Quadratic Functions. Graphing Quadratic Functions

Transcription

1 UNIT 3 Quadratic Functions CONTENTS COMMON CORE F-BF.B.3 F-BF.B.3 F-IF.B. MODULE Graphing Quadratic Functions Lesson.1 Understanding Quadratic Functions Lesson. Transforming Quadratic Functions Lesson.3 Interpreting Verte Form and Standard Form COMMON CORE F-IF.C.7a A-APR.B.3 A-REI.B. MODULE 7 Connecting Intercepts, Zeros, and Factors Lesson 7.1 Connecting Intercepts and Zeros Lesson 7. Connecting Intercepts and Linear Factors Lesson 7.3 Appling the Zero Product Propert to Solve Equations A Unit 3

2 Unit Pacing Guide UNIT 3 5-Minute Classes Module DAY 1 DAY DAY 3 DAY DAY 5 Lesson.1 Lesson. Lesson. Lesson.3 Lesson.3 DAY Module Review and Assessment Readiness Module 7 DAY 1 DAY DAY 3 DAY DAY 5 Lesson 7.1 Lesson 7. Lesson 7.3 Lesson 7.3 Module Review and Assessment Readiness DAY Unit Review and Assessment Readiness 9-Minute Classes Module DAY 1 DAY DAY 3 Lesson.1 Lesson. Lesson.3 Lesson. Lesson.3 Module Review and Assessment Readiness Module 7 DAY 1 DAY DAY 3 Lesson 7.1 Lesson 7.3 Module Review and Assessment Readiness Lesson 7. Unit Review and Assessment Readiness Unit 3 7B

3 Program Resources PLAN ENGAGE AND EXPLORE HMH Teacher App Access a full suite of teacher resources online and offline on a variet of devices. Plan present, and manage classes, assignments, and activities. Real-World Videos Engage students with interesting and relevant applications of the mathematical content of each module. Eplore Activities Students interactivel eplore new concepts using a variet of tools and approaches. eplanner Easil plan our classes, create and view assignments, and access all program resources with our online, customizable planning tool. Professional Development Videos Authors Juli Dion and Matt Larson model successful teaching practices and strategies in actual classroom settings. QR Codes Scan with our smart phone to jump directl from our print book to online videos and other resources. DO NOT EDIT--Changes must be made through "File info" CorrectionKe=NL-A;CA-A Teacher s Edition Support students with point-of-use Questioning Strategies, teaching tips, resources for differentiated instruction, additional activities, and more. NOT EDIT--Changes must made through "File info" DODO NOT EDIT--Changes must bebe made through "File info" CorrectionKe=NL-A;CA-A CorrectionKe=NL-A;CA-A Name Name Isosceles and Equilateral Triangles NOT EDIT--Changes must made through "File info" DODO NOT EDIT--Changes must bebe made through "File info" CorrectionKe=NL-A;CA-A CorrectionKe=NL-A;CA-A Class Class straightedge to draw segment to draw lineline segment BCBC.. CCUseUsethethestraightedge Investigating Isosceles Triangles G-CO.C.1 Eplore Eplore Prove theorems about triangles. INTEGRATE TECHNOLOGY Mathematical Practices COMMON CORE angles have base a side base angles. TheThe angles thatthat have thethe base as aasside areare thethe base angles. ENGAGE work in the space provided. a straightedge to draw angle. in the space provided. UseUse a straightedge to draw an an angle. AADoDoourourwork QUESTIONING STRATEGIES Possible answer Triangle m A 7 ; m B 55 ; m C 55. Possible answer forfor Triangle 1: 1: m A == 7 ; m B == 55 ; m C == 55. a different each time. is aisdifferent sizesize each time. A A Reflect Reflect Eplain Eplain 1 1 Proving Provingthe theisosceles Isosceles Triangle Theorem Triangle Theorem sides of the angle. Label points B and sides of the angle. Label thethe points B and C. C. anditsitsconverse Converse and A A In the Eplore, made a conjecture base angles of an isosceles triangle congruent. In the Eplore, ouou made a conjecture thatthat thethe base angles of an isosceles triangle areare congruent. This conjecture proven it can stated a theorem. This conjecture cancan be be proven so so it can be be stated as aastheorem. C C Isosceles Triangle Theorem Isosceles Triangle Theorem Houghton Mifflin Harcourt Publishing Compan Make a Conjecture Looking at our results, what conjecture made about base angles,.. Make a Conjecture Looking at our results, what conjecture cancan be be made about thethe base angles, C? B B andand C? The base angles congruent. The base angles areare congruent. Using a compass, place point verte draw intersects a compass, place thethe point onon thethe verte andand draw an an arcarc thatthat intersects thethe BBUsing B B If two sides a triangle congruent, then angles opposite sides If two sides of aoftriangle areare congruent, then thethe twotwo angles opposite thethe sides areare congruent. congruent. This theorem is sometimes called Base Angles Theorem stated as Base angles This theorem is sometimes called thethe Base Angles Theorem andand cancan alsoalso be be stated as Base angles of an isosceles triangle congruent. of an isosceles triangle areare congruent. Module Module must be EDIT--Changes A;CA-A DO NOT CorrectionKe=NL- made through Lesson Lesson "File info" Module Module Lesson Lesson Date Class al and Equilater. Isosceles Triangles Name Essential COMMON CORE IN1_MNLESE397_UML IN1_MNLESE397_UML Question: G-CO.C.1 relationships the special What are triangles? and equilateral Prove theorems triangle is The congruent The angle The side a triangle sides are formed b opposite with at least called the the legs is the verte among in isosceles Resource Locker HARDCOVER PAGES HARDCOVER PAGES PROFESSIONALDEVELOPMENT DEVELOPMENT PROFESSIONAL about triangles. Investigating Eplore An isosceles sides angles and legs of the the verte angle is the Isosceles two congruent Triangles sides. Legs Verte angle triangle. Base angle. Base angles base. the as a side are base angles. other potential the base and investigate that have triangles isosceles ou will construct special triangles. angle. es of these In this activit, to draw an characteristics/properti Use a straightedge space provided. figure. work in the in the Do our as shown angle A, A Label our The angles Check students construtions. The side opposite the verte angle is the base. The angles that have the base as a side are the base angles. In this activit, ou will construct isosceles triangles and investigate other potential characteristics/properties of these special triangles. How know triangles constructed isosceles triangles? How do do ouou know thethe triangles ouou constructed areare isosceles triangles?. The compass marks equal lengths both sides A; therefore, ABAB ACAC. The compass marks equal lengths onon both sides of of A; therefore, Check students construtions. Check students construtions. /19/1 1:1 /19/1 1:1 PM PM Watch the hardcover Watch forfor the hardcover student edition page student edition page numbers this lesson. numbers forfor this lesson. IN1_MNLESE397_UML IN1_MNLESE397_UML 1919 LearningProgressions Progressions Learning this lesson, students add their prior knowledge isosceles and equilateral InIn this lesson, students add toto their prior knowledge ofof isosceles and equilateral /19/1 1:1 /19/1 1:1 PM PM Legs The angle formed b the legs is the verte angle. How could ou draw isosceles triangles without using a compass? Possible answer: Draw A and plot point B on one side of A. Then _ use a ruler to measure AB and plot point C on the other side of A so that AC = AB. Repeat steps A D at least more times record results in the table. Make sure steps A D at least twotwo more times andand record thethe results in the table. Make sure A A EERepeat Verte angle The congruent sides are called the legs of the triangle. What must be true about the triangles ou construct in order for them to be isosceles triangles? The must have two congruent sides. m B m B Label our angle A, as shown in the figure. Label our angle A, as shown in the figure. Houghton Mifflin Harcourt Publishing Compan Triangle Triangle m C m C In this activit, construct isosceles triangles investigate other potential In this activit, ouou willwill construct isosceles triangles andand investigate other potential characteristics/properties of these special triangles. characteristics/properties of these special triangles. Houghton Mifflin Harcourt Publishing Compan Triangle Triangle 3 3 Houghton Mifflin Harcourt Publishing Compan View the Engage section online. Discuss the photo, eplaining that the instrument is a setant and that long ago it was used to measure the elevation of the sun and stars, allowing one s position on Earth s surface to be calculated. Then preview the Lesson Performance Task. Triangle Triangle m ma A Base Base Base angles Base angles opposite verte angle is the base. TheThe sideside opposite thethe verte angle is the base. PREVIEW: LESSON PERFORMANCE TASK Triangle Triangle 1 1 angle formed is the verte angle. TheThe angle formed b b thethe legslegs is the verte angle. Eplain to a partner what ou can deduce about a triangle if it has two sides with the same length. In an isosceles triangle, the angles opposite the congruent sides are congruent. In an equilateral triangle, all the sides and angles are congruent, and the measure of each angle is. Legs Legs congruent sides called of the triangle. TheThe congruent sides areare called thethe legslegs of the triangle. MP.3 Logic Language Objective Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles? DD Verte angle Verte angle Investigating Isosceles Triangles An isosceles triangle is a triangle with at least two congruent sides. Students have the option of completing the isosceles triangle activit either in the book or online. a protractor to measure each angle. Record measures in the table under column UseUse a protractor to measure each angle. Record thethe measures in the table under thethe column Triangle forfor Triangle InvestigatingIsosceles Isosceles Triangles Investigating Triangles isosceles triangle a triangle with at least congruent sides. AnAn isosceles triangle is aistriangle with at least twotwo congruent sides. G-CO.C.1 Prove theorems about triangles. Eplore C C B B Resource Resource Locker Locker The student is epected to: COMMON CORE Resource Locker EXPLORE A A Essential Question: What special relationships among angles and sides in isosceles Essential Question: What areare thethe special relationships among angles and sides in isosceles and equilateral triangles? and equilateral triangles? Common Core Math Standards COMMON CORE Date Date. Isosceles Isoscelesand andequilateral Equilateral. Triangles Triangles Date Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles? NOT EDIT--Changes must made through "File info" DODO NOT EDIT--Changes must bebe made through "File info" CorrectionKe=NL-A;CA-A CorrectionKe=NL-A;CA-A EXPLAIN 1 Proving the Isosceles Triangle Theorem and Its Converse Do our work in the space provided. Use a straightedge to draw an angle. Label our angle A, as shown in the figure. A CONNECT VOCABULARY Ask a volunteer to define isosceles triangle and have students give real-world eamples of them. If possible, show the class a baseball pennant or other flag in the shape of an isosceles triangle. Tell students the will be proving theorems about isosceles triangles and investigating their properties in this lesson. hing Compan. Class. Isosceles and Equilateral Triangles DONOT NOTEDIT--Changes EDIT--Changesmust mustbebemade madethrough through"file "Fileinfo" info" DO CorrectionKe=NL-A;CA-A CorrectionKe=NL-A;CA-A DO NOT EDIT--Changes must be made through "File info" CorrectionKe=NL-A;CA-A LESSON Name Base Base angles

4 Evaluate Lesson 19. Precision and Accurac PROFESSIONAL DEVELOPMENT TEACH ASSESSMENT AND INTERVENTION teacher Support Lesson XX.X Comparing Linear, Eponential, and Quadratic Models Lesson 19. Precision and Accurac 1 EXPLAIN Concept 1 Eplain Concept ComPLEtINg the SquArE with EXPrES- SIoNS Determining Precision Avoid Common Errors Some students ma not pa attention to whether b is positive or negative, since c is positive regardless of the sign of b. Have As ou have seen, measurements are given to a certain precision. Therefore, student change the sign of b in some problems and compare the factored forms of both the value reported does not necessaril represent the actual value of the epressions. measurement. For eample, a measurement of 5 centimeters, which is questioning Strategies In a perfect square trinomial, is the last term given to the nearest whole unit, can actuall range from.5 units below the alwas positive? Eplain. es, a perfect square trinomial can be reported rt value,.5 centimeters, up to, but not including,.5 units above either (a + b) or (a b) which can be factored as (a + b) = a + ab = b and (a b) it, centimeters. The actual length, l, is within a range of possible values: = a + ab = b. In both cases the last term is centimeters. Similarl, a length given to the nearest tenth can actuall range positive. reflect from.5 units below the reported value up to, but not including,.5 units 3. The sign of b has no effect on the sign of above it. So a length reported as.5 cm could actuall be as low as.5 cm or c because c = ( b ) and a nonzero number squared is alwas positive. Thus, c is as high as nearl.55 cm. alwas positive. c = ( b ) and a nonzero number c = ( b ) and a nonzero number Name Date Class LESSON Precision and Significant Digits 1-1 Success for English Learners The precision of a measurement is determined b the smallest unit or fraction of a unit used. Problem 1 Choose the more precise measurement..3 g is to the nearest tenth. Math On the Spot video tutorials, featuring program authors Dr. Edward Burger and Martha Sandoval-Martinez, accompan ever eample in the tetbook and give students step-b-step instructions and eplanations of ke math concepts. Interactive Teacher Edition Customize and present course materials with collaborative activities and integrated formative assessment. Differentiated Instruction Resources Support all learners with Differentiated Instruction Resources, including Leveled Practice and Problem Solving Reading Strategies Success for English Learners Challenge.7 g is to the nearest hundredth..3 g or.7 g Because a hundredth of a gram is smaller than a tenth of a gram,.7 g is more precise. Problem Choose the more precise measurement: 3 inches or 3 feet. Name Date Class LESSON -1 Linear Functions Reteach The graph of a linear function is a straight line. A + B + C = is the standard form for the equation of a linear function. A, B, and C are real numbers. A and B are not both zero. The variables and have eponents of 1 are not multiplied together are not in denominators, eponents or radical signs. Eamples These are NOT linear functions: + = no variable = 9 eponent on 1 = and multiplied together = 3 in denominator = in eponent = 5 in a square root Tell whether each function is linear or not. The Personal Math Trainer provides online practice, homework, assessments, and intervention. Monitor student progress through reports and alerts. Create and customize assignments aligned to specific lessons or Common Core standards. Practice With dnamic items and assignments, students get unlimited practice on ke concepts supported b guided eamples, step-b-step solutions, and video tutorials. Assessments Choose from course assignments or customize our own based on course content, Common Core standards, difficult levels, and more. Homework Students can complete online homework with a wide variet of problem tpes, including the abilit to enter epressions, equations, and graphs. Let the sstem automaticall grade homework, so ou can focus where our students need help the most! Intervention Let the Personal Math Trainer automaticall prescribe a targeted, personalized intervention path for our students. Elaborate Question 3 of 17 Save & Close Solve the quadratic equation b factoring. 7 + = 7 1 =, Personal Math Trainer View Step b Step Focus on Higher Order Thinking Raise the bar with homework and practice that incorporates higher-order thinking and mathematical practices in ever lesson. Assessment Readiness Prepare students for success on high stakes tests for Integrated Mathematics with practice at ever module and unit Assessment Resources Tailor assessments and response to intervention to meet the needs of all our classes and students, including Leveled Module Quizzes Leveled Unit Tests Unit Performance Tasks Placement, Diagnostic, and Quarterl Benchmark Tests Tier 1, Tier, and Tier 3 Resources Video Tutor?! Tetbook X Check Animated Math Turn It In Look Back 1. 1 =. 3 = =. = 1 The graph of = C is alwas a horizontal line. The graph of = C is alwas a vertical line. 1. When deciding which measurement is more precise, what should ou consider? Eamples. An object is weighed on three different scales. The results are shown in the table. Which scale is the most precise? Eplain our answer. = 1 = = 3 = 3 Unit 3 7D Scale Measurement T

5 Math Background Understanding Quadratic COMMON Functions F-BF.B.3 LESSON.1 A quadratic function is an function that can be written in the form = a + b + c, where a, b, and c are real numbers and a. This is called the standard form of a quadratic function. There are several was to identif a quadratic function. One wa is to use a table of values. Recall that linear functions have a constant rate of change. Therefore, in a table of values for a linear function, a constant change in the -values corresponds to a constant change in the -values. For quadratic functions, it is the rate of change itself that has a constant rate of change. In other words, when there is a constant change in -values, a quadratic function has constant second differences. The table illustrates this for the quadratic function = Y 7 First differences: Second differences: Developing Quadratic COMMON Functions CORE CORE F-BF.B.3 LESSONS. and.3 The simplest quadratic function is =. Its graph is a parabola that opens upward, has its verte at the origin, and is smmetric about the -ais. The graphs of all other quadratic functions ma be created b performing a series of transformations on the graph of =. For functions in the form = a (a ), the value of a determines the direction and shape of the parabola. If a >, the parabola opens upward; if a <, the parabola opens downward. If a > 1, the parabola is narrower than the graph of = ; if a < 1, the parabola is wider than the graph of =. The graph of a function in the form = a ( - h) is a horizontal translation of the graph of = a. For eample, the graph of = ( - 3) is identical to the graph of =, but shifted 3 units to the right. - = - 3 units = (- 3) 7E Unit 3

6 PROFESSIONAL DEVELOPMENT Finall, the constant k in = a ( - h) + k represents a vertical translation of the graph of = a ( - h). The translation is k units upward if k > and k units downward if k <. = h = a( - h) + k (h, k) The equation = a ( - h) + k is called the verte form of a quadratic function. This form makes it eas to identif the verte of the parabola, (h, k), and the ais of smmetr, = h. One wa to find the formula for the ais of smmetr of a parabola is b using verte form. As shown below, the basic idea is to transform verte form into standard form and then write the equation of the ais of smmetr, = h, in terms of a, b, and c. = a ( - h) + k = a ( - h + h ) + k = a - ah + ah + k Connecting Intercepts COMMON and Zeros CORE LESSON 7.1 F-IF.C.7a A quadratic equation is an equation that can be written in the form a + b + c =, where a, b, and c are real numbers and a. This is the standard form of a quadratic equation. Ever quadratic equation written in this form has a related quadratic function, = a + b + c. As with other equations, a solution of a quadratic equation is a value of the variable that makes the equation true. One wa to solve a quadratic equation in standard form is to graph the related quadratic function and find its -intercepts. For eample, for the quadratic equation + - =, the graph of the related function = + - is the set of all ordered pairs of the form (, + - ). It must be true that at an point where the graph intersects the -ais, the -coordinate is ; that is, + - =. In other words, the -intercepts of the graph are precisel the solutions of the quadratic equation. Students should understand that this is a general method that works for an equation with one side equal to, if the related function can be graphed. For eample, the linear equation 1 - = ma be solved b graphing the related linear function = 1 - and noticing that the graph intersects the -ais at =. In standard form, the value of b is the coefficient of, so b = -ah. Solving for h gives h = -. Thus, the a ais of smmetr is the vertical line = - b_ a. b_ Unit 3 7F

7 UNIT 3 Quadratic Functions MATH IN CAREERS Unit Activit Preview After completing this unit, students will complete a Math in Careers task b graphing and writing an equation for a function that fits a data set comparing gas mileage to speed. Critical skills include modeling real-world situati ons and fitting a function to a set of data. UNIT 3 Quadratic Functions MODULE Graphing Quadratic Functions MODULE 7 Connecting Intercepts, Zeros, and Factors For more information about careers in mathematics as well as various mathematics appreciation topics, visit The American Mathematical Societ at Houghton Mifflin Harcourt Publishing Compan Image Credits: UpperCut Images/Alam MATH IN CAREERS Transportation Engineer Transportation engineers design and modif plans for transportation sstems including airports, trains, highwas, and bridges. The use math when preparing budgets and project costs. The also use mathematical models to simulate traffic flow and analze engineering data. If ou are interested in a career as a transportation engineer, ou should stud these mathematical subjects: Algebra Geometr Trigonometr Calculus Differential Equations Research other careers that require developing and analzing mathematical models. Check out the career activit at the end of the unit to find ou how transportation engineers use math. Unit 3 7 TRACKING YOUR LEARNING PROGRESSION Before In this Unit After Students understand: adding and subtracting polnomial epressions multipling polnomial epressions special products of binomials Students will learn about: graphing quadratic functions interpreting verte and standard form of quadratic functions connecting intercepts and zeros solving quadratic equations using the Zero Product Propert Students will stud: solving quadratic equations b factoring solving quadratic equations b completing the square using the quadratic formula to solve equations comparing linear, quadratic, and eponential models 7 Unit 3

8 Reading Start -Up Visualize Vocabular Use the words to complete the graphic. Write the name of a form of linear equation that best fits each equation. A+ B= C standard form Find the - and -intercepts and plot the related points. Understand Vocabular linear equation - 1 = m(- 1 ) point-slope form Plot the point ( 1, 1 ) and use the slope m to find a second point on the line. Connect the points with a line. = m +b slope-intercept form Plot (, b) and use the slope m to find a second point on the line. Match the term on the left to the eample on the right. 1. C standard form of a quadratic equation A. = -( - ) + 9. B intercept form of a quadratic equation B. = -( + 1) ( - 5) 3. A verte form of a quadratic function C. = Active Reading Tri-Fold Before beginning the unit, create a tri-fold to help ou learn the concepts and vocabular in this unit. Fold the paper into three sections. Label the columns What I Know, What I Need to Know, and What I Learned. Complete the first two columns before ou read. After studing the unit, complete the third column. Vocabular Review Words point-slope form (forma de punto pendiente) slope-intercept form (forma de pendienteintersección) standard form (forma estándar) -intercept (intersección con el eje ) -intercept (intersección con el eje ) Preview Words intercept form of a quadratic equation (forma en intersección de una función cuadrática) standard form of a quadratic equation (forma estándar de una ecuación cuadrática) verte form of a quadratic function (forma en vértice de una función cuadrática) Houghton Mifflin Harcourt Publishing Compan Reading Start Up Have students complete the activities on this page b working alone or with others. VISUALIZE VOCABULARY The decision tree graphic helps students review vocabular associated with linear equations. If time allows, discuss the advantages of each form of a linear equation. UNDERSTAND VOCABULARY Use the following eplanations to help students learn the preview words. The intercept form of a quadratic equation shows the zeros, or -intercepts, on the graph of the function. The verte form of a quadratic equation shows the coordinates of the verte on the graph of the parabola. The standard form of a quadratic equation lists the terms in decreasing order of the eponents. ACTIVE READING Students can use these reading and note-taking strategies to help them organize and understand the new concepts and vocabular. Encourage them to be confident in their use of familiar vocabular and question an terms that are unfamiliar. ADDITIONAL RESOURCES Unit 3 Differentiated Instruction Reading Strategies Unit 3

9 MODULE Graphing Quadratic Functions ESSENTIAL QUESTION: Answer: If ou throw a ball, shoot an arrow, or fire a missile, it will go up into the air, slowing down as it goes, then come down again. If ou graph the path of the ball, arrow, or missile, ou will find that it is the graph of a quadratic function. PROFESSIONAL DEVELOPMENT VIDEO Graphing Quadratic Functions Essential Question: How can ou use the graph of a quadratic function to solve real-world problems? MODULE LESSON.1 Understanding Quadratic Functions LESSON. Transforming Quadratic Functions LESSON.3 Interpreting Verte Form and Standard Form Professional Development Video Author Juli Dion models successful teaching practices in an actual high-school classroom. Professional Development m.hrw.com Houghton Mifflin Harcourt Publishing Compan Image Credits: Image Source/Gett Images MODULE PERFORMANCE TASK PREVIEW Throwing for a Completion REAL WORLD VIDEO Projectile motion describes the height of an object thrown or fired into the air. The height of a football, volleball, or an projectile can be modeled b a quadratic equation. Do ou wonder how fast a football leaves the hands of a quarterback or how high up it goes? Some professionals can throw approimatel 5 miles per hour or faster. The height the ball reaches depends on the initial velocit as well as the angle at which it was thrown. You can use a mathematical model to see how high a football is at different times. Module 9 DIGITAL TEACHER EDITION Access a full suite of teaching resources when and where ou need them: Access content online or offline Customize lessons to share with our class Communicate with our students in real-time View student grades and data instantl to target our instruction where it is needed most PERSONAL MATH TRAINER Assessment and Intervention Assign automaticall graded homework, quizzes, tests, and intervention activities. Prepare our students with updated, Common Core-aligned practice tests. 9 Module

