UNIT 8 Quadratics Functions Part 2 - WELCOME TO MS. SMITH'S MATH … · 2018-12-14 · READY, SET,...

Developed by CHCCS and WCPSS

2018

NC Math 1

UNIT 8

Quadratics Functions Part 2

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MODULE 8 - TABLE OF CONTENTS

QUADRATIC FUNCTIONS, Part 2

8.1 Quanyka’s Quilts - A Develop Understanding Task An introduction to factoring using the area model. (NC.M1.F-IF.8a, NC.M1.A-SSE.3) READY, SET, GO Homework: Quadratic Functions Part 2 8.1

8.2 Quilt Blocks Galore! - A Solidify Understanding Task Continuing factoring with subtraction. (NC.M1.F-IF.8a, NC.M1.A-SSE.3) READY, SET, GO Homework: Quadratic Functions Part 2 8.2

8.3 Blocks are Multiplying Everywhere! - A Solidify Understanding Task Continuing factoring with a coefficient of 𝑥𝑥2that is greater than 1. (NC.M1.A-SSE.1a, NC.M1.A-SSE.3) READY, SET, GO Homework: Quadratic Functions Part 2 8.3

8.4 Comparing Methods - A Practice Understanding Task Compares methods for solving quadratics using square roots and factoring with the zero product property. (NC.M1.A-APR.3, NC.M1.A-REI.1, NC.M1.1-REI.4, NC.M1.A-SSE.3) READY, SET, GO Homework: Quadratic Functions Part 2 8.4

8.5 Giant Parthenon Wheel - A Develop Understanding Task Develops the idea of gravity to start projectile motion problems. (NC.M1.A-SSE.1a, NC.M1.F-BF.1b, NC.M1.F-IF.8a, NC.M1.A-SSE.3) READY, SET, GO Homework: Quadratic Functions Part 2 8.5

8.6 Fireworks - A Solidify Understanding Task Continues to work with projectile motion and compares different representations of functions. (NC.M1.F-IF.7, NC.M1.F-IF.8A, NC.M1.F-IF.9, NC.MA.A-SSE.3) READY, SET, GO Homework: Quadratic Functions Part 2 8.6

8.7 Next Year’s Gig - A Solidify Understanding Task Continues to work with quadratic functions in context, changing to profit questions. Solidifies the need for factored and standard form of quadratic functions. (NC.M1.F-IF.7, NC..M1.F-BF.1b, NC.MA.A-SSE.1b) READY, SET, GO Homework: Quadratic Functions Part 2 8.7

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8.8 Rainy Day - A Practice Understanding Task Identifying key features of quadratic functions in context while comparing multiple representations. (NC.M1.A-APR.3, NC.M1.F-IF.4NC.M1.F-IF.7, NC.M1.F-IF.9) READY, SET, GO Homework: Quadratic Functions Part 2 8.8

8.9 How do you want to be paid? - A Practice Understanding Task Compares linear, exponential, and quadratic functions and their end behavior. (NC.M1.F-LE.3) READY, SET, GO Homework: Quadratic Functions Part 2 8.9

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NC MATH 1//UNIT 8

QUADRATIC FUNCTIONS

Developed by CHCCS and WCPSS Adapted from Mathematics Vision Project

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8.1 Quanyka’s Quilts

A Develop Understanding Task

Quanyka is a quilt maker who has a booth in the Village of Yesteryear each year at the NC State Fair.

She demonstrates the art of quilting and has several blankets for sale. She has noticed that over the

past several years, interest in her craft has waned. In order to reach out to the younger generation, she

decides to make a craft table where children can create their own mini-quilts that can be used with their

dolls or stuffed animals.

At first, Quanyka wants to keep it simple by only having the children sew together square blocks of

material to make the quilts. She has a machine that can quickly cut the squares out of fabric material.

Quanyka charges based on the number of square inches of material in the block. If the length of the

side of a square block that is being made is x inches, then the area of the block can be found using the

formula 𝐴(𝑥) = 𝑥2.

Some of the children want to make designs that include rectangular blocks as well. Quanyka can use

sliders on her machine to change the shape being cut into a rectangle. To use the sliders to change the

shape, she must describe each side of the rectangle in terms of how it has been modified from the

original square shape. For example, one quilt block consists of starting with a square block and

extending one side length by 5 inches and the other side length by 2 inches to form a new rectangular

block. Quanyka determines how to modify her formula to calculate the area for these new blocks. In

this case, she knows that the area of this new block can be represented by the expression: 𝐴(𝑥) = (𝑥 +

5)(𝑥 + 2). However, she does not feel that this expression gives the children a real sense of how much

bigger this new block is (e.g., how much more area it has) when compared to the original square blocks.

1. Can you find a different expression to represent the area of this new rectangular block? You will

need to convince the customers that your formula is correct using a diagram.

Page 1

Developed by CHCCS and WCPSS adapted from Mathematics Vision Project

Here are some additional new rectangular block shapes that children have asked for. Find two different

algebraic expressions to represent each rectangle, and illustrate with a diagram why your

representations are correct.

2. The original square block was extended 2 inches on one side and 4 inches on the other.

3. The original square block was extended 3 inches on only one side.

Quanyka’s idea takes off and she has so many children wanting to make quilts at her booth that she calls

in her husband Quinn to help. Quinn starts taking orders for different rectangular block shapes. For

each order, he asks how much additional area they want beyond the original area of x2. Once an order is

taken for a certain type of block, Quanyka needs to have specific instructions on how to set the sliders

on her machine. The instructions need to explain how to extend the sides of a square block to create the

new rectangular block.

Quinn has placed the following orders on her table. For each, describe how to make the new blocks by

extending the sides of a square block with an initial side length of x. Your instructions should include

diagrams, written descriptions and algebraic descriptions of the area of the rectangles using expressions

representing the lengths of the sides.

