UNIT 7 Quadratics Functions Part 1

Developed by CHCCS and WCPSS

2018

NC Math 1

UNIT 7

Quadratics Functions Part 1

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MODULE 7 - TABLE OF CONTENTS

QUADRATIC FUNCTIONS, Part 1

7.1 Children’s Barnyard - A Develop/Solidify Understanding Task Building and interpreting linear functions. (NC.M1.A-APR.1) READY, SET, GO Homework: Quadratic Functions Part 1 7.1

7.2 Food at the Fair - A Practice Understanding Task Adding linear functions in context and out of context. (NC.M1.A-APR.1, NC.M1.F-IF.7NC.M1.F-BF.1b) READY, SET, GO Homework: Quadratic Functions Part 1 7.2

7.3 Children’s Barnyard Revisited - A Develop/Solidify Understanding Task Multiplying polynomials to find area. (NC.M1.A-APR.1, NC.M1.A-SSE.1b) READY, SET, GO Homework: Quadratic Functions Part 1 7.3

7.4 Ages and Wages - A Practice Understanding Task Multiplying binomials. (NC.M1.A-APR.1, NC.M1.A-SSE.1b) READY, SET, GO Homework: Quadratic Functions Part 1 7.4

7.5 Lines at the State Fair - A Develop Understanding Task Finding the second difference. (NC.M1.A-SSE.1b, NC.M1.A-CED.2) READY, SET, GO Homework: Quadratic Functions Part 1 7.5

7.6 Homegrown Music Fest - A Solidify Understanding Task Finding a maximum. (NC.M1.A-SSE.1b, NC.M1.F-BF.1b, NC.M1.F-IF.4, NC.M1.F-IF.5, NC.M1.F-IF.7) READY, SET, GO Homework: Quadratic Functions Part 1 7.6

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7.7 Circle “C” Racing Pigs - A Solidify Understanding Task Identifying key features of quadratics. (NC.M1.A-APR.3, NC.M1.F-IF.4, NC.M1.F-BF.1b) READY, SET, GO Homework: Quadratic Functions Part 1 7.7

7.8 Nothing Could Be Finer - A Practice Understanding Task Identifying key features of quadratic functions in context while comparing multiple representations. (NC.M1.A-APR.3, NC.M1.A-CED.2, NC.M1.F-IF.4) READY, SET, GO Homework: Quadratic Functions Part 1 7.8

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NC Math 1 Unit 7 Quadratic Functions


7.1 Children’s Barnyard

A Develop/Solidify Understanding Task The NC State Fair Planning Committee is working on the layout for the Children’s Barnyard, which is an exhibit that allows fair visitors to get up close and personal with farm animals. Their plan for the layout is shown in the figure below. Measurements are in feet.

1. Use your knowledge of geometric figures to find the perimeter of each animal pen.

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2. What does 𝑥 represent?

3. If 𝑥 = 7, what is the perimeter of the sheep pen?

4. If the perimeter of the sheep pen is 100 feet, what is 𝑥?

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NC Math 1 Unit 7 Quadratic Functions 7.1


READY Topic: Finding perimeter and area of figures

Find the perimeter of each figure below.

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9. 10.

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Go! Topic: Finding rate of change

Find the rate of change for each of the following functions.

13. rate of change = _______________________

14. rate of change = _______________________

15. rate of change = _______________________

16. rate of change = _______________________

532 += xy

( ) ( ) ( ) 5.71,211 +-== nfnff

( )

33527421315291nfn

11. 12.

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7.2 Food at the Fair

A Practice Understanding Task

At the NC State Fair, John is an owner of two food booths. One of his booths sells cotton candy and the other sells turkey legs.

The cotton candy booth pays $500 per day for food materials and $16 per hour for people to work the booth. The permit and materials for the turkey leg booth costs $700 for the day and labor costs $30 per hour. Both booths are open for 12 hours a day.

1. Model the total amount the owner is spending each day to operate the two booths. Usemultiple representations such as graphs, tables, and equations.

