To find the initial value (or y-intercept) of the linear function when given a slope \( m = -\frac{4}{7} \) and a point \( (x, y) = (14, 3) \), we can use the point-slope form of a linear equation:
\[ y - y_1 = m(x - x_1) \]
Here, \( (x_1, y_1) \) is the point \( (14, 3) \), and \( m = -\frac{4}{7} \). Plugging in the values:
\[ y - 3 = -\frac{4}{7}(x - 14) \]
Now, we can solve for \( y \) to put the equation in slope-intercept form \( y = mx + b \):
\[ y - 3 = -\frac{4}{7}x + \frac{4}{7} \cdot 14 \]
Calculating \( \frac{4}{7} \cdot 14 \):
\[ \frac{4 \cdot 14}{7} = \frac{56}{7} = 8 \]
So the equation becomes:
\[ y - 3 = -\frac{4}{7}x + 8 \]
Adding 3 to both sides:
\[ y = -\frac{4}{7}x + 8 + 3 \]
\[ y = -\frac{4}{7}x + 11 \]
In this equation, the y-intercept (initial value) is:
\[ b = 11 \]
Therefore, the initial value of the linear function is 11.