Applications and Asymptotes of Rational Functions

Study Notes

Rational functions have numerous applications in various fields, including:

Physics and Engineering: Modeling real-world phenomena, such as electrical circuits, mechanical systems, and population growth.
Economics: Analyzing economic systems, including supply and demand, and modeling the behavior of markets.
Computer Science: Algorithm design, data analysis, and machine learning.
Biology: Modeling population dynamics, chemical reactions, and epidemiology.

Asymptotes are crucial in understanding the behavior of rational functions. There are three types of asymptotes:

Occur when the degree of the numerator is less than or equal to the degree of the denominator.
Found by dividing the leading coefficients of the numerator and denominator.

Occur when the degree of the numerator is greater than the degree of the denominator.
Found by dividing the numerator by the denominator using polynomial long division or synthetic division.

Rules for Finding Asymptotes

If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y = 0.
If the degree of the numerator is equal to the degree of the denominator, there is a horizontal asymptote at y = leading coefficient of numerator / leading coefficient of denominator.
If the degree of the numerator is greater than the degree of the denominator, there is a slant asymptote.
Vertical asymptotes occur at the zeros of the denominator.