We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Rational Functions

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #382 : Pre Calculus

Find the vertical asymptotes of the following rational function.

$\small \frac{8x^2}{4x^2+31x-8}$

Possible Answers:

$\small x=-\frac{1}{8},4$

$\small x=\frac{1}{4}, -8$

No vertical asymptotes.

$\small x=4,\frac{1}{8}$

$\small \small x=-\frac{1}{4},8$

Correct answer:

$\small x=\frac{1}{4}, -8$

Explanation:

Finding the vertical asymptotes of the rational function $\small \frac{8x^2}{4x^2+31x-8}$ amounts to finding the roots of the denominator, $\small 4x^2+31x-8$ .

It is easy to check, using the quadratic formula,

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{-31\pm\sqrt {31^2-4(4)(-8)}}{2(4)}$

$x=\frac{-31\pm\sqrt {961+128}}{8}$

$x=\frac{-31\pm 33}{8}=\frac{1}{4}, -8$

that the roots, and thus the asymptotes, are $\small x=\frac{1}{4}, -8$ .

Report an Error

Example Question #383 : Pre Calculus

Find the y-intercept and asymptote, respectively, of the following function, if possible. $y=\frac{3x}{x}$

Possible Answers:

$\textup{There are no y-intercepts or asymptotes.}$

$\textup{y-intercept at: }(0,3);\:\textup{Asymptote at: }y=3$

$\textup{y-intercept at: }(0,3);\:\textup{Asymptote at: }x=0$

$\textup{The y-intercept does not exist. Asymptote at: }x=0$

$\textup{y-intercept at: }(0,3);\:\textup{Asymptote at: }y=0$

Correct answer:

$\textup{There are no y-intercepts or asymptotes.}$

Explanation:

Before we start to simplify the problem, it is crucial to immediately identify the domain of this function $y=\frac{3x}{x}$ .

The denominator cannot be zero, since it is undefined to divide numbers by this value. After simplification, the equation is:

y=3

The domain is $x\neq0$ and there is a hole at x=0 since there is a removable discontinuity. There are no asymptotes.

Since it's not possible to substitute x=0 into the original equation, the y-intercept also does not exist.

Therefore, the correct answer is:

$\textup{There are no y-intercepts or asymptotes.}$

Report an Error

Example Question #6 : Find Intercepts And Asymptotes

What is a vertical asymptote of the following function?

$\frac{x^2 + 3x + 6}{x + 2}$

Possible Answers:

y = 3

y = 2

x = 3

x = -3

x = -2

Correct answer:

x = -2

Explanation:

To find the vertical asymptote of a function, we set the denominator equal to .

With our function, we complete this process.

The denominator is x + 2 , so we begin:

x + 2 = 0

x = -2

Report an Error

Example Question #51 : Polynomial Functions

What is the -intercept of the following function?

$\frac{x^2 + 3x + 6}{x + 2}$

Possible Answers:

There is no -intercept.

Correct answer:

Explanation:

The y-intercept of a function is always found by substituting in x = 0 .

We can go through this process for our function.

$\frac{x^2 + 3x + 6}{x + 2}= \frac{0^2 + 3\cdot 0+ 6}{0 + 2} = \frac{ 6}{ 2} = 3$

Report an Error

Example Question #8 : Find Intercepts And Asymptotes

Which of these functions has a vertical asymptote of x = 3 and a slant asymptote of y = x - 8 ?

Possible Answers:

$y = \frac{x^2 - 11x + 24}{x - 3 }$

$y = \frac{x^2 + 11x + 9 }{x - 3 }$

$y = \frac{x^2 - 11x + 7 }{ x - 3 }$

$y = \frac{x^2 + 11x - 7 }{x + 3 }$

$y = \frac{x^2 - 11x + 9 }{x + 3 }$

Correct answer:

$y = \frac{x^2 - 11x + 7 }{ x - 3 }$

Explanation:

In order for the vertical asymptote to be x = 3 , we need the denominator to be x -3 . This gives us three choices of numerators:

x^2 - 11x + 24

x^2 - 11x + 7

x^2 + 11x + 9

If the slant asymptote is x - 8 , we will be able to divide our numerator by x - 3 and get x - 8 with a remainder.

Dividing the first one gives us x - 8 with no remainder.

Dividing the last one gives us x + 14 with a remainder.

The middle numerator would give us what we were after, x - 8 with a remainder of -17.

The answer is

$y = \frac{x^2 - 11x + 7 }{ x - 3 }$

Report an Error

Example Question #9 : Find Intercepts And Asymptotes

Find the zeros and asymptotes for

$y = \frac{2x^2 + 5 x - 3 }{2x^2 + 3x - 2 }$ .

Possible Answers:

Zero: x = -3 ; Asymptotes: $x = -2, \frac{1}{2}$

Zeros: $x = 2, \frac{1}{2}$ ; Asymptotes: $x = 3, \frac{1}{2}$

Zero: x = -3 ; Asymptote: x = -2

Zeros: x = 3, -1 ; Asymptote: x = 2

Zero: $x = \frac{1}{2}$ ; Asymptotes: $x = 2, -\frac{1}{2}$

Correct answer:

Zero: x = -3 ; Asymptote: x = -2

Explanation:

To find the information we're looking for, we should factor this equation:

$y = \frac{(2 x - 1 )(x + 3 )}{(2 x - 1 )(x+ 2 )}$

This means that it simplifies to $y = \frac{x + 3 }{ x + 2 }$ .

When the equation is in the form of a fraction, to find the zero of the function we need to set the numerator equal to zero and solve for the variable.

