Let's analyze the given function [tex]\( y = -6 - \tan \left(x + \frac{\pi}{3} \right) \)[/tex].
### (a) Finding the Period
The given function involves the tangent function, [tex]\( \tan(x) \)[/tex]. The basic property of the tangent function is that its period is [tex]\( \pi \)[/tex]. If you shift or translate the function horizontally or vertically, it does not affect its period. Therefore, the period of the given function remains the same as the period of the basic tangent function.
So, the period of [tex]\( y = -6 - \tan \left(x + \frac{\pi}{3} \right) \)[/tex] is:
[tex]\[ \boxed{\pi} \][/tex]
### (b) Finding the Phase Shift
The phase shift of a trigonometric function is determined by the horizontal translation inside the argument of the function.
For the function [tex]\( \tan \left( x + \frac{\pi}{3} \right) \)[/tex], the term [tex]\(\frac{\pi}{3}\)[/tex] represents a horizontal shift. In this case, given the form [tex]\( \tan(x + C) \)[/tex], the phase shift is [tex]\( -\frac{\pi}{3} \)[/tex] because adding [tex]\(\frac{\pi}{3}\)[/tex] inside the function shifts it to the left by [tex]\(\frac{\pi}{3}\)[/tex] units.
Thus, the phase shift of [tex]\( y = -6 - \tan \left(x + \frac{\pi}{3} \right) \)[/tex] is:
[tex]\[ \boxed{-\frac{\pi}{3}} \][/tex]
### (c) Finding the Range
The basic range of the tangent function [tex]\( \tan(x) \)[/tex] is all real numbers, [tex]\((-\infty, \infty)\)[/tex]. Even though the function is transformed by shifting and vertical translation, it does not change the range of the tangent function since tangent can still take any value from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].
Thus, the range of [tex]\( y = -6 - \tan \left(x + \frac{\pi}{3} \right) \)[/tex] remains:
[tex]\[ \boxed{(-\infty, \infty)} \][/tex]
In summary:
- (a) The period is [tex]\(\boxed{\pi}\)[/tex].
- (b) The phase shift is [tex]\(\boxed{-\frac{\pi}{3}}\)[/tex].
- (c) The range is [tex]\(\boxed{(-\infty, \infty)}\)[/tex].
