Graph the rational functions. Locate any asymptotes on the...

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Key Concepts

Rational Functions

Rational functions are defined as the ratio of two polynomials. Their properties such as domain, zeros, and asymptotes are determined by the behavior of the numerator and the denominator, making the analysis of their graphs a structured process.

Vertical Asymptote

A vertical asymptote occurs where the denominator of a rational function is zero and the numerator is non-zero, leading to values of the function that tend to infinity. Identifying these points is critical for understanding the behavior of the graph near undefined values.

Horizontal Asymptote

A horizontal asymptote describes the end behavior of a rational function as x approaches infinity or negative infinity. When the degrees of the numerator and the denominator are equal, the horizontal asymptote can be found by taking the ratio of the leading coefficients, which indicates the function’s limiting value at extreme x-values.

Graphing Techniques for Rational Functions

Graphing rational functions involves determining the domain, locating asymptotes (vertical, horizontal, and sometimes oblique), finding intercepts, and analyzing symmetry. These techniques help create an accurate visual representation of the function’s behavior across its entire domain.

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