Introduction to Modelling with Functions
Welcome! In this chapter, we are looking at Modelling. This is where the abstract algebra we learn meets the real world. Think of a mathematical model as a "translation." We take a real-life situation—like how fast a population grows or how a cup of coffee cools down—and translate it into a mathematical function. By doing this, we can make predictions about the future or understand the present better.
Don't worry if this seems a bit "wordy" at first; modelling is just about finding the right "math shape" to fit a "real-life story."
What is a Mathematical Model?
A mathematical model is a simplified representation of a real-world situation. We use functions to describe the relationship between different quantities.
Example: If you earn £10 per hour, the relationship between your pay (\(P\)) and hours worked (\(h\)) can be modelled by the function \(P = 10h\).
Key Terms to Know:
• Variables: The quantities that change (like time, distance, or cost).
• Parameters: The fixed values in your model (like the "10" in the pay example above).
• Domain: The set of possible input values (e.g., time cannot be negative).
• Range: The set of possible output values.
Quick Review: A model isn't the "perfect truth"—it’s a simplified version of reality that is "good enough" to be useful!
The Modelling Process
When you are asked to "model" something in your exam, you are usually following these steps:
1. Identify the variables: What are we measuring? (Usually time \(t\) is one of them).
2. Choose a function type: Does it look like a straight line (linear)? A curve (quadratic)? Or does it grow very fast (exponential)?
3. Formulate the equation: Write down the function \(f(x)\) or \(y = \dots\)
4. Solve and Interpret: Use the math to find an answer, then explain what that answer means in the real world.
5. Evaluate: Ask yourself, "Does this answer make sense?" and "What are the limitations?"
Common Types of Models
In the "Functions" section of your course, you will encounter a few main "shapes" of models:
1. Linear Models
Used when something changes at a constant rate.
Equation: \( y = mx + c \)
Example: A taxi charges a £3 flat fee plus £2 per mile. The model is \(C = 2m + 3\).
2. Quadratic Models
Used for things that go up and then come back down (like a ball thrown in the air) or have a minimum/maximum point.
Equation: \( y = ax^2 + bx + c \)
Analogy: Think of a fountain. The water goes up, reaches a peak, and falls in a symmetrical curve (a parabola).
3. Exponential Models
Used for things that grow or decay proportionally to their size.
Equation: \( y = Ae^{kt} \) or \( y = Ab^t \)
Did you know? Exponential models are used to track how viral videos spread on social media!
Limitations and Refinements
This is a vital part of the MEI syllabus (f8). You must be able to critique your model.
Understanding Limitations
A limitation is a reason why the model might stop working or be inaccurate.
• Example: If you model a person's height based on their age using a linear function, the model would eventually predict they become 10 feet tall! This is a limitation because humans stop growing.
Suggesting Refinements
A refinement is a way to make the model more realistic.
• Example: Instead of a simple linear model for a car's value, you might use an exponential decay model because cars lose value faster when they are new than when they are old.
Key Takeaway: Always check if your model's domain makes sense. If the math says a "length" is \(-5\), the model has hit a limitation!
Common Mistakes to Avoid
• Ignoring Units: If time is in minutes but the rate is in hours, your model will be wrong. Always check your units!
• Extrapolation: This is predicting values far outside your data range. Just because a plant grew 2cm this week doesn't mean it will grow 2cm every week for the next 10 years.
• Mixing up \(x\) and \(y\): Make sure you know which variable depends on the other (e.g., Cost depends on Time, so Cost is \(y\) and Time is \(x\)).
Summary Table: Choosing a Model
If the change is... | Use a... | Look for...
Steady / Constant | Linear Function | A straight line graph
U-shaped / Peak | Quadratic Function | A squared term (\(x^2\))
Very rapid growth/decay | Exponential Function | Variable in the power (\(e^x\))
Don't worry if this seems tricky at first! Modelling is a skill that gets much easier with practice. The more "real world" problems you see, the faster you'll recognise which function to use.