## GalacticPlunder one year ago 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. GalacticPlunder

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.

2. GalacticPlunder

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.

3. GalacticPlunder

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.

4. GalacticPlunder

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.

5. SolomonZelman

EZ! you just need to make an exponential function up - any exponential function you like. An exponential function (abstractly) is in a form of $$\large\color{black}{ \displaystyle y=a(b)^x }$$ I will assume you know: $$\large\color{black}{ \displaystyle {\rm C}^0=1 }$$ (for any non-zero number C)

6. SolomonZelman

$$\LARGE\color{black}{ \displaystyle \color{darkgoldenrod}{y}=\color{red}{a}(\color{blue}{b})^{\color{green}{x}} }$$ $$\normalsize \color{black}{ \displaystyle \color{darkgoldenrod}{\rm Number~of~snalis}=\color{red}{\rm Initial~population}\cdot (\color{blue}{\rm growth~rate})^{\color{green}{\large ~\rm \text{#}~of~years }} }$$

7. GalacticPlunder

So I just input random numbers?

8. SolomonZelman

yes, but a must be positive, and b>1

9. SolomonZelman

So that you don't end up getting a negative population.... and I would advise to choose Natural numbers for a and b.

10. SolomonZelman

natural numbers are: 1 , 2, 3, 4, 5, 6, 7, etc....

11. GalacticPlunder

I'm still kind of stuck.

12. SolomonZelman

try to put random values for a and b (once)

13. GalacticPlunder

Would y=10, a=5, b=2, x=2 work?

14. SolomonZelman

you don't need to plug in anything for x and y.

15. SolomonZelman

So you created a function: $$\large\color{black}{ \displaystyle y=5(2)^x }$$

16. SolomonZelman

Now lets explain to Iris (:O) what is what....

17. GalacticPlunder

Ok... so how would I explain it?

18. GalacticPlunder

Y= Number of snails a=Initial population⋅ b=growth rate x=number of years ?

19. SolomonZelman

Lets do that together. You need to explain what and why is the principal (initial) population of snails & what and why is the rate of growth of the population.

20. SolomonZelman

Ok, will you agree that the initial population is at year zero, or in other words when x=0?

21. GalacticPlunder

Ok. So that would mean a = 0?

22. SolomonZelman

no

23. GalacticPlunder

When x = 0

24. SolomonZelman

$$\large\color{black}{ \displaystyle y=5(2)^x }$$ Initially (at year zero - i.e. at x=0) $$\large\color{black}{ \displaystyle y=5(2)^0 }$$ $$\large\color{black}{ \displaystyle y=5(1) }$$ $$\large\color{black}{ \displaystyle y=5 }$$

25. SolomonZelman

See why the initial/principal population (when x=0) is (equivalent to) 5 (snails) ?

26. GalacticPlunder

So there was no increase?

27. SolomonZelman

There is going to be an increase as years pass by, BUT the population didn't increase right away when started.

28. SolomonZelman

This that we are discussing right now is just the principal population - or from what number of snails did we start.

29. GalacticPlunder

Oh ok. I get it now. So what would be the starting number? And how would I explain my function?

30. SolomonZelman

I will be gone. tag someone else. (I am constructing my basement and bath)

31. SolomonZelman

sorry

32. GalacticPlunder

Ok thanks for helping.