We are given two points: $(2, 336)$ and $(3, 2688)$, and we need to find an exponential function of the form:

$y = a \cdot b^x$

Step 1: Use the points to form a system of equations

From the general form $y = a \cdot b^x$, we can substitute the given points into this equation to form two equations.

1. For the point $(2, 336)$: $336 = a \cdot b^2$

2. For the point $(3, 2688)$: $2688 = a \cdot b^3$

Step 2: Solve the system of equations

Now, we can solve this system of equations.

Divide the second equation by the first to eliminate $a$:

$\frac{2688}{336} = \frac{a \cdot b^3}{a \cdot b^2}$

Simplifying: $8 = b$

So, $b = 8$.

Step 3: Solve for $a$

Now that we know $b = 8$, substitute this value back into one of the original equations. Let's use $336 = a \cdot b^2$ (from point $(2, 336)$):

$336 = a \cdot 8^2$ $336 = a \cdot 64$

Solving for $a$: $a = \frac{336}{64} = 5.25$

Step 4: Write the final equation

Now that we have $a = 5.25$ and $b = 8$, the exponential function is:

$y = 5.25 \cdot 8^x$

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Do you need further clarification or have any questions about this process?

Here are some related questions:

1. How do exponential growth and decay models differ?
2. How would the equation change if the base $b$ were a fraction?
3. What is the interpretation of the constant $a$ in this exponential function?
4. How can we model a real-world situation with this exponential function?
5. How can we find the value of the function at another point, say $x = 4$?

Tip: In exponential functions, the base $b$ determines how quickly the function grows or decays. If $b > 1$, the function grows exponentially.