Iris has been studying an invasive population of snails. This particular snail has no local predators, so the population grows wildly. She has observed that the population follows an exponential rate of growth for fifteen years.

1. Create your own exponential function, f(x), which models the snail population. You will need to identify the principal population of the snails and the rate of growth each year. Explain to Iris how your function shows the principal population and the rate of growth, in complete sentences.

This is the formula that is used to show exponential growth over time: f(x) = P(1 + r)x. Now I need to create an equation based off of the information given, and in the form of the equation above. Because I wasn’t told what the initial population of the snails was I need to make a value for the initial population of the snails; an appropriate value. So let’s say that the initial population of snails was 150. Let’s also say that the rate of growth is 50%; because we weren’t given this information to begin with. Now that we have a value for P and r we can plug in the values. F(x) = 150(1 + 0.5)x P represents the initial population. R represents the rate of growth. X represents the number of years.

2. A local snail population grows according to the function g(x) = 200(1.03)2x. Demonstrate the steps to convert g(x) into an equivalent function with only x as the exponent. Then, explain to Iris how the key features of this local snail population compares to the key features of the invasive population. G(x) = 200(1.03)2x G(x) = 200(1.032)x G(x) = 200(1.0609)x As you can see the local snails initial population is 200, whereas the initial population for the invasive snails is 150.

Also, if you take a look at the exponential function you may notice that g(x) = 200(1.0609)x can also mean g(x) = 200(1 + 0.0609)x. it means that the rate of growth for the local snails is 6.9%, whereas the growth rate for the invasive snails is 50%. So the local snails began with a larger population than the invasive snails, but the invasive snail’s growth rate is much larger than the local snail’s growth rate.

3. Iris wants to graph the invasive snail population to show the city council. Justify what the appropriate domain and range would be for the function f(x), what the y-intercept would be, and if the function is increasing or decreasing. F(x) = 150(1 + .5)x Let’s plug in some values for x! Earlier Iris mentioned that she observed the snail’s growth for 15 years. So we can graph the population of the snails for all 15 of those years using the exponential function we made earlier.

Starting at year 0, and ending at year 15. F(0) = 150(1 + .5)0 F(0) = 150(1.5)0 F(0) = 150(1) F(0) = 150 This means that at year 0, or the beginning, the snail population was 150. Now all we have to do is the same thing for year 1 to year 15. These should be our results: F(0) = 150 F(1) = 225 F(2) = 337 F(3) = 506 F(4) = 759 F(5) = 1139 F(6) = 1708 F(7) = 2562 F(8) = 3844 F(9) = 5766 F(10) = 8649 F(11) = 12974 F(12) = 19461 F(13) = 29192 F(14) = 43789 F(15) = 65684 What those mean is after 15 years [f(15)], the population of invasive snails is 65, 684. Now that we know what these represent and their values we can start to set their ordered pairs and plot them on a coordinate plane.

These are the ordered pairs: (0, 150) (1, 225) (2, 337) (3, 506) (4, 759) (5, 1139) (6, 1708) (7, 2562) (8, 3844) (9, 5766) (10, 8649) (11, 12974) (12, 19461) (13, 29192) (14, 43789) (15, 65684) The domain for this function would be 0≤x≤15 and the range for this function is 150≤y≤65684. The y-intercept of this function is 150. This function is increasing.

4. In five years, a garden festival plans on using the park where Iris has been studying the invasive snails. Explain to the garden festival committee how to find the average rate of change for the snail population between years 2 and 5. Describe what this average rate of change represents. So, to find the average rate of change for a function you use this formula. (F(b)-F(a))/(b-a) When f(b) and f(a) represent the output values, and b and a represent the input values. So this is how it would look for this scenario. (F(5)-F(2))/(5-2) (1139 -337)/3 802/3 267/1 This means that the average rate of change in this function from year 2 to year 5 is about 267. What the means is the population of invasive snails, on average, increased by 267 each year from year 2 to year 5.