To find the initial value (y-intercept) \( b \) of the linear function given the slope \( m \) and a point \( (x, y) \), you can use the point-slope form of a linear equation:
\[ y = mx + b \]
In this case, you have:
- \( m = -\frac{47}{1} \) (which simplifies to just \(-47\))
- The point \( (14, 3) \)
Substituting the point into the equation:
\[ 3 = -47(14) + b \]
Now calculate \( -47(14) \):
\[ -47 \times 14 = -658 \]
Now substitute this back into the equation:
\[ 3 = -658 + b \]
To solve for \( b \), add \( 658 \) to both sides:
\[ b = 3 + 658 \] \[ b = 661 \]
Now, let's reevaluate if any of the provided choices match \( b = 661 \). None of the provided options (15.71, 12.29, -5, 11) are correct.
Therefore, if you follow the typical expectations of linear equations, based on the calculations, the initial value (y-intercept) \( b \) of the linear function is actually 661. None of the provided answer options are correct or close to the computed value.