8'2 Rational Functions and Their Graphs

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8.2Rational Functionsand Their Graphs
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Essential Question How do I evaluate rational
functions? Objectives Identify and evaluate
rational functions. Graph a rational function,
find its domain, write equations for its
asymptotes, and identify any holes in the graph.
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Skill A - Finding the domain of a rational
function Recall - Division by zero is not
allowed it is undefined.
Solution The domain is all real numbers except
-5 and 2.
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Vertical Asymptote If (x a) is a factor of the
denominator of a rational function but not a
factor of its numerator, then x a is a vertical
asymptote of the graph of the function.
Hole in the Graph If (x b) is a factor of the
numerator and the denominator of a rational
function, then there is a hole in the graph of
the function when x b, unless x b is a
vertical asymptote.
PRACTICE - Identify all holes and vertical
asymptotes in each graph
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Horizontal Asymptote Let f(x) P / Q be a
rational function, where P and Q are
polynomials. If the degree of P is less than
the degree of Q, then y 0 is the equation of
the horizontal asymptote of the graph of R. If
the degree of P equals the degree of Q and a and
b are the leading coefficients of P and Q,
respectively, then y a / b is the equation of
the horizontal asymptote of the graph of R. If
the degree of P is greater than the degree of Q,
then the graph of R has no horizontal asymptote.
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Identify all excluded values, asymptotes, and
holes inthe graph of a rational function.
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Exercises Identify all asymptotes and holes in
the graph of each rational function.