To find the initial value \( b \) of the linear function, we can use the point-slope form of a linear equation, which is given by:
\[ y = mx + b \]
where:
- \( y \) is the output (dependent variable),
- \( m \) is the slope (rate of change),
- \( x \) is the input (independent variable), and
- \( b \) is the y-intercept (initial value).
We are given:
- the slope \( m = -47 \)
- the point \( (14, 3) \), meaning \( x = 14 \) and \( y = 3 \).
We can substitute these values into the equation to solve for \( b \):
\[ 3 = -47(14) + b \]
Calculating \( -47(14) \):
\[ -47 \times 14 = -658 \]
Now substitute this back into the equation:
\[ 3 = -658 + b \]
To solve for \( b \):
\[ b = 3 + 658 \]
\[ b = 661 \]
Upon checking the possible answers (11, 12.29, 15.71, and -5), it looks like none of those options match our calculated \( b \).
Please double-check the point given or confirm the value of the slope, as we derived \( b = 661 \) based on the information provided. If everything is accurate, the initial value (y-intercept) does not match the options given.