Non-Linear Relationships: When the Graph Isn't a Straight Line

A non-linear relationship is one where the rate of change isn't constant — the graph curves. The most common types at Stage 5 are parabolas (from quadratic equations like y = x²), hyperbolas (from inverse relationships like y = k/x), exponential curves (y = aˣ), and circle equations. Each has a distinctive shape and models a different type of real-world relationship. Recognising which type of curve fits a situation is as important as being able to draw it.

From ancient cones to orbiting planets

Apollonius of Perga catalogued the conic sections — the curves formed by slicing a cone at different angles — around 200 BC. He named and described the ellipse, parabola, and hyperbola with extraordinary precision. Galileo Galilei was the first to recognise, in 1638, that projectiles follow parabolic paths (assuming no air resistance) — a profound connection between geometry and physics. Isaac Newton showed in his Principia (1687) that the orbits of planets are ellipses, and that the same inverse-square law of gravity produces parabolic trajectories for slow-moving objects and hyperbolic paths for objects moving faster than escape velocity.

Non-linear curves in the real world

A ball thrown through the air follows a parabolic arc. Satellite dishes and car headlight reflectors are parabolic — this shape focuses incoming parallel signals or light to a single point. The cables of a suspension bridge hang in a shape close to a parabola. Supply and demand curves in economics are typically non-linear — price increases don't produce proportional decreases in demand. Population growth in an unconstrained environment is exponential. The relationship between a car's speed and its braking distance is quadratic — double the speed, four times the stopping distance. Non-linear thinking is needed whenever the world doesn't follow a straight line, which is most of the time.