Nonlinear Functions and their GraphsPolynomialsExtrema of Nonlinear FunctionsSlide 7Slide 8Slide 9Try It OutAssignmentEven and Odd FunctionsSlide 13Slide 14ApplicationsSlide 16Slide 17Nonlinear Functions and their GraphsLesson 4.1PolynomialsGeneral formulaa0, a1, … ,an are constant coefficientsn is the degree of the polynomialStandard form is for descending powers of xanxn is said to be the “leading term”11 1 0( ) ...n nn nP x a x a x a x a--= + + + +Extrema of Nonlinear FunctionsGiven the function for the Y= screeny1(x) = 0.1(x3 – 9x2)Use window -10 < x < 10 and -20 < y < 20Note the "top of the hill" and the"bottom of thevalley"These are localextrema••Extrema of Nonlinear FunctionsLocal maximumf(c) ≥ f(x) whenx is near cLocal minimumf(n) ≤ f(x) whenx is near n••cnExtrema of Nonlinear FunctionsAbsolute minimumf(c) ≤ f(x) for all xin the domain of fAbsolute maximumf(c) ≥ f(x) for all xin the domain of fDraw a function with an absolute maximum•Extrema of Nonlinear FunctionsThe calculator can find maximums and minimumsWhen viewing the graph, use the F5 key pulldown menuChoose Maximum or MinimumSpecify the upper and lower bound for x (the "near")Note resultsNote resultsTry It OutFind local extrema … absolute extrema( )21( ) 1 23f x x= - -5 4 3 2( ) 0.1 .02 .35 .36 1.8h x x x x x x= + - - +26( )2 2g xx x=+ +AssignmentLesson 4.1APage 256Exercises 1 – 45 oddEven and Odd FunctionsIf f(x) = f(-x) the graph is symmetric across the y-axis It is also an even functionEven and Odd FunctionsIf f(x) = -f(x) the graph is symmetric across the x-axis But ... is it a function ??Even and Odd FunctionsA graph can be symmetric about a point Called point symmetryIf f(-x) = -f(x) it is symmetric about the origin Also an odd functionApplicationsConsider the U.S. birthrate from 1900 to 2005(births per 1000 people)Can be modeled by where x = number of years since 1900Evaluate f(95)What does it mean?With domain 1900 ≤ x ≤ 2005Identify the absolute minimum and maximum3 2( ) .0000285 .0057 .48 34.4f x x x x=- + - +ApplicationsU.S. natural gas consumption from 1965 to 1980 can be modeled byx = 6 is 1966 and x = 20 is 1980Consumption measured in trillion cubic feetEvaluate f(10) …. What does it mean?Graph for 6 ≤ x ≤ 20 and 0.4 ≤ y ≤ 0.8Determine local extrema, interpret results4 3 2( ) .0001234 .005689 .08792 .5145 1.514f x x x x x= - + - +AssignmentLesson 4.1BPage 258Exercises 91 – 97