Solving Quadratic Equations by Taking Square Roots Square Taking by Equations Solving Quadratic B Lesson 1

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the values the values Factor the equation, the equation, Factor What are the roots of an equation? equation? an of the roots are What How can you use the factored form of a of form use the factored you can How the of find to the zeros equation quadratic of the variable that make the equation true the equation make that of the variable Discuss whether 1 can be used instead of 0 in Discuss be whether 1 can of used instead -values, x-values, non-integer occur zeros at When find be to them can difficult. a table using EXPLAIN 1 1 EXPLAIN EXPLORE EXPLORE INTEGRATE TECHNOLOGY INTEGRATE INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES QUESTIONING STRATEGIES QUESTIONING Show students that they can use a graphing calculator calculator they use a graphing can that students Show selecting the function and graphing by findto zeros menu. theCALCULATE from 2:zero apply the Zero Product Property, and then solve the and then solve Property, Product apply the Zero equation. related quadratic function? quadratic related Finding Real Solutions of Simple Finding Equations Quadratic Investigating Ways of Solving of Simple Ways Investigating Equations Quadratic Thinking on Critical Focus MP.3 the Zero Product Property to make a “One” Product Product a “One” make to Property the Zero Product if ab a that = 1, neither show students Have Property. other –1 and them consider equal 1. Have nor b must the property that lead conclude them to and numbers 0. for only holds 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 12:17 PM © Houghton Mifflin Harcourt Publishing Company may x Solving Quadratic Equations by Taking Square Roots Square Taking by Equations Solving Quadratic b Lesson 1 ,

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© Houghton Mifflin Harcourt Publishing Company methods. Studentsobserved. can continue to addnotes to cards the additional learn as they stepsthe of as well method the as any advantages or disadvantages that they have zeros tothe learned find of quadratic functions. Suggest that describe they Have pairs of students work together to make note cards for they eachmethod Peer-to-Peer Activity COLLABORATIVE LEARNING iiebt ie y2 Divide sides both by 2. functions. This means that for any equation of form the d represent before atime negative so began object falling, the values of tare excluded from domains the of these For models, is time both measured from instant the that toAnegative fall. begins object the value of twould Use product the property. Use quotient the property. Your Turn Use definition the of square root.  Simplify. Use definition the of square root.  oue 3 Module time atfeet) (in fallen Distance • Two commonly quadratic used models for near objects falling Earth’s are surface following: the Explain2 3. quadratic the Solve equation square by taking roots. • Height (in feet) at time time atfeet) Height(in • solution should be rejected. d 6t ohsds 2 Add 16to sides. both Example 1 2 x Subtract 9from sides. both Rationalize denominator. the x Simplify numerator. the x sides byDivide both -5 2

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esn 1 Lesson 19/03/14 12:17PM 116 AVOID COMMON ERRORS COMMON AVOID is often associated with with associated often is maximum Because the word assume incorrectly may students amounts, positive leading a positive function with a quadratic that in that, Stress a maximum. have coefficient should the and a minimum has parabola fact, the “positive” the maximum. with the one is parabola “negative” 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 12:17 PM © Houghton Mifflin Harcourt Publishing Company Solving Quadratic Equations by Taking Square Roots Square Taking by Equations Solving Quadratic Lesson Lesson 1 24 ft .

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DIFFERENTIATE INSTRUCTION DIFFERENTIATE tiles. Then demonstrate that arranging the tiles in a rectangle models the product the tiles the models product in a rectangle arranging that tiles. demonstrate Then ( Students may benefit from using algebra tiles to practice factoring quadratic quadratic tiles practice factoring to algebra using benefit from may Students model to how students show example, For expressions. Manipulatives Example 2 Example

the equation. Write the equation. Write Module Module 3 Reflect 5. window the third-story balloon passes by water t. The of value Reject the negative in root. square of the definition Use simplify. to property the quotient Use both sides. 50 from Subtract −16 . both Divide by sides root. square of the definition Use the d model Using   the model h Using property. the quotient Use . 16 by both sides Divide balloon falls in 4 feet water t. The of value Reject the negative

