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

*x*

0

1

2

*f*(*x*)

50

30

18

*f*(*x*) = 

To model the data using an exponential function \( f(x) = Ab^x \), we need to determine the constants \( A \) and \( b \) based on the values provided:

Given dataTo model the data using an exponential function \( f(x) = Ab^x \), we need to determine the constants \( A \) and \( b \) based on the values provided:

Given 
- When \( x = 0 \), \( f(x) = 50 \).
- When \( x = 1 \), \( f(x) = 30 \).
- When \( x = 2 \), \( f(x) = 18 \).

### Step 1: Determine \( A \)
Since \( f(0) = 50 \), and substituting \( x = 0 \) in \( f(x) = Ab^x \), we get:
\[
f(0) = A \cdot b^0 = A = 50
\]
So, \( A = 50 \).

### Step 2: Solve for \( b \)
Now that \( A = 50 \), we substitute this into the equations for \( f(1) \) and \( f(2) \) to find \( b \).

1. For \( f(1) = 30 \):
   \[
   50 \cdot b^1 = 30
   \]
   \[
   b = \frac{30}{50} = 0.6
   \]

2. Verifying with \( f(2) = 18 \):
   \[
   50 \cdot (0.6)^2 = 50 \cdot 0.36 = 18
   \]
   This matches the given value, confirming that \( b = 0.6 \) is correct.

### Final Model
The exponential function that models the data is:
\[
f(x) = 50 \cdot (0.6)^x
\]

Would you like further details on the steps, or have any other questions?

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Here are five related questions for further exploration:
1. How would the model change if the values of \( f(x) \) were different?
2. What does the constant \( A \) represent in an exponential model?
3. How can we interpret the decay factor \( b \) in the context of this function?
4. How would we use this function to predict \( f(x) \) for values of \( x \) not in the table?
5. How would the model differ if we had an exponential growth rather than decay?

**Tip:** Exponential models are useful for data showing rapid growth or decay and can be transformed to linear models by taking logarithms, simplifying calculations in some cases.

Model the data using an exponential function f(x) = Ab^x.
x
0
1
2
f(x)
50
30
18
f(x) = 

Learn how to model data points using the exponential function f(x) = Ab^x. This example involves data points where f(0) = 50, f(1) = 30, and f(2) = 18, providing a step-by-step approach to finding constants A and b for Grades 10-12.

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