Question

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.

I got - 200(1.03)^2x =200((1.03)^2)^x) =200(1.0609)^x

But i need help with the second part of the question.

the invasive function is - f(x) = 200(.5)^15x

Answer 1

is the invasive function
\[ f(x)= 200(0.5)^{15x} \]
?

Answer 2

yes (:

Answer 3

you know that
\[ 0.5 = \frac{1}{2} \]
and
\[ \frac{1}{2}= 2^{-1} \]
so you can re-write the invasive function as
\[ f(x)= 200(0.5)^{15x} \\ f(x)= 200(2^{-1})^{15x}\\
f(x)= 200(2^{-15x})
\]
notice that the exponent is negative... it makes the function get smaller as x gets bigger. (You get exponential decay)
on the other hand, your first function shows exponential growth.

Answer 4

So how do I compare the key features ?

Answer 5

the key features are whether you get growth or decay. If the base is bigger than 1, then positive exponents give growth, and negative exponents mean decay

Answer 6

you can also say the two populations start out the same size (200)

Answer 7

would it help if i gave you the whole assignment ? So you can see what the first question was since it relates to this one ?

Answer 8

The more info the better.

Answer 9

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
my answer - f(x)= a(b)^x
f(x) = 200(.5)^15x
15 = 15 Years
200 = the amount started with
.5 = the rate of growth each year
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.
200(1.03)^2x =200((1.03)^2)^x) =200(1.0609)^x


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.


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 10

ok, your invasive function needs to be tweaked.
normally, the *rate* is a percent, for example it grows (or decays) at a rate of 3%. the local population is growing at a rate of 6.09% per year.

what rate do you want the invasive species to grow (or decay) per year?

Answer 11

50%

Answer 12

you want it to grow or decay ?

Answer 13

grow

Answer 14

and do you want 50% growth per year ?

Answer 15

yes

Answer 16

that means you write 50% as 0.5, then add 1 to get 1.5

that is the number to use.

\[ f(x) = P(1.5)^x \]
where P is the starting population. I would pick a small number for P because it sounds more realistic that only a few of the critters show up.. rather than an army of them.

Answer 17

but where would the 15 years go ?

Answer 18

Is there more to this question? You put in 15 years as part of the answer, but it is not needed... unless there is more to this than you posted.