NATIONA MATH + SCIENCE Mathematics INITIATIVE · PDF fileAlgebra 1 or Math 1 in a unit on quadratic functions MODULE/CONNECTION TO AP* Position/Velocity/Acceleration ... and periodicity

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MathematicsNATIONALMATH + SCIENCEINITIATIVE

LEVELAlgebra 1 or Math 1 in a unit on quadratic functions

MODULE/CONNECTION TO AP*Position/Velocity/Acceleration*Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product.

MODALITYNMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding.

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Graphing Quadratic FunctionsABOUT THIS LESSONThis lesson presents real world situations involving quadratic functions. Students state domain and range in the context of the situation, analytically determine x-values, y-values, and maximums of position functions. In addition, students practice a variety of factoring skills and use symmetry of roots to determine the vertex. The lesson helps students connect the verbal questions to the functional notation needed to answer the questions.

OBJECTIVESStudents will

● translate questions from verbal to analytical form.

● determine zeros of quadratic functions.● analytically determine dependent and

independent values of quadratic functions.● apply symmetry of roots to determine the

vertex of a quadratic function.● analyze characteristics of quadratic functions

in the context of the situation.

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Mathematics—Graphing Quadratic Functions

COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENTThis lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introductiontothespecificskill.Thestarsymbol(★) attheendofaspecificstandardindicatesthatthehigh school standard is connected to modeling.

Targeted StandardsF-IF.4: For a function that models a relationship

between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ See questions 1a-e, 2b-e, 3b-e, 4c-f

Reinforced/Applied StandardsF-IF.2: Use function notation, evaluate functions

for inputs in their domains, and interpret statements that use function notation in terms of a context. See questions 1-4

F-IF.7a: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (a)Graphlinearandquadraticfunctionsand show intercepts, maxima, and minima.★ See questions 2e, 3e, 4f

F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★ See questions 1f, 2f, 3f, 4g

F-IF.8a: Writeafunctiondefinedbyan expression in different but equivalent forms to reveal and explain different properties of the function. (a)Usetheprocessoffactoringand completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. See questions 2b, 3b, 4c

A-REI.4b: Solve quadratic equations in one variable. (b)Solvequadraticequations byinspection(e.g.,forx2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. See questions 2a-b, 3a-b, 4b-c

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COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICEThese standards describe a variety of instructional practicesbasedonprocessesandproficienciesthat are critical for mathematics instruction. NMSI incorporates these important processes and proficienciestohelpstudentsdevelopknowledgeand understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice.

MP.3: Construct viable arguments and critique the reasoning of others. In question 1g, students defend whether the example given by Stella is correct or incorrect based on the graph.

FOUNDATIONAL SKILLSThe following skills lay the foundation for concepts included in this lesson:

● Graph quadratic functions● Solve quadratic equations by factoring● Use function notation● Identify domain and range

ASSESSMENTSThe following formative assessment is embedded in this lesson:

● Students engage in independent practice.

The following additional assessments are located on our website:

● Position/Velocity/Acceleration – Algebra 1 Free Response Questions

● Position/Velocity/Acceleration – Algebra 1 Multiple Choice Questions

MATERIALS AND RESOURCES● Student Activity pages● Interactive applet that allows students to

observe an object travel on a straight path while simultaneously watching the graphs of position, velocity, and acceleration http://www.calculusapplets.com/motionline.html

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TEACHING SUGGESTIONS

The teacher should lead the class in working questions 1 and 2. The student versions of the examples are included in the student

activity.Extensionquestions(g)–(j)forquestion2are provided in order to explore average velocity in the context of the situation.

Question 1Students answer these questions by reading the given graph. As the class works these questions, discuss the symmetry of the graph and ensure that students notice that the x-value that determines the maximum y-value is on the axis of symmetry.

Question 2The quadratic equations in this activity can be solved easily by factoring so that the students can practice this skill. a. To help students become comfortable with

function notation, relate this question to the notation h(t) = 528. After the students solve the equation 216224528 tt −= by factoring, write the answers in the form h(3)=528andh(11)=528.Askthestudentswhythereare two times when the rocket’s height is 528 feet.

b. Again write h(t) = 0 and have the students solve the equation 0 = 224t – 16t2 by factoring. Write the answers in the form h(0)=0andh(14)=0.Relate the answers to the zeros of the function h(t).

c. To determine the time when the rocket is at the maximum height, refer the students to question 1 and the symmetry of the graph. The students can determine the average of the times 0and14frompart(b),or3and11frompart(a),in order to determine that the x-value of the maximum point is 7 seconds. Have the students write the equation for the axis of symmetry, t = 7.

d. Now that the students know the t-coordinate of the maximum point, they can determine h(7).Ask students for the meaning of h(7)inthe

context of the situation.

e. The students now know enough points to graph the function. The students use the zeros of the function(whentherocketisontheground)tohelp set the scale for the horizontal axis on their graph and the maximum height to help set the scale for the vertical axis on their graph. This discussion helps students identify the reasonable domainandrangeforthissituationinpart(f).After the students plot the points they found in parts(a)–(d),askthemiftheyshouldconnectthe points.

f. The students can determine the domain and range by looking at the graph.

