Rational Functions

Front

What is a rational function?

Back

Front

What is a hole in a rational function?

Back

A hole occurs in a rational function at a value of x that makes both the numerator and denominator equal to zero, indicating that the function is undefined at that point.

Front

How do you find holes in a rational function?

Back

To find holes, factor both the numerator and denominator, and identify the common factors. The x-values that make these common factors zero are the locations of the holes.

Front

What is an asymptote?

Back

An asymptote is a line that a graph approaches but never touches. Asymptotes can be vertical, horizontal, or oblique.

Front

What is a vertical asymptote?

Back

A vertical asymptote occurs at values of x that make the denominator of a rational function equal to zero, indicating that the function approaches infinity.

Front

How do you find vertical asymptotes?

Back

To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.

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

How do you determine horizontal asymptotes for rational functions?

Back

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

Front

What is the significance of the leading coefficient in horizontal asymptotes?

Back

The leading coefficient of the numerator and denominator determines the value of the horizontal asymptote when the degrees of the polynomials are equal.

Front

What is the difference between a hole and a vertical asymptote?

Back

A hole indicates a removable discontinuity where the function is undefined, while a vertical asymptote indicates a non-removable discontinuity where the function approaches infinity.