Rational Functions and ModelsDefinitionLong Run BehaviorExampleSlide 5Try This OneWhen Numerator Has Larger DegreeSummarizeVertical AsymptotesSlide 10Zeros of Rational FunctionsSlide 12SummaryAssignmentRational Functions and ModelsLesson 4.6DefinitionConsider a function which is the quotient of two polynomialsExample: ( )( )( )P xR xQ x=Both polynomials2500 2( )xr xx+=Long Run BehaviorGivenThe long run (end) behavior is determined by the quotient of the leading termsLeading term dominates forlarge values of x for polynomialLeading 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 xExampleGivenGraph on calculatorSet window for -100 < x < 100, -5 < y < 5223 8( )5 2 1x xr xx x+=- +ExampleNote the value for a large xHow does this relate to the leading terms?2235xxTry This OneConsiderWhich terms dominate as x gets largeWhat happens to as x gets large?Note:Degree of denominator > degree numeratorPrevious example they were equal25( )2 6xr xx=+252xxWhen Numerator Has Larger DegreeTryAs x gets large, r(x) also gets largeBut it is asymptotic to the line22 6( )5xr xx+=25y x=SummarizeGiven a rational function with leading termsWhen m = nHorizontal asymptote atWhen m > nHorizontal asymptote at 0When n – m = 1Diagonal asymptote nnmma xb xabay xb=Vertical AsymptotesA vertical asymptote happens when the function R(x) is not definedThis 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-=+ -Zeros of Rational FunctionsWe know thatSo we look for the zeros of P(x), the numeratorConsiderWhat 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 = ± 3Summary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-=+ -AssignmentLesson 4.6Page 319Exercises 1 – 41 EOO 93, 95,