The minimum is the smallest possible $y$ value that can be obtained by plugging in some value of $x$ into the equation.

In the case of a parabola (that opens up), the minimum is indeed found by looking at the vertex. The $x$-coordinate of the vertex tells us which value of $x$ produces the minimum value, and the $y$-coordinate tells us what that minimum value is.

In this case we need to be careful too with that absolute value. It might be helpful noticing that the graph of $|f(x)|$ is almost the same as the graph of $f(x)$, but everything below the $x$-axis gets reflected.