TERRA Environmental Research Institute



Polynomial Functions-Class Notes-Summary-Section 5.1-REVISED 2015Cover Sheet 7 Name:_________________ Per_____ Assignment # 8 Read Section, do Got it problems, THEN complete this worksheet(Polynomial functions : Coefficients must be Real numbers and Exponents on the variable(s) must be whole numbers )Read p. 281 to fill in the blanks: Write the Standard Form for the polynomial P(x),P(x) =____________________________________________________________________(n>1),where the coefficients of P(x) are an, …..a0 , these coefficients must be _______________ numbers and the exponents of P(x) are n, n-1, n-2,……, 1,0, these exponents must be NON-_____________ integers, that is, the exponents on variables must be whole numbers, 0, 1,2,3, 4, 5, ……, n. The leading term of P(x) is _________ (term of highest degree, that is, term with greatest exponent). The leading coefficient is _________ ( coefficient of leading term, that is, number next to variable with greatest exponent). Polynomials are written in Standard Form, that is, in descending order by degree of its terms. Polynomials are classified by degree . as follows: constant (0th degree), linear (1st degree), quadratic (2nd degree), cubic (3rd degree), quartic (4th degree), quintic (5th degree) , 6th deg poly., 7th deg poly., etc and by number of terms [monomial, binomial , trinomial, polynomial of 4 terms, polynomial of 5 terms, etc….]. Determine the degree of each then classify using its proper terminology : a)The deg of 5x3 is _________ , therefore it’s a ____________ b) The deg of 5 is____________ ( because it can be written as ___________ ), therefore it’s a ________________ c) deg of 7 x2y3=_____________it’s a_________________ d) Tricky! deg of 32 x5 =_______________it’s a _____________________ e) Deg of polynomial 2x2 + 6x3-7x4 is _______________ , it’s a _________ f) Deg of polynomial 8 x2y5 + 3 x y4 - x21y3 is ________Polynomials (if applicable, write in S.F then classify them by degree and by number terms using the proper terminology) Not polynomials ( state reason(s) why they are not)3x4+5x3-7x8-x+ 2-1In Standard Form_______________________________________3x12+5x3-7x2+125 x7-7x2+12 x + 2713 x + 327 +9 x2In Standard Form_________________________________________5x + 6x2 - 1x511 can be re-written as ________________, the coefficient is ____11x5 can be re-written as 11 x? 11x-2+4x can be re-written as ___________________2X3 + 5 X2 + -6Listen to instructions about End behavior and Turning Points. Write your own annotations/notes for future reference.Fill in the blank with +∞ or-∞End Behavior Quadrant 2 (-, +) Quadrant 1 ( +, +) In Quadrant 1 as x→ + ∞ y →____ In Quadrant 2 as x→ ____ y → + ∞ Quadrant 3 (-, - ) Quadrant 4 ( + , -) In Quadrant 3 as x→ ____ y →____ In Quadrant 4 as x→ ____ y →____What is a turning point? PolynomialClassify by degreeLeading Coefficient.(Is it positive or negative?)Degree of polynomial(Is it even or odd?)End behavior of its graph (far left and far right, not around or near origin)As x →+∞, then y→As x→-∞ then y→# of Turning points, point where the graph changes direction”concavity”(there at most n-1 turning points)Use g.c to sketch the graph of the polynomial and verify End behavior and # of turning points ( listen to instructions and copy notes correctly)ExampleP(x)= 5x +3(this is the sameas y =5x+3, but using function notation)n=1 → LinearLeading Coeff = +5Circle one:Positive coefficientNegative coefficient Degree __1st___Circle one:Even DegreeOdd DegreeCircle one:Up and UpDown and DownDown and UpUp and downAs x →+∞, then y→__As x→-∞ then y→___How many turning points? n=1, so n-1 = 0 there are no turning pointsNone Quadrant 1 (up)Quadrant 3 (down)P(x)=-5x +3n=__ →____Leading Coeff=____Circle one:Positive coefficientNegative coefficient Deg is _____Circle one:Even DegreeOdd DegreeCircle one:Up and UpDown and DownDown and UpUp and downAs x →+∞, then y→__As x→-∞ then y→___How many turning points?Quadrant 2(up) Quadrant 4 (down)P(x)= +2x3n=3 →______Leading Coeff=____Circle one:Positive coefficientNegative coefficient Deg is _____Circle one:Even DegreeOdd DegreeCircle one:Up and UpDown and DownDown and UpUp and downAs x →+∞, then y→__As x→-∞ then y→___n=3Number of possibleTurning points?