Question

A marine biologist is studying the growth of a particular species of fish. She writes the following equation to show the length of the fish, f(m), in cm, after m months:

f(m) = 3(1.09)m

Part A: When the marine biologist concluded her study, the length of the fish was approximately 5.98 cm. What is a reasonable domain to plot the growth function? (4 points)

Part B: What does the y-intercept of the graph of the function f(m) represent? (2 points)

Part C: What is the average rate of change of the function f(m) from m = 2 to m = 8, and what does it represent? (4 points)

Answer 1

Part A: To plot the growth function, we need to choose a domain for the x-values (the number of months). Since the length of the fish is measured in months, a reasonable domain for the function would be the set of non-negative integers (0, 1, 2, 3, ...). This represents the number of months that have passed since the start of the study.

Part B: The y-intercept of a graph is the point at which the graph crosses the y-axis. In this case, the y-intercept of the graph of the function f(m) would be the point (0, f(0)), which represents the length of the fish at the start of the study (after 0 months).

Part C: To find the average rate of change of the function f(m) from m = 2 to m = 8, we need to first find the value of the function at each of these two points. We have f(2) = 3(1.09)^2 = 6.7207 and f(8) = 3(1.09)^8 = 11.3449. The average rate of change is then the difference in the y-values of these points divided by the difference in the x-values, which is (11.3449 - 6.7207) / (8 - 2) = 0.6242. This represents the average increase in the length of the fish over the 6-month period from m = 2 to m = 8.