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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 3 .1 Lesson 117 5 +6i≠. combined. Therefore, 5i+6=11,but of parts acompleximaginary number cannot be expression cannot combined, be real the and Emphasize that just as terms unlike inan algebraic For example, to write may 5+6ias 11i. they try by combining part. and real the part imaginary the Some students to simplify may acomplex try number Numbers Defining Imaginary numbers. invented—to square the define root of negative units numbers andimaginary imaginary were negative number Explain case). why (–1inthis unit its i,then is imaginary the square is original the root of positive square ifthe 1is. Likewise, root of –1 number units. Ask students to state what square the to real number unit orRelate whole imaginary kicked? after height the model it ball of is asoccer kicked. takes for thesoccer ballto hittheground itis after of asoccer ball, thezero isthetimeit ofthefunction x that makes theoutputf(x)equalzero. thecase In CONNECT VOCABULARY QUESTIONING STRATEGIES AVOID COMMON ERRORS EXPLAIN 3 translate you when quadratic use functions to What is zero the of How afunction? this does The zero isavalue oftheinput ofafunction CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U2M03L1.indd 117

© Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: NASA Students switch roles and repeat process the cards the until all have used. been represents. If agree, both first student the square the finds then student shows acard and other the student kindof the decides number it square roots negative of several and positive integers. Working inpairs, one Provide students with 4to 6“square root cards”—index cards displaying the Connect Vocabulary LANGUAGESUPPORT mathematicians allowed for solutions of equations like 7. 6. approximationirrational, adecimal (to nearest the tenth of asecond). Write and an solve equation to answer question. the Give answer exact the and, ifit’s Module 3 Module i that Given You know that quadratic the equation bi numbers Imaginary exponents: is anegative real number. When squaring number, an imaginary power the use of aproduct property of numbersimaginary are following: the and equation the

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50 Using themodeld Reject thenegativeReject value oft. The tool falls4feet in time time moon,On the distance the d(infeet) that in an falls object How long it inPart does water the take described Bto hit balloon ground? the does itdoes 4feet? tool the take to fall Suppose an astronaut on moon the drops atool. How long Reject thenegativeReject value oft. The water balloonhitstheground in Using themodelh d _

Explain 3 3 8 • • • • • ( t

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1 . You can use the You . -1 ―― √ What do you look for during the solving the solving during look for do you What equation a quadratic that indicate to process What is an imaginary number? imaginary an is number? What is defined as as unit i is defined EXPLAIN 4 4 EXPLAIN QUESTIONING STRATEGIES QUESTIONING QUESTIONING STRATEGIES QUESTIONING is equal to the number or x is equal to a negative equal to negative of a root square or negative positive number. might have imaginary solutions? imaginary solutions? have might Finding Imaginary SolutionsFinding of Equations Simple Quadratic imaginary unit to write the square root of any of any root imaginary square the write unit to Imaginary can be numbers number. negative real b is a nonzero bi, where in the form written number and i is the imaginary unit. 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 12:17 PM © Houghton Mifflin Harcourt Publishing Company Solving Quadratic Equations by Taking Square Roots Square Taking by Equations Solving Quadratic Lesson Lesson 1

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2 Subtract 12 from both sides. 12 from Subtract Use the definition of square root. square of the definition Use Write the equation. Write theUse product property. (

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Example 4 Example 3 Example

Module Module 3  Using imaginary numbers, you can solve simple quadratic equations that do not have real solutions. real have not do that equations quadratic simple solve can you imaginary numbers, Using Explain 4 Find the square of the imaginary of the square Find number. 9. Turn Your Reflect 8. 

