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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” LESSON esn 5 .2 Lesson 249 the box.the Then preview LessonPerformance the Task. sheetthe of cardboard and unknown the variables for asking students the to list known the quantities for View Engage the online. photo, section the Discuss and roughly where turningpoints occur. crosses thex-axis, where itistangent to the x-axis, x-intercepts. Sketch thegraph by showing where it function’s values determined ontheintervals by the Use to determine thefactors thesign ofthe constant factor. Identify andplotthex-intercepts. includesanegativeand whetherornotthefunction Determine theendbehavior basedonthedegree Essential Question: equations ordefinitions. Work orsmallgroup to to matchtheir withapartner types function Objective Language Practices Mathematical A-APR.B.3, F-IF.B.4 factorizations are available, andshowing endbehavior. Also A-SSE.A.1a, Graph identifying polynomialfunctions, zeros whensuitable The student isexpectedto: Common Core Math Standards Functions PolynomialGraphing function ininterceptfunction form? sketch thegraph ofapolynomial PERFORMANCE TASK PREVIEW: LESSON ENGAGE COMMON COMMON CORE CORE MP.7 Using Structure F-IF.C.7c 5 . 2 5.2 How doyou CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U3M05L2.indd 249

© Houghton Mifflin Harcourt Publishing Company “a polynomial sixth-degree function.” called a quintic A polynomial of function aquartic degree 4is called A In nhas standard the function of general, degree apolynomial form p quadratic functions are polynomial functions of degree 2,and cubic functions are polynomial functions of degree 3. functions Linear, quadratic, and cubic functions belong to amore general class of polynomial functions called HowEssential Question: doyou sketch inintercept thegraph ofapolynomialfunction form? 5.2 Name Module 5 Module x associated with it, but since and and where where

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when n x = rises without ) n falls No; this x n x ( x = = ) ) x x ( ( have end behavior where where behavior end have n x The statement says that as you move move as you that says statement The and the direction in which you are are in which you the direction and n x for all values of n. all values for n . What x) → +∞ as f( x → -∞. What Suppose the graph about you tell does this statement Does any function of the form the form function of Does any f(x) = Students have the option of completing the completing of the option have Students in the book either activity calculator graphing x Graphing Polynomial Functions Functions Polynomial Graphing = ) x ( EXPLORE 1 1 EXPLORE INTEGRATE TECHNOLOGY INTEGRATE QUESTIONING STRATEGIES QUESTIONING or online. or bound. This is true of the graph of f is true of the graph This bound. without bound, which is not true of the graph of which is not true of the graph without bound, f n is even. ? Explain. f(x) → -∞ as x → +∞? Explain. decreases x-axis so x decreases the leftto along the negative of f the graph without bound, of f(x) = the about say you can What the x-axis? on moving n? of value Investigating the End Behavior of the End Behavior Investigating Polynomial of Simple the Graphs Functions would mean that as you move right along the right move as you mean that would of f the graph x-axis, positive DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 19/03/14 3:48 PM

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Function f f f f f Integrate Mathematical Practices Mathematical Integrate = =

) ) x x ( ( Use a graphing calculator to graph the polynomial functions ƒ functions thepolynomial graph to calculator a graphing Use ƒ function’s domain, range, and end behavior. (Use interval notation for the domain and range.) and the domain for interval notation (Use behavior. end and range, domain, function’s f The domains of the functions domains of the The f is odd, the ranges of f the ranges If n is odd, or odd. depend on whether n is even behavior of f behavior both How can you generalize the results of this Explore for ƒ for this Explore of the results generalize you can How number?

Reflect

PROFESSIONAL DEVELOPMENT PROFESSIONAL , Practice MP.7 Mathematical address to opportunity an provides lesson This analyze Students structure.” use of make and “look for to students which calls for domain, they examine Specifically, functions. polynomial of the attributes of some the They also investigate behavior. end and points, turning intercepts, range, the factored to they relate how and functions, polynomial of graphs of x-intercepts polynomial also analyze Students expression. polynomial the related of form contexts. in real-world functions

B Module 5 1.

A2_MNLESE385894_U3M05L2.indd 250

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© Houghton Mifflin Harcourt Publishing Company and table asign for use graphs. their a different polynomial inintercept function form. Encourage students to display steps that were followed. Have students switch roles and repeat exercise the using instructions for eachstep. Then have student the sketched who graph the write the of apolynomial inintercept function form other the gives while verbal Have students work inpairs. Instruct one student ineachpair to sketch graph the Peer-to-Peer Activity COLLABORATIVE LEARNING is form said inintercept to be may maximum aglobal be) or minimum. global minimum relative than maximum the or less than minimum. the A point The graph must move toward x-axis the and away then from it near point the of tangency. x-intercept, graph the tangent can be which means that point graph the has aturning graphthe must move away from and move then back toward each pair x-axis of the between successivex-intercepts, A The cubic ƒ function Module 5 Module In general, apolynomial of function form the p with other linear factors inx,such as x The graph of p are real numbers (that nis number the are of not variable necessarily distinct)has factors. degree nwhere

y-coordinate of each turning point is amaximum or minimum value of at function the least near that turning . Amaximum or minimum global value is called Explore 2 the table. Use agraphing calculator to graph cubic the functions ƒ ƒ global? minimum values that are not values? How many local How many globalminimum global? maximum values that are not values? How many local How many globalmaximum does thegraph have? How many points turning x-axis at eachx-intercept? x-axis ordoesitcross the Is thegraph tangent to the x-intercepts? What are thegraph’s does f How manyfactors distinct ( x )

= ( x x ) ( ( have? x x Function ) , is amaximum or minimum within somearound interval turning the point that not need (but be

