Algebra: Unit 4

Unit 4 - Item 1
Kim needs to build a rectangular fence enclosure for her dogs with only 100 yards of fencing total. She will need 3 sides of fencing because the fourth side is the barn (see the image below). The length, 𝑥, and the width, 𝑦, has been marked.

1

Create a quadratic expression that represents the different possible lengths and the total area for Kim's fence enclosure.

1

What dimensions of the fence will create the largest possible area for the rectangular fence enclosure?

Unit 4 - Item 2
The girls’ basketball team at your school is playing for the regional title. The game comes down to the final seconds and your star player gets fouled and has an opportunity to win the game at the free-throw line. She makes the shot and wins the game!

The path of her shot can be modeled by the equation 𝑦 = −2𝑥2 + 8𝑥 + 3 .

1

What is the vertex of the parabola represented by the equation?

1

Determine the height of the player. What characteristic does this represent on the graph?

1

Identify the intervals where the ball's path is increasing and decreasing.

1

Determine the domain and range of this scenario.

Unit 4 - Item 3
The teacher puts two expressions on the board and asks students to discuss what is alike and what is different.

1

Which student is correct? Justify your reasoning.

Unit 4 - Item 4
One of the main events at this year’s Science Olympiad is the Water Rocket event which uses an air compressor for the flight. The picture shows an example of the flight of a rocket.


Jason and AJ have signed up for this event. Their math teacher overheard them strategizing
over how to win and asked them to represent the flight in the form of an equation.

1

Which student’s example could be true? How do you know?

1

Looking at Jason’s example, write this equation in factored form and vertex form.

1

Looking at AJ’s example, write this equation in factored form and vertex form.

1

Unit 4 - Item 5
Match the functions with the patterns:
(see image)

One of these patterns has a step with 199 tiles in it, one has a step with 576 tiles in it, and one of these patterns has a step with 801 tiles in it. Decide which is which and explain how you know.

1

Unit 4 - Item 6
A rectangle has side lengths of (x + 6) and (2x + 3). Write an expression in standard form that represents the area of the rectangle.

Unit 4 - Item 7
A local yogurt shop is looking for a poster design to help advertise their summer specials. Anyone can design a poster to be entered in the contest. The person’s poster that is chosen will win $100.00 cash and All You Can Eat Yogurt for one year.

The poster must be a rectangle with a side length of 2 more inches than two times
the width. The total area of the poster is 112 square inches.

1

Create a quadratic equation that represents the area of the poster

1

Determine the dimensions of the poster using the quadratic equation.

1

The shop decided to change the poster area to 144 square inches. Create the new quadratic equation and determine the new dimensions of the poster.

Unit 4 Item 8
Jaquaya is creating two design templates in inches for her friend’s jewelry boxes.

For the first design, she can make the width of the boxes any length she chooses but the length must be five more than twice the width.
 
For the second design, the width must be three times the width of the first design and the length must be 4 more than the width of the first design.

The total area of two rectangles can be represented by the expressions (x)(2x+5) and (3x)(x+4).

1

What is the area of Jaquaya’s first design?

1

What is the area of Jaquaya’s second design?

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Create an expression that represents the collective area of both designs.

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Unit 4 – Item 9
A city is planning to construct an Olympic-size pool in an existing city park. They are also planning to pour concrete in a uniform width around the pool to form a walkway. The standard size of an Olympic-size pool is 164𝑓𝑡 by 82𝑓𝑡 and they have enough concrete to pour 4192 𝑓𝑡2.
How wide can the walkway be?

1

Unit 4 – Item 10
Determine the values of 𝑎, ℎ, and 𝑘 that will make 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 be the equation
represented in the graph.

1

Unit 4 – Item 11
Irving launches a toy rocket from a platform. The height of the rocket is modeled by the function ℎ = −16𝑡2 + 64𝑡 + 36 where t represents the time in seconds after launch. What is the
appropriate domain for this situation?

1

Unit 4 – Item 12
A company sells baseball necklaces at several tournaments. The amount of profit f(x) is related to the selling price of each necklace, x, is modeled by the function below.
𝒇(𝒙) = −𝟗𝒙𝟐 + 𝟒𝟔𝟓𝒙 − 𝟑𝟎𝟔𝟎

Use the function, find the price the necklaces should be sold for to make the maximum profit.

