1. **Problem Statement:** We are given that $y$ is inversely proportional to $x$, and when $x=2$, $y=\frac{1}{2}$. We need to find the graph that represents this relationship and then find $y$ when $x=4$. 2. **Formula and Explanation:** If $y$ is inversely proportional to $x$, then: $$y = \frac{k}{x}$$ where $k$ is a constant. 3. **Find the constant $k$:** Given $x=2$ and $y=\frac{1}{2}$, substitute these values: $$\frac{1}{2} = \frac{k}{2}$$ Multiply both sides by 2: $$k = 1$$ 4. **Equation of the relationship:** $$y = \frac{1}{x}$$ 5. **Graph selection:** The graph of $y=\frac{1}{x}$ is a hyperbola that decreases as $x$ increases, approaching zero but never touching the axes. Among the given graphs, the one that shows this behavior with $k=1$ is the green curve (Graph A) starting near $y=4$ at $x=0$ and decreasing smoothly. 6. **Find $y$ when $x=4$:** Substitute $x=4$ into the equation: $$y = \frac{1}{4}$$ **Final answers:** - a) The graph that shows this relationship is Graph A (green curve). - b) When $x=4$, $y=\frac{1}{4}$.