# How do you graph, identify the domain, range, and asymptotes for y=(1/2)sec2(x-pi/2)+1?

Jul 21, 2018

See graph and explanation.

#### Explanation:

y = 1/2 sec( 2 ( x - pi/2 ) + 1 = 1/2 sec( 2 x - pi ) + 1

$= \frac{1}{2} \sec \left(\pi - 2 x\right) + 1$

$= \frac{1}{2} \sec \left(2 x\right) + 1$.

$= \frac{1}{2 \cos \left(2 x\right)} + 1$.

The period = period of cos( 2 x ) = 2pi )/2 = pi.

The asymptotes are given by $\cos \left(2 x\right) = 0$

$\Rightarrow 2 x = \left(2 k + 1\right) \frac{\pi}{2} \Rightarrow x = \left(2 k + 1\right) \frac{\pi}{4}$,

$k = = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$ So, the domain is

$x \in \ldots U \left(- \frac{5}{4} \pi , - \frac{3}{4} \pi\right) U \left(- \frac{3}{4} \pi , - \frac{\pi}{4}\right) U \left(- \frac{\pi}{4} , \frac{\pi}{4}\right)$

$U \left(\frac{\pi}{4} , 3 \frac{\pi}{4}\right) U \left(3 \frac{\pi}{4} , 5 \frac{\pi}{4}\right) U \ldots$

Range is given by

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

See illustrative graph, indicating range and two asymptotes,

near O..
graph{(2(y-1) cos (2x)-1)(y-1/2)(y-3/2)(x^2-(pi)^2/16 )=0[-8 8 -3 5]}