- 2 = ( a x ) ( ) ( x

x Investigating thex-intercepts and Turning Points of theGraphs ofPolynomial Functions = - + 2 x x 3

. Since graph the of p(x)intersects x-axis the only at its x-intercepts, 1

has three factors, of all which happen x.One to or be more of x’s the replaced can be ) ) ( Then use the graph .Then the use of each to function answer questions the in x - x to x-axis, the and point the of tangency becomes aturning point the because 2

) ... values No maximum values No minimum Crosses - 2,without changing that fact the is function the cubic. ( x - f (x) x between those x-intercepts. those between Also, instead of crossing x-axis the at an n

( ) 0 0 1 x = has has at at local maximum or

)

= x x

3 x a or absolute or

= 0 (

1 , x x 251 - 2

,...,and x maximum value values, butonelocal No globalmaximum minimum value values, butonelocal No globalminimum crosses at x Tangent at x 1

) ( x ( if the function never function ifthe takeson avalue that is greater f x - (x) x ) n

local minimum local x as its x-intercepts, which is why polynomial the = = 2

) ... x 0, 2 x 3

( ,ƒ 2 2 x

2 ( - = 2 ( x -2 = 0; x x )

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( x - 2 x-intercepts Crosses at allthree f minimum value values, butonelocal No globalminimum maximum value values, butonelocal No globalmaximum x

(x) )

1 and ,and , x =x 2

,...,and

0, 2,-2 ( x -2 3 2 x

)

( x +2 Lesson 2 Lesson ) 10/06/15 12:22AM

252

Between Between They are equal. are They How is the number of variable factors of a of factors variable of the number is How of the degree to function related polynomial a polynomial of points the turning are Where the local maxima are How function located? Graphing Polynomial Functions Functions Polynomial Graphing QUESTIONING STRATEGIES QUESTIONING -intercepts; the local maxima of x-intercepts; pairs consecutive points. the turning at and minima occur the polynomial? the polynomial? points? the turning to related minima and DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-C;CA-C 6/8/15 6:39 PM

)

4 x =

x + 3 )

( Lesson 2 x ) -3 ) ( 3 4 x - 2 x + 2 ( ( has no 0, 2, -2, 4

= x x (x) , = No global maximum No but one local values, maximum value

Crosses at all four all four at Crosses x-intercepts One global minimum value and one local minimum value f

) ) x (

- 2 ) x

( )

3 = 0; x = 2 x - 2 =

( ) 2

x + 2 3 3 -2 x x ( ( = 0, 2, -2 , ƒ =

4 x x

(x) f .Then use the graph of of use .Then the graph =

) No global No maximum values, but one local maximum value crosses at x at crosses and Tangent at x at Tangent One global minimum value (which occurs and no localtwice) minimum values ) x (

+ 3 ) x ( )

x - 2 (

+ 2

3 252 2 2 x x 0, 2 ( ) = - 2 (x) No maximum No values Crosses Crosses both at x-intercepts f x One global minimum no and value local minimum values ( x =

) 4

x x ( would have one global maximum value, no local maximum one global maximum value, have would 4

, and no minimum values. 1 = 1 0 x - (x) , and ƒ , and f ) =

= 0 ) Tangent at at Tangent x No maximum No values One global minimum no and value local minimum values x .) How would your answers to the questions about the functions and their graphs their graphs and the functions about the questions to answers your would .) How + 2

4 ( x ( ) -x 2 =

- ) x x ( ( 2 x that are not global are that , whereas f , whereas = Function

-axis. If the factor is raised to an even power, the graph will be tangent to the x-axis. to will be tangent the graph power, If an even the factor to x-axis. is raised ) x ( How many distinct many How factors? the are What x-intercepts? or cross to Tangent the x-axis at x-intercepts? turning many How points? global many How maximum values? local many How maximum values not are that global? global many How minimum values? local many How minimum values not are that global? Use a graphing calculator to graph the quartic functions ƒ the functions quartic graph to calculator a graphing Use ƒ each function to answer the questions in the in table. the questions answer each function to values values maximum values, one global minimum value, and no local minimum values that are not are and no local that minimum values one global minimum value, maximum values, global -axis, a maximum value would would a maximum value the x-axis, be reflected across would graph each function’s Since the questions to the answers change would This versa. and vice a minimum value become f instance, For about global and local maximum and minimum values. becomes ƒ becomes

1 into each of the quartic functions in Step B. (For instance, ƒ instance, (For B. in Step the functions quartic each of -1 into of factor a introduced you Suppose -axis or is is or the x-axis crosses form function in intercept a polynomial of whether determines the graph What x-intercept? an at it to tangent -intercepts the graph of a polynomial function in intercept form has? form function in intercept a polynomial of the graph x-intercepts many how determines What change? -intercept is raised to an odd power, the graph will cross will cross the graph an odd power, to is raised the x-intercept If the factor produces that the Each distinct one x-intercept. factor produces

Reflect

DIFFERENTIATE INSTRUCTION DIFFERENTIATE Suggest that all students create reference cards that show the general shapes of of shapes the general show that cards reference create all students that Suggest a as the think cards can of 5. Students 2 through degrees of functions polynomial functions. polynomial use them when graphing and graphs” “vocabulary of Cognitive Strategies Cognitive When discussing local maxima and minima, have students cover irrelevant parts irrelevant cover students have minima, local discussing and maxima When the local them value focus on extreme help to paper of a sheet with the graph of interest. of Visual Clues Visual

Module 5 Module 4. 3. 2.

