AHL 4.13—Non-linear regression

The data increases at an accelerating rate with no maxima, minima, or oscillation.

An accelerating, ever-increasing growth pattern with no bends is the signature of exponential growth, $y=a{e}^{bx}$.

The exponential model is the correct choice.

A quadratic has a single turning point (a maximum or minimum), which is not present here.

A sine model oscillates periodically around a mean, but this data never decreases.

A cubic model has an inflection with a change in curvature or up to two turning points, not a smooth accelerating increase.

Only the exponential model matches accelerating, non-oscillating, non-turning growth.

The context explicitly states the relationship follows $y=a{x}^b$, a proportional scaling law.

This equation form corresponds to the power model, used for physical scaling laws such as mass versus metabolic rate.

Power regression is the correct technique to fit $y=a{x}^b$.

Exponential regression fits $y=a{e}^{bx}$, where $x$ is in the exponent, not raised as a base to a power.

Quadratic regression fits $y=a{x}^2+bx+c$, a fixed power of 2, not a variable exponent $b$.

Sine regression is for periodic data, not scaling relationships.

Only power regression directly matches the stated model form $y=a{x}^b$.