Let's analyze the function [tex]\( y = -5 \sec \left( x + \frac{\pi}{4} \right) \)[/tex] to find its period, phase shift, and range.
### (a) Find the period
The secant function, [tex]\(\sec(x)\)[/tex], has a base period of [tex]\(2\pi\)[/tex]. The period of the transformed secant function [tex]\( \sec \left( x + \frac{\pi}{4} \right) \)[/tex] does not change due to the horizontal shift inside the function. Therefore, the period of the given function remains the same as the base period of the secant function.
So, the period of [tex]\( y = -5 \sec \left( x + \frac{\pi}{4} \right) \)[/tex] is:
[tex]\[ 2\pi \][/tex]
