Exponential Functions

A linear function adds a constant over equal input intervals. An exponential function multiplies by a constant instead.

The test

For equally spaced inputs, constant differences mean linear and constant ratios mean exponential.

The form

f(x) = a · bˣ, with a the initial value and b the growth factor.

Reading b

A factor above 1 grows and one between 0 and 1 decays. A 20% rise means b = 1.2, not 0.2.

The horizontal asymptote

A basic exponential approaches zero on one side without reaching it. Adding a constant shifts that asymptote.

It always wins

Exponential growth eventually exceeds any polynomial growth, however steep the polynomial.

Constant ratios, not constant differences

A linear function adds the same amount over equal input intervals; an exponential multiplies by the same factor. Checking differences and then ratios in a table identifies which model fits the data.

The general form

f(x) = abˣ, with a the initial value and b the growth factor. Growth when b > 1, decay when 0 < b < 1. A percentage change of r translates to b = 1 + r, which is the key modelling step.

Exponentials outgrow polynomials

Any exponential with base above 1 eventually exceeds any polynomial, however high its degree. The crossover may be far out, but it always comes — which is why exponential growth is so consistently underestimated.

The horizontal asymptote

An exponential approaches its horizontal asymptote at one end and grows without bound at the other. The asymptote is y = 0 unless a vertical shift has moved it, and identifying it is part of describing the function.