Worksheets · Algebra 2

Graphing rational functions worksheet for Algebra 2

A rational function has two lines it never quite reaches. Draw those first, in dashes, and each branch has only one place it can go.

Also taught in Precalculus, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Graphing Rational Functions

Date Period

Graph each function, or answer as asked.

  1. Graph a rational function

    Graph y=−3x+3+2−10−10−5−5551010
  2. Graph y=−1x−4−10−10−5−5551010
  3. Graph y=1x−10−10−5−5551010
  4. Graph y=2x+3−10−10−5−5551010
  5. Graph y=−3x−1+3−10−10−5−5551010
  6. Graph y=3x+3+1−10−10−5−5551010
  7. Graph y=−3x−3−3−10−10−5−5551010

  1. Find asymptotes, holes and xx-intercepts

    Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=x2+4xx2+4x+3
  2. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=x+4(x+1)(x−1)
  3. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=2x2−2x
  4. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=(x−5)(x+3)(x−5)(x−3)(x−1)
  5. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=2x2−25
  6. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=x+1x(x+1)
  7. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=−x2−10x−25x2−3x−4

  1. Rewrite in the form y=ax−h+ky = \dfrac{a}{x - h} + k

    Write y=−4x−9x+2 in the form y=ax−h+k
  2. Write y=x+2x+6 in the form y=ax−h+k
  3. Write y=3x−8x+2 in the form y=ax−h+k
  4. Write y=−x+6x+3 in the form y=ax−h+k
  5. Write y=3x+4x−5 in the form y=ax−h+k
  6. Write y=−5x+6x+6 in the form y=ax−h+k

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

Sigma Prep · sigmaprep.io/worksheets

Name

Graphing Rational FunctionsAnswer keyVersion 1

Date Period

Graph each function, or answer as asked.

  1. Graph a rational function

    Graph y=−3x+3+2−10−10−5−5551010Asymptotes x=−3 and y=2
  2. Graph y=−1x−4−10−10−5−5510105Asymptotes x=4 and y=0
  3. Graph y=1x−10−10−5−5551010Asymptotes x=0 and y=0
  4. Graph y=2x+3−10−10−5−5551010Asymptotes x=0 and y=3
  5. Graph y=−3x−1+3−10−10−5−5510105Asymptotes x=1 and y=3
  6. Graph y=3x+3+1−10−10−5551010−5Asymptotes x=−3 and y=1
  7. Graph y=−3x−3−3−10−10−5−5551010Asymptotes x=3 and y=−3

  1. Find asymptotes, holes and xx-intercepts

    Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=x2+4xx2+4x+3Vertical: x=−3,x=−1; holes: none; horizontal: y=1; x-intercepts: −4,0
  2. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=x+4(x+1)(x−1)Vertical: x=−1,x=1; holes: none; horizontal: y=0; x-intercepts: −4
  3. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=2x2−2xVertical: x=0; holes: none; horizontal: none; x-intercepts: −1,1
  4. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=(x−5)(x+3)(x−5)(x−3)(x−1)Vertical: x=1,x=3; holes: (5,1); horizontal: y=0; x-intercepts: −3
  5. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=2x2−25Vertical: x=−5,x=5; holes: none; horizontal: y=0; x-intercepts: none
  6. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=x+1x(x+1)Vertical: x=0; holes: (−1,−1); horizontal: y=0; x-intercepts: none
  7. Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=−x2−10x−25x2−3x−4Vertical: x=−1,x=4; holes: none; horizontal: y=−1; x-intercepts: −5

  1. Rewrite in the form y=ax−h+ky = \dfrac{a}{x - h} + k

    Write y=−4x−9x+2 in the form y=ax−h+ky=−1x+2−4
  2. Write y=x+2x+6 in the form y=ax−h+ky=−4x+6+1
  3. Write y=3x−8x+2 in the form y=ax−h+ky=−14x+2+3
  4. Write y=−x+6x+3 in the form y=ax−h+ky=9x+3−1
  5. Write y=3x+4x−5 in the form y=ax−h+ky=19x−5+3
  6. Write y=−5x+6x+6 in the form y=ax−h+ky=36x+6−5

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

How to do these

Graph y=2x−3+1

  1. The bottom is zero at x = 3, so that is the upright asymptote.
  2. Far out in either direction the fraction fades away and y settles at 1.
  3. One branch sits above and right of the crossing, the other below and left.

The answer is Asymptotes x=3 and y=1.

Where students go wrong

Reading the upright asymptote with the wrong sign. x - 3 makes the bottom zero at +3. The other slip is drawing one branch and stopping: there are always two, one on each side.

Also called rational graphs, asymptote graphs or plotting rational functions.

What you can put on this worksheet

A rational function
Graph y=3x+4+1 → Asymptotes x=−4 and y=1
Asymptotes, holes and intercepts
Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x)=(x−1)(x−3)x(x+5) → Vertical: x=−5,x=0; holes: none; horizontal: y=1; x-intercepts: 1,3
Rewrite as a/(x - h) + k
Write y=3xx−1 in the form y=ax−h+k → y=3x−1+3
Factor first, holes, domain and range, write one
Write a function of the form f(x)=ax−h+k with a vertical asymptote at x=29 and a horizontal asymptote at y=2 → f(x)=5x−29+2

Questions about these worksheets

Yes. Every graphing rational functions 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.

Algebra 2 is usually taken around 10th or 11th 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 graphing rational functions 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 two 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: a rational function, asymptotes, holes and intercepts, rewrite as a/(x - h) + k and factor first, holes, domain and range, write one. Tick as many as you want and set how many of each, or let it spread them evenly.

Reading the upright asymptote with the wrong sign. x - 3 makes the bottom zero at +3. The other slip is drawing one branch and stopping: there are always two, one on each side.