Determine whether the function below is exponential growth or exponential decay, and find the percentage rate of change.

Question

f(x)=4(0.72)^x



A
Exponential decay; 72%

B
Exponential growth; 72%

C
Exponential decay; 28%

D
Exponential growth; 28%

Answer 1

To determine whether the function \( f(x) = 4(0.72)^x \) represents exponential growth or decay, we need to examine the base of the exponent, which is \( 0.72 \).

1. Identify Growth or Decay:

   • If the base (the number being raised to the power of \( x \)) is between 0 and 1, it indicates exponential decay.
   • Here, \( 0.72 < 1 \), confirming that this function represents exponential decay.

2. Calculate the Percentage Rate of Change:

   • In the case of exponential decay, the percentage rate of change can be calculated as \( (1 - \text{base}) \times 100% \).
   • Here, the base is \( 0.72 \): \[ \text{Rate of Change} = (1 - 0.72) \times 100% = 0.28 \times 100% = 28% \]

Based on the analysis, the function \( f(x) = 4(0.72)^x \) represents exponential decay with a percentage rate of change of 28%.

Thus, the correct answer is:

C: Exponential decay; 28%