Question

How do you graph, identify the domain, range, and asymptotes for `y=sec(x-pi/2)`?

Answer 1

The domain is `x=`all real numbers except `x=n*pi`
The range is all real numbers.
The asymptotes occur at `x=n*pi`, where `n` is any integer.
To graph, see below.
graph{sec(x-pi/2) [-7.87, 6.177, -3.19, 3.833]}

Explanation:

Secant is the inverse of the cosine function, so it is asymptotal.
The range is all real numbers (remember secant, cosecant, tangent, and cotangent all hug their asymptotes).

The domain of the function is all the points except where the helper function (`y=cos(x-pi/2)`) crosses its midline, `0`. Since this function is phase-shifted `pi/2`, it is the same as `y=sin(x)`, so the zeros are at the beginning, 2nd quarter, and endpoints. Therefore they can be represented by `x=n*pi`. These are the asymptotes of the function, as well as the exclusions to the domain.

To graph, first graph the helper function `y=cos(x-pi/2)`. Then, sketch asymptotes at each point it crosses the midline. Finally, draw lines connecting to the min and max of the helper function and approaching the asymptotes (they kind of look like parabolas). See the graph above for reference.