rational functions and their asymptotes

Front

What is a rational function?

Back

A rational function is a function that can be expressed as the quotient of two polynomials, where the denominator is not zero.

Front

What is an asymptote?

Back

An asymptote is a line that a graph approaches but never touches or intersects.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line that the graph of a function approaches as the function's value approaches infinity or negative infinity.

Front

What is an oblique (slant) asymptote?

Back

An oblique asymptote occurs when the degree of the numerator is one greater than the degree of the denominator, resulting in a linear asymptote.

Front

How do you find horizontal asymptotes for rational functions?

Back

Compare the degrees of the numerator and denominator: If the degree of the numerator is less, the asymptote is y=0; if equal, y=leading coefficient of numerator/leading coefficient of denominator; if greater, no horizontal asymptote.

Front

What is the oblique asymptote of f(x) = (3x^3 - 2x + 1)/(x^2 + 4x + 2)?

Back

y = 3x - 12.

Front

What is the slant asymptote of f(x) = (x^2 - 4x + 2)/(x + 3)?

Back

y = x - 7.

Front

What happens to the graph of a rational function at vertical asymptotes?

Back

The graph approaches the vertical asymptote but never crosses it, leading to infinite behavior.

Front

What is the significance of the degrees of the polynomials in rational functions?

Back

The degrees determine the behavior of the function at infinity and the existence of horizontal or oblique asymptotes.