A2_MNLESE385894_U3M05L2 252

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© Houghton Mifflin Harcourt Publishing Company why it is amatch. from eachpile. If have they amatch, must they explain to rest the of group the cards, the shuffle place face down, them and turns take turning over one card Place name the cards inone pile and definition the cards inanother. Students definition (afunction with adegree of 3, for example, or an equation linear, writing one name card. per Have other the of half students the write a oftypes functions addressed lesson, inthis such as cubic, quartic, quadratic, and Distribute note cards to students. Have students the half write names the of the Connect Vocabulary LANGUAGE SUPPORT As As  factorsthose is. ofsign values function the by determining of sign the each factor and recognizing what of sign the product the of valuesfunction tells you graph the whether is above or below You interval. x-axis the on aparticular canthe find the Given apolynomial inintercept function form, you can sketch function’s the graph by using end the behavior, Explain1 oue 5 Module So, graph the of ƒ As As While you should precise be about where graph the crosses x-axis, the Sketch graph. the The The Identify graph’s the x-intercepts, and of sign the use then ƒ(x)on determined by intervals x-intercepts the Example 1 x-intercepts, and of sign the values function the on determined by intervals x-intercepts. the The of sign the

ƒ(x) has following the end behavior: negative. For given the degree f(x),the function is 3and constant the factor a,which is 1,is positive, so determined by degree the nis whether even or and constant odd the whether factor a is positive or Identify end the behavior. For p function the ƒ the graphthe lies above x-axis the and where it lies below x-axis. the graph that aren’t on x-axis. the Your sketch should simply show where you do not precise to need be about y-coordinates the of points on the on x intervals the to where find graph the is above x-axis the and where it’s below x-axis. the intervals: intervals: ( x -2

x <-2 = x-intercepts are x Sketch function. graph the of polynomial the x >3 → → x ( +∞, ƒ(x) -∞, ƒ(x) x x + 2 < -2, ) ( x Sketching theGraph ofPolynomial Functions in Intercept Form ( < -2and 0< − 3 x Sign of the Sign ofthe -2 → → Constant )

Factor is above x-axis the on -2 intervals the ) = 0,x +∞. -∞. <

+ + + + x < = 0, 0< -2, and x x < Sign ofx x 3. < + - - + 3, and x = three3. These x-intercepts dividex-axis the into four ( x )

= 253 a > ( x Sign of x +2 − 3. + + - + x 1

) ( < x − x <0 x 2

) Sign of . x -3 and and - - - + ( x − x x > - n

) 4 end, the behavior is 3, and it’s below x-axis the f = (x) ) on eachcard. x 0 Sign of ( x y - + - + +2 2 ) ( esn 2 Lesson x - 3 4 x 19/03/14 3:48PM )

the 254

? ) 0 ( When the When When sketching the graph of a polynomial a polynomial of the graph sketching When know you do how form, function in intercept find f when finding you you are What graph’s y-intercept graph’s Graphing Polynomial Functions Functions Polynomial Graphing QUESTIONING STRATEGIES QUESTIONING -axis? the x-axis? to tangent is when the graph same zero values occur an even number of times in number of times an even occur values same zero of the the graph the factorization of the polynomial, value. that function the x-axis at to tangent is DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 19/03/14 3:48 PM

x ) ) 4 Lesson 2 x + 2 x - 1 2 ( ( ) ) y - - + + - 0 x + 1 x - 4 ( ( Sign of 2 - = - 4 (x) f

- + + + - + of , Sign x + 2 . 1 4 -intercepts the x-intercepts intervals by determined on ) < - - + + + x of = ( x Sign x + 1 x , <

+ + + - - -1 of Sign 254 = x - 1 , x , , and -2 4 - - - - + -1 of

< ) Sign x - 4 < = + 2 x x x , ( <

) . . 1 -2 + 1 - - - - - x ( Factor -∞ -∞ ) = Constant Constant Sign of the and -axis on the intervals on the x-axis above is ) → → − 1 x ( x ( -1 ) 1 4 -1 . < − 4 4 4 -2 x

( -∞, ƒ(x) +∞, ƒ(x) − < x > < x < < x < < x < x < x > Interval → → -intercepts are x are x-intercepts x =

x x 1 ) -2 -1 x -2 ( and Sketch the graph. Sketch -axis on the intervalsx on the x-axis below it’s So, the graph of ƒ of the graph So, The -axis and where it’s below the x-axis. below it’s where and the x-axis above is the graph findto where -intercepts, and then use the sign of ƒ then use the sign of and x-intercepts, the graph’s Identify As Identify the behavior. end Identify As ƒ

Module 5

A2_MNLESE385894_U3M05L2.indd 254

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x+ b 1

)(a 2

x+ b 2

)(a 3

x+ b 3

is), which not CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U3M05L2.indd 255

© Houghton Mifflin Harcourt Publishing Company of box the is given interms of volume the x,then is acubic of function x. of abox (arectangular prism) by multiplying length, the width, and height. If each dimension You can cubic use functions to modelreal-world Explain2 situations. For example, you volume the find 5. Sketch function. graph the of polynomial the Your Turn oue 5 Module To nearest the tenth, value the find of xthat maximizes volume the of box. the fourthe sides of second the diagram), boxyou the (see glue each flapto side the it overlaps. of rectangle the indicates adashed line acut, segment while indicates afold.) After you fold up sides and folding along other the side. (In first diagram, the asolidline segment interior inthe you make asquare flapof side length xinches ineach corner by cutting along one of flap’s the To create an open-top box out of asheet of cardboard that is 9inches long and 5inches wide, So, thegraph off As The As x Example 2 x 0 ƒ (x) < 0 Interval x x < x x x = x-intercepts are x → →

