Characteristics of Rational Functions

Front

What is a rational function?

Back

A rational function is a function that can be expressed as the ratio of two polynomials, typically in the form f(x) = P(x)/Q(x), where P and Q are polynomials.

Front

What are vertical asymptotes?

Back

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

Front

How do you find vertical asymptotes?

Back

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

Front

What are horizontal asymptotes?

Back

Horizontal asymptotes are lines y = b that a rational function approaches as x approaches infinity or negative infinity.

Front

How do you find horizontal asymptotes for rational functions?

Back

To find horizontal asymptotes, compare the degrees of the numerator and denominator polynomials.

Front

What happens if the degree of the numerator is less than the degree of the denominator?

Back

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

Front

What happens if the degree of the numerator is equal to the degree of the denominator?

Back

If the degrees are equal, the horizontal asymptote is y = leading coefficient of numerator / leading coefficient of denominator.

Front

What happens if the degree of the numerator is greater than the degree of the denominator?

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If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

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 its graph.

Front

What is a hole in a rational function?

Back

A hole occurs in a rational function at a value of x that makes both the numerator and denominator zero, indicating a removable discontinuity.