Additional questions for discussion in Question 2:g. What is the average rate of change of the height

with respect to time for the interval of time 1 2t≤ ≤ ? What does this value represent? (The average rate of change, average velocity, is 176 feet per second, for 1 to 2 seconds inclusively.)

h. What is the average rate of change of the height with respect to time for the interval of time 3 4t≤ ≤ ? What does this value represent? (The average rate of change, average velocity, is 112 feet per second, for 3 to 4 seconds inclusively.)

i. Compare the values for the average rate of changefoundinquestions(g)and(h).Explainthe meaning of these values in the context of thissituation.(The rate of change is decreasing because the force of gravity is acting on the rocket.)

j. What is average rate of change for the interval of time 9 10t≤ ≤ ? What does this value represent? (The average rate of change is –80 feet per second, and represents the average velocity during this particular time period.) Explain the meaning of a positive rate of change and a negative rate of change in the context of this situation. (A positive rate (velocity) means that the rocket is climbing. A negative rate

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(velocity) means that the rocket is falling.)

You may wish to support this activity with TI-Nspire™ technology. See Finding Points of Interest in the NMSI TI-Nspire Skill Builders. Suggestedmodificationsforadditionalscaffoldinginclude the following:1 Provide a graph with the points labeled or a

tableofvaluesfortheidentifiedpoints.2-4 Allow the student to use a graphing calculator.

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NMSI CONTENT PROGRESSION CHART In the spirit of NMSI’s goal to connect mathematics across grade levels, a Content Progression Chart for eachmoduledemonstrateshowspecificskillsbuildanddevelopfromsixthgradethroughpre-calculusinanaccelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content.

The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence.

6th Grade Skills/Objectives

7th Grade Skills/Objectives

Algebra 1 Skills/Objectives

Geometry Skills/Objectives

Algebra 2 Skills/Objectives

Pre-Calculus Skills/Objectives

Relate real-life situations to graphs.






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MathematicsNATIONALMATH + SCIENCEINITIATIVE

Graphing Quadratic Functions

Answers1. a. When , the object is at ground level. This occurs at 0 seconds and 8 seconds.

b. The notation means at 4 seconds the object is 16 feet above ground level. This is the maximum point on the graph.

c. The object is in the air a total of 8 seconds.d. The notation means the height of the object is 7 feet at the times 1 and 7 seconds.e. The notation means the object is 15 feet in the air at 5 seconds.f. Domain: ; Range:

The domain means the object was projected at time 0 seconds and hits the ground at 8 seconds. The range means the ball was projected from the ground and reached a maximum height of 16 feet.

g. The example given by Stella is incorrect because the x-axis represents time not distance. The graph does not represent the path of an object, but instead the object’s height with respect to time.

h. The object starts at ground level, goes straight up to a maximum height of 16 feet, and falls straight back down to the ground.

2. a. 528 = 224t – 16t² 16t² – 224t + 528 = 0 16(t² – 14t+33)=0 16(t–3)(t – 11) = 0 t=3seconds;t = 11 seconds h(3)=528;h(11)=528 Therocketwillreachaheightof528feetat3secondsandagainat11seconds.

b. 0 = 224t – 16t² 0=16t(14–t) t = 0 seconds; t = 14 seconds h(14)=0 The rocket will reach the ground at 14 seconds.

c. 0 14 3 11 72 2+ += = seconds

d. (7) 784h = feete.

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f. Domain: 0 14t≤ ≤ Range: 0 ( ) 784h t≤ ≤

3. a. When is ( ) 192?h t = 2192 128 16t t= −

16( 6)( 2) 0t t− − = (2) 192h = ; (6) 192h =

The object will reach a height of 192 feet at 2 seconds and at 6 seconds.b. When is ( ) 0?h t =

0 16 (8 )t t= − (0) 0h = ; (8) 0h =

The object will return to the ground at 8 seconds.

c. 0 8 2 6 42 2+ += = . The object will reach its maximum height at 4 seconds.

d. What is (4)h ? The object will reach a maximum height of 256 feet at 4 seconds.e.

f. Domain: 0 8t≤ ≤ Range: 0 ( ) 256h t≤ ≤

4. a. What is the height of the object after 1 second? (1) 224h = . The height of the object at 1 second is 224 feet.

b. How long will it take the object to reach a height of 800 feet?; (5) 800h = ; (10) 800h = . The object is 800 feet in the air at 5 seconds and at 10 seconds.

c. When is the height of the object 0 feet above the ground?; (0) 0h = ; (15) 0h = . The object’s initial height above the ground is 0. The object returns to the ground at 15 seconds.

d. 0 15 5 10 7.52 2+ += = . The object will reach its maximum height at 7.5 seconds.

e. What is (7.5)h ? The object will reach a maximum height of 900 feet.

