Graph of Tangent

Intuition and core definition

The tangent function is $\tan x=\sin x/\cos x$. Its graph has period $\pi$, zeros where $\sin x=0$, and vertical asymptotes where $\cos x=0$. On each interval between asymptotes, $y=\tan x$ increases from negative to positive infinity.

Notation, language, and conditions

For $y=a\tan(b(x-h))+k$ with $b\ne0$, period is $\pi/|b|$, phase shift is $h$, midline is $y=k$, and vertical scale or reflection comes from $a$. Asymptotes occur when $b(x-h)=\pi/2+n\pi$. The range is all real numbers when $a\ne0$; if $a=0$, the graph is the constant $y=k$ on the tangent expression's domain.

Why this idea matters

The tangent graph records repeating slopes, with zeros at horizontal directions and vertical asymptotes where cosine vanishes.

A dependable method

1. Identify $a,b,h,k$ from a grouped form.
2. Compute period $\pi/|b|$ and locate a central crossing at $(h,k)$.
3. Place nearest asymptotes half a period on either side of $h$.
4. Use quarter-period points where base tangent is $\pm1$, scaled by $a$.
5. Repeat periodically and check asymptote/domain behaviour.

Worked example

Common mistakes