ACTIVITY 5 - Unauthorizedmedia.collegeboard.com/...Lesson_Algebra_1_Teacher_Edition_3.16.1… ·...

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Unit 5 • Quadratic Functions 317

1 Look for a Pattern, Quickwrite Students may notice that the data appears quadratic based on their prior knowledge of quadratic functions and by analyzing the patterns in the fi rst and second differences in the table.

ACTIVITY 5.5 Investigative

Applying Quadratic Equations

Activity Focus• Solving problems that involve

quadratic functions• Solving quadratic equations

including piecewise defi ned functions

Materials• Graphing calculator

Chunking the Activity#1 #7 #15–16#2 #8–10 #17#3–4 #11 #18#5 #12–13#6 #14

Introduction Shared Reading, Questioning the Text, Marking the Text The purpose of this Activity is to have students apply their previous knowledge of quadratic functions and solving quadratic equations to a tangible context. Model rocketry is the context used and students will determine the height of rockets in different settings by solving quadratic equations using factoring and the quadratic formula.

Differentiating Instruction

Students may have trouble with the language in the fi rst paragraphs.

• Review the pronunciation and meaning of ignited, accelerate, free fall, acceleration, gravity and any other words causing diffi culty.

• Use the visual model on the page to describe the stages of rocket fl ight matching the vocabulary words with the appropriate stage of fl ight.

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ACTIVITY

My Notes


Applying Quadratic EquationsRockets in Flight 5.5

Homer H. Hickam Jr. is a coal miner’s son, who lived in West Virginia during the 1950s. Aft er the Russians launched the Sputnik satellite, Homer was inspired to learn about model rocketry. Aft er many tries, Homer and his friends discovered how to launch and control the fl ight path of their model rockets. Homer went on to college and then worked for NASA.

Cooper is a model rocket fan. Cooper’s model rockets have single engines and, when launched, can rise as high as 1000 ft depending upon the engine size. Aft er the engine is ignited, it will burn for 3–5 seconds and the rocket will accelerate upward. Once the engine burns out, the rocket will be in free fall, because the only acceleration is due to gravity. Th e rocket has a parachute that will open as the rocket begins to fall back to Earth.

Cooper wanted to track one of his rockets, the Eagle, so that he could investigate its time and height while in fl ight. He installed a device into the nose of the Eagle to measure the time and height of the rocket as it fell back to Earth. Th e device started measuring when the parachute opened. Th e data for one fl ight of the Eagle is shown in the table below.

Th e EagleTime Since Parachute Opened

(seconds)

0 1 2 3 4 5 6 7 8 9

Height (feet) 625 618 597 562 513 450 373 282 177 58

1. Use the data in the table above. Determine whether the height of the Eagle can be modeled by a linear function of time. Explain your reasoning.

CONNECT TO SCIENCESCIENCE

NASA is the National Aeronautics and Space Administration and is responsible for all space exploration.

SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Text, Questioning the Text, Look for a Pattern, Quickwrite

The data in the table for the Eagle are not linear because there is a constant increase in the time, but not a constant decrease in the height.

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318 SpringBoard® Mathematics with Meaning™ Algebra 1

ACTIVITY 5.5 Continued

Create Representation, Debriefi ng It is important

to note that the quadratic model produces a graph of the “height versus time” and not a graph of the path of the rocket. A common misconception of students is that the “height versus time” graph represents the path of the rocket through the sky.

A good example of this is a ball that is dropped from a height above the fl oor will have a “height versus time” graph similar to the graph they created in Item 2. The ball will physically fall in a straight line down to the fl oor, while the graph will show a curve because it is tracking height versus time.

3 Think/Pair/Share, Close Reading, Debriefi ng This question presents projectile motion using a simple quadratic model for the height of the Eagleafter the parachute opens.

3a Activating Prior Knowledge Substituting 0 for t and 625 for h(t) yields the equation 625 = k(0 ) 2 + h 0 . Be sure students understand how to correctly substitute values for the variables.

3b Any other point in the table can be used to solve for k. For example, using (1, 618) yields the equation 618 = k(1 ) 2 + h 0 .

CD

to note tha

TEACHER TO

TEACHER

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SCIENCE

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My Notes

318 SpringBoard® Mathematics with MeaningTM Algebra 1

Applying Quadratic EquationsACTIVITY 5.5continued Rockets in FlightRockets in Flight

SUGGESTED LEARNING STRATEGIES: Create Representations, Close Reading, Activating Prior Knowledge

2. Graph the data from the table above Item 1 on the grid below.

3. Th e height of the Eagle can be modeled by the quadratic function h(t) = k t 2 + h0, where k is a constant and h0 is the initial height of the rocket. You can use the table data to fi nd the values of k and h0 in the Eagle’s height function.

a. Solve for the value of h0 using the point (0, 625). Include the appropriate units for h0 in your solution.

b. Use a diff erent ordered pair from the table above Item 1 to fi nd the value of k. Write a function for the height of the rocket as a function of time.

