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To solve the problem of identifying the correct equation from each graph, we need to analyze the behavior of the graphs and match them with the given exponential functions. Let's go through each graph step by step.

Graph 1:

- The graph shows a decreasing exponential function.
- As \( x \) increases, the value of \( f(x) \) decreases.
- The base of the exponential function must be between 0 and 1 (i.e., \( 0 < b < 1 \)) for the function to be decreasing.
- The options are:
- \( f(x) = \left(\frac{1}{3}\right)^x \)
- \( f(x) = (3)^x \)
- \( f(x) = \left(\frac{1}{6}\right)^x \)
- \( f(x) = (6)^x \)

Since the function is decreasing, the correct choices are \( f(x) = \left(\frac{1}{3}\right)^x \) or \( f(x) = \left(\frac{1}{6}\right)^x \). To determine which one, we can look at the initial value (when \( x = 0 \)):
- When \( x = 0 \), \( f(0) = 9 \).
- For \( f(x) = \left(\frac{1}{3}\right)^x \), when \( x = 0 \), \( f(0) = \left(\frac{1}{3}\right)^0 = 1 \), which is not correct.
- For \( f(x) = \left(\frac{1}{6}\right)^x \), when \( x = 0 \), \( f(0) = \left(\frac{1}{6}\right)^0 = 1 \), which is not correct either.

However, the graph shows that the function starts at 9 when \( x = -2 \):
- For \( f(x) = \left(\frac{1}{3}\right)^x \), when \( x = -2 \), \( f(-2) = \left(\frac{1}{3}\right)^{-2} = 3^2 = 9 \), which matches the graph.

Thus, the correct equation is:
\[ \boxed{f(x) = \left(\frac{1}{3}\right)^x} \]

Graph 2:

- The graph shows an increasing exponential function.
- As \( x \) increases, the value of \( f(x) \) increases.
- The base of the exponential function must be greater than 1 (i.e., \( b > 1 \)) for the function to be increasing.
- The options are:
- \( f(x) = (1)^x \)
- \( f(x) = \left(\frac{1}{2}\right)^x \)
- \( f(x) = (2)^x \)

Since the function is increasing, the correct choice is \( f(x) = (2)^x \). To confirm, we can look at the initial value (when \( x = 0 \)):
- When \( x = 0 \), \( f(0) = 1 \).
- For \( f(x) = (2)^x \), when \( x = 0 \), \( f(0) = 2^0 = 1 \), which matches the graph.

Thus, the correct equation is:
\[ \boxed{f(x) = (2)^x} \]

Graph 3:

- The graph shows an increasing exponential function.
- As \( x \) increases, the value of \( f(x) \) increases.
- The base of the exponential function must be greater than 1 (i.e., \( b > 1 \)) for the function to be increasing.
- The options are:
- \( f(x) = (1)^x \)
- \( f(x) = (2)^x \)
- \( f(x) = \left(\frac{1}{2}\right)^x \)

Since the function is increasing, the correct choice is \( f(x) = (2)^x \). To confirm, we can look at the initial value (when \( x = 0 \)):
- When \( x = 0 \), \( f(0) = 1 \).
- For \( f(x) = (2)^x \), when \( x = 0 \), \( f(0) = 2^0 = 1 \), which matches the graph.

Thus, the correct equation is:
\[ \boxed{f(x) = (2)^x} \]

Graph 4:

- The graph shows a decreasing exponential function.
- As \( x \) increases, the value of \( f(x) \) decreases.
- The base of the exponential function must be between 0 and 1 (i.e., \( 0 < b < 1 \)) for the function to be decreasing.
- The options are:
- \( f(x) = (5)^x \)
- \( f(x) = (10)^x \)
- \( f(x) = \left(\frac{1}{5}\right)^x \)
- \( f(x) = \left(\frac{1}{10}\right)^x \)

Since the function is decreasing, the correct choices are \( f(x) = \left(\frac{1}{5}\right)^x \) or \( f(x) = \left(\frac{1}{10}\right)^x \). To determine which one, we can look at the initial value (when \( x = 0 \)):
- When \( x = 0 \), \( f(0) = 1 \).
- For \( f(x) = \left(\frac{1}{5}\right)^x \), when \( x = 0 \), \( f(0) = \left(\frac{1}{5}\right)^0 = 1 \), which matches the graph.

Thus, the correct equation is:
\[ \boxed{f(x) = \left(\frac{1}{5}\right)^x} \]

Final Answers:

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