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Learn about rational expressions, vertical asymptotes, holes in graphs, and graphing techniques to identify domains and asymptotes while analyzing rational functions.
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Rational Expression • It is the quotient of two polynomials. • A rational function is a function defined by a rational expression. Examples: Not Rational:
Find the domain: Graph it:
Find the domain: Graph it:
Find the domain: Graph it:
Find the domain: Graph it:
Vertical Asymptote • If x – a is a factor of the denominator of a rational function but not a factor of the numerator, then x = a is a vertical asymptote of the graph of the function.
Find the domain: Graph it using the graphing calculator. What do you see?
Find the domain: Graph it using the graphing calculator. What do you see?
Hole (in the graph) • If x – b is a factor of both the numerator and denominator of a rational function, then there is a hole in the graph of the function where x = b, unless x = b is a vertical asymptote. • The exact point of the hole can be found by plugging b into the function after it has been simplified.
Horizontal Asymptotes • Degree of numerator = Degree of denominator • Degree of numerator < Degree of denominator • Degree of numerator > Degree of denominator Horizontal Asymptote: Horizontal Asymptote: Horizontal Asymptote:
