Question

exponential decay functions
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(0.97x) )

( f(x) = 1,200(0.03)^x )

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

Explanation:

Step1: Recall exponential decay formula

The general form of an exponential decay function is \( f(x) = a(1 - r)^x \), where \( a \) is the initial amount, \( r \) is the rate of decay (as a decimal), and \( x \) is the time.

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

Here, the initial population \( a = 1200 \), and the decay rate \( r = 3\% = 0.03 \). So \( 1 - r = 1 - 0.03 = 0.97 \).

Step3: Substitute into the formula

Substituting \( a = 1200 \) and \( 1 - r = 0.97 \) into the exponential decay formula, we get \( f(x) = 1200(0.97)^x \).

Answer:

\( f(x) = 1,200(0.97)^x \) (the last option)