Algebra II : Asymptotes

Study concepts, example questions & explanations for Algebra II

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Asymptotes

What is the horizontal asymptote of the graph of the equation  ?

Possible Answers:

Correct answer:

Explanation:

The asymptote of this equation can be found by observing that  regardless of . We are thus solving for the value of as approaches zero.

So the value that  cannot exceed is , and the line  is the asymptote.

Example Question #2 : Asymptotes

What is/are the asymptote(s) of the graph of the function

 ?

Possible Answers:

 

Correct answer:

Explanation:

An exponential equation of the form  has only one asymptote - a horizontal one at . In the given function, , so its one and only asymptote is .

 

 

Example Question #27 : Exponents

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 #3 : Asymptotes

Consider the exponential function . Determine if there are any asymptotes and where they lie on the graph.

Possible Answers:

One vertical asymptote at .

One horizontal asymptote at .

No asymptotes.  goes to positive  infinity in both the  and  directions.

One vertical asymptote at .

Correct answer:

One horizontal asymptote at .

Explanation:

For positive  values,  increases exponentially in the  direction and goes to positive infinity, so there is no asymptote on the positive -axis. For negative  values, as  decreases, the term  becomes closer and closer to zero so  approaches  as we move along the negative  axis. As the graph below shows, this is forms a horizontal asymptote.

Exp_asymp

Example Question #4 : Asymptotes

Determine the asymptotes, if any:  

Possible Answers:

Correct answer:

Explanation:

Factorize both the numerator and denominator.

Notice that one of the binomials will cancel.

The domain of this equation cannot include .

The simplified equation is:

Since the  term canceled, the  term will have a hole instead of an asymptote.   

Set the denominator equal to zero.

Subtract one from both sides.

There will be an asymptote at only:  

The answer is:  

Example Question #5 : Asymptotes

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 #6 : Asymptotes

Where is an asymptote located, if any?   

Possible Answers:

Correct answer:

Explanation:

Factor the numerator and denominator.

Rewrite the equation.

Notice that the  will cancel.  This means that the root of  will be a hole instead of an asymptote.

Set the denominator equal to zero and solve for x.

An asymptote is located at:  

The answer is:  

Learning Tools by Varsity Tutors

Incompatible Browser

Please upgrade or download one of the following browsers to use Instant Tutoring: