To derive the equation of the line \( y = mx \) through the origin from a proportional relationship, you need to determine the slope \( m \). The slope \( m \) represents how much \( y \) changes for a given change in \( x \).
To choose the correct response from the options you provided, you would examine the graph to find the values of \( y \) corresponding to specific values of \( x \). The slope \( m \) can also be found by selecting two points on the line and using the slope formula:
\[ m = \frac{\Delta y}{\Delta x} \]
In this case, without the specific graph, you would select the option that fits the proportional relationship represented in the graph. If, for example, you find from the graph that:
- At \( x = 1 \), \( y = 8 \), then \( m = 8 \); so the equation would be \( y = 8x \).
- At \( x = 1 \), \( y = 2 \), then \( m = 2 \); so the equation would be \( y = 2x \).
- At \( x = 1 \), \( y = 4 \), then \( m = 4 \); so the equation would be \( y = 4x \).
- At \( x = 1 \), \( y = 12 \), then \( m = 12 \); so the equation would be \( y = 12x \).
Please examine the specific graph you have for the values and choose the corresponding equation. If you provide those specific point values from the graph, I could help identify the correct equation.