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

Una funzione razionale è il rapporto di due polinomi, con un denominatore diverso da zero.Una caratteristica chiave delle funzioni razionali sono i loro asintoti, linee a cui il grafo si avvicina ma non tocca mai.Questi asintoti possono essere verticali o orizzontali.Gli asintoti verticali si verificano dove il denominatore è uguale a zero, creando interruzioni nel grafo. Questi si trovano risolvendo il denominatore per zero.Gli asintoti orizzontali descrivono il comportamento finale di una funzione razionale quando l'input diventa molto grande o molto piccolo.Le funzioni razionali possono anche attraversare asintoti orizzontali, a differenza di quelle verticali, che agiscono come confini stretti.La posizione degli asintoti orizzontali è determinata confrontando i gradi del numeratore e del denominatore.Quando il grado del numeratore è minore di quello del denominatore, l'asintoto orizzontale è y uguale a zero.Un grado maggiore nel numeratore rispetto al denominatore significa che la funzione non ha asintoto orizzontale.Si consideri, ad esempio, la concentrazione di un inquinante nell'acqua, modellata come una funzione razionale. All'aumentare del volume dell'acqua, la concentrazione di un inquinante diminuisce e si avvicina allo zero, formando un asintoto orizzontale a y uguale a zero.

asintoti nelle funzioni razionali

Una funzione razionale &#232; definita come il quoziente di due polinomi:<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>dove Q(x) &#8800; 0. Queste funzioni presentano spesso asintoti, ovvero le rette a cui il grafico si avvicina senza mai toccarle. Gli asintoti sono classificati in base al comportamento della funzione vicino a valori specifici della variabile indipendente.Gli asintoti verticali si verificano quando il denominatore &#232; zero mentre il numeratore &#232; diverso da zero, rendendo la funzione indefinita. Questi asintoti si trovano risolvendo Q(x) = 0. Ad esempio:<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>ha un asintoto verticale per x = 3, dove il grafico diverge all'infinito.Gli asintoti orizzontali descrivono il comportamento della funzione quando |x| cresce indefinitamente. La loro posizione dipende dai gradi del numeratore e del denominatore:<ul><li>Se il grado del numeratore &#232; minore del grado del denominatore, l'asintoto orizzontale &#232; y = 3.</li><li>Se i gradi del numeratore e del denominatore sono uguali, l'asintoto orizzontale &#232; il rapporto tra i coefficienti dei termini di grado massimo.</li><li>Se il grado del numeratore &#232; maggiore di quello del denominatore, non esiste un asintoto orizzontale.</li></ul>Quando il grado del numeratore &#232; esattamente di uno superiore a quello del denominatore, esiste un asintoto obliquo. Lo si determina mediante la divisione polinomiale. Ad esempio:<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>ha l'asintoto obliquo y = x - 1.Le funzioni razionali possono attraversare asintoti orizzontali o obliqui, ma mai quelli verticali.

Asintoti nelle funzioni razionali

Una funzione razionale &#232; definita come il quoziente di due polinomi:dove Q(x) &#8800; 0. Queste funzioni presentano spesso asintoti, ovvero le rette ...

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Math Fundamentals

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Polynomial and Rational Functions

Funzioni polinomiali e razionali

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

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...

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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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