Rational Functions and Asymptotes

Front

What is a rational function?

Back

Front

What is an asymptote?

Back

An asymptote is a line that a graph approaches but never touches. It can be vertical, horizontal, or oblique.

Front

How do you find vertical asymptotes of a rational function?

Back

Vertical asymptotes occur where the denominator of the rational function is zero and the numerator is not zero at those points.

Front

What is a horizontal asymptote?

Back

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

Front

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

Back

1. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. 2. If the degrees are equal, the horizontal asymptote is y = \frac{a}{b} where a and b are the leading coefficients. 3. If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What is the domain of a rational function?

Back

The domain of a rational function is all real numbers except where the denominator is zero.

Front

How do you find the x-intercepts of a rational function?

Back

The x-intercepts occur where the numerator is zero (i.e., set P(x) = 0) and the denominator is not zero.

Front

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

Back

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

Front

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

Back

A removable discontinuity occurs when a factor cancels in the rational function, while a non-removable discontinuity occurs at vertical asymptotes where the function is undefined.

Front

How do you find the range of a rational function?

Back

To find the range, analyze the behavior of the function as x approaches the vertical asymptotes and infinity, and consider any horizontal asymptotes.