Question
Determine whether the function below is exponential growth or exponential decay, and find the percentage rate of change.
f(x)=4(0.72)^x
A
Exponential decay; 72%
B
Exponential growth; 72%
C
Exponential decay; 28%
D
Exponential growth; 28%
f(x)=4(0.72)^x
A
Exponential decay; 72%
B
Exponential growth; 72%
C
Exponential decay; 28%
D
Exponential growth; 28%
Answers
GPT-4o mini
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%**
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%**