Write the following sentence as an equation: y varies directly as x.

Part A) A reasonable domain  is the interval [0,7]

Part B) see  the explanation

Part C) The average rate of change is

see the explanation

Step-by-step explanation:

Let

m ---> the number of months

f(m) ---> the length of the fish in cm

we have

This is a exponential function of the form

where

a is the initial value or y-intercept

b is the base of the exponential function

b=1+r

r is the percent average of change

In this problem we have

Part A)  When the marine biologist concluded her study, the length of the fish was approximately 6.86 cm. What is a reasonable domain to plot the growth function?  

For f(m)=6.86 cm

substitute the given value in the function

solve for m

Apply log both sides

therefore

A reasonable domain  is the interval [0,7]

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

we know that

The y-intercept is the value of the function when the value of m is equal to zero

In this context the y intercept is the initial value of the function, or the starting size of the fish  at month 0

For m=0

therefore

At the beginning of the study the length of the fish was 4 cm

Part C) What is the average rate of change of the function f(m) from m = 3 to m = 7, and what does it represent?

we know that

To find the average rate of change, we divide the change in the output value by the change in the input value

the average rate of change is equal to

In this problem we have

Substitute

The average rate of change is how fast the growth changed over the time period