In relationships where the change is exponential, a quantity is repeatedly multiplied by the same amount. The multiplier is called the growth factor.

Suppose a population of cells starts at 500 and triples every day. The number of cells each day can be calculated as follows:

We can see that the number of cells ($p$) is changing exponentially, and that $p$ can be found by multiplying 500 by 3 as many times as the number of days ($d$) since the 500 cells were observed. The growth factor is 3. To model this situation, we can write this equation: $\displaystyle p = 500 \boldcdot 3^d$.

The equation can be used to find the population on any day, including day 0, when the population was first measured. On day 0, the population is $500 \boldcdot 3^0$. Since $3^0 = 1$, this is $500 \boldcdot 1$ or 500.

Here is a graph of the daily cell population. The point $(0,500)$ on the graph means that on day 0, the population starts at 500.

Graph of an exponential function, origin O. Horizontal axis, number days, scale 0 to 4, by 1’s Vertical axis, cell population, scale 0 to 20,000, by 5,000’s. The function is discrete and has these points: (0 comma 500), (1 comma 1,500), (2 comma 4,500) and (3 comma 13,500).

Each point is 3 times higher on the graph than the previous point. $(1,1500)$ is 3 times higher than $(0,500)$, and $(2,4500)$ is 3 times higher than $(1,1500)$.