# How do you find the domain, range, and asymptote for y = 2 - sec ( 2x - pi/2 )?

Aug 9, 2018

See explanation and graph.

#### Explanation:

$y = 2 - \sec \left(2 x - \frac{\pi}{2}\right) = 2 - \sec \left(\frac{\pi}{2} - 2 x\right)$

$= 2 - \csc 2 x , 2 x \ne$ asymptotic kpi, k =0, 1, 2, 3, ...#

$\Rightarrow x \ne k \left(\frac{\pi}{2}\right)$

Also, as csc values $\notin \left(- 1 , 1\right)$,

$y \notin \left[- 1 + 2 , 1 + 2\right) = \left(1 , 3\right)$.

The period is period of $\sin 2 x = \frac{2 \pi}{2} = \pi$.

Vertical shift = 2, giving midline $y = 2$.

See graph, depicting all these aspects.
graph{((2-y)sin (2x )-1)(x+0.0001y)(x-pi/2 + 0.0001y)(x+pi/2 + 0.0001y)(y-1+0x)(y-2+0x)(y-3+0x)=0[-10 10 -3 7]}