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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” that sum their is 0. that solutions the are opposites and should surmise and what sum their might Students be. should see quotient properties ofsquarequotient roots properties asnecessary. the radical expression, and usingtheproduct positive andanegative square root, andsimplify thatpositive every realthe fact numberhasbotha equation, thentake square roots ofbothsides. Use squared variable isisolated ononesideofthe Powers andsimplify. property the Product ofPowers orQuotient of property quadratic equation of form the ax Focus onCritical Thinking esn 3 .1 Lesson 119 square roots? How can you solve aquadratic equation by taking transformations of parent their function. Functions that are same inthe family are graphs have characteristics basic incommon. A family of functions of is aset functions whose MP.3 look like andhelpyou fillinthemissingparts. canhelpyou what predict thegraphfunction will PRACTICES INTEGRATE MATHEMATICAL SUMMARIZE THE LESSON SUMMARIZE THE QUESTIONING STRATEGIES CONNECT VOCABULARY ELABORATE Ask how solutions imaginary the of a useful foruseful graphing. Explain why recognizing parent functions is exponents the when base are rational? How doyou multiply powers with same the Rewrite theequation sothat the Recognizing theparent 2

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Your Turn

square roots ofthecorresponding positive number, a. The square roots ofanegative number, -a,are given unititimesthe by theimaginary theinstant thatafter thefallbegins. A negative value oftrepresents timebefore thefall, butbothmodelsdealonlywithtime isanegativeIt real number. uniti. solutions willonlydiffer oftheimaginary by thefactor The solutions, firstequationwhilethesecond hasimaginary hasreal solutions, butthe The quadratic equations 4 _ Check-InEssential Question Earth’s surface, d Why do you negative reject values of solving twhen equations on models the for based near object afalling What kindof anumber is square the of number? an imaginary Without solving actually equations, the what can you say about relationship the solutions? their between 4 1 Elaborate 4 Use qoutient the property. Use definition the of square root. sides byDivide both Write equation. the Subtract 11from sides. both x

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4 1

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Lesson 1 Lesson 19/03/14 12:17PM 120 Exercises 1–2 Exercises 2–6 Exercises 10 7– Exercises –13 11 Exercises –18 15 Exercises Practice Concepts and Skills Concepts Explore Solving of Ways Investigating Equations Simple Quadratic Example 1 Real Solutions of Simple Finding Equations Quadratic Example 2 Using Problem Solving a Real-World Equation a Simple Quadratic Example 3 Defining Imaginary Numbers Example 4 ImaginaryFinding Solutions of Equations Simple Quadratic EVALUATE EVALUATE ASSIGNMENT GUIDE ASSIGNMENT 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 12:17 PM © Houghton Mifflin Harcourt Publishing Company Solving Quadratic Equations by Taking Square Roots Square Taking by Equations Solving Quadratic Lesson Lesson 1 • Online Homework • Hints and Help • Extra Practice = 1. = 3.

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― 9 - 2 and √

x 2

√ 2 1 ± ±1 x - 1 and x -3 and x Mathematical Practices Mathematical ± ±3 9 = 2 =

======

Modeling Modeling Reasoning Modeling Using Tools )

) = = = 2 2

x x x

x ( x

2 ( x x CORE 2 x x COMMON MP.4 MP.4 MP.2 MP.4 MP.5 Solve by taking square roots. taking square by Solve Solve by taking square roots. taking square by Solve c. c. 120 5, on the same coordinate plane. = 5, on the same coordinate 7, on the same coordinate plane. = 7, on the same coordinate

-coordinates of the two points where where points of the two x-coordinates -coordinates of the two points where where points of the two x-coordinates ) ) = 1 x x ( ( x = 3 Treat each side of the equation as a each side of the equation Treat functions,function, the two and graph ƒ which in this case are Treat each side of the equation as a of the equation each side Treat functions,function, the two graph and f which in this case are the graphs intersect are x intersect are the graphs the graphs intersect are x intersect are the graphs The The The g g x - 1 = 0 - 3 = 0 x x 0 0 or x 1 or - 0 4 4 -3 or 0 0 or x 0 5 using the indicated method. the indicated + 3 = 5 using = = = =