4. 𝑥2 + 3𝑥 + 6𝑥 + 18 6. 𝑥2 + 7𝑥

5. 𝑥2 + 12𝑥 + 2𝑥 + 24 7. 𝑥2 + 10𝑥

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Some of the orders are written in an even more simplified algebraic code. Figure out what these entries

mean by finding the sides of the rectangles that have this area. Use the sides of the rectangle to write

equivalent expressions for the area.

8. 𝑥2 + 8𝑥 + 12

9. 𝑥2 + 6𝑥 + 5

10. 𝑥2 + 12𝑥

11. What relationships or patterns do you notice when you find the lengths of the sides of the

rectangles for a given area?

12. One of the orders that Quinn wrote down was the following: 𝑥2 + 7𝑥 + 9. Quanyka says that she

cannot make a rectangle with this area. Do you agree or disagree? How can you tell if a rectangle

can be constructed from a given area?

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NC Math 1 Unit 8 Quadratic Functions 8.1


READY

Topic: Multiplying Linear Binomials

Find the product of the binomials and write the equivalent expression in standard form, given

that standard form for a trinomial is 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄.

1. (𝑥 + 1)(𝑥 + 4) 2. (𝑥 + 9)(𝑥 + 4) 3. (𝑥 − 2)(𝑥 − 3)

4. (𝑥 + 10)(𝑥 + 3) 5. (𝑥 + 7)(𝑥 − 7) 6. (𝑥 − 6)(𝑥 − 6)

SET

Topic: Factoring Polynomials.

Re-write each expression as a product of two linear factors.

7. 𝑥2 + 8𝑥 + 16 8. 𝑥2 + 12𝑥 + 32 9. 𝑥2 + 8𝑥

10. 𝑥2 + 9𝑥 + 14 11. 𝑥2 + 12𝑥 12. 𝑥2 + 12𝑥 + 27

13. 𝑥2 + 11𝑥 + 24 14. 𝑥2 + 3𝑥 + 2 15. 𝑥2 + 4𝑥

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Go!

Topic: Writing Area Expressions

Find the area of each of the shapes below.

16. 17.

18. 19.

20. Find an expression for the area of the shaded region:

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QUADRATIC FUNCTIONS


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8.2 Quilt Blocks Galore!

A Solidify Understanding Task

Now that Quanyka is custom-making different-sized rectangular blocks for the children to use, her

business is growing by leaps and bounds. Some of the children have asked for rectangular blocks that

are bigger than the standard square block on one side. Sometimes they want one side of the block to be

the standard length, 𝑥, with the other side of the block 2 inches bigger.

1. Draw and label this block. Write two different expressions for the area of the block.

Sometimes they want blocks with one side that is the standard length, 𝑥, and one side that is 2 inches

less than the standard size.

2. Draw and label this block. Write two different expressions for the area of the block. Use your

diagram and verify algebraically that the two expressions are equivalent.

There are many other size blocks requested, with the side lengths all based on the standard length, 𝑥.

Draw and label each of the following blocks. Use your diagrams to write two equivalent expressions for

the area. Verify algebraically that the expressions are equal.

3. One side is 1” less than the standard size and the other side is 3” more than the standard size.

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4. One side is 4” more than the standard size and the other side is 2” less than the standard size.

5. One side is 1” more than the standard size and the other side is 3” less than the standard size.

6. An expression that has 3 terms in the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐: is called a trinomial. Look back at the

trinomials you wrote in questions 3-7. How can you tell if the middle term (𝑏𝑥) is going to be

positive or negative?

7. One child has an unusual request. She wants a block that is extended 3 inches on one side and

decreased by 3 inches on the other. Quinn thinks that this rectangle will have the same area as the

original square since one side was decreased by the same amount as the other side was increased.

What do you think? Use a diagram to find two expressions for the area of this block.

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8. The result of the unusual request makes Quinn curious. Is there a pattern or a way to predict the

two expressions for area when one side is increased and the other side is decreased by the same

number? Try modeling these two problems, look at your answer to #9, and see if you can find a

pattern in the result.

a. (𝑥 + 1)(𝑥 − 1)

b. (𝑥 + 5)(𝑥 − 5)

9. What pattern do you notice? What is the result of (𝑥 + 𝑛)(𝑥 − 𝑛)?

10. Some children want both sides of the block reduced. Draw the diagram for the following blocks and

find a trinomial expression for the area of each block. Use algebra to verify the trinomial expression

that you found from the diagram.

a. (𝑥 − 2)(𝑥 − 4)

b. (𝑥 − 1)(𝑥 − 3)

11. Look back over all the equivalent expressions that you have written so far, and explain how to tell if

the third term, 𝑐, in the trinomial expression 𝑥2 + 𝑏𝑥 + 𝑐 will be positive or negative.

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12. Quinn gives Quanyka a number of orders that are given as rectangular areas using a trinomial

expression. Find the equivalent expression that shows the lengths of the two sides of the rectangles.

a. 𝑥2 + 7𝑥 + 10

b. 𝑥2 + 3𝑥 − 10

c. 𝑥2 − 2𝑥 − 24

d. 𝑥2 − 10𝑥 + 24

e. 𝑥2 − 6𝑥 + 5

13. Write an explanation of how to factor a trinomial in the form: 𝑥2 + 𝑏𝑥 + 𝑐.

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READY

Topic: Number Sense.

Find two numbers that meet the requirements of the situation in each question.

