What is the tangent graph?

The tangent graph is the graph of a trigonometric function.

$\tan x$ is a trigonometric ratio. If we substitute $x$ values into the function $y=\tan x$ then we generate a $\textbf{tangent graph}$.

Like many graphs, the tangent graph has distinct characteristics.

First we note that the trigonometric ratios, including tangent, have periodicity (i.e. they repeat a pattern over a certain period). For the standard tangent graph, the period is $\pi$ which means every $\pi$ units along the $x$-axis the pattern repeats itself.

Another characteristic of the tangent graph is that there are vertical asymptotes. Tangent is the ratio of the sine function to the cosine function

$$\begin{align*} \tan x=\frac{\sin x}{\cos x} \end{align*}$$

so whenever the cosine function equals zero $(\cos x=0)$ then the tangent curve has a vertical asymptote, because we are not able to divide by $0$.

Due to this, we cannot substitute every $x$ value. The domain of tangent is $\{x\in\mathbb{R}, x\neq\frac{\pi}{2},\frac{3\pi}{2}, \frac{5\pi}{2}...\}$.

Between the asymptotes, the tangent graph starts from negative infinity, passes through the $x$-axis halfway between the asymptotes, and then rapidly increases to positive infinity as it approaches the next asymptote. Hence the range of values of tangent is $\{y\in\mathbb{R}\}$. It is not a straight diagonal line but increases rapidly near the asymptotes and has a more shallow gradient as it passes through the $x$-axis.