To determine whether the change represented by the exponential function [tex]\( y = 72(0.913)^x \)[/tex] indicates growth or decay, and to calculate the percentage rate of change, we follow these steps:
### Step 1: Identify the Base of the Exponential Function
The base of the exponential function is the value that is raised to the power of [tex]\( x \)[/tex]. In the given function [tex]\( y = 72(0.913)^x \)[/tex], the base is [tex]\( 0.913 \)[/tex].
### Step 2: Determine Growth or Decay
To identify whether the function represents growth or decay, we examine the base:
- If the base is greater than 1, the function represents exponential growth.
- If the base is between 0 and 1, the function represents exponential decay.
In this case, the base is [tex]\( 0.913 \)[/tex]. Since [tex]\( 0.913 \)[/tex] is less than 1 (but greater than 0), the function represents exponential decay.
### Step 3: Calculate the Percentage Rate of Change
The percentage rate of change can be determined by calculating how much the base deviates from 1:
1. Subtract the base from 1:
[tex]\[
1 - 0.913 = 0.087
\][/tex]
2. Convert the deviation to a percentage by multiplying by 100:
[tex]\[
0.087 \times 100 = 8.7\%
\][/tex]
Since the function represents decay, this 8.7% is the rate of decrease.
### Final Answer
- The change represented by the function is exponential decay.
- The percentage rate of decrease is [tex]\( 8.7\% \)[/tex].
Thus, the correct answer is:
- Decay
- 8.7% decrease
