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.