Question
say growth or decay and percent increase or decay y=9^(-3x)
Answers
GPT-4o mini
To analyze the function \( y = 9^{-3x} \), we need to determine if it represents growth or decay, as well as the percent increase or decay.
1. **Growth or Decay:**
- The function \( y = 9^{-3x} \) can be rewritten as \( y = \frac{1}{9^{3x}} \). This indicates that as \( x \) increases, the value of \( y \) decreases. Therefore, the function represents **decay**.
2. **Percent Decay:**
- To find the rate of decay, we can express \( y = 9^{-3x} \) in a form that looks like \( y = a(1 - r)^x \), where \( r \) is the decay rate. In this case, we can determine that the base \( 9^{-3} \) can be expressed in terms of \( e \) or as a specific decay percentage.
- Since \( 9 = 3^2 \), we can rewrite \( y \) as \( y = (3^2)^{-3x} = 3^{-6x} \).
To get the decay percentage, note that each increase in \( x \) results in a factor of \( 9^{-3} \) for every unit change in \( x \).
Specifically:
\[
y = 9^{-3x} = e^{(-3x \ln(9))}
\]
Therefore, if we analyze the decay factor:
\[
y = a e^{kt}
\]
From this, \( k \) represents the rate of decay. Here, the decay constant over one unit of \( x \) (or each increment in \( x \)) can be approximated.
- The decay factor for each unit change in \( x \) is calculated from \( 9^{-3} \):
\[
9^{-3} = \frac{1}{729}
\]
This indicates that for each 1 unit increase in \( x \), \( y \) becomes approximately \( \frac{1}{729} \) of its previous value.
To find the percent decay:
\[
\text{Percent Decay} = \left(1 - \frac{1}{729}\right) \times 100 \approx 99.86\%
\]
Thus, the conclusion is:
- The function \( y = 9^{-3x} \) represents **decay**.
- It has a **percent decay** of approximately **99.86%** for each unit increase in \( x \).
1. **Growth or Decay:**
- The function \( y = 9^{-3x} \) can be rewritten as \( y = \frac{1}{9^{3x}} \). This indicates that as \( x \) increases, the value of \( y \) decreases. Therefore, the function represents **decay**.
2. **Percent Decay:**
- To find the rate of decay, we can express \( y = 9^{-3x} \) in a form that looks like \( y = a(1 - r)^x \), where \( r \) is the decay rate. In this case, we can determine that the base \( 9^{-3} \) can be expressed in terms of \( e \) or as a specific decay percentage.
- Since \( 9 = 3^2 \), we can rewrite \( y \) as \( y = (3^2)^{-3x} = 3^{-6x} \).
To get the decay percentage, note that each increase in \( x \) results in a factor of \( 9^{-3} \) for every unit change in \( x \).
Specifically:
\[
y = 9^{-3x} = e^{(-3x \ln(9))}
\]
Therefore, if we analyze the decay factor:
\[
y = a e^{kt}
\]
From this, \( k \) represents the rate of decay. Here, the decay constant over one unit of \( x \) (or each increment in \( x \)) can be approximated.
- The decay factor for each unit change in \( x \) is calculated from \( 9^{-3} \):
\[
9^{-3} = \frac{1}{729}
\]
This indicates that for each 1 unit increase in \( x \), \( y \) becomes approximately \( \frac{1}{729} \) of its previous value.
To find the percent decay:
\[
\text{Percent Decay} = \left(1 - \frac{1}{729}\right) \times 100 \approx 99.86\%
\]
Thus, the conclusion is:
- The function \( y = 9^{-3x} \) represents **decay**.
- It has a **percent decay** of approximately **99.86%** for each unit increase in \( x \).