Question

ginny is studying a population of frogs. she determines that the population is decreasing at an average rate of 3% per year. when she began her study, the frog population was estimated at 1,200. which function represents the frog population after x years?
\\( f(x) = 1,200(1.03)^x \\)
\\( f(x) = 1,200(0.03)^x \\)
\\( f(x) = 1,200(0.97)^x \\)
\\( f(x) = 1,200(0.97x) \\)

Explanation:

Step1: Recall exponential decay formula

The general formula for exponential decay is \( f(x)=a(1 - r)^x \), where \( a \) is the initial amount, \( r \) is the rate of decrease (in decimal), and \( x \) is the time.

Step2: Identify values of \( a \) and \( r \)

Here, the initial population \( a = 1200 \), and the rate of decrease \( r=3\% = 0.03 \).

Step3: Substitute values into the formula

Substitute \( a = 1200 \) and \( r = 0.03 \) into the exponential decay formula: \( f(x)=1200(1 - 0.03)^x=1200(0.97)^x \).

Answer:

\( f(x) = 1,200(0.97)^x \)