The problem asks to find the constant of proportionality $k$ based on the graph provided. Here's the solution:

Step 1: Understand the Relationship

The graph shows a straight line passing through the origin, which suggests a proportional relationship of the form:

$y = kx$

Here, $k$ is the constant of proportionality (the slope of the line).

Step 2: Determine the Slope $k$

To find $k$, choose two points on the line. From the graph:

• First point: $(2, 6)$
• Second point: $(6, 2)$

The slope $k$ is calculated using the formula:

$k = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the coordinates: $k = \frac{2 - 6}{6 - 2} = \frac{-4}{4} = -1$

Final Answer:

The constant of proportionality is: $k = -1$

Let me know if you'd like a detailed explanation of any step!

5 Related Questions:

1. How does the constant of proportionality relate to the slope of a line?
2. What happens if $k$ is positive instead of negative?
3. How can you check if two variables are directly proportional?
4. Why does the graph of proportional relationships always pass through the origin?
5. How do you interpret the meaning of $k = -1$ in a real-world context?

Tip:

When working with proportional relationships, always verify the points by plugging them back into the equation $y = kx$.