Chapter 6.5, Problem 3AYU

The graph of $y=\tan x$ is symmetric with respect to the ______ and has vertical asymptotes at ________________.

To determine

To find: The graph of the tangent function is symmetric with respect to the origin and has vertical asymptotes.

Answer to Problem 3AYU

Solution:

Origin; odd multiples of $\frac{\pi }{2}$.

Explanation of Solution

Given:

$y=\tan x$

Calculation:

The graph of the tangent function can be plotted using a graphing utility. This is shown below.

We note that the graph of the tangent function is symmetric with respect to the origin.

According to one of the properties for the tangent function, the vertical asymptotes occur at ,

$x=\dots -\frac{3\pi }{2},-\frac{\pi }{2},\frac{\pi }{2} ,\frac{3\pi }{2}\dots .$

Thus, the vertical asymptotes of the tangent function at odd multiples of $\frac{\pi }{2}$.

Therefore, the sentence can be completed as follows:

The graph of the tangent function is symmetric with respect to the origin and has vertical asymptotes at odd multiples of $\frac{\pi }{2}$.