To determine which of the given functions illustrates a phase shift, let's analyze each one step by step.
A phase shift in a trigonometric function is indicated by a horizontal shift, which usually occurs inside the argument of the trigonometric function in the form of [tex]\( (x - c) \)[/tex] or [tex]\( (x + c) \)[/tex].
### Option A: [tex]\( y = 3 \cos(4x) \)[/tex]
- In this function, the argument [tex]\(4x\)[/tex] indicates a change in frequency (period) but does not include a phase shift.
- Since there is no horizontal translation included in the argument (nothing of the form [tex]\( x - c \)[/tex] or [tex]\( x + c \)[/tex]), there is no phase shift here.
### Option B: [tex]\( y = -2 - \cos(x - \pi) \)[/tex]
- Here, the argument [tex]\((x - \pi)\)[/tex] includes a horizontal translation by [tex]\(\pi\)[/tex] units to the right.
- This is the form [tex]\( (x - c) \)[/tex], where [tex]\( c = \pi \)[/tex], indicating a phase shift.
### Option C: [tex]\( y = 1 + \sin(x) \)[/tex]
- The argument [tex]\( x \)[/tex] does not include a horizontal translation. It is simply [tex]\( x \)[/tex], which implies no phase shift.
- There is a vertical shift by 1, but no horizontal shift is involved.
### Option D: [tex]\( y = \tan(2x) \)[/tex]
- In this case, the argument [tex]\( 2x \)[/tex] again indicates a change in frequency (period) but not a horizontal translation.
- Therefore, no phase shift is present in this function.
Based on this analysis, the function that illustrates a phase shift is [tex]\( y = -2 - \cos(x - \pi) \)[/tex]. The phase shift occurs due to the [tex]\( (x - \pi) \)[/tex] term, shifting the function horizontally by [tex]\(\pi\)[/tex] units.
Thus, the correct answer is:
B. [tex]\( y = -2 - \cos(x - \pi) \)[/tex]
