To find the value of $f(6.5)$ for the exponential function $f(x) = ab^x$, we need to determine the values of $a$ and $b$ using the given points $f(-5) = 9$ and $f(4.5) = 56$.

Step 1: Set up the equations

Using the given points:

1. $f(-5) = ab^{-5} = 9$
2. $f(4.5) = ab^{4.5} = 56$

Step 2: Solve for $b$

Divide the second equation by the first equation to eliminate $a$:

$\frac{ab^{4.5}}{ab^{-5}} = \frac{56}{9}$

This simplifies to:

$b^{4.5 + 5} = \frac{56}{9}$

$b^{9.5} = \frac{56}{9}$

Now, solve for $b$:

$b = \left(\frac{56}{9}\right)^{\frac{1}{9.5}}$

Step 3: Calculate $b$

Let's compute this value:

$b \approx \left(\frac{56}{9}\right)^{\frac{1}{9.5}} \approx 1.1772$

Step 4: Solve for $a$

Now that we have $b$, use one of the original equations to solve for $a$. Let's use $f(-5) = 9$:

$9 = a \times 1.1772^{-5}$

Solve for $a$:

$a \approx 9 \times (1.1772^{-5}) \approx 5.146$

Step 5: Find $f(6.5)$

Finally, substitute $a$ and $b$ back into the function to find $f(6.5)$:

$f(6.5) = 5.146 \times 1.1772^{6.5}$

$f(6.5) \approx 5.146 \times 2.367 \approx 12.18$

So, the value of $f(6.5)$ to the nearest hundredth is 12.18.

Would you like further details on any of these steps, or do you have any other questions? Here are some related questions you might find interesting:

1. How does the growth rate affect the value of an exponential function?
2. What are the properties of exponential functions?
3. How can we verify the solution using logarithms?
4. What are some real-life examples of exponential functions?
5. How does changing the base $b$ affect the graph of the exponential function?

Tip: When solving exponential functions, dividing equations is a useful method to eliminate one of the variables, simplifying the problem significantly.