How do you graph #y=sec(x-pi/4)#?

graph{sec(x - pi/4) [-10, 10, -5, 5]}

To graph ( y = \sec(x - \frac{\pi}{4}) ), follow these steps:

1. Identify the key features:

   • The parent function of secant is ( y = \sec(x) ).
   • Shifting ( \frac{\pi}{4} ) units to the right from the origin will shift the graph of ( \sec(x) ) horizontally.

2. Plot the asymptotes:

   • The graph of ( \sec(x) ) has vertical asymptotes where ( \cos(x) = 0 ).
   • The vertical asymptotes occur at ( x = \frac{\pi}{2} + k\pi ) for integer values of ( k ).
   • Since ( \sec(x - \frac{\pi}{4}) ) is shifted ( \frac{\pi}{4} ) units to the right, the vertical asymptotes will occur at ( x = \frac{\pi}{2} + \frac{\pi}{4} + k\pi ) for integer values of ( k ).

3. Determine the behavior around the asymptotes:

   • As ( x ) approaches a vertical asymptote, ( \sec(x) ) approaches positive or negative infinity depending on the sign of ( \cos(x) ).

4. Plot additional points and sketch the graph:

   • Choose values of ( x ) to evaluate ( y = \sec(x - \frac{\pi}{4}) ).
   • Plot these points and connect them smoothly, considering the behavior around the asymptotes.

By following these steps, you can accurately graph the function ( y = \sec(x - \frac{\pi}{4}) ).