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

有理関数は、分母がゼロではない2つの多項式の比率です。有理関数の重要な特徴は、漸近線、つまりグラフが近づいても決して触れない線です。これらの漸近線は、垂直または水平にすることができます。垂直方向の漸近線は、分母がゼロに等しい場所で発生し、グラフに区切りが生じます。これらは、分母をゼロに解くことによって求められます。水平漸近線は、入力が非常に大きくなったり、非常に小さくなったりするときの有理関数の最終動作を表します。有理関数は、厳密な境界として機能する垂直関数とは異なり、水平漸近線を横切ることもあります。水平漸近線の位置は、分子と分母の次数を比較することによって決定されます。分子の次数が分母の次数より小さい場合、水平漸近線は y はゼロに等しい。分子の次数が分母よりも大きいということは、関数に水平漸近線がないことを意味します。たとえば、有理関数としてモデル化された水中の汚染物質の濃度を考えてみましょう。水量が増加すると、汚染物質の濃度は減少してゼロに近づき、yで水平漸近線が形成され、ゼロに等しくなります。

有理関数の漸近線

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Q(x) = 0 &#12434;&#35299;&#12367;&#12371;&#12392;&#12391;&#27714;&#12417;&#12425;&#12428;&#12414;&#12377;&#12290;&#20363;&#12360;&#12400;&#12289;<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>&#12399; x = 3 &#12395;&#12362;&#12356;&#12390;&#22402;&#30452;&#28472;&#36817;&#32218;&#12434;&#25345;&#12385;&#12289;&#12464;&#12521;&#12501;&#12399;&#12381;&#12398;&#36817;&#20621;&#12391;&#28961;&#38480;&#22823;&#12395;&#30330;&#25955;&#12375;&#12414;&#12377;&#12290;&#27700;&#24179;&#28472;&#36817;&#32218;&#12399;&#12289;&#20837;&#21147;&#20516;&#12398;&#32118;&#23550;&#20516;&#12364;&#38750;&#24120;&#12395;&#22823;&#12365;&#12367;&#12394;&#12387;&#12383;&#12392;&#12365;&#12398;&#38306;&#25968;&#12398;&#25369;&#21205;&#12434;&#31034;&#12375;&#12414;&#12377;&#12290;&#12381;&#12398;&#20301;&#32622;&#12399;&#20998;&#23376;&#12392;&#20998;&#27597;&#12398;&#27425;&#25968;&#12395;&#12424;&#12387;&#12390;&#27770;&#12414;&#12426;&#12414;&#12377;&#12290;<ul><li>&#20998;&#23376;&#12398;&#27425;&#25968;&#12364;&#20998;&#27597;&#12398;&#27425;&#25968;&#12424;&#12426;&#23567;&#12373;&#12356;&#22580;&#21512;&#12289;&#27700;&#24179;&#28472;&#36817;&#32218;&#12399; y = 3 &#12391;&#12377;&#12290;</li><li>&#20998;&#23376;&#12392;&#20998;&#27597;&#12398;&#27425;&#25968;&#12364;&#31561;&#12375;&#12356;&#22580;&#21512;&#12289;&#27700;&#24179;&#28472;&#36817;&#32218;&#12399;&#20027;&#20418;&#25968;&#12398;&#27604;&#12395;&#12394;&#12426;&#12414;&#12377;&#12290;</li><li>&#20998;&#23376;&#12398;&#27425;&#25968;&#12364;&#20998;&#27597;&#12398;&#27425;&#25968;&#12424;&#12426;&#22823;&#12365;&#12356;&#22580;&#21512;&#12289;&#27700;&#24179;&#28472;&#36817;&#32218;&#12399;&#23384;&#22312;&#12375;&#12414;&#12379;&#12435;&#12290;</li></ul>&#12373;&#12425;&#12395;&#12289;&#20998;&#23376;&#12398;&#27425;&#25968;&#12364;&#20998;&#27597;&#12398;&#27425;&#25968;&#12424;&#12426;&#12385;&#12423;&#12358;&#12393; 1 &#22823;&#12365;&#12356;&#22580;&#21512;&#12395;&#12399;&#12289;&#20670;&#26012;&#28472;&#36817;&#32218;&#65288;&#26012;&#28472;&#36817;&#32218;&#65289;&#12364;&#23384;&#22312;&#12375;&#12414;&#12377;&#12290;&#12371;&#12428;&#12399;&#22810;&#38917;&#24335;&#12398;&#38500;&#27861;&#12395;&#12424;&#12387;&#12390;&#27714;&#12417;&#12425;&#12428;&#12414;&#12377;&#12290;&#20363;&#12360;&#12400;&#12289;<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>&#12398;&#20670;&#26012;&#28472;&#36817;&#32218;&#12399; y = x - 1 &#12391;&#12377;&#12290;&#26377;&#29702;&#38306;&#25968;&#12399;&#12289;&#27700;&#24179;&#28472;&#36817;&#32218;&#12414;&#12383;&#12399;&#20670;&#26012;&#28472;&#36817;&#32218;&#12392;&#20132;&#24046;&#12377;&#12427;&#12371;&#12392;&#12364;&#12354;&#12426;&#12414;&#12377;&#12364;&#12289;&#22402;&#30452;&#28472;&#36817;&#32218;&#12392;&#20132;&#12431;&#12427;&#12371;&#12392;&#12399;&#12354;&#12426;&#12414;&#12379;&#12435;&#12290;

