Rational Functions and End Behavior

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 is a vertical asymptote?

Back

A vertical asymptote is a vertical line 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 is a horizontal asymptote (H.A.)?

Back

A horizontal asymptote is a horizontal line y = b that the graph of a function approaches as x approaches infinity or negative infinity.

Front

How do you determine the horizontal asymptote when 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 to the horizontal asymptote if the degrees of the numerator and denominator are equal?

Back

If the degrees of the numerator and denominator are equal, the horizontal asymptote is y = a/b, where a and b are the leading coefficients of the numerator and denominator.

Front

What indicates that a factor in the denominator is a hole?

Back

A factor in the denominator is a hole if it cancels out with a corresponding factor in the numerator.

Front

How do you find the y-intercept of a rational function?

Back

To find the y-intercept, replace all x-values with 0 in the function and solve for y.

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 end behavior and the existence of horizontal asymptotes.

Front

What does it mean if a rational function has no horizontal asymptote?

Back

If the degree of the numerator is greater than the degree of the denominator, the rational function has no horizontal asymptote.