How do I find the vertical asymptotes of a rational function?

Cancel any factor the numerator and the denominator share, then find where the reduced denominator is 0: each real root there is a vertical asymptote. Type 2x^3 + 15x^2 + 22x - 11 over x^2 + 8x + 15 and the page reports x = -5 and x = -3, with the direction of the graph on each side: at x = -5 it goes to ∞ from the left and to -∞ from the right.

What is the difference between a hole and a vertical asymptote?

A hole is a root of the original denominator that cancels completely, so the graph is missing a single point. A vertical asymptote is a root that is still in the denominator after cancelling. (x^2 - 1)/(x - 1) cancels x - 1 and leaves a hole at (1, 2). (x - 1)/(x - 1)^2 cancels x - 1 once too, but one copy is left below, so x = 1 is still a vertical asymptote and there is no hole. The cancellation table shows how many copies of each root sit in the numerator and the denominator and how many are left.

How does it decide between a horizontal and a slant asymptote?

It divides the numerator by the denominator exactly and looks at the quotient. With a nonzero remainder, a constant quotient is a horizontal asymptote: (3x^2 + 1)/(x^2 - 4) gives y = 3, approached from above on both sides. A quotient of degree 1 is a slant asymptote: 2x^3 + 15x^2 + 22x - 11 divided by x^2 + 8x + 15 is 2x - 1 with remainder 4, so the slant asymptote is y = 2x - 1. A quotient of degree 2 or more is not a line, so (x^4 + 1)/(x - 1) is reported with the end behavior y = x³ + x² + x + 1 instead.

Why does (x^2 - 1)/(x - 1) show an end behavior line instead of a slant asymptote?

Because the remainder is 0. After cancelling x - 1 the function is exactly x + 1 everywhere except x = 1, so the graph lies on y = x + 1 rather than approaching it. The page labels that an end behavior line and marks the hole at (1, 2). Some textbooks still call y = x + 1 a slant asymptote, since the difference between the function and the line tends to 0; here that difference is 0 wherever the function is defined.

What happens when a pole is irrational, like the roots of x^2 - 2?

It is never given as a bare rounded decimal. When the polynomial that defines the root is quadratic, as x^2 - 2 is, or as a quadratic you type as its own bracketed factor is, the page writes the closed form as long as it fits in 60 characters, so x^2 over x^2 - 2 has the vertical asymptotes x = -√2 and x = √2. Every irrational root also gets a certificate such as the only root of x² - 2 in (1.4142, 1.4143), an interval a Sturm count proves contains exactly one root. The real root of x^3 - x - 1 has no quadratic factor, so it is shown as ≈ 1.3247 with the interval (1.3247, 1.3248). Approximations can show up to 12 decimal places.

What can I type in the boxes?

Each box takes one polynomial in one letter: numbers, the letter, + - * / ^ and brackets, with products such as (x - 1)^2(x + 2) expanded for you. Fractions such as 2/3 and decimals such as 0.5 are read as exact rationals, pasted x² and minus signs work, and a list such as 1, 0, -1 is read as coefficients from the highest power down. Each box can reach degree 8 after expanding, with coefficients up to 100 digits, in up to 4000 characters. A square root, a trigonometric function, a logarithm, an absolute value, a power with the variable in it such as 2^x, or a fractional power is refused with the reason, because the result would not be a rational function. A negative power such as x^-1 is refused with a pointer to the Denominator box, where that factor belongs.

How is the graph window chosen?

By default the window frames every pole and hole with room on each side, and the y range follows most of the curve. Type any of x from, x to, y from and y to to set them yourself, within ±1,000,000; a blank field keeps the automatic value shown in it. The table under the graph lists the function at nine evenly spaced points of the window and says not defined where a point lands on a pole or a hole.

Can I save the graph or the working?

Download SVG hands the browser the graph as drawn, with its asymptotes, holes and window labels and a text description inside the file. Copy analysis puts the domain, the poles with their one-sided behavior, the holes, the asymptote, the division and the certificates on the clipboard as plain text. Copy LaTeX gives the function, common factor, reduced form, domain, asymptotes, holes and division in an aligned environment. If the browser blocks the clipboard or the download, the exact text appears in a box you can select and copy by hand.