10 Are YOU Read? Complete these eercises to review skills ou will need for this chapter. Linear Functions Eample 1 Tell whether - = 9 represents a linear function. When a linear equation is written in standard form, the following are true. and both have eponents of 1. and are not multiplied together. and do not appear in denominators, eponents, or radicands. - = 9 represents a linear function. Tell whether the equation represents a linear function. 1. = no. 3 = 1-1_ es 3. = + 5 Online Homework Hints and Help Etra Practice es Are You Read? ASSESS READINESS Use the assessment on this page to determine if students need strategic or intensive intervention for the module s prerequisite skills. ASSESSMENT AND INTERVENTION Algebraic Representations of Transformations Eample The vertices of a triangle are A (-3, 1), B (, -), and C (-, ). Find the vertices if the figure is translated b the rule (, ) ( +, - 3). A (-3, 1) A (-3 +, 1-3), so A (1, -) Add to each -coordinate B (, -) B ( +, - - 3), so B (, -5) and subtract 3 from each -coordinate. C (-, ) C (- +, - 3), so C (, -1) The vertices of a triangle are A (, 3), B (-, -), and C (1, 5). Find the new vertices.. Use the rule (, ) ( -, + ) to translate 5. Use the rule (, ) ( + 1, - ) to translate each verte. each verte. A (-, 7), B (-, ), C (-1, 9) A (1, 1), B (-1, -), C (, 3) Algebraic Epressions Eample 3 Find the value of when =. Find the value () + 5 () Substitute for. Follow the order of operations when = when = -3-1 Houghton Mifflin Harcourt Publishing Compan 3 1 TIER 1, TIER, TIER 3 SKILLS Personal Math Trainer will automaticall create a standards-based, personalized intervention assignment for our students, targeting each student s individual needs! ADDITIONAL RESOURCES See the table below for a full list of intervention resources available for this module. Response to Intervention Resources also includes: Tier Skill Pre-Tests for each Module Tier Skill Post-Tests for each skill Module 1 Response to Intervention Differentiated Instruction Tier 1 Lesson Intervention Worksheets Tier Strategic Intervention Skills Intervention Worksheets Tier 3 Intensive Intervention Worksheets available online Reteach.1 Reteach. Reteach.3 5 Algebraic Epressions Algebraic Representations of Transformations 1 Eponents 1 Linear Functions 37 Properties of Translations Building Block Skills 19,, 3,, 7, 9, 3,,, 51, 59, 9, 7, 1, 1 Challenge worksheets Etend the Math Lesson Activities in TE Module 1

11 COMMON CORE Locker LESSON Common Core Math Standards The student is epected to: COMMON CORE F-BF.B.3 Identif the effect on the graph of replacing f() b... f(k)... for specific values of k (both positive and negative)... Also F-IF.A., F-IF.B., F-IF.C.7a Mathematical Practices COMMON CORE.1 Understanding Quadratic Functions MP. Reasoning Language Objective Describe terms associated with quadratic functions. ENGAGE Essential Question: What is the effect of the constant a on the graph of f() =a? Possible answer: The effect of the constant a is either to verticall stretch or to shrink the graph of g () = a. A negative value of a reflects the graph across the -ais. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the path of a real helicopter landing would be different from the path of a paper helicopter being dropped. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing Compan Name Class Date.1 Understanding Quadratic Functions Essential Question: What is the effect of the constant a on the graph of f () = a? Eplore Understanding the Parent Quadratic Function A function that can be represented in the form of ƒ () = a + b + c is called a quadratic function. The terms a, b, and c, are constants where a. The greatest eponent of the variable is. The most basic quadratic function is ƒ () =, which is the parent quadratic function. A Here is an incomplete table of values for the parent quadratic function. Complete it. Reflect f () = 3 ƒ () = = ( 3) = Discussion What is the domain of ƒ () =? The domain is the set of all real numbers.. Discussion What is the range of ƒ () =? The range is the set. 1 9 B Plot the ordered pairs as points on the graph, and connect the points to sketch a curve. - - The curve is called a parabola. The point through which the parabola turns direction is called its verte. The verte occurs at (, ) for this function. A vertical line that passes through the verte and divides the parabola into two smmetrical halves is called the ais of smmetr. For this function, the ais of smmetr is the -ais. Resource Locker Module 11 Lesson 1 Name Class Date.1 Understanding Quadratic Functions Essential Question: What is the effect of the constant a on the graph of f () = a? F-BF.B.3 Identif the effect on the graph of replacing f() b f(k) for specific values of k (both positive and negative) Also F-IF.A., F-IF.B., F-IF.C.7a Eplore Understanding the Parent Quadratic Function A function that can be represented in the form of ƒ () = a + b + c is called a quadratic function. The terms a, b, and c, are constants where a. The greatest eponent of the variable is. The most basic quadratic function is ƒ () =, which is the parent quadratic function. Here is an incomplete table of values for the parent quadratic function. Complete it. f () = 3 ƒ () = = ( 3) = Resource Plot the ordered pairs as points on the graph, and connect the points to sketch a curve. The curve is called a parabola. The point through which the parabola turns direction is called its verte. The verte occurs at (, ) for this function. A vertical line that passes through the verte and divides the parabola into two smmetrical halves is called the ais of smmetr. For this function, the ais of smmetr is the -ais. HARDCOVER PAGES 11 Watch for the hardcover student edition page numbers for this lesson. Houghton Mifflin Harcourt Publishing Compan Reflect 1. Discussion What is the domain of ƒ () =? The domain is the set of all real numbers.. Discussion What is the range of ƒ () =? The range is the set. Module 11 Lesson 1 11 Lesson.1

12 Eplain 1 Graphing g () = a when a > The graph g () = a, is a vertical stretch or compression of its parent function ƒ () =. The graph opens upward when a >. Vertical Stretch g () = a with a > 1. The graph of g () is narrower than the parent function ƒ (). - f() - g() - - Vertical Compression g () = a with < a < 1. The graph of g () is wider than the parent function ƒ (). The domain of a quadratic function is all real numbers. When a >, the graph of g () = a opens upward, and the function has a minimum value that occurs at the verte of the parabola. So, the range of g () = a, where a >, is the set of real numbers greater than or equal to the minimum value. Eample 1 g () = Graph each quadratic function b plotting points and sketching the curve. State the domain and range. g () = Domain: all real numbers Range: Module 1 Lesson 1 PROFESSIONAL DEVELOPMENT g() Integrate Mathematical Practices This lesson provides an opportunit to address Mathematical Practice MP., which calls for students to reason abstractl and quantitativel. Students learn to recognize the parent function of a quadratic equation and then to analze the relationship between the value of a and the graph of the quadratic function. - f() - Houghton Mifflin Harcourt Publishing Compan EXPLORE Understanding the Parent Quadratic Function INTEGRATE TECHNOLOGY Students have the option of completing the activit either in the book or online. CONNECT VOCABULARY This lesson tells students that if a is positive, the graph opens upward, and if a is negative, the graph opens downward. The suffi -ward means in the direction of. Students ma know the word toward; point out also the words outward and inward, used in the FOIL method. EXPLAIN 1 Graphing g() = a when a > QUESTIONING STRATEGIES How are the graphs of g() = and g() = 1 similar? How are the different? The graphs are similar because both have the same parent graph, f () = ; both are smmetric in the -ais; and both have the same verte, (, ). The are different in that the graph of g () = is a vertical stretch of the graph of f () =, while the graph of g () = 1 is a vertical compression of the graph of f () =. If (, ) and (3, 7) are two points on the graph of g(), where (, ) is the verte, what is another point on the graph? (-3, 7), because the graph is smmetrical about the -ais Understanding Quadratic Functions 1

13 EXPLAIN Graphing g() = a when a < AVOID COMMON ERRORS Students ma think that increasing a increases the width of the graph. Make sure that students understand that the value of a tells how the parent function ƒ () = is stretched or compressed verticall. When the absolute value of a is greater than 1, the graph is stretched, awa from the -ais. When the absolute value of a is greater than but less than 1, the graph is compressed toward the -ais. When a graph is stretched verticall, it appears to shrink horizontall. QUESTIONING STRATEGIES How would the graph of the function g () = -15 differ from the graph of ƒ () =? It would reflect the curve across the -ais and stretch it downward b a factor of 15. Houghton Mifflin Harcourt Publishing Compan B g () = 1 Reflect g () = 1 3. For a graph that has a vertical compression or stretch, does the ais of smmetr change? No, the ais of smmetr does not change. Your Turn Graph each quadratic function. State the domain and range.. g () = 3 D: all real; R: 5. g () = 1_ 3 D: all real; R: f() g() Eplain Graphing g () = a when a < The graph of = opens downward. It is a reflection of the graph of = across the -ais. So, When a <, the graph of g () = a opens downward, and the function has a maimum value that occurs at the verte of the parabola. In this case, the range is the set of real numbers less than or equal to the maimum value. Vertical Stretch 1 1_ g () = a with a > 1. The graph of g () is narrower than the parent function f() g() - Vertical Compression g () = a with < a < 1. Domain: all real numbers Range: The graph of g () is wider than the parent function f (). f() Module 13 Lesson 1 IN_MNLESE393_U3ML1.indd 13 COLLABORATIVE LEARNING Peer to Peer Activit Have students work in pairs. Ask one student to draw a graph of = a for some value of a. The second student then writes the equation for the graph. Finall, the pairs determine whether the equation is correct for the given graph. Students then switch roles and repeat. //1 1:35 AM 13 Lesson.1

14 Eample g () = Graph each quadratic function b plotting points and sketching the curve. State the domain and range. g () = g () = 1_ g () = 1_ 1_ Domain: all real numbers Range: INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. As students reflect a function across the -ais and stretch or compress the function verticall, make sure the understand that making the function negative reflects it across the -ais and changing the value of a affects its width. CONNECT VOCABULARY You ma wish to have English language learners epress terms from the lesson in their own words. For eample, ask students: An ais is an imaginar line. Use this information and our understanding of smmetr to define the term ais of smmetr. an imaginar line that splits a figure so that one side of the figure is a mirror image of the other _ 1_ Domain: all real numbers Range: Reflect. Does reflecting the parabola across the -ais (a < ) change the ais of smmetr? No, the ais of smmetr is a line that etends both up and down and does not change upon reflection. Houghton Mifflin Harcourt Publishing Compan Module 1 Lesson 1 DIFFERENTIATE INSTRUCTION Visual Cues Have students draw the graph of a function that is wider than the graph of ƒ () = and one for which the graph is narrower. Have them use a different color for each graph. Then have them write the function above each graph in black, ecept for the value of a, which should be the same color as the corresponding graph. This ma help students to remember which values of a lead to wider graphs and which lead to narrower graphs. Understanding Quadratic Functions 1

15 EXPLAIN 3 Writing a Quadratic Function Given a Graph QUESTIONING STRATEGIES How can ou immediatel tell from the graph that the value of a in the quadratic function is negative? The graph opens downward. How does having a point of the parabola other than the verte help ou find the value of a? If ou have a point other than (, ), ou can substitute the - and -values into = a and solve for a. AVOID COMMON ERRORS Students ma forget that the square of a negative number is positive. Remind students to check whether their calculated values of a are reasonable for the graph. A positive value of a for a graph that opens downward indicates a sign error. Houghton Mifflin Harcourt Publishing Compan Your Turn Graph each function. State the domain and range. 7. g () = 3 D: all real; R:. g () = 1 3 D: all real; R: Eplain (-, -) - - (, ) Writing a Quadratic Function Given a Graph You can determine a function rule for a parabola with its verte at the origin b substituting and values for an other point on the parabola into g () = a and solving for a. Eample 3 Write the rule for the quadratic functions shown on the graph Use the point (, ). Start with the functional form. g () = a Replace and g () with point values. = a () Evaluate. Divide both sides b to isolate a. = a 1 = a Write the function rule. g () = 1 Use the point (-, ). Start with the functional form. g () = a Replace and g () with point values. = a ( ) Evaluate. = a Divide both sides b to isolate a. = a Write the function rule. g () = Module 15 Lesson 1 IN_MNLESE393_U3ML1.indd 15 LANGUAGE SUPPORT Communicate Math English learners ma find the spelling patterns in English somewhat puzzling, especiall if the are literate in a language that has a ver consistent set of spelling and pronunciation rules. Some words the need to understand and use for algebra ma require additional support and review; for eample, focusing on the final sounds of: graph /-f/, stretch /-ch/, verte /-/, and intercept /-pt/ Have students work in small, mied language-proficienc groups to generate a word bank of English and mathematical terms students can use to describe quadratic function graphs (such as upward, downward, smmetr, verte). //1 1:35 AM 15 Lesson.1

16 Your Turn (1, ) - - Use the point (1, ). Eplain Depth (ards) (-, -3) (, -3) Time (seconds) Modeling with a Quadratic Function Real-world situations can be modeled b parabolas. Eample g () = a = a (1) = a g () = (-1, 1) For each model, describe what the verte, -intercept, and endpoint(s) represent in the situation it models, and then determine the equation of the function. This graph models the depth in ards below the water s surface of a dolphin before and after it rises to take a breath and descends again. The depth d is relative to time t, in seconds, and t = is when dolphin reaches a depth of ards at the surface. Use the point (1, 1). g () = a 1 = a (1) 1 = a g () = The -intercept occurs at the verte of the parabola at (, ), where the dolphin is at the surface to breathe. The endpoint (-, -3) represents a depth of 3 ards below the surface at seconds before the dolphin reaches the surface to breathe. The endpoint (, -3) represents a depth of 3 ards below the surface at seconds after the dolphin reaches the surface to breathe. The graph is smmetric about the -ais with the verte at the origin, so the function will be of the form = a, or d (t) = at. Use a point to determine the equation. d (t) = at -3 = a () -3 = a 1 - = a The function is d (t) = - t. Module 1 Lesson 1 Houghton Mifflin Harcourt Publishing Compan Image Credits: Malcolm Schul/Alam EXPLAIN Modeling with a Quadratic Function QUESTIONING STRATEGIES What would happen to the graph if the dolphin ascends at the same rate, but starts at a shallower depth? The starting and ending points would be closer to the -ais. What would happen to the graph if the dolphin ascends and descends at a slower speed? The graph would be wider. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. When students have to write an equation for the quadratic function shown in a graph, make sure the understand that the need to pick a point on the graph to substitute in the equation = a to find a Understanding Quadratic Functions 1

17 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 Have students use graphing calculators to graph g () = a for several different values of a on the same screen b using the graphing functions Y 1, Y, and so on. This will allow them to create man graphs in a short period of time and to compare the effects of different values of a. SUMMARIZE How does the value of a affect the graph of ƒ () = a in relation to the graph of the quadratic parent function? Use the whiteboard and have students complete the table below. Value of a Tpe of transformation Graph opens a > 1 stretch up < a < 1 compression up -1 < a < compression down a = -1 reflection down a < -1 stretch down Houghton Mifflin Harcourt Publishing Compan B Satellite dishes reflect radio waves onto a collector b using a reflector (the dish) shaped like a parabola. (, 1) The graph shows the height h in feet of the reflector 1 relative to the distance in feet from the center of the satellite dish The -intercept occurs at the verte, which represents Distance from Center (feet) the distance = feet from the center of the dish. The left end-point represents the height h = feet at the center of the dish. The right end-point represents the height h = 1 feet at the distance = feet from the center of the dish. h () = a The function will be of the form. Use, 1 to determine the equation. Your Turn h () = a 1 = a ( ) 1 = 3 a a = 1_ 3 1 h () = The graph shows the height h in feet of a rock dropped down a deep well as a function of time t in seconds. Height (feet) The -intercept occurs at the left end-point, which is also the verte, and represents the height h = at which the rock was released at ground level. The right end point represents the height h = - feet at which the rock hits the bottom of the well t = seconds after it was released. Using the point (, -) to determine the equation: h (t) = a t - = a () - = a a = -1 d (t) = -1 t Height (feet) (, -) Time Seconds Module 17 Lesson 1 17 Lesson.1

18 Elaborate 1. Discussion In eample 1A the points (3, 1) and (-3, 1) did not fit on the grid. Describe some strategies for selecting points used to guide the shape of the curve. Sample Answers: 1. Start with point (, ) and work outwards.. Skip values that don t place the values on the grid (if ou can spot the pattern). 3. Change the grid limits so that more calculated points fit in the plot. 13. Describe how the ais of smmetr of the parabola sitting on the -ais can be used to help plot the graph of ƒ () = a. The graph should look the same on either side of the ais, so once the positive portion of the parabola has been drawn, points on the parabola can be duplicated across the ais. If the point (, 5) is on the plot, for eample, then so is the point (-, 5). 1. Essential Question Check-In How can ou use the value of a to predict the shape of ƒ () = a without plotting points? The effect of the constant a is to either verticall stretch (if a > 1) or shrink (if a < 1) the graph compared to the parent function. A negative value of a causes the graph to open downwards. 1. Plot the function ƒ () = and g () = - on the grid. - Evaluate: Homework and Practice f() g() Which of the following features are the same and which are different for the two functions? a. Domain b. Range c. Verte d. Ais of smmetr e. Minimum f. Maimum Domain, verte, and ais of smmetr are the same. Range, maimum, and minimum are different. (f () doesn t have a maimum, and g () doesn t have a minimum.) Online Homework Hints and Help Etra Practice Houghton Mifflin Harcourt Publishing Compan EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore Understanding the Parent Quadratic Function Eample 1 Graphing g () = a when a > Eample Graphing g () = a when a < Eample 3 Writing a Quadratic Function Given a Graph Eample Modeling with a Quadratic Function Practice Eercises 1, 5 Eercises 5, 3 Eercises 9, Eercises 1 13 Eercises 1 1 INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Review the characteristics of a parabola with students. Show students how to use these characteristics to graph a parabola. Then show them how to interpret the graph of a quadratic function when it models a real-world relationship such as height over time. Module 1 Lesson 1 Eercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 1 1 Recall of Information MP. Modeling 9 Skills/Concepts MP. Modeling 1 13 Skills/Concepts MP. Reasoning 1 17 Skills/Concepts MP. Modeling 1 19 Skills/Concepts MP. Reasoning 1 Skills/Concepts MP. Reasoning Understanding Quadratic Functions 1

19 AVOID COMMON ERRORS Watch for students who do not graph points on both sides of the verte of the parabola. Remind these students that a parabola is U-shaped and smmetric, and the can use that smmetr to locate points on both sides of the verte. Graph each quadratic function. State the domain and range.. g () = 3. g () = 1_ D: all real; R: D: all real; R:. g () = 3 _ 5. g () = Houghton Mifflin Harcourt Publishing Compan D: all real; R:. g () = - 1_ 7. g () = D: all real; R: D: all real; R: D: all real; R: Module 19 Lesson 1 Eercise IN_MNLESE393_U3ML1.indd 19 Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices //1 1:3 AM 3 3 Strategic Thinking MP.3 Logic 5 3 Strategic Thinking MP. Reasoning 19 Lesson.1

20 . g () = - 3_ D: all real; R: 9. g () = -5 D: all real; R: Determine the equation of the parabola graphed (-1, 3) (, -) MULTIPLE REPRESENTATIONS Show students the graph of the function g () = -1.5, and instruct them to make a table showing several - and -values for both the given graph and the parent function ƒ () =. Have them compare the values in the table and describe how the values of g () are related to the values of ƒ () for an value of. Then have them write the equation for g (). The should find that for a given value of, the value of g () is -1.5 times the value of the parent function, so the can conclude that g () = Use the point (-1, 3). g () = a 3 = a (-1) 3 = a g () = (3, -) Use the point (3, -). g () = a - = a (3) - = 9a - _ 9 = a g () = - _ 9 Use the point (, -). g () = a - = a () - = 1a - 1_ = a g () = - 1_ - - Use the point (, 5). g () = a 5 = a () 5 = a 5_ = a g () = 5_ (, 5) Houghton Mifflin Harcourt Publishing Compan Module Lesson 1 IN_MNLESE393_U3ML1.indd //1 1:3 AM Understanding Quadratic Functions

21 INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Give students the graphs of two parabolas, f () = a and g () = a, that are smmetric about the -ais. Have them consider the sum of their a-values. Students should see that the values of a are opposites, so the sum of these values is. Have them verif that for an value of the sum of the function values is. A cannonball fired horizontall appears to travel in a straight line, but drops to earth due to gravit, just like an other object in freefall. The height of the cannonball in freefall is parabolic. The graph shows the change in height of the cannonball (in meters) as a function of distance traveled (in kilometers). Refer to this graph for questions 1 and 15. Height (m) - - h (., -5) Distance (km) d Houghton Mifflin Harcourt Publishing Compan Image Credits: (t) Brandon Alms/Alam; (b) Olegusk/Shutterstock 1. Describe what the verte, -intercept, and endpoint represent. 15. Find the function h (d) that describes these coordinates. The verte is the position of the cannon that Use the point (., -5). fired the cannonball. h (d) = ad The -intercept represents the height of the -5 = a (.) cannonball relative to the cannon at d =. -5 =.1a The endpoint is the end of the cannonball s = a trajector. h (d) = d A slingshot stores energ in the stretched elastic band when it is pulled back. The amount of stored energ versus the pull length is approimatel parabolic. Questions 1 and 17 refer to this graph of the stored energ in millijoules versus pull length in centimeters. Energ (mj) E (, ) Pull Length (cm) 1. Describe what the verte, -intercept, and endpoint represent. The verte is the point at which the slingshot is relaed and stores no energ. The -intercept is the energ, mj, when the pull length is cm at the beginning. The endpoint is at the maimum etent the slingshot is pulled back and the maimum stored energ. d 17. Determine the function, E (d), that describes this plot. Use the point (, ). E (d) = ad = a () = a 1_ = a E (d) = 1_ d Module 1 Lesson 1 1 Lesson.1

22 Newer clean energ sources like solar and wind suffer from unstead availabilit of energ. This makes it impractical to eliminate more traditional nuclear and fossil fuel plants without finding a wa to store etra energ when it is not available. One solution being investigated is storing energ in mechanical flwheels. Mechanical flwheels are heav disks that store energ b spinning rapidl. The graph shows how much energ is in a flwheel, as a function of revolution speed. Energ (kwh) 1 E (1, 1) 1 Rotation Speed (rps) r QUESTIONING STRATEGIES For a function in the form = a, where a, what is the relationship between the value of a and the graph of the function? The value of a determines the direction and shape of the parabola. If a >, the parabola opens upward; if a <, the parabola opens downward. If a > 1, the parabola is narrower than the graph of = ; if a < 1, the parabola is wider than the graph of =. 1. Describe what the verte, -intercept, and 19. Determine the function, E (r), endpoint represent. that describes this plot. The verte represents the flwheel at rest Use the point (1, 1). with no rotation and no stored energ. E (r) = ar The -intercept is the energ, kwh, when 1 = a (1) the rotation speed is rps at the beginning. 1 = 1,a The endpoint represents the maimum.1 = a rotation speed and energ storage. E (r) =.1 r Phineas is building a homemade skate ramp and wants to model the shape as a parabola. He sketches out a cross section shown in the graph. Height (feet) h (1, ) 5 1 Length (feet). Describe what the verte -intercept, and endpoint represent. The verte is the bottom of the ramp. The -intercept represents the height of the ramp relative to the length at l =. The endpoint is the highest point on the curved portion. l 1. Determine the function, h (l), that describes this plot. Use the point (1, ). h (l) = al = a (1). = a E (l) =. l Houghton Mifflin Harcourt Publishing Compan Image Credits: Tusumaru/Shutterstock Module Lesson 1 Understanding Quadratic Functions

23 JOURNAL In their journals, have students eplain how the graphs of = 5 and 1 5 compare to the graph of =. The graph of = 1 5 is compressed verticall and is the widest. The graph of = 5 is stretched verticall and is the narrowest. H.O.T. Focus on Higher Order Thinking. Multipart Classification f() g() - Mark the following statements about ƒ () = and g () = a as true or false. a. a > 1 b. a < c. a > d. a < e. a < 1 false true false false true f. The graphs of ƒ () and g () share a verte. true g. The graphs share an ais of smmetr. true h. The graphs share a minimum. false i. The graphs share a maimum. false 3. Check for Reasonableness The graph of g () = a is a parabola that passes through the point (-, ). Kle sas the value of a must be - 1. Eplain wh this value of a is not reasonable. When a is negative, the values cannot be positive. Since the value is positive, a must be positive. Houghton Mifflin Harcourt Publishing Compan. Communicate Mathematical Ideas Eplain how ou know, without graphing, what the graph of g () = 1 1 looks like. Compared to the graph of the parent function, f () =, the graph of g () would be wider (verticall compressed). It will open upwards because a is positive. 5. Critical Thinking A quadratic function has a minimum value when the function s graph opens upward, and it has a maimum value when the function s graph opens downward. In each case, the minimum or maimum value is the -coordinate of the verte of the function s graph. What can ou sa about a when the function ƒ () = a has a minimum value? A maimum value? What is the minimum or maimum value in each case? When f () has a minimum value, it means a >. When it has a maimum value, a <. In either case, the minimum or the maimum value will be. Module 3 Lesson 1 3 Lesson.1

