Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x) = (x^2 + 4x + 3)/(x^2 + 5x + 4)

Vertical: x = -4; holes: (-1, 2/3); horizontal: y = 1; x-intercepts: -3

Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x) = (-x - 3)/(x^2 - x - 2)

Vertical: x = -1, x = 2; holes: none; horizontal: y = 0; x-intercepts: -3

Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x) = (-x^2 - 3x - 2)/(x^2 - 7x + 10)

Vertical: x = 2, x = 5; holes: none; horizontal: y = -1; x-intercepts: -2, -1

Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x) = (-x)/(x^2 + 2x - 15)

Vertical: x = -5, x = 3; holes: none; horizontal: y = 0; x-intercepts: 0

Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x) = (-x + 3)/(x^2 - 2x - 15)

Vertical: x = -3, x = 5; holes: none; horizontal: y = 0; x-intercepts: 3

Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x) = (3x + 3)/(x^2 + 9x + 20)

Vertical: x = -5, x = -4; holes: none; horizontal: y = 0; x-intercepts: -1

Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x) = (2x + 8)/(x^2 - 4x - 5)

Vertical: x = -1, x = 5; holes: none; horizontal: y = 0; x-intercepts: -4

Find the vertical asymptotes, holes, horizontal asymptote and x-intercepts of f(x) = (x - 3)/((x + 5)(x))

Vertical: x = -5, x = 0; holes: none; horizontal: y = 0; x-intercepts: 3

Worksheets · Precalculus

Precalculus graphing rational functions worksheet

Precalculus rational graphs need a full feature list before anything is drawn: holes, vertical and horizontal or slant asymptotes, and intercepts. Factor the top and bottom first, because a factor that cancels makes a hole, not an asymptote.

Also taught in Algebra 2, 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.



Sigma Prep · sigmaprep.io/worksheets Name Graphing Rational Functions Date Period Graph each function, or answer as asked. Graph $y = \frac{2}{x - 1} + 2$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{x^{2} + 4 x + 3}{x^{2} + 5 x + 4}$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{- 4 x - 9}{x + 2}$ Graph $y = \frac{- 2}{x + 1} + 3$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x - 3}{x^{2} - x - 2}$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{2 x - 3}{x - 1}$ Graph $y = \frac{3}{x + 2} - 2$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x^{2} - 3 x - 2}{x^{2} - 7 x + 10}$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{- 5 x}{x + 6}$ Graph $y = \frac{- 1}{x - 4} - 2$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x}{x^{2} + 2 x - 15}$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{- x - 4}{x + 6}$ Graph $y = \frac{1}{x + 3} + 2$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x + 3}{x^{2} - 2 x - 15}$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{x + 9}{x + 1}$ Graph $y = \frac{3}{x - 4} - 3$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{3 x + 3}{x^{2} + 9 x + 20}$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{- 4 x + 9}{x + 5}$ Graph $y = \frac{- 2}{x + 2} - 3$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{2 x + 8}{x^{2} - 4 x - 5}$ 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.

Graph $y = \frac{2}{x - 1} + 2$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{x^{2} + 4 x + 3}{x^{2} + 5 x + 4}$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{- 4 x - 9}{x + 2}$
Graph $y = \frac{- 2}{x + 1} + 3$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x - 3}{x^{2} - x - 2}$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{2 x - 3}{x - 1}$
Graph $y = \frac{3}{x + 2} - 2$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x^{2} - 3 x - 2}{x^{2} - 7 x + 10}$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{- 5 x}{x + 6}$
Graph $y = \frac{- 1}{x - 4} - 2$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x}{x^{2} + 2 x - 15}$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{- x - 4}{x + 6}$
Graph $y = \frac{1}{x + 3} + 2$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x + 3}{x^{2} - 2 x - 15}$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{x + 9}{x + 1}$
Graph $y = \frac{3}{x - 4} - 3$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{3 x + 3}{x^{2} + 9 x + 20}$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{- 4 x + 9}{x + 5}$
Graph $y = \frac{- 2}{x + 2} - 3$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{2 x + 8}{x^{2} - 4 x - 5}$



