Question

The function f(x) = 4(3)x represents the growth of a dragonfly population every year in a remote swamp. Erin wants to manipulate the formula to an equivalent form that calculates four times a year, not just once a year. Which function is correct for Erin's purpose, and what is the new growth rate?

Answer 1

There are 12 months a year, and if the growth is to be calculated four times a year, the interval would be every three months.

We use the variable
z as the number of 3 months per year

So,
x = z/4

Substituting, the new growth rate would be:
f(z) = 4(3)^(z/4)

Answer 2

`f(x) = 4(1.32)^x` function is correct for Erin's purpose and the new growth rate is 32%

Explanation:

Growth rate function:
`a(1+r)^x`--A

where r is the rate of growth

We are given that The function
`f(x) = 4(3)^x` represents the growth of a dragonfly population every year in a remote swamp.

Now Erin wants to manipulate the formula to an equivalent form that calculates four times a year, not just once a year

So, equation becomes:
`f(x) = 4(3^(1)/(4))^x`

`f(x) = 4(1.32)^x`

Now on comparing with A

1.32=1+r

0.32=r

So, New growth rate = 0.32=32%

Hence
`f(x) = 4(1.32)^x` function is correct for Erin's purpose and the new growth rate is 32%