Short Run Behavior of Rational FunctionsZeros of Rational FunctionsSlide 3Vertical AsymptotesSlide 5SummaryDrawing the Graph of a Rational FunctionGiven the Graph, Find the FunctionLook for the HoleSlide 10AssignmentShort Run Behavior of Rational FunctionsLesson 11.5Zeros of Rational Functions•We know that•So we look for the zeros of P(x), the numerator•ConsiderWhat are the roots of the numerator?Graph the function to double check( )( ) 0 ( ) 0( )P xR x P xQ x= = � =229( )5 6xr xx x-=+ -Zeros of Rational Functions•Note the zeros of thefunction whengraphed•r(x) = 0 whenx = ± 3Vertical Asymptotes•A vertical asymptote happens when the function R(x) is not definedThis happens when thedenominator is zero•Thus we look for the roots of the denominator•Where does this happen for r(x)?( )( )( )P xR xQ x=229( )5 6xr xx x-=+ -Vertical Asymptotes•Finding the roots ofthe denominator•View the graphto verify25 6 0( 6)( 1) 06 or 1x xx xx x+ - =+ - ==- =229( )5 6xr xx x-=+ -Summary•The zeros of r(x) arewhere the numeratorhas zeros•The vertical asymptotes of r(x)are where the denominator has zeros229( )5 6xr xx x-=+ -Drawing the Graph of a Rational Function•Check the long run behaviorBased on leading termsAsymptotic to 0, to a/b, or to y=(a/b)x•Determine zeros of the numeratorThese will be the zeros of the function•Determine the zeros of the denominatorThis gives the vertical asymptotes•Consider ( )235xyx+=+Given the Graph, Find the Function•Consider the graphgiven with tic marks = 1•What are the zeros of the function?•What vertical asymptotes exist?•What horizontal asymptotes exist?•Now … what is the rational function?Look for the Hole•What happens when both the numerator and denominator are 0 at the same place?•Consider •We end up with which is indeterminateThus the function has a point for which it is not defined … a “hole”22( )2x xr xx+ -=+00y =Look for the Hole•Note that when graphed and traced at x = -2, the calculator shows no value•Note also, that it does not display a gap in the lineAssignment•Lesson 11.5•Page 465•Exercises 1 – 41
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