Graphing the Tangent Function

The tangent function can be graphed by plotting $\left(x,f(x)\right)$ points. The shape of the function can be created by finding the values of the tangent at special angles. However, it is not possible to find the tangent functions for these special angles with the unit circle. We apply the formula, $\displaystyle{ \tan x = \frac{\sin x}{\cos x} }$ to determine the tangent for each value.

We can analyze the graphical behavior of the tangent function by looking at values for some of the special angles. Consider the points below, for which the $x$-coordinates are angles in radians, and the $y$-coordinates are $\tan x$: 

$\displaystyle{ (-\frac{\pi}{2}, \text{undefined}) \quad (-\frac{\pi}{3}, -\sqrt{3}) \quad (-\frac{\pi}{4}, -1) \quad (-\frac{\pi}{6}, -\frac{\sqrt{3}}{3}) \quad (0, 0) \\ (\frac{\pi}{6}, \frac{\sqrt{3}}{3}) \quad (\frac{\pi}{4}, 1) \quad (\frac{\pi}{3}, \sqrt{3}) \quad (\frac{\pi}{2}, \text{undefined}) }$

Notice that $\tan x$ is undefined at $\displaystyle{x = -\frac{\pi}{2}}$ and $\displaystyle{x = \frac{\pi}{2}}$. The above points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. Let's consider the last four points. We can identify that the values of $y$ are increasing as $x$ increases and approaches $\displaystyle{\frac{\pi}{2}}$. We could consider additional points between $\displaystyle{x=0}$ and $\displaystyle{x = \frac{\pi}{2}}$, and we would see that this holds. Likewise, we can see that $y$ decreases as $x$ approaches $\displaystyle{-\frac{\pi}{2}}$, because the outputs get smaller and smaller.

Recall that there are multiple values of $x$ that can give $\cos x = 0$. At any such point, $\tan x$ is undefined because $\displaystyle{\tan x = \frac{\sin x}{\cos x}}$. At values where the tangent function is undefined, there are discontinuities in its graph. At these values, the graph of the tangent has vertical asymptotes.

Graph of the tangent function

The tangent function has vertical asymptotes at $\displaystyle{x = \frac{\pi}{2}}$ and $\displaystyle{x = -\frac{\pi}{2}}$.

Characteristics of the Graph of the Tangent Function

As with the sine and cosine functions, tangent is a periodic function. This means that its values repeat at regular intervals. The period of the tangent function is $\pi$ because the graph repeats itself on $x$-axis intervals of $k\pi$, where $k$ is a constant. In the graph of the tangent function on the interval $\displaystyle{-\frac{\pi}{2}}$ to $\displaystyle{\frac{\pi}{2}}$, we can see the behavior of the graph over one complete cycle of the function. If we look at any larger interval, we will see that the characteristics of the graph repeat.

The graph of the tangent function is symmetric around the origin, and thus is an odd function. In other words, $\text{tan}(-x) = - \text{tan } x$ for any value of $x$. Any two points with opposite values of $x$ produce opposite values of $y$. We can see that this is true by considering the $y$ values of the graph at any opposite values of $x$. Consider $\displaystyle{x=\frac{\pi}{3}}$ and $\displaystyle{x=-\frac{\pi}{3}}$. We already determined above that $\displaystyle{\tan (\frac{\pi}{3}) = \sqrt{3}}$, and $\displaystyle{\tan (-\frac{\pi}{3}) = -\sqrt{3}}$.