Let the function $f(x)$ be defined by the equation:
$$ f(x) = 2x^2 - 4x + k $$
where $k$ is a constant. If the function has a minimum value at $x = 1$, what is the value of $k$?
Note: The minimum value of a quadratic function in the form $ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$.