Graphing Rational Functions

Front

What is a rational function?

Back

A rational function is a function that can be expressed as the quotient of two polynomials, i.e., \( f(x) = \frac{P(x)}{Q(x)} \) where P and Q are polynomials.

Tags

CCSS.HSF-IF.C.7D

Front

What is an asymptote?

Back

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

Tags

CCSS.HSF-IF.C.7D

Front

What is a vertical asymptote?

Back

A vertical asymptote occurs where the function approaches infinity or negative infinity, typically at values of x that make the denominator zero.

Tags

CCSS.HSF-IF.C.7D

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 found by comparing the degrees of the numerator and denominator.

Tags

CCSS.HSF-IF.C.7D

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.

Tags

CCSS.HSF-IF.C.7D

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.

Tags

CCSS.HSF-IF.C.7D

Front

How do you find horizontal asymptotes?

Back

To find horizontal asymptotes, compare the degrees of the numerator and denominator: If the degree of the numerator is less, y=0; if equal, y=\( \frac{a}{b} \); if greater, no horizontal asymptote.

Tags

CCSS.HSF-IF.C.7D

Front

What is a point of discontinuity (POD)?

Back

A point of discontinuity is a point where a function is not continuous. In rational functions, this can occur at vertical asymptotes or holes.

Tags

CCSS.HSF-IF.C.7D

Front

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

Back

A hole occurs when a factor in the numerator and denominator cancels out, indicating a point where the function is undefined.

Tags

CCSS.HSF-IF.C.7D

Front

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

Back

Perform polynomial long division of the numerator by the denominator. The quotient (without the remainder) gives the equation of the slant asymptote.

Tags

CCSS.HSF-IF.C.7D