A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.

1. Write an equation of the form representing the relationship, where is the time in years since the coral was measured, and is the volume of coral in cubic centimeters. (You need to figure out what numbers and are in this situation.)
2. Find the volume of the coral when is 5, 1, 0, -1, and -2.
3. What does it mean, in this situation, when is -2?
4. In a certain year, the volume of the coral is 37.5 cubic centimeters. Which year is this? Explain your reasoning.

The volume, , of coral in cubic centimeters is modeled by the equation where is the number of years since the coral was measured. Three students used graphing technology to graph the equation that represents the volume of coral as a function of time.

Two points on coordinate plane, origin O. Horizontal axis, scale is -2 to 4 by 1’s. Vertical axis, scale is 0 to 2,000 by 500’s. Y-intercept (0 comma 12,000) and (-1 comma 6,000).

Graph of a function on grid, origin O.  Horizontal axis, x, from negative 2 to 10 by 1's. Vertical axis, y, from 0 to 1,000,000, by 100,000's. Plotted coordinates appear as follows: negative 1 comma 0, 0 comma 0, 1 comma 0, 2 comma 0, 3 comma 10,000, 4 comma 20,000, 5 comma 30,000, 6 comma 75,000, 7 comma 125,000, 8 comma 250,000, 9 comma 600,000.  

Graph of a function on grid, origin O.  Horizontal axis, x, from negative 2 to 6 by 0 point 5's. Vertical axis, y, from 0 to 60,000, by 10,000's. Given plotted coordinates as follows: negative 1 comma 600, 0 comma 1,200, 1 comma 2,400, 1 comma 4,800, 3 comma 9,600, 4 comma 19,200, 5 comma 38,400.

For each graph:

1. Describe how well each graphing window does, or does not, show the behavior of the function.
2. For each graphing window that you think does not show the behavior of the function well, describe how you would change it.
3. Make the change(s) you suggested, and sketch the revised graph using graphing technology.

A town’s population decreased exponentially from the late 1800’s until the mid 1900’s, when the last residents left the town, leaving it a ghost town.

1. 1. The population of the town, , for a number of years since 1900, , is shown in the table. What is the growth factor? That is, what is in an equation of the form ? What is ?
   2. Find the population of the town when is -1 and -3. Record them in the table.

   , years since 1900	, population
   0	1,500
   1	1,350
   2	1,215

2. What do , , and mean in this situation?
3. The town’s population was at its greatest in 1895. What was that population? Explain your reasoning.
   
4. Plot the points whose coordinates are shown in the table.
   

   

5. Based on your graph:

   1. When did the town have about 2,000 people?
   2. What was the population in 1905?