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.