Name Date ______________ HSF.IF.B.4 Class

Real World QuadraticsKey Takeaways:

Standard: 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.

The key features of the graph of a quadratic function represent different parts of a projectile’s path. 

The y-intercept tells you the launch height.  You can calculate the vertex to determine the maximum height. The x-intercepts can be used to find the total time the projectile was in the air.

Vocabulary: Projectile, maximum, minimum, vertex, turning point, parabola, x-intercept, y-intercept, launch height____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Part 1: Activation of Prior Knowledge

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Quallique throws a baseball straight up into the air. The relationship between the height of the baseball in feet, f(x), and the time spent in the air, x, is represented by the equation f ( x )=−x2+10 x+2. Below is a graph of the relationship.

Part A: Circle the baseball that tells you the maximum height the baseball reached. What do we call that point?

Part B: What does the x-value of that point represent?_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Part C: What does the y-value of that point represent?_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Part D: The table below represents the same relationship, f (x)=−x2+10 x+2.

x 0 1 2 3 4 5 6 7 8 9 10 11f (x) 2 11 18 23 26 27 26 23 18 11 2 -9

How many seconds does the ball spend in the air? How can you tell?

_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Part 2: Guided Practice

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All quadratics of the form y=a x2+bx+chave either a maximum or minimum value. Those maximum or minimum values are represented by the y-valueof the _______________________________.

Which of the graphs below has a maximum? Which has a minimum?

Graph A Graph B:

Example 1: A baseball was thrown into the air. The height of the ball after t seconds is given by the equation h (t )=−1

2t2

+4 t

A) Graph the height of the ball over time on the coordinate plane below.

B) What are the x-intercepts of your graph? Explain what each one means in the context_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________C) What is the vertex of the graph? Explain what each coordinate of the vertex means in the context of the baseball.__________________________________________________________________________________________________________________________________________________________Example 2: At the Nets game, an employee shoots a t-shirt out of a cannon. The height of the shirt, h ( x ), based on the number of seconds, x, is given by the function h ( x )=−4 x2+24 x+10.

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Part A: What is the starting height of the t-shirt? How can you tell?

Part B: After how many seconds does the t-shirt reach its greatest height? Solve algebraically. Remember to include units.

Part C: What is the t-shirt’s greatest height? Remember to include units.

Part D: Approximately how many seconds does the t-shirt stay in the air?(1) Between 4 and 5 seconds(2) Between 5 and 6 seconds(3) Between 6 and 7 seconds(4) Between 7 and 8 seconds

On the lines below please explain how you determined your answer.

________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Part 3: Independent Practice (MILD)

1. A ball in thrown into the air from the edge of a 48-foot-high cliff so that it eventually lands on the ground. The graph below shows the height in feet, y, of the ball from the ground after x seconds.

Part A: After how many seconds does the ball reach its maximum height?

(a) 2(b) 2.5(c) 3(d) 150

Part B: What is the ball’s maximum height in feet?

(a) 2 feet (b) 3 feet(c) 144 feet(d) 150 feet

Part C: Approximately how many seconds does the ball spend in the air? Justify your answer. Remember to include units.

2) A baseball was thrown into the air. The height, y, in feet of the baseball after s seconds is given by the equation y=−2 s2+12 s.

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Part A: On the coordinate plane below, sketch a graph that shows how the height of the ball changes over time.

Part B: What are the x-intercepts of your graph? Explain the significance of each one in the context of the baseball being thrown._______________________________________________________________________________________________________________________________________________________________________________________________________________________________________C) What is the vertex of your graph? ______________________________D) Explain the significance of the x-coordinate of your vertex in the context of the baseball’s path._______________________________________________________________________________________________________________________________________________________________________________________________________________________________________

E) Explain the significance of the y-coordinate of your vertex in the context of the baseball’s path._______________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Part 4: Independent Practice (MEDIUM)

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1. The amount of fans in a stadium, y, is based on the number of hours, x, into the game. This can be represented by the equation y=−100 x2+400 x+5,000.

Part A: How many fans are in the stadium to start the game? Remember to include units.

