How do you identify the period and asympotes for #y=-2tan(pitheta)#?

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Answer 1 · 2018-04-11T12:43:16

See below.

If we express the tangent function in the following way:

#y=atan(bx+c)+d#

Then:

# \ \ \bba \ \ \ ="the amplitude"#

#bb((pi)/b) \ \="the period"#

#bb((-c)/b)= "the phase shift"#

# \ \ \ \bbd \ \ \="the vertical shift"#

For given function we have:

#b=pi#

So period is:

#(pi)/pi=color(blue)(1)#

The function will have vertical asymptotes everywhere it is undefined.

We know:

#tan(theta)# is undefined at #pi/2# , #pi/2+pi# and so on. We can write this in a general way as follows:

#pi/2+npi#

Where #n# in an integer.

We now solve:

#pitheta=pi/2+npi#

#theta=1/2+n#

So vertical asymptotes occur everywhere #theta=1/2+n#

For:

#n in ZZ#

The graph confirms these findings:

enter image source here