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Example Question #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions
Given the function , determine the equation of all the vertical asymptotes across the domain. Let
be an integer.
For the function , it is not necessary to graph the function. The y-intercept does not affect the location of the asymptotes.
Recall that the parent function has an asymptote at
for every
period.
Set the inner quantity of equal to zero to determine the shift of the asymptote.
This indicates that there is a zero at , and the tangent graph has shifted
units to the right. As a result, the asymptotes must all shift
units to the right as well. The period of the tangent graph is
.
Example Question #2 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions
Which of the choices represents asymptote(s), if any?
Factor the numerator and denominator.
Notice that the terms will cancel. The hole will be located at
because this is a removable discontinuity.
The denominator cannot be equal to zero. Set the denominator to find the location where the x-variable cannot exist.
The asymptote is located at .
Example Question #3 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions
Find the vertical asymptote of the equation.
There are no vertical asymptotes.
To find the vertical asymptotes, we set the denominator of the function equal to zero and solve.
Example Question #4 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions
Given the function , determine the equation of all vertical asymptotes across the domain. Let
be any integer.
Example Question #5 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions
Given the function , determine the equation of all vertical asymptotes across the domain. Let
be any integer.
Example Question #6 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions
Which of the following represents the asymptotes for the general parent function ?
If you do not have these asymptotes memorized, they can be easily derived. Write in terms of
.
Now we need to solve for since it is the denominator of the function. When the denominator of a function is equal to zero, there is a vertical asymptote because that function is then undefined.
when
. So for any integer
, we say that there is a vertical asymptote for
when
.
Example Question #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions
Assume that there is a vertical asymptote for the function at
, solve for
from the equation of all vertical asymptotes at
.
We know that the parent function has vertical asymptotes at
. So now we will set the inner quantity of the
function equal to zero to find the shift of the asymptote.
Now we will add this to the parent function equation for vertical asymptotes
Now we will set this equation for the given vertical asymptote at
Example Question #8 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions
True or false: There is a vertical asymptote for at
There is insufficient information to answer the question.
The statement is false.
The statement is true.
The statement is true.
We know that the parent function of has asymptotes at
where
is any integer. Considering
we can set the asymptotic equation equal to this one and solve for
to see if
is an integer. If
is an integer, then there is a vertical asymptote here.
So when , then
, so the given statement is true.
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