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How to Find Slant Asymptotes Download Article A step-by-step math guide to help you calculate and graph an asymptote Explore this Article Steps Steps Other Sections Questions & Answers Video Tips and Warnings Related Articles References Co-authored by David Jia
Last Updated: January 8, 2025 References
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This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 139,845 times.
An asymptote of a polynomial is any straight line that a graph approaches but never touches. It can be vertical or horizontal, or it can be a slant asymptote – an asymptote with a slope.[1]XResearch source A slant asymptote of a polynomial exists whenever the degree of the numerator is higher than the degree of the denominator.[2]XResearch source
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1 Check the numerator and denominator of your polynomial. Make sure that the degree of the numerator (in other words, the highest exponent in the numerator) is greater than the degree of the denominator.[3]XResearch source If it is, a slant asymptote exists and can be found. .
As an example, look at the polynomial x^2 + 5x + 2 / x + 3. The degree of its numerator is greater than the degree of its denominator because the numerator has a power of 2 (x^2) while the denominator has a power of only 1. Therefore, you can find the slant asymptote. The graph of this polynomial is shown in the picture.
2 Create a long division problem. Place the numerator (the dividend) inside the division box, and place the denominator (the divisor) on the outside.[4]XResearch source
For the example above, set up a long division problem with x^2 + 5x + 2 as the dividend and x + 3 as the divisor.
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3 Find the first factor. Look for a factor that, when multiplied by the highest degree term in the denominator, will result in the same term as the highest degree term of the dividend. Write that factor above the division box.
In the example above, you would look for a factor that, when multiplied by x, would result in the same term as the highest degree of x^2. In this case, that’s x.Write the x above the division box.
4 Find the product of the factor and the whole divisor. Multiply to get your product, and write it beneath the dividend.
In the example above, the product of x and x + 3 is x^2 + 3x. Write it under the dividend, as shown.
5 Subtract. Take the lower expression under the division box and subtract it from the upper expression. Draw a line and note the result of your subtraction underneath it.
In the example above, subtract x^2 + 3x from x^2 + 5x + 2. Draw a line and note the result, 2x + 2, underneath it, as shown.
6 Continue dividing. Repeat these steps, using the result of your subtraction problem as your new dividend.
In the example above, note that if you multiply 2 by the highest term of the divisor (x), you get the highest degree term of the dividend, which is now 2x + 2. Write the 2 on top of the division box by adding it to first factor, making it x + 2. Write the product of the factor and the divisor beneath the dividend, and subtract again, as shown.
7 Stop when you get an equation of a line. You do not have to perform the long division all the way to the end. Continue only until you get the equation of a line in the form ax + b, where a and b can be any numbers.
In the example above, you can now stop. The equation of your line is x + 2.
8 Draw the line alongside the graph of the polynomial. Graph your line to verify that it is actually an asymptote.
In the example above, you would need to graph x + 2 to see that the line moves alongside the graph of your polynomial but never touches it, as shown below. So x + 2 is indeed a slant asymptote of your polynomial.
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Question Where did the two polynomials come from? Technist Community Answer The (x^2 + 5x + 2) / (x+3) is part of an example. It is possible to tell that there will be a slant asymptote because the polynomial in the numerator has a degree than the polynomial in the denominator. With this in mind, you can make up as many problems or examples as you want. Thanks! We're glad this was helpful.Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 4 Helpful 9
Question What does the remainder, once you've divided, mean in terms of the asymptote? Community Answer It represents the vertical distance between the curve and the asymptote. It should approach zero as |x| approaches infinity. You might investigate whether the remainder is positive or negative, since that tells you whether the curve approaches the asymptote from above or from below. Thanks! We're glad this was helpful.Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 0 Helpful 11
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In engineering, asymptotes are tremendously helpful, as they create approximate linear behaviors, which are easy to analyze, for non-linear behaviors. Thanks Helpful 0 Not Helpful 1
The length of your x-axis should be kept small, so you can clearly see that the asymptote does not touch the graph of your polynomial. Thanks Helpful 0 Not Helpful 0
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Co-authored by: David Jia Math Tutor This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 139,845 times. 50 votes - 86% Co-authors: 6Updated: January 8, 2025 Views: 139,845 Categories: Algebra In other languages Russian Spanish French Indonesian
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Download Article A step-by-step math guide to help you calculate and graph an asymptote Explore this Article Steps Steps Other Sections Questions & Answers Video Tips and Warnings Related Articles References Co-authored by David Jia 
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Co-authored by: David Jia Math Tutor This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 139,845 times. 50 votes - 86% Co-authors: 6 Updated: January 8, 2025 Views: 139,845 Categories: Algebra In other languages Russian Spanish French Indonesian 
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1 Check the numerator and denominator of your polynomial. Make sure that the degree of the numerator (in other words, the highest exponent in the numerator) is greater than the degree of the denominator.[3] If it is, a slant asymptote exists and can be found. . 
2 Create a long division problem. Place the numerator (the dividend) inside the division box, and place the denominator (the divisor) on the outside.[4] 
3 Find the first factor. Look for a factor that, when multiplied by the highest degree term in the denominator, will result in the same term as the highest degree term of the dividend. Write that factor above the division box. 
4 Find the product of the factor and the whole divisor. Multiply to get your product, and write it beneath the dividend. 
5 Subtract. Take the lower expression under the division box and subtract it from the upper expression. Draw a line and note the result of your subtraction underneath it. 
6 Continue dividing. Repeat these steps, using the result of your subtraction problem as your new dividend. 
7 Stop when you get an equation of a line. You do not have to perform the long division all the way to the end. Continue only until you get the equation of a line in the form ax + b, where a and b can be any numbers. 
8 Draw the line alongside the graph of the polynomial. Graph your line to verify that it is actually an asymptote. 
Technist Community Answer The (x^2 + 5x + 2) / (x+3) is part of an example. It is possible to tell that there will be a slant asymptote because the polynomial in the numerator has a degree than the polynomial in the denominator. With this in mind, you can make up as many problems or examples as you want. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 4 Helpful 9 
Community Answer It represents the vertical distance between the curve and the asymptote. It should approach zero as |x| approaches infinity. You might investigate whether the remainder is positive or negative, since that tells you whether the curve approaches the asymptote from above or from below. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 0 Helpful 11 
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