1

Unit 4 – Item 13
A soccer player kicks a ball into the air. The height of the ball above the ground can be modeled by the quadratic function ℎ(𝑡) = −5𝑡2 + 20𝑡 + 2, where h(t) represents the height of the ball in meters at time t seconds after the kick.
At what time will the ball hit the ground?

Unit 4 – Item 14
A company sells headphones for $20 per unit. The company finds that the revenue R(x) from
selling x headphones can be modeled by the quadratic function 𝑅(𝑥) = −2𝑥2 + 60𝑥, where x is
the number of headphones sold.

1

How many headphones should the company sell to maximize revenue?

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What is the maximum revenue that can be achieved?

Unit 4 – Item 15
Jaquez threw a tennis ball from a certain height. The height in feet is given by the function 𝑓(𝑡) = −16𝑡2 + 400 where t represents the time in seconds after the launch.

1

Sketch a graph to represent the time in seconds after the launch.

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What is the ball’s initial height?

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After how many seconds does the ball hit the ground?

1

Unit 4 – Item 16
Harry kicks a football. It’s height in feet is given by ℎ(𝑡) = −16𝑡2 + 48𝑡 where t represents the time in seconds after kick.

Study the graph below that represents this scenario and determine which of the statements are true statements.

Create a quadratic expression that represents the different possible lengths and the total area for Kim's fence enclosure.

What dimensions of the fence will create the largest possible area for the rectangular fence enclosure?

What is the vertex of the parabola represented by the equation?

Determine the height of the player. What characteristic does this represent on the graph?

Identify the intervals where the ball's path is increasing and decreasing.

Determine the domain and range of this scenario.

Which student is correct? Justify your reasoning.

Which student’s example could be true? How do you know?

Looking at Jason’s example, write this equation in factored form and vertex form.

Looking at AJ’s example, write this equation in factored form and vertex form.

Unit 4 - Item 5Match the functions with the patterns:(see image)One of these patterns has a step with 199 tiles in it, one has a step with 576 tiles in it, and one of these patterns has a step with 801 tiles in it. Decide which is which and explain how you know.

Unit 4 - Item 6A rectangle has side lengths of (x + 6) and (2x + 3). Write an expression in standard form that represents the area of the rectangle.

Create a quadratic equation that represents the area of the poster

Determine the dimensions of the poster using the quadratic equation.

The shop decided to change the poster area to 144 square inches. Create the new quadratic equation and determine the new dimensions of the poster.

What is the area of Jaquaya’s first design?

What is the area of Jaquaya’s second design?

Create an expression that represents the collective area of both designs.

Unit 4 – Item 10Determine the values of 𝑎, ℎ, and 𝑘 that will make 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 be the equationrepresented in the graph.

Unit 4 – Item 11Irving launches a toy rocket from a platform. The height of the rocket is modeled by the function ℎ = −16𝑡2 + 64𝑡 + 36 where t represents the time in seconds after launch. What is theappropriate domain for this situation?

How many headphones should the company sell to maximize revenue?

What is the maximum revenue that can be achieved?

Sketch a graph to represent the time in seconds after the launch.

What is the ball’s initial height?

After how many seconds does the ball hit the ground?

Unit 4 – Item 16Harry kicks a football. It’s height in feet is given by ℎ(𝑡) = −16𝑡2 + 48𝑡 where t represents the time in seconds after kick.Study the graph below that represents this scenario and determine which of the statements are true statements.

Unit 4 - Item 5
Match the functions with the patterns:
(see image)

One of these patterns has a step with 199 tiles in it, one has a step with 576 tiles in it, and one of these patterns has a step with 801 tiles in it. Decide which is which and explain how you know.

Unit 4 - Item 6
A rectangle has side lengths of (x + 6) and (2x + 3). Write an expression in standard form that represents the area of the rectangle.

Unit 4 – Item 10
Determine the values of 𝑎, ℎ, and 𝑘 that will make 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 be the equation
represented in the graph.

Unit 4 – Item 11
Irving launches a toy rocket from a platform. The height of the rocket is modeled by the function ℎ = −16𝑡2 + 64𝑡 + 36 where t represents the time in seconds after launch. What is the
appropriate domain for this situation?

Unit 4 – Item 16
Harry kicks a football. It’s height in feet is given by ℎ(𝑡) = −16𝑡2 + 48𝑡 where t represents the time in seconds after kick.

Study the graph below that represents this scenario and determine which of the statements are true statements.