READING MATH

A variable with a zero subscript such as h0 is read “h naught,” or “h sub zero.” This means that it is the initial value.

Time (in seconds)

Hei

ght o

f the

Eag

le (i

n fe

et)

61 85432 97 10

100

200

300

400

500

600

700

h0 = 625 feet

k = -7; h(t) = -7 t 2 + 625

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4 Think/Pair/Share, Group Presentation, Debriefi ng Have students work in groups and then ask for a volunteer to do a presentation. Debrief as necessary.

4a Activating Prior Knowledge Students should set the function equal to 450 and solve the equation. This approach is similar to the process of solving a linear equation with the addition of fi nding the square root of both sides of the equation.

4b Activating Prior Knowledge This question has two purposes: fi rst, to fi nd the zeros of a quadratic function, and second, to work with answers that are irrational. For the remainder of this activity, all irrational answers will be rounded to three decimal places. Only the positive value has meaning in the context of these problems.

5 Create Representations, Visualize, Questioning the Text Some students may question why the rocket increases in height when the engine cuts off. Remind students that due to inertia the rocket will continue in the direction in which it was traveling before the engine cut off. After a time it will begin the free fall.

Suggested Assignment

CHECK YOUR UNDERSTANDING p. 326, #1

UNIT 5 PRACTICEp. 330, #29

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My Notes

ACTIVITY 5.5continued

Applying Quadratic EquationsRockets in FlightRockets in Flight

4. Use the function from Item 3 to answer these questions.

a. At what time was the rocket’s height 450 ft above Earth? Verify that your result agrees with the data in the table.

b. Aft er the parachute opened, how long did it take for the rocket to hit Earth?

Cooper wanted to investigate the fl ight of a rocket from the time the engine burns out until the rocket lands. He set a device in a second rocket, named Spirit, to begin collecting data the moment the engine shut off . Unfortunately, the parachute failed to open. When the rocket began to descend it was in free fall.

5. Graph the data for the height of the Spirit versus time on the grid.

Th e SpiritTime Since the Engine

Burned Out (s) Height (ft )

0 5121 5602 5763 5604 5125 4326 320

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Visualization, Questioning the Text, Create Representations, Think/Pair/Share, Group Presentation

CONNECT TO SCIENCESCIENCE

A free falling object is an object which is falling under the sole infl uence of gravity.

Time (in seconds)

Hei

ght o

f the

Spirit

(in

feet

)

61 85432 97 10

100

200

300

400

500

600

700

t = 5 seconds

t ≈ 9.449 seconds

The curve on the graph is the answer to Item 7(b) on the next page. The dashed horizontal line is part of the answer to Item 12.The dashed horizontal line is part of the answer to Item 16.

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6 Think/Pair/Share, Debriefi ng

6a–b Close Reading Students can study the table of values as well as the graph to answer these items.

6c Think/Pair/Share Various answers may be given and should be accepted. Some students may give 2 < t ≤ 6 as the time of the free fall since these are the times shown in the table. Other students may give 2 < t < 8, because a curve that approximates the graph will intersect the horizontal axis at t = 8. Still others may give t > 2.

7a Create Representations Students will use a graphing calculator to compute a quadratic regression equation. If students are not familiar with fi nding a regression equation on a graphing calculator, you will need to guide them through this skill prior to this item. See the Mini-Lesson. There are also demos on the Internet.

Students should be encouraged to write the equation as a function of t.

If graphing calculators are not available, be sure to give students this function. Students will need this function to answer subsequent questions throughout the activity.

7b Create Representations, Debriefi ng Students can use the Graph feature of the graphing calculator fi rst to fi nd the graph of the function they determined in Item 7a. Then use that display to sketch the graph on the grid in Item 5. Have them use a different color than the color they used to plot the points from the table of values.

MINI-LESSON: Math Tip on Quadratic Regression

Check the manual for your calculator to adapt the steps below, if necessary.

Press . Press 1 to select 1:Edit. This takes you to the data editor. Enter the data in L1 (list 1), L2 (list 2). If there are old data in a column, use the arrows to highlight a list header (such as L1), then press

Enter to clear the column.