Education

JoVE Core

Math Fundamentals

数学の基礎

Unit

ユニット

Polynomial and Rational Functions

多項式関数と有理関数

KEY TERMS AND CONCEPTS

主要用語と概念

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

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...

introduction to polynomial functions

Polynomial Graphs, Zeros, and Degree

Polynomial functions are an important part of algebra and calculus. They are made from variables and constants combined by addition, subtraction, and multiplication. The variable is raised to a nonnegative integer exponent. A general polynomial function of degree n has a leading term aₙxⁿ, where aₙ is not equal to 0. The coefficient aₙ is the leading coefficient, and a₀ is the constant term. Polynomials are grouped by degree. A constant function has degree zero, a linear function has degree...

long division of polynomials

Polynomial Division: Quotient and Remainder

Polynomial division breaks a polynomial into a quotient and a remainder. It works like numerical division, but the dividend is a polynomial and the divisor is an algebraic expression. The result shows how much of the divisor fits into the dividend and what is left over. The first step is to write both polynomials in standard form, with terms in descending powers of the variable. Then divide the leading term of the dividend by the leading term of the divisor. That first result becomes the first...

synthetic disvision of polynomials

Using Synthetic Division to Find Factors

Synthetic division is a fast way to divide a polynomial by a linear binomial of the form x - c, where c is a real number. It gives a shorter method than polynomial long division and is useful for checking values and finding factors. The setup starts with the coefficients of the polynomial written in descending order of powers. If a term is missing, a zero must be used to keep the columns lined up. The constant c from the divisor x - c is placed to the left of the synthetic division table. The...

real zeros of polynomials

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...

fundamental theorem of algebra

Polynomial Roots in Complex Numbers

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...

complex zeros

Finding Polynomial Roots With Complex Numbers

Complex numbers help describe every root of a polynomial equation. A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit, with i^2 = -1. A complex zero is any value that makes P(x) = 0 for a polynomial P(x). The complex number system includes the real numbers, so it gives a complete way to study polynomial roots. Every polynomial of degree n  1 can be written as a product of n linear factors over the complex numbers. In that factorization, each c_i...

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...

introduction to nonlinear inequalities

Graphing and Solving Nonlinear Inequalities

Linear and nonlinear inequalities help describe variable relationships and the ranges that satisfy a condition. A linear inequality uses variables raised only to the first power. Its graph is a straight line that divides the coordinate plane into two regions. One region satisfies the inequality, and the other does not. Nonlinear inequalities use variables raised to powers greater than one. These include quadratic inequalities and higher-degree polynomial inequalities. Their graphs are curved,...

application of nonlinear inequalities

Solving Nonlinear Inequalities on a Number Line

Nonlinear inequalities can be solved by checking where an expression is positive or negative on a number line. These inequalities often include squares, products, or variables in the denominator, so they do not follow the same simple pattern as linear inequalities. The first step is to rewrite the inequality so that zero is on one side. After rewriting, factor the expression if possible. Factoring helps you find the key values where the expression equals zero. In one example, the values 3 and...

key terms and concepts

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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