shapes" of graphs: straight lines, parabolas, and waves. But why do we learn them? In the real world, mathematical modelling is the process of using these functions to describe, explain, and predict real-life situations. Whether it is predicting the path of a ball, the growth of a population, or the cooling of a cup of coffee, modelling is the bridge between numbers and reality.

In this chapter, we focus on the skills required to take a set of data and turn it into a working model. Don't worry if this seems abstract at first; modelling is as much an "art" as it is a "science"!

1. Choosing an Appropriate Model

The first step in modelling is looking at your data (usually on a scatter diagram) and deciding which type of function fits the shape best. You need to justify your choice based on the shape of the data and the context of the problem.

Common "Shapes" to Look For:
  • Linear \( (y = mx + c) \): The data points follow a roughly straight line. This suggests a constant rate of change.
  • Quadratic \( (y = ax^2 + bx + c) \): The data shows a single "turning point" (a peak or a valley) and is roughly symmetric. Think of a ball being thrown through the air.
  • Exponential \( (y = k a^x + c) \): The data grows or decays very rapidly. It usually has a horizontal asymptote (a level that the graph approaches but never crosses).
  • Power/Inverse \( (y = a x^n) \): If \( n \) is negative, the graph looks like a curve that crashes towards the axes (like how the gravity between two objects gets weaker as they move apart).
  • Sinusoidal \( (y = a \sin(bx) + d) \): The data repeats in a regular, wave-like pattern (cycles), like tides or temperatures over a year.

Quick Tip: Always ask yourself about the long-term behavior. For example, if a model predicts a population will keep growing forever at an increasing rate, an exponential model might be better than a linear one.

2. Determining Parameters and Fitting

Once you have chosen a model (e.g., "I think this is quadratic"), you need to find the parameters. Parameters are the specific numbers (\( a \), \( b \), \( c \), etc.) that make the function fit your specific data.

In the IB Applications and Interpretation course, you will almost always use your Graphic Display Calculator (GDC) to do this through Regression. Your GDC calculates the "line of best fit" or "curve of best fit" automatically.

Step-by-Step GDC Approach:
1. Enter your data into the lists (usually \( L1 \) and \( L2 \)).
2. Create a scatter plot to visualize the data.
3. Run the appropriate regression analysis (e.g., LinReg for linear, QuadReg for quadratic, ExpReg for exponential).
4. Write down the resulting equation, ensuring you use the variables given in the question (like \( t \) for time instead of \( x \)).

3. Defining a Reasonable Domain and Range

In pure math, functions often go on forever in both directions. In the real world, they don't! A reasonable domain represents the "real-world limits" of your input variable.

Example: If you are modelling the height of a plant over time \( t \):
- The domain cannot include \( t < 0 \) because time doesn't go backwards from the start of the experiment.
- The range cannot include negative heights because a plant cannot have a height of \( -10 \) cm.

Key Takeaway: Always check if your model makes sense at the boundaries. If your model predicts a car is traveling at \( -20 \) km/h, you have likely moved outside the "reasonable domain."

4. Testing and Commenting on the Model

Once you have an equation, you must comment on its appropriateness. Is it a "good" model? You can justify this in three ways:

  • Visual Fit: Does the curve actually pass through or near the majority of the data points on your scatter plot?
  • Contextual Sense: Does the model's behavior match reality? (e.g., "A quadratic model is appropriate here because the projectile must eventually hit the ground.")
  • Reliability: Does the model have a horizontal asymptote where we expect one? (e.g., "A cooling cup of tea should never drop below room temperature.")

Common Mistake: Don't just say "it looks right." Use mathematical language like "The data shows a clear cyclical pattern, which justifies the use of a sinusoidal model."

5. Using the Model: Interpretation and Prediction

The whole point of a model is to use it to find missing information. This usually involves Interpolation and Extrapolation.

Interpolation

This is predicting a value inside the range of your original data points.
Example: You have data for 2010 and 2012, and you use the model to estimate a value for 2011.
Reliability: Usually very high.

Extrapolation

This is predicting a value outside the range of your original data points.
Example: You have data for 2010–2020, and you use the model to predict the value for 2050.
Reliability: Often low. Trends can change over time (e.g., a population might stop growing due to limited food), so extrapolation is risky.

Did you know? Many economic crashes happen because people "extrapolate" a rising trend and assume it will go up forever!

Summary Checklist for Modelling Questions

When you face a modelling problem in Paper 2 or Paper 3, run through this mental list:

  • Identify: What is the general shape? (Linear, Quadratic, Exponential, etc.)
  • Fit: Use the GDC to find the equation.
  • Restrict: What are the real-life limits for \( x \) and \( y \)? (Domain and Range)
  • Justify: Why did I pick this? Does it match the graph and the context?
  • Predict: Use the equation to find values, but be careful when extrapolating!

Note: For more details on specific types of functions used in these models, see the chapters on "Modelling with standard functions" or "Logarithmic, sinusoidal, and logistic models".