Press and use to select CALC. Press 5 for QuadReg. Press Enter . This yields an equation of the form y = a x 2 + bx + c.

Select and use to select Y-VARS. Then press 1 for Y1 = and press Enter .

STAT

STAT

VARS

CLEAR

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My Notes



6. Use the table and graph in Item 5.

a. How high was the Spirit when the engine burned out?

b. How long did it take the rocket to reach its maximum height aft er the engine cut out?

c. Estimate the time the rocket was in free fall before it reached the earth.

7. Use the table and graph in Item 5.

a. Use a graphing calculator to determine a quadratic h(t) function for the data.

b. Sketch the graph of the function on the grid in Item 5.

8. Use the function found in Item 7 to verify the height of the Spirit when the engine burned out.

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Close Reading, Create Representations

For Item 7(a), enter the data from the table in Item 5 into a graphing calculator. Use the calculator’s quadratic regression feature to fi nd a representative function.

TECHNOLOGY

512 feet

2 seconds

Answer may vary. See notes on Item 6c to the left.

h(t) = -16 t 2 + 64t + 512

See graph on previous page.

h(0) = -16 (0) 2 + 64(0) + 512 = 512

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9 Create Representation, Predict and Confi rm The intent of this question is to use technology to better estimate the time interval during which the Spirit was falling to Earth. Students can enter the function into a graphing calculator and trace it on the graph to fi nd the time when the height of the Spirit is at 0 ft.

0 Activating Prior Knowledge, Quickwrite Students can use what they learned about solving quadratic equations to fi nd the time the rocket was in the air after the engine burned out.

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SUGGESTED LEARNING STRATEGIES: Create Representations, Predict and Confi rm, Activating Prior Knowledge, Quickwrite

9. Graph the function found in Item 7 on your graphing calculator. Use the graph to approximate the time interval in which the Spirit was in free fall. Explain how you determined your answer.

10. Th e total time that the Spirit was in the air aft er the engine burned out is determined by fi nding the values of t that makes h(t) = 0.

a. Set the equation found in Item 7(a) equal to 0.

b. Completely factor the equation.

c. Identify and use the appropriate property to fi nd the time that the Spirit took to strike Earth aft er the engine burned out.

The Spirit was in free fall for about 6 seconds. Sample explanation: By entering the function into a graphing calculator and tracing it on the graph, I can see that the maximum value appears to occur when t is 2. The time it took the Spirit to reach the ground at 0 feet was close to 8 seconds.

-16 t 2 + 64t + 512 = 0

-16(t - 8)(t + 4) = 0

The zero product property of multiplication can be used.

(t - 8) = 0 (t + 4) = 0

t = 8 t = −4

The solution t = 8 seconds is the only solution in the domain of the function.

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SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Think/Pair/Share

11. Th e quadratic formula can also be used to solve the equation from Item 10(a).

a. State the quadratic formula.

b. Use the quadratic formula to determine the total time that the Spirit was in the air aft er the engine burned out. Show your work.

12. Draw a horizontal line on the graph in Item 5 to indicate a height of 544 ft above Earth. Estimate the approximate time(s) that the Spirit was 544 ft above Earth.

13. Th e time(s) that the Spirit was 544 ft above Earth can be determined exactly by fi nding the values of t that make h(t) = 544.

a. Set the equation from Item 7(a) equal to 544.

x = -b ± √

________ b 2 - 4ac ______________

2a

t = -(-4) ± √

_______________ ( -4) 2 - 4(1) (-32) ________________________

2(1) = 4 ± √

____

144 _________ 2

t = 4 + 12 ______ 2 = 8 or t = 4 - 12 ______

2 = -4

The Spirit was in the air for 8 seconds after the engine burned out.

-16 t 2 + 64t + 512 = 544

Estimates may vary, but should be approximately 0.5 seconds and 3.5 seconds.

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a Think/Pair/Share, Activating Prior Knowledge Students should recall that the quadratic

formula is x =-b ± √

________ b 2 - 4ac _____________

2a,

which can be used to solve the equations of the form a x 2 + bx +

c = 0, when a ≠ 0.

When solving equations using the Quadratic

Formula, it is acceptable to divide out the monomial factors to make the computations easier. In Item 11b, students can use the equation -16 t 2 + 64t - 512 = 0 as is and apply the Quadratic Formula. Or, they can factor out -16 to get -16( t 2 - 4t - 32) = 0. Dividing both sides by -16 and applying the Quadratic Formula to the equation t 2 - 4t - 32 = 0 gives the same results as the original equation and is easier to compute.

b Create Representations The horizontal line will help students see that the intersection between the line y = 544 and the curve occurs between 0 and 1 second and that the other intersection occurs between 3 and 4 seconds. These are only estimates. Students will fi nd the exact answers by doing Item 13.

c Activating Prior Knowledge, Think/Pair/Share Have students work in pairs to determine the algebraic solution to the problem.

