Name: graphing rational functions with asymptotes
Uploaded: 2025-11-26T05:43:50.000Z
Duration: 1 min 30 s
Channel: JoVE

Una función racional es la razón de dos polinomios, con un denominador distinto de cero.Una característica clave de las funciones racionales son sus asíntotas, líneas a las que el grafo se acerca pero nunca toca.Estas asíntotas pueden ser verticales u horizontales.Las asíntotas verticales ocurren donde el denominador es igual a cero, creando rupturas en el gráfico. Estos se encuentran resolviendo el denominador para cero.Las asíntotas horizontales describen el comportamiento final de una función racional a medida que la entrada se vuelve muy grande o muy pequeña.Las funciones racionales también pueden cruzar asíntotas horizontales, a diferencia de las verticales, que actúan como límites estrictos.La posición de las asíntotas horizontales se determina comparando los grados del numerador y el denominador.Cuando el grado del numerador es menor que el del denominador, la asíntota horizontal es y igual a cero.Un grado mayor en el numerador que en el denominador significa que la función no tiene asíntota horizontal.Por ejemplo, considere la concentración de un contaminante en el agua, modelada como una función racional. A medida que aumenta el volumen de agua, la concentración de un contaminante disminuye y se acerca a cero, formando una asíntota horizontal en y igual a cero.

asíntotas en funciones racionales

Una funci&#243;n racional se define como el cociente de dos polinomios:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/191;' src='https://cloudfront.jove.com/files/ftp_upload/40059/40059_Equation_1_editor_40059.svg' alt='40059_Equation_1.svg' /></figure>donde Q(x) &#8800; 0, estas funciones suelen presentar as&#237;ntotas, que son las rectas a las que la gr&#225;fica se aproxima pero nunca llega a tocar. Estas as&#237;ntotas se clasifican seg&#250;n el comportamiento de la funci&#243;n cerca de valores espec&#237;ficos de la variable.Las as&#237;ntotas verticales se producen cuando el denominador se anula y el numerador no, lo que hace que la funci&#243;n no est&#233; definida. Estas se obtienen resolviendo Q(x) = 0. Por ejemplo:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/128;' src='https://cloudfront.jove.com/files/ftp_upload/40059/40059_Equation_2_editor_40059.svg' alt='40059_Equation_2.svg' /></figure>tiene una as&#237;ntota vertical en x = 3, donde la gr&#225;fica diverge hacia el infinito.Las as&#237;ntotas horizontales describen el comportamiento de la funci&#243;n cuando la variable independiente crece en valor absoluto. Su posici&#243;n depende de los grados del numerador y del denominador:<ul><li>Si el grado del numerador es menor que el del denominador, la as&#237;ntota horizontal es y = 3.</li><li>Si ambos grados son iguales, la as&#237;ntota horizontal es el cociente de los coeficientes principales.</li><li>Si el grado del numerador es mayor que el del denominador, no existe as&#237;ntota horizontal.</li></ul>Cuando el grado del numerador es exactamente uno mayor que el del denominador, existe una as&#237;ntota oblicua. Se calcula mediante divisi&#243;n de polinomios. Por ejemplo:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:300/21;' src='https://cloudfront.jove.com/files/ftp_upload/40059/40059_Equation_3_editor_40059.svg' alt='40059_Equation_3.svg' /></figure>tiene la as&#237;ntota oblicua y = x &#8722; 1.Las funciones racionales pueden cruzar as&#237;ntotas horizontales u oblicuas, pero nunca las verticales.

Asíntotas en funciones racionales

Una funci&#243;n racional se define como el cociente de dos polinomios:donde Q(x) &#8800; 0, estas funciones suelen presentar as&#237;ntotas, que son las ...

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Math Fundamentals

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Polynomial and Rational Functions

Funciones polinómicas y racionales

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Asymptotes in Rational Functions

graphing rational functions with asymptotes

Rational functions are quotients of two polynomials, with the denominator Q(x) not equal to zero. Their graphs often include asymptotes, which are lines the graph gets close to but does not touch. These lines help show how the function behaves near certain input values and at large values of x.

A vertical asymptote appears when the denominator is zero and the numerator is not zero. At those x-values, the function is undefined. To find a vertical asymptote, solve Q(x) = 0. For example, a rational function can have a vertical asymptote at x = 3, where the graph rises or falls without bound.

Horizontal asymptotes describe what happens as x becomes very large or very negative. Their position depends on the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 3. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater, there is no horizontal asymptote.

When the degree of the numerator is exactly one more than the degree of the denominator, the graph has a slant asymptote. A slant asymptote is found using polynomial division. For example, a rational function can have the slant asymptote y = x - 1. Rational functions may cross horizontal or slant asymptotes, but they never cross vertical ones.

Graphing Rational Functions with Asymptotes

graphing-rational-functions-with-asymptotes

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asymptotes in rational functions

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Learn how to identify vertical, horizontal, and slant asymptotes in rational functions. See how the numerator and denominator degrees affect the graph.

Graphing Rational Functions with Asymptotes | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

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Graphing Rational Functions with Asymptotes

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