2. How much more does the turkey leg booth cost per day? Model the situation usingmultiple representations.

3. If cotton candy sells for $4 per bag, how many bags does the owner have to sell to breakeven?

4. If turkey legs sell for $5 per leg, how many turkey legs does he have to sell to break even?

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Given the functions below, find the sums.

f(x) = 7x – 9 g(x) = -2x + 1 h(x) = 3x + 5 j(x) = -x – 11

5. What is the sum of f(x) and g(x)?

6. What is the sum of h(x) and j(x)?

7. What is the sum of f(x) and j(x)?

8. What is the sum of g(x) and h(x)?

9. f(x) + h(x) = ?

10. g(x) + j(x) = ?

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READY

Topic: Add and subtract linear expressions

Write an equivalent expression in the form ax + b.

1)

2)

3) 4)

5) Marcie is shopping for a movie streaming service. Flixnet charges a flat $4.50 per movie withno monthly membership fee. Choyko charges a monthly membership fee of $4, and then$3.25 per movie.

a. Write a function to describe the total monthly cost for each service

Flixnet: _____________________

Choyko: _____________________

b. Write an expression for the difference between the monthly costs for the two services.

6) Xian is planning to sell souvenirs at one of the upcoming county fairs. At the Walch CountyFair, a vendor’s permit would cost $75, and he could expect to pay his seller $9 per hour. Atthe Pickaway County Fair, a vendor’s permit costs $60, but since the fair is busier and furtheraway, he would have to pay his seller $11 per hour.

a. Write a function to describe Xian’s total costs for each fair

Walch County _______________________

Pickaway County _______________________

7853

+++xx ( ) ( )8725 ++- xx

( ) ( )3114 +-+ xx ( ) ( )xx 3523 ---

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b. If Xian decides to send a seller to both fairs, write an expression for the total cost of

permits and sellers for x hours ____________________________

i. What is the rate of change (total hourly wage cost) if Xian decides to send

sellers to both fairs? _____________

c. Write an expression for the difference in the costs between the two fairs

_________________________

SET Topic: Combine linear functions and determine features of the graphs

Given the functions for questions 7 & 8:

𝑓(𝑥) = '(𝑥 + 2 7) Find 𝑓(𝑥) + 𝑔(𝑥)

𝑔(𝑥) = − -(𝑥 + 5

8) How does the y-intercept of this new function compare to the y-intercept of each of theoriginal functions?

Given the functions: for questions 9 & 10:

𝑚(𝑥) = 0'𝑥 + 3 9) What do you notice about the slopes of the linear

𝑝(𝑥) = 0'𝑥 − 4 functions given? What does this mean for their graphs?

10) Make a prediction: When you find the sum of these functions, what will the new graph looklike? Specify what you think the y-intercept, the x-intercept, the slope will be.

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GO! Topic: Write a linear equation based on a context

Write an explicit linear equation to model the situations.

11) Sarah is going to put on a charity concert. She gets a total donation of $3,500 from herbiggest supporter, and then also makes money when people purchase a ticket for $3 each.Write an equation for her total income depending on the number of tickets she sells.

12) Micah is a car salesperson who made $4,000 in commission this week. He also makes $9.50per hour. Write an equation that represents the total amount of money made that weekdepending on the number of hours he worked.

13) Miraya is watching the concert venue fill up with people. The stadium holds 35,000 peopleand she notices that every minute 300 more people sit down. Write an equation that willhelp her predict how many seats will be still empty at any given minute.

14) Jamal and his friends are going to put on a concert. They have to pay $2,000 in start-upcosts for their venue, electricity, security, and concessions. They will make $12 per ticketthat they sell. Write an equation for their profit depending on the number of tickets sold.

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7.3 Children’s Barnyard RevisitedA Develop/Solidify Understanding Task

The NC State Fair Planning Committee is still working on plans for the Children’s Barnyard.

They need to rebuild the barnyard using new dimensions.

1. In order to begin, the Committee will start with the duck pen. Let x represent thedimensions of each side of the duck pen.

x

x

2. Now draw a diagram to represent the area of the cow pen if it starts at the same size asthe duck pen, but is increased by 4 feet on one side and 3 on the other.

