Write an exponential function in the form y, equals, a, b, to the power xy=ab^x
that goes through the points (0,12) and (2,108)

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 \(0 < b < 1\).

Learn how to find the exponential function y = 12 * 3^x that passes through the points (0, 12) and (2, 108). Understand the process of determining values for a and b using exponential function properties. Suitable for high school students studying exponential functions.

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