Precalculus : Find the Equations of Vertical Asymptotes of Tangent, Cosecant, Secant, and Cotangent Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

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.

Possible Answers:

Correct answer:

Explanation:

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 #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions

Which of the choices represents asymptote(s), if any?   

Possible Answers:

Correct answer:

Explanation:

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 #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions

Find the vertical asymptote of the equation.

Possible Answers:

There are no vertical asymptotes.

Correct answer:

Explanation:

To find the vertical asymptotes, we set the denominator of the function equal to zero and solve.

Example Question #1 : 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.

Possible Answers:

Correct answer:

Explanation:

We know that the parent function  has vertical asymptotes at  where  is any integer. We will set the quantity inside the  function equal to zero to solve for the shift of the asymptote.

 

Now we must add this to the asymptotes of the parent function:

Example Question #1 : 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.

Possible Answers:

Correct answer:

Explanation:

We know that the parent function  has asymptotes at  where  is any integer.  We will set the quantity within the  function equal to zero in order to find the shift of the asymptote. 

 

Now we must add this to the asymptotes of the parent function:

Example Question #1 : 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 ?

Possible Answers:

Correct answer:

Explanation:

 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 .

Possible Answers:

Correct answer:

Explanation:

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 #1 : Find The Equations Of Vertical Asymptotes Of Tangent, Cosecant, Secant, And Cotangent Functions

True or false: There is a vertical asymptote for  at 

Possible Answers:

There is insufficient information to answer the question.

The statement is false.

The statement is true.

Correct answer:

The statement is true.

Explanation:

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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