3. Continue to find the dimensions of each pen by drawing a diagram and writingexpressions given the following scenarios. Use x to represent the length of the sides of theoriginal square.a. Sheep pen is increased by 7 feet on one side and decreased by 2 on the other.

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b. Goat pen is decreased by 5 on one side and increased by 3 on the other.

4. Using your new dimensions of each pen, find the area of each new lot as the product ofthe length and width.

5. If x = 5, what would the area be for each pen?

6. Write an expression to represent the total area of the cow’s pen and sheep’s pentogether.

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READY Topic: Using Distributive Property and Exponent Rules to simplify.

Simplify the following expressions.

1. (3x)(x – 7) 2. x2 + 5 + 2x2 – 6 + x

3. x(x + 1) – x(3x + 8) 4. (x2 + 7 + x) – (x2 + x – 3)

5. x(x + 12) – 3(x2 – 3) 6. (x – 11) – x(x + 2)

SET

Topic: Evaluating Functions

Given the functions below, find the products identified.

𝑓(𝑥) = 7𝑥 − 9

𝑔(𝑥) = −2𝑥 + 1

ℎ(𝑥) = 3𝑥 + 5

𝑗(𝑥) = −𝑥 − 11

7. What is the product of 𝑓(𝑥) and 𝑔(𝑥)?

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8. What is the product of ℎ(𝑥)and 𝑗(𝑥)?

9. What is the product of 𝑓(𝑥)and 𝑗(𝑥)?

10. 𝑔(𝑥) ∙ ℎ(𝑥) 11. 𝑓(𝑥) ∙ ℎ(𝑥) 12. (𝑔(𝑥))(𝑗(𝑥))

GO! Topic: Evaluate a quadratic function for a given value of 𝑥

The graph to the right 𝒇(𝒙) represents a ball being shot straight up in the air from the ground, traveling up through the air, and then coming back down.

13. What is 𝑓(0) in this graph?

14. What does 𝑓(0) represent?

15. What is 𝑓(5)in this graph?


17. What is 𝑓(10) in this graph?


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Evaluate the functions 𝒇(𝒙) = 𝟐𝒙𝟐 − 𝟑𝒙 + 𝟗 and 𝒈(𝒙) = −𝟑𝒙𝟐 + 𝟖𝒙 − 𝟏 for the given values of 𝒙.

19. What is 𝑓(3) ?

20. What is 𝑔(−2) ?

21. What is 𝑓(−5)?

22. What is 𝑔(8)?

23. What is𝑓(0)?

24. What is 𝑔(0)?

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7.4 Ages and WagesA Practice Understanding Task

1. Edgar is six years older than Patty, and Isabella is two years younger than Patty. Write twoexpressions for the product of Edgar and Isabella’s age, based on Patty’s unknown age.

2. Fran is twice Patty’s age, and Roberto is four years less than three times Patty’s age. Write twoexpressions for the product of Fran and Roberto’s age, based on Patty’s unknown age.

3. Write and simplify an expression that represents the sum of all of their ages, including Patty’s, allbased on Patty’s unknown age.

4. Edgar makes $2 per hour more than Patty, and Isabella makes $0.50 less per hour than Patty. Writetwo expressions for the product of Edgar and Isabella’s money that they make per hour, based onPatty’s unknown wage.

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5. Fran makes four times as much money as Patty does, and Roberto makes one dollar more thantwice as much money as Patty. Write two expressions for the product of Fran and Roberto’s moneythat they make per hour, based on Patty’s unknown wage.

6. Write and simplify an expression that represents the sum of all of their wages, including Patty’s, allbased on Patty’s unknown wage.

7. If Patty is a 25 year old waitress who makes $8 per hour, find the ages and wages of all of herfriends.

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READY Topic: Graphing Quadratics.

Using the quadratic function, complete the table and then graph.

1. y = x2 – 4x – 5

2. y = x2 + 4x + 3

x y -2 -1 0 1 2 3 4 5

x y

x y -2 -1 0 1 2 3 4 5

x y -2 -1 0 1 2 3 4 5

x y -2 -1 0 1 2 3 4 5

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SET Topic: Multiplying Polynomials.

