Asymptotes of Rational Functions

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as the input (x) approaches positive or negative infinity.

Front

How do you find the horizontal asymptote of a rational function?

Back

To find the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the denominator, the asymptote is y=0. If they are equal, the asymptote is y=\frac{a}{b}, where a and b are the leading coefficients.

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line that the graph of a function approaches as the input (x) approaches a certain value, typically where the function is undefined.

Front

How do you find vertical asymptotes in a rational function?

Back

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

Front

What does it mean if a function has a non-removable discontinuity?

Back

A non-removable discontinuity occurs at a vertical asymptote where the function approaches infinity or negative infinity, and cannot be 'fixed' by redefining the function at that point.

Front

What is the end behavior of a function?

Back

The end behavior of a function describes how the function behaves as x approaches positive or negative infinity.

Front

What is the significance of the leading coefficient in determining horizontal asymptotes?

Back

The leading coefficient of the numerator and denominator helps determine the horizontal asymptote when the degrees of both are equal.

Front

What happens to the graph of a function at a vertical asymptote?

Back

At a vertical asymptote, the graph of the function will approach infinity or negative infinity, indicating a discontinuity.

Front

If a rational function has a horizontal asymptote at y=4, what does this indicate about the function's end behavior?

Back

It indicates that as x approaches positive or negative infinity, the value of the function approaches 4.

Front

What is the relationship between the degrees of the numerator and denominator in determining horizontal asymptotes?

Back

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. If they are equal, the asymptote is y=\frac{a}{b}, where a and b are the leading coefficients.