The problem is asking which graph represents the function $y = \left( \frac{1}{2} \right)^x$.

Key Steps to Analyze:

1. Function Characteristics:

   • The function $y = \left( \frac{1}{2} \right)^x$ is an exponential decay function.
   • When $x = 0$: $y = 1$, so the graph must pass through the point $(0, 1)$.
   • As $x \to \infty$: $y \to 0$, meaning the graph approaches the $x$-axis but never touches it (asymptote at $y = 0$).
   • As $x \to -\infty$: $y \to \infty$, so the graph grows rapidly.

2. Visual Inspection of Graphs:

   • Graph A matches the behavior of $y = \left( \frac{1}{2} \right)^x$:

     • It passes through $(0, 1)$.
     • It decays toward $y = 0$ for $x > 0$.
     • It grows for $x < 0$.

   • Graph B has incorrect behavior. It does not grow as $x \to -\infty$.

   • Graph C and D are not exponential functions at all. They appear parabolic or have incorrect curvature.

Answer:

The correct graph is A.

Would you like further details or explanations on exponential functions?

5 Related Questions:

1. What is the general form of an exponential function and how do its parameters affect the graph?
2. How do exponential growth and decay differ in terms of their equations and graphs?
3. What happens to the graph of $y = \left( \frac{1}{a} \right)^x$ when $a > 1$?
4. How can you find the horizontal asymptote of an exponential function?
5. How does the base of the exponential function (e.g., $\frac{1}{2}$, $e$, or 2) impact its rate of growth or decay?

Tip:

When identifying exponential decay, remember that the base $0 < b < 1$ causes the graph to decay as $x$ increases.