7 using the indicated method. the indicated - 2 = 7 using

= = = =

2 ) x x

2 2 x ) x Recall of Information + 1 - 2

- 1 Skills/Concepts Skills/Concepts Skills/Concepts Skills/Concepts y

2 y + 3 - 9 x

- 3 Depth of Knowledge (D.O.K.) Depth of Knowledge 2 2 1 2 2

2 x x ( 2 x 0 0 x x ) 1 7 6 5 4 3 2 1 1 7 6 5 4 3 2 1 ( ) - - 2 2 + 1 - - + 3 x ( x Solve by factoring. by Solve Solve by graphing. by Solve Solve by factoring. by Solve Solve by graphing. by Solve 2

( 1–6 4 4 7–10 11–16 17–20 21–22 - - Evaluate: Homework and Practice and Homework Evaluate: Solve the equation 2 the equation Solve Solve the equation the equation Solve Exercise

2. b. a. b. a. 1. Module Module 3

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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 3 .1 Lesson 121 points rather than asmoothcurve. graph of isaset function actually the discrete whole-number (or values. decimal) Therefore, the thatthem areasonable domain of F(x)consists of through points the plotted.curve they Remind quadratic may F(x),they function draw asmooth When students draw graph the of of amodel a AVOID COMMON ERRORS CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U2M03L1.indd 121 Exercise © Houghton Mifflin Harcourt Publishing Company 24–25 23 26 8. 7. nearest tenth of asecond). answer and, ifit’s approximation irrational, adecimal (to the Write and an solve equation to answer question. the Give exact the 3. quadratic the Solve equation square by taking roots. 5. Module 3 Module

10 Using themodelh Using themodeld 16 working at aheight of 20feet? of 60feet. How long it does squeegeeto the take pass by another window washer A person washing windows the of an building office drops asqueegeefrom aheight A squirrel inatree drops an acorn. How long it does acorn 20feet? the take to fall 4 2 60 x ( x x 2 5 2

- 10

t t -10

- 16 = = = 24 t t

2 2

- 5 -16 3 3 1 Depth ofKnowledge (D.O.K.)

= = = = of Information Recall 6 ± Strategic Thinking Strategic Thinking x x x x ± ± _ 20 2

x x 4 5 t t t √

2 2 2

) t t

2 2 2

= 5 √ = = = = = _

6 ― = = = = =

√ 2

_ ―

4 5 ± ± _ -5 5 5 ―

2 1 _ -40 20 ± ±

_ √

√ √

√

_ ―

1 2 2 ―

―― _

2 10 ( ( 10 ―

t t

) )

= 16 =

t - 16

2 ,solve theequation d t

2 ,solve theequation h

washer in squeegee passesby theotherwindow thenegativeReject value oft.The

falls 20feet in thenegativeacorn Reject value oft.The

121 MP.3 MP.4 MP.2 _

√ COMMON 2 6. 4. CORE 10 ―

(

Logic Modeling Reasoning

t _ ≈ 1.6seconds.

3 - √ )

x 2 = 20. _

2 x 5 ―

5 Mathematical Practices

- 8=12 3 ( 2

t x x ≈ 1.1seconds. + 15=0 - ) x x

2 2

= 20. _ x = = = = 5 x x x

2 2

= = = = _ ± ± 20 20 3 

_ -15 ± 75 ±5

√

√ _ ―

20 √ 20 ― 3

3 ―

√

75 ―

3 ―

± _

√ 3 60 ―

= ± _ 2 Lesson 1 Lesson √ 3

15 ―

19/03/14 12:17PM

122

A The The What is meant by reasonable domain? domain? reasonable by meant is What might a function? Why of the domain is What domain? the reasonable differ from it reasonable domain consists of the values of of the values consists domain reasonable QUESTIONING STRATEGIES QUESTIONING domain of a function of the is all the values which the function for is variable independent represent that It values include defined. may a nearly such as situations, impossible physically number of minutes. infinite the independent variables that make sense in the sense make that variables the independent situation. of the real-world context 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 12:17 PM © Houghton Mifflin Harcourt Publishing Company

Solving Quadratic Equations by Taking Square Roots Square Taking by Equations Solving Quadratic i

Lesson 1

⋅ )

i 6

25 ___ )

― 6

)

― 2

- ――

- 40 √ 2 - 10

√

2 5

2 25 ______√ ___ -1 1 2 x x __ (

( - -25 - 4 ± ± 1 2

- ( __

======4

= = =

)

2 2 3x x x 2 x x )

― 2 - 3

― 2

- 15 2 2

√ √

x _ i

____ 2 x ( ≈ 3.9 cm, ≈ 4.2 -

- i 5

( 10

― 2

― 15 11.6 cm. ≈ 12.7 cm.

