Exponential Growth and Decay

A linear function adds the same amount each step. An exponential one multiplies by the same factor.

The form

y = a · bˣ, where a is the starting amount and b is the factor per step.

Growth or decay

b greater than 1 grows. b between 0 and 1 decays. A 5% rise means b = 1.05.

Exponential always wins eventually

However steep a line is, an exponential with b > 1 overtakes it. That is worth seeing once.

The exponential form

An exponential function is y = a·bˣ, where a is the starting value and b is the growth factor. Each step in x multiplies the output by b. That repeated multiplication is what distinguishes it from linear growth.

What the base tells you

If b > 1 the function grows; if 0 < b < 1 it decays. A 5% annual increase gives b = 1.05; a 5% decrease gives b = 0.95. Reading a percentage change into a growth factor is the key modelling step.

Adding against multiplying

Linear growth adds a fixed amount each step; exponential growth multiplies by a fixed factor. Over a few steps they can look similar; over many, exponential growth outstrips any linear function, however steep.

Where it appears

Compound interest, population growth, radioactive decay, drug concentration, and the early stages of epidemics are all exponential. So is anything where the change depends on how much is already there.