TECHNOLOGYCreate Representations Students can also get solutions of the equation by using a graphing calculator. See Mini-Lesson to the right for an example.

Wu

Formula i

TEACHER TO

TEACHER

MINI-LESSON: Solving a Quadratic Equation by Graphing

Solve the equation -16 t 2 + 64t + 512 = 544.

Press • Y = . In the Y1 row, type the expression -16 t 2 + 64t + 112. In Y2, type 544. Graph the equations so the intersections are visible.

Press • 2nd and Trace . Press 5 to select 5:intersection.

Move the cursor near the intersection point and press • Enter twice to select the curves.

Press • Enter again for the Guess?

Repeat the steps for the second intersection point.


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SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Quickwrite, Create Representations, Think/Pair/Share

b. Is the method of factoring eff ective in solving this equation? Explain your reasoning.

c. Is the quadratic formula eff ective in solving this problem? Explain your reasoning.

d. Determine the time(s) that the rocket was 544 ft above Earth. Round your answer to the nearest hundredth of a second. Verify that this solution is reasonable compared to the estimated times from the graph, Item 12.

14. Cooper could not see the Spirit when it was higher than 528 ft above Earth.

a. Find the values of t for which h(t) = 528.

b. Write an inequality for the values of t that are between the two times that the rocket was not within Cooper’s sight.

Answers may vary. Sample explanation: Factoring is not an effective method for solving this problem, because the quadratic that remains after factoring out -16 cannot be factored easily.

Answers may vary. Sample explanation: The quadratic formula is a more effective method for solving this problem, because the quadratic formula can be used to solve any quadratic equation.

t = 4 ± √

__ 8 ______

2 = 2 ± √

__ 2 ≈ 2 ± 1.414; t ≈ 0.586 seconds and

t ≈ 3.414 seconds.

t = 2 + √

__ 3 ≈ 3.732 seconds and t = 2 - √

__ 3 ≈ 0.268 seconds

2 - √

__ 3 < t < 2 + √

__ 3 or 0.268 < t < 3.732

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CHECK YOUR UNDERSTANDING p. 326, #2–3

UNIT 5 PRACTICEp. 330, #30, 31

da Think/Pair/Share, Debriefi ng Students need to use the Quadratic Formula to fi nd the two values of t.


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MINI-LESSON: Finding the Maximum Using the Calculator

Press 2nd . Then press Trace . Press 4 to select 4:maximum.

Move the cursor to the left of the maximum value on the graph. Press Enter for the LeftBound?

Move the cursor to the right of the maximum value on the graph. Press Enter for the RightBound?

Press Enter again for the Guess?

The calculator will then give the point at which the maximum occurs on the chosen interval.


Introduction Close Reading, Marking the Text Two equations are needed to represent the fl ight of the Speedy. One equation models the fl ight when the engine is burning and the other equation models the fl ight when the engine has burned out. Each equation represents a model of the situation. In reality, a graph showing the height of the rocket versus time would be a smooth curve, without the bend at t = 4 seconds.

e Visualization, Group Presentation, Debriefi ng By using the graph, students should recognize that at t = 4, the shape of the graph changes. This shape change indicates the time at which the engine burned out.

TECHNOLOGYCreate Representations If students know the function, they can use the capabilities of a graphing calculator to fi nd the maximum on the graph. See the Mini-Lesson for an example.

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SUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Quickwrite, Visualization, Group Presentation

Cooper has a third rocket named Speedy. He decided to fi re the rocket without a parachute to investigate a rocket’s motion in free fall. Cooper represented the launch time as t = 0. Th e graph of the height of the Speedy is a piecewise function, shown below.

15. Use the graph above to estimate the answer to each question.

a. At what time did the Speedy’s engine burn out?

b. What was the maximum height of the rocket and at what time did the rocket reach that height?

c. At what time did the rocket hit Earth?

Time (in seconds)

Hei

ght o

f the

Spe

edy

(in

feet

)

61 85432 97 15

100

200

300

400

500

600

700

10 12 1311 14

t = 4 seconds

The maximum height of the rocket was at about 680 feet and occurred at approximately 7.5 seconds.

The rocket hit Earth at about 14 seconds after it was launched.

