# How do you find the period for y = -tan(x - pi/2)?

The period is always $\pi$ in this case:
if $f$ has period $T$:
1)$f \left(x + T\right) = f \left(x\right) \implies f \left(x + T + a\right) = f \left(\left(x + a\right) + T\right) = f \left(x + a\right)$ so the period of $f \left(x + a\right)$ is still $T$ (you know it's minimal because it's minimal in $f$)
2) $f \left(x + T\right) = f \left(x\right) \implies a f \left(x + T\right) = a f \left(x\right)$, and still it's minimal because it's minimal in $f$
so the period is $\pi$, the same as in $\tan \left(x\right)$