$y=\tan x$ Tangent has no amplitude, maximum or minimum. Its range is all real numbers. Function$0^\circ$ ($0$)$90^\circ$ ($\frac{\pi}{2}$)$180^\circ$ ($\pi$)$270^\circ$ ($\frac{3\pi}{2}$)$360^\circ$ ($2\pi$) $\tan x$ $y=a\tan(bx)$, $a,b>0$ $T=\dfrac{180^\circ}{b}\text{ or}\hspace{0.5em}\dfrac\pi b$ $\text{Zeros:}\hspace{0.5em}x=\dfrac{k\pi}{b}\qquad\text{Asymptotes:}\hspace{0.5em}x=\dfrac{\pi/2+k\pi}{b}\quad(k\in\mathbb Z)$ EquationMidlineAmplitudeMaximum and minimumPeriod $y=3\tan 2x-1$ FunctionPeriod (degrees)Period (radians) $y=5\tan 3x$ $y=3\tan 2x$ $y=\tan\frac{3x}{4}$ $y=\tan(\pi x)$ Draw dashed vertical asymptotes, mark the zeros midway between them, then sketch an increasing branch in each interval. Branches never cross vertical asymptotes. Example: $y=\tan\left(\dfrac32x\right)$, $0\le x\le2\pi$ Period $\dfrac{2\pi}{3}$. Zeros: $0,\dfrac{2\pi}{3},\dfrac{4\pi}{3},2\pi$. Vertical asymptotes: $\dfrac\pi3,\pi,\dfrac{5\pi}{3}$. Each branch increases between its asymptotes.

Sketching of $y=a\tan \left( bx \right)$

$y=\tan x$

Tangent has no amplitude, maximum or minimum. Its range is all real numbers.

Function	$0^\circ$ ($0$)	$90^\circ$ ($\frac{\pi}{2}$)	$180^\circ$ ($\pi$)	$270^\circ$ ($\frac{3\pi}{2}$)	$360^\circ$ ($2\pi$)
$\tan x$

$y=a\tan(bx)$, $a,b>0$

$T=\dfrac{180^\circ}{b}\text{ or}\hspace{0.5em}\dfrac\pi b$

$\text{Zeros:}\hspace{0.5em}x=\dfrac{k\pi}{b}\qquad\text{Asymptotes:}\hspace{0.5em}x=\dfrac{\pi/2+k\pi}{b}\quad(k\in\mathbb Z)$

Equation	Midline	Amplitude	Maximum and minimum	Period
$y=3\tan 2x-1$

Function	Period (degrees)	Period (radians)
$y=5\tan 3x$
$y=3\tan 2x$
$y=\tan\frac{3x}{4}$
$y=\tan(\pi x)$

Draw dashed vertical asymptotes, mark the zeros midway between them, then sketch an increasing branch in each interval. Branches never cross vertical asymptotes.

Example: $y=\tan\left(\dfrac32x\right)$, $0\le x\le2\pi$

Period $\dfrac{2\pi}{3}$. Zeros: $0,\dfrac{2\pi}{3},\dfrac{4\pi}{3},2\pi$. Vertical asymptotes: $\dfrac\pi3,\pi,\dfrac{5\pi}{3}$. Each branch increases between its asymptotes.