24 Lesson Performance Task Klie made a paper helicopter and is testing its flight time from two different heights. The graph compares the height of the helicopter during the two drops. The graph of the first drop is labeled g () and the graph of the second drop is labeled h (). a. At what heights did Klie drop the helicopter? What is the helicopter s flight time during each drop? The intercepts are (, 1) and (, 3) so Klie dropped the helicopter from 1 feet and 3 feet. The intercepts are (1, ) and (1.5, ) so the flight times of the helicopter are 1 second and 1.5 seconds, respectivel. b. If each graph is represented b a function of the form ƒ () = a, are the coefficients positive or negative? Eplain. Both graphs open downward, so the coefficients are both negative. c. Estimate the functions for each graph. Height (ft) The graph of g () is the graph of f () = translated up 1 units. The parabola opens downward, so the coefficient a is negative. The intercept of the graph is (1, ) Solve for a. g () = a + b = a (1) + 1 = a = a The function for the first drop is g () = The graph of h () is the graph of f () = translated up 3 units. The parabola opens downward, so the coefficient a is negative. The intercept of the graph is (1.5, ) Solve for a. h () = a + b = a (1.5) + 3 =.5a =.5a -1 = a The function for the second drop is h () = Helicopter s Height Time (s) Houghton Mifflin Harcourt Publishing Compan QUESTIONING STRATEGIES What information to help ou write the functions do ou get when ou look at the graph? The graphs give the - and -intercepts of both curves and, because both curves open downward, a will have a negative value. INTEGRATED MATHEMATICAL PRACTICES Focus on Modeling MP. Discuss the difference between the two models shown. h =.9 t + h h = 1 t + h The first equation is given in metric units (h in meters) while the second is in English units (h in feet). Students should recognize that time in seconds is the onl unit of measure that is the same for both models. INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 Have students use graphing calculators to graph their two equations, ƒ () = and h () = , as a wa to check that the functions are good representations of the situation described. Module Lesson 1 EXTENSION ACTIVITY Have small groups of students investigate how to make a paper helicopter and then make their own. Then have students set up their own helicopter flight test to test two different height drops and their flight times. Finall, have students graph the results of their helicopter drops and write functions to represent each graph. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. points: Student does not demonstrate understanding of the problem. Understanding Quadratic Functions

25 COMMON CORE Locker LESSON Common Core Math Standards The student is epected to: COMMON CORE F-BF.B.3 Identif the effect on the graph of replacing f() b f() + k, kf(), f(k), and f( + k) for specific values of k (both positive and negative)... Also F-BF.A.1, F-BF.B., F-IF.B., F-IF.A. Mathematical Practices COMMON CORE. Transforming Quadratic Functions MP. Patterns Language Objective Students work in pairs or small groups to both give and listen to oral clues about graphs of quadratic functions. ENGAGE Essential Question: How can ou obtain the graph of g ( ) = a ( - h) + k from the graph of f () =? Possible answer: Identif the verte (h, k) and the sign of a to determine whether the graph opens up or down. Generate a few points on one side of the verte and sketch the graph using those points and smmetr. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the path of a ball used in sports can be modeled b a quadratic function. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing Compan Name Class Date. Transforming Quadratic Functions Essential Question: How can ou obtain the graph of g () = a ( - h) + k from the graph of f() =? Eplore Understanding Quadratic Functions of the Form g () = a ( - h) + k Ever quadratic function can be represented b an equation of the form g () = a ( - h) + k. The values of the parameters a, h, and k determine how the graph of the function compares to the graph of the parent function, =. Use the method shown to graph g () = ( - 3) + 1 b transforming the graph of ƒ () =. Graph ƒ () =. Stretch the graph verticall b a factor of graph of =. Graph =. Notice that point (, ) moves to point (, ). to obtain the Translate the graph of = right 3 units and up 1 unit to obtain the graph of g () = ( - 3) + 1. Graph g () = ( - 3) + 1. Notice that point (, ) moves to point (5, 9) The verte of the graph of ƒ () = (, ) is while the verte of the graph of g () = ( - 3) + 1 is (3, 1). - Resource Locker Module 5 Lesson Name Class Date. Transforming Quadratic Functions Essential Question: How can ou obtain the graph of g () = a ( - h) + k from the graph of f() =? Houghton Mifflin Harcourt Publishing Compan F-BF.B.3 Identif the effect on the graph of replacing f() b f() + k, kf(), f(k), and f( + k) for specific values of k (both positive and negative) Also F-BF.A.1, F-BF.B., F-IF.B., F-IF.A. Eplore Understanding Quadratic Functions of the Form g () = a ( - h) + k + k. The values of the Ever quadratic function can be represented b an equation of the form g () = a ( - h) parameters a, h, and k determine how the graph of the function compares to the graph of the parent function, =. Use the method shown to graph g () = ( - 3) + 1 b transforming the graph of ƒ () =. Graph ƒ () =. Stretch the graph verticall b a factor of to obtain the graph of =. Graph =. Notice that point (, ) moves to point. Translate the graph of = right 3 units and up 1 unit to obtain the graph of g () = ( - 3) + 1. Graph g () = ( - 3) + 1. Notice that point (, ) moves to point. The verte of the graph of ƒ () = is while the verte of the graph of g () = (3, 1) ( - 3) + 1 is. (, ) (5, 9) (, ) Resource HARDCOVER PAGES 5 3 Watch for the hardcover student edition page numbers for this lesson. Module 5 Lesson 5 Lesson.

26 Reflect 1. Discussion Compare the minimum values of ƒ () = and g () = ( - 3) + 1. How is the minimum value related to the verte? The minimum value of f () = is and the minimum value g () = ( - 3) +1 is 1. The minimum value is the -coordinate of the verte.. Discussion What is the ais of smmetr of the function g () = ( - 3) + 1? How is the ais of smmetr related to the verte? The ais of smmetr of g () = ( - 3) + 1 is = 3. The ais of smmetr alwas passes through the verte of the parabola. The -coordinate of the verte gives the equation of the ais of smmetr of the parabola. Eplain 1 Understanding Vertical Translations A vertical translation of a parabola is a shift of the parabola up or down, with no change in the shape of the parabola. Vertical Translations of a Parabola The graph of the function ƒ () = + k is the graph of ƒ () = translated verticall. If k >, the graph ƒ () = is translated k units up. If k <, the graph ƒ () = is translated k units down. Eample 1 g () = + Graph each quadratic function. Give the minimum or maimum value and the ais of smmetr. Make a table of values for the parent function f () = and for g () = +. Graph the functions together. f () = g () = The function g () = + has a minimum value of. The ais of smmetr of g () = + is =. - f() = Module Lesson PROFESSIONAL DEVELOPMENT Math Background In this lesson, students graph the famil of quadratic functions of the form g () = a ( - h) + k and compare those graphs to the graph of the parent function f () =. Some ke understandings are: The function f () = is the parent of the famil of all quadratic functions. To graph a quadratic function of the form g () = a ( - h) + k, identif the verte (h, k). Then determine whether the graph opens upward or downward. Then generate points on either side of the verte and sketch the graph of the function. Houghton Mifflin Harcourt Publishing Compan EXPLORE Understanding Quadratic Functions of the Form g ( ) = a ( - h) + k INTEGRATE TECHNOLOGY Students have the option of completing the activit either in the book or online. CONNECT VOCABULARY This lesson discusses translation in terms of a transformation of a function graph. English learners ma know about language translation. Discuss with students how the two meanings of translate are different. EXPLAIN 1 Understanding Vertical Translations QUESTIONING STRATEGIES How is the graph of g () = + related to the graph of g () = 5? Both are translated graphs of the same parent function, f () =, but g () = + is translated units up and g () = - 5 is translated 5 units down. So, the graph of g () = + is 7 units higher than the graph of g () = - 5. Is the verte of the graph of g() = + the same as the verte of the graph of g () = 5? No; g () = + has verte (, ), and g () = - has verte (, -5). Transforming Quadratic Functions

27 B g () = - 5 Make a table of values for the parent function ƒ () = and for g () = - 5. Graph the functions together. 1 f () = g () = The function g () = - 5 has a minimum value of -5. The ais of smmetr of g () = - 5 is =. Reflect 3. How do the values in the table for g () = + compare with the values in the table for the parent function ƒ () =? For each in the table, g () is greater than f (). Houghton Mifflin Harcourt Publishing Compan. How do the values in the table for g () = - 5 compare with the values in the table for the parent function ƒ () =? For each in the table, g () is 5 less than f (). Your Turn Graph each quadratic function. Give the minimum or maimum value and the ais of smmetr. 5. g () = + The function g () = + has a minimum value of. The ais of smmetr for g () = + is =. - - Module 7 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Have one student draw a graph of = + k for some value of k. The second student then writes the equation for the graph. Students then compare their results and determine whether the equation is correct for the given graph. Have students should take turns in the two roles. 7 Lesson.

28 . g () = - 7 The function g () = - 7 has a minimum value of -7. The ais of smmetr for g () = - 7 is = EXPLAIN Understanding Horizontal Translations Eplain Understanding Horizontal Translations A horizontal translation of a parabola is a shift of the parabola left or right, with no change in the shape of the parabola. Horizontal Translations of a Parabola The graph of the function ƒ () = ( h) is the graph of ƒ () = translated horizontall. If h >, the graph ƒ () = is translated k units right. If h <, the graph ƒ () = is translated k units left. Eample g () = ( - 1) Graph each quadratic function. Give the minimum or maimum value and the ais of smmetr. Make a table of values for the parent function ƒ () = and for g () = ( - 1). Graph the functions together. 1 - QUESTIONING STRATEGIES How is the graph of g () = ( 1) related to the graph of g () = ( + )? Both are translated graphs of the same parent function, f () =, but the graph of g () = ( - 1) is translated 1 unit to the right and has verte (1, ), while the graph of g () = ( + ) is translated units to the left and has verte (-, ). So, the graph of g () = ( - 1) is 3 units to the right of the graph of g() = ( - 1). What is the verte of the graph of g () = ( h)? (h, ) f () = g () = ( -1) The function g () = ( - 1) has a minimum value of Houghton Mifflin Harcourt Publishing Compan The ais of smmetr of g () = ( - 1) is = 1. Module Lesson DIFFERENTIATE INSTRUCTION Visual Cues Have students take a coordinate grid and label it Verte of g () = ( h) + k. Have them place these points, labels, and functions into the four quadrants. (, 3) h =, k = 3 g () = ( - ) + 3 (-, 3) h = -, k = 3 g () = ( + ) + 3 (, 3) h =, k = 3 g () = ( + ) 3 (, 3) h =, k = 3 g () = ( ) 3 Students can use this graph as a reminder of how the location of the verte and the function are related. Transforming Quadratic Functions

29 AVOID COMMON ERRORS Students ma forget that the can use a pattern to write equations from graphs. Remind students that adding k to moves the graph up for k > or down for k < and that subtracting h from moves the graph left for h < or right for h >. This is true for all nonzero values of k and h. B g () = ( +1) Make a table of values and graph the functions together. f () = g () = ( + 1) The function g () = ( +1) has a minimum value of. The ais of smmetr of g () = ( + 1) is = -1. Reflect 7. How do the values in the table for g () = ( - 1) compare with the values in the table for the parent function ƒ () =? For each in the table, g () is the same as f ( - 1).. How do the values in the table for g () = ( + 1) compare with the values in the table for the parent function ƒ () =? For each in the table, g () is the same as f ( + 1). Houghton Mifflin Harcourt Publishing Compan Your Turn Graph each quadratic function. Give the minimum or maimum value and the ais of smmetr. 9. g () = ( - ) g () = ( + 3) Module 9 Lesson LANGUAGE SUPPORT Communicate Math Have each student sketch a graph of a parabola on a card, and write a quadratic function in an form on another card. Then have them write clues about the graph and about the function. For eample, The parabola opens upward/downward. Its ais of smmetr is ; the verte is at the point. The function s graph will open downward/upward. Provide sentence stems if needed to help students begin their clues. Collect the graph and function cards in one pile, and the clue cards in another. Have other students match graph and function cards to fit the clues. 9 Lesson.

30 Eplain 3 Graphing g () = a ( - h) + k The verte form of a quadratic function is g () = a ( - h) + k, where the point (h, k) is the verte. The ais of smmetr of a quadratic function in this form is the vertical line = h. To graph a quadratic function in the form g () = a ( - h) + k, first identif the verte ( h, k ). Net, consider the sign of a to determine whether the graph opens upward or downward. If a is positive, the graph opens upward. If a is negative, the graph opens downward. Then generate two points on each side of the verte. Using those points, sketch the graph of the function. Eample 3 Graph each quadratic function. g () = -3 ( + 1) - Identif the verte. The verte is at (-1, -). Make a table for the function. Find two points on each side of the verte g () Plot the points and draw a parabola through them. g () = ( - 1) - 7 Identif the verte. The verte is at (1, -7). Make a table for the function. Find two points on each side of the verte. - 1 g () Plot the points and draw a parabola through them. Reflect How do ou tell from the equation whether the verte is a maimum value or a minimum value? If the value of a is positive, the verte is a minimum value. If the value of a is negative, the verte is a maimum value. Module 3 Lesson 3 Houghton Mifflin Harcourt Publishing Compan EXPLAIN 3 Graphing g () = a ( - h) + k QUESTIONING STRATEGIES What can ou tell about the graph of g () from its equation? the location of the verte and the fact that the graph opens downward Are the domain and range both the set of all real numbers? The domain is all real numbers, but the range is the set of all function values less than or equal to the maimum value. AVOID COMMON ERRORS Students ma tr to graph a quadratic function of the form g () = a ( - h) + k b using a value other than = h. Remind them that the need to first identif and plot the verte. Then the should identif and plot other points and use the plotted points to draw a parabola. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Tell students that a transformed quadratic function models the height of an object dropped from a given height, based upon the time since it was dropped. Sketch a quadratic function that models the situation, and draw students attention to the verte (, k) being the maimum point of the graph. Ask about the sign of a in the function g () = a + k, and note that the values to the left of the -ais are not considered. Transforming Quadratic Functions 3

31 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 Give students a function in the form g () = a ( h) + k. Have students use the whiteboard to identif and plot the verte and then identif and plot other points on the graph before drawing the graph of the function. SUMMARIZE How do ou graph a quadratic function of the form g () = a ( h) + k? First, identif and plot the verte. Then, identif and plot other points on the graph. Finall, draw the graph. Your Turn Graph each quadratic function. 1. g () = -( - ) g () = ( + 3) g () -5-5 g () Elaborate 1. How does the value of k in g () = + k affect the translation of ƒ () =? If k >, the graph f () = is translated k units up. If k <, the graph f () = is translated k units down. Houghton Mifflin Harcourt Publishing Compan 15. How does the value of h in g () = ( - h) affect the translation of ƒ () =? If h >, the graph f () = is translated h units right. If h <, the graph f () = is translated h units left. 1. In g () = a ( - h) + k, what are the coordinates of the verte? (h, k) 17. Essential Question Check-In How can ou use the values of a, h, and k, to obtain the graph of g () = a ( - h) + k from the graph ƒ () =? The graph of f () = is stretched or compressed b a factor of a, and reflected across the -ais if a is negative; it is translated h units horizontall and k units verticall. Module 31 Lesson 31 Lesson.

32 Evaluate: Homework and Practice Graph each quadratic function b transforming the graph of ƒ () =. Describe the transformations. 1. g () = ( - ) + 5. g () = ( + 3) - Online Homework Hints and Help Etra Practice EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore Understanding Quadratic Functions of the Form g ( ) = a ( - h) + k Practice Eercises The parent function has been translated units right and 5 units up. It has been stretched verticall b a factor of The parent function has been translated 3 units left and units down. It has been stretched verticall b a factor of. Eample 1 Understanding Vertical Translations Eample Understanding Horizontal Translations Eample 3 Graphing g () = a ( - h) + k Eercises 5 1 Eercises 11 1 Eercises g () = - 1_ ( - 3) -. g () = 3 ( - ) Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Make sure that students understand where h, k, and a come from. Give coordinates for a verte and have students substitute the - and -values of the verte into the equation of h and k, determine the value of a, and then write the equation of the function. - The parent function has been translated 3 units right and units down. It has been stretched horizontall b a factor of. The parent function has been translated units right and units down. It has been stretched verticall b a factor of 3. Module 3 Lesson Eercise IN_MNLESE393_U3ML.indd 3 Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices //1 1:3 AM 1 1 Recall of Information MP. Precision 9 1 Skills/Concepts MP.5 Using Tools 13 1 Skills/Concepts MP. Reasoning 17 Skills/Concepts MP. Precision 1 Skills/Concepts MP. Reasoning Strategic Thinking MP.3 Logic 7 3 Strategic Thinking MP. Reasoning Transforming Quadratic Functions 3

33 VISUAL CUES Graph each quadratic function. 5. g () = -. g () = + 5 Have students create a design made of transformed parabolas and keep a record of each parabola s function. Encourage students to use their imaginations to add colors and patterns to the design. INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 Allow students to use graphing calculators to check their work. Some students will be motivated to eplore additional tpes of transformations g () = -. g () = Graph g () = - 9. Give the minimum or maimum value and the ais of smmetr. Houghton Mifflin Harcourt Publishing Compan The function has a minimum value of -9. The ais of smmetr is =. 1. How is the graph of g () = + 1 related to the graph of ƒ () =? The graph of g () = + 1 is the graph of f () = translated 1 units up. Module 33 Lesson 33 Lesson.

34 Graph each quadratic function. Give the minimum or maimum value and the ais of smmetr. 11. g () = ( - 3) AVOID COMMON ERRORS Some students ma automaticall sa that the function has a minimum when a parabola opens downward, and a maimum when a parabola opens upward, because of word association. Tell students to visualize the graph before determining whether it has a minimum or maimum The function has a minimum value of. The ais of smmetr is = g () = ( + ) The function has a minimum value of. The ais of smmetr is = -. Houghton Mifflin Harcourt Publishing Compan 13. How is the graph of g () = ( + 1) related to the graph of ƒ () =? The graph of g () = ( + 1) is the graph of f () = translated 1 units left. 1. How is the graph of g () = ( - 1) related to the graph of ƒ () =? The graph of g () = ( - 1) is the graph of f () = translated 1 units right. Module 3 Lesson Transforming Quadratic Functions 3

35 KINESTHETIC EXPERIENCE Displa each function below, one at a time. Have students discuss, in pairs, whether to lift their arms up in the shape of a U to signal the graph opens upward, or move them downward in the shape of an upside-down U, to signal that the graph opens downward. Then have students demonstrate their decisions. 15. Compare the given graph to the graph of the parent function ƒ () =. Describe how the parent function must be translated to get the graph shown here. Translate the graph of the parent function units to the right = down = down - + = -5 up 3 - = - up 1. For the function g () = ( - 9) give the minimum or maimum value and the ais of smmetr. The minimum value is. The ais of smmetr is = 9. Graph each quadratic function. Give the minimum or maimum value and the ais of smmetr. 17. g () = ( - 1) g () = - ( + ) Houghton Mifflin Harcourt Publishing Compan g () The function has a minimum value of -5. The ais of smmetr is = g () The function has a maimum value of 5. The ais of smmetr is = -. Module 35 Lesson 35 Lesson.

36 19. g () = 1_ ( + 1) - 7. g () = - 1_ 3 ( + 3) PEER-TO-PEER ACTIVITY Have students work in pairs. Have each student change one or more of the parameters in f () = a ( h) + k and graph the function. Then have students trade graphs and tr to write the function equation for the other student s graph. Have each student justif the function equation and discuss it with the partner g () The function has a minimum value of -7. The ais of smmetr is = g () The function has a maimum value of. The ais of smmetr is = Compare the given graph to the graph of the parent function ƒ () =. Describe how the parent function must be translated to get the graph shown here. Translate the graph of the parent function 3 units to the right and units up Multiple Representations Write an equation for the function represented b the graph of a parabola that is a translation of ƒ () =. The graph has been translated 11 units to the left and 5 units down. a. g () = ( - 11) - 5 b. g () = ( + 11) - 5 c. g () = ( + 11) + 5 d. g () = ( - 11) + 5 e. g () = ( - 5) - 11 f. g () = ( - 5) + 11 g. g () = ( + 5) - 11 h. g () = ( + 5) + 11 Houghton Mifflin Harcourt Publishing Compan Module 3 Lesson IN_MNLESE393_U3ML.indd 3 //1 1:3 AM Transforming Quadratic Functions 3

37 JOURNAL H.O.T. Focus on Higher Order Thinking Critical Thinking Use a graphing calculator to compare the graphs of = (), = (3), and = () with the graph of the parent function =. Then compare In their journals, have students eplain how to use the values of a, h, and k to obtain the graph of g() = a( h) + k from the graph of f() =. The graph of f() = is stretched or compressed b a factor of a, and reflected across the -ais if a is negative; it is translated h units horizontall and k units verticall. ( ) ( ) ( ) the graphs of = 1, = 13, and = 1 with the graph of the parent function =. 3. Eplain how the parameter b horizontall stretches or compresses the graph of = (b) when b > 1? When b > 1, the graph of = (b) is compressed horizontall b a factor 1. of b. Eplain how the parameter b horizontall stretches or compresses the graph of = (b) when < b < 1? When < b < 1, the graph of = (b) is stretched horizontall b a 1. factor of b 5. Eplain the Error Nina is tring to write an equation for the function represented b the graph of a parabola that is a translation of ƒ() =. The graph has been translated units to the right and units up. She writes the function as g() = ( + ) +. Eplain the error. Nina should have subtracted from in the equation instead of adding it.. Multiple Representations A group of engineers drop an eperimental tennis ball from a catwalk and let it fall to the ground. The tennis ball s height above the ground (in feet) is given b a function of the form ƒ(t) = a(t - h) + k where t is the time (in seconds) after the tennis ball was dropped. Use the graph to find the equation for ƒ(t). 3 Houghton Mifflin Harcourt Publishing Compan 1 t 1 3 f(t) = a(t - ) + 3, or f(t) = at = a(1) = a The equation for the function is f(t) = -1t Make a Prediction For what values of a and c will the graph of ƒ() = a + c have one -intercept? For an real value of a with a, the function will have one -intercept when c =. Module IN_MNLESE393_U3ML.indd Lesson. 37 Lesson //1 1:3 AM

38 Lesson Performance Task The path a baseball takes after it has been hit is modeled b the graph. The baseball s height above the ground is given b a function of the form ƒ (t) = a (t - h) + k, where t is the time in seconds since the baseball was hit. a. What is the baseball s maimum height? At what time was the baseball at its maimum height? b. When does the baseball hit the ground? c. Find an equation for ƒ (t). d. A plaer hits a second baseball. The second baseball s path is modeled b the function g (t) = -1 (t - ) + 5. Which baseball has a greater maimum height? Which baseball is in the air for the longest? a. The verte of the parabola is (3, 1). So, the baseball is at its maimum height of 1 feet after 3 seconds. b. The second -intercept of the graph is (, ). So, the baseball hits the ground after seconds. c. The verte of the parabola is (3, 1) and one intercept of the graph is (, ). Solve for a. Height (ft) Baseball s Height Time (s) INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP. Discuss with students which tpe of hit a ground ball, a pop-up, or a line drive would likel make a baseball have the path shown in the graph. Ask how knowing the maimum height of the ball and the time it takes the ball to hit the ground help ou write an equation to represent the path of the baseball. The highest height of the ball, the verte of the path, is the ordered pair (h, k), and the time it takes the ball to hit the ground is the ordered pair (t, f(t)), so the h, k, t, and f(t) values can be substituted into the standard form f (t) = a (t - h) + k to find the value of a. f (t) = a (t - h) + k = a ( - 3) = 9a -1 = a So, f (t) = -1 (t - 3) + 1. d. The verte is (,5) so the baseball was at its maimum height of 5 feet after seconds. g (t) = -1 (t-) + 5 = -1 (t-) = -1 (t-) 1 = (t-) 1 = t- ± + = t, = t The ball hits the ground after seconds. So, the second baseball has a greater maimum height and it traveled longer in the air than the first. Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Point out that the model students write in the form f (t) = a (t - h) + k to represent the path the baseball takes can be used to approimate the height h in feet above the ground after t seconds because it does not account for air resistance, wind, or other real-world factors. Module 3 Lesson EXTENSION ACTIVITY Have groups of students draw or tape a large, first-quadrant coordinate grid on the chalkboard. Have one student toss a tennis ball in front of the grid, making sure that the path of the ball stas within the grid s borders, while another student videotapes the toss at a rate of about 15 frames per second. Then have students pla back the video, marking points on the grid to show the path of the ball. Finall, have students use the model f (t) = a (t - h) + k to write a function that models the path of the tennis ball. S tudents ma discover that the angle at which the ball is tossed affects the height and width of the curved path the ball follows. Have students save their data for Part of the Etension Activit in the following lesson. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. points: Student does not demonstrate understanding of the problem. Transforming Quadratic Functions 3