Asymptote Calculator

Type a numerator and a denominator and the page finds every vertical asymptote and hole of the fraction, its horizontal or slant asymptote, and its end behavior. It cancels the greatest common factor exactly while keeping the original domain, so a factor that appears twice below and once above still leaves a vertical asymptote, and one that cancels completely leaves a hole whose height is worked out for you. The numerator is divided by the denominator with every step shown: with a nonzero remainder, a constant quotient is a horizontal asymptote, a quotient of degree 1 is a slant asymptote, and degree 2 or more is reported as end behavior rather than a line, while a zero remainder means the graph is the quotient itself apart from its holes. Every irrational pole or hole position comes with an interval that a Sturm count proves contains exactly one root, and an irrational hole height comes with certified bounds. The graph splits its branches at those certified poles instead of guessing from where the curve jumps, marks holes with open circles and downloads as an SVG, and the analysis copies as plain text or LaTeX.

Your function

Example input

Numerator (degree up to 8)Denominator

Plot window, blank for automatic

x from

x to

y from

y to

Decimal places for approximations

Numbers, one letter, + - * / ^ and brackets. Fractions such as 2/3, decimals, pasted x² and a coefficient list such as 1, 0, -1 are read exactly. Each box can reach degree 8 after expanding, with coefficients up to 100 digits.

Result

f(x) = (2x³ + 15x² + 22x - 11)/(x² + 8x + 15)

vertical x = -5 and x = -3; slant y = 2x - 1

Text equivalent: Graph of f(x) = (2x³ + 15x² + 22x - 11)/(x² + 8x + 15) for x from -8 to 0 and y from -20 to 5. 3 branches split at the vertical asymptotes x = -5 and x = -3, drawn as dashed amber lines. The slant asymptote y = 2x - 1 is a dashed light line. No holes.

The example is already analyzed. Edit the numerator or the denominator to analyze your own function.

Vertical asymptotes: x = -5, x = -3

Holes: none

Slant asymptote: y = 2x - 1

Remainder: 4

Cancellation and domain

Common factor: 1
Reduced: f(x) = (2x³ + 15x² + 22x - 11)/(x² + 8x + 15)
Domain: all real x except x = -5 and x = -3

Root	In denominator	In numerator	Cancelled	Left in reduced denominator	Result	Certificate
x = -5	1	0	0	1	vertical asymptote x = -5, order 1: f(x) → ∞ from the left and -∞ from the right	exact rational
x = -3	1	0	0	1	vertical asymptote x = -3, order 1: f(x) → -∞ from the left and ∞ from the right	exact rational

Division

Step	Quotient term	Subtract	Remainder
1	2x	2x³ + 16x² + 30x	-x² - 8x - 11
2	-1	-x² - 8x - 15	4

Check: (2x - 1)(x² + 8x + 15) + 4 = 2x³ + 15x² + 22x - 11

End behavior and poles

Slant asymptote: y = 2x - 1
As x → ∞, f(x) → ∞ and approaches y = 2x - 1 from above.
As x → -∞, f(x) → -∞ and approaches y = 2x - 1 from above.
x = -5: f(x) → ∞ from the left and -∞ from the right
x = -3: f(x) → -∞ from the left and ∞ from the right

Values in the graph window

x	f(x)
-8	≈ -16.7333
-7	-14.5
-6	≈ -11.6667
-5	not defined (vertical asymptote)
-4	-13
-3	not defined (vertical asymptote)
-2	≈ -3.6667
-1	-2.5
0	≈ -0.7333

Worked examples

Hole(x^2 - 1)/(x - 1)
hole at (1, 2); no vertical asymptote; end behavior line y = x + 1
Simple pole(1)/(x - 2)
vertical x = 2; horizontal y = 0
Slant(2x^3 + 15x^2 + 22x - 11)/(x^2 + 8x + 15)
vertical x = -5 and x = -3; slant y = 2x - 1
Pole kept(x - 1)/((x - 1)^2)
vertical x = 1; horizontal y = 0
Irrational poles(x^2)/(x^2 - 2)
vertical x = -√2 and x = √2; horizontal y = 1

