To find the exponential function in the form $y = ab^x$ that passes through the points $(0, 12)$ and $(2, 108)$, we need to determine the values of $a$ and $b$.

1. Using the first point $(0, 12)$:

   Substitute $x = 0$ and $y = 12$ into the equation $y = ab^x$:

   $12 = ab^0$

   Since $b^0 = 1$:

   $12 = a \times 1 \implies a = 12$

2. Using the second point $(2, 108)$:

   Substitute $x = 2$ and $y = 108$ into the equation $y = ab^x$:

   $108 = 12b^2$

   Now, solve for $b^2$:

   $b^2 = \frac{108}{12} = 9$

   Take the square root of both sides:

   $b = 3$

3. Final Exponential Function:

   Substitute the values of $a = 12$ and $b = 3$ back into the equation:

   $y = 12 \cdot 3^x$

So, the exponential function that passes through the points $(0, 12)$ and $(2, 108)$ is:

$\boxed{y = 12 \cdot 3^x}$

Would you like more details or have any questions? Here are 5 related questions you might find interesting:

1. What is the value of $y$ when $x = 1$ in the function $y = 12 \cdot 3^x$?
2. How would you graph the function $y = 12 \cdot 3^x$?
3. What happens to the function $y = 12 \cdot 3^x$ as $x$ approaches infinity?
4. How would you modify the function if it needed to pass through the point $(1, 36)$?
5. How can you verify that the function $y = 12 \cdot 3^x$ is correct for the given points?

Tip: When working with exponential functions, the value of $a$ often represents the initial value (when $x = 0$), and $b$ represents the growth or decay factor depending on whether $b > 1$ or $0 < b < 1$.