To analyze the relationship between the number of parts produced (y) and the number of hours (x), we can look at the data given in the table:
- When \( x = 0 \), \( y = 0 \)
- When \( x = 1 \), \( y = 125 \)
- When \( x = 2 \), \( y = 250 \)
- When \( x = 3 \), \( y = 375 \)
- When \( x = 4 \), \( y = 500 \)
From these observations, we can see that for each additional hour (1 unit increase of \( x \)), the number of parts produced increases by 125. This indicates a constant rate of change.
To find the equation that describes this relationship, we can represent it in the form \( y = mx + b \), where \( m \) is the slope (the rate of change) and \( b \) is the y-intercept.
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The slope \( m \) can be determined from the increase in \( y \) over the increase in \( x \): \[ m = \frac{\Delta y}{\Delta x} = \frac{125}{1} = 125 \]
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Since the machine produces 0 parts when 0 hours are worked, the y-intercept \( b \) is 0.
Putting it all together, the equation becomes: \[ y = 125x \]
This shows that the relationship between \( y \) and \( x \) is proportional because it passes through the origin (0,0) and maintains a constant ratio.
Therefore, the correct answer is: y = 125x, proportional