Part B: After the game, the Fire Marshall fined the stadium owner and said, “The maximum occupancy of our stadium is 5,500. You exceeded the maximum occupancy.” What would you say to the Fire Marshall if you were the stadium owner?

________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Part C: You need to pay each employee $8 an hour for the hours that they are in the stadium. All employees must stay until all fans leave the stadium. What range of pay should each employee expect?

(1) $56 and $64(2) $64 and $72(3) $72 and $80(4) $80 and $88

On the lines below, please justify your answer.

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Part 5: Independent Practice (SPICY)

1. The relationship between the number of users of two different graphing calculator iPhone apps can be seen below, where y represents the number of users, in hundreds, and x represents the number of months starting at the beginning of 2016.

Graphs-A-Lot I graph 4 U

X 4 5 6 7 8Y 85 95 105 115 125

y=5 x+100

2. Which app is gaining users more quickly? How much more quickly are they gaining users? Remember to include units.

3. How many users did Graphs –A-Lot begin the year with? How can you tell? Remember to include units.

__________________________________________________________________________________________________________________________________________________________________________________________________________________

4. If they started their calculations on January 1st (x=0) during what month will the two have the same number of users?

Mathletes

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“”I’m a mathlete

1) A model of a statue is built to a scale of 1:5 from the same material as the real statue and weighs 4 pounds. How many pounds does the real statue weigh? (The answer is NOT 20 pounds).

2) Two students are selected at random from a group of 3 girls and 3 boys. What is the probability that exactly one of each gender was selected?

3) Imagine a chessboard. Start by placing a coin in each one of the 64 squares.

Then drop another coin in each square, except for the a1 square.Then drop a third coin in each square except for those in the 2x2 square from a1 to b2.Then drop a fourth coin in each square except for those in the 3 x3 square from a1 to c3.…Keep going until an 8th coin is placed in each square in row 8 and column h.How many coins are used altogether?

Name Date ______________ HSF.IF.B.4 Class

Real World QuadraticsExit Ticket

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Directions: Complete each problem by showing ALL work. Don’t forget to use MOLE!

1. Mr. Thomas is trying to throw a ball over a fence. The height of the ball, f (t), in feet based on the number of seconds, t , is represented by the equation:

f (t )=−3.5t 2+14 t+3

Part A: The fence that Mr. Thomas is trying to throw it over is 19 feet tall. Would Mr. Thomas’ throw make it over the fence? Justify your answer.

Part B: After how many seconds does the ball reach its maximum height? How can you tell? Remember to include units.

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Name Date ______________ HSF.IF.B.4 Class

Real World QuadraticsHomework

Directions: Solve each problem. Show all work using MOLE.

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1. The equationy=2 x2−50 x+400 represents the height in meters, y, of a balloon over time, x, in minutes. What is the initial height of the balloon?

(a) 2 meters(b) −50 meters(c) 400 meters(d) 25 meters

2. The equation below represents the height in feet, h ( x ) , of a mountain climber over a certain number of minutes x.

h ( x )=−5 x2+20x+50

Part A: What is the maximum height in feet that the climber achieved?

Answer;______________________

Part B: After how many minutes did the climber achieve his maximum height?

Answer:____________________

Part C: What was the starting height of the climber?

Answer: ___________________3. Which of the following would have a graph that opens downward?

(1) y=x2

(2) y=−8 x2

(3) y=2x2

4. Evaluate the difference:(6 x2−7 x)−(8 x2+3 x−2)

(1) −2 x2−4 x(2) −2 x2−10x(3) −2 x2−10x+2(4) −2 x2−10x−2

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(4) y=52 x2

5. What is the solution to the system of equations below?

y=3 x+5y=−x−7

(a) (−3 ,4 )(b) (−3 ,−4 )(c) (−3 ,−10 )(d) (3 ,14)

6. What is the value of k in the following equation?

−18−6k=6(1+3k )

a) k=1b) k=−1c) k=3d) k=0

7. The height of a rectangular prism is 4 x feet and the width is x+5 feet and the length is 3 x feet. Write a simplified expression that could be used to calculate the volume of the rectangular prism. Answer must include units.

V=lwh

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