1. Multiply to give you 10, and adds to give you 7 _____________ and ______________




5. Multiply to give you −80, and adds to give you −2 _____________ and ______________


7. Multiply to give you −36, and adds to give you 0 _____________ and ______________


SET



9. 𝑥2 − 4𝑥 − 5 10. 𝑥2 + 13𝑥 + 22 11. 𝑥2 − 15𝑥 + 26

12. 𝑥2 − 9 13. 𝑥2 − 81 14. 𝑥2 − 6𝑥 − 16

15. 𝑥2 + 4𝑥 − 21 16. 𝑥2 + 3𝑥 − 10 17. 𝑥2 − 49

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18. 19.

20. 21.

22. 23.

Go!

Topic: Graphing Quadratic Functions.

Match the graph of the quadratic function with the equations below.

A. 𝑦 = −(𝑥 − 5)(𝑥 − 9) D. 𝑦 = −𝑥(𝑥 − 5)

B. 𝑦 = (𝑥 + 5)(𝑥 + 2) E. 𝑦 = (𝑥 + 2)(𝑥 − 1)

C. 𝑦 = −𝑥(𝑥 + 3) F. 𝑦 = (𝑥 + 3)(𝑥 − 2)

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NC MATH 1//UNIT 8

QUADRATIC FUNCTIONS


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8.3 Blocks are Multiplying Everywhere!


Quanyka sometimes gets orders for blocks that are multiples of a given block. For instance, she got an

order for a block that was exactly twice as big as the rectangular block that has a side that is 1” longer

than the basic size, 𝑥, and one side that is 3” longer than the basic size.

1. Draw and label this block. Write two equivalent expressions for the area of the block.

2. Oh dear! This order was scrambled and the pieces are shown here. Put the pieces together to make

a rectangular block and write two equivalent expressions for the area of the block.

3. What do you notice when you compare the two equivalent expressions in problems #1 and #2?

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4. Quanyka has a lot of new requests. Use diagrams to help you find equivalent expressions for each of

the following:

a. 6𝑥2 + 12𝑥

b. 3𝑥2 + 15𝑥 + 18

c. 2𝑥2 + 4𝑥 – 6

d. 2𝑥2 − 18

The children keep coming up with new and interesting requests. One option is to have rectangles that

have side lengths that are more than one x. For instance, one child made a drawing of this cool block:

5. Write two equivalent expressions for this block. Use the distributive property to verify that your

answer is correct.

6. Here we have some partial requests. We have one of the expressions for the area of the block and

we know the length of one of the sides. Use a diagram to find the length of the other side and write

a second expression for the area of the block. Verify your two expressions for the area of the block

are equivalent using algebra.

a. Area: 2𝑥2 + 7𝑥 + 3 Side: (𝑥 + 3)

Equivalent expression for area:

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b. Area: 5𝑥2 + 8𝑥 + 3 Side: (𝑥 + 1)


c. Area: 2𝑥2 + 7𝑥 + 3 Side: (2𝑥 + 1)


What are some patterns you see in the two equivalent expressions for area that might help you to

factor?

7. Business is booming! More and more requests are coming in! Use diagrams or number patterns (or

both) to write each of the following orders in factored form:

a. 3𝑥2 + 16𝑥 + 5 c. 3𝑥2 + 𝑥 − 10

b. 2𝑥2 − 13𝑥 + 15 d. 2𝑥2 + 9𝑥 − 5

8. In “Quilt Blocks Galore!” you wrote some rules for deciding about the signs inside the factors. Do

those rules still work in factoring these types of expressions? Explain your answer.

9. Explain how Quanyka can tell if the block is a multiple of another block or if one side has a multiple

of 𝑥 in the side length.

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READY

Topic: Greatest Common Factor.

Find the Greatest Common Factor for each of the number pairs below.

1. 30 𝑎𝑛𝑑 45 2. 12 𝑎𝑛𝑑 48 3. 16 𝑎𝑛𝑑 64

4. 9 𝑎𝑛𝑑 30 5. 20 𝑎𝑛𝑑 125 6. 22 𝑎𝑛𝑑 55

7. 80 𝑎𝑛𝑑 24 8. 32 𝑎𝑛𝑑 36 9. 18 𝑎𝑛𝑑 27

Find the Greatest Common Factor for each of the pairs of terms.

10. 8𝑥 𝑎𝑛𝑑 20 11. 𝑥2 𝑎𝑛𝑑 7𝑥 12. 4𝑥2 𝑎𝑛𝑑 6𝑥

13. 9𝑥2𝑎𝑛𝑑 9 14. 5𝑥2 𝑎𝑛𝑑 10𝑥2 15. 50𝑥 𝑎𝑛𝑑 9𝑥

SET



16. 2𝑥2 − 15𝑥 − 8 17. 4𝑥2 − 13𝑥 + 3 18. 3𝑥2 + 7𝑥 + 4

19. 3𝑥2 + 10𝑥 − 25 20. 2𝑥2 − 2𝑥 − 12 21. 5𝑥2 + 29𝑥 − 42

22. 2𝑥2 − 𝑥 − 15 23. 3𝑥2 + 34𝑥 + 63 24. 4𝑥2 + 12𝑥 + 9

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Go!

Topic: Evaluating exponential functions in context.

Answer the following questions about the exponential function that has been graphed.

The graph 𝑏(𝑡) shows the growth of bacteria over time (in days).

25. What does 𝑏(0) = _________ and what does it

mean in context?

26. What does 𝑏(3) = _________ and what does it

mean in context?

27. When 𝑏(𝑡) = 4 what does t = ________ and

what does this mean in context?

The graph ℎ(𝑡) shows the half-life of a radioactive element over time (in days). Originally, there was 80

mg of this substance present.

28. What does ℎ(2) = ________ and what does

it mean in context?

29. What does ℎ(4) = ________ and what does

it mean in context?

30. When ℎ(𝑡) = 5 what does t = ________

and what does this mean in context?

