To find the initial value (y-intercept) of the linear function given a rate of change (slope) and a point on the line, we can use the slope-intercept form of a linear equation, which is:
\[ y = mx + b \]
Where:
- \( m \) is the slope
- \( b \) is the y-intercept
- \( (x, y) \) is a point on the line
You are given:
- \( m = -\frac{4}{7} \)
- The point \( (14, 3) \)
We can plug these values into the equation to solve for \( b \):
\[ 3 = -\frac{4}{7}(14) + b \]
Now, calculate \( -\frac{4}{7} \times 14 \):
\[ -\frac{4}{7} \times 14 = -\frac{4 \times 14}{7} = -\frac{56}{7} = -8 \]
Now substitute that back into the equation:
\[ 3 = -8 + b \]
To solve for \( b \), add 8 to both sides:
\[ b = 3 + 8 \] \[ b = 11 \]
So the initial value (y-intercept) of the linear function is:
\[ \boxed{11} \]