Write an exponential function in the form y, equals, a, b, to the power x*y*=*ab^x* that goes through the points (0,10) and (2,250)

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.

Learn how to find an exponential function y = ab^x that passes through the points (0, 10) and (2, 250). Step-by-step solution provided, including determination of constants a and b. Suitable for Grades 10-12 students.

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