31. Describe the common ratio in this context.

32. What is the average rate of change over the interval [0, 2]?

33. What is the average rate of change over the interval [4, 6]?

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NC MATH 1//UNIT 8

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8.4 Comparing Methods

A Practice Understanding Task

Solve the following equations for x using two different methods.

Traditional inverse operations: Factor and then solve:

1. 𝑥2 − 25 = 0 2. 𝑥2 − 25 = 0

3. 𝑥2 − 21 = 100 4. 𝑥2 − 21 = 100

5. 2𝑥2 − 98 = 0 6. 3𝑥2 − 27 = 0

7. 4𝑥2 + 20 = 56 8. 5𝑥2 − 86 = −66

What do you notice is similar and different about the processes above?

The following equations can only be solved using one of the two methods above. Identify which method

you can use, and then solve the equation.

9. 𝑥2 + 2𝑥 − 3 = 0

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NC MATH 1//UNIT 8

QUADRATIC FUNCTIONS

10. 10𝑥2 − 20𝑥 = 0

11. 10𝑥 − 20 = 0

12. 3𝑥 − 12 = −18

13. 3𝑥2 + 13𝑥 = 10

14. 4𝑥2 − 36 = 0

15. 4𝑥 = 20

16. 𝑥2 = 𝑥 + 42

17. How can you decide when to use which method for solving equations?

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READY

Topic: Rewrite Quadratic Expressions in Factored Form.

Re-write each of the expressions as a product of two linear factors.

1. 𝑥2 − 36 2. 𝑥2 + 2𝑥 − 8 3. 6𝑥2 − 24𝑥

4. 2𝑥2 − 3𝑥 − 14 5. 𝑥2 − 100 6. 20𝑥 − 40𝑥2

SET

Topic: Solving Quadratic Equations.

Find the solutions for the variable in each quadratic equation.

7. 𝑝2 − 2𝑝 − 24 = 0 8. 𝑛2 − 3𝑛 = 0

9. 2𝑥2 − 7𝑥 − 49 = 0 10. 3𝑟2 + 8𝑟 − 3 = 0

11. 4𝑥2 − 17𝑥 + 10 = −5 12. 𝑥(𝑥 − 2) = 15

13. (𝑥 − 8)2 = 0 14. 𝑥2 + 4𝑥 = −4

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Go!

Topic: Finding x-intercepts

Find the x-intercept(s) for each function below.

15. 84 xxy

16. 75 xxy

17. 24 xy

18. 29xy

19. xxy 453

20. 𝑦 = 𝑥2 + 5𝑥 + 6

21. 59 xxy

22. 212xy

23. 3415 xxy

24. 1534 xxy

25. 283 xy

26. 𝑦 = 𝑥2 − 4𝑥 − 5

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8.5 Giant Parthenon Wheel

A Develop Understanding Task

Ignoring air resistance, the following graph shows how far an object – a bowling ball, basketball,

anvil, etc. – will fall after 0, 1, 2, and 3 seconds.

1. What patterns do you notice with the distance in feet that the object has fallen over the

three seconds?

2. Use these patterns to predict how far the object will fall after 4 seconds and 5 seconds.

3. Would this pattern be a linear, exponential, or quadratic function? How can you tell?

4. Write an explicit equation to model how far an object will fall after t seconds.

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NC MATH 1//UNIT 8

QUADRATIC FUNCTIONS


While sitting at the very top of the Giant Parthenon Wheel, Sam accidentally drops his

sunglasses. His sunglasses will fall given the same rate as the graph on the previous page. The

top of the Giant Parthenon Wheel is 256 feet above ground.

5. Model the height of his sunglasses from the ground over time as they fall. Use multiple

representations.

6. Write two explicit equations for this situation, one in standard form, and one in factored

form. What do you notice about these forms when compared to the new table, graph, and

situation?

7. If the Giant Parthenon Wheel makes one full turn in 60 seconds, will Sam make it to the

bottom before his sunglasses do? Explain.

8. Suppose Sam didn’t drop his sunglasses. In fact, he was having an argument with his

girlfriend who took his sunglasses and threw them downward with a velocity of 96 feet per

second. What effect(s) would this initial velocity (𝑣0) have on the equation, table, and graph?

How long will it take his sunglasses to hit the ground?

ℎ(𝑡) = 256 − 16𝑡2 − 𝑣0𝑡

9. Rewrite this explicit equation in factored form. What do you notice about the connections

between the factored form and the table, graph, and situation?

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READY

Topic: Evaluating different types of functions modeled in different ways.

Evaluate each function for the given inputs.

1. 𝑔(𝑥) = 7𝑥 − 10 2. ℎ(𝑡) = −16𝑡2 + 32𝑡 + 20

a. 𝑔(−2)= a. ℎ(0)=

b. 𝑔(0)= b. ℎ(2)=

c. 𝑔(9)= c. ℎ(5)=

3. 𝑝(𝑡) = 2000(1.12)𝑥 4. 𝑟(𝑥) = 500 − 25𝑥

a. 𝑝(−3)= a. 𝑟(−10)=

b. 𝑝(0)= b. 𝑟(0)=

c. 𝑝(13)= c. 𝑟(10)=

5. The graph below is the function 𝑓(𝑥) 6. The graph below is the function 𝑤(𝑥)

a. 𝑓(0)= a. 𝑤(−4)=

b. 𝑓(10)= b. 𝑤(−2)=

c. 𝑓(−6)= c. 𝑤(0)=

7. The table below is the function 𝑚(𝑥) 8. The table below is the function 𝑝(𝑡)

a. 𝑚(2)= a. 𝑝(0)=

b. 𝑚(−1)= b. 𝑝(3)=

c. 𝑚(4)= c. 𝑝(8)=

𝒙 𝒎(𝒙) 0 6

1 12

2 24

3 48

𝒕 𝒑(𝒕) 0 100

1 90

2 80

5 50

7 30

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SET

Topic: Identifying Key Features of a Quadratic Function graph.

