Question

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

Answer 1

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

Firstly we know that,
`\tan x=(\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) (\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) (\pi)/(2)`, n is the set of integers.

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