If we have enough information about a graph representing an exponential function $f$, we can write a corresponding equation.

Here is a graph of $y = f(x)$.

An equation defining an exponential function has the form $f(x) = a \boldcdot b^x$. The value of $a$ is the starting value or $f(0)$, so it is the $y$-intercept of the graph. We can see that $f(0)$ is 500 and that the function is decreasing.

The value of $b$ is the growth factor. It is the number by which we multiply the function’s output at $x$ to get the output at $x+1$. To find this growth factor for $f$, we can calculate $\frac{f(1)}{f(0)}$, which is $\frac{300}{500}$ (or $\frac35$).

Graph of a function on grid, origin O. Horizontal axis, x, from 0 to 4, by 2's. Vertical axis, y. from 0 to 600, by 200's. Plotted points as follows: 0 comma 500, 1 comma 300. Line is drawn through the points, moving downward and to the right.  

So an equation that defines $f$ is: $f(x) = 500 \boldcdot \left(\frac{3}{5}\right)^x$

We can also use graphs to compare functions. Here are graphs representing two different exponential functions, labeled $g$ and $h$. Each one represents the area of algae (in square meters) in a pond, $x$ days after certain fish were introduced.

• Pond A had 40 square meters of algae. Its area shrinks to $\frac{8}{10}$ of the area on the previous day.
• Pond B had 50 square meters of algae. Its area shrinks to $\frac 25$ of the area on the previous day.

Graph of 2 functions on grid, origin O. Horizontal axis, x, Vertical axis, y. Function h, blue line, starts on the vertical axis,  moves downward and to the right, almost linearly.. Function g, green line, starts above function h, moves downward and to the right, exponentially.  

Can you tell which graph corresponds to which algae population?

We can see that the $y$-intercept of $g$'s graph is greater than the $y$-intercept of $h$'s graph. We can also see that $g$ has a smaller growth factor than $h$ because as $x$ increases by the same amount, $g$ is retaining a smaller fraction of its value compared to $h$. This suggests that $g$ corresponds to Pond B, and $h$ corresponds to Pond A.