The following is a graph of an object that has been thrown into the air from the ground. It shows the

height (in meters) of the object over time (in seconds).

9. What is the maximum?

10. What is the y-intercept?

11. What are the x-intercepts?

12. What is the axis of symmetry?

13. What interval is increasing?

14. What interval is decreasing?

15. What is the domain?

16. What is the range?

17. What does the maximum of this graph mean in context of the projectile? (Discuss both x and y)

18. What does the y-intercept mean in context?

19. What do the x-intercepts represent?

20. What would the practical domain and range be for this situation?

The graph to the right shows Juliette throwing her locket necklace out of her bedroom window to

Romeo.

21. What interval is her necklace above 90 feet?

22. What is the maximum height of the necklace and when does it

reach this height?

23. What is the y-intercept, and what does it mean in context to

this situation?

24. What is the positive x-intercept, and what does it mean in

context to this situation?

Page 26



Go!

Topic: Solving Equations

Find the value of x in each of the equations below.

25. 𝑥2 = −12𝑥 − 27 26. 4𝑥3 = 108 27. −2𝑥2 = −98

28. 8 = 𝑥2 − 7𝑥 29. 2

3𝑥2 = 6 30. −

1

2𝑥 = 8

31. 2𝑥(𝑥 − 3) + 6𝑥 = 162 32. 3(𝑥2 + 7) = 96 33. 3𝑥2 − 10𝑥 − 14 = 9𝑥

Page 27


NC MATH 1//UNIT 8

QUADRATIC FUNCTIONS


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Crossette https://www.vulcaneurope.eu/product-images/product-135/VS5-5082A.jpg

Waterfall https://upload.wikimedia.org/wikipedia/commons/3/34/Canada_Day_2016_Fireworks_%2827929885402%29.jpg

8.6 Fireworks


Every night at 9:45, fairgoers get to experience a magnificent fireworks show.

Each firework is carried by a rocket that behaves like a projectile. Each rocket is shot into the air with an initial velocity. The firework is designed to explode when the rocket reaches its highest point in the air. Once the rocket explodes, the sparkles of different colors fly out of the rocket and the rocket falls back to the ground.

Below, you are given information about the flight of each type of firework.

Glitter Palm (graph of ft. vs. s) Crossette

Rocket height above ground after 𝑡 seconds is given by

𝑐(𝑡) = −16𝑡2 + 80𝑡

Waterfall

The rocket has an initial height of 64 feet, and an upward initial velocity of 48 feet per second.

Page 29

NC MATH 1//UNIT 8

QUADRATIC FUNCTIONS


Answer the following questions. Be sure to explain your thinking and make connections between representations. 1. Model each of the types of fireworks using a different representation than the one given on

the previous page. 2. Describe all three fireworks, including their initial height, initial velocity, maximum height,

how long it takes to get to its maximum height, and when it hits the ground.

True or False: If the fireworks are all set off simultaneously, which of these statements are true, and which are false? Be prepared to justify your response with a representation for each firework.

3. The Crossette has the maximum height. 4. The Waterfall is the only firework that doesn’t take off from the ground. 5. The Glitter Palm and the Waterfall take the same amount of time to hit the ground. 6. The Waterfall is the firework that reaches its maximum height the fastest. 7. Since the Crossette and the Waterfall have the same maximum height, this must mean they

hit the ground at the same time. 8. After 1 second, the Crossette is the closest firework to the ground. 9. At 4 seconds, the Glitter Palm is the highest off the ground. 10. The Waterfall is at 96 feet twice, at 1 second, and at 3 seconds. 11. The Crosette is the last firework to reach its maximum height. 12. The Glitter Palm hits the ground two seconds after the Waterfall hits the ground. 13. The firework that is the slowest when it takes off is the Waterfall. 14. The practical domain for the Waterfall is [0, 4]. 15. The practical range for the Glitter Palm is (-∞, 144]. 16. The Waterfall is increasing in height from 0 to 1.5 seconds. 17. The Crossette is decreasing in height from 5 to 6 seconds.

Page 30



READY

Topic: Graphing Quadratic Functions and Key Features of Quadratic Functions.

Graph the quadratic functions, and identify the increasing interval, decreasing interval, the

vertex, the x-intercepts and the y-intercept.

1. 𝑓(𝑥) = (𝑥 − 3)(𝑥 + 1) 2. 𝑔(𝑥) = 3𝑥(𝑥 − 2)

3. 𝑓(𝑥) = 𝑥2 − 4𝑥 4. 𝑔(𝑥) = 𝑥2 + 6𝑥

Page 31



SET

Topic: Interpret Key Features of a Quadratic Function in context.

Herman is a stunt devil for a movie, and has been asked to jump out of a 50 story building, which is 320

meters above the ground (Yikes!). His first attempt he just lets himself gently fall out of the window to

get comfortable with the landing situation. His path from the window to the ground can be modeled

with the equation 𝑓(𝑡) = −5𝑡2 + 320. Once he is comfortable with the height, the directors set the

stage for his scene, where a villain actually throws him out of the window towards the ground! This

equation will include a −60 meters per second velocity since he is being thrown downwards.

5. In which situation do you think Herman will hit the ground first?

6. Find how long it takes for Herman to hit the ground when he just falls out of the window.

7. Write a new function for his movie scene where he is thrown down out of the window.

8. Find how long it takes for Herman to hit the ground when he is thrown.

9. What is the difference in the amount of seconds it takes for him to hit the ground when he falls out,

verses when he is thrown down?

