Alright, let's analyze the function [tex]\( y = -6 - \tan\left(x + \frac{\pi}{3}\right) \)[/tex] to determine its characteristics.
(a) Period:
To find the period of this tangent function, we recognize that the general form of the tangent function is [tex]\( \tan(bx + c) \)[/tex], where [tex]\( b \)[/tex] affects the period.
The period of [tex]\( \tan(x) \)[/tex] is [tex]\( \pi \)[/tex]. For a function [tex]\( \tan(bx + c) \)[/tex], the period is given by [tex]\( \frac{\pi}{|b|} \)[/tex].
In the given function [tex]\( \tan\left(x + \frac{\pi}{3}\right) \)[/tex], we compare it with the general form and see that [tex]\( b = 1 \)[/tex]. Therefore, the period is:
[tex]\[ \text{Period} = \frac{\pi}{|1|} = \pi \][/tex]
(b) Phase Shift:
The phase shift of the function [tex]\( \tan(bx + c) \)[/tex] is determined by [tex]\( \frac{-c}{b} \)[/tex].
Here, in [tex]\( \tan\left(x + \frac{\pi}{3}\right) \)[/tex], [tex]\( c = \frac{\pi}{3} \)[/tex] and [tex]\( b = 1 \)[/tex]. The phase shift is:
[tex]\[ \text{Phase Shift} = \frac{-\left(\frac{\pi}{3}\right)}{1} = -\frac{\pi}{3} \][/tex]
(c) Range:
The range of the tangent function [tex]\( \tan(x) \)[/tex] is all real numbers [tex]\( (-\infty, \infty) \)[/tex].
When we introduce a vertical shift or a constant, like in [tex]\( y = -6 - \tan\left(x + \frac{\pi}{3}\right) \)[/tex], the range of the function remains all real numbers because the vertical translation does not affect the range of the tangent function. Thus, the range of this function is also:
[tex]\[ \text{Range} = (-\infty, \infty) \][/tex]
To summarize, for the function [tex]\( y = -6 - \tan\left(x + \frac{\pi}{3}\right) \)[/tex]:
- The period is [tex]\( \pi \)[/tex].
- The phase shift is [tex]\( -\frac{\pi}{3} \)[/tex].
- The range is all real numbers [tex]\((- \infty, \infty)\)[/tex].
