To analyze the given exponential function [tex]\( y = 500(1.3)^{-3x} \)[/tex], follow these steps:
### Step 1: Identify the Type of Change
First, understand the general form of an exponential function:
[tex]\[ y = ab^{kx} \][/tex]
where [tex]\( a \)[/tex] is the initial value, [tex]\( b \)[/tex] is the base of the exponential, [tex]\( k \)[/tex] is the coefficient of [tex]\( x \)[/tex], and [tex]\( x \)[/tex] is the variable.
In our given function:
[tex]\[ y = 500(1.3)^{-3x} \][/tex]
we can identify [tex]\( a = 500 \)[/tex], [tex]\( b = 1.3 \)[/tex], and [tex]\( k = -3 \)[/tex].
### Step 2: Analyze the Base and the Exponent
- The base [tex]\( b = 1.3 \)[/tex] is greater than 1, which would typically imply growth if the exponent were positive.
- However, the exponent [tex]\( -3x \)[/tex] contains a negative coefficient.
When the exponent is negative, it inversely affects the growth factor, causing the function to represent decay rather than growth.
### Step 3: Determine the Percentage Rate of Decrease
To find the percentage rate of decrease:
- The base [tex]\( 1.3 \)[/tex] suggests a 30% increase per unit of [tex]\( x \)[/tex] without considering the negative exponent.
- However, with the exponent being negative, it turns this into a 30% decrease per unit of [tex]\( x \)[/tex].
Calculate the percentage rate of decrease:
[tex]\[ (1.3 - 1) \times 100 = 0.3 \times 100 = 30 \][/tex]
Hence, the function represents exponential decay, with a 30% decrease per unit of [tex]\( x \)[/tex].
### Final Answer
Therefore, the change represents exponential decay with a 30.0% decrease per unit of [tex]\( x \)[/tex].
[tex]\[
\boxed{\text{Decay} \quad 30.0\% \text{ decrease}}
\][/tex]
