Linear Models: Represented by , these are used for situations with a constant rate of change. The gradient represents the rate (e.g., dollars per month), while the y-intercept represents the initial value at .
Quadratic Models: Often taking the form , these are essential for modelling motion under gravity. The parabolic shape naturally represents the rise and fall of an object.
Critical Points: In quadratic models, the vertex represents the maximum or minimum value of the system. This is found by completing the square or using the formula .
| Feature | Linear Model | Quadratic Model |
|---|---|---|
| Rate of Change | Constant | Variable (Changing linearly) |
| Shape | Straight line | Parabola (U-shape or n-shape) |
| Common Use | Depreciation, constant production | Projectiles, area optimization |
| Key Feature | Gradient () | Vertex (Turning point) |
Discrete vs. Continuous: Some models apply to continuous data (like time), while others apply to discrete data (like the number of items sold). It is vital to ensure the function type matches the data type.
Domain Constraints: Mathematical functions often extend to infinity, but real-world models have a restricted domain. For example, time cannot be negative, and a projectile's height cannot be negative once it hits the ground.
Contextual Interpretation: Always relate your mathematical findings back to the question's context. If the math says , but represents the number of toys produced, you must reject that solution as invalid.
Sketching is Essential: Even if not asked, sketch the function. A visual representation helps identify if you are looking for an intercept, a gradient, or a maximum point, preventing simple calculation errors.
Check the Units: Ensure that the units in your final answer match the units defined in the model (e.g., converting seconds to minutes if the model is defined in minutes).
Reasonableness Check: Evaluate if your answer makes sense. A stone thrown by a human reaching a height of meters suggests a calculation error or a flawed model.
Confusing Gradient with Value: Students often mistake the value of the function at a point for the rate of change. Remember that the gradient represents 'how fast' something is changing, not 'how much' there is.
Ignoring the Constant Term: In many models, the constant represents the starting condition. Forgetting to include it or misinterpreting it as the 'total' is a frequent error.
Extrapolation Errors: Using a model to predict values far outside the range of the original data is dangerous. A model that works for the first years of a business may not work for year .
