An exponential function can give us information about a graph that represents it.

For example, suppose that function $q$ represents a bacteria population $t$ hours after it is first measured, and $q(t) = 5,000 \boldcdot (1.5)^t$. The number 5,000 is the bacteria population measured, when $t$ is 0. The number 1.5 indicates that the bacteria population increases by a factor of 1.5 each hour.

A graph can help us see how the starting population (5,000) and growth factor (1.5) influence the population. Suppose functions $p$ and $r$ represent two other bacteria populations and are given by $p(t) = 5,000 \boldcdot 2^t$ and  $r(t) =5,000 \boldcdot (1.2)^t$. Here are the graphs of $p$, $q$, and $r$.

Graph of 3 functions labeled, p, q, and r on grid, origin O. Horizontal axis, time in hours, from 0 to 10, by 2's. Vertical axis, number of bacteria, from 0 to 40,000 by 20,000's. All three functions start at 0 comma 5,000 and trend upward and to the right. Function p passes through 2 comma 20,000 and trends rapidly upward and right. Function q passes through 4 comma11,250 and trends steadily upward and right. Function r passes through 6 comma 7,200 and trends more slowly upward and right.

All three graphs start at $5,000$, but the graph of $r$ grows more slowly than does the graph of $q$, while the graph of $p$ grows more quickly. This makes sense because a population that doubles every hour is growing more quickly than one that increases by a factor of 1.5 each hour, and both grow more quickly than a population that increases by a factor of 1.2 each hour.