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SUGGESTED LEARNING STRATEGIES: Marking the Text, Simplify the Problem, Create Representations, Think/Pair/Share

Cooper is entering the Speedy into a contest. Th e winner is the owner of the rocket that stays above 400 ft for the longest period of time.

16. Draw a horizontal line on the graph above Item 15 to indicate a height of 400 ft above Earth. Estimate when the rocket will be more than 400 ft above Earth.

While Speedy’s engine is burning for the fi rst 4 seconds, the height is given by the function h1(t) = 30 t 2 . Aft er the engine burns out, the height is given by h2(t) = -224 + 240t - 16 t 2 .

17. Write a piecewise function h(t) that expresses the height of the Speedy above Earth as a function of time.

18. For the contest, Cooper needs to determine the length of time that the Speedy will be 400 ft above Earth. Use the piecewise function for the height of the rocket from Item 17 to determine the exact time(s) that the rocket will be exactly 400 ft above Earth.

a. Explain why two diff erent equations must be solved to determine the time(s) that the rocket will be 400 ft above Earth.

The rocket will be more than 400 feet above the Earth between approximately 3.5 seconds and 11.5 seconds after launch.

h(t ) = { 30 t 2 , 0 ≤ t ≤ 4

-224 + 240t - 16 t 2 , 4 < t ≤ 14

The fi rst time that the rocket will be 400 feet above Earth occurs during the fi rst 4 seconds of fl ight, when the height is given by the equation h1(t) = 30 t 2 . The second time occurs after 4 seconds of fl ight, when the height is given by the equation h2(t) = -224 + 240t - 16 t 2 .

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f Simplify the Problem The answers are approximations and reasonable variations in student answers should be expected.

TECHNOLOGYCreate Representations If students know the function, they can get more exact answers for the solutions of the equation by using a graphing calculator. See the Mini-Lesson for an example.

g Debriefi ng Students who have a different approximation for the time when the Speedy will hit Earth might not have used 14 for the endpoint of the second equation’s time interval.

h Think/Pair/Share, Simplify the Problem, Debriefi ng Students realize that a piecewise function is needed to model this situation because the rocket moves differently when the engine is burning and when it is burned out.


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SUGGESTED LEARNING STRATEGIES: Simplify the Problem, Think/Pair/Share, Debriefing

b. Write and solve an equation to determine the time that the rocket will be 400 ft above Earth during the time interval for which 0 ≤ t ≤ 4.

c. Write and solve an equation to determine the time that the rocket will be 400 ft above Earth during the time interval for which t > 4.

d. Determine the length of time that the rocket will be 400 ft above Earth.

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper or grid paper. Show your work.

Th e equation h(t) = -16 t 2 + 128t + 320 represents the fl ight of a model rocket aft er the rocket’s engine burns out.

1. Determine when the rocket hits Earth. 2. Graph the equation. Determine the time

at which the rocket reaches its maximum height.

3. Determine the times when the rocket is higher than 423 ft . Explain how you arrived at your solution.

A model rocket burns for 3.5 s. Th e rocket will be 286 ft in the air when the engine burns out. Aft er the rocket’s engine burns out, the rocket’s height is given by the function h(t) = 286 + 190t - 16 t 2 .

4. Determine the total time aft er the rocket is launched that it will be in the air.

5. Determine the times aft er the rocket is launched that it will be 450 ft in the air.

6. MATHEMATICAL R E F L E C T I O N

Why are quadratic functions used to model

free-fall motion instead of linear functions?

400 = 30 t 2 , and t = ± √ ___

40 ___ 3 ≈ ±3.651 seconds;

t = 3.651 seconds is the only value in the domain.

400 = -224 + 240t - 16 t 2 , t = 15 + √

___ 69 ________

2 ≈ 11.653 seconds

approximately 8 seconds

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CHECK YOUR UNDERSTANDING p. 326, #4–6

UNIT 5 PRACTICEp. 331, #32, 33

1. t = 10 seconds

2. See below.

3. The rocket is higher than 423 feet between about 0.91 second and 7.09 seconds after the rocket engine burns out. Sample explanation: I solved the equation -16t 2 + 128t + 320 = 423 and found that the solutions are approximately 0.91 second and 7.09 seconds. I can see from the graph that the rocket is higher than 423 feet between those two times.

4. about 13.226 seconds.

5. about 0.937 second and about 10.938 seconds

6. Answers may vary. Sample answer: In linear functions, there is a constant rate of change. Free-fall motion does not have a constant rate of change.

2. The rocket reaches its maximum height in about 4 seconds.

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