To determine which equations represent decreasing exponential functions, we need to examine the base of the exponential term in each equation. A decreasing exponential function occurs when the base of the exponential term is between 0 and 1 (exclusive). Let’s analyze each option step by step:

1. $f(x) = 2(1.85)^x$

   • The base of the exponential term is $1.85$, which is greater than 1.
   • This represents an increasing exponential function.
   • Not a decreasing function.

2. $f(x) = 10(64)^{\frac{-1}{3}x}$

   • Rewrite the base: $64^{-\frac{1}{3}} = (64^{\frac{1}{3}})^{-1} = 4^{-1} = \frac{1}{4}$.
   • So the base of the exponential term is $\frac{1}{4}$, which is between 0 and 1.
   • This is a decreasing exponential function.

3. $f(x) = \frac{1}{4}(2)^x$

   • The base of the exponential term is $2$, which is greater than 1.
   • This represents an increasing exponential function.
   • Not a decreasing function.

4. $f(x) = 5\left(\frac{1}{3}\right)^{2x}$

   • The base of the exponential term is $\frac{1}{3}$, which is between 0 and 1.
   • This is a decreasing exponential function.

5. $f(x) = 3\left(\frac{5}{2}\right)^x$

   • The base of the exponential term is $\frac{5}{2} = 2.5$, which is greater than 1.
   • This represents an increasing exponential function.
   • Not a decreasing function.

6. $f(x) = 0.25^{0.5x}$

   • The base of the exponential term is $0.25$, which is between 0 and 1.
   • This is a decreasing exponential function.

Final Answer:

The equations that represent decreasing exponential functions are:

• $f(x) = 10(64)^{\frac{-1}{3}x}$
• $f(x) = 5\left(\frac{1}{3}\right)^{2x}$
• $f(x) = 0.25^{0.5x}$

Would you like further clarification or help with any step? Here are some related questions:

1. What is the significance of the base in exponential functions?
2. How do you determine whether a function is increasing or decreasing?
3. Can an exponential function with a base greater than 1 ever decrease?
4. How does the exponent's coefficient affect the growth/decay of an exponential function?
5. What are some real-life examples of decreasing exponential functions?

Tip: Always rewrite the base of exponential terms in a simpler form (e.g., roots or fractions) to make analysis easier.