To find the exponential function in the form $y = ab^x$ that passes through the points $(0, 10)$ and $(2, 250)$, follow these steps:

1. Using the first point $(0, 10)$:

   Substitute $x = 0$ and $y = 10$ into the equation $y = ab^x$:

   $10 = ab^0$

   Since $b^0 = 1$:

   $10 = a \times 1 \implies a = 10$

2. Using the second point $(2, 250)$:

   Substitute $x = 2$ and $y = 250$ into the equation $y = ab^x$:

   $250 = 10b^2$

   Now, solve for $b^2$:

   $b^2 = \frac{250}{10} = 25$

   Take the square root of both sides:

   $b = 5$

3. Final Exponential Function:

   Substitute the values of $a = 10$ and $b = 5$ back into the equation:

   $y = 10 \cdot 5^x$

So, the exponential function that passes through the points $(0, 10)$ and $(2, 250)$ is:

$\boxed{y = 10 \cdot 5^x}$

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 $y = 10 \cdot 5^x$?
2. How does the base $b = 5$ affect the rate of growth in the function $y = 10 \cdot 5^x$?
3. What is the doubling time for the function $y = 10 \cdot 5^x$?
4. How would you modify the function if it needed to pass through the point $(1, 50)$?
5. How can you use logarithms to solve for $x$ given a specific $y$ value in the function $y = 10 \cdot 5^x$?

Tip: When solving for $b$ in exponential functions, you may need to take the square root or logarithm, depending on the situation. Be careful with the algebraic manipulations to ensure accuracy.