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ALGEBRA 2 HELP! 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?


f(x) = 4(3)^x; growth rate 300%


f(x) = 4(3)^x, growth rate 4%


f(x) = 4(1.32)^x; growth rate 4%


f(x) = 4(1.32)4^x; growth rate 32%

Respuesta :

mreijepatmat
f(x) = 4.(3)ˣ. This is the growth of a dragonfly population i ONE year. 

4 is a constant and 3 is the growth pattern over ONE year, then we can write it:

f(x) = 4.(3¹)ˣ , where the exponent 1 (on 3) represents the cycle of growth with is ( once a year).

Now if we want to calculate the growth 4 times a year, then the equation becomes:
 f(x) = 4.[3^(1/4)]ˣ

f(x) = 4[1.3160]ˣ  OR rounding up f(x) = 4.(1.32)ˣ with a growth rate of 32%

The correct function for Erin's purpose is [tex]\bold{f(x) = 4(1.32)^{4x}}[/tex] and the new growth rate is 32%

What is exponential growth?

" In exponential growth a value increases in proportion to its current value. Such as always doubling."

Exponential growth equation:

[tex]f(x) = a(1 + r)^{t}[/tex]

where a is the initial value

r is the growth rate

t is the time

For given example,

The function [tex]f(x) = 4(3)^x[/tex] 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.

So the function becomes,

[tex]\Rightarrow f(x) = 4(3^{\frac{1}{4} })^{4x}\\\\\Rightarrow f(x) = 4(1.316)^{4x}\\\\\Rightarrow f(x) = 4(1+0.316)^{4x}\\[/tex]

By comparing above function with exponential growth function we have,

⇒ 1 + r = 1 + 0.316

⇒ r = 0.316

⇒ r ≈ 0.32

The percentage growth rate is,

⇒ r = 32%

Therefore, the correct function is [tex]\bold{f(x) = 4(1.32)^{4x}}[/tex] with the new growth rate of 32%

Learn more about the growth rate here:

https://brainly.com/question/9883435

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