Rational Functions and ModelsDefinitionLong Run BehaviorExampleSlide 5Try This OneWhen Numerator Has Larger DegreeSummarizeVertical AsymptotesSlide 10Zeros of Rational FunctionsSlide 12SummaryAssignmentRational Functions and ModelsLesson 4.6DefinitionConsider a function which is the quotient of two polynomialsExample: ( )( )( )P xR xQ x=Both polynomials2500 2( )xr xx+=Long Run BehaviorGivenThe long run (end) behavior is determined by the quotient of the leading termsLeading term dominates forlarge values of x for polynomialLeading terms dominate forthe quotient for extreme x11 1 011 1 0...( )...n nn nm mm ma x a x a x aR xb x b x b x b----+ + + +=+ + + +nnmma xb xExampleGivenGraph on calculatorSet window for -100 < x < 100, -5 < y < 5223 8( )5 2 1x xr xx x+=- +ExampleNote the value for a large xHow does this relate to the leading terms?2235xxTry This OneConsiderWhich terms dominate as x gets largeWhat happens to as x gets large?Note:Degree of denominator > degree numeratorPrevious example they were equal25( )2 6xr xx=+252xxWhen Numerator Has Larger DegreeTryAs x gets large, r(x) also gets largeBut it is asymptotic to the line22 6( )5xr xx+=25y x=SummarizeGiven a rational function with leading termsWhen m = nHorizontal asymptote atWhen m > nHorizontal asymptote at 0When n – m = 1Diagonal asymptote nnmma xb xabay xb=Vertical AsymptotesA vertical asymptote happens when the function R(x) is not definedThis happens when thedenominator is zeroThus we look for the roots of the denominatorWhere does this happen for r(x)?( )( )( )P xR xQ x=229( )5 6xr xx x-=+ -Vertical AsymptotesFinding the roots ofthe denominatorView the graphto verify25 6 0( 6)( 1) 06 or 1x xx xx x+ - =+ - ==- =229( )5 6xr xx x-=+ -Zeros of Rational FunctionsWe know thatSo we look for the zeros of P(x), the numeratorConsiderWhat 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 FunctionsNote the zeros of thefunction whengraphedr(x) = 0 whenx = ± 3SummaryThe zeros of r(x) arewhere the numeratorhas zerosThe vertical asymptotes of r(x)are where the denominator has zeros229( )5 6xr xx x-=+ -AssignmentLesson 4.6Page 319Exercises 1 – 41 EOO 93, 95,
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