graphing y=2cot(2pix)

Thanks for that analysis, geno....here's the graph........https://www.desmos.com/calculator/ehqq7efbzs

cot(x) = 1/tan(x)

The period of tan(x) and cot(x) is the same; it equals π.

If there is a multiplier of x, so that you have cot(ax), the period will be π/a; this means that the period of this function is π/(2π) = 1/2. The entire function will make one period between 0 and 1/2.

At 0, tan(x) = 0, so cot(x) is undefined; because the period is 1/2, it will also be undefined at 1/2.

Halfway between, at 1/4, 2cot(2π(1/4)  =  2cot(π/2) = 0.

The graph starts with a y-value of +∞ at x = 0, waves down through the point (1/4,0), and continues down to   -∞ at x = 1/2.

Graph it on a calculator, restrict the x-values to between -1/2 and +1/2.