graph one cycle of each transformed csc or sec function

Question

1) y= 3 cot ( x- pi/6) + 1

2) y = -2 cot (3x+ pi/2) - 3

3) y= 5 tan (1/4x- pi/2)

4) y = - tan (2x + pi) + 3

Answer 1

y = 3cot(x - pi/6) + 1

·        The graph starts at (pi/6, 1)

·        It then goes downward to negative infinity at (pi/2, -inf)

·        The graph has a vertical asymptote at x = pi/2

·        It then goes upward to positive infinity at (5pi/6, inf)

·        The graph ends at (pi, 1)

y = -2cot(3x + pi/2) – 3

·        The graph starts at (-pi/6, -3)

·        It then goes upward to positive infinity at (pi/6, inf)

·        The graph has a vertical asymptote at x = -pi/6

·        It then goes downward to negative infinity at (pi/2, -inf)

·        The graph has a vertical asymptote at x = pi/2

·        The graph ends at (5pi/6, -3)

y = 5tan(1/4x - pi/2)

·        The graph starts at (pi/2, -inf)

·        It then goes upward to positive infinity at (pi/2 + 4pi, inf)

·        The graph has a vertical asymptote at x = pi/2 + 2k*pi, where k is an integer

·        It then goes downward to negative infinity at (pi/2 + 8pi, -inf)

·        The graph has a vertical asymptote at x = pi/2 + (2k + 1)*pi/2, where k is an integer

·        The graph ends at (pi/2 + 16pi, -inf)

y = -tan(2x + pi) + 3

·        The graph starts at (-3pi/4, 3)

·        It then goes downward to negative infinity at (-pi/2, -inf)

·        The graph has a vertical asymptote at x = -pi/2

·        It then goes upward to positive infinity at (-pi/4, inf)

·        The graph has a vertical asymptote at x = -pi/4

·        The graph ends at (pi/4, 3)