To determine the value of \(k\) in the function \(g(y) = y + k\) that describes the translation from the point \((4, -1)\) to the point \((4, 2)\), we need to analyze the change in the \(y\)-coordinate.
The initial \(y\)-coordinate is \(-1\) and the final \(y\)-coordinate is \(2\). The translation can be described by identifying how the \(y\)-coordinate changes:
\[ \text{Change in } y = \text{final } y - \text{initial } y = 2 - (-1) = 2 + 1 = 3 \]
This indicates that the \(y\)-coordinate increases by \(3\).
The function \(g(y) = y + k\) represents this change. If we apply it to the initial point's \(y\)-coordinate:
\[ g(-1) = -1 + k \]
For the translation to yield a final \(y\)-coordinate of \(2\), we set up the equation:
\[ -1 + k = 2 \]
Solving for \(k\):
\[ k = 2 + 1 = 3 \]
Thus, the value of \(k\) is \(3\). Therefore, the correct answer is:
C. \(3\)