Quick answer: A linear model has the form y = mx + c. The gradient m is the change in y for each increase of 1 in x, so in context it is a rate, such as euros per kilometre. The intercept c is the value of y when x = 0, such as a fixed charge. Find m from two data points, then c from either one.
Slope, intercept, and interpreting the gradient in context. SL AI dresses linear models as taxi fares, phone plans, and rental costs — always ask what the gradient MEANS in the real-world context.
Practise Linear functions and modelling with AI marking
Open the full SL AI engine — every question in this unit is worked through with instant AI-graded feedback on your handwritten workings.
Gradient from two points: m = (y₂ − y₁) ÷ (x₂ − x₁). Give it units in context.
The y-intercept is the starting value or fixed cost; say what it means, not just its value.
The point-slope form y − y₁ = m(x − x₁) is in the formula booklet and is often quicker than finding c first.
To compare two plans, solve the two linear equations together (on the GDC or by hand): the intersection is where they cost the same.
A linear model may only be sensible for some values of x; say so when a question asks about a value far outside the data.
Worked example
A taxi ride of 4 km costs €9.80 and a ride of 10 km costs €18.20. The cost, C euros, is a linear function of the distance, d km. Find the model, interpret the gradient and the intercept, and find the distance you can travel for €25.
It is the rate of change: how much y changes for each increase of 1 in x, with units of y per unit of x. In a cost model it is the cost per item or per kilometre; in a temperature model it might be degrees per hour.
Where can I practise linear functions and modelling questions?
On the Direct/Inverse Variation and Cubic Models questions page: 14 exam-style SL AI questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for linear functions and modelling?
Yes: a GDC is allowed on every SL AI paper, so practise the calculator method as well as the working. The Tables of values guide gives the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.