39 COMMON CORE Locker LESSON.3 Interpreting Verte Form and Standard Form Name Class Date.3 Interpreting Verte Form and Standard Form Essential Question: How can ou change the verte form of a quadratic function to standard form? Resource Locker Common Core Math Standards The student is epected to: COMMON CORE F-IF.B. For a function that models a relationship between two quantities, interpret ke features of graphs Also F-IF.C., F-IF.A., F-IF.B., F-BF.A.1 Mathematical Practices COMMON CORE MP. Reasoning Language Objective Work with a partner to describe how to write quadratic functions in verte form and standard form. Eplore Identifing Quadratic Functions from Their Graphs Determine whether a function is a quadratic function b looking at its graph. If the graph of a function is a parabola, then the function is a quadratic function. If the graph of a function is not a parabola, then the function is not a quadratic function. Use a graphing calculator to graph each of the functions. Set the viewing window to show -1 to 1 on both aes. Determine whether each function is a quadratic function. Use a graphing calculator to graph ƒ () = + 1. Determine whether the function ƒ () = + 1 is a quadratic function. ENGAGE Essential Question: How can ou change the verte form of a quadratic function to standard form? Sample answer: You can rewrite the quadratic epression in the verte form b multipling and then combining like terms so that the function rule is written in descending order of the eponents. Houghton Mifflin Harcourt Publishing Compan The function ƒ () = + 1 is not a quadratic function. Use a graphing calculator to graph ƒ () = + -. Determine whether the function ƒ () = + - is a quadratic function. The function ƒ () = + - Use a graphing calculator to graph ƒ () =. is a quadratic function. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and the tpe of data that is needed in order to represent the path of the tennis ball as a function. Then preview the Lesson Performance Task. Determine whether the function ƒ () = is a quadratic function. The function ƒ () = is not a quadratic function. Module 39 Lesson 3 Name Class Date.3 Interpreting Verte Form and Standard Form Essential Question: How can ou change the verte form of a quadratic function to standard form? F-IF.B. For a function that models a relationship between two quantities, interpret ke features of graphs Also F-IF.C., F-IF.A., F-IF.B., F-BF.A.1 Eplore Identifing Quadratic Functions from Their Graphs Determine whether a function is a quadratic function b looking at its graph. If the graph of a function is a parabola, then the function is a quadratic function. If the graph of a function is not a parabola, then the function is not a quadratic function. Use a graphing calculator to graph each of the functions. Set the viewing window to show -1 to 1 on both aes. Determine whether each function is a quadratic function. Use a graphing calculator to graph ƒ () = + 1. Determine whether the function ƒ () = + 1 is a quadratic function. is not The function ƒ () = + 1 a quadratic function. Use a graphing calculator to graph ƒ () = + -. Resource HARDCOVER PAGES 39 5 Watch for the hardcover student edition page numbers for this lesson. Houghton Mifflin Harcourt Publishing Compan Determine whether the function ƒ () = + - is a quadratic function. The function ƒ () = + - a quadratic function. function. Use a graphing calculator to graph ƒ () =. Determine whether the function ƒ () = is a quadratic is not The function ƒ () = a quadratic function. is Module 39 Lesson 3 39 Lesson.3

40 Use a graphing calculator to graph ƒ () = - 3. Determine whether the function ƒ () = - 3 is a quadratic function. The function ƒ () = - 3 a quadratic function. Use a graphing calculator to graph ƒ () = - ( - 3) + 7. Determine whether the function ƒ () = -( - 3) + 7 is a quadratic function. The function ƒ () = -( - 3) + 7 is a quadratic function. Use a graphing calculator to graph ƒ () =. Determine whether the function ƒ () = is a quadratic function. The function ƒ () = is not is = a quadratic function. EXPLORE Identifing Quadratic Functions from Their Graphs INTEGRATE TECHNOLOGY Students have the option of completing the activit either in the book or online. INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP. Point out to students that the graph of a parabola is smmetric about a vertical line through its verte. Reflect 1. How can ou determine whether a function is quadratic or not b looking at its graph? Sample answer: All quadratic functions have graphs that are parabolas, opening either upward or downward. If the graph of a function does not have this shape, then the function is not quadratic.. Discussion How can ou tell if a function is a quadratic function b looking at the equation? Sample answer: If the highest power of the variable is, then the function is quadratic. Houghton Mifflin Harcourt Publishing Compan Module Lesson 3 PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunit to address Mathematical Practice MP., which calls for students to reason abstractl and quantitativel. Students analze the relationship between quadratic functions in standard and verte forms and convert between verte form and standard form. Interpreting Verte Form and Standard Form

41 EXPLAIN 1 Identifing Quadratic Functions in Standard Form QUESTIONING STRATEGIES How can ou tell b looking at an equation whether the value of a in = a + b + c is? If there is no squared term, a =. If the values of a, b, and c in = a + b + c are all positive, what do ou know about the graph? The graph opens up because a is positive; and the verte is to the left of the -ais because b - is negative. a INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Point out to students that an equation must be solved for in terms of before the determine whether the equation represents a quadratic function. For eample + = does not represent a quadratic function. Even though there is an term in the equation, when the equation is solved for and simplified, there is no term. Houghton Mifflin Harcourt Publishing Compan Eplain 1 Identifing Quadratic Functions in Standard Form If a function is quadratic, it can be represented b an equation of the form = a + b + c, where a, b, and c are real numbers and a. This is called the standard form of a quadratic equation. The ais of smmetr for a quadratic equation in standard form is given b the equation = - b_. The verte of a a quadratic equation in standard form is given b the coordinates ( - b_, ƒ ( - b_ Eample 1 = - + Determine whether the function represented b each equation is quadratic. If so, give the ais of smmetr and the coordinates of the verte. = - + Compare to = a + b + c. This is not a quadratic function because a =. + 3 = - Rewrite the function in the form = a + b + c. = -3 - Compare to = a + b + c. This is a quadratic function. If + 3 = - is a quadratic function, give the ais of smmetr. = If + 3 = - is a quadratic function, give the coordinates of the verte. (, -) Reflect 3. Eplain wh the function represented b the equation = a + b + c is quadratic onl when a. If a =, there is no -squared term and all quadratic equations include an -squared term.. Wh might it be easier to determine whether a function is quadratic when it is epressed in function notation? All the terms are alread on one side of the equal sign; the onl need to be arranged in standard form. 5. How is the ais of smmetr related to standard form? The ais of smmetr for a quadratic equation in standard form is given b the equation b = - a, where a and b are constant in = a + b + c. Your Turn Determine whether the function represented b each equation is quadratic.. - = = 7. + = 1 + = - + = 13_ = is a quadratic function. ) ). Compare = 13_ + 3 to = a + b + c. This is not a quadratic function because a =. Module 1 Lesson 3 COLLABORATIVE LEARNING Peer to Peer Activit Have students work with a partner. Have each partner make up a quadratic function in verte form. Then have students echange papers and convert the verte form to standard form, showing each step in the solution. Encourage students to repeat this several times, using different signs and reorganizing the equation to make it more difficult. For eample, Convert - 3 = ( - 5) to standard form. = ( - 5) + 3; = ; = Lesson.3

42 Eplain Changing from Verte Form to Standard Form It is possible to write quadratic equations in various forms. Eample Rewrite a quadratic function from verte form, = a ( - h) + k, to standard form, = a + b + c. = ( - ) + 3 = ( ) + 3 Epand ( - ). = = Multipl. Simplif. The standard form of = ( - ) + 3 is = = -3 ( + ) - 1 = -3 ( ) - 1 Epand ( + ). = Multipl. = Simplif. The standard form of = ( - ) + 3 is = Reflect If in = a ( - h) + k, a = 1, what is the simplified form of the standard form, = a + b + c? = + b + c EXPLAIN Changing from Verte Form to Standard Form QUESTIONING STRATEGIES When is the verte form of a quadratic function useful? When is standard form useful? Verte form makes it eas to identif the verte, (h, k), and the ais of smmetr, = h. Standard form makes it eas to identif the -intercept, c. How do ou change the verte form to standard form? Use the FOIL method to epand and simplif the binomial square. What value is the same in verte form as in standard form? Wh? The value of a. If ou epand a ( - h) + k, the coefficient of is a. Your Turn Rewrite a quadratic function from verte form, = a ( - h) + k, to standard form, = a + b + c. 9. = ( + 5) = -3( - 7) + = ( ) + 3 = = The standard form of = ( - ) + 3 is = = -3 ( ) + = = The standard form of = ( - ) + 3 is = Houghton Mifflin Harcourt Publishing Compan MODELING Have students consider what value is the same in verte form as in standard form. Students comfortable with manipulating variables and constants might tr to epand a ( - h) + k to determine the values of a, b, and c in terms of a, h, and k. The value of a is the same. If ou epand a ( - h ) + k, the coefficient of is a: Module Lesson 3 DIFFERENTIATE INSTRUCTION Technolog Have students set Xmin = and Ymin = when graphing the eample on a graphing calculator. Yma should be set to slightl above the maimum height. To generate a table from the function, set TblStart to. a ( - h ) + k a ( - h + h ) + k a - ah + ah + k a - ah + ah + k So, a = a, b = ah, and c = ah + k. Interpreting Verte Form and Standard Form

43 EXPLAIN 3 Writing a Quadratic Function Given a Table of Values QUESTIONING STRATEGIES What do ou look for in a table of values in order to decide that it is quadratic? A minimum or maimum value of, indicating the verte and ais of smmetr, and values of that are equal for -values that are the same distance from the verte. Eplain 3 Writing a Quadratic Function Given a Table of Values You can write a quadratic function from a table of values. Eample 3 Use each table to write a quadratic function in verte form, = a ( - h ) + k. Then rewrite the function in standard form, = a + b + c. The verte of the function is (-3, ) The verte of the parabola is (-3, ). Substitute the values for h and k into = a ( - h) + k. = a ( - (-3)) +, or = a ( + 3) Use an point from the table to find a. = a ( + 3) 1 = a (- + 3) = a The verte form of the function is = 1 ( - (-3)) + or = ( + 3). Rewrite the function = ( + 3) in standard form, = a + b + c = ( + 3) = AVOID COMMON ERRORS b Some students ma overlook the negative sign in - a when calculating the ais of smmetr. Suggest that students double check that the included the negative sign before finding the verte of the graph. The standard form of the function is = The verte of the function is (-, -3). The verte of the parabola is is (-, -3). Substitute the values for h and k into = a ( -h) + k. = a ( + ) - 3 Use an point from the table to find a. a = Houghton Mifflin Harcourt Publishing Compan The verte form of the function is = ( + ) - 3. Rewrite the resulting function in standard form, = a + b + c. = Reflect 11. How man points are needed to find an equation of a quadratic function? two points, the verte and one other point Module 3 Lesson 3 LANGUAGE SUPPORT Communicate Math Hand out cards to pairs of students with three numbers between -5 and 5 for a, h, and k; for eample a =, h = -, and k = 1. Have one student describe to the other how to write the function in verte form, and then convert it to standard form. Then have them reverse roles and use the opposites of the numbers (in this eample, a = -, h =, and k = -1) to write a new function in verte form, and convert that to standard form. Then have students describe how the graphs of these functions would be related to each other. 3 Lesson.3

44 Your Turn Use each table to write a quadratic function in verte form, = a ( - h) + k. Then rewrite the function in standard form, = a + b + c. 1. The verte of the function is (, 5) = a ( - ) = a (1 - ) + 5 = a (-1) = a verte form = ( - ) + 5. = ( - + ) + 5 = = standard form = The verte of the function is (-, -7). = a ( + ) = a (-3 + ) = a (-1) -5 = a verte form = -5 ( + ) - 7. = -5 ( + + ) - 7 = = standard form = EXPLAIN Writing a Quadratic Function Given a Graph QUESTIONING STRATEGIES In order to find the equation of the graph of an object that is dropped and falls from a height over time, what values can ou use? the coordinates of the -intercept, which represents the verte, and the coordinates of another point on the curve Eplain Writing a Quadratic Function Given a Graph The graph of a parabola can be used to determine the corresponding function. Eample Use each graph to find an equation for ƒ (t). A house painter standing on a ladder drops a paintbrush, which falls to the ground. The paintbrush s height above the ground (in feet) is given b a function of the form ƒ (t) = a (t - h) where t is the time (in seconds) after the paintbrush is dropped. The verte of the parabola is (h, k) = (, 5). ƒ (t) = a ( - h) + k ƒ (t) = a (t - ) + 5 ƒ (t) = at + 5 Use the point (1, 9) to find a. Height (feet) 3 1 (, 5) (1, 9) 1 3 Time (seconds) Houghton Mifflin Harcourt Publishing Compan ƒ (t) = at = a (1) = a The equation for the function is ƒ (t) = -1t + 5. Module Lesson 3 Interpreting Verte Form and Standard Form

45 VISUAL CUES Remind students that, when graphing a curve of an object in free fall, the graph represents the distance the object is above the surface and not the path of the object. A rock is knocked off a cliff into the water far below. The falling rock s height above the water (in feet) is given b a function of the form ƒ (t) = a (t - h) + k where t is the time (in seconds) after the rock begins to fall. Height (feet) 5 3 (, ) (1, ) Time (seconds) The verte of the parabola is (h, k) = (, ). ƒ (t) = a (t - h) + k ƒ (t) = a ( t - ) +. ƒ (t) = -a t + Use the point (1, ) to find a. Houghton Mifflin Harcourt Publishing Compan Image Credits: Radius Images/Corbis ƒ (t) = at + = a 1 + a = -1 The equation for the function is ƒ (t) = -1t +. Reflect 1. Identif the domain and eplain wh it makes sense for this problem. The domain is t. Time cannot be negative. 15. Identif the range and eplain wh it makes sense for this problem. The range is f (t). Height cannot be negative. Module 5 Lesson 3 5 Lesson.3

46 Your Turn 1. The graph of a function in the form ƒ () = a ( - h) + k, is shown. Use the graph to find an equation for ƒ (). The verte of the parabola is (h, k) = (1, 1). f () = a ( - 1) + 1 From the graph f (3) = -3 Substitute 3 for and -3 for f () and solve for a. -3 = a (3-1) = a () - = a -1 = a The equation for the function is f () = -( - 1) + 1. (1, 1) (3, -3) A roofer accidentall drops a nail, which falls to the ground. The nail s height above the ground (in feet) is given b a function of the form ƒ (t) = a (t - h) + k, where t is the time (in seconds) after the nail drops. Use the graph to find an equation for ƒ (t). Height (feet) (, 5) (1, 9) t Time (seconds) The verte of the parabola is (h, k) = (, 5). f (t) = a (t - ) + 5, or f (t) = a t + 5 From the graph f (1) = 9. Substitute 1 for t and 9 for f (t) and solve for a. 9 = a (1) = a The equation for the function is f (t) = -1 t + 5. Houghton Mifflin Harcourt Publishing Compan Image Credits: Minik/ Shutterstock Module Lesson 3 Interpreting Verte Form and Standard Form

47 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Have two students hold the ends of a jump rope at the same height from the ground. Use a measuring tape to place the students feet, feet, and feet apart. Have the students verif whether the lowest point of the jump rope is alwas halfwa between the students holding the rope. Students can graph the height of the rope based on the distance from either end and determine appropriate quadratic functions. Mention that the actual shape of the curve is called a catenar, and that a quadratic model is appropriate. Elaborate 1. Describe the graph of a quadratic function. parabola 19. What is the standard form of the quadratic function? = a + b + c. Can an quadratic function in verte form be written in standard form? es 1. How man points are needed to write a quadratic function in verte form, given the table of values? two points; verte and another point. If a graph of the quadratic function is given, how do ou find the verte? Look for the minimum or maimum value on the graph. 3. Essential Question Check-In What can ou do to change the verte form of a quadratic function to standard form? Sample answer: You can rewrite the quadratic epression in the verte form b multipling and then combine like terms so that the function rule is written in descending order of the eponents. Evaluate: Homework and Practice SUMMARIZE THE LESSON How can ou change the verte form of a quadratic function to standard form? You can rewrite the quadratic epression in the verte form b multipling and then combining like terms so that the function rule is written in descending order of the eponents. Houghton Mifflin Harcourt Publishing Compan Determine whether each function is a quadratic function b graphing. Online Homework Hints and Help 1. ƒ () = ƒ () = 1_ - Etra Practice The graph is a parabola. It is quadratic. The graph is not a parabola. It is not quadratic. 3. ƒ () = ƒ () = The graph is a parabola. It is quadratic. The graph is not a parabola. It is not quadratic. Determine whether the function represented b each equation is quadratic. 5. = = No, a = = + Yes, a, b, and c are real numbers =. + = 1 = = Yes, a, b, and c are real numbers. No, a = 9. Which of the following functions is a quadratic function? Select all that appl. a. = + 3 d. + = b. + = 3-1 e. - = c. 5 = For ƒ () = + - 1, give the ais of smmetr and the coordinates of the verte. b - a = - (1) = - f (-) = -3 The verte is at (-, -3). The ais of smmetr is = -. Module 7 Lesson 3 7 Lesson.3

48 11. Describe the ais of smmetr of the graph of the quadratic function represented b the equation = a + b + c. when b =. The ais of smmetr is the -ais, or =. EVALUATE Rewrite each quadratic function from verte form, = a ( - h) + k, to standard form, = a + b + c. 1. = 5 ( - ) = - ( + ) - 11 = 5 ( - + ) + 7 = = The standard form of = 5 ( - ) + 7 is = = 3 ( + 1) = - ( - 3) - 9 = 3 ( + + 1) + 1 = = The standard form of = 3 ( + 1) + 1 is = Eplain the Error Tim wrote = - ( + ) - 1 in standard form as = Find his error. = - ( + + ) - 1 = = = - ( + + 1) - 11 = = How do ou change from verte form, ƒ () = a ( - h) + k, to standard form, = a + b + c? The standard form of = - ( + ) - 11 is = = - ( - + 9) - 9 = = The standard form of = - ( + ) - 1 is = Tim multiplied b instead of -. Sample answer: Epand the squared term. Then distribute the a-value. Finall, combine like terms. The standard form of = - ( - 3) - 9 is = Houghton Mifflin Harcourt Publishing Compan ASSIGNMENT GUIDE Concepts and Skills Eplore Identifing Quadratic Functions from Their Graphs Eample 1 Identifing Quadratic Functions in Standard Form Eample Changing from Verte Form to Standard Form Eample 3 Writing a Quadratic Function Given a Table of Values Eample Writing a Quadratic Function Given a Graph QUESTIONING STRATEGIES Practice Eercises 1 Eercises 5 11 Eercises 1 17 Eercises 1 Eercises 3 5 When ou graph the points of a quadratic function, wh are the points connected b a curve rather than straight line segments? The values of a quadratic function change constantl and smoothl, but the rate of change is not constant. A quadratic function results in a U-shaped graph. Module Lesson 3 Eercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 1 Skills/Concepts MP.5 Using Tools 5 11 Skills/Concepts MP. Precision 1 1 Skills/Concepts MP. Reasoning 3 Strategic Thinking MP. Modeling 3 3 Strategic Thinking MP. Precision Skills/Concepts MP. Modeling 5 3 Strategic Thinking MP. Reasoning Interpreting Verte Form and Standard Form

49 QUESTIONING STRATEGIES Is the function represented b the equation - 1 = 9 quadratic? Eplain. Yes, because the equation can be written in the form = a + b + c, where a = 1, b =, and c = 9. INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP. Remind students that in a table of quadratic values, the verte can be identified b looking for the single least or greatest function value. If function values appear more than once, the represent points that are smmetric about the verte; the verte will be halfwa between them. AVOID COMMON ERRORS Students ma make a sign error in converting from verte form when the value of a is negative and the squared value is in the form - h where h is positive. Remind students that when squaring a term like - 3, the result has a negative coefficient of and a positive constant term, so multipling b a negative value for a reverses those signs. Houghton Mifflin Harcourt Publishing Compan Use each table to write a quadratic function in verte form, = a ( - h) + k. Then rewrite the function in standard form, = a + b + c. 1. The verte of the function is (, -). = a ( - ) - = a ( - ) - = a () = a 1 - verte form = ( - ) -. = ( ) - = = - + standard form = The verte of the function is (-, -1) = a ( - (-) ) + (-1), or = a ( + ) - 1 = a ( + ) = a () = a verte form = ( + ) - 1. = ( + + ) - 1 = = standard form = The verte of the function is (, 7). = a ( - ) = a ( - ) = a (1) - 1_ = a verte form = - 1_ ( - ) + 7. = - 1_ ( - + 1) + 7 = - 1_ = - 1_ standard form = - 1_ The verte of the function is (-3, 1) = a ( - (-3) ) + 1, or = a ( + 3) = a (-1 + 3) = a () - = a verte form = - ( + 3) + 1. = - ( + + 9) + 1 = = standard form = Module 9 Lesson 3 9 Lesson.3

50 H.O.T. Focus on Higher Order Thinking. Make a Prediction A ball was thrown off a bridge. The table relates the height of the ball above the ground in feet to the time in seconds after it was thrown. Use the data to write a quadratic model in verte form and convert it to standard form. Use the model to find the height of the ball at 1.5 seconds. The verte is at (1, 1). Time (seconds) Height (feet) = a ( - 1) = a ( - 1) = a (1) -1 = a verte form = -1 ( - 1) + 1. = -1 ( - + 1) + 1. = = standard form = = -1 (1.5) + 3 (1.5) + 1 = -1 (.5) = = 1 The height of the ball is 1 feet after 1.5 seconds. MULTIPLE REPRESENTATIONS Give students the graph of a quadratic function. Have students give the verte of the graph, identif the verte as a maimum or minimum value of the graph, find the equation of the ais of smmetr, and find an equation of the parabola. 3. Multiple Representations A performer slips and falls into a safet net below. The function ƒ (t) = a (t - h) + k, where t represents time (in seconds), gives the performer s height above the ground (in feet) as he falls. Use the graph to find an equation for ƒ (t). Height (feet) (, ) (1, ) Time (seconds) f (t) = a (t - ) + or f (t) = at + = a(1) + -1 = a (1) -1 = a The equation for the function is f (t) = -1t +. Module 5 Lesson 3 Houghton Mifflin Harcourt Publishing Compan Image Credits: PhotoStock-Israel/Alam Interpreting Verte Form and Standard Form 5

51 INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Ask whether a quadratic function in the form f () = ( - h ) + k could be the same function as one in the form f () = + b + c in standard form. Students should recognize that the values of a are not equal, and that the is not squared in the first function. JOURNAL In their journals, have students eplain where the letters a, b, c, h, and k come from in the standard and verte forms of a quadratic function.. Represent Real-World Problems After a heav snowfall, Ken and Karin made an igloo. The dome of the igloo is in the shape of a parabola, and the height of the igloo in inches is given b the function ƒ () = a ( - h) + k. Use the graph to find an equation for ƒ (). Height (feet) f () = a ( - ) + 3 = a ( - ) + -1 = a () -.3 = a (, ) (, 3) Width (feet) The equation for the function is f (t) = Houghton Mifflin Harcourt Publishing Compan 5. Check for Reasonableness Tim hits a softball. The function ƒ (t) = a (t - h) + k describes the height (in feet) of the softball, and t is the time (in seconds). Use the graph to find an equation for ƒ (t). Estimate how much time elapses before the ball hits the ground. Use the equation for the function and our estimate to eplain whether the equation is reasonable. Height (feet) (1.5, ) (, 3) Time (seconds) f (t) = a (t - 1.5) + 3 = a ( - 1.5) + - = a (.5) -1 = a The equation for the function is f (t) = -1 (t - 1.5) -. The curve appears to intersect the -ais at 3.1. So, evaluate the function f (t) at 3.1; f (3.1) = -1 ( ) + = is close to the epected value of, so the equation is reasonable. Module 51 Lesson 3 51 Lesson.3

