Online Function Grapher

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rational function asymptotes graph

Graph of a rational function: vertical, horizontal and oblique asymptotes

Type a fraction of polynomials, for example (2x + 1)/(x - 3). The tool draws each branch of the curve without connecting the two sides of the asymptote, dashes the asymptotic lines and lists the verticals, the horizontal or oblique one, the holes and the roots in the panel.

How to find the asymptotes of p(x)/q(x)

A rational function is a quotient of two polynomials, f(x) = p(x)/q(x). The domain excludes the roots of the denominator, and the roots of the function are the roots of p(x) that do not zero q(x). In f(x) = (2x + 1)/(x - 3), the denominator vanishes at x = 3, so the domain is all reals except 3. The numerator vanishes at x = -0.5, which is the root, and the intercept is f(0) = 1/(-3), about -0.333.
A vertical asymptote appears at every root of the denominator that does not zero the numerator. At x = 3 the numerator equals 7, so near 3 the value blows up: f tends to plus infinity from the right and to minus infinity from the left. The graph has two branches and the tool never draws the straight line joining them. Repeated roots of the denominator change the behavior: in 1/x² both sides go up to plus infinity, and in 1/x the sides go to infinities of opposite sign.
The horizontal or oblique asymptote depends on the degrees. If the numerator degree is lower than the denominator degree, y = 0: this happens in 1/x and in x/(x² - 4). If the degrees are equal, y is the ratio of the leading coefficients: in (2x + 1)/(x - 3) it is y = 2/1 = 2. If the numerator has one degree more, there is an oblique asymptote, the quotient of the division: (x² + 1)/x = x + 1/x has y = x. If the degree difference exceeds 1, there is no asymptotic line. Unlike the vertical one, the curve can cross the horizontal: x/(x² - 4) passes through y = 0 at x = 0.
When the numerator and the denominator share a factor, the denominator root does not create an asymptote but a hole. (x² - 1)/(x - 1) equals x + 1 for any x other than 1, so the graph is the line y = x + 1 with a gap at (1, 2): the limit exists but the function is not defined there. The tool marks the hole with an open circle and does not list it as an asymptote. Because polynomials and fractions of polynomials are handled with algebra, these answers are exact and hold on the whole real line, even outside the window of the graph.

Checked examples

Vertical and horizontal asymptotes

Input: (2x + 1)/(x - 3)
Expected output: x = 3; y = 2; f(-0.5) = 0

Vertical x = 3; horizontal y = 2; root -0.5. Equal degrees: 2/1.

Hyperbola

Input: 1/x
Expected output: x = 0; y = 0

Vertical x = 0; horizontal y = 0; no root; odd function. Numerator degree is lower.

Hole in the graph

Input: (x^2 - 1)/(x - 1)
Expected output: (1, 2); f(-1) = 0

Hole at (1, 2); root -1; no vertical asymptote. The factor (x - 1) cancels.

Oblique asymptote

Input: x + 1/x
Expected output: x = 0; y = x; (-1, -2); (1, 2)

Vertical x = 0; oblique y = x; local maximum (-1, -2) and local minimum (1, 2). Numerator of degree 2, denominator of degree 1.

Two vertical asymptotes

Input: x/(x^2 - 4)
Expected output: x = -2, x = 2; y = 0; f(0) = 0

Verticals x = -2 and x = 2; horizontal y = 0; root 0. x² - 4 = (x - 2)(x + 2).

Even-order pole

Input: 1/x^2
Expected output: x = 0; y = 0

Vertical x = 0; horizontal y = 0; always positive. Both branches go up.

Full tool FAQ

How do I graph a quadratic function?

Type the function, for example x^2 - 4x + 3. The parabola opens upward if a is positive and downward if it is negative. The vertex is at x = -b/2a: here x = 2 and y = -1. The roots come from (-b ± √Δ)/2a with Δ = b² - 4ac = 4, so x = 1 and x = 3, and the parabola crosses the y axis at c = 3. The tool marks all of these points and shows their exact form.

How do I find the roots of a function on the graph?

How do I identify the asymptotes of a function?

What is the difference between an asymptote and a hole in a graph?

How do I plot several graphs and find where they cross?

What are the parameters a, b and c?

Which functions and symbols can I use?

Does the panel show the important points even off screen?

How do I save or share the graph?

Why does my function not show up or give an error?

Frequently asked questions

Can a function cross its horizontal asymptote?

It can. The horizontal asymptote describes the behavior for very large x and does not stop the curve from crossing it earlier. The function x/(x² - 4) has asymptote y = 0 and passes through (0, 0). A vertical asymptote is never crossed, because the function is not defined at that x.

How do I know if a denominator root is an asymptote or a hole?

Substitute the root into the numerator. If the numerator is not zero, it is a vertical asymptote. If it is also zero and the factor cancels, it is a hole. In (x² - 1)/(x - 1) the numerator is 0 at x = 1, and after canceling you get x + 1, with a hole at (1, 2). The tool does this check using the multiplicity of the roots.

How do I find the oblique asymptote?

Divide the numerator by the denominator; the linear quotient is the asymptote and the remainder vanishes at infinity. In (x² + 3x + 1)/(x + 1) the division gives x + 2 with remainder -1, so the asymptote is y = x + 2. It only exists when the numerator degree is exactly one more than the denominator degree.

