Consider the equation of a parabola given by the function:
$$f(x) = ax^2 + bx + c.$$
If the parabola has its vertex at the point (3, -2) and passes through the point (1, 2), determine the value of $b$ given that $a = 1$ and $c = -5$.
Recall that the x-coordinate of the vertex of a parabola described by the equation $$y = ax^2 + bx + c$$ is given by the formula $$x_v = -\f\frac{b}{2a}.$$ You will need to establish the relationship between $b$ and the given parameters.