Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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)

Answer 6

\(\LARGE\color{black}{ \displaystyle \color{darkgoldenrod}{y}=\color{red}{a}(\color{blue}{b})^{\color{green}{x}} }\)

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

Answer 7

So I just input random numbers?

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

I'm still kind of stuck.

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

Ok... so how would I explain it?

Answer 18

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

?

Answer 19

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.

Answer 20

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

Answer 21

Ok. So that would mean a = 0?

Answer 22

no

Answer 23

When x = 0

Answer 24

\(\large\color{black}{ \displaystyle y=5(2)^x }\)

Initially (at year zero - i.e. at x=0)

\(\large\color{black}{ \displaystyle y=5(2)^0 }\)
\(\large\color{black}{ \displaystyle y=5(1) }\)
\(\large\color{black}{ \displaystyle y=5 }\)

Answer 25

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

Answer 26

So there was no increase?

Answer 27

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

Answer 28

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

Answer 29

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

Answer 30

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

Answer 31

sorry

Answer 32

Ok thanks for helping.