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.

Answer 2

Tangent is sine over cosine. Since sine and cosine are periodic, then tangent has to be, as well.–π to –π/2: The tangent will be zero wherever its numerator (the sine) is zero. This happens at 0, π, 2π, 3π, etc, and at –π, –2π, –3π, etc. The tangent will be undefined wherever its denominator (the cosine) is zero. A zero in the denominator means you'll have a vertical asymptote. So the tangent will have vertical asymptotes wherever the cosine is zero.