To find the function that represents the frog population after [tex]\( x \)[/tex] years, let's go through the necessary steps to understand the problem:
1. Initial Population: The problem states that the initial population of frogs when Ginny began her study was 1,200.
2. Rate of Decrease: The population is decreasing at an average rate of 3% per year.
3. Understanding the Decrease Rate:
- A decrease of 3% per year means that each year the population is 97% of what it was the previous year.
- Mathematically, a 3% decrease implies multiplying the population by [tex]\( 1 - 0.03 = 0.97 \)[/tex] each year.
4. Finding the Function:
- Let the population after [tex]\( x \)[/tex] years be represented by the function [tex]\( f(x) \)[/tex].
- Initially (at [tex]\( x = 0 \)[/tex]), the population is 1200.
- After 1 year, the population is [tex]\( 1200 \times 0.97 \)[/tex].
- After 2 years, the population is [tex]\( 1200 \times 0.97^2 \)[/tex].
- Therefore, after [tex]\( x \)[/tex] years, the population is [tex]\( 1200 \times 0.97^x \)[/tex].
Putting this information together, the function that represents the frog population after [tex]\( x \)[/tex] years is:
[tex]\[ f(x) = 1200 \cdot 0.97^x \][/tex]
From the given options, the correct function is:
[tex]\[ f(x) = 1200(0.97)^x \][/tex]
This matches the option:
[tex]\[ f(x) = 1200(0.97)^x \][/tex]