Simplify.

3. (3w + 4)(2w – 1) 4. (y + 3)2

5. (6x – 11)(x + 3) 6. (d2 + 1)(d – 1)

7. (4x – 6)2 8. (x + y)(x – y)

GO! Topic: Solving Systems of Equations.

Solve.

9. y = 3x – 1y = -2x + 4

10. 5x + 7y = 775x + 3y = 53

11. x = -2y + 200x = y + 50

12. -6x = 4y + 113x + 2y – 21 = 0

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7.5 Lines at the State Fair

A Develop Understanding Task

Waiting in lines at the state fair – for better or worse – is part of the experience! As more people join the line, the waiting area expands. The set of figures below describes how large the waiting area is in each stage of growth. Each block represents 1 square meter.

1. What patterns do you notice in the set of figures?

2. Is there a linear relationship between the figure number and the total number of blocks?Why or why not?

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3. Sketch the next figure in the sequence.

4. Determine an equation for the total number of blocks in any figure in the sequence.Explain your equation and show how it relates to the visual diagram of the figures.

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READY

Topic: Finding dimensions or lengths using perfect squares.

List the dimensions of each square.

1. Area = 144 in2 2. Area = 4𝑥# ft2

3. Area = 36 m2 4. Area = 49𝑥% cm2

List the area of square given a side length.

5. Side = 11 ft 6. Side = 13𝑥 mi

7. Side = 5 yd 8. Side = 10y in

SET Topic: Quadratic patterns

9. Match the following three patterns with their equations in explicit form.

I. 𝑓(𝑛) = 3𝑛#

II. 𝑔(𝑛) = 𝑛# + 4

III. ℎ(𝑛) = 𝑛# − 1

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GO! Topic: Solve a literal equation for a given variable and apply the distributive property to simplify expressions

Use steps for solving equations to solve each equation for x.

10. 𝑎𝑥# + 𝑐 = 𝑑

11. 𝑚𝑥# − 𝑛 = 𝑝

Use the distributive property to simplify each of the following expressions.

12. 8(2𝑥 − 9) 13. −2𝑥(5𝑥 + 6)

14. 3𝑥#(4𝑥 − 11) 15. −8𝑥(3𝑥# − 9𝑥)

16. 8(3𝑥 + 1) + 𝑥(2𝑥 − 4) 17. −2(5𝑥 − 2) + 𝑥(7𝑥 + 6)

18. 𝑥(2𝑥 − 4) − 3(𝑥 + 8)

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7.6 Homegrown Music Fest

A Solidify Understanding Task

The State fair puts on a ton of concerts over the 10 days it is open. The concert promoter has hired you to help decide how much to charge for some of the tickets! If tickets for Maurice and the O’s were $6, the fair would make a profit of $12,000, but they could actually make more money! If they raise their ticket prices by $1, their profit would then be $13,300. If they raise their ticket prices by $2, their profit would be $14,400!

1. If this pattern continues, how much should they raise their ticket prices to ensure they will make amaximum profit? Will there ever be a maximum profit, or will the profit continue to grow if you continue to increase the ticket price? Use multiple representations to defend your answer.

2. At a ticket price of $14 (which is raising the ticket price by $8) what is happening to the profit? Whatwill happen to the profit at a ticket price of $15? How can you explain this?

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3. Describe the pattern/relationship between the ticket price and the profit. Is there a pattern thatexists with the number of people attending the concert as well?

4. If the ticket price is FREE ($0) the profit should be $0, can you use your representations to prove thistheory?

5. Is there another ticket price that would result in $0 profit? How could you find this amount? Howcould this possible?

Page 28 Developed by CHCCS and WCPSS



READY Topic: Find an expression for area

Write and simplify an expression for the area of each figure below.

1. ______________________ square miles

2. ______________________ square inches

3. Area of shaded region: ________________ ft2

4. Area of shaded region: _______________ cm2

SET Topic: Using quadratic patterns to answer questions.

