Name Date _________ HSF.IF.B.4 HSA.CED.A.3 Class

Write and solve quadratics that model real-world scenariosKey Takeaways:

Standard(s):

HSF.IF.B.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

HSA.CED.A.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Quadratic equations can be used to model real-world scenarios such as in finding dimensions or area in a geometry problem. When we solve real-world quadratics, the x-value represents a real-world quantity, and there will be times when we only consider one of the solutions of x. For example, the side length of a rectangle cannot be negative.

Vocabulary: Dimension, length, width, area, perimeter, consecutive, reject________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Part 1: Activation of Prior Knowledge (Mini Explore!)

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1. There is a rectangular garden that has a length that is 6 meters more than the width. The area of the garden is 55 m2. To model this situation, Ms. Hills set up the equation below:

( x ) (x+6 )=55Part A: What does x represent? ____________________________________________________________________________________________________________________________________________

Part B: How does this equation model the situation?__________________________________________________________________________________________________________________________________________________________________________________________________________________

Part C: Solve for x. Remember to include units.

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Part 2: Guided Practice

Example 1: The area of a rooftop can be expressed as 4 x2+4 x+1square meters. The rooftop is a quadrilateral.

Part A: What expression describes the length of one side of the rooftop?

Part B: What type of quadrilateral is the rooftop? How do you know? __________________________________________________________________________________________________________________________________________________________________________________________________________________

Part C: If the area of the rooftop is 441 m2 ,what is the value of x?

Part D: What is the length of the rooftop? Remember to include units.

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Example 2: A rectangular garden that has a width 12 meters and a length 16 meters is going to have a walkway installed around it. The walkway will have a width of x meters, as shown in the diagram below. Together, the walkway and the garden have an area of 396 square meters.

Part A: Write an equation that can be used to find x, the width of the walkway.

Answer:__________________________________________

Part B: Determine and state the width of the garden with the walkway included. Remember to include units.

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Part 3: Independent Practice (MILD)

1. The width of a rectangle is 2 x+1 feet and the length of the rectangle is x−5 feet. The area of the rectangle is 51 square feet.

Part A: Which of the following expressions represents the area of the rectangle? Circle all that apply.

(a) (2 x+1 ) ( x−5 )=51

(b) (2 x+1 ) ( x−5 )

(c) 2(2 x+1)+2(x –5)

(d) 2 x2−9 x−5

(e) 2 x2−10x+x−5

Part B: Knowing that the area is 51 square feet, write an equation that could be used to solve for x.

Part C: On the lines below, explain how that equation represents x.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Part D: Find the length and width of the rectangle. Remember to include units.

2. You can use the formula V=lwh to find the volume of a rectangular prism. A rectangular prism has sides of 2 feet, (3x – 5) feet, and (2x – 1) feet. Write a quadratic equation in standard form that represents the volume of the box.

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3. The length of a rectangle is four less than twice its width. The area of the rectangle is 96 square meters. Find the perimeter of the rectangle. Remember to include units.

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Part 3: Independent Practice (MEDIUM)

4. A rectangle has an area of 180 ft2 . Find the width of the rectangle if the length of the rectangle is 8 more than the width.

Part A: Draw and label the rectangle that models this situation.

Part B: Write an equation that models this situation.

Part C: What is the length of the rectangle? Remember to include units.

Part D: Using your width and length, prove that your answer is correct.

5. Use the information below to answer Parts A-C.

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Mrs. Hills’ vegetable garden is currently a rectangle measuring 8 feet by 4 feet. She wants to increase each side by 2 x feet so that her garden has an area of 140 square feet.

Part A: Write an equation she could use to determine the value of x.

Part B: In the space below describe why your equation models this situation.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Part C: Solve for the value(s) of x.

Part D: What are the width and length of her new vegetable garden?

Answer _____________________________________

Part 6: Independent Practice (SPICY)6. The height of a triangle is 2 inches less than twice the length of its base. If the total area of the triangle is 56 square inches, find the lengths of the base and height. Remember to include units.

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7. On the graph below, graph the quadratic equation y=x2+6 x+2

What is the axis of symmetry of the graph above? ______________What is the vertex of the graph above? ______________What is the y-intercept of the graph above? ______________

Mathletes

1.

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“”I’m a mathlete

2.

3.

Name Date _________ HSF.IF.B.4 HSA.CED.A.3 Class

Write and solve quadratics that model real-world scenariosExit Ticket

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

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1. A square is altered such that one dimension is increased by 4 meters whereas the other dimension is decreased by 2 meters. The new area of the resulting rectangle is 55 square meters.

Part A: Draw a diagram that represents this situation.

Part B: What are the length and width of the rectangle? Remember to include units.

Part C: What is the original area of the square? Remember to include units.

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Name Date _________ HSF.IF.B.4 HSA.CED.A.3 Class

Write and solve quadratics that model real-world scenariosHomework

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

1. The length of Mr. Marcus’ front porch is 9 feet and the width is 10 feet. He wanted to increase each side length by 3x feet. Write a trinomial expression that could be used to calculate the length and width of his new porch.

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Answer:____________________________

2. If his new porch has an area of 240square feet, what is the value of x?

Answer:_______________________

3. Which of the values of x didn’t make sense in the context of this problem? Why?

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________4. What is the length and width of his deck after the extension is built?

Answer:___________________________5. What is the area of a rectangle with a width of (4 x+1¿and length of (x+3)?

a) x2+13 x+3b) 4 x2+12 x+3c) 4 x2+13 x+3d) 4 x2+13 x+4

6. A man is offered a gym membership. It costs $85 for five months and $130 for eight months. What equation represents the cost g(x ) of the membership based on the number of months, x?

(a) g ( x )=15 x+85

(b)g ( x )=17 x

(c) g ( x )=15 x+20

(d)g ( x )=15 x+10

7. Which of the factored quadratics 8. Which value of c will make this a perfect square trinomial?

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below has zeroes of 34 and −2

3?

(a) (4 x−3 ) (3x−2 )(b)(4 x−3 ) (3x+2 )(c) (4 x+3 ) (3 x+2 )(d)(4 x+3 )(3 x−2)

n2−14n+c

a) - 49b) 49c) 9d) 64

9. What value(s) of x will make the equation below true? Complete the square.

x2 + 14x + 4 = 0

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