To determine which function best models the relationship between the number of months since the start of the build (x) and the percentage of the house still left to build (y), we need to analyze the expected behavior of the data.
Since we are tracking the percentage of the house still left to build, we would expect that as time (in months) increases, the percentage of the house still left to build decreases. This indicates a negative relationship, meaning the slope of the linear function should be negative.
From the options given:
a. \( y = -13.5x + 97.8 \) - This has a negative slope, suggesting a decrease in percentage as x increases, which is consistent with our expectations.
b. \( y = -13.5x + 7.3 \) - This also has a negative slope, suggesting the same decrease as x increases.
c. \( y = 97.8x - 13.5 \) - This has a positive slope, which is not consistent with the expected behavior (percentage should decrease).
d. \( y = 7.3x - 97.8 \) - This also has a positive slope, which again is not consistent with the expected behavior.
Now, between the two valid choices (a and b), we should consider the y-intercept. The y-intercept represents the percentage left to build when x (time in months) is 0.
- Option a suggests that when the build starts (0 months), 97.8% of the house is still left to build.
- Option b suggests that only 7.3% would be left at start, which would not make sense.
Given that 97.8% of the house should still be left to build at the starting point, option a seems the most logical.
Thus, the function that best models the data is:
a. \( y = -13.5x + 97.8 \)