To model the given data using an exponential function of the form $f(x) = Ab^x$, we need to determine the constants $A$ and $b$. The provided data points are:

$x = 0, f(x) = 600$ $x = 1, f(x) = 1800$ $x = 2, f(x) = 5400$

Step 1: Find $A$

When $x = 0$, the exponential function simplifies to:

$f(0) = A \cdot b^0 = A$

Given that $f(0) = 600$, we have:

$A = 600$

Step 2: Find $b$

Next, we can use the second data point $x = 1, f(1) = 1800$ to solve for $b$. Substituting into the equation $f(x) = 600 \cdot b^x$, we get:

$1800 = 600 \cdot b^1$

Dividing both sides by 600:

$b = \frac{1800}{600} = 3$

Step 3: Verify the function

Now that we have $A = 600$ and $b = 3$, the function is:

$f(x) = 600 \cdot 3^x$

We can verify this with the third data point $x = 2, f(2) = 5400$:

$f(2) = 600 \cdot 3^2 = 600 \cdot 9 = 5400$

This matches the given data.

Final Model:

The exponential function that models the data is:

$f(x) = 600 \cdot 3^x$

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the constant $A$ in an exponential function affect the graph?
2. What role does the base $b$ play in determining the rate of growth?
3. How would you modify this model if the growth was exponential decay?
4. How can you determine an exponential function from a graph without given points?
5. What are the real-world applications of exponential growth functions?

Tip: In exponential functions, the base $b$ determines the growth factor. If $b > 1$, the function represents exponential growth. If $0 < b < 1$, it represents exponential decay.