Next steps

Properties

f₁(x) = x^2 - 4x + 3Quadratic

Quadratic function, f(x) = ax² + bx + c with a not equal to 0. The graph is a parabola, symmetric about the vertical line through the vertex. Here a = 1, b = -4 and c = 3; the parabola opens upward.

Exact result (algebra)

Domain: ℝ
Range: [-1, +∞)
Roots (zeros): x = 1
x = 3
y intercept: (0, 3)
Vertex: (2, -1) minimum point
Axis of symmetry: x = 2
Discriminant Δ: 4 (two real roots)
Increasing and decreasing: Decreasing on (-∞, 2]
Increasing on [2, +∞)
Sign: f₁(x) > 0 on (-∞, 1) ∪ (3, +∞)
f₁(x) < 0 on (1, 3)
Asymptotes: None found
Parity: Neither even nor odd.

f₂(x) = x/2 + 1Linear (1st degree)

Linear function, f(x) = ax + b. The graph is a line: a is the slope (increasing if a is positive) and b is where the line crosses the y axis. Here a = 0.5 (increasing) and b = 1.

Exact result (algebra)

Domain: ℝ
Range: ℝ
Roots (zeros): x = -2
y intercept: (0, 1)
Increasing and decreasing: Increasing on (-∞, +∞)
Sign: f₂(x) > 0 on (-2, +∞)
f₂(x) < 0 on (-∞, -2)
Asymptotes: None found
Parity: Neither even nor odd.

Intersections between the functions

f₁ ∩ f₂ (0.5, 1.25)
f₁ ∩ f₂ (4, 3)

x	f₁(x)	f₂(x)
-5	48	-1.5
-4	35	-1
-3	24	-0.5
-2	15	0
-1	8	0.5
0	3	1
1	0	1.5
2	-1	2
3	0	2.5
4	3	3
5	8	3.5

f₁(x)

f₂(x)

-5

-1.5

-4

-1

-3

-0.5

-2

-1

0.5

1.5

-1

2.5

3.5

How to find the asymptotes of p(x)/q(x)

A rational function is a quotient of two polynomials, f(x) = p(x)/q(x). The domain excludes the roots of the denominator, and the roots of the function are the roots of p(x) that do not zero q(x). In f(x) = (2x + 1)/(x - 3), the denominator vanishes at x = 3, so the domain is all reals except 3. The numerator vanishes at x = -0.5, which is the root, and the intercept is f(0) = 1/(-3), about -0.333.

A vertical asymptote appears at every root of the denominator that does not zero the numerator. At x = 3 the numerator equals 7, so near 3 the value blows up: f tends to plus infinity from the right and to minus infinity from the left. The graph has two branches and the tool never draws the straight line joining them. Repeated roots of the denominator change the behavior: in 1/x² both sides go up to plus infinity, and in 1/x the sides go to infinities of opposite sign.

The horizontal or oblique asymptote depends on the degrees. If the numerator degree is lower than the denominator degree, y = 0: this happens in 1/x and in x/(x² - 4). If the degrees are equal, y is the ratio of the leading coefficients: in (2x + 1)/(x - 3) it is y = 2/1 = 2. If the numerator has one degree more, there is an oblique asymptote, the quotient of the division: (x² + 1)/x = x + 1/x has y = x. If the degree difference exceeds 1, there is no asymptotic line. Unlike the vertical one, the curve can cross the horizontal: x/(x² - 4) passes through y = 0 at x = 0.

When the numerator and the denominator share a factor, the denominator root does not create an asymptote but a hole. (x² - 1)/(x - 1) equals x + 1 for any x other than 1, so the graph is the line y = x + 1 with a gap at (1, 2): the limit exists but the function is not defined there. The tool marks the hole with an open circle and does not list it as an asymptote. Because polynomials and fractions of polynomials are handled with algebra, these answers are exact and hold on the whole real line, even outside the window of the graph.

Checked examples

Vertical and horizontal asymptotes

Input: (2x + 1)/(x - 3)
Expected output: x = 3; y = 2; f(-0.5) = 0

Vertical x = 3; horizontal y = 2; root -0.5. Equal degrees: 2/1.

Hyperbola

Input: 1/x
Expected output: x = 0; y = 0

Vertical x = 0; horizontal y = 0; no root; odd function. Numerator degree is lower.

Hole in the graph

Input: (x^2 - 1)/(x - 1)
Expected output: (1, 2); f(-1) = 0

Hole at (1, 2); root -1; no vertical asymptote. The factor (x - 1) cancels.

Oblique asymptote

Input: x + 1/x
Expected output: x = 0; y = x; (-1, -2); (1, 2)

Vertical x = 0; oblique y = x; local maximum (-1, -2) and local minimum (1, 2). Numerator of degree 2, denominator of degree 1.

Two vertical asymptotes

Input: x/(x^2 - 4)
Expected output: x = -2, x = 2; y = 0; f(0) = 0

Verticals x = -2 and x = 2; horizontal y = 0; root 0. x² - 4 = (x - 2)(x + 2).

Even-order pole

Input: 1/x^2
Expected output: x = 0; y = 0

Vertical x = 0; horizontal y = 0; always positive. Both branches go up.

Full tool FAQ

How do I graph a quadratic function?

How do I find the roots of a function on the graph?

How do I identify the asymptotes of a function?

What is the difference between an asymptote and a hole in a graph?

How do I plot several graphs and find where they cross?

What are the parameters a, b and c?

Which functions and symbols can I use?

Does the panel show the important points even off screen?

How do I save or share the graph?

Why does my function not show up or give an error?

Frequently asked questions