Let's tackle each part of the question in detail:
### Part (a): Period of the Function
The function given is [tex]\( y = -5 \sec \left(x + \frac{\pi}{4}\right) \)[/tex].
The period of the secant function, [tex]\( \sec(x) \)[/tex], is [tex]\( 2\pi \)[/tex]. This property is retained even with modifications inside the argument of the secant function, such as horizontal shifts and reflections.
Therefore, the period of [tex]\( y = -5 \sec \left(x + \frac{\pi}{4}\right) \)[/tex] is:
[tex]\[ \boxed{2\pi} \][/tex]
### Part (b): Phase Shift
To find the phase shift, we look at the term inside the secant function. Specifically, we consider the function in the form [tex]\( \sec(x + C) \)[/tex], where [tex]\( C \)[/tex] is a constant that indicates the phase shift.
In our function, [tex]\( y = -5 \sec \left(x + \frac{\pi}{4}\right) \)[/tex], the term inside the secant is [tex]\((x + \frac{\pi}{4})\)[/tex].
Setting [tex]\((x + \frac{\pi}{4}) = 0\)[/tex] to determine the phase shift:
[tex]\[ x + \frac{\pi}{4} = 0 \][/tex]
[tex]\[ x = -\frac{\pi}{4} \][/tex]
This means the phase shift is:
[tex]\[ \boxed{-\frac{\pi}{4}} \][/tex]
### Part (c): Range of the Function
The range of the secant function, [tex]\( \sec(x) \)[/tex], is [tex]\((-\infty, -1] \cup [1, \infty)\)[/tex].
When multiplying [tex]\(\sec(x + \frac{\pi}{4})\)[/tex] by -5, the range notation changes. We need to examine the transformation of each segment of the range.
1. For the segment [tex]\((- \infty, -1]\)[/tex]:
- Multiplying by -5, which inverts the sign and stretches by a factor of 5:
[tex]\[
-5 \times \sec(x + \frac{\pi}{4}) \Rightarrow (-5, - \infty) \text{ (Switching endpoints directionally gives } -\infty \text{ to } -5\text{)}
\][/tex]
This segment becomes:
[tex]\[
(- \infty, -5]
\][/tex]
2. For the segment [tex]\([1, \infty)\)[/tex]:
- Multiplying by -5 similarly inverts and stretches:
[tex]\[
-5 \times \sec(x + \frac{\pi}{4}) \Rightarrow [-5, \infty] \text{ (Switching endpoints directionally gives } -5 \text{ to } \infty\text{)}
\][/tex]
This segment becomes:
[tex]\[
[5, \infty)
\][/tex]
Thus, the total range of the function [tex]\( y = -5 \sec \left(x + \frac{\pi}{4}\right) \)[/tex] is the union of these intervals:
[tex]\[ \boxed{(-\infty, -5] \cup [5, \infty)} \][/tex]
To summarize, the solutions to each part are:
(a) The period is [tex]\( 2\pi \)[/tex].
(b) The phase shift is [tex]\( -\frac{\pi}{4} \)[/tex] units.
(c) The range is [tex]\( (-\infty, -5] \cup [5, \infty) \)[/tex].
