To find the initial value (y-intercept) of the linear function given the rate of change (slope) \( m = -47 \) and a point on the line \( (14, 3) \), we can use the point-slope form of a linear equation:
\[ y = mx + b \]
Where:
- \( y \) is the output (dependent variable),
- \( m \) is the slope,
- \( x \) is the input (independent variable),
- \( b \) is the y-intercept.
We know \( m = -47 \) and the point \( (14, 3) \), where \( x = 14 \) and \( y = 3 \).
We can substitute these values into the equation:
\[ 3 = -47(14) + b \]
Calculating \( -47(14) \):
\[ -47(14) = -658 \]
So, substituting this into the equation gives:
\[ 3 = -658 + b \]
Now we solve for \( b \):
\[ b = 3 + 658 \] \[ b = 661 \]
Therefore, the initial value (y-intercept) of the linear function \( b \) is 661, which is not listed in your responses. Please verify the given data or options as it seems there may be an error in what options were provided.