Sigma Prep · sigmaprep.io/worksheets Name Graphing Rational FunctionsAnswer keyVersion 1 Date Period Graph each function, or answer as asked. Graph $y = \frac{2}{x - 1} + 2$ Asymptotes $x = 1$ and $y = 2$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{x^{2} + 4 x + 3}{x^{2} + 5 x + 4}$ Vertical: $x = - 4$ ; holes: $(- 1, \frac{2}{3})$ ; horizontal: $y = 1$ ; $x$ -intercepts: $- 3$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{- 4 x - 9}{x + 2}$ $y = \frac{- 1}{x + 2} - 4$ Graph $y = \frac{- 2}{x + 1} + 3$ Asymptotes $x = - 1$ and $y = 3$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x - 3}{x^{2} - x - 2}$ Vertical: $x = - 1, x = 2$ ; holes: none; horizontal: $y = 0$ ; $x$ -intercepts: $- 3$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{2 x - 3}{x - 1}$ $y = \frac{- 1}{x - 1} + 2$ Graph $y = \frac{3}{x + 2} - 2$ Asymptotes $x = - 2$ and $y = - 2$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x^{2} - 3 x - 2}{x^{2} - 7 x + 10}$ Vertical: $x = 2, x = 5$ ; holes: none; horizontal: $y = - 1$ ; $x$ -intercepts: $- 2, - 1$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{- 5 x}{x + 6}$ $y = \frac{30}{x + 6} - 5$ Graph $y = \frac{- 1}{x - 4} - 2$ Asymptotes $x = 4$ and $y = - 2$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x}{x^{2} + 2 x - 15}$ Vertical: $x = - 5, x = 3$ ; holes: none; horizontal: $y = 0$ ; $x$ -intercepts: $0$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{- x - 4}{x + 6}$ $y = \frac{2}{x + 6} - 1$ Graph $y = \frac{1}{x + 3} + 2$ Asymptotes $x = - 3$ and $y = 2$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x + 3}{x^{2} - 2 x - 15}$ Vertical: $x = - 3, x = 5$ ; holes: none; horizontal: $y = 0$ ; $x$ -intercepts: $3$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{x + 9}{x + 1}$ $y = \frac{8}{x + 1} + 1$ Graph $y = \frac{3}{x - 4} - 3$ Asymptotes $x = 4$ and $y = - 3$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{3 x + 3}{x^{2} + 9 x + 20}$ Vertical: $x = - 5, x = - 4$ ; holes: none; horizontal: $y = 0$ ; $x$ -intercepts: $- 1$ Rewrite in the form $y = \frac{a}{x - h} + k$ : $y = \frac{- 4 x + 9}{x + 5}$ $y = \frac{29}{x + 5} - 4$ Graph $y = \frac{- 2}{x + 2} - 3$ Asymptotes $x = - 2$ and $y = - 3$ Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{2 x + 8}{x^{2} - 4 x - 5}$ Vertical: $x = - 1, x = 5$ ; holes: none; horizontal: $y = 0$ ; $x$ -intercepts: $- 4$ 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.

Graph $y = \frac{2}{x - 1} + 2$ Asymptotes $x = 1$ and $y = 2$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{x^{2} + 4 x + 3}{x^{2} + 5 x + 4}$ Vertical: $x = - 4$ ; holes:
$(- 1, \frac{2}{3})$ ; horizontal: $y = 1$ ; $x$ -intercepts: $- 3$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{- 4 x - 9}{x + 2}$ $y = \frac{- 1}{x + 2} - 4$
Graph $y = \frac{- 2}{x + 1} + 3$ Asymptotes $x = - 1$ and $y = 3$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x - 3}{x^{2} - x - 2}$ Vertical: $x = - 1, x = 2$ ; holes: none; horizontal: $y = 0$ ; $x$ -intercepts: $- 3$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{2 x - 3}{x - 1}$ $y = \frac{- 1}{x - 1} + 2$
Graph $y = \frac{3}{x + 2} - 2$ Asymptotes $x = - 2$ and $y = - 2$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x^{2} - 3 x - 2}{x^{2} - 7 x + 10}$ Vertical: $x = 2, x = 5$ ; holes: none; horizontal: $y = - 1$ ; $x$ -intercepts: $- 2, - 1$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{- 5 x}{x + 6}$ $y = \frac{30}{x + 6} - 5$
Graph $y = \frac{- 1}{x - 4} - 2$ Asymptotes $x = 4$ and $y = - 2$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x}{x^{2} + 2 x - 15}$ Vertical: $x = - 5, x = 3$ ; holes: none; horizontal: $y = 0$ ; $x$ -intercepts: $0$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{- x - 4}{x + 6}$ $y = \frac{2}{x + 6} - 1$
Graph $y = \frac{1}{x + 3} + 2$ Asymptotes $x = - 3$ and $y = 2$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{- x + 3}{x^{2} - 2 x - 15}$ Vertical: $x = - 3, x = 5$ ; holes: none; horizontal: $y = 0$ ; $x$ -intercepts: $3$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{x + 9}{x + 1}$ $y = \frac{8}{x + 1} + 1$
Graph $y = \frac{3}{x - 4} - 3$ Asymptotes $x = 4$ and $y = - 3$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{3 x + 3}{x^{2} + 9 x + 20}$ Vertical: $x = - 5, x = - 4$ ; holes: none; horizontal: $y = 0$ ; $x$ -intercepts: $- 1$
Rewrite in the form $y = \frac{a}{x - h} + k$ :
$y = \frac{- 4 x + 9}{x + 5}$ $y = \frac{29}{x + 5} - 4$
Graph $y = \frac{- 2}{x + 2} - 3$ Asymptotes $x = - 2$ and $y = - 3$
Find the vertical asymptotes, holes, horizontal asymptote and $x$ -intercepts of $f (x) = \frac{2 x + 8}{x^{2} - 4 x - 5}$ Vertical: $x = - 1, x = 5$ ; holes: none; horizontal: $y = 0$ ; $x$ -intercepts: $- 4$

How to do these

Find the features of $f (x) = \frac{x^{2} - x - 2}{x^{2} - 4}$

Factor: (x - 2)(x + 1) / ((x - 2)(x + 2)).
x - 2 cancels, so x = 2 is a hole. The simplified form (x + 1)/(x + 2) gives y = 3/4 there.
x = -2 is the vertical asymptote. Equal degrees give y = 1, and the top is zero at x = -1.

The answer is Hole $(2, \frac{3}{4})$ , VA $x = - 2$ , HA $y = 1$ , $x$ -int $(- 1, 0)$ .

Where students go wrong

Calling every zero of the denominator a vertical asymptote. Factor first: if the factor cancels with the top, the graph has a hole there instead.

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

What you can put on this worksheet

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

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.

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 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.

Precalculus graphing rational functions worksheet

Select your difficulty

Graphing Rational Functions

Graphing Rational FunctionsAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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