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 observe the change in the \( y \)-coordinate during the translation.
The initial \( y \)-coordinate is \( -1 \) and the final \( y \)-coordinate is \( 2 \).
The translation can be described as follows: \[ g(-1) = 2 \]
We can substitute \( y = -1 \) into the function \( g(y) \): \[ g(-1) = -1 + k \] Setting this equal to the final \( y \)-value: \[ -1 + k = 2 \] Now, we can solve for \( k \): \[ k = 2 + 1 = 3 \]
Thus, the value of \( k \) is \( \boxed{3} \).