9 in. < 4,andit’s below thex-axisoninterval > 4 < 0 x - -∞, +∞, < 4 x 2

( x - 4 f f ( ( x x Sign ofthe Modeling withaPolynomial Function ) ) ) Constant

( 5 in. → → Factor x ) - - -

-∞. +∞. is above thex-axisonintervals = 0andx x x Sign of = 4. + + + 2

Sign of x -4 255 + - - f > 4. ( x ) = Sign of < 0and x - + + 2

( x -4 )

- 4 - 2 0 y 2 esn 2 Lesson 4 19/03/14 3:48PM x 256 describe those on x describe constraints The Since length, height, and width will length, height, Since for length and and length x for on do the constraints Why x be that require simply not width to this situation from generalize you can How volume any for the finding domain Graphing Polynomial Functions Functions Polynomial Graphing QUESTIONING STRATEGIES QUESTIONING always be nonnegative, the domain of a volume of a volume the domain be nonnegative, always the independent that require function always will each make that on only those values take variable dimension nonnegative. nonnegative? nonnegative? function? that make expressions for length and for expressions make of x that values values. width nonnegative DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 3/27/14 3:59 PM

) 1.0 2x ≈ - 2x 2.5 4.5 5 2x ( ) < < x 2x - 9 ≈ 21.0 when x (

0, or x > 0, or 0, or x > 0, or ) x = (

) V x ( - 2x - 2x > 0 V Adjust the viewing window so you can see the maximum. From the From see can the maximum. so you the viewing window Adjust select CALC menu, the locate to 4: maximum calculator’s graphing occurs. value the maximum where point Length of box: 9 - box: Length of Taken together, these constraints give a domain of 0 < of a domain give these constraints together, Taken x 5 - 5 box: of Width box: of Height 9 So, Because the length, width, and height of the box must all be positive, the volume the volume all be positive, must the box of height Because and width, the length, constraints: three the following by determined is domain function’s maximum volume of about 21 cubic inches when square flaps with a side a side with flaps whensquare inches 21 cubic about of volume maximum cardboard. of the sheet of in the corners made are 1 inch of length 3. domain. its function on the volume graph to calculator a graphing Use 2. domain. its determine function and the volume Write 1. the box. of the dimensions for expressions Write Find the dimensions of the box once the flaps have been made and the sides the sides been and made the have flaps once the box of the dimensions Find the function graph the box, function for a volume Create been up. folded have maximizes that x find of to the value use the graph and calculator, a graphing on the volume. Identify the important information. the important Identify a rectangular of corner in each made is x inches length side of flap A square sheet cardboard. of 5 inches. by inches 9 measures cardboard of sheet The Solve Formulate a Plan Formulate Analyze Information Analyze

Module Module 5

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” be represented ofdegree by 3. afunction related to afigure withthree dimensions, soitwill even orodd, andpositive ornegative. and whetherthedegree oftheleadingcoefficient is maximum andminimumvalues, theendbehavior, of theturningpoints. You canalsodetermine the zeros andtheapproximate ofthefunction, location determine thex-intercepts andtherefore thereal location of turning the points. encourage students to predict number the and If function. the is function inintercept the form, onshould degree the and based look end behavior of functions, encourage to them predict how graph the Focus onCritical Thinking esn 5 .2 Lesson 257 MP.3 determine from graph the of function? the PRACTICES INTEGRATE MATHEMATICAL SUMMARIZE THE LESSON SUMMARIZE THE QUESTIONING STRATEGIES ELABORATE Before students graph any polynomial polynomial of function degree nthat you can What are some of key attributes the of a Explain. What is degree the of any volume function? Three; geometrically, volume is You can CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U3M05L2 257

© Houghton Mifflin Harcourt Publishing Company 9. 7. Your Turn 6. Reflect oue 5 Module 8. Elaborate is The lengthofthebox is25-2x,thewidth132x,andheightx is when square flapswithasidelengthof2.7inchesare madeinthecorners. three dimensionsofthebox nonnegative. mustbe volume Maximum 402cubicinches isabout even, greater thanorequalto 0,whereas whenn isodd, therange isallreal numbers. Whenn is The domainsare always thesame, butwhenniseven, therange contains onlynumbers minimum values. onwhetherthey-coordinates ofturningpointshas animpact represent maximumor also It The ontheendbehavior ofthefunction. signoftheconstant hasanimpact factor f also satisfies x All three satisfied inequalitiesmustbe simultaneously. Any x-value that satisfies x important graphing when function? the only two of Why? them. Check-InEssential Question volume of box. the box, you glue eachflapto sideit the overlaps. To nearest the tenth, value the of find xthat maximizes the dashed line segment indicates afold.) Once you fold up four the sidesof the linea solid segment interior inthe of rectangle the indicates a acut, while cutting along one of flap’s the sidesand folding along other. the (In diagram, the 13 inches wide, you make asquare flapof sidelength xinches ineachcorner by To create an open-top box out of asheet of cardboard that is 25inches long and Discussion Compare and contrast domain, the range, and end behavior of ƒ (x Justify andEvaluate ( inches is maximum the volume. volumesBoth are slightly less than 21cubic inches, suggests which that 21cubic V V square flapswith asidelength of 0.9inch and 1.1inches: wide, and volume the 1inch so high, is 21cubic inches. As acheck on result, this consider making Making square flapswith asidelength of 1inch means 7inches that be long, box the will 3inches x V ( ( ) ( 1.1 0.9

approaches x ) f ) )

