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

Funkcja wymierna to stosunek dwóch wielomianów o niezerowym mianowniku.Kluczową cechą funkcji wymiernych są ich asymptoty — linie, do których wykres się zbliża, ale nigdy ich nie dotyka.Te asymptoty mogą być pionowe lub poziome.Asymptoty pionowe występują, gdy mianownik jest równy zero, tworząc przerwy w wykresie. Znajdują się one przez rozwiązanie mianownika dla zera.Asymptoty poziome opisują końcowe zachowanie funkcji wymiernej, gdy dane wejściowe stają się bardzo duże lub bardzo małe.Funkcje wymierne mogą również przecinać asymptoty poziome, w przeciwieństwie do funkcji pionowych, które działają jak ścisłe granice.Położenie asymptot poziomych określa się przez porównanie stopni licznika i mianownika.Gdy stopień licznika jest mniejszy niż mianownik, asymptota pozioma wynosi y równe zero.Większy stopień w liczniku niż mianownik oznacza, że funkcja nie ma asymptoty poziomej.Weźmy na przykład pod uwagę stężenie zanieczyszczenia w wodzie, modelowane jako funkcja racjonalna. Wraz ze wzrostem objętości wody stężenie zanieczyszczenia zmniejsza się i zbliża się do zera, tworząc poziomą asymptotę w punkcie y równym zero.

asymptoty w funkcjach wymiernych

Funkcja wymierna jest definiowana jako iloraz dw&#243;ch wielomian&#243;w:<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>gdzie Q(x) &#8800; 0. Funkcje te cz&#281;sto maj&#261; asymptoty, czyli linie, do kt&#243;rych wykres zbli&#380;a si&#281;, lecz kt&#243;rych nigdy nie dotyka. Asymptoty klasyfikuje si&#281; na podstawie zachowania funkcji w pobli&#380;u okre&#347;lonych warto&#347;ci argumentu.Asymptoty pionowe wyst&#281;puj&#261; wtedy, gdy mianownik przyjmuje warto&#347;&#263; r&#243;wn&#261; zeru, a licznik nie, co sprawia, &#380;e funkcja jest niezdefiniowana. Wyznacza si&#281; je, rozwi&#261;zuj&#261;c r&#243;wnanie Q(x) = 0. Na przyk&#322;ad:<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>ma asymptot&#281; pionow&#261; przy x = 3, gdzie wykres rozbiega si&#281; do niesko&#324;czono&#347;ci.Asymptoty poziome opisuj&#261; zachowanie funkcji, gdy warto&#347;&#263; argumentu ro&#347;nie (co do modu&#322;u) do bardzo du&#380;ych warto&#347;ci. Ich po&#322;o&#380;enie zale&#380;y od stopni licznika i mianownika:<ul><li>Je&#347;li stopie&#324; licznika jest mniejszy ni&#380; stopie&#324; mianownika, asymptota pozioma wynosi y = 3.</li><li>Je&#347;li oba stopnie s&#261; r&#243;wne, asymptota pozioma jest ilorazem wsp&#243;&#322;czynnik&#243;w wiod&#261;cych.</li><li>Je&#347;li stopie&#324; licznika jest wi&#281;kszy ni&#380; stopie&#324; mianownika, asymptota pozioma nie wyst&#281;puje.</li></ul>Gdy stopie&#324; licznika jest dok&#322;adnie o jeden wi&#281;kszy od stopnia mianownika, istnieje asymptota sko&#347;na. Wyznacza si&#281; j&#261; za pomoc&#261; dzielenia wielomianowego. Na przyk&#322;ad:<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>ma asymptot&#281; sko&#347;n&#261; y = x &#8722; 1.Funkcje wymierne mog&#261; przecina&#263; asymptoty poziome lub sko&#347;ne, ale nigdy asymptot pionowych.

Asymptoty w funkcjach wymiernych

Funkcja wymierna jest definiowana jako iloraz dw&#243;ch wielomian&#243;w:gdzie Q(x) &#8800; 0. Funkcje te cz&#281;sto maj&#261; asymptoty, czyli linie, ...

Education

JoVE Core

Math Fundamentals

Podstawy matematyki

Unit

Jednostka

Polynomial and Rational Functions

Funkcje wielomianowe i wymierne

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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.

math fundamentals

Graphing Rational Functions with Asymptotes

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