To determine whether the given equation represents exponential growth or exponential decay, let's take a detailed step-by-step approach.
The given equation is:
[tex]\[
y = \left(\frac{11}{30}\right) \cdot \left(\frac{22}{19}\right)^{-x}
\][/tex]
1. Identify the base of the exponent:
The base of the exponent in the given equation is [tex]\(\frac{22}{19}\)[/tex]. However, the equation has a negative exponent, [tex]\(-x\)[/tex].
2. Understand the effect of the negative exponent:
A negative exponent means that we take the reciprocal of the base. Thus, the term [tex]\(\left(\frac{22}{19}\right)^{-x}\)[/tex] can be rewritten as:
[tex]\[
\left(\frac{22}{19}\right)^{-x} = \left(\frac{19}{22}\right)^{x}
\][/tex]
3. Determine the value of the new base:
After taking the reciprocal, the new base is [tex]\(\frac{19}{22}\)[/tex].
4. Compare the new base to 1:
- If the base is greater than 1, it indicates exponential growth.
- If the base is less than 1, it indicates exponential decay.
Since [tex]\(\frac{19}{22} < 1\)[/tex], we see that the base is less than 1.
Hence, the equation:
[tex]\[
y = \left(\frac{11}{30}\right) \cdot \left(\frac{19}{22}\right)^{x}
\][/tex]
With a base ([tex]\(\frac{19}{22}\)[/tex]) less than 1, this equation represents exponential decay.
Conclusion:
The given equation represents exponential decay.
