The problem shows the function:

$y = \left(\frac{1}{3}\right)^x$

This is an exponential function with a base of $\frac{1}{3}$, which is less than 1. Here are some details to help with graphing this type of function:

Key Characteristics of $y = \left( \frac{1}{3} \right)^x$:

1. Decay Behavior: Since the base $\frac{1}{3}$ is between 0 and 1, this is an exponential decay function. As $x$ increases, $y$ will get closer to 0 but never reach it, approaching the x-axis asymptotically.
2. Intercept: When $x = 0$, $y = 1$ (since $\left(\frac{1}{3}\right)^0 = 1$). This means the graph will pass through the point $(0, 1)$.
3. As $x$ Decreases: As $x$ becomes more negative, $y$ will increase rapidly because raising $\frac{1}{3}$ to a negative power results in a larger number. For example, when $x = -1$, $y = 3$, and when $x = -2$, $y = 9$.

General Shape:

The graph will start high on the left (as $x$ approaches $-\infty$), pass through the point $(0, 1)$, and decay toward 0 as $x$ approaches $\infty$.

Choosing the Correct Graph:

1. Look for a graph with a curve that passes through $(0, 1)$.
2. The graph should decrease and approach 0 as $x$ increases, showing exponential decay.
3. It should increase sharply as $x$ becomes negative.

Would you like a detailed explanation of why exponential decay behaves this way, or do you have any specific questions about graphing it?

Related Questions

1. What happens to the graph of $y = b^x$ when $b > 1$ instead of $b < 1$?
2. How does changing the base of an exponential function affect its rate of growth or decay?
3. What are some real-world examples of exponential decay?
4. How would the graph of $y = \left(\frac{1}{3}\right)^{x+1}$ differ from $y = \left(\frac{1}{3}\right)^x$?
5. How does the graph of $y = -\left(\frac{1}{3}\right)^x$ compare to the original function?

Tip:

For exponential functions, remember that bases less than 1 cause decay, while bases greater than 1 cause growth.