( =

x = = ) (

( ( approaches 25 Although volume the has function three constraints on its domain, domain the involves 9 9 - 2.2 - 1.8 - 2x < 4.5,sotheconstraint x -∞ asxapproaches -∞andf ) ) ) ( ( ( 5 5 13 - 2.2 - 1.8 +∞ asxapproaches -∞and+∞,whereas both whennisodd, - 2x ) ) ( ( ) 1.1 0.9 For apolynomial inintercept function form, why is constant the factor xwithadomainof0< ) )

= = 20.736 20.944 < 257 4.5 has no impact onthedomain. 4.5 hasnoimpact ( x )

approaches x < 6.5determined by theconstraints that all ) =

x n

when n when +∞ asxapproaches +∞. is evenand n when x . So, thevolume function x 25 in. < 2.5 is odd. esn 2 Lesson 13 in. 6/8/15 6:39PM 258 Exercises 1–4 Exercises 5–8 Exercises 9–11 Exercises 12–13 Exercises Practice Students should recognize that the that recognize should Students Graphing Polynomial Functions Functions Polynomial Graphing Concepts and Skills Concepts 1 Explore of End Behavior the Investigating of Simple Polynomial the Graphs Functions 2 Explore and the x-intercepts Investigating of of the Graphs Points Turning Functions Polynomial Example 1 of Polynomial the Graph Sketching Form in Intercept Functions Example 2 Modeling with a Polynomial Function EVALUATE EVALUATE INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES ASSIGNMENT GUIDE ASSIGNMENT Focus on Reasoning Focus MP.2 characteristics of the expression that defines a that the expression of characteristics thus and the function nature, determine polynomial the function. Have of the graph of the attributes, their graphing use grid and/or paper students types of different how explore to calculators similar with graphs produce functions polynomial attributes. differing and DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-C;CA-C 6/8/15 6:39 PM

) ) x x ( ( • Online Homework • Hints and Help • Extra Practice f f

) 2 -

+∞, +∞,

3 )

x ) ) +∞. -∞. (

) → → ) → → + 2

- 1 ) ) ] x x x ( 0 x ) ( ( ( f f 2

) Mathematical Practices Mathematical - 1

-∞, +∞ - ∞, +∞

9 8 x ( ( + 1 -∞, -∞, +∞ ( -∞, -∞, ( ( x Logic Reasoning Modeling Using Tools Modeling Reasoning - - x - x ( → → CORE = = = =

x x ) ) ) ) COMMON x x x x MP.3 MP.2 MP.4 MP.5 MP.4 MP.2 ( ( ( ( ƒ ƒ ƒ End behavior: As x As End behavior: Range: Range: As As End behavior: As x As End behavior: Domain: Domain: ƒ The graph has three turning points. turning points. has three graph The functionThe has one local maximum and one global minimum value, value, one local minimum value. has one turning point. graph The functionThe has one global maximum value. 2. 6. 4. 8. 258 +∞. + ∞. → →

) ) x x ( ( f f +∞, +∞,

) ) +∞. -∞.

) → → ) → →

+ 3 ) ) 2

) x x ) x ( ( ( Thinking Strategic Skills/Concepts Thinking Strategic Skills/Concepts Skills/Concepts Thinking Strategic f f ) Depth of Knowledge (D.O.K.) Depth of Knowledge 3 2 3 2 2 3 - 2 +∞ -∞, +∞ -∞, +∞ x + 1 ( ( (

-∞, +∞ 0, x

-∞, -∞,

10 7 [ ( ( -x → → = = x = x = x

Evaluate: Homework and Practice and Homework Evaluate: x x 18 19 ) ) ) ) 1-8 x x x x 9–11 12–13 14–17 ( ( ( ( Range: Range: End behavior: As x As End behavior: As As End behavior: As x As End behavior: Domain: Domain: ƒ ƒ ƒ ƒ The graph has two turning points. turning points. has two graph The functionThe has one local maximum one local and value minimum value. turning points. has two graph The functionThe has one local maximum one local and value minimum value. Exercise

7. 5. Module Module 5 Use a graphing calculator to graph the function. Then use the graph to determine the determine use Then to the function. the graph graph calculator to a graphing Use type and the number (global, and points local or global) not turning of but ofnumber values. minimum or maximum any 1. Use a graphing calculator to graph the polynomial graph calculator use to Then graphing a function. Use behavior. end and range, domain, the function’s determine to the graph range.) and the domain for interval notation (Use 3.

A2_MNLESE385894_U3M05L2 258

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” how eachstep would applied. be polynomial from function lesson this and describe polynomial one can They function. choose illustrating process the of graphing afactorable Focus onModeling esn 5 .2 Lesson 259 MP.4 polynomial is expressed inintercept form. terms of factors the of polynomial, the ifthe with highest the power first, or multiply variable the must either write polynomial the instandard form of apolynomial Emphasize function. that students andcoefficient therefore end behavior expected the Students may have identifyingleading the difficulty PRACTICES INTEGRATE MATHEMATICAL AVOID COMMON ERRORS Suggest that students make aflowchart CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U3M05L2.indd 259

= = = x 4 4 - - x - - (

2 ( x 3 3 Depth ofKnowledge (D.O.K.) ( x 2 2 x Strategic Thinking Strategic Thinking + 2 + 1 - 2 0 0 ) 0 )

2 ) y ( (

x x y y - - 2 2 2 2 1 ) ) (

x 4 - 4 4 x x x 3 )

intervals 0< and As So, thegraph of f on theintervalsx The As -1 and it’s below thex-axisonintervals on theintervalsx As So, thegraph of f The So, thegraph off the on theintervalx The As -2 -1 0 x x x 2 Interval x x 0 < x-intercepts are x Interval x 259 → → < x x-intercepts are x x-axis ontheintervals Interval x → x-intercepts are x < < → < x < > 1,it’s below thex-axison x x < < x > 1 x > 3 +∞, -∞, x > 2 < 0 x -2 < 2andx +∞, x < 1 +∞, x -1 < 3 < 0 < 2 < 2 MP.4 MP.2 COMMON CORE f f x f ( ( f ( x x ( < 1. Modeling Reasoning x x ) ) )

