Asymptotes of Rational Functions

Front

What is a rational function?

Back

Front

What is an asymptote?

Back

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

Front

What is a vertical asymptote?

Back

A vertical asymptote occurs at values of x where the function approaches infinity, typically where the denominator of a rational function equals zero.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote indicates the behavior of a function as x approaches infinity or negative infinity. It is determined by the degrees of the numerator and denominator.

Front

How do you find horizontal asymptotes for rational functions?

Back

1. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. 2. If the degrees are equal, the horizontal asymptote is y=\frac{a}{b} where a and b are the leading coefficients. 3. If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What is a hole in a graph of a rational function?

Back

A hole occurs in the graph of a rational function at a point where a factor in the numerator cancels with a factor in the denominator.

Front

How do you identify holes in rational functions?

Back

To identify holes, factor both the numerator and denominator, then find values of x that make both equal to zero.

Front

What is the significance of the degree of a polynomial in determining asymptotes?

Back

The degree of the polynomial in the numerator and denominator helps determine the type and number of asymptotes present in the rational function.

Front

What is the relationship between vertical asymptotes and the factors of the denominator?

Back

Vertical asymptotes occur at the values of x that make the denominator zero, provided those values do not also make the numerator zero.

Front

How do you find vertical asymptotes for rational functions?

Back

Set the denominator equal to zero and solve for x. The solutions are the x-values of the vertical asymptotes.