8.1.4 Rational Functions Key Objectives • Graph rational functions. • Transform rational functions by changing parameters. • Determine properties of hyperbolas. Key Terms
• A rational function is a quotient of polynomials, where the denominator has a degree of at least 1. • A hole in a graph is an omitted point. If a rational function has the same factor x − b in both the numera- tor and the denominator, and the line x = b is not a vertical asymptote, then there is a hole in the graph at the point where x = b . The rule of a rational function can be written as a ratio of two polynomials. The parent rational function is f ( x ) = 1/ x . Its graph is a hyperbola, which has two separate branches. Like logarithmic and exponential functions, rational functions may have asymptotes. The function f ( x ) = 1/ x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The rational function f ( x ) = 1/ x can be transformed by using methods similar to those used to transform other types of functions.
Example 1 Transforming Rational Functions To graph a rational function of the form f ( x ) = 1/( x − h ) + k , translate the graph of f ( x ) = 1/ x vertically k units and horizontally h units.
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