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 the function approaches infinity or negative infinity as x approaches a.

Front

What are horizontal asymptotes?

Back

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

Front

How do you find vertical asymptotes?

Back

Vertical asymptotes occur at values of x that make the denominator zero, provided those values do not cancel with the numerator.

Front

What is a hole in a rational function?

Back

A hole occurs at a value of x where both the numerator and denominator are zero, indicating a removable discontinuity.

Front

How do you find horizontal asymptotes for rational functions?

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 = leading coefficient of numerator/leading coefficient of denominator; if greater, no horizontal asymptote.

Front

What is the significance of the degree of a polynomial in a rational function?

Back

The degree of the polynomial affects the behavior of the function, including the number of asymptotes and the end behavior of the graph.

Front

What happens to a rational function as x approaches infinity?

Back

As x approaches infinity, the function approaches its horizontal asymptote, if one exists.

Front

What is the relationship between zeros of the numerator and vertical asymptotes?

Back

Zeros of the numerator indicate x-intercepts, while vertical asymptotes are determined by zeros of the denominator.

Front

What is the formula for finding the horizontal asymptote when degrees are equal?

Back

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