52 Lesson Performance Task The table gives the height of a tennis ball t after it has been hit, where the maimum height is feet. Height (ft) Time (s) a. Use the data in the table to write the quadratic function ƒ (t) in verte form, where t is the time in seconds for the height of the tennis ball. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 When distributing the value of a, -1, to the binomial square, (t -.5 ) in -1 (t -.5 ) +, some students ma err b distributing 1 to. Have students analze their results. If the distributed -1 incorrectl, the ma have gotten -1t + t - 5, in which case the original height of the ball (when hit) would have been 5 feet above the ground. This is not a likel scenario, so it is an indication that an error was made in calculation. b. Rewrite the function found in part a in standard form. c. At what height was the ball originall hit? Eplain. a. The maimum height is feet, so the verte of the parabola is (.5, ). Using a second point from the table, solve for a in the function f (t) = a (t - h) + k. f (t) = a (t - h) + k = a ( ) + - = a (.5) - =.5a -1 = a The function for the height of the tennis ball is f (t) = -1 (t -.5) +. b. f (t) = -1 (t -.5) + = -1 ( t -.5t +.5) + = -1 t + t = -1 t + t + 3 c. The ball was originall hit at a height of t =, or the -intercept. In a function of the form f () = a + b + c, the -intercept is c. In f (t) = -1 t + t + 3, the value of c is 3. So, the ball was originall hit 3 feet above the ground. Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Have students suppose that the are told that a = 1. Have students justif how the know wh it makes sense that the given value of a is incorrect and that this situation makes sense onl if it is modeled with a downward opening parabola with a as a negative number. Module 5 Lesson 3 EXTENSION ACTIVITY For students who completed Part 1 of the activit in the previous Lesson, videotaping the path of a tennis ball against a grid and writing a quadratic function for the path of the ball, ask them to rewrite the function the wrote in standard form, determine the height of the ball using each function after it has gone a distance of feet, and compare these distances to the actual height. Students should discuss the two functions and tell what information each gives about the situation. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. points: Student does not demonstrate understanding of the problem. Interpreting Verte Form and Standard Form 5

53 MODULE Stud Guide Review ASSESSMENT AND INTERVENTION Assign or customize module reviews. MODULE PERFORMANCE TASK COMMON CORE Mathematical Practices: MP.1, MP., MP.5 F-IF.B., F-IF.C.7a, A-CED.A. SUPPORTING STUDENT REASONING Students should begin this problem b focusing on what information the will need. The can then do research, or ou can provide them with specific information. Here is some of the information the ma ask for. What units are being used in the formula? feet for distance, and seconds for time. What do the variables stand for? The h represents the height of the ball, t represents the time the ball is in the air, v represents the initial vertical velocit of the ball, and h represents the initial height of the ball at the time it is thrown. What is the initial height of the throw? Give students a reasonable initial height of 5 ft or a range of heights from ft to 5.5 ft. What is the height of the receiver? Have students choose a receiving height anwhere from ft to ft. 53 Module Houghton Mifflin Harcourt Publishing Compan STUDY GUIDE REVIEW Graphing Quadratic Functions Essential Question: How can ou use the graph of a quadratic function to solve real-world problems? KEY EXAMPLE (Lesson.3) The graph of a function in the form ƒ () = a ( - h) + k is shown. Use the graph to find an equation for ƒ () The verte of the parabola is (h, k) = (, -3). ƒ () = a ( - ) - 3 From the graph, ƒ (3) =. Substitute 3 for and for ƒ () and solve for a. = a (3 - ) = a The equation for the function is ƒ () = 3 ( - ) - 3. KEY EXAMPLE (Lesson.) Graph g () = -( + ) +. The verte is at (-, ). Make a table for the function. Find two points on each side of the verte g () - - Plot the points and draw a parabola through them. - - Ke Vocabular ais of smmetr (eje de simetría) parabola (parábola) MODULE quadratic function (función cuadrática) standard form of a quadratic equation (forma estándar de una ecuación cuadrática) verte (vértice) Module 53 Stud Guide Review SCAFFOLDING SUPPORT Watch for students who use the initial velocit of feet per second in their formulas. Have them carefull reread the problem to verif that velocit. Once students find the maimum height of the ball, the ma think the ve answered the question. Remind them to reread the problem to ensure that the ve done all necessar work. Watch for students who graph the ball s height over time thinking the are graphing the ball s horizontal and vertical positions. For students who need more structure, name an initial and end height. Challenge students b having them compare results of two different velocities or two different initial heights.

54 EXERCISES Graph each quadratic function. Give the minimum or maimum value and the ais of smmetr. (Lessons.1,.) 1. ƒ () =. g () = - ( + ) minimum = ; = Write the equation for the function in each graph, in verte form. (Lesson.3) f () = ( - 1) - 3 MODULE PERFORMANCE TASK Throwing for a Completion Professional quarterbacks can throw a football to a receiver with a velocit of feet per second or greater. If a quarterback throws a pass with that velocit at a 3 angle with the ground, then the initial vertical velocit is 33 feet per second. How can ou use the formula h = -1 t + vt + h to describe the quarterback s pass? Find the maimum height that the football reaches, and then find the total amount of time that the pass is in the air f () = - 1 maimum = ; = - Houghton Mifflin Harcourt Publishing Compan SAMPLE SOLUTION Assume that the initial height is 5 ft and that the receiver catches the ball at a height of 3 ft. Substitute 5 ft for h and 33 for the initial vertical velocit v into the formula, creating the equation. h = -1 t + 33t + 5. Then graph it. Height (feet) Time (seconds) The graph shows the maimum height is about ft. It also shows that the total time the ball is in the air is about. seconds if the ball is allowed to hit the ground. If the ball were caught b a receiver at a height of 3 ft, the ball would be in the air about.1 seconds. Module 5 Stud Guide Review DISCUSSION OPPORTUNITIES Have students compare their answers. Ask them wh their graphs and answer might be similar but not eactl the same. IN_MNLESE393_U3MMC 5 Have students discuss an obstacles the needed to overcome to answer the question. /1/1 :7 AM Assessment Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain. points: Student does not demonstrate understanding of the problem. Stud Guide Review 5

55 Read to Go On? ASSESS MASTERY Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module. Read to Go On?.1.3 Graphing Quadratic Functions Graph each quadratic function. (Lesson.1) 1. ƒ () = -. g () = 1_ Online Homework Hints and Help Etra Practice ASSESSMENT AND INTERVENTION Access Read to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment. ADDITIONAL RESOURCES Response to Intervention Resources Reteach Worksheets Differentiated Instruction Resources Reading Strategies Success for English Learners Challenge Worksheets Assessment Resources Leveled Module Quizzes Houghton Mifflin Harcourt Publishing Compan Describe the transformations necessar to get from the graph of the parent function ƒ () = to the graph of each of the given functions. (Lesson.) 3. g () = ( + ) - 7. g () = 5 ( - ) + 9 Translate the graph units to the left and 7 units down. Rewrite each function in standard form. (Lesson.3) 5. ƒ () = ( + 3) -. ƒ () = 3 ( - ) + 3 ESSENTIAL QUESTION Stretch the graph verticall b a factor of 5, and then translate the graph units to the right and 9 units up. f () = f () = If the onl information ou have about a parabola is the location of its verte, what other characteristics of the graph do ou know? Possible Answer: You know the ais of smmetr, which goes through the -coordinate of the verte. You also know that the -coordinate of the verte is either the maimum or minimum of the parabola, though ou don t know which. Module 55 Stud Guide Review COMMON CORE Common Core Standards Lesson Items Content Standards Mathematical Practices.1 1 F-BF.B.3, F-IF.A., F-IF.C.7.1 F-BF.B.3, F-IF.A., F-IF.C.7 MP.1 MP.1. 3 F-BF.B.3, F-IF.B. MP.. F-BF.B.3, F-IF.B. MP..3 5 F-IF.B., F-IF.C., F-IF.B. MP.7 55 Module.3 F-IF.B., F-IF.C., F-IF.B. MP.7

56 MODULE MIXED REVIEW Assessment Readiness 1. Consider the graph of f () = 3 ( - 3). Choose True or False for each statement about the graph. A. The verte is (3, ). True False B. The minimum value is. True False C. The ais of smmetr is =. 3 True False MODULE MIXED REVIEW Assessment Readiness ASSESSMENT AND INTERVENTION 3 5. Is the given epression equivalent to 1 + 3? Select Yes or No for each epression. A. B. 1 1_ + 3 1_ Yes No 5 3 Yes No C. 3 + Yes No Assign read-made or customized practice tests to prepare students for high-stakes tests. 3. Write the slope-intercept equation of the line that has the same slope as - 3 = 1 ( + 3) and contains the point (, ). Eplain how ou wrote the equation. 1 = ; I used the general form of a line in slope-intercept form, or = m + b, to get = The slope, m, is 1. Then, I substituted for and for in + b and solved for b, which is. = 1. Write f () = - ( - 5) + 3 in standard form. In which form is it easier to determine the maimum value of the graph? Eplain. f () = ; It is easier to find the maimum value when the equation is written as f () = - ( - 5) + 3. The verte is (5, 3), so the maimum value is 3. Houghton Mifflin Harcourt Publishing Compan ADDITIONAL RESOURCES Assessment Resources Leveled Module Quizzes: Modified, B AVOID COMMON ERRORS Item 3 Some students ma not notice that the are looking for a line parallel to the given line, and the will instead stop after finding the slope-intercept form of the given equation. Remind students to highlight or underline important parts of the problem to help them focus on all the necessar information. Module 5 Stud Guide Review COMMON CORE Common Core Standards Lesson Items Content Standards Mathematical Practices.1 1 F-IF.B. MP. 3.1 * N-RN.A.1 MP. IM1.1, IM1. 3* A-CED.A. MP..3 F-IF.C. MP.3 * Item integrates mied review concepts from previous modules or a previous course. Stud Guide Review 5

57 MODULE 7 Connecting Intercepts, Zeros, and Factors ESSENTIAL QUESTION: Answer: You can use quadratic functions to solve real-world area problems. Each dimension can represent a linear factor of the quadratic function. The intercepts ma represent the solution to th e problem. PROFESSIONAL DEVELOPMENT VIDEO Connecting Intercepts, Zeros, and Factors Essential Question: How can ou use intercepts of a quadratic function to solve real-world problems? MODULE 7 LESSON 7.1 Connecting Intercepts and Zeros LESSON 7. Connecting Intercepts and Linear Factors LESSON 7.3 Appling the Zero Product Propert to Solve Equations Professional Development Video Author Juli Dion models successful teaching practices in an actual high-school classroom. Professional Development m.hrw.com Houghton Mifflin Harcourt Publishing Compan Image Credits: Vladimir Ivanovich Danilov/Shutterstock MODULE PERFORMANCE TASK PREVIEW Skateboard Ramp REAL WORLD VIDEO Skateboard ramps come in man shapes and sizes. The iconic half-pipe ramp has a flat section in the middle and curved, raised sides. Skateboarders can use a half-pipe ramp to perform tricks, turns, and flips. Skateboard riders often use curved ramps to perform difficult tricks and have fun. In this module, ou will imagine that ou are a design engineer hired b the local government to help construct a new skateboard ramp for the skateboard riders in the area. How do ou model the curve of the ramp? Let s find out! Module 7 57 DIGITAL TEACHER EDITION Access a full suite of teaching resources when and where ou need them: Access content online or offline Customize lessons to share with our class Communicate with our students in real-time View student grades and data instantl to target our instruction where it is needed most PERSONAL MATH TRAINER Assessment and Intervention Assign automaticall graded homework, quizzes, tests, and intervention activities. Prepare our students with updated, Common Core-aligned practice tests. 57 Module 7

58 Are YOU Read? Complete these eercises to review skills ou will need for this chapter. Eponents Eample 1 Simplif. Simplif. 5 3 = = _ 9 = 9 - = 5 1. b b. b Algebraic Epressions The bases are the same. Add the eponents. The bases are the same. Subtract the eponents. _ a 1 3. _ n n 7 a 7 5 n a 5 n Eample Find the value of 3 when =. Find the value. 3-3 () when = - 1_ Linear Functions Eample 3 Substitute for. Follow the order of operations. Online Homework Hints and Help Etra Practice when = 5_. 9 + when = - _ 3 Tell whether = 7 represents a linear function. = 7 does not represent a linear function because has an eponent of. Tell whether the equation represents a linear function. When a linear equation is written in standard form, the following are true. and both have eponents of 1. and are not multiplied together. and do not appear in denominators, eponents, or radicands. 7. = = = no es no Houghton Mifflin Harcourt Publishing Compan Are You Read? ASSESS READINESS Use the assessment on this page to determine if students need strategic or intensive intervention for the module s prerequisite skills. ASSESSMENT AND INTERVENTION 3 1 TIER 1, TIER, TIER 3 SKILLS Personal Math Trainer will automaticall create a standards-based, personalized intervention assignment for our students, targeting each student s individual needs! ADDITIONAL RESOURCES See the table below for a full list of intervention resources available for this module. Response to Intervention Resources also includes: Tier Skill Pre-Tests for each Module Tier Skill Post-Tests for each skill Module 7 5 Response to Intervention Differentiated Instruction Tier 1 Lesson Intervention Worksheets Tier Strategic Intervention Skills Intervention Worksheets Tier 3 Intensive Intervention Worksheets available online Reteach 7.1 Reteach 7. Reteach Algebraic Epressions 1 Eponents (G..1) 1 Linear Functions Building Block Skills 19,, 3,, 7, 9, 3,, 59, 9, 7, 1, 1 Challenge worksheets Etend the Math Lesson Activities in TE Module 7 5

59 COMMON CORE = = = = = = = Locker LESSON Common Core Math Standards The student is epected to: COMMON CORE F-IF.C.7a Graph...quadratic functions and show intercepts, maima, and minima. Also A-REI.B., A-APR.B.3, A-REI.D.11 Mathematical Practices COMMON CORE 7.1 Connecting Intercepts and Zeros MP.5 Using Tools Name Class Date 7.1 Connecting Intercepts and Zeros Essential Question: How can ou use the graph of a quadratic function to solve its related quadratic equation? Eplore Graphing Quadratic Functions in Standard Form A parabola can be graphed using its verte and ais of smmetr. Use these characteristics, the -intercept, and smmetr to graph a quadratic function. Graph = b completing the steps. Resource Locker Language Objective Given a quadratic function modeling a real-world situation, eplain to a partner what the zeros of the function represent. ENGAGE Essential Question: How can ou use the graph of a quadratic function to solve its related quadratic equation? You can graph the two sides of the equation as two function rules and find the intersection of their graphs, or ou can write the equation with one side equal to, graph the related function, and find the zeros of the function. PREVIEW: LESSON PERFORMANCE TASK Houghton Mifflin Harcourt Publishing Compan Find the ais of smmetr. = - b _ a = - _ - = The ais of smmetr is =. Find the -intercept. = = - + ( -5 ) The -intercept is -5 ; the graph passes through (, -5 ). Find the verte. = = = = -9 Th e verte is (, ). Find two more points on the same side of the ais of smmetr as the -intercept. a. Find when = 1. = = = View the Engage section online. Discuss what tpe of = - path is made b a diver diving into the water when The first point is ( 1, - ). her initial height is the height of the diving platform. 59 Then preview the Lesson Performance Task. b. Find when = -1. = = = = -9-1 The second point is (, ). Module 7 59 Lesson 1 Name Class Date 7.1 Connecting Intercepts and Zeros Essential Question: How can ou use the graph of a quadratic function to solve its related quadratic equation? F-IF.C.7a Graph quadratic functions and show intercepts, maima, and minima. Also A-REI.B., A-APR.B.3, A-REI.D.11 Houghton Mifflin Harcourt Publishing Compan Eplore Graphing Quadratic Functions Find the ais of smmetr. a in Standard Form A parabola can be graphed using its verte and ais of smmetr. Use these characteristics, the -intercept, and smmetr to graph a quadratic function. Graph = b completing the steps. = - b_ = = - _ = The ais of smmetr is =. - 1 Find the -intercept. = = - + ( ) -5-5 Find the verte. Th e verte is (, ). The -intercept is ; the graph passes through (, ). Find two more points on the same side of the ais of smmetr as the -intercept. a. Find when = 1. b. Find when = -1. = = The first point is (, ) Resource = = The second point is (, ). Module 7 59 Lesson 1 HARDCOVER PAGES 59 7 Watch for the hardcover student edition page numbers for this lesson. 59 Lesson 7.1

60 Graph the ais of smmetr, the verte, the -intercept, and the two etra points on the same coordinate plane. Then reflect the graphed points over the ais of smmetr to create three more points, and sketch the graph. Reflect (-1, ) (, -5) ( = ) - (1, -) (, -9) 1. Discussion Wh is it important to find additional points before graphing a quadratic function? Additional points provide more information about the shape of the parabola, and the sketch of the quadratic function will be more accurate. Eplain 1 Using Zeros to Solve Quadratic Equations Graphicall A zero of a function is an -value that makes the value of the function. The zeros of a function are the -intercepts of a function. A quadratic function ma have one, two, or no zeros. Quadratic equations can be solved b graphing the related function of the equation. To write the related function, rewrite the quadratic equation so that it equals zero on one side. Replace the zero with. Graph the related function. Find the -intercepts of the graph, which are the zeros of the function. The zeros of the function are the solutions to the original equation. Eample 1-5 = -3 Solve b graphing the related function. a. Write the related function. Add 3 to both sides to get - =. The related function is = -. b. Make a table of values for the related function c. Graph the points represented b the table and connect the points. d. The zeros of the function are -1 and 1, so the solutions of the equation - 5 = -3 are = -1 and = 1. (-, ) (, ) (-1, ) (1, ) (, -) Module 7 Lesson 1 PROFESSIONAL DEVELOPMENT Math Background Quadratic equations are often used to model the motion of falling objects. The general formula for this motion is h = -1t + v t + h, where h represents the height of the object in feet, t is the number of seconds the object has been falling, v is the initial vertical velocit of the object in feet per second, and h is the initial height of the object in feet. The coefficient -1 is equal to half of the constant acceleration due to gravit, -3 ft/s. Students can compare the quadratic equations given for falling objects in this lesson to the general formula to determine the values of v and h in each case. - Houghton Mifflin Harcourt Publishing Compan EXPLORE Graphing Quadratic Functions in Standard Form QUESTIONING STRATEGIES How man points on the graph of a quadratic function are on the ais of smmetr? Eplain. One; the onl point on the graph of a quadratic function that is on the ais of smmetr is the verte of the function. Wh is it helpful to find the ais of smmetr when graphing a quadratic function? After ou find the ais of smmetr and a few points on one side of it, ou can use smmetr to quickl and easil find an equal number of points on the other side. EXPLAIN 1 Using Zeros to Solve Quadratic Equations Graphicall AVOID COMMON ERRORS Students ma think that the zeros of a quadratic function can be found b substituting for in the function. Make sure students understand that the zeros of a function are the values of when =, not the values of when =. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 For cases in which a quadratic function has two zeros, discuss with students how to use the zeros to determine the function s ais of smmetr. Students should realize that the ais of smmetr will be the -value of the point that is midwa between the two zeros. Connecting Intercepts and Zeros

61 QUESTIONING STRATEGIES When a quadratic function has two zeros that are opposites, what must be true about the function? The ais of smmetr must be =, and the verte must not be (, ). B + = - a. Write the related function. Add to get + + =. The related function is = + +. b. Make a table of values for the related function c. Graph the points represented b the table and connect the points. d. The zeros of the function are - and -, so the solutions of the equation = - are = - and = -. (-5, 3) (-1, 3) (-, ) (-, ) - - (-3, -1) - - Reflect. How would the graph of a quadratic equation look if the equation has one zero? If the quadratic equation has one zero, the graph will intersect the -ais at its verte. Your Turn Houghton Mifflin Harcourt Publishing Compan 3. - = -3 - = -3-1 = The related function is = - 1. Graph: The zeros of the function are 1 and -1, so the solutions of the equation are = -1 and = 1. (-, 3) (, 3) (-1, ) (1, ) - - (, -1) - - Module 7 1 Lesson 1 COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Have each student find the solution to the same quadratic equation b graphing. One student in each pair then solves the equation b finding the zeros of the function, and the other student uses points of intersection to find the solution. Students compare their answers; while the graphs ma be different, the solution should be the same regardless of the method used. 1 Lesson 7.1

62 Eplain Using Points of Intersection to Solve Quadratic Equations Graphicall You can solve a quadratic equation b rewriting the equation in the form a + b = c or a ( - h) = k and then using the epressions on each side of the equals sign to define a function. Graph both functions and find the points of intersection. The -coordinates are the points of intersection on the graph. As with using zeros, there ma be two, one, or no points of intersection. Eample Solve each equation b finding points of intersection of two related functions. ( - ) - = Write in verte form. ( - ) = Graph each side as related function. a. Let ƒ () = ( - ). Let g () =. b. Graph ƒ () and g () on the same graph. c. Determine the points at which the graphs of ƒ () and g () intersect. The graphs intersect at two locations: (3, ) and (5, ). This means ƒ () = g () when = 3 and = 5. So the solutions of ( - ) - = are = 3 and = 5. 3 ( - 5) -1 = 3 ( - 5) -1 = 3 ( - 5) = 1 a. Let ƒ () = 3 ( - 5). Let g () = 1. b. Graph ƒ () and g () on the same graph. c. Determine the points at which the graphs of ƒ () and g () intersect. The graphs intersect at two locations: ( 3, 1 ) and ( 7, 1 ). This means ƒ () = g () when = 3 and = 7. Therefore, the solutions of the equation ƒ () = g () are 3 and 7. So the solutions of 3 ( - 5) - 1 = are = 3 and = 7. g() (3, ) (5, ) f() 1 (7, 1) (3, 1) Module 7 Lesson 1-1 Houghton Mifflin Harcourt Publishing Compan EXPLAIN Using Points of Intersection to Solve Quadratic Equations Graphicall INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Remind students that quadratic equations in the form a ( - h) + k will have the verte at (h, k). After identifing the verte, students can use substitution to find enough points to graph the quadratic function. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. When using points of intersection to find solutions to quadratic equations, make sure that students understand that onl the -values of the points of intersection are solutions to the equation. The -values come from the function that was created to find the solutions; the are not part of the solution. QUESTIONING STRATEGIES How can ou use graphing to determine that a quadratic equation has no solutions? After rewriting the equation in the form a + b = c, if the graphs of the functions f () = a + b and g () = c do not intersect, then the quadratic function has no solutions. DIFFERENTIATE INSTRUCTION Communicating Math Review the parameters of a parabola that students can use when graphing a function. Students should understand that being able to identif the verte of a parabola will make it easier to graph the function, but the ma have different ideas about how to find enough other points in order to make an accurate graph. Discuss how students can be sure that the have plotted enough points to sketch the function on a coordinate grid. Connecting Intercepts and Zeros

