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The problem states that \( y \) is inversely proportional to \( x \), and when \( x = 2 \), \( y = \frac{1}{2} \).

### Step-by-step breakdown:

1. **Inverse proportionality**: Since \( y \) is inversely proportional to \( x \), we can write the relationship as:
   \[
   y = \frac{k}{x}
   \]
   where \( k \) is the constant of proportionality.

2. **Finding \( k \)**: We know that when \( x = 2 \), \( y = \frac{1}{2} \). Substituting these values into the equation:
   \[
   \frac{1}{2} = \frac{k}{2}
   \]
   Solving for \( k \):
   \[
   k = 1
   \]
   So, the equation becomes:
   \[
   y = \frac{1}{x}
   \]

### Part (a): Selecting the correct graph

The equation \( y = \frac{1}{x} \) is a hyperbolic graph. From the options in the image:

- **The correct graph** is the one where \( y \) decreases as \( x \) increases, showing an inverse relationship. That matches **graph C**, which shows this hyperbolic behavior.

### Part (b): Finding \( y \) when \( x = 4 \)

Substitute \( x = 4 \) into the equation \( y = \frac{1}{x} \):
\[
y = \frac{1}{4}
\]
So, \( y = 0.25 \) when \( x = 4 \).

### Summary of answers:
- **a)** The correct graph is **C**.
- **b)** The value of \( y \) when \( x = 4 \) is \( 0.25 \).

Would you like more details or further clarifications?

Here are some questions to expand on this topic:
1. What does it mean for two quantities to be inversely proportional?
2. How would the graph change if the constant \( k \) were different?
3. What are some real-life examples of inverse proportionality?
4. How do you determine whether a relationship is linear or inverse from a graph?
5. What happens to \( y \) as \( x \) approaches 0 in an inverse relationship?

**Tip**: Inverse proportionality can be identified graphically by a hyperbolic curve where one variable increases as the other decreases.

y is inversely proportional to x and when x = 2, y = 1/2. Select the graph that shows this relationship correctly and find the value of y when x = 4.

Understand the concept of inverse proportionality through graph selection and calculations. This problem helps students interpret graphs and apply the inverse proportional formula y = k/x. Suitable for Grade 8-9 students.

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