√ The solutions are imaginary. solutions are The 13. 16.

― 2 √ ≈

√

― 15 √ x

i ⋅

―― -16 ) )

― 5 √

-1 ( ± ± 4 -16 -8 √

122 5

( -5 = = = =

= = =

2 2 x x

x 2 x

) 1 2

+ 12 = 4

― 5 Reject the negative value of x because value Reject the negative the So, length cannot be negative. width of the rectangle is 3 Reject the negative value of x because value Reject the negative the So, length cannot be negative. width of the rectangle is and the length is 3 cm, and the length is 9

― 5

√ √ 1 2 i

_ i

( The solutions are imaginary. solutions are The . . 2 2 12. 15.

― 2 cm cm

― 18

― 15 √ √ √ ± area 54 54 18 ± 3 ± 45 45 15 area

======

2 3

― 6

2 2 ― __

2 2 ) ) ) ) x x x 3 x x x x x x

√

( √ ____ ( ) 3

) ) i 2 3 __ ± ± 10 ⋅ -1 3x

3x width width 2 (

(

( (

( = = = =

-9 3 9

) )

2 2 x x Determine the lengths of the sides of the of the sides of theDetermine lengths x x

= = =

- 10 = 0

2 ) x length length 3i

( ( (

The area of the rectangle is 54 is the rectangle of area The 15 3i The area of the rectangle is 45 is the rectangle of area The

The solutions are real. solutions are The

10. Module 3 14. Determine whether the quadratic equation has real solutions or imaginary or has real solutions whether equation Determine the quadratic solving by the equation. solutions Find the square of the imaginary of the square Find number. 11. Geometry rectangle using the given area. Give answers both exactly and both exactly and answers Give area. the given using rectangle tenth). the nearest (to approximately 9.

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© Houghton Mifflin Harcourt Publishing Company 21. nearest tenth). answers and exactly both approximately (to the of each square using given the information. Give Geometry 17. solutions. imaginary quadratic the Solve equation square for by taking roots. Allow Module 3 Module 22. 19. (

( The area of larger the square is 42 x the smaller square. smaller the If area the of larger the square by is decreased 28c than area the of square. smaller the 5 Use theformula for thearea Aofasquare withsidelengths: Use theformula for thearea Aofasquare withsidelengths: x x 2x 2x x 2

3 4 4 2

= = =

x x x ) 5 x ) - 4= x 2

2 2 2 2

x x -81

± ±9i

x x - - 28 2 2

= = = = =

Determine the lengthDetermine the of sides the __

√ 2 7 = = = = 28

x x ± 14 42 x x -81 ―― x

-8 2 2

2 2

± ± - -4

√ + 42 + 42 = = = = = ______2 √ 5 4 14 ―

√

__ __ ± 8 28 - ――

2 1 2 1

5 ―

__ x x √

5 4

2 2

8 ―

= ±2 √ 2 ―

cm 2 has asidelengthof4 a sidelengthof2 thenegativeReject value ofx. The smallersquare has more more has asidelengthof2 a sidelengthof thenegativeReject value ofx. The smallersquare has 18. 20.

x 7 x 2

+ 64=

123 + 10=0 x 7 m x x

2 x x 2

x x

2 2

, the result ,the is of half area the of = = =

= = = = x 0 -64 ±8i ± √ √ ± ± - -10 √ 14 ―

______√ 2 ― 10

√ 7 ―― -64

7 √

≈ 2.8cm,andthelarger square

70 ― - ―― √

≈ 3.7cm,andthelarger square

2 ― ___

14 ―

7 i

≈ 5.7cm.