5. If you want to sell your cupcakes for $2 per dozen, you can make a profit of $160. If raise the cupcake price by $1, you can make a profit of $216. If you raise the cupcake price by $2 (which makes them $4 per dozen) you can make $256 profit. What is the maximum profit you can make? What should you charge for your cupcakes?

(3x) mi

(5x + 3) mi

(0.6x) in (2x + 3) in (5.8x) in

(5x – 1) in

(2w) ft

(7w – 3) ft

(3x + 2) cm

(5x) cm

(w) ft (w + 3) ft

(x) cm



(3, -3)

6. Given the table below, identify the pattern. Continue the pattern to find the maximum.

x y -4 0 -3 65 -2 120 -1 165

7. Given the table below, identify the pattern. Will there be a maximum value? How can you tell?

x y 4 8 5 12 6 18 7 26

8. Given the graph below, find the maximum as an ordered pair.

(1, 13)

(2, 6)

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GO! Topic: Find the x-intercept of a linear function

Find the coordinates of the x-intercept for each line.

9. ( ______, ______)

10. ( ______, ______)

11. ( ______, ______)

12. ( ______, ______)

13. ( ______, ______)

2443 =+ yx

1243 -= xy

155184213242271yx

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7.7 Circle “C” Racing Pigs

A Solidify Understanding Task

Dennis Cook runs the Circle “C” Racing Pigs on the Hogway Speedway at the N.C. State Fair. Everyone loves to watch the swift swine rush around the track to get a cheese puff, which is what they are awarded at the end of the race.

Mr. Cook wants to create a pen outside of the pig trailer so that the pigs can meet and greet their fans. He will use one side of the trailer for part of the pen and has enough money for 80 feet of fencing material.

1. What are some possible dimensions for the pen? Use multiple representations to show yourpossibilities.

2. Would a long skinny rectangular pen have more or less area than a square pen? Explainhow you know.

3. Write a function to represent the areas of this pen in terms of its width. Be sure to clearlylabel what each of your variables represents.

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4. What is the maximum area for the pen? What are the dimensions that would give the penthe maximum area? (The pigs deserve the biggest and best!) Show this maximum usingboth a table and a graph.

5. What are the x-intercepts of the function? How do the x-intercepts relate to the context?

6. Describe where the table/graph is increasing and decreasing. Explain what each means incontext.

7. What is the relationship between the x-intercepts and the vertex of the function?

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READY

Topic: Naming Conventions of Polynomials

Classifying Polynomials: Write each polynomial in the correct cell below based on its name.

4𝑥 − 3 4𝑥% − 2𝑥 + 1 465

−11𝑧 5𝑑% − 7 2𝑥% − 5𝑥 + 1

5𝑎 −121 5𝑘 − 5

3𝑦% 0 −15𝑎% − 21

Constant Monomial Linear Monomial Quadratic Monomial

Linear Binomial Quadratic Binomial Quadratic Trinomial

Write each polynomial in simplified, standard form. Then, name each polynomial. 1. 4𝑥(𝑥 − 2)

Simplified, standard form: ____________________________________

Name of polynomial: _______________________________________

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2. 5𝑥 − 3 + 2𝑥 + 7



3. −2𝑥(𝑥 − 4) + 4𝑥 − 5



SET!

Topic: Find the x-intercepts and the vertex of a quadratic function given the graph.

Identify the x-intercepts and the vertex.

4.

x-intercepts:

Vertex:

5.

x-intercepts:

Vertex:

6.

x-intercepts:

Vertex:

7.

x-intercepts:

Vertex:

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8. If the x-intercepts of a quadratic function are (8,0) and (6, 0), what is the x-coordinate of thevertex?

9. If the x-intercepts of a quadratic function are (-4,0) and (4, 0), what is the x-coordinate ofthe vertex?

10. Given a vertex with an x-coordinate of 7, what are two possible x-intercepts for thisquadratic?

GO!

Topic: Properties of Exponents

Evaluate each expression.

11. 12. 13.

Simplify each expression.

14. 15. 16.

17. 18. 19.

20. 21. 22.

