Modelling with Functions

Proportionality: Many models begin with direct proportion ($y = kx$) or inverse proportion ($y = \frac{k}{x}$), where $k$ is the constant of proportionality.

Solving for Constants: To find the value of $k$, substitute a known pair of values (coordinates) from the scenario into the general equation.

Domain Restrictions: In real-world models, variables often have physical constraints, such as time $t \ge 0$ or length $L > 0$, which must be explicitly stated.

Interpreting Intercepts: The y-intercept ($x=0$) often represents the 'initial' value of the system before any change has occurred.

Check the Context: Always verify if your mathematical answer makes sense in the real world; for example, a negative time or a negative number of objects is usually invalid.

Units and Precision: Ensure all variables are in consistent units (e.g., converting minutes to hours) and round final answers to a sensible degree of accuracy based on the context.

Limit Analysis: Test what happens to the function as the input variable becomes very large ($x \to \infty$) or very small ($x \to 0$) to see if the model remains plausible.

Graph Sketching: Sketching the function can help identify asymptotes or turning points that represent physical maximums or minimums.

Ignoring the Domain: Students often solve equations and provide all mathematical roots, forgetting that only positive values might be relevant in a physical context.

Confusing Proportions: Mistaking inverse proportion for a linear decrease is a common error; inverse proportion ($y \propto \frac{1}{x}$) creates a curve, not a straight line.

Over-modelling: Attempting to include every possible variable can make the mathematics impossible to solve without significantly improving the accuracy of the prediction.