) ( → →

( < > 2,andit’s below Sign off

→ x ( > 3. → f < x Sign off ( x ) Mathematical Practices ) x -2,

) +∞. +∞. is above thex-axis -1 and2<

-∞. Asx ) is above thex-axis

+∞. Asx is above thex-axis = =- = = 0andx -2, -2 -1, ( x ( ( < x +1 ) x x < 0and x =x ) - + - Sign of = 0,andx = x = 2andx → → < 0, + + + - + - - + x = 2. x ) ( -∞, ( x +2 2

< 3, -∞, (x-2) x -2 )

f f

)

( 2

= 3. ( = 1. ( ( x x x x -1 x -3 ) ) < 2.

→ → +∞. ) ) -∞. Lesson 2 Lesson

19/03/14 3:48PM 260

Suggest that students work in small groups to to small in groups work students that Suggest Graphing Polynomial Functions Functions Polynomial Graphing INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES MP.1 Focus on Math Connections on Math Focus discuss why the sign of the leading coefficient the leading the sign of discusswhy function, polynomial a of affects behavior the end to related is the polynomial of the degree how and behavior. the end DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 19/03/14 3:48 PM

x 2 _ x ( 6 in. 36 in. y 36. So, the < 36. So, = 36, so y x x + x x x x 2 in. 2 2 x, and 36 - 2 > 0, or x + x y + x 1.5. Maximum volume is 5.2 volume < 1.5. Maximum . The domain of the functionThe is . x ) inches

x 2 x __ - < 17; and 36 - 36 ( ) x -

17 > 0, or x 260 ( x x =

) , you find that the box has a maximum volume of about has a maximum volume the box find that , you x with a domain of 0 < x with a domain ) ( ) 17. Using the graphing calculator to locate the graph’s the graph’s locate to calculator the graphing < 17. Using > 0; 17 - 0, 17 x - 2x ( 3 ( ) . Then the dimensions of the box are x, 17 - are the dimensions of the box Then x. - 2x 6 (

= = 17 -

) x ( . The volume function volume is V The x. = 17, or y y + highest point on the interval highest point domain of the function is 0 < determine by the constraints x the constraints by determine The template shows how to create a box from a square sheet of cardboard cardboard of sheet a square from a box create to how shows template The segments solid line the template, In 36 inches. of length a side has that rectangles grayed and folds, indicate segments line dashed cuts, indicate the left 2 inches is wide on vertical The that strip removed. pieces indicate it that will the box of be theflap that a side is to glued the template of side are that strips horizontal The up. folded is when the box overlaps To create an open-top box out of a sheet of cardboard that is 6 inches long and and long 6 inches is that cardboard of a sheet of out box open-top an create To corner in each x inches length side flap of a square make you wide, 3 inches you Once the other. along folding and sides the flap’s of one along cutting by To overlaps. it the side to flap each glue you the box, of sides the four up fold the box. of the volume maximizes x that of find the value tenth, the nearest 2032 cubic inches when the dimensions of the box are 7.3 inches, 9.7 inches, and 28.7 inches. 9.7 inches, 7.3 inches, are box 2032 cubic inches when the dimensions of the wide at the top and bottom of the template are also flaps that will overlap will also overlap that flaps are the template of bottom and the top wide at a Write up. folded is when the box the box of bottom and the top form to will need determine to (You x only. of terms in the box function for volume find the tenth, between x and the nearest to y first.) Then, a relationship volume. maximum with the box of dimensions or 36 - , use that fact x and y, use that 2 + that between find the relationship To x . So, the volume the volume is x. So, is 3 - 2x, and the height is 6 - 2x, the width of the box length The function is V cubic inches when square flaps with a side length of 0.6 inch are made in the corners. are side length of 0.6 inch flaps with a when square cubic inches

13. 12. Module 5

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” the graph.the at least two points eachzero between to sketch behavior before starting agraph, and choose then Suggest function. the that determine end the they points zeros the between to accurately represent should donot or look, they because graph enough donotthey have an overall of sense how graph the graph apolynomial This function. may because be Focus onMath Connections esn 5 .2 Lesson 261 MP.1 PRACTICES INTEGRATE MATHEMATICAL Students may have asked when to difficulty CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U3M05L2.indd 261

© Houghton Mifflin Harcourt Publishing Company are integers. Assume that constant the factor ais either 1or -1. Write function aquartic inintercept form for x-intercepts given the graph, whose are integers. Assume that constant the factor ais either 1or -1. Write acubic function inintercept form for x-intercepts given the graph, whose Module 5 Module 14. 16. f power. So, is thegeneral function mustberaisedeach factor to thefirst isquartic, andthefunction factors x, a positive. So, with thespecificfunction function’s endbehavior, amustbe some constant a.Given factor the and The

( general function isf general function must beraised to thefirstpower. So, the iscubic, eachfactor and thefunction and x ( The a positive. So, with thespecific function the function’s endbehavior, amustbe x = 1isf x x - - = 1. The related are x factors = 1isf ) -1

- 2,andx 4 4 x-intercepts are x = x x-intercepts are x x = 4. The related are x factors ax ) - 1.Since there are three factors - - for someconstant a.Given factor ( ( 2 2 x ( x x ) + 3

