To determine the function that represents the frog population after [tex]\( x \)[/tex] years, we need to account for the fact that the population is decreasing at an average rate of 3% per year.
Here's a step-by-step breakdown:
1. Initial Population: The initial population of frogs when Ginny began her study is 1,200.
2. Annual Decrease Rate: The population is decreasing by 3% per year, which means the population retains 97% of its size each year. Mathematically, this means that each year, the population is multiplied by [tex]\( 1 - 0.03 = 0.97 \)[/tex].
3. Population After One Year: After one year, the population would be:
[tex]\[
\text{Population after 1 year} = 1,200 \times 0.97
\][/tex]
4. Population After Two Years: The population after two years would be:
[tex]\[
\text{Population after 2 years} = 1,200 \times 0.97 \times 0.97 = 1,200 \times (0.97)^2
\][/tex]
5. Generalizing for [tex]\( x \)[/tex] Years: To generalize, after [tex]\( x \)[/tex] years, the population can be calculated using the formula:
[tex]\[
\text{Population after } x \text{ years} = 1,200 \times (0.97)^x
\][/tex]
Given the choices:
- [tex]\( f(x) = 1,200 (1.03)^x \)[/tex]: This function represents a population increasing by 3%, which is incorrect.
- [tex]\( f(x) = 1,200 (0.03)^x \)[/tex]: This function incorrectly represents just the decrease rate, not the remaining population.
- [tex]\( f(x) = 1,200 (0.97)^x \)[/tex]: This function correctly represents the population decreasing by 3% per year.
- [tex]\( f(x) = 1,200 (0.97x) \)[/tex]: This function is incorrect because the multiplier [tex]\( 0.97 \)[/tex] should be raised to the power of [tex]\( x \)[/tex], not multiplied by [tex]\( x \)[/tex].
The correct function that represents the frog population after [tex]\( x \)[/tex] years is:
[tex]\[
f(x) = 1,200 (0.97)^x
\][/tex]
