Rational Functions - VA, HA, & Slant Asymptotes

Front

What is a rational function?

Back

A rational function is a function that can be expressed as the quotient of two polynomials, typically in the form f(x) = P(x)/Q(x), where P and Q are polynomials.

Front

What is a vertical asymptote (VA)?

Back

A vertical asymptote is a line x = a where a rational function approaches infinity or negative infinity as x approaches a.

Front

How do you find vertical asymptotes of a rational function?

Back

Vertical asymptotes occur at the values of x that make the denominator Q(x) = 0, provided that P(a) is not also 0.

Front

What is a horizontal asymptote (HA)?

Back

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

Front

How do you determine horizontal asymptotes for rational functions?

Back

1. If the degree of the numerator is less than the degree of the denominator, HA is y = 0. 2. If degrees are equal, HA is y = leading coefficient of P / leading coefficient of Q. 3. If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What is a slant (oblique) asymptote?

Back

A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator.

Front

How do you find a slant asymptote?

Back

To find a slant asymptote, perform polynomial long division of the numerator by the denominator. The quotient (ignoring the remainder) gives the equation of the slant asymptote.

Front

What is the significance of the degrees of the numerator and denominator in determining asymptotes?

Back

The degrees of the numerator and denominator determine the type of asymptotes present: vertical, horizontal, or slant.

Front

What happens to the function as x approaches a vertical asymptote?

Back

As x approaches a vertical asymptote, the function's value approaches infinity or negative infinity.

Front

Provide an example of a rational function with a vertical asymptote.

Back

f(x) = 1/(x-2) has a vertical asymptote at x = 2.