63 EXPLAIN 3 Modeling a Real-World Problem INTEGRATE TECHNOLOGY Before students are familiar with using the quadratic formula, graphing calculators offer the most accessible wa to find the solutions to quadratic equations that do not have simple whole-number solutions. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. The functions used to determine the height of a thrown object can be difficult to understand. Discuss different was to rewrite the functions used to determine height. Students ma wish to rewrite a function like h (t) = -1t + 1 as h (t) = 1 1 t to make it more evident that the object loses height for each second of time. Houghton Mifflin Harcourt Publishing Compan Image Credits: USBFCO/ Shutterstock Reflect. In Part B above, wh is the -coordinates the answer to the equation and not the -coordinates? The -coordinates are the solution because the -values are the unknown amount being solved for in the original equation. The -values are used to create a related function to find the values, but since the are not part of the original equation, the values are not part of the solution. Your Turn 5. 3 ( - ) -3 = 3( - ) - 3 = Eplain 3 Modeling a Real-World Problem Man real-world problems can be modeled b quadratic functions. Eample 3 3( - ) = 3 Let f () = 3 ( - ) and let g () = 3. The graphs intersect at two locations: (1, 3) and (3, 3). This means f () = g () when = 1 and = 3. So the solutions of 3 (- ) - 3 = are 3 and 5. Create a quadratic function for each problem and then solve it b using a graphing calculator. Nature A squirrel is in a tree holding a chestnut at a height of feet above the ground. It drops the chestnut, which lands on top of a bush that is 3 feet below the tree. The function h (t) = -1t + gives the height in feet of the chestnut as it falls, where t represents time. When will the chestnut reach the top of the bush? Analze Information Identif the important information. The chestnut is feet above the ground, and the top of the bush is 3 feet below the chestnut. The chestnut s height as a function of time can be represented b h (t) = -1 t +, where (h) t is the height of the chestnut in feet as it is falling. (1, 3) (3, 3) Formulate a Plan Create a related quadratic equation to find the height of the chestnut in relation to time. Use h (t) = -1 t + and insert the known value for h. Module 7 3 Lesson 1 LANGUAGE SUPPORT Connect Vocabular Students ma epect that problems involving points of intersection will alwas have solutions. Relate the word intersection to its uses outside the math classroom. Just as two streets in a town ma or ma not intersect, the graphs of two functions ma or ma not intersect. When dealing with the intersection of a quadratic function and a linear function, remind students that there ma be, 1, or points of intersection. 3 Lesson 7.1

64 Solve Write the equation that needs to be solved. Since the top of the bush is 3 feet below the squirrel, it is 1 feet above the ground. -1t + = 1 Separate the function into = f (t) and = g (t). f (t) = -1 t + and g (t) = 1. To graph each function on a graphing calculator, rewrite them in terms of and. = -1 + and = 1 Graph both functions. Use the intersect feature to find the point in amount of time it takes for the chestnut to hit the top of the bush. The chestnut will reach the top of the bush in Justif and Evaluate -1 ( 1.5 ) + = = 1 1 = 1 When t is replaced b seconds. in the original equation, -1 t + = 1 is true. QUESTIONING STRATEGIES When finding the time it takes for an object to fall to the ground, wh is onl the positive zero of the function used as an answer? Since the value for time will alwas be a positive number, the negative zero of the function can be ignored as a solution. INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 Students who have the minimum and maimum - and -values set incorrectl on their graphing calculators ma not be able to see the point of intersection when the graph two functions. Discuss how to change the dimensions of the graph b accessing the WINDOW menu. Reflect. In Eample B above, the graphs also intersect to the left of the -ais. Wh is that point irrelevant to the problem? That point is irrelevant to the problem since negative time has no meaning in this problem. Your Turn 7. Nature An egg falls from a nest in a tree 5 feet off the ground and lands on a potted plant that is feet below the nest. The function h (t) = -1t + 5 gives the height in feet of the egg as it drops, where t represents time. When will the egg land on the plant? = 5 = and = 5. The egg will hit the plant after about 1.1 seconds. Houghton Mifflin Harcourt Publishing Compan Module 7 Lesson 1 Connecting Intercepts and Zeros

65 EXPLAIN Interpreting a Quadratic Model AVOID COMMON ERRORS When viewing graphs of quadratic functions modeling height, students ma believe that the shape of the graph represents the path an object takes while in the air. Remind students that while the -ais represents the height of the object, the -ais does not show distance, but rather time. Eplain Interpreting a Quadratic Model The solutions of a quadratic equation can be used to find other information about the situation modeled b the related function. Eample Use the given quadratic function model to answer questions about the situation it models. Nature A dolphin jumps out of the water. The quadratic function h (t) = -1 t + 5t models the dolphin s height above the water in feet after t seconds. How long is the dolphin out of the water? Use the level of the water as a height of feet. When the dolphin leaves and then reenters the water again, its height is feet. Solve = -1 t + 5t to find the times when the dolphin both leaves the water and then reenters. The difference between the times is the amount of time the dolphin is out of the water. a. Write the related function for = = -1 + QUESTIONING STRATEGIES For quadratic functions that model the height of a thrown object, how can ou use the zeros of the function to determine when the object is at its maimum height? Eplain. Since the ais of smmetr for a quadratic function lies midwa between the two zeros of the function, the maimum height is reached at a time halfwa between the two zeros. For a quadratic function modeling the height of a thrown object, when one of the zeros of the function is, what must be true about the object? The object must have been thrown from ground level, so that the height of the object equals at time. Houghton Mifflin Harcourt Publishing Compan b. Graph the function on a graphing calculator. Use the trace feature to estimate the zeros. The zeros appear to be and 1.5. When =, the equation reduces to =, which is true. So = is a solution. Check = (1.5) + (1.5) = -1 (1.55) + 5 = = so 1.5 is a solution. Since = 1.5, the dolphin is out of the water for 1.5 seconds. Sports A baseball coach uses a pitching machine to simulate pop flies during practice. The quadratic function = 1 t + t + 5 models the height in feet of the baseball after seconds. The ball leaves the pitching machine and is caught at a height of 5 feet. How long is the baseball in the air? Solve = 1 t + t + 5 to find the times when the baseball enters the air and when it is caught. a. Write the related function for = 1 t + t + 5. = b. Graph the function on a graphing calculator. Use the trace feature to find the zeros. The zeros appear to be and 5. Since = makes the right side of the equation equal to 5, which is the height Module 7 5 Lesson 1 5 Lesson 7.1

66 of the baseball when it is released b the pitching machine, it is a solution. Check to see if 5 is a solution = -1( 5 ) + ( 5 ) + 5 = = 5, so 5 is a solution. The ball is in the air for 5 seconds. Your Turn. Nature The quadratic function = models the height, in feet, of a fling fish above the water after seconds. How long is the fling fish out of the water? The graph of = shows a zero at about.315. The fish is out of the water for.315 second. ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Discuss with students what kinds of realworld problems can be solved b identifing the zeros of a quadratic function. Students should understand that the zeros of a quadratic function will not be the solution for ever real-world problem. Elaborate 9. How is graphing quadratic functions in standard form similar to using zeros to solve quadratic equations graphicall? If there are two solutions for a quadratic function, the reflection of that point across the parabola will be the other point, which is the second solution. Both methods use the value of the function at to find the second point. 1. How can graphing calculators be used to solve real-world problems represented b quadratic equations? Graphing calculators can be used to solve real-world quadratic equations b writing the equation and then finding either an intersection or the zeros of the equation s graph. 11. Essential Question Check-In How can ou use the graph of a quadratic function to solve a related quadratic equation b wa of intersection? You can graph the two sides of the equation as two functions and find their points of intersection. The -coordinate or coordinates of the intersection(s) will be the solution to the quadratic equation. Houghton Mifflin Harcourt Publishing Compan SUMMARIZE THE LESSON How do ou solve equations graphicall using zeros and points of intersection? To solve quadratic equations graphicall using zeros, rewrite the equation so one side is equal to zero, then graph the other side of the equation and identif the zeros. To solve quadratic equations graphicall using points of intersection, rewrite the equation in the form a + b = c, graph both sides of the equation, then find the -values of the points of intersection. Module 7 Lesson 1 Connecting Intercepts and Zeros

67 EVALUATE Evaluate: Homework and Practice Solve each equation b graphing the related function and finding its zeros = = -1 Online Homework Hints and Help Etra Practice ASSIGNMENT GUIDE Concepts and Skills Eplore 1 Graphing Quadratic Functions in Standard Form Eample 1 Using Zeros to Solve Quadratic Equations Graphicall Eample Using Points of Intersection to Solve Quadratic Equations Graphicall Eample 3 Modeling a Real-World Problem Eample Interpreting a Quadratic Model Practice Eercise Eercises 1, 5 Eercises 7 1, 1, 3 Eercises 13 1, Eercises 17 1 (, 9) (-, 9) (1, ) - - (-1, ) -1 (, -3) - 1 (, 1) (-, 1) (1, ) - - (-1, ) - (, -) -1 The zeros of = 3-3 are 1 and -1, so = -1 or = 1. 3 (-, ) (, ) 1 (, ) - - (-, ) (, -) = = - The zeros = - are and -, so = - or =. (-, ) (-5, ) (-, ) (-3, -) (-, -) - Houghton Mifflin Harcourt Publishing Compan The zeros of = - are 1 and -1, so = -1 or = = = - (-3, ) (1, ) - - (-, -3) - (, -3) (-1, -) - The zeros of = are -5 and -, so = -5 or = -. (-, 3) (, 3) (-1, ) (1, ) (, -1) - The zeros of = are -3 and -1, so = -3 or = 1. The zeros of = - 1 are -1 and 1, = -1 or = 1. Module 7 7 Lesson 1 Eercise IN_MNLESE393_U3M7L1.indd 7 Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices //1 3:3 PM Recall of Information MP. Modeling 13 1 Recall of Information MP. Modeling 1 1 Recall of Information MP.5 Using Tools 1 Recall of Information MP. Modeling 3 5 Skills/Concepts MP.3 Logic 7 Lesson 7.1

68 Solve each equation b finding points of intersection of two related functions. 7. ( - 3) - =. ( + ) - = (1.59, ) (.1, ) (-, ) (, ) INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Students ma feel that certain equations must be solved using certain methods. Discuss the fact that most problems can be solved using more than one of the methods outlined in this lesson, and that the will learn additional methods as the continue in algebra. The graphs of f () = ( - 3) and g () = intersect at (1.59, ) and (.1, ). So = 1.59 or =.1. The graphs of f () = ( + ) and g () = intersect at (-, ) and (, ). So = or = ( - 3) + = 1. - ( + ) - = (1, ) (5, ) The graphs of f () = ( - 3) and g () = intersect at (5, ) and (1, ). So = 5 or = The graphs of f () = ( + ) and g () = - do not intersect. So there is no solution. 11. ( + 1) - 1 = 1. ( + ) - = (-3.1, ) (-.59, ) (-, 1) (, 1) Houghton Mifflin Harcourt Publishing Compan The graphs of f () = ( + 1) and g () = 1 intersect at (-, 1) and (, 1). So = - or =. The graphs of f () = ( + ) and g () = intersect at (-3.5, ) and (-.5, ). So -3.5 or -.5. Module 7 Lesson 1 IN_MNLESE393_U3M7L1.indd //1 3:3 PM Connecting Intercepts and Zeros

69 AVOID COMMON ERRORS When solving quadratic equations graphicall, students ma mistake the zeros of a quadratic function for the places where the function intersects the -ais. Remind students that the zeros of a function are the -values at the points where the function intersects the -ais. Create a quadratic equation for each problem and then solve the equation with a related function using a graphing calculator. 13. Nature A bird is in a tree 3 feet off the ground and drops a twig that lands on a rosebush 5 feet below. The function h (t) = -1 t + 3, where t represents the time in seconds, h gives the height, in feet, of the twig above the ground as it falls. When will the twig land on the bush? The graphs of = and = 5 intersect at about (1.5, 5). The twig will hit the rosebush after about 1.5 seconds. 1. Nature A monke is in a tree 5 feet off the ground and drops a banana, which lands on a shrub 3 feet below. The function h (t) = -1 t + 5, where t represents the time in seconds, h gives the height, in feet, of the banana above the ground as it falls. When will the banana land on the shrub? The graphs of = and = intersect at about (1., ). The banana will hit the shrub after about 1. seconds. 15. Sports A trampolinist jumps inches in the air off a trampoline inches off the ground. The function h (t) = -1 t +, where t represents the time in seconds, h gives the height, in inches, of the trampolinist above the ground as he falls. When will the trampolinist land on the trampoline? Houghton Mifflin Harcourt Publishing Compan Dmitr Laudin/ Shutterstock The graphs of = -1 + and = 5 intersect at about (., 5). The trampolinist will land back on the trampoline after about. seconds. 1. Phsics A ball is dropped from 1 feet above the ground. The function h (t) = -1 t + 1, where t represents the time in seconds, h gives the height, in feet, of the ball above the ground. When will the ball be feet above the ground? The graphs of = and = intersect at about (., ). The ball will land on the roof of the trash shed in about. seconds. Use the given quadratic function model to answer questions about the situation it models. 17. Nature A shark jumps out of the water. The quadratic function ƒ () = models the shark s height, in feet, above the water after seconds. How long is the shark out of the water? The graph of f () = shows a zero of about 1.5. The shark is out of the water for about 1.15 seconds. Module 7 9 Lesson 1 9 Lesson 7.1

70 1. Sports A baseball coach uses a pitching machine to simulate pop flies during practice. The quadratic function ƒ () = models the height in feet of the baseball after seconds. How long is the baseball in the air? The graph of f () = shows a zero of about.5. The ball is in the air for.5 seconds. 19. The quadratic function ƒ () = models the height, in feet, of a fish above the water after seconds. How long is the fish out of the water? The graph of f () = shows a zero of about.7. The seal is in the air for about.7 seconds. CRITICAL THINKING Challenge students to solve a quadratic equation graphicall using two different methods: b finding the zeros of the function and b finding points of intersection. The should find that the graphs look similar, but that one is translated verticall relative to the other. Students should understand that when solving one problem using two different methods, the graphs ma look different, but the final solutions will be the same.. A football coach uses a passing machine to simulate 5-ard passes during practice. The quadratic function ƒ () = models the height in feet of the football after seconds. How long is the football in the air? The graph of f () = shows a zero of about 3.. The football is in the air for about 3. seconds. Houghton Mifflin Harcourt Publishing Compan Radharc Images/Alam Module 7 7 Lesson 1 Connecting Intercepts and Zeros 7

71 JOURNAL Have students write a paragraph outlining the steps the take when graphing a quadratic function b hand. Students should be sure to describe how to find enough points so that the function can be accuratel graphed on a coordinate grid. 1. In each polnomial function in standard form, identif a, b, and c. a. = b. = + 1 c. = d. = 5 e. = a. b. c. d. e. a = 3, b =, c = a =, b =, c = 1 a = 1, b =, c = a =, b =, c = 5 a = 3, b =, c = 11. Identif the ais of smmetr, -intercept, and verte of the quadratic function = + - and then graph the function on a graphing calculator to confirm. Ais of smmetr: = - b_ 1_ a = - -Intercept: = + - = - Verte: ( - 1_, -5 1_ ) because = ( - 1_ ) - 1_ 1_ - = - H.O.T. Focus on Higher Order Thinking 3. Countereamples Pamela sas that if the graph of a function opens upward, then the related quadratic equation has two solutions. Provide a countereample to revoke Pamela s claim. Sample answer: f () =, but the related equation, =, has onl one solution.. Eplain the Error Rodne was given the function h (t) = -1 t + 5 representing the height above the ground (in feet) of a water balloon t seconds after being dropped from a roof 5 feet above the ground. He was asked to find how long it took the balloon to fall feet. Rodne used the equation -1 t + 5 = to solve the problem. What was his error? Houghton Mifflin Harcourt Publishing Compan Falling feet means that the balloon would have been 5 - = 3 feet above the ground, so Rodne should have solved the equation -1 t + 5 = 3 instead. 5. Critical Thinking If Jamie is given the graph of a quadratic equation with onl the zeros and a random point labeled, could she determine an equation for the graph? Jamie could create an equation of the graph b knowing onl the zeros if she rewrote the zeros as factors of the polnomial and then solved b inserting the values from the random point to find a. Module 7 71 Lesson 1 71 Lesson 7.1

72 Lesson Performance Task QUESTIONING STRATEGIES Stella is competing in a diving competition. Her height in feet above the water is modeled b the function ƒ () = -1 t + 1t + 3, where t is the time in seconds after she jumps from the diving board. Graph the function and solve. What do the solutions mean in the contet of the problem? Are there solutions that do not make sense? Eplain. The graph crosses the -ais at about (1.9, ) so the -intercept is about 1.9. So, one solution of the equation = is about 1.9. This solution means that Stella reaches the surface of the water almost seconds after she starts the dive. If the graph is etended on the left, it will cross the -intercept once more. Since time cannot be negative, this solution does not make sense. Height (ft) 3 1 Height of Dive 1 Time (s) How can ou find the height at which the diver starts her dive? Find the value of the function when t =. How can ou find the maimum height that the diver reaches? Use the graph to find the -value of the verte of the parabola, or find the -value halfwa between the two zeros and calculate the corresponding -value. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Have students eamine how the zeros, intercepts, function, and graph modeling the dive would be different if the diver had started the dive at the level of the water, that is, at height. Houghton Mifflin Harcourt Publishing Compan Module 7 7 Lesson 1 EXTENSION ACTIVITY Give students the functions g (t) = - 1t + t + and h (t) = - 1t + 1 and eplain that the represent the height in feet of two different divers, where t is time in seconds from the start of each dive. Have students compare the initial height, maimum height, and time to reach the water for the two divers. Students should find that g (t) represents a dive starting at feet, reaching a maimum of 5 feet, and hitting the water after 1.5 seconds, and h (t) represents a dive starting at 1 feet, which is also its maimum height, and hitting the water after 1 second. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. points: Student does not demonstrate understanding of the problem. Connecting Intercepts and Zeros 7

73 COMMON CORE Locker LESSON Common Core Math Standards The student is epected to: COMMON CORE A-APR.B.3 Identif zeros of polnomials and use the zeros to construct a rough graph of the function defined b the polnomial. Also F-IF.C.7c, A-APR.A.1, A-SSE.A. Mathematical Practices COMMON CORE 7. Connecting Intercepts and Linear Factors MP. Reasoning Language Objective Eplain to a partner how to determine the zeros of a quadratic function from an equation written in factored form. ENGAGE Essential Question: How are -intercepts of a quadratic function and its linear factors related? The -intercepts of a quadratic function are the same as the -intercepts of its linear factors. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss how engineers might be able to use a quadratic function to model the shape of an arched underpass. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing Compan Name Class Date 7. Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Eplore Connecting Factors and Intercepts Use graphs and linear factors to find the intercepts of a parabola. A B C D E Graph = + and = - using a graphing calculator. Then sketch the graphs on the grid. Identif the -intercept of each line. The -intercepts are - and. The quadratic function = ( + ) ( - ) is the product of the two linear factors that have been graphed. Use a graphing calculator to graph the function = ( + ) ( - ). Then sketch a graph of the quadratic function on the same grid with the linear factors that have been graphed. Identif the -intercepts of the parabola. The -intercepts are - and. What do ou notice about the intercepts of the parabola? Reflect 1. Use a graph to determine whether is the product of the linear factors - 3 and +. Yes. The -intercepts of = are the same. Discussion Make a conjecture about the linear factors and -intercepts of a quadratic function. The -intercepts of the parabola are the same as those - - The intercepts of the parabola are the same as those of the two linear factors. as those of = - 3 and = +. of the two linear factors Resource Locker Module 7 73 Lesson DO NOT EDIT--Changes must be made through File info CorrectionKe=NL-A;CA-A Name Class Date 7. Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? A-APR.B.3 Identif zeros of polnomials and use the zeros to construct a rough graph of the function defined b the polnomial. Also F-IF.C.7c, A-APR.A.1, A-SSE.A. Houghton Mifflin Harcourt Publishing Compan Eplore Connecting Factors and Intercepts Use graphs and linear factors to find the intercepts of a parabola. Graph = + and = - using a graphing calculator. Then sketch the graphs on the grid. Identif the -intercept of each line. The -intercepts are and. The quadratic function = ( + ) ( - ) is the product of the two linear factors that have been graphed. Use a graphing calculator to graph the function = ( + ) ( - ). Then sketch a graph of the quadratic function on the same grid with the linear factors that have been graphed. Identif the -intercepts of the parabola. The -intercepts are and. What do ou notice about the intercepts of the parabola? Reflect - 1. Use a graph to determine whether is the product of the linear factors - 3 and +.. Discussion Make a conjecture about the linear factors and -intercepts of a quadratic function. Resource The intercepts of the parabola are the same as those of the two linear factors. Yes. The -intercepts of = are the same as those of = - 3 and = +. The -intercepts of the parabola are the same as those of the two linear factors Module 7 73 Lesson IN_MNLESE393_U3M7L 73 /9/1 : PM HARDCOVER PAGES 73 Watch for the hardcover student edition page numbers for this lesson. 73 Lesson 7.

74 Eplain 1 Rewriting from Factored Form to Standard Form A quadratic function is in factored form when it is written as = k ( - a) ( - b) where k. Eample 1 Write each function in standard form. = ( + 1) ( - ) Multipl the two linear factors. = ( ) = ( ) Multipl the resulting trinomial b. = - - The standard form of = ( + 1) ( - ) is = - -. = 3 ( - 5) ( - ) Multipl the two linear factors. = 3 ( - 5 )( - ) = 3 ( ) Multipl the resulting trinomial b 3. = The standard form of = 3 ( - 5) ( - ) is = EXPLORE Connecting Factors and -Intercepts INTEGRATE TECHNOLOGY Students will use graphing calculators to graph quadratic functions on the same grid as the linear functions representing their factors. B comparing the graphs, the can discover relationships between a quadratic function and the -intercepts of its factors. Reflect 3. How do the signs in the factors affect the sign of the term in the resulting trinomial? If both signs in the factors are negative, then the -term will be negative. If both signs in the factors are positive, then the -term will be positive.. How do the signs in the factors affect the sign of the constant term in the resulting trinomial? If both signs in the factors are different, then the constant term will be negative. If both signs in the factors are the same, then the constant term will be positive. Your Turn Write each function in standard form. 5. = ( - 7) ( - 1). = ( - 1) ( + 3) = = = = ( ) = ( + - 3) = Houghton Mifflin Harcourt Publishing Compan QUESTIONING STRATEGIES If a quadratic function and a linear function share the same -intercept, must the epression in the linear function be a factor of the quadratic function? Eplain. No; the epression could be a factor of the quadratic equation, but since man linear functions have the same -intercept, the epression does not have to be a factor of the quadratic equation. EXPLAIN 1 Rewriting from Factored Form to Standard Form Module 7 7 Lesson PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunit to address Mathematical Practice MP., which calls for students to reason abstractl and quantitativel. Students will epand quadratic functions in factored form so that the are in standard form. Students will then graph the standard form of the function and analze the graph to determine the relationship between the function in factored form and the -intercepts of the function. AVOID COMMON ERRORS When multipling two binomials, it can be eas to forget to multipl both parts of one binomial b both parts of the other binomial. Review the FOIL method with students so the can remember to multipl the First, Outer, Inner, and Last terms together. QUESTIONING STRATEGIES When the coefficient of the term is negative for a quadratic function in standard form, what must be true about the function in factored form? When factored as, = k ( - a) ( - b), k will be negative. Connecting Intercepts and Linear Factors 7

75 INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Remind students that the Commutative and Associative Properties of Multiplication state that factors can be rearranged and regrouped without changing the product. Therefore, when rewriting an equation from factored form into standard form, the factors can be multiplied in an order. Eplain Connecting Factors and Zeros In the Eplore ou learned that the factors in factored form indicate the -intercepts of a function. In a previous lesson ou learned that the -intercepts of a graph are the zeros of the function. Eample Write each function in standard form. Determine -intercepts and zeros of each function. = ( - 1) ( - 3) Write the function in standard form. The factors indicate the intercepts. * Factor ( 1) indicates an intercept of 1. * Factor ( 3) indicates an intercept of 3. = ( ) = ( - + 3) = - + EXPLAIN Connecting Factors and Zeros INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Review the formula for finding the ais of smmetr for a quadratic function written in standard form. Emphasize the importance of using the correct sign for each number in the equation = - b ( a ). AVOID COMMON ERRORS When epanding a quadratic function written in factored form, students ma forget to combine like terms after multipling factors. To help students check their work, remind them that a quadratic equation can have at most three terms. Houghton Mifflin Harcourt Publishing Compan The -intercepts of a graph are the zeros of the function. * An intercept of 1 indicates that the function has a zero of 1. * An intercept of 3 indicates that the function has a zero of 3. = ( + ) ( + ) Write the function in standard form. The factors indicate the intercepts * Factor ( + ) indicates an intercept of. * Factor ( - 3) indicates an intercept of. The intercepts of a graph are the zeros of the function. = = = ( ) ( ) * An intercept of indicates that the function has a zero of -. - * An intercept of indicates that the function has a zero of. Reflect 7. Discussion What are the zeros of a function? A zero of a function is an value that makes the -value equal to zero. The zeros of a function are the intercepts.. How man -intercepts can quadratic functions have?,1, or QUESTIONING STRATEGIES When a quadratic function has two zeros, how do the zeros relate to the factored form of the function? When the function is in the form, = k ( - a) ( - b) the zeros will be a and b. How can ou find the ais of smmetr for a function in factored form without rewriting it in standard form? First find the -intercepts, then average them. The ais of smmetr is at the -value that is halfwa between the -intercepts. Module 7 75 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Have each student write, then graph, a quadratic equation in factored form, = k (-a) (-b). Each student then rewrites the equation in standard form. Students trade the equations in standard form and graph the partners equations. After completing both graphs, students compare their work and discuss whether standard form or factored form was easier to graph. 75 Lesson 7.

