How Do You Model a Real-World Situation With a Linear Function?

A real-world situation can be modelled by a linear function when one quantity changes at a constant rate relative to another. Represent the relationship as $y=mx+c$, where $m$ is the constant rate of change and $c$ is the initial value.

Building the model

First define the variables, including their units. Then identify the gradient and vertical intercept from the context or calculate them from two data points.

Suppose a taxi fare consists of a fixed charge of 4 currency units plus 2.50 currency units per kilometre. Let $d$ be the distance travelled in kilometres and $C(d)$ be the total cost.

The fixed charge gives $c=4$, while the cost per kilometre gives $m=2.5$. Therefore,

$$C(d)=2.5d+4.$$

For a journey of $8$ km,

$$C(8)=2.5(8)+4=24.$$

Thus, the predicted fare is 24 currency units. A reasonable domain is $d\geq 0$, although a maximum distance may be imposed by the context.

The model assumes that the fare per kilometre remains constant. Before using it for prediction, check whether the situation includes distance bands, surcharges, or a maximum journey length; these would require a piecewise model or a restricted domain.

If two points $(x_1,y_1)$ and $(x_2,y_2)$ are given instead, calculate

$$m=\frac{y_2-y_1}{x_2-x_1},$$

then substitute one point into $y=mx+c$ to find $c$.

A common misconception is that any increasing relationship is linear. A linear model requires a constant absolute change, not a constant percentage change; constant percentage change is exponential.

Exam technique

This is IB Mathematics: Applications and Interpretation SL 2.5, suitable for Paper 1 or Paper 2. Define variables, show how $m$ and $c$ are obtained, state a realistic domain, and interpret the final answer in context rather than giving only an equation. Use interpolation where possible and identify unreliable extrapolation.