Characteristics of 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, where the denominator is not zero.

Front

What are vertical asymptotes?

Back

Vertical asymptotes are lines x = a where a rational function approaches infinity or negative infinity as the input approaches a.

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 line y = b that a function approaches as x approaches infinity or negative infinity.

Front

How do you find 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, y = 0; if equal, y = leading coefficient of numerator/leading coefficient of denominator; if greater, no horizontal asymptote.

Front

What is a hole in a rational function?

Back

A hole occurs at a point where both the numerator and denominator of a rational function are zero, indicating a removable discontinuity.

Front

How do you find holes in rational functions?

Back

To find holes, factor the numerator and denominator, and identify common factors that can be canceled.

Front

What is the significance of asymptotes in rational functions?

Back

Asymptotes indicate the behavior of the function as it approaches certain values, helping to understand limits and graphing.

Front

What is the domain of a rational function?

Back

The domain of a rational function includes all real numbers except where the denominator is zero.

Front

What is the range of a rational function?

Back

The range of a rational function is the set of all possible output values, which can be determined by analyzing horizontal asymptotes and holes.