76 Your Turn Write each function in standard form. Determine intercepts and zeros of each function. 9. = - ( + 5) ( + 1) 1. = 5 ( - 3) ( - 1) = - ( + 5) ( + 1) = - ( + + 5) = The factored form shows that intercepts are -5 and -1.; the zeros are -5 and -1. Eplain 3 Writing Quadratic Functions Given -Intercepts Given two quadratic functions ƒ () = ( - a) ( - b) and g () = k ( - a) ( - b), where k is an non-zero real constant, eamine the intercepts for each quadratic function. f () = ( - a) ( - b) = ( - a) ( - b) - a = or - b = = a = b g () = k ( - a) ( - b) = k ( - a) ( - b) = ( - a) ( - b) - a = or - b = = a = b = 5 ( - 3) ( - 1) = 5 ( - + 3) = The factored form shows that -intercepts are 1 and 3; the zeros are 1 and 3. Notice that ƒ () = ( - a) ( - b) and g () = k ( - a) ( - b) have the same -intercepts. You can use the factored form to construct a quadratic function given the intercepts and the value of k. EXPLAIN 3 Writing Quadratic Functions Given -Intercepts INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. When writing multiple quadratic functions that have the same -intercepts but different values for k, students ma have trouble visualizing how the functions are alike and how the are different. Graphing the functions on the same coordinate grid can make it easier to identif the functions similarities and differences. Eample 3 For the two given intercepts, use the factored form to generate a quadratic function for each given constant k. Write the function in standard form. -intercepts: and 5; k = 1, k = -, k =3 Write the quadratic function with k = 1. ƒ () = k ( - a) ( - b) ƒ () = 1 ( - ) ( - 5) ƒ () = ( - ) ( - 5) ƒ () = Write the quadratic function with k = -. ƒ () = -( - ) ( - 5) ƒ () = -( ) ƒ () = Houghton Mifflin Harcourt Publishing Compan Write the quadratic function with k = 3. ƒ () = 3 ( - ) ( - 5) ƒ () = 3 ( ) ƒ () = Module 7 7 Lesson DIFFERENTIATE INSTRUCTION Critical Thinking When working with a function in factored form, students can identif the -intercepts b finding -values that will make =. For a problem like, = 5 ( - ) ( + ) challenge students to find a value for that will make =, and a value for that will make + =. Review the Zero Propert of Multiplication so students understand that when one factor equals, the entire function will be equal to. Connecting Intercepts and Linear Factors 7

77 QUESTIONING STRATEGIES If ou know the two -intercepts for a quadratic function and want to write the function in factored form, does it matter which value is assigned to a and which is assigned to b? Eplain. No; it does not matter because the factors are being multiplied b each other, and b the Commutative Propert of Multiplication, the can be multiplied in either order and give the same result. B -intercepts: -3 and ; k = 1, k = -3, k = Write the quadratic function with k = 1. ƒ () = ( + 3) ( - ) ƒ () = Write the quadratic function with k = -3. ƒ () = ƒ () = -3( + 3) ( - ) Write the quadratic function with k =. ƒ () = ( + 3) ( - ) ƒ () = - - Reflect 11. How are the functions with same intercepts but different constant factors the same? How are the different? Same intercepts and zeros, same ais of smmetr; The have different vertices. Your Turn For the given two intercepts and three values of k generate three quadratic functions. Write the functions in factored form and standard form. 1. -intercepts: 1 and ; k = 1, k = -, k = intercepts: -7 and 3; k = 1, k = -5, k = 7 Houghton Mifflin Harcourt Publishing Compan k = 1: f () = 1 ( - 1) ( - ) f () = ( - 1) ( - ) f () = k = -: f () = -( - 1) ( - ) f () = - ( -9 + ) f () = k = 5: f () 5 ( - 1) ( - ) f () = 5 ( ) f () = k = 1: f () = (1 + 7) ( - 3) f () = ( + 7) ( - 3) f () = k = -5: f () = -5( +7) ( - 3) f () = -5( + - 1) f () = k = 7: f () = f () = f () = Module 7 77 Lesson LANGUAGE SUPPORT Graphic Organizers Help students to complete the following table. Caution students that a, b, and k do not represent the same values in the different forms. Standard Form Verte Form Factored Form = a + b + c = a ( - h) + k = k ( - a) ( - b) or Intercept Form ais of smmetr -b = _ a verte (h, k) ais of smmetr = h factors ( - a) and ( - b) -intercepts a and b a + b ais of smmetr = _ 77 Lesson 7.

78 Elaborate 1. If the intercepts of a quadratic function are 3 and, what can be said about the intercepts of its linear factors? The intercepts of its linear factors are 3 and. 15. If a quadratic function has onl one zero, it has to occur at the verte of the parabola. Using the graph of a quadratic function, eplain wh. If there is onl one zero, then the graph of the function has onl one intercept. This can onl occur when the verte of the parabola lies on the ais because a parabola is smmetric. 1. How are intercepts and zeros related? The intercepts of a quadratic function are the same as the zeros of a quadratic function. 17. What would the factored form look like if there were onl one intercept? f () = k ( - a) where a is the intercept. 1. Essential Question Check-In How can ou find intercepts of a quadratic function if its linear factors are known? The intercepts of a quadratic function are the same as the intercepts of its linear factors. ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Compare the factored form for a quadratic function with one -intercept, ƒ () = k ( - a ), with the verte form of a quadratic function, ƒ () = a ( - h ) + k. Students should realize that when the verte form has k =, both forms will look the same. SUMMARIZE THE LESSON How do ou identif the -intercepts of a quadratic function in factored form? For a function in the form, = k ( - a) ( - b) the -intercepts will be a and b. Houghton Mifflin Harcourt Publishing Compan Module 7 7 Lesson Connecting Intercepts and Linear Factors 7

79 EVALUATE Evaluate: Homework and Practice Graph each quadratic function and each of its linear factors. Then identif the -intercepts and the ais of smmetr of each parabola. 1. = ( - ) ( - ). = ( + 3) ( - 1) Online Homework Hints and Help Etra Practice ASSIGNMENT GUIDE Concepts and Skills Eplore Activit Connecting Factors and -Intercepts Eample 1 Rewriting from Factored Form to Standard Form Eample Connecting Factors and Zeros Eample 3 Writing Quadratic Functions Given -Intercepts Practice Eercises 1 Eercises 5 1 Eercises 11 1 Eercises 17 1 INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 Students ma want to epand a quadratic equation in factored form before entering it into a graphing calculator. Remind students that quadratic equations can be entered into a calculator in either standard or factored form. Houghton Mifflin Harcourt Publishing Compan The -intercepts are and. The -intercepts are 3 and 1. The ais of smmetr is =. The ais of smmetr is = = ( - 5) ( + ). = ( - 5) ( - 5) The -intercepts are - and 5. The intercept is 5. The ais of smmetr is = 1.5. The ais of smmetr is = 5. Module 7 79 Lesson Eercise IN_MNLESE393_U3M7L.indd 79 Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices //1 1: AM 1 Skills/Concepts MP. Precision Recall of Information MP. Precision 11 1 Skills/Concepts MP. Reasoning 15 1 Skills/Concepts MP.1 Problem Solving 17 1 Skills/Concepts MP. Precision Strategic Thinking MP. Reasoning 79 Lesson 7.

80 Write each function in standard form. 5. = 5 ( - ) ( + 1). = ( + ) ( + 3) = 5 ( - - ) = = - ( + ) ( - 5). = - ( + ) ( + 3) = - ( - - ) = Which of the following is the correct standard form of = 3 ( - ) ( - 5)? a. = b. = c. = d. = e. = The area of a Japanese rock garden is = 7 ( - 3) ( + 1). Write = 7 ( - 3) ( + 1) in standard form. = 7 ( - - 3) = = 3 ( ) = = ( ) = = - ( ) = CRITICAL THINKING Discuss with students wh the graphs of the factors of a quadratic equation in factored form are alwas parallel. Note that when a function is in the form = k ( - a) ( - b), the graphs for the factors are the lines = - a and = - b. Students should realize that these lines have the same slope, so the must be parallel. AVOID COMMON ERRORS When epanding a quadratic function written in factored form, students ma forget to combine the results from multipling the inner and outer terms of the factors. Remind students that a quadratic equation in standard form should have at most one term, one term, and one constant. Write each function in standard form. Determine intercepts and zeros of each function. 11. = - ( - ) ( - ) 1. = ( + ) ( - ) = - ( - + ) = The factored form shows that -intercepts are and. The zeros are and. 13. = -3 ( + 1) ( - 3) 1. = ( + ) ( - 1) = -3 ( - - 3) = The factored form shows that -intercepts are -1 and 3. The zeros are -1 and 3. = ( + - ) = The factored form shows that intercepts are - and. The zeros are - and. = ( + - ) = + - The factored form shows that intercepts are - and 1. The zeros are - and 1. Module 7 Lesson Houghton Mifflin Harcourt Publishing Compan Image Credits: David Maska/Shutterstock INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Relate the factored form of a quadratic function to the factors of an integer. Students should understand that just as an integer can have man factor pairs, a quadratic function can be factored in man different was. The factored form is presented as = k ( - a) ( - b) where k so that the -intercepts will be eas to identif as a and b. When a quadratic equation is written so that the terms in the factors have coefficients other than 1, the -intercepts are not as readil apparent. For eample, in the equation = (3 - ) ( + ), the -intercepts are not and -. Discuss with students how to divide each factor b a constant so that the -intercepts can be determined more easil. Connecting Intercepts and Linear Factors

81 JOURNAL Have students eplain how a quadratic equation looks when written in factored form, and how the can use the information in the factored form of a quadratic equation to find the zeros, -intercepts, and ais of smmetr of a function. 15. A soccer ball is kicked from ground level. The function = -1 ( - ) gives the height (in feet) of the ball, where is time (in seconds). After how man seconds will the ball hit the ground? Use a graphing calculator to verif our answer. The -intercepts are and. The ball will hit the ground in seconds. 1. A tennis ball is tossed upward from a balcon. The height of the ball in feet can be modeled b the function = -( + 1) ( - 3) where is the time in seconds after the ball is released. Find the maimum height of the ball and the time it takes the ball to reach this height. Graph the function to determine approimatel how long it takes the ball to hit the ground. The -intercepts are -.5 and 1.5. It will take 1.5 seconds to hit the ground. For the two given intercepts, use the factored form to generate a quadratic function for each given constant k. Write the function in standard form intercepts: -5 and 3; k = 1, k = -, k = intercepts: and 7; k = 1, k = -3, k = 5 k = 1: f () = 1 ( + 5) ( - 3) k = 1: f () = 1 ( - ) ( - 7) f () = k = -: f () = - ( + 5) ( - 3) f () = k = 5: f () = 5 ( + 5) ( - 3) f () = f () = k = -3: f () = -3 ( - ) ( - 7) f () = k = 5: f () = 5 ( - ) ( - 7) f () = H.O.T. Focus on Higher Order Thinking 19. Eplain the Error For the given two intercepts, 3 and 9, k =, Kell wrote a quadratic function in factored form, ƒ () = ( + 3) ( + 9), and in standard form, f () = What error did she make? The factored form is f () = ( - 3) ( - 9) and the standard form is f () = Kell substituted negative values for the intercepts. Houghton Mifflin Harcourt Publishing Compan. Critical Thinking How is the graph of ƒ () = 7 ( + 3) ( - ) different from the graph of ƒ () = ? The graph of f () = 7 ( + 3) ( - ) opens upward, and has the minimum value for the function, and the graph of f () = opens downward, and has the maimum value for the function. 1. Make a Prediction How could ou find an equation of a quadratic function with zeros at 3 and at 1? Find the product ( - (-3) ) ( - 1), or ( + 3) ( - 1). Then write a quadratic function using this product. So a possible quadratic function is = Module 7 1 Lesson 1 Lesson 7.

82 Lesson Performance Task The cross-sectional shape of the archwa of a bridge (measured in feet) is modeled b the function ƒ () = where is the distance between the two sides of the arch. How wide is the arch at its base? Will a bo truck that is feet wide and 13.5 feet tall fit under the arch? If not, what is the maimum height a truck feet wide passing under the bridge can be? Write the function in intercept form, and then identif the intercepts of the graph. f () = Height (ft) = -.5 ( - 1) = -.5 ( - 1) = -.5 ( - ) ( - 1) The intercepts are and 1. The width of the arch at its base is the horizontal distance between the two intercepts. The horizontal distance between and 1 is 1 units. So, the arch is 1 units wide at its base. Find the verte. The -coordinate of the verte is halfwa between and 1, or. f () = -.5 ( - ) ( - 1) = -.5 () (-) = 1 The coordinate of the verte is 1. So, the verte is (, 1) Graph the function and sketch the bo truck centered on the graph to determine if it fits under the arch. 1 1 Height of Arch Distance from left end (ft) The upper corners of the bo truck are higher than the arch, so the bo truck does not fit under the arch. The bo truck is feet wide. Since it is centered on the arch, the corners are an equal distance from the ais of smmetr, =. So, the corners are located at = and = 1 The maimum height a truck can be to fit under the arch is equal to the value of the function evaluated at one of these two values. f () = -.5 ( - ) ( - 1) = -.5 ( - ) ( - 1) = -.5 () (-1) = 1 So, the maimum height a truck can be to fit under the arch is 1 feet. Houghton Mifflin Harcourt Publishing Compan LANGUAGE SUPPORT Some students ma not understand the terms cross-section, bo truck, and arch as used in the problem in the Lesson Performance Task. Demonstrate what a cross-section of a tennis ball or other object looks like. Have volunteers describe a bo truck and an arch. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Have students construct a graphic organizer showing the graph of the archwa of the bridge and its equation in verte form, standard form, and factored form. Verte Form: f() = -.5( - ) + 1 V = (, ) Standard Form: f() = Factored Form: f() = -.5( - )( - 1) Height (ft) 1 1 Height of Arch Distance from left end (ft) -intercepts and 1 Module 7 Lesson EXTENSION ACTIVITY Have students use a coordinate grid to design an arched entrancewa tall enough and wide enough so that the can walk through the archwa with their heads just touching the top and their arms etended straight out from side to side. Have students model the arched entrance b a quadratic function. Students ma find that the can use three points representing the top of the head and the tips of the fingers to define a parabola, then translate the parabola verticall to make it the correct total height. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. points: Student does not demonstrate understanding of the problem. Connecting Intercepts and Linear Factors

83 COMMON CORE Locker LESSON 7.3 Appling the Zero Product Propert to Solve Equations Name Class Date 7.3 Appling the Zero Product Propert to Solve Equations Essential Question: How can ou use the Zero Product Propert to solve quadratic equations in factored form? Resource Locker Common Core Math Standards The student is epected to: COMMON CORE A-REI.B. Solve quadratic equations in one variable. Also A-APR.B.3, A-SSE.A., A-SSE.B.3 Mathematical Practices COMMON CORE MP. Reasoning Language Objective Eplain what the Zero Product Propert sas, and give an eample. ENGAGE Essential Question: How can ou use the Zero Product Propert to solve quadratic equations in factored form? You can use the Zero Product Propert to set each linear factor equal to and then solve each resulting linear equation. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss wh it might be a good idea for a pole vaulter to understand the shape of her path as she sails over the bar. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing Compan Eplore Understanding the Zero Product Propert For all real numbers a and b, if the product of the two quantities equals zero, then at least one of the quantities equals zero. Zero Product Propert For all real numbers a and b, the following is true. Words Sample Numbers Algebra If the product of two quantities equals zero, at least one of the quantities equals zero. A B C D E 9 ( ) = () = If ab =, then a = or b =. Consider the equation ( - 3) ( + ) =. Let a = - 3 and b = +. Since ab =, ou know that a = or b =. - 3 = or + = Solve for. - 3 = or + = = 3 = - So, the solutions of the equation ( - 3) ( + ) = are = 3 and = -. Recall that the solutions of an equation are the zeros of the related function. So, the solutions of the equation ( - 3) ( + ) = are the zeros of the related function ( - 3) ƒ () = ( + ) because the satisf the equation ƒ () =. The solutions of the ( - 3) ( + ) 3 related function ƒ () = are and -. Reflect 1. Describe how ou can find the solutions of the equation ( - a) ( - b) = using the Zero Product Propert. Let - a = and - b =, and then solve each equation for. The solutions are a and b. Module 7 3 Lesson 3 Name Class Date 7.3 Appling the Zero Product Propert to Solve Equations Essential Question: How can ou use the Zero Product Propert to solve quadratic equations in factored form? A-REI.B. Solve quadratic equations in one variable. Also A-APR.B.3, A-SSE.A., A-SSE.B.3 Houghton Mifflin Harcourt Publishing Compan Eplore Understanding the Zero Product Propert For all real numbers a and b, if the product of the two quantities equals zero, then at least one of the quantities equals zero. For all real numbers a and b, the following is true. If the product of two quantities equals zero, at least one of the quantities equals zero. Zero Product Propert Words Sample Numbers Algebra a + 9 ( ) = () = If ab =, then = or b =. Consider the equation ( - 3)( + ) =. Let a = - 3 and b =. Since ab =, ou know that a = or b =. = or + = Solve for. - 3 = or + = = = So, the solutions of the equation ( - 3)( + ) = are = and =. Recall that the solutions of an equation are the zeros of the related function. So, the solutions of the equation ( - 3)( + ) = are the zeros of the related function ƒ() = because the satisf the equation ƒ() =. The solutions of the related function ƒ() = are and. Reflect 3 - ( - 3) ( + ) - 3 ( - 3)( + ) 3 - Resource 1. Describe how ou can find the solutions of the equation ( - a) ( - b) = using the Zero Product Propert. Let - a = and - b =, and then solve each equation for. The solutions are a and b. Module 7 3 Lesson HARDCOVER PAGES 3 9 Watch for the hardcover student edition page numbers for this lesson. 3 Lesson 7.3

84 Eplain 1 Appling the Zero Product Propert to Functions When given a function of the form ƒ () = ( + a) ( + b), ou can use the Zero Product Propert to find the zeros of the function. Eample 1 Find the zeros of each function. ƒ () = ( - 15) ( + 7) Set ƒ () equal to zero. ( - 15) ( + 7) = Appl the Zero Product Propert = or + 7 = Solve each equation for. = 15 = -17 The zeros are 15 and -7. ƒ () = ( + 1) ( + 3) Set ƒ () equal to zero. ( + 1) ( + 3) = Appl the Zero Product Propert. + 1 = or + 3 = Solve for. = -1 = -3 The zeros are -1 and -3. Reflect. Discussion Jordie was asked to identif the zeros of the function ƒ () = ( - 5) ( + 3). Her answers were = -5 and = 3. Do ou agree or disagree? Eplain. Disagree; If ( - 5) ( + 3) =, then - 5 = or + 3 = ; so = 5 or = -3 EXPLORE Understanding the Zero Product Propert INTEGRATE TECHNOLOGY To check their answers for the zeros of a function, students can enter the equation into a graphing calculator and use the TABLE function to find the -values for which =. QUESTIONING STRATEGIES If an equation has two factors, a and b, and ab =, can both a and b equal? Eplain. Yes; the Zero Product Propert states that at least one of the factors a and b must equal zero. It is possible that both factors are equal to zero. 3. How would ou find the zeros of the function ƒ () = -( - )? The -values that make f() = are zeros of the function, so solve -( - ) =. The onl solution is.. What are the zeros of the function ƒ () = ( - 1)? Eplain. and 1 because the are the solutions of ( - 1) =. Your Turn Find the zeros of each function. 5. ƒ () = ( - 1) ( - ). ƒ () = 7 ( - 13) ( + 1) ( - 1) ( - ) = 7 ( - 13) ( + 1) = - 1 =, so = 1, or Since 7, set onl the factors ( - 13) - =, so =. and ( + 1) equal to and solve. The zeros are 1 and =, so = 13, or + 1 =, so = -1. The zeros are 13 and -1. Module 7 Lesson 3 Houghton Mifflin Harcourt Publishing Compan EXPLAIN 1 Appling the Zero Product Propert to Functions AVOID COMMON ERRORS When finding the zeros of a quadratic function, students sometimes choose the constants in the binomial factors as the solutions. For eample, students might sa that the zeros of f () = ( - ) ( - 9) are - and -9. Remind students to find the value of that would make each factor equal. For this eample, the zeros are and 9. PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunit to address Mathematical Practice MP., which calls for students to reason abstractl and quantitativel. Students will rewrite quadratic equations so that the can appl the Zero Product Propert to find the solutions. The build on their previous understanding of the connection between zeros, -intercepts, and factors of quadratic functions, as well as the relationship between different forms of quadratic equations. In working with real-world problems, students interpret the solutions of equations to connect them to the real-world contet. QUESTIONING STRATEGIES When using the Zero Product Propert to find the zeros of a function, wh do ou set each factor equal to zero? You set each factor equal to zero because when at least one of the factors is zero, the product is equal to zero. Appling the Zero Product Propert to Solve Equations

85 EXPLAIN Solving Quadratic Equations Using the Distributive Propert and the Zero Product Propert AVOID COMMON ERRORS When instructed to use the Distributive Propert to rewrite an equation, students ma think that the should epand the equation. Remind students that if the want to use the Zero Product Propert to find the solutions to the equation, the should use the Distributive Propert in the other direction, that is, to rewrite the equation as the product of factors. QUESTIONING STRATEGIES What must be true about an equation for it to be possible to use the Distributive Propert to rewrite that equation as the product of factors? The same factor must appear in more than one term of the equation. How can ou verif that the solutions to an equation are correct? You can substitute the values into the original equation to check that the make the equation true. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 For some equations, it is necessar to factor part of the equation before using the Distributive Propert to rewrite the equation in factored form. Remind students to look for terms with a common factor. Houghton Mifflin Harcourt Publishing Compan Eplain Solving Quadratic Equations Using the Distributive Propert and the Zero Product Propert The Distributive Propert states that, for real numbers a, b, and c, a (b + c) = ab + ac and ab + ac = a (b + c). The Distributive Propert applies to polnomials, as well. For instance, 3 ( - ) + 5 ( - ) = (3 + 5) ( - ). You can use the Distributive Propert along with the Zero Product Propert to solve certain equations. Eample Solve each equation using the Distributive Propert and the Zero Product Propert. 3 ( - ) + 5 ( - ) = Use the Distributive Propert to rewrite 3 ( - ) + 5 ( - ) = (3 + 5) ( - ) the epression 3 ( - ) + 5 ( - ) as a product. Rewrite the equation. (3 + 5) ( - ) = Appl the Zero Product Propert = or - = Solve each equation for. 3 = -5 = = - 5_ 3 The solutions are = - 5_ and =. 3-9 ( + ) + 3 ( + ) = Use the Distributive Propert to rewrite -9( + ) + 3 ( + ) = ( ) ( ) + the epression -9( + ) + 3 ( + ) as a product. -9 Rewrite the equation. ( + 3) ( + ) = Appl the Zero Product Propert = or + = Solve each equation for. 3 = 9 = - = 3 The solutions are = 3 and = -. Reflect 7. How can ou solve the equation 5 ( - 3) = using the Distributive Propert? Factor - 1 as ( - 3) first. Then 5 ( - 3) + ( - 3) = becomes (5 + ) ( - 3) =. Your Turn Solve each equation using the Distributive Propert and the Zero Product Propert.. 7 ( - 11) - ( - 11) = 9. - ( + ) = (7 - ) ( - 11) = 7 - = or - 11 = = 7 = 11 - ( + ) + 3 ( + ) = (- + 3) ( + ) = = or + = = 3 = - Module 7 5 Lesson 3 COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Have each student write two factored equations: one in the form = k ( - a) ( - b), and one that is factored so that the terms have coefficients other than 1. Students solve their equations, then trade equations and solve both equations written b their partners. Have partners discuss how the solved each equation and tell which equations the found easiest to solve. 5 Lesson 7.3

