Model the data using an exponential function *f*(*x*) = *Abx*.

*x*

0

1

2

*f*(*x*)

600

1,800

5,400

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.

Model the data using an exponential function f(x) = Ab^x.

x:
0, 1, 2

f(x):
600, 1800, 5400

This example demonstrates how to model data using an exponential function f(x) = Ab^x. The given data points are (0, 600), (1, 1800), and (2, 5400). Follow along as we calculate the constants A = 600 and b = 3, creating the final model f(x) = 600 * 3^x. Ideal for students in Grades 9-11.

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