To determine whether the given equation [tex]\( y = 0.54 \cdot (0.52)^{-1.7x} \)[/tex] represents exponential growth or exponential decay, we should carefully analyze the components of the equation and their implications.
1. Identify the Base of the Exponential Expression:
The given equation is [tex]\( y = 0.54 \cdot (0.52)^{-1.7x} \)[/tex].
Here, the base of the exponential term is [tex]\( 0.52 \)[/tex].
2. Consider the Exponent Factor:
The exponent is given by [tex]\( -1.7x \)[/tex].
Analyzing this exponent involves understanding how it affects the base:
- If the base of the exponentiation (here [tex]\( 0.52 \)[/tex]) is between 0 and 1, and the exponent is positive, the term [tex]\( (0.52)^{-1.7x} \)[/tex] will grow smaller, representing exponential decay.
3. Evaluate the Entire Exponential Term:
Since the exponent is [tex]\( -1.7x \)[/tex], let's rewrite it to make the analysis clearer:
- [tex]\( (0.52)^{-1.7x} \)[/tex] can be rewritten using properties of exponents as [tex]\( \left(\frac{1}{0.52}\right)^{1.7x} \)[/tex].
- The term [tex]\( \frac{1}{0.52} > 1 \)[/tex]. Hence, the term [tex]\(\left(\frac{1}{0.52}\right)^{1.7x} \)[/tex] represents a growing function as [tex]\( x \)[/tex] increases.
4. Combine with the Coefficient:
The coefficient [tex]\( 0.54 \)[/tex] is a positive multiplier. It does not change the nature of growth or decay but scales the function.
5. Conclusion on the Nature:
Since the base [tex]\( 0.52 \)[/tex] with the exponent [tex]\( -1.7x \)[/tex] ultimately translates to [tex]\( \left(\frac{1}{0.52}\right)^{1.7x} \)[/tex], which grows as [tex]\( x \)[/tex] increases, it indicates an increasing function when transformed due to the negative exponent context.
Therefore, the given equation [tex]\( y = 0.54 \cdot (0.52)^{-1.7x} \)[/tex] represents exponential decay.
