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 that makes both the numerator and denominator zero, indicating a removable discontinuity.

Front

How do you find the coordinates of a hole in a rational function?

Back

To find the coordinates of a hole, factor the numerator and denominator, cancel the common factors, and substitute the x-value of the canceled factor into the simplified function.

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.

Front

How do you find vertical asymptotes in a rational function?

Back

Vertical asymptotes occur at values of x that make the denominator zero after canceling any common factors.

Front

What is a horizontal asymptote?

Back

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

Front

How do you determine 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 leading coefficient in a rational function?

Back

The leading coefficient affects the end behavior of the graph, determining how the function behaves as x approaches positive or negative infinity.

Front

What is the difference between a removable and non-removable discontinuity?

Back

A removable discontinuity occurs at a hole in the graph, while a non-removable discontinuity occurs at a vertical asymptote.

Front

How can you identify the end behavior of a rational function?

Back

The end behavior can be identified by analyzing the degrees of the numerator and denominator and their leading coefficients.