Let the function is y = g(x) = 3 cot (x + π/6) + 2.

Compare the equation y = 2 + 3 cot [ 1 (x + π/6) ] with y = k + A cot [ B (x - h) ] where B > 0.

k = 2, A = 3, B = 1 and h = - π/6.

Period = π/B = π/1 = π.

Horizontal translation = Phase shift = h = - π/6.

Vertical translation = k = 2.

For cotangent functions, there is no concept of amplitude since the range of the cotangent function is (- ∞ , ∞) or the set of all real numbers. The value of | A | is the factor by which the basic graphs are expand or contracted vertically. If A < 0 the graph will be reflected about the x - axis.

The domain of cotangent function, y = cot (x) is - ∞ < x < ∞, where x not equal to integer multiplies of π.

The period of y = tan x is π, so the period of y = k + A cot [ B (x - h) ] is π/b = π/(1) = π, caused by the horizontal compression of the graph by a factor of 1/b = 1/1 = 1.

Two consecutive vertical asymptotes can be found by solving the equations B (x - h) = 0 and B (x - h) = π.

x + π/6 = 0 ------> x = - π/6 and

x + π/6 = π -------> x = π - π/6 -----> x = (6π - π)/6 -----> x = 5π/6.

The two consecutive vertical asymptotes occur at x = - π/6 and x = 5π/6.

The interval [- π/6, 5π/6] corresponds to one cycle of the graph. Dividing this interval into four equal parts produces the key points.

one fourth of cycle is [5π/6 - (- π/6) ]/4 = (6π/6)(1/4) = π/4.

The x - coordinates of the five key points are

x = - 13π/3.

x = - 13π/3 + π/2 = (- 26π + 3π)/6 = - 23π/6.

x = - 23π/6 + π/2 = (- 23π + 3π)/6 = - 20π/6 = - 10/3π.

x = - 10π/3 + π/2 = (- 20π + 3π)/6 = - 17π/6.

x = - 17π/6 + π/2 = (- 17π + 3π)/6 = - 14π/6 = - 7π/3.

Between these two asymptotes, plot a few points, including the -intercept, as shown in the table.

First ploting the asymptotes.

The midpoint between two consecutive vertical asymptotes is an x - intercept of the graph. The period of the function y = a tan(bx - c) is the distance between two consecutive vertical asymptotes. The amplitude of a tangent function is not defined.

After plotting the asymptotes and the x - intercept, plot a few additional points between the two asymptotes and sketch one cycle. Finally, sketch one or two additional cycles to the left and right.

Plot these five points and fill in the graph of the tangent function as shown in Figure.