To determine the constant of proportionality for a line, we need to verify if the line passes through the origin and has a consistent ratio between $y$ and $x$ coordinates. The constant of proportionality is the slope of the line, calculated as $k = \frac{y}{x}$.

Steps:

1. Confirm the line passes through the origin $(0, 0)$. If it does not, the line does not have a constant proportionality.
2. Calculate the slope ($k$) by selecting any point on the line (apart from the origin, if applicable), and computing $k = \frac{y}{x}$.

Without the actual graph, I can't confirm these values. However, the options suggest possible slopes:

• $k = 2$
• $k = -2$
• $k = \frac{1}{2}$
• $k = -\frac{1}{2}$

To solve:

• Use two points from the graph to calculate $k = \frac{y_2 - y_1}{x_2 - x_1}$ if the origin is not clear.

Would you like me to explain how to interpret the graph or provide a detailed worked example?

Related Questions:

1. What is the geometric meaning of the constant of proportionality?
2. How do you find the slope if given a table of values instead of a graph?
3. How is the constant of proportionality related to direct variation equations?
4. What happens to the constant if the line is horizontal or vertical?
5. Can the constant of proportionality be negative?

Tip:

Always verify the origin's inclusion for proportional relationships—this ensures the ratio $k = \frac{y}{x}$ remains consistent.