To solve the equation for the exponential function $f(x) = a \cdot b^x$ using the given table data, follow these steps:

Given the points:

• $(0, 9)$
• $(1, 15)$

1. Find $a$: Since $f(0) = a \cdot b^0 = a$, and from the table, $f(0) = 9$: $a = 9$

2. Find $b$: Use the second point $(1, 15)$: $f(1) = a \cdot b^1 = 9 \cdot b = 15$ Solve for $b$: $b = \frac{15}{9} = \frac{5}{3}$

Now that we have $a = 9$ and $b = \frac{5}{3}$, the equation for $f(x)$ is: $f(x) = 9 \cdot \left( \frac{5}{3} \right)^x$

Would you like more details or have any questions?

Here are some related questions you might consider:

1. How do you solve for $b$ in an exponential function given two points?
2. What are the steps to verify if an exponential function fits given data points?
3. How can you determine if a function is exponential based on a table of values?
4. What are common mistakes to avoid when solving exponential functions?
5. How can exponential functions be applied in real-world scenarios?

Tip: When solving for $b$ in an exponential function, always check your arithmetic carefully to ensure accuracy.