86 Eplain 3 Eample 3 Solving Real-World Problems Using the Zero Product Propert The height of one diver above the water during a dive can be modeled b the equation h = - (t + 5) (t - 3), where h is height in feet and t is time in seconds. Find the time it takes for the diver to reach the water. EXPLAIN 3 Solving Real-World Problems Using the Zero Product Propert Analze Information Identif the important information. The height of the diver is given b the equation h = -(t + 5) (t - 3). The diver reaches the water when h =. Formulate a Plan To find the time it takes for the diver to reach the water, set the equation equal to and use the Propert to solve for t. Solve Set the equation equal to zero. -(t + 5) (t - 3) = Appl the Zero Product Propert. t + 5 = or t - 3 = Since -, set the other factors equal to. Solve each equation for. t + 5 = or t - 3 = t = -5 t = 3 t = The zeros are t = - 5 and t = 3. Since time cannot be negative, the time it takes for the diver to reach the the water is 3 seconds. Justif and Evaluate Check to see that the answer is reasonable b substituting 3 for t in the equation -(t + 5) (t - 3) = ((3) + 5) ((3) - 3) = -( + 5) ( - 3) = -( 17 )( ) - 5 Houghton Mifflin Harcourt Publishing Compan Image Credits: Paul Souders/Corbis INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 When working with real-world problems that give the factored form of a quadratic function modeling the height of a falling object, have students multipl the factors and write the function in standard form. The should find that all the functions fit the same standard form: ƒ (t) = 1 t + v t + h, where ƒ (t) is the height in feet at time t, t is the time in seconds, v is the initial vertical velocit, and h is the initial height. (The coefficient of t will differ if the units are different.) Eplain that this general form is found in all problems modeling falling objects because the force of gravit affects the motion of all objects in the same wa. = Since the equation is equal to for t = 3, the solution is reasonable. The diver will reach the water after 3 seconds. Module 7 Lesson 3 DIFFERENTIATE INSTRUCTION Multiple Representations For problems involving quadratic functions that model the height of a falling object, have students create a table of values for the height of the object. Continue the table until the object reaches a height of. Point out to students that the time when h = represents the time it takes for the object to reach the ground. Discuss the differences between using this method and using the Zero Product Propert to find the solution to a falling object problem. Appling the Zero Product Propert to Solve Equations

87 QUESTIONING STRATEGIES Not all quadratic functions can be written in a factored form. Can quadratic functions that model the height of a falling object alwas be written in factored form? Eplain. Yes; a function modeling the height of a falling object will have a value of when the object reaches the ground, so it will have -intercepts. An quadratic function that has -intercepts can be written in factored form. ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Discuss with students whether the Zero Product Propert can be applied onl to quadratic functions. Students should realize that the Zero Product Propert can be used to find the -intercept of an function. Reflect 1. If ou were to graph the function ƒ (t) = -(t + 5) (t - 3), what points would be associated with the zeros of the function? The -intercepts of the graph of the function, 5 (-, ) and (3, ) Your Turn 11. The height of a golf ball after it has been hit from the top of a hill can be modeled b the equation h = - (t - ) (t + 1), where h is height in feet and t is time in seconds. How long is the ball in the air? The ball is in the air from the time it leaves the ground to the time it returns to the ground. The height of the ball when it returns to the ground is h =. Set the equation equal to. -(t - ) (t + 1) = Since -, set the factors (t - ) and (t + 1) equal to and solve each equation. t - =, so t =, or t + 1 =, so t = -1. The zeros are t = and t = -1. Since time cannot be negative, the ball is in the air for seconds. Elaborate 1. Can ou use the Zero Product Propert to find the zeros of the function ƒ () = ( - 1) + ( - 9)? Eplain. No; the epression ( - 1) + ( - 9) is not a product of factors, so the Zero Product Propert cannot be used. SUMMARIZE THE LESSON How do ou solve a quadratic equation using the Zero Product Propert? If the function is in the form f () = ( + a) ( + b), ou can use the Zero Product Propert b setting both + a = and + b =, and finding both solutions for. Houghton Mifflin Harcourt Publishing Compan 13. Suppose a and b are the zeros of a function. Name two points on the graph of the function and eplain how ou know the are on the graph. What are the points called? (a, ) and (b, ) ; zeros of a function are values of that make the function value ; (a, ) and (b, ) are the -intercepts of the graph of the function. 1. Essential Question Check-In Suppose ou are given a quadratic function in factored form that is set equal to. Wh can ou solve it b setting each factor equal to? The Zero Product Propert states that if the product of two numbers is, then at least one of the numbers must be. A quadratic function in factored form is the product of two linear factors, so at least one of the factors must be. Module 7 7 Lesson 3 LANGUAGE SUPPORT Connect Vocabular When encountering the Zero Product Propert for the first time, the formal construction of the sentences in the propert ma be difficult for English learners to understand. Point out that the phrase the following is true directs the reader s attention to the information that follows and emphasizes that it is true. 7 Lesson 7.3

88 Evaluate: Homework and Practice Find the solutions of each equation. 1. ( - 15) ( - ) =. ( + ) ( - 1) = - 15 = or - = + = or - 1 = = 15 = = - = 1 Online Homework Hints and Help Etra Practice EVALUATE Find the zeros of each function. 9) ( + 1_ ) 9 )( 1_ + ) = 3. ƒ () = ( + 15) ( + 17). ƒ () = ( - _ ( + 15) ( + 17) = + 15 = or + 17 = = -15 = -17 ( _ - _ - 1_ 9 = or + = = _ = - 1_ 9 5. ƒ () = -. ( - 1.9) ( - 3.5). ƒ () = ( + ) -. ( - 1.9) ( - 3.5) = = or = = 1.9 = 3.5 ( - ) ( + ) = - = or - = = = ASSIGNMENT GUIDE Concepts and Skills Eplore Understanding the Zero Product Propert Eample 1 Appling the Zero Product Propert to Functions Eample Solving Quadratic Equations Using the Distributive Propert and the Zero Product Propert Eample 3 Solving Real-World Problems Using the Zero Product Propert Practice Eercises 1 Eercises 3, Eercises 9 1, 1, 5 Eercises 15, 3 7. ƒ () = 3_ ( - 3_ 3 ( - 3 ) = - 3 = = 3 ). ƒ () = ( + ) ( + ) ( + ) ( + ) = + = or + = = - = - Houghton Mifflin Harcourt Publishing Compan Module 7 Lesson 3 Eercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices Recall of Information MP. Reasoning 1 Skills/Concepts MP. Reasoning 15 Skills/Concepts MP. Modeling 1 Skills/Concepts MP. Reasoning 3 Strategic Thinking MP.3 Logic 3 3 Strategic Thinking MP. Modeling Appling the Zero Product Propert to Solve Equations

89 INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Discuss with students whether the Zero Product Propert can be used to find the zero of a function that has onl one zero. Students should understand that a function with onl one zero will have the form f () = ( + a) ( + a) or f () = ( + a). Solve each equation using the Distributive Propert and the Zero Product Propert ( + 1) - 15 ( + 1) = 1. 1 ( - 3) - ( - 3) = (- - 15) ( + 1) = - = 15 or = = ( + 3) _ + ( + 3) _ (5 + 1) ( + _ 3 ) = 5 = -1 or = - _ 3 = - 1_ 5 (1 - ) ( - 3) = = 1 or = 3. = 1. - ( + ) + ( + ) = (-1 + ) ( + ) = -1 = - or = - 1 = (9 - ) + 1_ (9 - ) = 1. -( - 3) = 3 ( 1_ )(9 - ) = -( - 3) + ( - 3) = 7 = - 1_ 3 or 9 = (- + ) ( - 3) = = - 1_ - = - or = 3 1 = Houghton Mifflin Harcourt Publishing Compan Solve using the Zero Product Propert. 15. The height of a football after it has been kicked from the top of a hill can be modeled b the equation h = (- -t) (t - 5), where h is the height of the football in feet and t is the time in seconds. How long is the football in the air? (- - t) (t - 5) = - = t or t = 5-1_ 5_ = t t = Module 7 9 Lesson 3 Eercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 3 Strategic Thinking MP.3 Logic 5 3 Strategic Thinking MP.1 Problem Solving 9 Lesson 7.3

90 1. Archer Klie shoots an arrow during an archer lesson at camp. The height of the arrow can be modeled b the equation h = -t (t - ), where h is the height in feet of the arrow and t is the time in seconds. How long is the arrow in the air? -t (t - ) = -t = or t - = t = t = t = 3 The solutions are t = and t = 3. Since the starting time is, the arrow is in the air for 3 seconds. CRITICAL THINKING Not all quadratic equations are shown in the factored form = k (-a) (-b), where a and b are the zeros of the function. After finding the zeros of a function not in this form, discuss with students was to rewrite the function so that the zeros will be more evident. 17. Phsics The height of a flare fired from a platform can be modeled b the equation h = t (-t + 1) + (-t + 1), where h is the height of the flare in meters and t is the time in seconds. Find the time it takes for the flare to reach the ground. -t (t + 1) + (-t + 1) = (-t + ) (-t + 1) = t + = or -t + 1 = t = - -t = -1 t = -.5 t = 5 The solutions are t = -.5 and t = 5. Since time cannot be negative, the flare is in the air for 5 seconds. 1. Diving The depth of a scuba diver can be modeled b the equation d =.5t (3.5t -.5), where d is the depth in meters of the diver and t is the time in minutes. Find the time it takes for the diver to reach the surface. Give our answer to the nearest minute..5t (3.5t -.5) =.5t = or 3.5t -.5 = t = 3.5t =.5 t The solutions are t = and t. Since the starting time is, the time it takes for the diver to reach the surface is about minutes. Houghton Mifflin Harcourt Publishing Compan Image Credits: Appl Picture/Alam Module 7 9 Lesson 3 Appling the Zero Product Propert to Solve Equations 9

91 AVOID COMMON ERRORS In a factored quadratic equation, it is eas to mistake the constants for the zeros. Remind students that finding the zeros of the function requires finding the values of that make each factor equal to. 19. A group of friends tries to keep a small beanbag from touching the ground b kicking it. On one kick, the beanbag s height can be modeled b the equation h = - (t - 1) - 1t (t - 1), where h is the height of the beanbag in feet and t is the time in seconds. Find the time it takes the beanbag to reach the ground. (t - 1) - 1t (t - 1) = ( - 1t) (t - 1) = = 1t or t = 1_ = t =.15 The solutions are t = -.15 and t = 1. Since time cannot be negative, the time it takes for the beanbag to reach the ground is 1 second.. Elizabeth and Markus are plaing catch. Elizabeth throws the ball first. The height of the ball can be modeled b the equation h = -1t (t - 5), where h is the height of the ball in feet and t is the time in seconds. Markus is distracted at the last minute and looks awa. The ball lands at his feet. If the ball travels horizontall at an average rate of 3.5 feet per second, how far is Markus standing from Elizabeth when the ball hits the ground? -1t (t - 5) = -1t = or t - 5 = t = t = 5 The solutions are t = and t = 5. Since the starting time is, the ball is in the air for 5 seconds. Find the distance the ball travels in that time. d = rt = (3.5)(5) = 17.5 Markus is standing 17.5 feet awa from Elizabeth when the ball hits the ground. 1. Match the function on the left with its zeros on the right. Indicate a match b writing the letter for a function on the line in front of the corresponding values of and. A. f () = 11 ( - 9) + ( - 9) E a. = -11 and = -9 (11 + ) ( - 9) = Houghton Mifflin Harcourt Publishing Compan B. f () = ( + 9) ( - 11) A, D b. = 9 and = -11 ( + 9) ( - 11) = C. f () = 11 ( - 9) - ( - 9) C c. = 9 and = 11 (11 - ) ( - 9) = D. f () = ( - 9) ( + 11) B d. = -9 and = 11 ( - 9) ( + 11) = E. f () = - ( + 9) - 11 ( + 9) (- - 11) ( + 9) = Module 7 91 Lesson 3 91 Lesson 7.3

92 H.O.T. Focus on Higher Order Thinking. Eplain the Error A student found the zeros of the function f () = ( - 5) + ( -5). Eplain what the student did wrong. Then give the correct answer. ( - 5) + ( - 5) = ( - 5) =, so =, and =, or - 5 =, so = 5, or ( - 5) =, so = 5 Zeros:, 5 and 5 The student should have used the Distributive Propert to rewrite the equation as ( + ) ( - 5) =. Then + =, so = - 3, or - 5 =, so = 5 Zeros: and 5 3. Draw Conclusions A ball is kicked into the air from ground level. The height h in meters that the ball reaches at a distance d in meters from the point where it was kicked is given b h = - d (d - ). The graph of the equation is a parabola. a. At what distance from the point where it is kicked does the ball reach its maimum height? Eplain. The ball is on the ground when h =. - d (d - ) = - d =, so d =, or d - =, so d =. The solutions are d = and d =. Since the ball starts at d =, it will hit the ground at d =. Since the curve is smmetric, the ball will be at its maimum height at half this distance, or meters. b. Find the maimum height. What is the point (, h) on the graph of the function called? The ball is at its maimum height when d =. Substitute for d in the equation andsolve for h. h = -()( - ) = - (- ) = The maimum height is meters. The point (, h), or (, ), is the verte of the graph of the function. Houghton Mifflin Harcourt Publishing Compan Module 7 9 Lesson 3 Appling the Zero Product Propert to Solve Equations 9

93 JOURNAL Have students write a paragraph that compares finding the zeros of a function using the Zero Product Propert to other methods the have learned for finding the zeros of a function. Ask students to tell what form the function should take in order for the Zero Product Propert to be used.. Justif Reasoning Can ou solve ( - ) ( + 3) = 5 b solving - = 5 and + 3 = 5? Eplain. No. You can t use the Zero Product Propert because the product is not equal to. If - = 5 then = 7, and if + 3 = 5, then =. Neither 7 nor is a solution of ( - ) ( + 3) = 5. Instead, (7 - ) (7 + 3) = 5 and ( - ) ( + 3) =. 5. Persevere in Problem Solving Write an equation to find three numbers with the following properties. Let be the first number. The second number is 3 more than the first number. The third number is times the second number. The sum of the third number and the product of the first and second numbers is. Solve the equation and give the three numbers. First number: Houghton Mifflin Harcourt Publishing Compan Second number: + 3 Third number: ( + 3) The sum of the third number and the product of the first and second numbers is. ( + 3) + ( + 3) = ( + ) ( + 3) = + =, so = -, or + 3 =, so = - 3 Suppose = - First number: - Second number: = -1 Third number: (- + 3) = (-1) = - Suppose = -3 First number: -3 Second number: = - Third number: (-3 + 3) = () = So, the three numbers are either -, -1, and -, or -3,, and Module 7 93 Lesson 3 93 Lesson 7.3

94 Lesson Performance Task The height of a pole vaulter as she jumps over the bar is modeled b the function ƒ (t) = (t - ) (t - 3.5), where t is the time at which the pole vaulter leaves the ground. a. Find the solutions of the function when t = using the Zero Product Propert. What do these solutions mean in the contet of the problem? b. If the bar is feet high, will the pole vaulter make it over? a. f (t) = (t - ) (t - 3.5) t - = t = t = t = 3.5 The solutions are t = and t = 3.5 This means that the pole vaulter left the ground at seconds and returned to the ground after 3.5 seconds. So, the pole vaulter was in the air for 3.5 seconds. b. Find the verte of the parabola. First, rewrite the function in standard form. f (t) = -1.75(t - ) (t - 3.5) = ( t - 3.5t) = t +.15t a) ). The verte is ( - b _ a, f ( - b _ - _ b a = - _.15 = 1.75 (- 1.75) f ( - b_ a) = f (1.75) = -1.75(1.75) +.15 (1.75) 5.3 The pole vaulter s maimum height is 5.3 feet, so she will not make it over the -foot high bar. Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 To check their results, have students graph the function using a graphing calculator, then use the TRACE feature or the TABLE feature to check the maimum height and the -intercepts that the calculated. INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have students eplain wh one zero of the function is. The should understand that a pole vaulter is at height both when starting to jump and when landing. Module 7 9 Lesson 3 EXTENSION ACTIVITY Have students research the Fosbur Flop and investigate how a high jumper tries to control the location of his or her center of gravit in order to reach the maimum height possible. Students ma find that an athlete s center of mass is about two-thirds of the wa up the bod when standing or running. However, b curling themselves into an inverted U-shape as the sail over the bar, high jumpers cause their centers of gravit to drop below the bar. In this wa the are able to minimize the energ needed to jump over the bar. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. points: Student does not demonstrate understanding of the problem. Appling the Zero Product Propert to Solve Equations 9

95 MODULE 7 Stud Guide Review ASSESSMENT AND INTERVENTION STUDY GUIDE AND REVIEW Connecting Intercepts, Zeros, and Factors Essential Question: How can ou use intercepts of a quadratic function to solve real-world problems? KEY EXAMPLE (Lesson 7.) Generate the quadratic function with -intercepts 3 and - and k = 3. Write the function in factored form and standard form. Ke Vocabular Zero Product Propert (Propiedad del producto cero) zero of a function (cero de una función) MODULE 7 Assign or customize module reviews. MODULE PERFORMANCE TASK COMMON CORE Mathematical Practices: MP.1, MP., MP. A-CED.A., F-BF.A.1 SUPPORTING STUDENT REASONING Students should begin this problem b focusing on what information the will need. Here is some of the information the ma ask for. The horizontal distance between point A and point B: This information is not given but is up to the student. Students must choose a value between 3 meters and meters. The coordinates of the points on the diagram: The coordinates of point A can be seen on the diagram. The -coordinates of points B and C will depend on the width chosen. The location of the verte: From the given information and the diagram, students should be able to deduce the location of the verte themselves. Houghton Mifflin Harcourt Publishing Compan Write the quadratic function with k = 3. Substitute the given values of the -intercepts and k into ƒ () = k ( - a) ( - b) and simplif. ƒ () = 3 ( + ) ( - 3) Write ƒ () = 3 ( + ) ( - 3) in standard form. ƒ () = 3 ( + ) ( - 3) = KEY EXAMPLE (Lesson 7.3) Solve ( - ) + ( - ) = b using the Distributive Propert and the Zero Product Propert. Use the Distributive Propert to rewrite ( - ) + ( - ) = ( + ) ( - ) the epression ( - ) + ( - ) as a product of binomials. Rewrite the equation. ( + ) ( - ) = Appl the Zero Product Propert. + = and - = Solve each equation for. = - = The solutions are = - and =. = - Module 7 95 Stud Guide Review SCAFFOLDING SUPPORT Although it is not strictl necessar to solve the problem, ou ma wish to remind students of the verte form of a quadratic equation, = ( - h) + k. Solution methods ma include using the verte form of a quadratic equation, using quadratic regression, or other methods. An method that leads to a correct equation is acceptable. 95 Module 7

96 EXERCISES Solve each equation b graphing. (Lesson 7.1) =. ( + ) - = SAMPLE SOLUTION Assumptions The width of the skateboard ramp is 5 m The verte (point B) is half the ramp width, or.5 m. The diagram shows the verte is.75 m above the ground. Substitute (.5,.75) ) for (h, k) into the verte form of a quadratic equation. = -1 and 3 = - and Write a function in factored and standard form for each k and set of -intercepts. (Lesson 7.) 3. -intercepts: -3 and ; k = -. -intercepts: 7 and ; k = 3 f () = -( + 3) ( -) f () = 3 ( - ) ( -7) = a ( - h) + k = a ( -.5) +.75 The diagram shows that one point on the parabola is (, 3). Substitute (, 3) for (, ) and solve for a. f () = Find the zeros of each function. (Lesson 7.3) f () = = a ( -.5) = a (.5) 5. ƒ () = ( + 17). ƒ () = - ( -.3) ( -.).3 = a = -17 or MODULE PERFORMANCE TASK Designing a Skateboard Ramp =.3 or. In verte form, the equation that models the parabola is =.3 ( -.5) The standard form of this equation that models the ramp curve is = The local government has made partial plans for the construction of a skateboard ramp, which are shown here. Your task is to complete the plans b modeling the parabolic curve of the ramp itself. First, choose the total width of the parabola from point A to point C, which should be between 3 meters and meters. Then, create an equation that models the parabola that starts at point A, reaches a minimum at point B, and ends at point C. Note that the - and -aes are marked in the diagram. Epress our equation in standard form. 3 m A B?.75 m C Houghton Mifflin Harcourt Publishing Compan Use our own paper to complete the task, using graphs, numbers, or algebra to eplain how ou reached our conclusion. Module 7 9 Stud Guide Review DISCUSSION OPPORTUNITIES Some ramps use semicircular curves, or have a flat portion at the bottom of the ramp. Discuss how closel the curve of a parabola matches an actual half-pipe ramps students have seen. Students ma be interested in making additional recommendations to the local government about the construction of future half-pipe skateboard ramps. Assessment Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain. points: Student does not demonstrate understanding of the problem. Stud Guide Review 9

97 Read to Go On? ASSESS MASTERY Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module. Read to Go On? Connecting Intercepts, Zeros, and Factors Solve each equation b graphing. (Lesson 7.1) Online Homework Hints and Help Etra Practice = = ASSESSMENT AND INTERVENTION Access Read to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment. ADDITIONAL RESOURCES Response to Intervention Resources Reteach Worksheets Differentiated Instruction Resources Reading Strategies Success for English Learners Challenge Worksheets Assessment Resources Leveled Module Quizzes Houghton Mifflin Harcourt Publishing Compan = Write a function in factored and standard form for each k and set of -intercepts. (Lesson 7.) 3. -intercepts: 5 and -7; k = -3. -intercepts: -1 and -; k = f () = -3 ( - 5) ( + 7) f () = Find the zeros of each function. (Lesson 7.3) 5. ƒ () = - ( + 7) ( -.). ƒ () = ( - ) = -7 or. 7. ƒ () = 9 ( - ) + 3 ( - ). ƒ () = -( + ) + + = or = - or 3 ESSENTIAL QUESTION = -1 or 1 f () = ( + 1) ( + ) f () = = or 9. How can ou use factoring to solve quadratic equations in standard form? Factor the right side of the equation. Set each linear factor equal to, and then solve each linear equation. These are the solutions of the original quadratic equation. Module 7 97 Stud Guide Review COMMON CORE Common Core Standards 97 Module 7 Lesson Items Content Standards Mathematical Practices F-IF.C.7, A-APR.B.3 MP A-APR.B.3, A-APR.A.1, A-SSE.A. MP A-REI.B., A-APR.B.3 MP. 7.3 A-REI.B., A-APR.B.3 MP A-REI.B., A-APR.B.3 MP. 7.3 A-REI.B., A-APR.B.3 MP.

98 MODULE 7 MIXED REVIEW Assessment Readiness 1. Solve (7 - ) + 1_ (7 - ) =. 5 A. = -7 Yes No B. = - _ 5 C. = - 1_ Yes Yes No No MODULE 7 MIXED REVIEW Assessment Readiness ASSESSMENT AND INTERVENTION. For each statement, determine if it is True or False for the graph of - 3 = 1. A. The -intercept is 3 1_. Yes No B. The -intercept is -7. Yes No C. The slope is 3. Yes No 3. Is the sequence, 3,, -3, -, -9, arithmetic, geometric, or neither? Eplain our answer. Write a recursive rule for the sequence. The sequence is arithmetic because it has a common difference of -3. f (1) = ; f (n) = f (n - 1) - 3. Graph f () = ( - ) -. Describe the relationship between the -intercepts of the graph and the solutions of ( - ) - = The -intercepts are the solution of the equation. Houghton Mifflin Harcourt Publishing Compan Assign read-made or customized practice tests to prepare students for high-stakes tests. ADDITIONAL RESOURCES Assessment Resources Leveled Module Quizzes: Modified, B AVOID COMMON ERRORS Item Some students ma have a hard time finding the slope of an equation that is not in slope-intercept form. Remind students that the can rewrite equations in new forms whenever needed, and the can also find the slope using the intercepts. Module 7 9 Stud Guide Review COMMON CORE Common Core Standards Lesson Items Content Standards Mathematical Practices A-REI.B. MP. IM1 5. * F-IF.B. MP. IM1. 3* F-BF.A. MP A-REI.D.11 MP.7 * Item integrates mied review concepts from previous modules or a previous course. Stud Guide Review 9