https://file.aiquickdraw.com/mathbot/file/1730081930993bobixve2.jpg

To match the given exponential function \( f(x) = 3^{1 - x} \) with its graph, we should analyze the properties of the function. The general form of an exponential function is \( f(x) = a \cdot b^{x} \), where \( a \) is a constant and \( b \) is the base.

### Step-by-Step Analysis:

1. **Rewrite the Function**:
   The function can be rewritten as:
   \[
   f(x) = 3^{1 - x} = 3 \cdot 3^{-x} = 3 \cdot \left(\frac{1}{3}\right)^x
   \]
   This indicates that the function is an exponential decay, where the base \( b \) is \( \frac{1}{3} \), and the initial value (when \( x = 0 \)) is \( 3 \).

2. **Identify Key Features**:
   - At \( x = 0 \), the function value is \( f(0) = 3^{1-0} = 3 \), giving the point \( (0, 3) \).
   - Since \( b = \frac{1}{3} \), the function decreases as \( x \) increases, which indicates a decay behavior.
   - As \( x \to \infty \), \( f(x) \to 0 \), and as \( x \to -\infty \), \( f(x) \to \infty \).

3. **Match with the Graphs**:
   - The graph that starts at \( (0, 3) \) and exhibits exponential decay matches the properties described.

Thus, the correct graph is the second one in the image (top right), which passes through the point \( (0, 3) \).

Would you like further details or have any questions about the solution?

### Related Questions:
1. How does changing the base in an exponential function affect its graph?
2. What happens to the graph of \( f(x) = 3^{1 - x} \) if we change the exponent to \( 3^{1 + x} \)?
3. How would the graph change if the initial value was altered, such as \( f(x) = 2 \cdot 3^{-x} \)?
4. How can you determine whether an exponential function is increasing or decreasing?
5. What are the characteristics of the horizontal asymptote in an exponential decay function?

### Tip:
To quickly identify if an exponential function represents growth or decay, look at the base: if it is greater than 1, it’s growth; if it's between 0 and 1, it’s decay.

Match the exponential function with its graph. Given: f(x) = 3^(1 - x).

Learn how to match the exponential decay function f(x) = 3^(1 - x) to its graph. This solution explains the steps for rewriting and analyzing exponential functions and identifying graph characteristics such as decay and asymptotes. Suitable for high school students studying advanced algebra.

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