Common questions

How do I find the vertical asymptotes of a rational function?: Cancel any factor the numerator and the denominator share, then find where the reduced denominator is 0: each real root there is a vertical asymptote. Type 2x^3 + 15x^2 + 22x - 11 over x^2 + 8x + 15 and the page reports x = -5 and x = -3, with the direction of the graph on each side: at x = -5 it goes to ∞ from the left and to -∞ from the right.
What is the difference between a hole and a vertical asymptote?: A hole is a root of the original denominator that cancels completely, so the graph is missing a single point. A vertical asymptote is a root that is still in the denominator after cancelling. (x^2 - 1)/(x - 1) cancels x - 1 and leaves a hole at (1, 2). (x - 1)/(x - 1)^2 cancels x - 1 once too, but one copy is left below, so x = 1 is still a vertical asymptote and there is no hole. The cancellation table shows how many copies of each root sit in the numerator and the denominator and how many are left.
How does it decide between a horizontal and a slant asymptote?: It divides the numerator by the denominator exactly and looks at the quotient. With a nonzero remainder, a constant quotient is a horizontal asymptote: (3x^2 + 1)/(x^2 - 4) gives y = 3, approached from above on both sides. A quotient of degree 1 is a slant asymptote: 2x^3 + 15x^2 + 22x - 11 divided by x^2 + 8x + 15 is 2x - 1 with remainder 4, so the slant asymptote is y = 2x - 1. A quotient of degree 2 or more is not a line, so (x^4 + 1)/(x - 1) is reported with the end behavior y = x³ + x² + x + 1 instead.
Why does (x^2 - 1)/(x - 1) show an end behavior line instead of a slant asymptote?: Because the remainder is 0. After cancelling x - 1 the function is exactly x + 1 everywhere except x = 1, so the graph lies on y = x + 1 rather than approaching it. The page labels that an end behavior line and marks the hole at (1, 2). Some textbooks still call y = x + 1 a slant asymptote, since the difference between the function and the line tends to 0; here that difference is 0 wherever the function is defined.
What happens when a pole is irrational, like the roots of x^2 - 2?: It is never given as a bare rounded decimal. When the polynomial that defines the root is quadratic, as x^2 - 2 is, or as a quadratic you type as its own bracketed factor is, the page writes the closed form as long as it fits in 60 characters, so x^2 over x^2 - 2 has the vertical asymptotes x = -√2 and x = √2. Every irrational root also gets a certificate such as the only root of x² - 2 in (1.4142, 1.4143), an interval a Sturm count proves contains exactly one root. The real root of x^3 - x - 1 has no quadratic factor, so it is shown as ≈ 1.3247 with the interval (1.3247, 1.3248). Approximations can show up to 12 decimal places.
What can I type in the boxes?: Each box takes one polynomial in one letter: numbers, the letter, + - * / ^ and brackets, with products such as (x - 1)^2(x + 2) expanded for you. Fractions such as 2/3 and decimals such as 0.5 are read as exact rationals, pasted x² and minus signs work, and a list such as 1, 0, -1 is read as coefficients from the highest power down. Each box can reach degree 8 after expanding, with coefficients up to 100 digits, in up to 4000 characters. A square root, a trigonometric function, a logarithm, an absolute value, a power with the variable in it such as 2^x, or a fractional power is refused with the reason, because the result would not be a rational function. A negative power such as x^-1 is refused with a pointer to the Denominator box, where that factor belongs.
How is the graph window chosen?: By default the window frames every pole and hole with room on each side, and the y range follows most of the curve. Type any of x from, x to, y from and y to to set them yourself, within ±1,000,000; a blank field keeps the automatic value shown in it. The table under the graph lists the function at nine evenly spaced points of the window and says not defined where a point lands on a pole or a hole.
Can I save the graph or the working?: Download SVG hands the browser the graph as drawn, with its asymptotes, holes and window labels and a text description inside the file. Copy analysis puts the domain, the poles with their one-sided behavior, the holes, the asymptote, the division and the certificates on the clipboard as plain text. Copy LaTeX gives the function, common factor, reduced form, domain, asymptotes, holes and division in an aligned environment. If the browser blocks the clipboard or the download, the exact text appears in a box you can select and copy by hand.

Rational functions only: one polynomial over another in one variable, up to degree 8. Common factors are cancelled exactly while the original domain is kept, so a root still in the reduced denominator is a pole even if it was also a common factor, an original denominator root that cancels completely is a hole, and an irrational pole or hole is given by its defining polynomial and a certified interval rather than a rounded guess. Horizontal and slant asymptotes are classified by the degree of the exact quotient, a quotient of degree 2 or more is reported as end behavior rather than a line, and nothing is inferred from a jump between plotted pixels.