Graph of a tangent function

Text

Sketch the graph of $y=1-\dfrac{1}{2}\tan\left(2x+\dfrac{\pi}{2}\right)$.

Solution

From $c=2$ the period is $T=\dfrac{\pi}{2}$. Writing the argument as $2x+\dfrac\pi2=2\left(x+\dfrac\pi4\right)$ shows a horizontal shift to the left by $\dfrac\pi4$. The vertical asymptotes occur where the argument of the tangent equals $\dfrac\pi2+k\pi$, i.e. $2x+\frac\pi2=\frac\pi2+k\pi \;\Longrightarrow\; x=\frac{k\pi}{2},\quad k\in\mathbb{Z}.$ The coefficient $-\dfrac12$ produces a reflection about the horizontal axis together with a vertical compression, while the term $+1$ shifts the branches’ center to $y=1$; each branch of the tangent is therefore decreasing.

$\boxed{T=\frac\pi2,\ \ \text{asymptotes } x=\frac{k\pi}{2},\ \ \text{branch center } y=1}$