Worksheets · Precalculus

Asymptotes worksheet

A factor left in the bottom is an asymptote; a factor that cancels is a hole. The horizontal one is a race between the degrees, and when the top wins by exactly one you get a slant asymptote instead.

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Pick more than one for a sheet that mixes them.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Asymptotes of Rational Functions

Date Period

Find each asymptote or hole.

  1. Find the vertical asymptotes: f(x)=x−3(x−8)(x−1)
  2. Find the horizontal asymptote: f(x)=−5x2−x−35x3+8x2−x
  3. Find the hole: f(x)=(x−2)(x+2)(x−4)(x+2)
  4. Find the vertical asymptotes: f(x)=x+7(x−8)(x−4)
  5. Find the horizontal asymptote: f(x)=56x−1
  6. Find the hole: f(x)=(x−7)(x−3)(x−7)(x+7)
  7. Find the vertical asymptotes: f(x)=x+1(x−8)(x+6)
  8. Find the horizontal asymptote: f(x)=6x3−4x2+2x+95x2+x
  9. Find the hole: f(x)=(x−6)(x+7)(x−6)(x+2)
  10. Find the vertical asymptotes: f(x)=x+1(x−5)(x+2)
  11. Find the horizontal asymptote: f(x)=−6x3+9x2+26x3+8x2−1
  12. Find the hole: f(x)=(x−2)(x+2)(x−6)(x+2)
  13. Find the vertical asymptotes: f(x)=x+8(x+2)(x+5)
  14. Find the horizontal asymptote: f(x)=−6x3−x2+9x−25x2−2x
  15. Find the hole: f(x)=(x−3)(x+5)(x−4)(x−3)
  16. Find the vertical asymptotes: f(x)=x−1(x−7)(x+2)
  17. Find the horizontal asymptote: f(x)=−56x
  18. Find the hole: f(x)=(x−4)(x+3)(x−1)(x+3)
  19. Find the vertical asymptotes: f(x)=x−2(x−6)(x+1)
  20. Find the horizontal asymptote: f(x)=65x−7

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Asymptotes of Rational FunctionsAnswer keyVersion 1

Date Period

Find each asymptote or hole.

  1. Find the vertical asymptotes: f(x)=x−3(x−8)(x−1)x=1, x=8
  2. Find the horizontal asymptote: f(x)=−5x2−x−35x3+8x2−xy=0
  3. Find the hole: f(x)=(x−2)(x+2)(x−4)(x+2)x=−2
  4. Find the vertical asymptotes: f(x)=x+7(x−8)(x−4)x=4, x=8
  5. Find the horizontal asymptote: f(x)=56x−1y=0
  6. Find the hole: f(x)=(x−7)(x−3)(x−7)(x+7)x=7
  7. Find the vertical asymptotes: f(x)=x+1(x−8)(x+6)x=−6, x=8
  8. Find the horizontal asymptote: f(x)=6x3−4x2+2x+95x2+xNone
  9. Find the hole: f(x)=(x−6)(x+7)(x−6)(x+2)x=6
  10. Find the vertical asymptotes: f(x)=x+1(x−5)(x+2)x=−2, x=5
  11. Find the horizontal asymptote: f(x)=−6x3+9x2+26x3+8x2−1y=−1
  12. Find the hole: f(x)=(x−2)(x+2)(x−6)(x+2)x=−2
  13. Find the vertical asymptotes: f(x)=x+8(x+2)(x+5)x=−5, x=−2
  14. Find the horizontal asymptote: f(x)=−6x3−x2+9x−25x2−2xNone
  15. Find the hole: f(x)=(x−3)(x+5)(x−4)(x−3)x=3
  16. Find the vertical asymptotes: f(x)=x−1(x−7)(x+2)x=−2, x=7
  17. Find the horizontal asymptote: f(x)=−56xy=0
  18. Find the hole: f(x)=(x−4)(x+3)(x−1)(x+3)x=−3
  19. Find the vertical asymptotes: f(x)=x−2(x−6)(x+1)x=−1, x=6
  20. Find the horizontal asymptote: f(x)=65x−7y=0

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

f(x)=x−1(x+2)(x−3)

  1. The bottom is zero at -2 and 3.
  2. The top is not zero at either.
  3. So both are vertical asymptotes.

The answer is x=−2, x=3.

Where students go wrong

Calling a canceled factor an asymptote. If it divides out of the top as well, the graph has a hole there, not a wall, and the sheet is testing whether you noticed.

Also called vertical asymptotes, horizontal asymptotes, slant asymptote, holes in a graph or graphs of rational functions.

What you can put on this worksheet

Vertical asymptotes
Find the vertical asymptotes: f(x)=x−1x−4 → x=4
The horizontal asymptote
Find the horizontal asymptote: f(x)=−3x2+32x2+x → y=−32
Holes
Find the hole: f(x)=x−3(x−3)(x−1) → x=3
Slant asymptotes
Find the slant asymptote: f(x)=2x2+10x+14x+3 → y=2x+4

Questions about these worksheets

Yes. Every asymptotes sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the asymptotes questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: vertical asymptotes, the horizontal asymptote, holes and slant asymptotes. Tick as many as you want and set how many of each, or let it spread them evenly.

Calling a canceled factor an asymptote. If it divides out of the top as well, the graph has a hole there, not a wall, and the sheet is testing whether you noticed.