We are tasked with finding an exponential function of the form:

$y = ab^x$

that passes through the points $(0, 17)$ and $(3, 8704)$.

Step 1: Use the point $(0, 17)$

When $x = 0$, we substitute into the exponential equation:

$17 = ab^0$

Since $b^0 = 1$, we get:

$17 = a$

Thus, the equation becomes:

$y = 17b^x$

Step 2: Use the point $(3, 8704)$

Now, substitute $x = 3$ and $y = 8704$ into the updated equation:

$8704 = 17b^3$

Step 3: Solve for $b$

To find $b$, divide both sides by 17:

$b^3 = \frac{8704}{17} = 512$

Now, solve for $b$ by taking the cube root of 512:

$b = \sqrt[3]{512} = 8$

Step 4: Write the final equation

Now that we know $a = 17$ and $b = 8$, the exponential function is:

$y = 17 \cdot 8^x$

This is the exponential function that passes through the points $(0, 17)$ and $(3, 8704)$.

---

Would you like any further details or have any questions?

Here are 5 related questions to expand on this concept:

1. How do you graph an exponential function like $y = 17 \cdot 8^x$?
2. What is the general effect of increasing or decreasing $b$ in the equation $y = ab^x$?
3. How would the equation change if the point $(3, 8704)$ were different?
4. Can exponential functions model real-world situations? What are some examples?
5. How does an exponential function compare to a linear function in terms of growth?

Tip: When solving for parameters like $b$, always remember that exponentiation is reversible through roots or logarithms.