To find the exponential function in the form $y = ab^x$ that goes through the points $(0, 13)$ and $(3, 2808)$, follow these steps:

Step 1: Use the point $(0, 13)$

When $x = 0$: $y = ab^0 = a \cdot 1 = a$ This means $a = 13$. The equation now becomes: $y = 13b^x$

Step 2: Use the point $(3, 2808)$ to solve for $b$

Substitute $x = 3$ and $y = 2808$ into the equation: $2808 = 13b^3$ Divide both sides by 13: $b^3 = \frac{2808}{13} = 216$ Take the cube root of both sides: $b = \sqrt[3]{216} = 6$

Step 3: Write the final equation

Now that $a = 13$ and $b = 6$, the exponential function is: $y = 13 \cdot 6^x$

Final Answer:

$y = 13 \cdot 6^x$

Would you like a detailed explanation of any step or further examples?

Relative Questions:

1. How do we verify if the exponential function passes through other points?
2. What happens if the points are not on the same exponential curve?
3. Can we generalize this process for other pairs of points?
4. What if the value of $b$ is not a whole number?
5. How can exponential functions model real-world problems?

Tip: Always use one point to find $a$, and the second point to solve for $b$ when determining exponential equations.