Question #41b35

3 Answers
Apr 19, 2015

You know the period is \piπ because you know the period of tan(x)tan(x) is piπ, and you know that if a function ff is periodic with period TT, also is alphaf, alpha != 0αf,α0 because alphaf(x) = alphaf(x+T)αf(x)=αf(x+T), this is trivial (it's just multiplication on both sides of the == sign).
For the graph, you know it's zero for x=kpi, k \in \ZZx=kπ,kZ and in every zero the tangent has 4 as angular coefficient, an it is 4 for x=pi/4 + kpi, -4 for x=-pi/4+kpi4forx=π4+kπ,4forx=π4+kπ, and has a polar singularity in x=pi/2 + kpix=π2+kπ
graph{4*tan(x) [-10, 10, -5, 5]}

Apr 19, 2015

The period is piπ, because tanx repeats it self in the interval (-pi/2, pi/2)(π2,π2) and would repeat in successive intervals to (pi/2,(3pi)/2π2,3π2).. and so on to the right of the origin and similarly to the left of it.

Apr 19, 2015

I am not sure it helps, but also you can "see" the period of your function considering the coefficient of x (the number in front of it called kk);
In this case is k=1k=1 so you have that:
k=(2pi)/(period)k=2πperiod
If k=1k=1 then
period=2piperiod=2π