Sketch the graph of each rational function. See Examples 5-9...

Key Concepts

Horizontal Asymptote

The horizontal asymptote of a rational function is determined by the degrees of the numerator and denominator polynomials. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. This asymptote describes the long-term behavior of the function as x approaches positive or negative infinity.

Graphical Analysis Techniques

Graphing a rational function involves several steps: identifying the domain, locating zeros and vertical asymptotes, determining the horizontal (or sometimes oblique) asymptote, and analyzing the function's behavior near these asymptotes. Together, these elements help in sketching an accurate and comprehensive graph that illustrates the function’s key features.

Vertical Asymptotes

Vertical asymptotes occur at the values of x that make the denominator zero, provided these factors do not cancel with factors in the numerator. They represent lines which the graph of the function approaches but never intersects, indicating a type of discontinuity in the function.

Domain Restrictions

The domain of a rational function consists of all real numbers for which the function is defined. Specifically, values that make the denominator equal to zero are excluded from the domain, as they would result in division by zero, leading to undefined behavior in the function.

Rational Functions

A rational function is a ratio of two polynomials. The properties of rational functions, including their intercepts, asymptotes, and domain restrictions, are determined by the factors and degrees of these polynomials. They often exhibit complex behavior such as asymptotic trends and discontinuities which are key in graphing and understanding their overall behavior.

Zeros (x-intercepts)

The zeros, or x-intercepts, of a rational function are found by setting the numerator equal to zero and solving for x. These are the points where the function crosses the x-axis. They provide critical points that help in understanding and sketching the graph's overall behavior.

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