____ or _____How many turning points? Sketch it________P(x)= -x3 + 5x2n=__ →______Leading Coeff=____Circle one:Positive coefficientNegative coefficient Deg is _____Circle one:Even DegreeOdd DegreeCircle one:Up and UpDown and DownDown and UpUp and downAs x →+∞, then y→__As x→-∞ then y→___n=_____Number of possibleTurning points?____ or _____How many turning points? Sketch it________P(x)= + x2n=2 →______Leading Coeff=____Circle one:Positive coefficientNegative coefficient Deg is _____Circle one:Even DegreeOdd DegreeCircle one:Up and UpDown and DownDown and UpUp and downAs x →+∞, then y→__As x→-∞ then y→___n=_____Number of possibleTurning points?____ How many turning points? Sketch it________P(x)= - x2n=2Leading Coeff=____Circle one:Positive coefficientNegative coefficient Deg is _____Circle one:Even DegreeOdd DegreeCircle one:Up and UpDown and DownDown and UpUp and downAs x →+∞, then y→__As x→-∞ then y→___n=_____Number of possibleTurning points?____ How many turning points? Sketch it________Px= 3x4+x3-8x2-2x+4Leading Coeff=____Circle one:Positive coefficientNegative coefficient Deg is _____Circle one:Even DegreeOdd DegreeCircle one:Up and UpDown and DownDown and UpUp and downAs x →+∞, then y→__As x→-∞ then y→___How many turning points? n=_____Number of possibleTurning points?____ How many turning points? Sketch it________P(x)= -x4+2x3Leading Coeff=____Circle one:Positive coefficientNegative coefficient Deg is _____Circle one:Even DegreeOdd DegreeCircle one:Up and UpDown and DownDown and UpUp and downAs x →+∞, then y→__As x→-∞ then y→___How many turning points? n=_____Number of possibleTurning points?____ How many turning points? Sketch it________Write a brief summary ODD DEGREE ( THINK LINES!) EVEN DEGREE (THINK PARABOLAS!) Down/Up Up/Down Up/Up Down/Down Q3 & Q1 Q2 & Q4 Q2& Q1 Q3 & Q4END BEHAVIORTURNING POINTS : # of possible turning points (maximum number is n-1 turning points, then “down by 2”)Degree of Polynomialn=1linearn=2quadraticn=3cubic n=4quarticn=5quinticn=6n=7# of possible turning pointsNONEONE2 or NONE3 or ____ 4 or ___ or ___5 or ___ or ___ 6 or ___ or ___ or___ Use your g.c to come up with an example for each case of a 6th degree polynomial, POSSIBLE TURNING POINTS ARE 5,3 OR 1. Challenge: Use your g.c. Find one polynomial equation for each case: a 6th degree poly with 5 turning points→ y=_______________________________, a 6th degree poly with 3 turning points→y= _________________________ and a 6th degree poly with 1 turning pt → y=____________________Make a quick Sketch of each (use g.c or online grapher) 5 turning points 3 turning pts 1 turning ptUsing Differences to Determine Degree of a polynomial function with the given “points” (data)? See p. 284 Fill in the blanks. xyy-value of data(function values) 1st difference 2nd difference 3rd difference 4th difference How far do you go? -3-1-1 (-7) – (-1)= -6 ( -3 ) - ( )= ( ) – ( )= ( ) – ( )= ( ) – ( )= ( ) – ( )=-2-7-7-1-3-305511111299 3-7-7Conclusion: According to your calculations , this set of data points “belongs” to a polynomial function of degree _________ because _____________________________________________________________________________________________________________________________________________________________Now use g.c. to confirm your conclusion above: Use your g.c. to enter the data points in g.c. (STAT, 1:EDIT, enter data on List), calculate each of the following regressions, list the regression equation and its regression coefficient r2 in the table below . Which one has the best correlation, that is, which equation has the r-value closest to 1 or -1? Circle one: 1st degree, Linear 2nd degree, Quadratic 3rd degree, Cubic 4th degree, QuarticLinRegQuadRegCubicRegQuartReg. (Ooops!)Equation: y=Equation: y=Equation: y=Equation: y=r2=r2=r2=r2=Input the best Reg equation in y= then check table of values, “ are the data points in the table of values”? Check it, if all of them are then it is a perfect fit!!! What was the degree of the equation of bet fit? ________Does it agree with your conclusion above?______ SELF-NOTES: Write additional notes below (observations/things to remember/ study before test/warnings to myself, etc…..) ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download