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.