To do this, we can analyze the graphs of the following parent functions.
y_1 = tan x y_2 = cot x y_3 = sec x y_4 = csc x
Let's start by taking a look at the parent tangent function.
Note that tan x increases without bound as x approaches π/2 from the left, and it decreases without bound as x approaches - π/2 from the right. That is why y_1=tan x has vertical asymptotes at x=π/2 and x=- π/2. In addition, since the period of this function is π, vertical asymptotes also occur at x=π/2+nπ, where n is an integer. Next, let's analyze y_2, y_3 and y_4.
The parent cotangent, secant, and cosecant functions have vertical asymptotes as well. This is because these functions are undefined for some values of x. In general, all the tangent, cotangent, secant, and cosecant functions have vertical asymptotes. With this in mind, we can complete the given sentence.
The graphs of the tangent, cotangent, secant, and cosecant functions have vertical asymptotes.
