Fitting Quadratic and Exponential Models

A straight line adds the same amount every time x increases by 1. Plenty of real data doesn't behave that way: a thrown ball rises then falls, and a bacteria culture piles on more new cells each hour than the hour before. Both curve, but they curve for different reasons, so a single line can't fit either one. You need a model whose shape matches the reason the data bends.

A quadratic model, like h = -16x² + vx + h0 for a ball's height, bends because it has one turning point, called the vertex. At the vertex the quantity switches from increasing to decreasing, and its x-coordinate is -b/(2a). For the ball, that's the exact second it stops rising and starts falling back down. An exponential model, like N(x) = N0·k^x, never turns around that way. Instead it grows (or shrinks) by multiplying by the same factor k every time x increases by 1 — that's what doubling every 3 hours means: multiply by 2 every 3-hour step. The giveaway in a table: constant second differences mean quadratic, a constant ratio between consecutive outputs means exponential.

This matters because both curves bend upward on a graph, so it's tempting to call any upward curve exponential. It isn't — a quadratic bends upward too, but it bends back down eventually, and its growth comes from adding a changing amount, not multiplying by a fixed one. Checking whether consecutive outputs share a common ratio versus a common second difference settles it, and this same instinct — matching an equation's structure to the shape and rate of change in the data — carries over to reading scatterplots and judging other models in this unit.