Asymptotes of Rational Functions

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.

Tags

CCSS.HSF-IF.C.7D

Front

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

Back

To find the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the asymptote is y=0. If they are equal, the asymptote is y=\frac{leading coefficient of numerator}{leading coefficient of denominator}. If the numerator's degree is greater, there is no horizontal asymptote.

Tags

CCSS.HSF-IF.C.7D

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line that the graph of a function approaches as the input approaches a certain value, typically where the function is undefined.

Tags

CCSS.HSF-IF.C.7D

Front

How do you find vertical asymptotes in a rational function?

Back

Vertical asymptotes can be found by setting the denominator of the rational function equal to zero and solving for x.

Tags

CCSS.HSF-IF.C.7D

Front

What does it mean if a function has a vertical asymptote at x = 2?

Back

It means that as x approaches 2, the function's value approaches infinity or negative infinity, indicating a discontinuity at that point.

Tags

CCSS.HSF-IF.C.7D

Front

What is the significance of the degrees of the numerator and denominator in rational functions?

Back

The degrees of the numerator and denominator determine the behavior of the function as x approaches infinity and help identify horizontal asymptotes.

Tags

CCSS.HSF-IF.C.7D

Front

What is a non-vertical asymptote?

Back

A non-vertical asymptote refers to any asymptote that is not vertical, typically horizontal or oblique (slant) asymptotes.

Tags

CCSS.HSF-IF.C.7D

Front

How do you find the non-vertical asymptote of a rational function?

Back

To find a non-vertical asymptote, perform polynomial long division if the degree of the numerator is greater than the degree of the denominator.

Tags

CCSS.HSF-IF.C.7D

Front

What is the horizontal asymptote of the function f(x) = \frac{3x^2 + 2}{5x^2 + 1}?

Back

The horizontal asymptote is y = \frac{3}{5}.

Tags

CCSS.HSF-IF.C.7D

Front

What happens to the graph of a function at a vertical asymptote?

Back

The graph approaches the vertical asymptote but never touches or crosses it, indicating a discontinuity.

Tags

CCSS.HSF-IF.C.7D