Show that loop in $S^1$ is not nullhomotopic

Question

Let $q$ be the covering map $z\mapsto z^2$ from $S^1$ into itself. Let $\alpha$ be a path in $S^1$ from a point $x$ to its antipodal. I need to show that the path $q\circ \alpha$ is not null homotopic, i.e. here is not homotopic with a constant function. I can see how this path should “wrap” the circle around, but I can not argue this formally.