To derive the equation of the line in slope-intercept form \( y = mx + b \) using the provided points (1, -1) and (0, -4), we need to calculate the slope \( m \) first.
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (1, -1) and (0, -4):
\((x_1, y_1) = (1, -1)\)
\((x_2, y_2) = (0, -4)\)
Calculating the slope:
\[ m = \frac{-4 - (-1)}{0 - 1} = \frac{-4 + 1}{-1} = \frac{-3}{-1} = 3 \]
Now that we have the slope \( m = 3 \), we can use one of the points to find \( b \), the y-intercept. We can use the point (0, -4):
Since the y-intercept \( b \) occurs where \( x = 0 \), we already have that \( y = -4\) when \( x = 0 \).
Thus, we can write the equation with the slope and y-intercept:
\[ y = 3x - 4 \]
This corresponds to option D: \( y = 3x - 4 \).