) =

0 0 = - x ) y y

( ( ( x x x 4. Since there are four + 3 + 3 - 2 2 2 ( ) ) x = ) ( ( = ) ( x 4 4 x

x -3, = + 1 -3, x x - 2 - 4 a ( x ) ) x ) x ( ( = 0,x for + 3 x x = + 3,x -1 - 4 -1, and ) + 3, ( )

) x = 2, . . + 1 + 1, )

261 15. 17.

f witha the specificfunction end behavior, amustbenegative. So, constant a.Given factor thefunction’s The function isf function x tangent to thex-axisat x must besquared. Given that thegraph is iscubic, oneofthefactors the function Since there are and onlytwo factors The related are x factors ( The The related are x factors x-axis at bothx Given that thegraph istangent to the must beraised to apower otherthan1. oneorbothfactors isquartic, function Since there are andthe onlytwo factors with a mustbenegative. So, thespecific function factor f be squared. So, is thegeneral function factor factor ( + x - - x ) ) 4

4 x-intercepts are x = 2 mustbesquared. So, thegeneral

x-intercepts are x = a - - a. Given thefunction’s end behavior, x a = ( 2 ( 2 and the factor x + 2andthefactor x x -1 isf + 2 + 2 ( 0 x 0 ) )

) y y = 2 2 ( ( = ( x x x a ) 2 2 -2 andx - 3 ( - 3

= x + 2

= - ) ) = 4

. 4 2 ( -2 andx for someconstant + 2andx x x -2 andx ) x = + 2andx 2 + 2 ( = 2, the factor -2, thefactor x = 3,boththe - 3must - 3 -1 is ) 2 ( x ) = 3. for some - 3 = 3. - - 3. Lesson 2 Lesson 3. ) 2

. 19/03/14 3:48PM 262

Graphing Polynomial Functions Functions Polynomial Graphing AVOID COMMON ERRORS COMMON AVOID Students may write the wrong power for a factor of a of a factor for power the wrong write may Students the if they know that function. Explain polynomial the know they automatically a zero, of multiplicity the told they are If factor. thecorresponding of power the that then quartic, they know or cubic function is 4. or 3 is power DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-C;CA-C 6/8/15 6:39 PM

x , , ( , , 2 ) - 1 x

( -2. -2. = 2. = 2. = 0 and is = = =

) x ( x 2, and a local< 2, and 1, and a local< 1, and 2, and a local< 2, and 1, and a local< 1, and 4 x x x x < < < < 2 y 0

2 262 - = 3. -axis at x at the x-axis crosses -1 and 4 - = -axis at x at the x-axis crosses = 1 and = 2. = 1. = 2. = 1. -axis at x at thex-axis to tangent is -1 and -2. = 2. = x at the x-axis to tangent is = 1 and = x 3. Instead, the graph should be tangent to the to should be tangent the graph = 3. Instead, 1 and x -1 and 4 1 and x = 1 and = 2 Select all statements that apply to the graph of ƒ of the graph to apply that Select all statements A student was asked to sketch the graph of the function of the graph sketch to asked was A student

y . Describe what the student did wrong. Then sketch the correct graph. the correct sketch Then did wrong. . Describe the student what ) 0 - 3 x the x-axis at = 0 and cross 2 , but not global, maximum occurs on the occurs interval -1 on global, maximum not , but , but not global, minimum occurs on the occurs interval -1 on global, minimum not , but , but not global, maximum occurs on the interval occurs -2 on global, maximum not , but , but not global, minimum occurs on the occurs interval -2 on global, minimum not , but x ( -

2 x are x-intercepts x are x-intercepts x 4 =

Focus on Higher Order Thinking on Higher Order Focus - but not global, minimum occurs at x occurs at global, minimum not but A local but not global, maximum occurs at x occurs at global, maximum not but A local but not global, minimum occurs at x occurs at global, minimum not but A local but not global, maximum occurs at x occurs at global, maximum not but A local -axis at x at the x-axis to tangent is graph The x at the x-axis to tangent is graph The -axis at x at the x-axis crosses graph The The The x at the-axis x crosses graph The The The )

( J. I. H. G. E. F. D. A. B. C. Explain the Error x the x-axis at it crosses so that the graph sketched student The -axis at x the x-axis at to tangent x x-axis at Multiple ResponseMultiple ƒ

H.O.T. H.O.T.

19. 18. Module 5 Module

A2_MNLESE385894_U3M05L2 262

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” esn 5 .2 Lesson 263 drawing graph the of function. the information about attributes these is helpful in writtenfunction inintercept form, and how polynomial are function from determined the Have students how attributes describe the of a formthe f each pair to sketch graph the of acubic of function Have students work inpairs. Instruct one student in attributes included as labels on graph. the degree three or of with higher all applicable the make aposter showing graph the of apolynomial of behavior descriptions. intheir Then have students Make sure proper the use they notation for end attribute graph that particular this may have. also polynomial and function its graph, and list any other write adescription of attribute, the show an example maxima and minima, behavior. end and domain,for range, x-intercepts, turning points, make note cards for eachattribute, including cards attributesthe of apolynomial have function, them To help students remember words the associated with f exercisethe using of aquartic function form the of graph. the Have students switch roles and repeat steps that were followed and determine attributes the have student the sketched who graph the write the other gives instructions verbal for eachstep. Then CONNECT VOCABULARY JOURNAL PEER TO PEER DISCUSSION ( x )

= ( x + ( x ) b

= 1

) ( ( x + x + b b 2

)

) ( ( x + x + b b 3

)

) ( ( x + x + b b 4

Ask to them 3

)

) . the , while CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U3M05L2.indd 263