≈ 7.5cm. A A = = s s 2x

2 2

. . Lesson 1 Lesson 19/03/14 12:17PM 124

PEER TO PEER DISCUSSION PEER TO PEER Ask students to discuss with a partner a topic they discuss a partner with to a topic students Ask set find time as with a data to research to like would variable. the independent the or length as such find examples may Students an of the populations grows; it as animal an of weight species in a city; a particular of the cost endangered of pairs Have percentage. winning a team’s or item; the appropriate on then find information students set. the data model for 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 12:17 PM

) t ( 2 seconds, is = 3, -2 seconds,

) t Imaginary Imaginary Imaginary Imaginary Imaginary Imaginary (

X X 124

Real Real Real Real Real Real

, solve the equation h the equation , solve . The of t. The value Rejectoutfielder the negative after its it reached the ball 2 seconds caught maximum height. 2

t X X X X 16

h =

) t

( When a batter hits a baseball, you a baseball, hits you a batter When

― 4

√ (in feet) at time t at h (in feet) height find to the ball’s ± ±2 4 3 -64

2 t = = = = =

2 2 2

t t

- 16 t t t 2

) 10 2 _ h ) √ -3 _ 2i -5 -16 = Focus on Higher Order Thinking on Higher Order Focus - 16 - A square root of 5 of root A square i ( √ ( )

( Using the model h Using 67 Can you determine (without writing or solving any equations) the total time the ball the total equations) any solving or writing (without determine Can you make. you assumptions any state and reasoning your Explain in the air? was Suppose a fly ball reaches a maximum height of 67 feet and an outfielder catches catches outfielder an and 67 feet of height a fly a maximum ballSuppose reaches theball after begins descend does to the long How the ground. the ball above 3 feet the ball? catch outfielder (in seconds) as it falls the ground. to it as (in seconds) c. d. e. f. b. Critical Thinking Determine whether each of the following numbers is real or imaginary. or real is numbers the following whetherDetermine of each a. can model the ball’s height using a quadratic function that function that a quadratic using height model the ball’s can once However, vertical initial velocity. the ball’s for accounts is verticalits velocity height, maximum its the ball reaches use the can model you per and feet 0 second, momentarily h The other solution to the quadratic equation h equation the quadratic to other solution The another time when the ball would have been 3 feet above the ground. the ground. above been 3 feet have another time when the ball would its before the ball reached happened 2 seconds have would This a height at hit the ball the batter assume that If you maximum height. of a total in the air for the ball was that can conclude then you of 3 feet, 4 seconds.

H.O.T. H.O.T. b. a.

Module 3 24. 23.

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© Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: (t) ©Levent Konuk/Shutterstock; (b) ©Robert D. Young/Shutterstock 25. oue 3 Module Russell’sDoes answer makeIf sense? not, and find correct Russell’s error. 26. (width)(height) up following the equation and to solve tries it. to bottom the of ahole thatdeep. is 32 feet He sets from acrane’s clamshell bucket at aheight of 16feet amount of it time takesfor of aload to fall dirt Explain theError Represent Real-World Situations what are dimensions the of screen? the images ratio with an aspect of 16:9.If area the ofscreen is 864 an HDTV in screen is ratio the of image width to image screen shows height. An HDTV is 9m,andtheratio ofwidthto height is common (positive) multiplier, sothat thewidthis16m,height height ofscreen mustbethesamemultiple of9.Let mbethe The widthofthescreen mustbesomemultipleof16,andthe

16 16 the bottom ofthehole. Reject thenegativeReject value oft. took The dirt ground level. the hole. theholeis32feet If deep, thebottom isat -32feet relative to was usingapositive numberto represent the “height” ofthebottom of No, numberofseconds. thetimeshouldnotbeanimaginary Hiserror - 16 -16 - 16 -16 16m t t t t t

2 2 2

t t t m 144 m t

2 2 2

= = = = =

= = = = ⋅ ± -1 16 32 ± i m 9m 3 -48 -32 ± m √ 2 2 _ -1

√ = = = = = Russell wants to calculate the

3 ―

6 864 ± 864 area

√ 6 ―

9 inches, andtheheight ofthescreen is width ofthescreen is16 thenegativeReject value ofm.The The aspect ratioThe aspect of an image on a √ 6 6 ―