NASA has successfully landed a robot on a new unknown planet. The gravitational pull on this planet is

much stronger than earth. The first test this robot will conduct is launching a ball upwards away from

the surface of the earth at a velocity of 400 feet per second. The function 𝑑(𝑡) = −50𝑡2 + 400𝑡

models the new gravitational pull.

10. How long will it take for the ball to come back down and hit the surface of the planet?

11. What is the maximum height that the ball will reach?

12. The gravitation pull on earth can be calculated using −16𝑓𝑡

𝑠2⁄ If this same situation occurred on

earth, how long would it take for the ball to come back down and hit the ground?

13. What would the maximum height of the ball be on earth?

14. Which planet has a greater maximum height, and by how many feet?

Page 32



Go!

Topic: Parallel and Perpendicular Lines

Identify whether the given linear functions will generate parallel lines, perpendicular lines, or just

intersecting lines.

15. {𝑦 = −9𝑥 + 8

2𝑥 − 18𝑦 = 3616. {

𝑦 =1

2𝑥 + 8

2 − 𝑦 = 3𝑥17. {

𝑦 = −3

2𝑥 + 6

2𝑦 = 10 − 3𝑥

18. {3𝑥 + 4𝑦 = 123𝑥 + 4𝑦 = 9

19. {𝑦 = −

5

6𝑥

6𝑥 − 5𝑦 = 2020. {

𝑦 = −1

2𝑥 + 8

7 + 𝑦 = 2𝑥

21. {2𝑥 + 4𝑦 = 13𝑥 − 4𝑦 = 3

22. {−2𝑥 + 3𝑦 = 183𝑥 + 2𝑦 = 22

23. {4𝑦 = 12𝑥 − 5

3𝑥 = 9 − 𝑦

Page 33


NC MATH 1//UNIT 8

QUADRATIC FUNCTIONS

http://www.lesonunique.com/sites/default/files/concert.jpg

8.7 Next Year’s Gig


One reality of the NC State Fair is that planning for next year begins before this year’s fair is

over. Fair officials have already started looking at new bands to entertain next year’s fairgoers.

Johnny Baldridge and the Piedmont Boys is one of the acts the NC State Fair planners are

considering for next year’s fair. At a recent concert, 1000 fans spent $20 per ticket. Market

research suggests that for every dollar a ticket price goes up, 10 fewer people would come to

the concert. The profit 𝐽(𝑥) from the concert can be modeled by the function below, where 𝑥

represents the number of $1 increases in the price of a ticket.

𝐽(𝑥) = (1000 10𝑥)(20 + 𝑥)

1. What does the expression (1000 − 10𝑥) represent in the context of this situation?

2. What does the expression (20 + 𝑥) represent in the context of this situation?

3. 𝐽(0) means they have not increased or decreased the price per ticket (there have been zero

$1 changes). What would the profit be if they kept the ticket prices at exactly $20 per ticket?

4. What would 𝐽(30) mean in context, and what would their profit be if they change their

ticket prices in this way?

5. What would 𝐽(90) mean in context, and what would their profit be if they change their

ticket prices in this way?

Page 35


6. Type this function into your graphing calculator, using the profits and $1 adjustments above

to help you determine an appropriate window. If 𝐽(𝑥) = 0 find the solutions for x. What do

these solutions mean in context and where are they on the graph of the function?

7. What would the maximum profit be for the NC State Fair, and how many $1 increases wouldthey have to include to get this maximum profit? What is the resulting “best ticket price” for this concert?

8. Write the function for Johnny Baldridge and the Piedmont Boys in standard form. Whatinformation can we easily gather from the standard form of this function?

Another band the Fair planners are looking at for next year’s Fair is Anna’s Animals. Their recent concert also sold 1000 tickets, but fans spent $35 per ticket. Market research indicates that an increase of $1 in the price of a ticket would lead to a drop in ticket sales of 20 tickets. The income 𝐴(𝑥) from this performance can be modeled by the following function, where 𝑥 represents the number of $1 increases in the price of a ticket.

𝐴(𝑥) = 35,000 + 300𝑥 − 20𝑥2

9. What does the 35,000 represent in the context of this problem?

10. Type this function into your graphing calculator. What are the x-intercepts of this graph,

and what do they mean in the context of this problem?

11. What is the maximum profit that Anna’s Animals will make for the NC State Fair? How

many $1 increases would they have to include to get this maximum profit?

12. Which band would you choose to hire for the NC State Fair next year and why?

Page 36



READY

Topic: Evaluate functions using f(x) notation.

Evaluate each function for the given value.

1. Given: 3112 2 xxxf

2f _____________________

2. 852 xxxf

0f _____________________

3. 127 2 xxxf

0f _____________________

4. 3511 2 xxxf

0f _____________________

5. 1216 2 xxf

0f _____________________

6. xxxf 86 2

0f _____________________

7. Briefly discuss any relationship you notice about each function and the value of f(0)?

SET

Topic: Identifying and Interpreting Key Features

Dorton Arena at the State Fairgrounds in Raleigh, North Carolina, has

the shape of two intersecting parabolas. An aerial model of Dorton

Arena has been created and put on a set of axes. The shape of one of

the parabolas can be modeled by the equation 𝑓(𝑥) = 𝑥2 − 127𝑥,

where 𝑥 and 𝑓(𝑥) are measured in feet.

8. Which parabola is modeled by this equation? How do you know?

Page 37



9. What are the features of the parabola modeled by the equation 𝑓(𝑥) = 𝑥2 − 127𝑥? Include

domain, range, intercepts, maxima or minima, intervals of increase and decrease.

10. Can the other parabola be modeled by the equation 𝑓(𝑥) = −𝑥(𝑥 − 127)? How do you

know?