$\\x+3=0\\x=-3$

To find the asymptote of an equation with a fraction we need to set the denominator of the fraction equal to zero and solve for the variable.

$\\x+2=0\\x=-2$

Therefore our equation has a zero at -3 and an asymptote at -2.

Report an Error

Example Question #10 : Find Intercepts And Asymptotes

Find the slant and vertical asymptotes for the equation

$y = \frac{2x^2 - 5 x + 1 }{x + 3 }$ .

Possible Answers:

Vertical asymptote: x = -3 ; Slant asymptote: y = 2 x + 1

Vertical asymptote: x = -3 ; Slant asymptote: y = 2x^2 - 11 x + 34

Vertical asymptote: x = -3 ; Slant asymptote: y = 2x -1

Vertical asymptote: x = -3 ; Slant asymptote: y = 2x - 8

Vertical asymptote: x = 3 ; Slant asymptote: y = 2x + 1

Correct answer:

Vertical asymptote: x = -3 ; Slant asymptote: y = 2x - 8

Explanation:

To find the vertical asymptote, just set the denominator equal to 0:

x + 3 = 0

x = -3

To find the slant asymptote, divide the numerator by the denominator, but ignore any remainder. You can use long division or synthetic division.

2x-8

$\\x+3\ |\overline{2 x^ 2 - 5 x + 1 }$

$\\ \underline{-(2x^2+3x) }$

-8x+1

$\underline{-(-8x-24)}$

The slant asymptote is

y = 2x - 8 .

Report an Error

Example Question #11 : Find Intercepts And Asymptotes

Find the slant asymptote for

$y = \frac{2x^2 - 13 x - 7 }{ x - 7 }$ .

Possible Answers:

y = 2x - 27

x = -7

y = 2x + 1

x = 7

This graph does not have a slant asymptote.

Correct answer:

This graph does not have a slant asymptote.

Explanation:

By factoring the numerator, we see that this equation is equivalent to

$y = \frac{(2x+1)(x - 7 )}{x - 7}$ .

That means that we can simplify this equation to y = 2x + 1 .

That means that y = 2x + 1 isn't the slant asymptote, but the equation itself.

x = 7 is definitely an asymptote, but a vertical asymptote, not a slant asymptote.

Report an Error

Example Question #391 : Pre Calculus

Find the y-intercept of $y=\frac{3-x}{x-3}+3$ , if any.

Possible Answers:

$\infty$

$\textup{The y-intercept does not exist.}$

Correct answer:

Explanation:

Be careful not to confuse this equation with the linear slope-intercept form. The y-intercept of an equation is the y-value when the x-value is zero.

Substitute the value of x=0 into the equation.

$y=\frac{3-0}{0-3}+3$

Simplify the equation.

$y=\frac{3-0}{0-3}+3 = \frac{3}{-3}+3 = -1+3 = 2$

The y-intercept is:

Report an Error

Example Question #31 : Rational Functions

Find the horizontal asymptote of the function:

$f(x) = \frac{x^2+6x+9}{x^2+5x+6}$

Possible Answers:

y = 2

y = 3

$y = \frac{1}{2}$

y = 0

y = 1

Correct answer:

y = 1

Explanation:

To find the horizontal asymptote, take the leading term of the numerator and the denominator and divide. In this case:

$f(x) = \frac{x^2}{x^2}$

f(x) = 1

Report an Error

View Pre-Calculus Tutors

Kimone
Certified Tutor

Texas Tech University, Bachelor of Science, Mechanical Engineering. Florida State University, Master of Science, Mechanical E...

View Pre-Calculus Tutors

Dean
Certified Tutor

CUNY New York City College of Technology, Bachelor of Engineering, Electrical Engineering. University of Phoenix-Online Campu...

View Pre-Calculus Tutors

Fateme
Certified Tutor

niversity of Bonab, Doctor of Science, Mathematics.

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Algebra Tutors in Seattle, MCAT Tutors in Miami, Computer Science Tutors in Seattle, Math Tutors in San Francisco-Bay Area, Reading Tutors in San Francisco-Bay Area, Chemistry Tutors in New York City, GMAT Tutors in Washington DC, SSAT Tutors in Chicago, LSAT Tutors in Boston, Reading Tutors in Houston

Popular Courses & Classes

SAT Courses & Classes in Boston, SSAT Courses & Classes in Chicago, GMAT Courses & Classes in Boston, MCAT Courses & Classes in Miami, LSAT Courses & Classes in Denver, SAT Courses & Classes in Phoenix, Spanish Courses & Classes in New York City, GRE Courses & Classes in Phoenix, LSAT Courses & Classes in Houston, GRE Courses & Classes in Denver

Popular Test Prep

GMAT Test Prep in Philadelphia, MCAT Test Prep in Denver, ISEE Test Prep in Denver, ISEE Test Prep in Dallas Fort Worth, ACT Test Prep in Denver, ISEE Test Prep in Miami, GRE Test Prep in Seattle, GMAT Test Prep in Chicago, GMAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Washington DC

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Precalculus : Rational Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #382 : Pre Calculus

Example Question #383 : Pre Calculus

Example Question #6 : Find Intercepts And Asymptotes

Example Question #51 : Polynomial Functions

Example Question #8 : Find Intercepts And Asymptotes

Example Question #9 : Find Intercepts And Asymptotes

Example Question #10 : Find Intercepts And Asymptotes

Example Question #11 : Find Intercepts And Asymptotes

Example Question #391 : Pre Calculus

Example Question #31 : Rational Functions

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: