8-3 Rational Functions and Their Graphs

Objectives

To identify properties of rational functions

To graph rational functions

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You use a ratio of polynomial functions to form a rational function, like y = x + 3 x + 16 . y equals . fraction x plus 3 , over x plus 16 end fraction . .

Essential Understanding If a function has a polynomial in its denominator, its graph has a gap at each zero of the polynomial. The gap could be a one-point hole in the graph, or it could be the location of a vertical asymptote for the graph.

A rational function is a function that you can write in the form f ( x ) = P ( x ) Q ( x ) f , open x close , equals . fraction p , open x close , over q , open x close end fraction  where P(x) and Q(x) are polynomial functions. The domain of f(x) is all real numbers except those values for which Q(x) = 0.

Here are graphs of three rational functions:

y = x 2 x 2 + 1 y equals . fraction x squared , over x squared , plus 1 end fraction  
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y = ( x + 3 ) ( x + 2 ) ( x + 2 ) y equals . fraction open , x plus 3 , close . open , x plus 2 , close , over open , x plus 2 , close end fraction  

y = x + 4 x − 2 y equals . fraction x plus 4 , over x minus 2 end fraction  
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