To determine the linear relationship between the number of roses blooming and the number of days that pass, we can first calculate the slope of the line using the two given points: (2, 30) and (6, 18).
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Calculate the slope (m): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{18 - 30}{6 - 2} = \frac{-12}{4} = -3 \]
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Use the point-slope form of the equation of a line: Using the point (2, 30): \[ y - 30 = -3(x - 2) \] Expanding this: \[ y - 30 = -3x + 6 \ y = -3x + 36 \]
The y-intercept of the equation \( y = -3x + 36 \) is 36. To find the number of blooms at \( x = 0 \) (the y-intercept), we see that when \( x = 0 \):
\[ y = 36 \]
Now, let's find the value of \( y \) when \( x = 8 \):
\[ y = -3(8) + 36 = -24 + 36 = 12 \]
Thus, the equation describes the line that goes through the points \( (0, 36) \) and \( (8, 12) \).
Now we can match the derived points to the options given:
- The first option goes through points \( (0, 34) \) and \( (8, 10) \).
- The second option goes through points \( (0, 36) \) and \( (8, 12) \) (this matches our derived line).
- The third option goes through points \( (0, 38) \) and \( (8, 14) \).
- The fourth option goes through points \( (0, 40) \) and \( (8, 16) \).
Therefore, the correct graph that represents the relationship is the second option: "a coordinate plane with the x-axis labeled time in days and the y-axis labeled number of roses blooming, with a line segment that passes through the points (0, 36) and (8, 12)."