Let's determine which function correctly represents the frog population after [tex]\( x \)[/tex] years, given the conditions outlined.
### Step-by-Step Solution:
1. Identify Initial Population:
Ginny started her study with an initial frog population of 1,200. This means at [tex]\( x = 0 \)[/tex] years, the population was [tex]\( 1,200 \)[/tex].
2. Determine Rate of Decrease:
The population decreases at an average rate of [tex]\( 3\% \)[/tex] per year. A [tex]\( 3\% \)[/tex] decrease is equivalent to keeping [tex]\( 97\% \)[/tex] of the population every year. Thus, the decay rate per year is [tex]\( 0.97 \)[/tex].
3. Formulate the Population Function:
In mathematical terms, a population decreasing by a fixed percentage each year can be described with an exponential decay function. The general form of an exponential decay function is:
[tex]\[
f(x) = a \cdot (b)^x
\][/tex]
where:
- [tex]\( a \)[/tex] is the initial population
- [tex]\( b \)[/tex] is the decay factor (in this case, [tex]\( 0.97 \)[/tex], representing [tex]\( 97\% \)[/tex] of the population)
- [tex]\( x \)[/tex] is the number of years
4. Apply Values to the Function:
Plugging the values into the formula, we get:
[tex]\[
f(x) = 1,200 \cdot (0.97)^x
\][/tex]
5. Select the Correct Choice:
We now look at the listed options to find the one that matches our derived function:
- Option 1: [tex]\( f(x) = 1,200(1.03)^x \)[/tex]
- Option 2: [tex]\( f(x) = 1,200(0.03)^x \)[/tex]
- Option 3: [tex]\( f(x) = 1,200(0.97)^x \)[/tex]
- Option 4: [tex]\( f(x) = 1,200(0.97x) \)[/tex]
As per our derivation, the function should be [tex]\( f(x) = 1,200(0.97)^x \)[/tex], which corresponds to Option 3.
### Conclusion:
The function that represents the frog population after [tex]\( x \)[/tex] years is:
[tex]\[
f(x) = 1,200(0.97)^x
\][/tex]
Therefore, the correct choice is Option 3.