© Houghton Mifflin Harcourt Publishing Company 21. 20. Module 5 Module ƒ quartic functions inintercept form, sketch graph the of quintic the function Represent Real-World Situations Make aPrediction of of twothe ends of tube the has alength of xinches, but has of aperimeter metal 36inches. of Each two the sides of rectangle the that form and riveted to form acircular tube that is open at ends, both as shown. The sheet maximizes volume the of tube. the forfunction tube the interms of x.Then, to nearest the tenth, value the find of xthat inches. thatlength ofthesidesrectangle form theendsoftubeis12.3 that thetubehasamaximumvolume ofabout57.9cubicincheswhenthe Using thegraphing calculator to locate thegraph’s highestpoint, you find and 2πr Since inches, you that know 2x the otherdimension.Since is36 theperimeter oftherectangle Given that xrepresents onedimensionoftherectangle, letyrepresent V withradius randheighttube isacylinder is y,thevolume function the function isdetermined by theconstraintsthe function x ( ( x x x ) )

- 1inches an because overlap of 1inch for is needed rivets. the Write avolume = = x- 18

= x x π - - 1represents thecircumference ofthetube, you that know 2

( r x x 2 + 2

1 where ristheradius ofthetube, sor y x > 0,orx = ) ( π x ( - 2

( ______

x Knowing characteristics the of graphs the of cubic and 4 - 1 π ) < 18.So, is1 thedomainoffunction 2 .

) 2 -

) ( + 2y 4 18 - - 2 = 36,sox - x -

) 12 12 A rectangular piece of is rolled sheet metal

6 6 = 0

__ 4π 1

( x 2 + - 1 263 y = 18,andy 4 the tube the has acircumference ) 2 x ( > 0; x -1 18 = -

x ____ x 2π - 1 x - 1>0,orx )

. The domainof

= 18-x. Since the < x < 18. > 1;

Lesson 2 Lesson 10/17/14 12:49AM 264

and ) is negative V(x) is negative the length of one Those are the possible are Those It would appear to be a piece of be a piece to appear It would The length of the narrow end of the length of the narrow The . Have a student volunteer describe card volunteer a student . Have . Why do we use the interval do we find (0, 5) to the Why function maximum? 6 be between used x-values and 5 to can’t Why find a function maximum? very if x were look like a box would What 0? close to is very when x is the box to close happens What 5? to Ask students how they the determine can how students Ask represents on the final on box. represents Graphing Polynomial Functions Functions Polynomial Graphing x (a sturdy paper used to make cards) and and cards) used make paper to (a sturdy QUESTIONING STRATEGIES QUESTIONING CONNECT VOCABULARY VOCABULARY CONNECT MATHEMATICAL INTEGRATE PRACTICES cardboard with no height. cardboard Some students may not be familiar with the term with be familiar not may students Some stockcard stock cardboard as such materials other to it compare or paper MP.4 Focus on Modeling Focus values of the length x. values box shrinks to 0, and there is no flap to fold up. fold is no flap to 0, and there shrinks to box length, width, and height of the box from the two- from the box of height and width, length, of a 3D picture them draw Have diagram. dimensional x edge thethe appropriate final labeling box, writing formulas for the other two sides. Ask students students Ask sides. two the other for formulas writing what between these values, and a volume cannot be and a volume these values, between negative. side of the box, or the height of a short, or the height box flat side of the box, Scoring Rubric his/her reasoning. and explains the problem solves correctly Student 2 points: but does not fully good understanding of the problem shows Student 1 point: his/her reasoning. or explain solve understanding of the problem. does not demonstrate Student 0 points: 19/03/14 3:48 PM

, or 3 2

_ ) 10 in. ) - 3x 24 box from a sheet of cardstock. Have Have cardstock. of a sheet from box (

1 2 _ (

) < 5; and 12 - 264 - 2x < 5. 2 so that the dimensions of the box the box the dimensions of = 2 so that , you find that the box has a maximum the box find that , you ) x 10 ( x 0, 5 > 0, or x ( 24 in. =

. The domain of the function by The is determined . ) ) x x

( 3 2 _

- 12 ( > 0; 10 - 2x ) - 2x x 10 x ( x =

) x ( V are 2 inches, 6 inches, and 9 inches. and 9 inches. 6 inches, 2 inches, are volume of 108 cubic inches when x volume the domain of the function is 0 < the constraints x the constraints The volume function volume is V The Using a graphing calculator to graph the function and locate the graph’s the function the graph’s graph to and locate calculator a graphing Using on the interval highest point that will produce the box with with the box will produce x that of find to the value calculator a graphing Use box? that of the dimensions are What volume. maximum Write a polynomial function that represents the volume of the box, and state its domain. its state and the box, of the volume represents function that a polynomial Write

a. b. b. a.

EXTENSION ACTIVITY EXTENSION The box in this Performance Task was an open box, that is, it had no top. Ask Ask top. no had it is, that box, open an was Task in this Performance box The a closed create to ways research to students two-dimensional to they disassemble can that lives in their daily them find boxes they geometry, two-dimensional based the resulting on how, students Ask form. whether be they and would the box, of the volume for equation an set up would equation. based that on volume a maximum determine to able

The template shows how to create a box with a lid from a sheet of card stock that is 10 inches wide 10 inches is that stock card of a sheet from a lid with a box create to how shows template The segments line dashed and cuts, indicate segments line solid the template, In long. 24 inches and they the sides to overlap glued are x inches, of length a side with each flaps, square The folds. indicate to attached is which lid, The sides. upright four and a bottom has box The up. folded is when the box the can lid of sides the three that Assume own. its of sides upright three has sides, the upright of one when closed. the is lid the box inside be tucked Lesson Lesson Task Performance Module 5

A2_MNLESE385894_U3M05L2.indd 264

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