2

922- ( )325-

2

555

6 3a a• ( )25x ( )52 34a b

8

6xx

23

442ab

æ öç ÷è ø

4 03x y--

( ) 34 7x y - 2

126x yxy -

3 9

2 2520

x yx y -

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Page 38 This page intentionally left blank.



7.8 Nothing Could Be Finer

A Practice Understanding Task

Fairgoers can take home some official State Fair souvenirs from The Nest in the Expo Center

lobby. This year, The Nest has a new hat design to celebrate 150 years of the State Fair and

they want to decide on a price that maximizes profit. The graph below represents profit 𝑃(𝑥)

in hundreds of dollars generated by each hat price, 𝑥.

1. If The Nest wants to make a maximum profit, what should the price of the hat be?

2. What is the minimum price of a hat that produces profit for The Nest? Explain your

answer.

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62

Page 39



3. Estimate the value of 𝑃(0), and explain what the value means in the context of the

problem and how this may be possible.

4. If The Nest wants to make a profit of $13,700, how much should they charge per hat?

5. Find the portion of the domain that results in a profit for The Nest, and find the

corresponding range of profit.

6. What hat price(s) give us an increasing profit? Decreasing profit?

7. Choose the interval where the profit is increasing the fastest: (2,3), (4,5), (5.5, 6.5), or (6,7).

Explain your reasoning.

8. The Nest owner believes that selling the hat at a higher price results in a greater profit.

Explain to the owner how selling the hat at a higher price affects the profit.

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Part 2: Match each quadratic to the appropriate graph.

1. 𝑦 = (𝑥 − 2)(𝑥 + 4) 4. 𝑦 = (𝑥 − 1)(𝑥 + 3)

2. 𝑦 = −(𝑥 − 2)(𝑥 + 2) 5. 𝑦 = (𝑥 + 1)(𝑥 − 3)

3. 𝑦 = −2(𝑥 − 2)(𝑥 + 1) 6. 𝑦 = −(𝑥 − 2)(𝑥 + 3)

Identify which key features were helpful in matching these equations with their graphs. Given

this pattern, could you write a possible equation to match the profit graph 𝑃(𝑥) above?

A B C

D E F

Page 41

NC Math 1 Unit 7 Quadratic functions 7.8


READY

Topic: Key Features from a Quadratic.

1. Define the following characteristics of the quadratic function 𝒇(𝒙) = −𝒙𝟐 − 𝟒𝒙

Vertex: ______________ Minimum or Maximum? ________________

Axis of symmetry: ______________

X-intercepts: ______________

Y-intercept: ______________

Increasing: ________________

Decreasing: ________________

Domain: ________________

Range: ________________

2. Define the following characteristics of the quadratic function 𝒚 = 𝒙𝟐 − 𝟒𝒙 − 𝟓

Vertex: ______________ Minimum or Maximum? ___________________

Axis of symmetry: ______________

X-intercepts: ______________

Y-intercept: ______________

Increasing: ________________

Decreasing: ________________

Domain: ________________

Range: ________________

3

6

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SET

Topic: Use key features to interpret a quadratic in context.

Consider the graph of the quadratic function shown below and answer the questions.

A company is comparing their toy price (x) and their profits(y). Use the graph to help answer the following questions.

3. If the company wantsto maximize profit,how much should thetoy cost?

4. What is the minimumprice a toy should costto make any profit?

5. If the company wants to make a profit of $400, how much should the toy be sold?

6. Find the domain and range that result in a profit for the company.

Toy Price

Profit

Page 43



GO!

Topic: Solve a system of equations

Solve the following system of equations.

7. 𝑦 = 3– 𝑥 5𝑥 + 3𝑦 = −1

8. 2𝑥 − 3𝑦 = 8 3𝑥 − 7𝑦 = 7

9. The perimeter of a rectangular wooden deckis 90 feet. The deck's length, l, is 5 feet less than 4 times its width, w. Which system of linear equations can be used to determine the dimensions, in feet, of the wooden deck?

10. A movie theater charges $5 for an adult’sticket and $2 for a child’s ticket. OneSaturday, the theater sold 785 tickets for$3280. How many of each type of ticketwere sold?

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UNIT 7 Quadratics Functions Part 1

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