Mathematics

The tangent function is periodic and undefined wherever its denominator (cosine) equals zero—those are your vertical asymptotes. Between asymptotes, $\tan x$ is continuous and strictly increasing, passing through the origin and climbing steeply toward $\pm\infty$. The key is recognizing the symmetry: $\tan x$ is an odd function, so the graph mirrors through the origin, and you only need to understand one period to sketch the whole interval.

The tangent function is periodic and undefined wherever its denominator (cosine) equals zero—those are your vertical asymptotes. Between asymptotes, $\tan x$ is continuous and strictly increasing, passing through the origin and climbing steeply toward $\pm\infty$. The key is recognizing the symmetry: $\tan x$ is an odd function, so the graph mirrors through the origin, and you only need to understand one period to sketch the whole interval.