Graphs 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 is a hole in the graph of a rational function?

Back

A hole occurs in the graph of a rational function at a value of x where both the numerator and denominator are zero, indicating that the function is undefined at that point.

Front

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

Back

The horizontal asymptote can be found by comparing the degrees of the numerator and denominator polynomials. 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 = leading coefficient of numerator / leading coefficient of denominator. If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where the function approaches infinity or negative infinity as x approaches a. It occurs at values of x that make the denominator zero, provided the numerator is not also zero at that point.

Front

How do you determine the vertical asymptotes of a rational function?

Back

To find the vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.

Front

What is the significance of the horizontal asymptote in a rational function?

Back

The horizontal asymptote indicates the behavior of the function as x approaches positive or negative infinity, showing the value that the function approaches.

Front

What happens to the graph of a rational function at a hole?

Back

At a hole, the graph will not be defined, and there will be a gap in the graph at that x-value.

Front

Can a rational function have more than one vertical asymptote?

Back

Yes, a rational function can have multiple vertical asymptotes, depending on the number of factors in the denominator that can be set to zero.

Front

What is the difference between a hole and a vertical asymptote?

Back

A hole indicates a point where the function is undefined due to a common factor in the numerator and denominator, while a vertical asymptote indicates a point where the function approaches infinity.

Front

What is the end behavior of a rational function?

Back

The end behavior of a rational function describes how the function behaves as x approaches positive or negative infinity, often determined by the horizontal asymptote.