Question

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 1

f(x) = principal population (1 + growth rate as decimal)^x

Pick a number for the principal population (aka initial amount) and pick a number for the growth rate

Answer 2

i got this but im not sure if im right:
f(x) = 2(5)^x , because i chose 2 to start off with, and the number of growth rate is 5...

Answer 3

or would i have to do it like this f(x) = 2(b)^5 ?

Answer 4

ok that's great. So you started off with 2 snails and your growth rate is actually 400%.
(5 = 1 + 4. The one is for the original amount, and the 4 is for the growth rate.
4 is the decimal form of 400%)

f(x) = 2(5)^x is correct

Answer 5

can you help me with the other parts too please?

Answer 6

what are they?

Answer 7

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 8

You can use this rule
\[(x^m)^n=x^{mn}\]
So \[200(1.03)^{2x}=200(1.03^2)^x\]

Answer 9

then do 1.03² to find the growth rate for this one

Answer 10

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 11

y is the amount of snails and I assume x is time, so they both have to be greater than 0, and since the population is growing the function must be increasing. The y-intercept will be the principal amount (population when x = 0)

Answer 12

last one, 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 13

average rate of change is basically the slope between two point on the function. This is the formula between 2 and 5.
\[aroc=\frac{ f(5)-f(2) }{ 5-2}\]
It would give the change in population over the 3 years.

Answer 14

you just saved my life! and can you make the 3d one a little more clear im struggling to solve the whole thing

Answer 15

which one?

Answer 16

200(1.03)^2x=200(1.032)^x

Answer 17

you have to do 1.03^2 to get the number in parentheses
Then subtract 1 to get the growth rate as a decimal

Answer 18

so would it be f(x) = 200(1.0609)^x

Answer 19

yes

Answer 20

and for the third one since it is growth in this situation, as the number heightens higher than it originally was, The range is [200, infinity) if the domain is [0, infinity) right?

Answer 21

to graph it..

Answer 22

yes. The domain is [0, inf). I suppose there's nothing wrong with making the range start at 200. I'd make is [0, inf) as well because population could technically drop below 200. But I think that's a personal choice.

Answer 23

thank youuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu

Answer 24

you're welcome