Test Review: Graphs of Rational Functions

Front

What are horizontal asymptotes in rational functions?

Back

Horizontal asymptotes are lines that the graph of a rational function approaches as x approaches positive or negative infinity. They are determined by comparing the degrees of the numerator and denominator.

Front

How do you determine the horizontal asymptote of a rational function?

Back

To determine the horizontal asymptote, compare the degrees of the numerator and denominator: if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0; if they are equal, the asymptote is y = leading coefficient of numerator / leading coefficient of denominator; 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. It can be found using polynomial long division.

Front

How do you find the slant asymptote of a rational function?

Back

To find the slant asymptote, perform polynomial long division on the rational function. The quotient (ignoring the remainder) will give the equation of the slant asymptote.

Front

What is the equation of the slant asymptote for the function y=(x^2-4x-5)/(x+3)?

Back

The equation of the slant asymptote for the function y=(x^2-4x-5)/(x+3) is y = x - 7.

Front

What are vertical asymptotes in rational functions?

Back

Vertical asymptotes are lines that the graph of a rational function approaches as the function approaches infinity or negative infinity. They occur at values of x that make the denominator zero (provided the numerator is not also zero at those points).

Front

How do you find vertical asymptotes of a rational function?

Back

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

Front

What does it mean if a rational function has no vertical asymptotes?

Back

If a rational function has no vertical asymptotes, it means that the denominator does not have any real roots, or any values that would make the denominator zero.

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 horizontal asymptote and whether a slant asymptote exists. They help classify the behavior of the function as x approaches infinity.

Front

What is the horizontal asymptote of the function y=2x/(x^2-5)?

Back

The horizontal asymptote of the function y=2x/(x^2-5) is y=0.