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f.

g. Domain: 0 15,t≤ ≤ Range: 0 ( ) 900h t≤ ≤

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1Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.

Mathematics NATIONALMATH + SCIENCEINITIATIVE

Graphing Quadratic Functions

1. Fillintheblanksfor(a)-(e)andanswerquestions(f)-(h)basedonthegraph.From ground level, an object is projected upward. The graph represents the height, h, of the object at time, t.

a. When the object is at ground level. This occurs at ______ seconds and

______ seconds.

b. The notation means at______ seconds the object is ______ feet above ground level.

This is the maximum point on the graph.

c. The object is in the air a total of ___________ seconds.

d. The notation means the height of the object is _______ feet at the times ______ and

______ seconds.

e. The notation means the object is _______ feet in the air at ________ seconds.

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f. What are the domain and the range of the graph? Explain what the domain and range mean within the context of the situation.

g. Stella was asked to give an example of what this graph could represent in the real world. Stella wrote “This graph could represent a punt in a football game where the punter kicked the football standing on the goal line and a player caught it 80 yards away. The ball was 16 feet in the air at the highest point.” Is Stella correct or incorrect with her example? Explain your reasoning.

h. Describe the path and sketch a diagram that illustrates the actual path the object travels.



2. A rocket is launched from ground level with an initial velocity of 224 ft

sec. The height, h, in feet of

the rocket at any given time, t, in seconds is given by h(t) = 224t – 16t2.

a. Whenwilltherocketreachaheightof528feet?(Whenish(t) = 528?) Determine an appropriate equation and then solve it by factoring. Write the answer in function notation and then write the answer as a sentence in terms of the situation.

b. Whenwilltherocketreachtheground?(Atwhattimet is ( ) 0h t = ?) Determine an appropriate equation and then solve it by factoring. Write the answer in function notation and then write the answer as a sentence in terms of the situation.

c. When will the rocket reach its maximum height? Determine the answer using the parabola’s symmetry and its roots. Show the work that leads to your answer.

d. What is the maximum height of the rocket? Write the question in function notation using the answer frompart(c)andthendeterminetheanswer.

e. Usetheanswersfromparts(a)–(d) to graph this situation.

f. What are the domain and range of the graph?



3. From ground level, an object travels upward with an initial velocity of 128 ft

sec . The height, h, in feet of the object at any given time, t, in seconds is given by h(t) = 128t – 16t2.a. When will the object reach a height of 192 feet? Write the question in function notation. Determine

an appropriate equation and then solve it by factoring. Write the answer in function notation and then write the answer as a sentence in terms of the situation.

b. When will the object reach the ground? Write the question in function notation. Determine an appropriate equation and then solve it by factoring. Write the answer in function notation and then write the answer as a sentence in terms of the situation.

c. When will the object reach its maximum height? Determine the answer using the parabola’s symmetry and its roots. Show the work that leads to your answer.

d. What is the maximum height of the object? Write the question in function notation using the answerfrompart(c)andthendeterminetheanswer.

e. Usetheanswersfromparts(a)–(d)tograph this situation.

f. What are the domain and the range of the graph?



4. From ground level, an object travels upward with an initial velocity of 240 ft

sec. The height, h, in feet of

the object at any given time, t, in seconds is given by h(t) = 240t – 16t2.a. What is h(1)?Explainthemeaningofthisquestioninthecontextofthesituation.Writethesolution

in function notation and then explain the meaning of the answer in terms of the situation.

b. What is the value of t when h(t) = 800? Explain the meaning of this question in the context of the situation. Write the solution in function notation and then explain the meaning of the answer in terms of the situation.

c. What is the value of t when h(t) = 0? Explain the meaning of this question in the context of the situation. Write the solution in function notation and then explain the meaning of the answer in terms of the situation.

d. When will the object reach its maximum height? Determine the answer using the parabola’s symmetry and its roots. Show the work that leads to your answer.

e. What is the maximum height of the object? Write the question in function notation using the answer frompart(d)andthendeterminetheanswer.

f. Usetheanswersfromparts(a)–(e)tographthissituation.

g. What are the domain and the range of the graph?



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NATIONA MATH + SCIENCE Mathematics INITIATIVE · PDF fileAlgebra 1 or Math 1 in a unit on quadratic functions MODULE/CONNECTION TO AP* Position/Velocity/Acceleration ... and periodicity