Find Intercepts and Asymptotes

Find intercepts and asymptotes of each trig function.

(1) y = tan[x - (pi/3)]

(2) y = (-1/2)cot(x/2)

1.
y intercept will occur when x = 0
y = tan(0 - pi/3) = tan(-pi/3) = -sqrt(3)
so y intercept at (0,-sqrt(3))

the curve crosses the y axis at y = 0
0 = tan(x - pi/3)
x-pi/3 = arctan(0) = 0
x = pi/3
so x intercept at (pi/3,0)

asymptote will occur when (x - pi/3) = pi/2 [tan(90) is infinite]
x = pi/2 + pi/3 = (3pi+2pi)/6 = 5pi/6
vertical asymptote at x = 5pi/6

2. cot(a) = 1/tan(a) so
y = (-1/2)cot(x/2) = -1/(2*tan(x/2))

y intercept at x = 0
y = -1/(2*tan(0)) = infinite, i.e. there is no y intercept

x intercept at y = 0
0 = -(1/2)cot(x/2)
in the limit, cot(a) = 0 when tan(a) is infinite, i.e. at pi/2
so for cot(x/2) = 0, x/2 = pi/2
i.e. x = pi
x intercept at (pi,0)

asympote is at tan(x/2) = tan(0)
i.e. asymptote at y = 0