11. What are the features of this other parabola?

12. Based on these features, how long is Dorton Arena? How wide is the arena?

13. How do the different forms (standard and factored form) help you find different features of the

graph?

GO! Use the changes in the data to determine what kind of function (linear, exponential, or

quadratic) is demonstrated in the data. Justify your answer.

14. Stuart began an exercise similar to Growing Dots from Module 1.2. His results are shown in

Table 1 below.

Table 1

Minutes past 0 1 2 3 4 5 6

Number of dots in the pattern 1 7 13 19 25 31 37

Change from the last stage --

Linear Exponential Quadratic

How do you know? ____________________________________________________

Page 38



15. Marion graphed some of the results from his dot-growing experiment. Each point is labeled

with the number of dots.


How do you know? ____________________________________________________

16. Sonia built a recursive function to describe the dots in her growing experiment.

13

21

nfnf

f


How do you know? ____________________________________________________

2 3

6

11

18

27

0

5

10

15

20

25

30

0 2 4 6

Number of dots in the pattern

Page 39


NC MATH 1//UNIT 8 QUADRATIC FUNCTIONS


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8.8 Rainy Day


If it rains at the fair, the vendors and games can have a hard time making a profit.

At the turkey leg booth, the profit on a typical day can be modeled by the function:

𝑃(𝑡) = −.065𝑡2 + 29.25𝑡 − 1300

where 𝑃(𝑡) is the profit in dollars, and 𝑡 is the number of turkey legs sold.

1. What is this form of a quadratic equation called?

2. What does the -1,300 mean in context of the problem if 𝑃(𝑡) represents profit in dollars?

3. Using your graphing calculator, list a few values for the number of turkey legs (t) that will

make the booth profit, and that will not make the booth profit. How can you summarize

your findings for any number of turkey legs?

4. What is the number of turkey legs that will make the stand the most amount of money?

What is that maximum profit?

5. If the booth wants to make at least $1,000 approximately how many turkey legs does the

stand need to sell?

Page 41



The graph below shows the same turkey leg booth, but on a rainy fair day. Since not as many

people attend the fair when it is raining, the turkey leg booth doesn’t get the same amount of

business.

6. Compare this graph with the information you gathered on a typical day at the fair for this

turkey leg booth. What similarities and differences do you notice? Be sure to compare the

maximum profit, the number of turkey legs needed to make a profit, and the places where

you see the profit increasing and decreasing.

7. Discuss what 𝑃(100) means in context for each day, rainy or not. Which day will the stand

make a higher profit?

8. Discuss what 𝑃(𝑡) ≥ 500 means in context for each day, rainy or not. For what values of 𝑡

will this be true for each day?

9. The vendors believe that you will always make a higher profit when it is NOT raining if you

sell the same number of turkey legs. Do you think this is true? Justify your answer using

examples.

Page 42


Developed by CHCCS and WCPSS adapted from Illustrative Mathematics

READY

Topic: Interpreting Quadratic Expressions and Functions.

1. A ball thrown vertically upward at an initial velocity of 𝑣0 feet/second rises a distance of d

feet in t seconds, given by 𝑑 = 6 + 𝑣0𝑡 − 16𝑡2.

a. Write what this equation would look like if the ball is thrown at 88 feet per second.

b. Write what this equation would look like if the ball is thrown to rise a distance of 20

feet in 2 seconds. Can you solve for this missing velocity in this case to find how fast

the ball was thrown?

2. The expression −4.9𝑡2 + 17𝑡 + 0.6 describes the height in meters of a basketball 𝑡 seconds

after it has been thrown vertically into the air. Interpret the coefficients of each term of

this expression in the context of the situation.

3. Three equivalent equations for 𝑓(𝑥) are shown below.

𝑓(𝑥) = −2𝑥2 + 24𝑥 − 54

𝑓(𝑥) = −2(𝑥 − 3)(𝑥 − 9)

𝑓(𝑥) = −2(𝑥 − 6)2 + 18

a. Which form currently reveals the x-intercepts of the function?

b. Circle all values of 𝑥 for which 𝑓(𝑥) = 0.

−54 − 18 − 9 − 6 − 3 0 3 6 9 18 54

4. The expression −4𝑥2 + 8𝑥 + 12 represents the height in feet of an apple thrown from a

person on a ladder near an apple tree to a basket on the ground, where 𝑥 is time in

seconds. Write an equivalent expression for this situation that is in factored form. What

information can you find easily from each form?

Page 43


Developed by CHCCS and WCPSS adapted from Illustrative Mathematics

SET

Topic: Analyzing and Practicing Projectile Motion problems.

5. A baseball is “popped” straight up by a batter with an initial velocity of 64 ft/sec. The height of the ball above ground is given by a function where t is time in seconds after the ball leaves the bat and h(t) is the height in feet above the ground. The batter hit the ball at an original height of 3 feet off of the ground, and the acceleration due to gravity is -16ft/se𝑐2.

a. Write the function that will model this situation: ____________________________

b. Make a table of values for 0 seconds – 4 seconds.

c. What is the maximum height that the baseball will reach?

d. At what time will the baseball be back down to its original height?

6. A bottle rocket that is originally on the ground is launched vertically with an initial velocity of 128 feet per second. The acceleration due to gravity is -16 feet per second squared. The formula gives the height, h(t), of the rocket after t seconds.

a. Write the function that will model this situation: ________________________________

b. How long is the rocket in the air?

c. A balloonist sees the rocket go by 4 seconds after it leaves the ground. How high is the balloonist from the ground when he sees the rocket?

d. A helicopter is 600 feet in the air. The helicopter sees the rocket approaching the helicopter. Is it possible for the rocket to hit the helicopter if the pilot remains at an altitude of 600 feet? Explain.

e. The rocket will pass a sight-seer on a ledge 240 feet high. When does the sight-seer see the rocket pass on its way up? When does the sight-seer see the rocket pass on its way down?

