# How do you sketch the graph of y = sec(3x + pi/2)?

Jul 22, 2018

See explanation and graph.

#### Explanation:

$y = \sec \left(3 x + \frac{\pi}{2}\right) = - \csc 3 x \notin \left(- 1 , 1\right)$

and $3 x \ne$ an integer multiple of $\pi$

$\Rightarrow x \ne$ integer multiple of $\frac{\pi}{3}$

The asymptotes are $x = k \left(\frac{\pi}{3}\right) , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

The period = period of sin 3x = (2pi)/3

Rewrite $y \sin 3 x + 1 = 0$, by cross multiplication..

For this form, the Socratic graph is immediate.
graph{(ysin(3x) + 1)(y^2-1)(x^2-1.0966)= 0[-10 10 -5 5]}

Observe

$y \notin \left(- 1 , 1\right)$ and the asymptotes $x = \pm \frac{\pi}{3}$, near O and also the period $\frac{2 \pi}{3}$.