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%
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 exponential decay, we look at the base of the exponent.
In the function \( f(x) = 4(0.72)^x \):
- The base is \( 0.72 \), which is less than 1.
Since the base is between 0 and 1, this indicates that the function represents exponential decay.
Next, we find the percentage rate of change. For an exponential function of the form \( f(x) = a(b)^x \), where \( 0 < b < 1 \), the percentage rate of change can be calculated as:
\[
\text{Rate of change} = (1 - b) \times 100\%
\]
For our function, \( b = 0.72 \):
\[
\text{Rate of change} = (1 - 0.72) \times 100\% = 0.28 \times 100\% = 28\%
\]
Hence, the function \( f(x) = 4(0.72)^x \) is **exponential decay** with a percentage rate of change of **28%**.
Thus, the correct answer is:
**C: Exponential decay; 28%**.
In the function \( f(x) = 4(0.72)^x \):
- The base is \( 0.72 \), which is less than 1.
Since the base is between 0 and 1, this indicates that the function represents exponential decay.
Next, we find the percentage rate of change. For an exponential function of the form \( f(x) = a(b)^x \), where \( 0 < b < 1 \), the percentage rate of change can be calculated as:
\[
\text{Rate of change} = (1 - b) \times 100\%
\]
For our function, \( b = 0.72 \):
\[
\text{Rate of change} = (1 - 0.72) \times 100\% = 0.28 \times 100\% = 28\%
\]
Hence, the function \( f(x) = 4(0.72)^x \) is **exponential decay** with a percentage rate of change of **28%**.
Thus, the correct answer is:
**C: Exponential decay; 28%**.