a population of bees is decreasing. the population in a particular region this year is 1,250. after 1 year, it is estimated that the population will be 1,000. after 3 years, it is estimated that the population will be 640. a. write a function to model this scenario. b. create a graph to show the bee population over the next 10 years. c. identify the key features of the function. identify the x- and y-intercepts. determine the maximum, the minimum, whether the function is increasing or decreasing, the rate of change of the function over the interval [0, 10], and any asymptotes.

To model this situation, we are going to use the decay formula: 
where 
is the final pupolation
is the initial population 
is the Euler's constant
is the decay rate 
is the time in years

A. We know for our problem that the initial population is 1,250, so ; we also know that after a year the population is 1000, so  and . Lets replace those values in our formula to find :








Now that we have , we can write a function to model this scenario:
.

B. Here we are going to use a graphing utility to graph the function we derived in the previous point. Please check the attached image.

C. 
- The function is decreasing
- The function doe snot have a x-intercept 
- The function has a y-intercept at (0,1250)
- Since the function is decaying, it will have a maximum at t=0: 



- Over the interval [0,10], the function will have a minimum at t=10:



D. To find the rate of change of the function over the interval [0,10], we are going to use the formula: 
where 
is the rate of change 
is the function evaluated at 10
is the function evaluated at 0
We know from previous calculations that  and , so lets replace those values in our formula to find :



We can conclude that the rate of change of the function over the interval [0,10] is -111.572.