7. Jenny is practicing her diving off of a spring board. She follows the function: ℎ(𝑡) = −𝑡2 + 6𝑡 + 7

where ℎ(𝑡) is her height from the pool over (t) time in seconds.

a. How long will it take for Jenny to hit the water?

b. What is her maximum height and when will she reach it?

c. How high is Jenny at the start (how high is the diving board?)

d. What is ℎ(8) and what does it represent in context of this problem?

Page 44



Go!

Topic: Other Applications of Quadratics.

8. A rectangle has a length that is 2 units longer than the width. If the width is increased by 4 units

and the length increased by 3 units, write two equivalent expressions for the area of this new

rectangle.

9. A vacant rectangular lot is being turned into a community vegetable garden with a uniform path

around it. The area of the lot is represented by the equation 4𝑥2 + 40𝑥 = 44 where 𝑥 is the width

of the path in meters. Find the width of the path surrounding the garden.

10. The area of a trapezoid is found using the formula 𝐴 =1

2 ℎ(𝑏1 + 𝑏2), where 𝐴 is the area, ℎ is the

height, and 𝑏1 and 𝑏2 are the lengths of the bases. Consider the trapezoid below.

Write two equivalent expressions for the area of the trapezoid.

11. A town council plans to build a public parking lot. The outline below represents the proposed shape

of the parking lot.

Write an expression for the area, in square yards, of this proposed parking lot.

Page 45




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8.9 How would you want to be paid?


The boss wants to offer an incentive for excellent work at the Fair venue. She believes that workers should be rewarded for excellent work. She has three possible pay options in mind.

Option 1: The employee earns $100 per day for the 10-day Fair.

Option 2: The employee makes $2 for the first day, $4 total for the first two days, $8 total for the first three days, and so on, doubling each day of the fair.

Option 3: The employee makes $60 on the first day. If he/she does a really good job, then on the second day, the daily pay is increased by $10 to $70. On the third day, the daily pay is increased by another $10 to be $80, and so on, increasing the daily pay by $10 for each day of the fair, assuming the employee continues to do a good job. The total amount of money made from choosing this option can be modeled using the function 𝑓(𝑑) = 5𝑑2 + 55𝑑

1. Without doing any calculations which pay scheme would you choose for the 10-day Fair?Why?

2. Model the total amount of money you would earn for working d days at the fair under eachpayment scheme. You can use equations, tables, graphs, etc.

Under which option would a worker make the most money for the 10-day NC State Fair?

3. What type of function is represented by the total pay for each pay option?

Page 47



4. The boss is so impressed by your work that she wants you to work the South Carolina State

Fair too! She agrees to continue to keep paying you with the scheme you selected. The

South Carolina Fair also runs for 10 days. Continue your models to calculate the total pay

earned for each option after a total of 20 days.

Under which option would a worker make the most money for the two State Fairs?

5. Compare the growth of the Total Earned for each of the three options. Be sure to consider

the short-term and long-term information. Which function has the greatest growth in the

long term? Do you think this is always the case? Why or why not?

Page 48



READY

Topic: Data and Statistics

Jason wants to compare the mean height of the players on his favorite basketball and soccer

teams. He thinks the mean height of the players on the basketball team will be greater but

doesn’t know how much greater. He used the rosters and player statistics from the team

websites to generate the following lists.

Basketball Team – Height of Players in inches for 2017-18 Season

75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84

Soccer Team – Height of Players in inches for 2017

73, 73, 73, 72, 69, 76, 72, 73, 74, 70, 65, 71, 74, 76, 70, 72, 71, 74, 71, 74, 73, 67

1. Create a graphical display for each set of data.

2. Find the mean, median, mode, and range of each set of data.

3. Compare the mean heights of the players on the Basketball Team and the Soccer Team.

Was Jason’s assumption correct? What is the difference between their mean heights?

Page 49



SET

Topic: Comparing Linear, Exponential, and Quadratic Functions.

4. The following tables show the values of linear, quadratic, and exponential functions for

various values of x. Indicate which function type corresponds to each table. Justify your choice.

A B C D

x y x y x y x y

1 6 1 7 1 6 1 56

2 9 2 14 2 9 2 28

3 12 3 28 3 14 3 14

4 15 4 56 4 21 4 7

A _________________ B __________________ C __________________ D __________________

Given the three functions: 𝑓(𝑥) = 60𝑥 + 20, 𝑔(𝑥) = 3(2)𝑥 , and ℎ(𝑥) = 4𝑥2 + 18𝑥

5. Label each function as quadratic, linear, or exponential.

6. At x = 3, which function has the greatest value?



9. At any x value higher than 8, which function will have the greatest value? Do you think this

type of function will always eventually have the greatest value?

Page 50



Go!

Topic: Identify the initial value and average rate of change for linear and exponential functions

in different forms.

Find the initial value and rate of change for the given functions.

10. 𝑔(𝑥) = −4(𝑥 − 1) + 10 11. ℎ(𝑡) = 2(6)𝑡

a. Find the initial value: a. Find the initial value:

b. Find the average rate of change: b. Find the average rate of change on the

interval [2,3]:

12. Below is the graph of the function 𝑓(𝑥)

a. Find the initial value:

b. Find the average rate of change on the interval [1,2]

13. The table below is the function 𝑚(𝑥) 14. The table below is the function 𝑝(𝑡)



b. Find the average rate of change

on the interval [1, 3]: b. Find the average rate of change:

𝒙 𝒎(𝒙) 1 9

2 18

3 36

𝒕 𝒑(𝒕) 3 90

4 75

6 45

7 30

Page 51

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