Question

Using complete sentences, explain the key features of the graph of the tangent function.

Answer 1

Answer:

We are given the tangent function $f(x)=\tan x$.

Firstly we know that, $\tan x=\frac{\sin x}{\cos x}$, where $\sin x$ is the sine function and $\cos x$ is the cosine function.

Now, tangent function will be zero when its numerator is zero.

i.e. $\tan x=0$ when $\sin x=0$.

i.e. $\tan x=0$ when $x=n \pi$, where n is the set of integers.

So, tangent function crosses x-axis at $n \pi$, n is the set of integers.

Further, tangent function will be undefined when its denominator is zero.

i.e. $\tan x=0$ when $\cos x=0$.

i.e. $\tan x=0$ when $x=(2n-1) \frac{\pi}{2}$, where n is the set of integers.

Moreover, a zero in the denominator gives vertical asymptotes.

So, tangent function will have vertical asymptotes at $(2n-1) \frac{\pi}{2}$, n is the set of integers.

Therefore, these key features gives us the graph of a tangent function as shown below.