In y=atan(bx)+c, a vertical shift changes the height of each branch but does not move the vertical asymptotes or change the period. Tangent has no amplitude. For b 0, the period is |b| radians or 180 |b|. Locate asymptotes by finding where cos(bx)=0 in the given interval. Example: y=2tan x+1, 0 <= x <= pi. The period is pi, the intercept is (0,1), and the vertical asymptote is x= 2. The right-hand endpoint is (pi,1). The curve has no point on the asymptote, even if that value is included in the stated interval. A vertical shift can change the zeros: here tan x=- 12.

In \(y=a\tan(bx)+c\), a vertical shift changes the height of each branch but does not move the vertical asymptotes or change the period. Tangent has no amplitude.

For \(b\ne0\), the period is \(\dfrac\pi{|b|}\) radians or \(\dfrac{180^\circ}{|b|}\). Locate asymptotes by finding where \(\cos(bx)=0\) in the given interval.

Example: \(y=2\tan x+1\), \(0\le x\le\pi\). The period is \(\pi\), the intercept is \((0,1)\), and the vertical asymptote is \(x=\dfrac\pi2\).

The right-hand endpoint is \((\pi,1)\). The curve has no point on the asymptote, even if that value is included in the stated interval. A vertical shift can change the zeros: here \(\tan x=-\dfrac12\).

Vertical shifts of tangent

Vertical shifts of tangent