Functions and graphs

A straight line describes a constant change in one variable for each unit change in another. In y = mx + c, x is the independent variable, y is the dependent variable, m is the gradient and c is the vertical intercept. Calculate gradient as change in y divided by change in x: m = (y₂ − y₁)/(x₂ − x₁). Keep the order of the points consistent in both differences.

A positive gradient means the graph rises from left to right; a negative gradient means it falls. Gradient has units of output per unit of input, so it is more informative than an unlabelled number.

If a point and gradient are known, use y − y₁ = m(x − x₁), then rearrange if required. Distinct parallel nonvertical lines have equal gradients. Perpendicular nonvertical lines have gradients whose product is −1. Thus a line perpendicular to gradient 2 has gradient −1/2. Through (2, 4), its equation is y − 4 = −(x − 2)/2, or y = −x/2 + 5.

Horizontal and vertical lines are perpendicular too, although a vertical line has undefined gradient and cannot be written as y = mx + c.

For a delivery service charging $18 plus $3 per kilometre, C = 18 + 3d. The vertical intercept represents the fixed charge; the gradient is $3 per kilometre. At six kilometres the model predicts $36. A charge of $42 corresponds to eight kilometres. Although the algebraic line extends indefinitely, the context requires d ≥ 0, and the service may restrict the distances available. The horizontal intercept would be negative and has no practical meaning here.

When constructing a model from data, plot the measurements, label axes and inspect whether a constant rate is plausible. Two points determine a line but do not prove that all future observations follow it. Use additional data to test predictions, distinguish interpolation from extrapolation, and report assumptions such as unchanged pricing and no extra fees.