≈ 22.0inches. ____ 16m 9m 125 √

= 3 3 ―

___

16 ≈ 1.7seconds to reach 9

. √ 6 6 ―

≈ 39.2 2 , esn 1 Lesson 19/03/14 12:17PM 126 Graph the equation, and if the graph does and if the graph the equation, Graph How can you tell without calculating whether without tell you can How imaginary has equation a quadratic A graphing calculator or spreadsheet can also can spreadsheet or calculator A graphing QUESTIONING STRATEGIES QUESTIONING MATHEMATICAL INTEGRATE PRACTICES roots? roots? MP.5 Focus Focus on Technology be used to quickly evaluate expressions for many many for expressions evaluate be used quickly to a of feature the table Use the variable. of values for expression an evaluate to calculator graphing values. unknown different -axis, the solutions are imaginary. solutions are the not intersect the x-axis, 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: 5/21/14 8:17 AM

© Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©John Wang/Getty Images Solving Quadratic Equations by Taking Square Roots Square Taking by Equations Solving Quadratic Lesson 1 cosh = a cosh

)

= 3 and

- x/a e . Have students students . Have + x/a e 3 for a. + 3 for (

2 a x _ a =

) x ( 3.45. =

) 3 (

and solve h and solve 126 ) x ( 3.2 and h =

3.05 3.2 3.45 ) = 60 m.

2 ) ( Cable (m) Cable -30 Height Height of Vertical ( 3 . Since the height of the shortest the height verticalk. Since cablem, k is 3

―― 900 a 3. Confirm that h that + 3. Confirm

√ =

x ) ± ±30 48 45 900 48 1 2 3 0 x + 3 (

======0.05 + 3. + 3

1 2 2

2 2 ) x x ⋅ + 3 x x x x = x

( a a a a a ) + 3

h x -axis to the vertex. Another topic of of topic Another the vertex. to the x-axis the vertical from a is , where distance

2 = = = = = ( )

x 0.05 h ) ) ) x a __ x 1 x

EXTENSION EXTENSION ACTIVITY research online to compare the shapes of a parabola and catenary. Some students students Some catenary. and a parabola of the shapes compare to online research y = a catenary, for in the equation be interested might ( The supporting cable on a suspension bridge is in the shape of a parabola, but a but a parabola, of in the shape is bridge a suspension on cable supporting The a catenary of takes the both shape ends from suspended cable interest related to suspension bridges is the Tacoma Narrows Bridge collapse. collapse. Bridge Narrows the Tacoma is bridges suspension to related interest event. this dramatic of find to footage search Internet do an students Have ( ( ( Displacement from the from Displacement (m) Cable Vertical Shortest

Find a quadratic function that describes the height (in meters) of a parabolic cable above above cable a parabolic of (in meters) describes the function height that a quadratic Find lowest the cable’s from (in meters) displacement the horizontal a function of as the road cable if the parabolic between the towers the distance predict the function to Use point. tower. each at the road 48 m above of height a maximum reaches The two towers are at 30 m in opposite directions the shortest 30 m in opposite from at so the vertical are cable, towers two The is 30 - towers between distance Set the height function equal to 48 and solve for x. for Set function the height 48 and solve equal to Module 3 y-axis is and the of the road, the level at the x-axis is located plane where a coordinate Use of the height form the general system, the shortest at In this coordinate verticallocated cable. function is h A suspension bridge uses two thick cables, one on each side of the of side each on one cables, thick uses two bridge A suspension between suspended two are cables The the road. up hold to road, the connect vertical Smaller cables shape. a parabolic have and towers the first of the lengths gives table The the road. to cables parabolic the shortest one. with starting verticalfew cables Lesson Lesson Task Performance h , use the data for one of the vertical cables for (other than the shortest of a, use the data find the value one). To h x and 3.05 for 1 for substitute instance, For h 3.05 h 0.05 So, 0.05

A2_MNLESE385894_U2M03L1 126

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