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

Uma função racional é a razão de dois polinômios, com um denominador diferente de zero.Uma característica fundamental das funções racionais são suas assíntotas - linhas que o gráfico se aproxima, mas nunca toca.Essas assíntotas podem ser verticais ou horizontais.Assíntotas verticais ocorrem onde o denominador é igual a zero, criando quebras no gráfico. Estes são encontrados resolvendo o denominador para zero.As assíntotas horizontais descrevem o comportamento final de uma função racional à medida que a entrada se torna muito grande ou muito pequena.As funções racionais também podem cruzar assíntotas horizontais, ao contrário das verticais, que atuam como limites estritos.A posição das assíntotas horizontais é determinada comparando os graus do numerador e do denominador.Quando o grau do numerador é menor que o do denominador, a assíntota horizontal é y igual a zero.Um grau maior no numerador do que o denominador significa que a função não tem assíntota horizontal.Por exemplo, considere a concentração de um poluente na água, modelada como uma função racional. À medida que o volume de água aumenta, a concentração de um poluente diminui e se aproxima de zero, formando uma assíntota horizontal em y igual a zero.

assíntotas em funções racionais

Uma fun&#231;&#227;o racional &#233; definida como o quociente de dois polin&#244;mios:<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>onde Q(x) &#8800; 0. Essas fun&#231;&#245;es frequentemente apresentam ass&#237;ntotas, que s&#227;o retas &#224;s quais o gr&#225;fico se aproxima, mas nunca as toca. Essas ass&#237;ntotas s&#227;o classificadas com base no comportamento da fun&#231;&#227;o em torno de valores espec&#237;ficos da vari&#225;vel de entrada.Ass&#237;ntotas verticais ocorrem quando o denominador &#233; igual a zero e o numerador n&#227;o, tornando a fun&#231;&#227;o indefinida. Elas s&#227;o determinadas resolvendo Q(x) = 0. Por exemplo:<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>tem uma ass&#237;ntota vertical em x = 3, onde o gr&#225;fico diverge para o infinito.Ass&#237;ntotas horizontais descrevem o comportamento da fun&#231;&#227;o quando a vari&#225;vel de entrada cresce em magnitude. Sua posi&#231;&#227;o depende dos graus do numerador e do denominador:<ul><li>Se o grau do numerador for menor que o grau do denominador, a ass&#237;ntota horizontal &#233; y = 3.</li><li>Se ambos os graus forem iguais, a ass&#237;ntota horizontal &#233; a raz&#227;o dos coeficientes l&#237;deres.</li><li>Se o grau do numerador for maior que o do denominador, n&#227;o existe ass&#237;ntota horizontal.</li></ul>Quando o grau do numerador for exatamente uma unidade maior que o grau do denominador, existe uma ass&#237;ntota obl&#237;qua, que se obt&#233;m pela divis&#227;o polinomial. Por exemplo:<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>tem a ass&#237;ntota obl&#237;qua y = x &#8722; 1.Fun&#231;&#245;es racionais podem cruzar ass&#237;ntotas horizontais ou obl&#237;quas, mas nunca as verticais.

Assíntotas em Funções Racionais

Uma fun&#231;&#227;o racional &#233; definida como o quociente de dois polin&#244;mios:onde Q(x) &#8800; 0. Essas fun&#231;&#245;es frequentemente ...

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Math Fundamentals

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Polynomial and Rational Functions

Funções Polinomiais e Racionais

KEY TERMS AND CONCEPTS

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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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