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

유리함수는 분모가 0이 아닌 두 다항식의 비율입니다.유리함수의 주요 특징은 점근선, 즉 그래프가 접근하지만 결코 건드리지 않는 선입니다.이러한 점근선은 수직 또는 수평일 수 있습니다.수직 점근선은 분모가 0과 같은 곳에서 발생하여 그래프에 중단이 발생합니다. 이들은 분모를 0으로 풀어 찾습니다.수평 점근선은 입력이 매우 크거나 매우 작아질 때 유리함수의 최종 동작을 설명합니다.합리적 함수는 엄격한 경계 역할을 하는 수직 함수와 달리 수평 점근선을 교차할 수도 있습니다.수평 점근선의 위치는 분자와 분모의 각도를 비교하여 결정됩니다.분자의 차수가 분모의 차수보다 작을 때 수평 점근선은 y 가 0과 같습니다.분자의 차수가 분모보다 크다는 것은 함수에 수평 점근선이 없음을 의미합니다.예를 들어, 유리함수로 모델링된 물 속의 오염 물질 농도를 생각해 보십시오. 물의 부피가 증가함에 따라 오염 물질의 농도가 감소하고 0에 가까워지며 y에서 수평 점근선을 형성하면 0과 같습니다.

유리 함수의 점근선

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&#50976;&#47532; &#54632;&#49688;&#45716; &#46160; &#45796;&#54637;&#49885;&#51032; &#47787;&#51004;&#47196; ...

Education

교육

JoVE Core

Math Fundamentals

수학 기초

Unit

단위

Polynomial and Rational Functions

다항함수와 유리함수

KEY TERMS AND CONCEPTS

주요 용어 및 개념

Asymptotes in Rational Functions

유리함수의 점근

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

quadratic models

Quadratic Motion and Parabolas

Quadratic models describe relationships where the rate of change changes at a constant rate. In math and science, this often shows up as a curved pattern called a parabola. These models help explain how a system changes over time or under a set force. A common example is the vertical motion of an object moving under gravity, such as a ball thrown into the air. The object rises to a maximum height and then falls back down. Gravity slows the ascent and speeds up the descent, which creates the...

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Using Synthetic Division to Find Factors

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Finding Polynomial X-Intercepts

Polynomials can be analyzed by finding their real zeros, which are the values of x that make the polynomial equal to zero. These points are also the x-intercepts of the graph. Understanding them helps students see where a polynomial crosses the x-axis. The Rational Zeros Theorem gives a list of possible rational solutions for a polynomial with integer coefficients. If f(x)=anx^n+...+a0, then every rational zero has the form p/q. In that form, p divides the constant term a0 and q divides the...

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Polynomial roots in complex numbers are a key result in algebra. The Fundamental Theorem of Algebra says that every non-constant polynomial with complex coefficients has at least one complex zero. For a polynomial of degree n, where n is at least 1 and the leading coefficient a n is not 0, there is at least one solution in the complex number system. This result also applies to polynomials with real coefficients, because the real numbers are a subset of the complex numbers. The Complete...

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Finding Polynomial Roots With Complex Numbers

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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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