Given the points (2,5) and (6,9) write an Exponential equation

To find the exponential equation of the form \( y = ab^x \) that passes through two given points \((2, 5)\) and \((6, 9)\), follow these steps:

1. **Substitute the points into the exponential form**:
   Using \((2, 5)\), we get:
   \[
   5 = ab^2
   \]
   Using \((6, 9)\), we get:
   \[
   9 = ab^6
   \]

2. **Divide the two equations to eliminate \(a\)**:
   \[
   \frac{9}{5} = \frac{ab^6}{ab^2} = b^4
   \]
   So, 
   \[
   b^4 = \frac{9}{5}
   \]
   Taking the 4th root of both sides:
   \[
   b = \left(\frac{9}{5}\right)^{\frac{1}{4}} \approx 1.1665
   \]

3. **Substitute \(b\) into one of the original equations to find \(a\)**:
   Using \(5 = ab^2\) and \(b \approx 1.1665\):
   \[
   5 = a(1.1665)^2 \approx a(1.3607)
   \]
   Solving for \(a\):
   \[
   a = \frac{5}{1.3607} \approx 3.673
   \]

4. **Final Exponential Equation**:
   The exponential equation is approximately:
   \[
   y = 3.673 \cdot (1.1665)^x
   \]

Would you like any further details or explanations?

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Here are 5 related questions:
1. How would the equation change if the points were different?
2. How do you graph an exponential function like this one?
3. What does the base \(b\) represent in an exponential function?
4. How can you use logarithms to solve exponential equations?
5. What is the significance of the constant \(a\) in the equation?

**Tip**: Always check your results by substituting the points back into the final equation to verify.

Discover how to find an exponential equation of the form y = ab^x passing through the points (2,5) and (6,9). This step-by-step solution includes substitution, dividing equations, and solving for constants. Ideal for students in Grades 9-12 learning about exponential functions.

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