Step-by-Step Solution:

a) Select the graph that shows this relationship:

1. Since $y$ is inversely proportional to $x$, the relationship is given by: $y = \frac{k}{x}$ where $k$ is a constant.

2. From the problem, when $x = 2$, $y = \frac{1}{2}$. Substitute these values into the equation to solve for $k$: $\frac{1}{2} = \frac{k}{2}$ Multiply through by $2$: $k = 1$ The equation becomes: $y = \frac{1}{x}$

3. The graph of $y = \frac{1}{x}$ is a hyperbola. Among the given options, Graph A correctly represents this relationship.

b) Find the value of $y$ when $x = 4$:

1. Use the equation $y = \frac{1}{x}$: $y = \frac{1}{4}$

2. Therefore, $y = 0.25$.

Final Answers:

• a) The correct graph is Graph A.
• b) The value of $y$ when $x = 4$ is 0.25.

Would you like more details about how graphs for inverse proportions are analyzed, or have any other questions?

5 Related Questions:

1. What are the properties of graphs representing inverse proportionality?
2. How can you distinguish inverse proportionality from direct proportionality in equations and graphs?
3. What happens to $y$ as $x \to \infty$ in an inversely proportional relationship?
4. How can you determine the constant $k$ in proportionality relationships from different points?
5. How do you solve problems involving joint proportionality (e.g., $y \propto \frac{1}{xz}$)?

Tip:

Always verify the proportionality constant $k$ from given points before analyzing or graphing proportionality relationships!