Modelling with Functions

Step 1: Identify the Relationship

• Determine if the variables share a linear, quadratic, exponential, or reciprocal relationship based on the context or provided data trends.

Step 2: Formulate the Equation

• Translate the relationship into a general function form, such as $y = mx + c$ for linear trends or $y = \frac{k}{x}$ for inverse proportions.

Step 3: Determine Constants

• Use known data points (initial conditions or specific observations) to solve for unknown parameters like the rate of change ($m$) or the constant of proportionality ($k$).

Step 4: Interpret and Predict

• Use the completed function to calculate values for the dependent variable at specific points or to find the 'break-even' or 'limit' points of the system.

It is vital to distinguish between the mathematical domain of a function and the practical domain of the model.

For example, a reciprocal function $f(x) = \frac{1}{x}$ mathematically exists for all $x \neq 0$, but in a model for 'time taken to complete a task,' $x$ must be positive and likely has a maximum realistic value.

• Check the Intercepts: Always interpret what the $y$-intercept represents in context. It usually signifies the 'initial state' or 'starting value' before any change has occurred.

• Verify Units: Ensure that the units of your final answer match the context of the question (e.g., if the model outputs pressure in Pascals, do not provide an answer in atmospheres).

• Sanity Checks: Evaluate if your answer is realistic. If a model for the height of a person yields 50 meters, there is likely an error in the calculation or the model's parameters.

• Asymptotic Behavior: For reciprocal or exponential models, consider what happens as the independent variable becomes very large. Does the model approach a logical limit?