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ALGEBRA 1

A Learning Cycle Approach

MODULE 7

Quadratic Equations

The Mathematics Vision Project

Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2016 Mathematics Vision Project

Prior work done in partnership with the Utah State Office of Education © 2013 Licensed under the Creative Commons Attribution-NonCommercial ShareAlike 4.0 Unported license

1

7.1 CW: Rational Exponents

Refer to your “Book of Exponent Rules.”

Write each exponential expression in radical form.

1. x

3

10

2. 5

1

x 3. 3

1

3n

4. 7

2

6

5. 3

5

7 6. 5

4

t

Write each radical expression in exponential form.

7. 5 3

8. 56 7a 9. 3x

10. 3 5n

11. xyn 12.

p qn

Simplify each of the radical expressions. Use rational exponents if desired.

13. 4 12881 yx

14. 3

107

a

ba

15. 5 12625x

16. 5n 17. 3 27 18. 238 2

2

7.1 HW: Rational Exponents

Fill in the table so each expression is written in radical form and with rational exponents.

Radical Form Exponential Form

19. 4 38

20. 4

3

256

21. 4 57 42

22. 2

1

2

3

416

23. 10 3123 yx

24. 5 18964 ba

3

7.2 CW: Experimenting with Exponents

[This task was adopted from the Illustrative Mathematics Project:

http://www.illustrativemathematics.org/illustrations/385 Content on this site is

licensed under a Creative Commons Attribution-NonCommercial-ShareAlike

3.0 Unported License]

Travis and Miriam are studying bacterial growth. They were surprised to find

that the population of the bacteria doubled every hour.

1. Complete the following table and plot the data on the graph provided on page 5.

Hours into the study 0 1 2 3 4

Bacteria population

(in thousands) 4

2. Write an equation for P, the population of the bacteria, as a function of time, t, and verify that it produces

correct populations for t = 1, 2, 3, and 4 hours.

Travis and Miriam want to create a table with more entries; specifically, they want to fill in the population at

each half hour. Unfortunately, they forgot to make these measurements so they decide to estimate the values.

Travis makes the following claim:

“If the population doubles in 1 hour, then half that growth occurs in the first half-hour and the other half in the

second half-hour. So for example, we can find the population at t = 1

2 by finding the average of the

populations at t = 0 and t = 1.”

3. Fill in the parts of the table below that you've already computed, and then decide how you might use Travis’

strategy to fill in the missing data. Also plot Travis’ data on the graph.

Hours 0 1

2 1

3

2 2

5

2 3

7

2 4

Bacteria (in thousands)

4 8

4. Comment on Travis’ idea. How does it compare to the table generated in problem 1? For what kind of

function would this reasoning work?

http://www.illustrativemathematics.org/illustrations/385

4

Miriam suggests they should fill in the data in the table in the following way:

“To make the estimates, I noticed that the population increases by the same factor each hour, and I think

that this property should hold over each half-hour interval as well.”

5. Fill in the parts of the table below that you've already computed in problem 1, and then decide how you

might use Miriam’s new strategy to fill in the missing data. As in the table in problem 1, each entry should

be multiplied by some constant factor in order to produce consistent results (hint: look at your ISN).

a. What page are we looking at in your ISN?

b. Use this constant factor to complete the table. Plot Miriam’s data on the graph.

Hours 0 1

2 1

3

2 2

5

2 3

7

2 4


4 8

6. Now Miriam wants to estimate the population every 20 minutes instead of every 30 minutes.

a. What multiplier would she use for every third of an hour to consistent with the population doubling every

hour?

b. Use this multiplier to complete the following table.

Hours 0 1

3

2

3 1

4

3

5

3 2

7

3

8

3 3


4 8

c. Give a detailed description of how you would estimate the population, P, at 5

3t hours.

5

Hours vs Bacteria Population (in thousands)

6

7.2 HW: Experimenting with Exponents

READY

Topic: Additive and multiplicative patterns

The sequences below exemplify either an additive (arithmetic) or a multiplicative (geometric) pattern. Identify

the type of sequence, fill in the missing values on the table and write an explicit equation.

1.

Term 1 2 3 4 5 6 7 8

Value 2 4 8 16 32

Equation: Type of sequence:

2.

Term 1 2 3 4 5 6 7 8

Value 66 50 34 18


3.

Term 1 2 3 4 5 6 7 8

Value -3 9 -27 81


4.

Term 1 2 3 4 5 6 7 8

Value 160 80 40 20


7

5.

Term 1 2 3 4 5 6 7 8

Value -9 -2 5 12


Use the graph of the function to find the desired values of the function.

6. Find the value of 𝑓(2) 7. Find where 𝑓(𝑥) = 4

8. Find the value of 𝑓(6) 9. Find where 𝑓(𝑥) = 16

10. What do you notice about the way that inputs and outputs for this

function relate?

11. What is the explicit equation for this function?

8

SET

Topic: Fill in the missing values of the table based on the growth that is described.

12. The growth in the table is by a factor of four each whole year.

Years 0 1

2 1

3

2 2

5

2 3

7

2 4


2 8

13. The growth in the table is tripled at each whole year.

Years 0 1

2 1

3

2 2

5

2 3

7

2 4


2 6

14. The growth in the table is tripled at each whole year.

Years 0 1

3

2

3 1

4

3

5

3 2

7

3

8

3 3


2 6

9

7.3 CW: Factor Fixin’

At first, Optima’s Quilts only made square blocks for quilters and

Optima spent her time making perfect squares. Customer service

representatives were trained to ask for the length of the side of the

block, x, that was being ordered, and they would let the customer

know the area of the block to be quilted using the formula:

2A x x

Optima found that many customers that came into the store were making designs that required a combination

of squares and rectangles. So, Optima’s Quilts has decided to produce several new lines of rectangular quilt

blocks. Each new line is described in terms of how the rectangular block has been modified from the original

square block. For example, one line of quilt blocks 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. The design

department knows that the area of this new block can be represented by the expression: 5 2A x x x

but they do not feel that this expression gives the customer 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 your customers that your formula is correct using a diagram (hint: use your Algebra

tiles to find the new formula and create the diagram)

10

Here are some additional new lines of blocks that Optima’s Quilts has introduced. 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 3 inches on one side and 4 inches on the other.

Diagram: Two algebraic expressions

3. The original square block was extended 4 inches on only one side.


4. The original square block was extended 5 inches on each side.


11

5. The original square block was extended 2 inches on one side and 6 inches on the other


Customers start ordering custom-made block designs by requesting how much additional area they want

beyond the original area of 2x . Once an order is taken for a certain type of block, customer service needs to

have specific instructions on how to make the new design for the manufacturing team. The instructions need to

explain how to extend the sides of a square block to create the new line of rectangular blocks.

The customer service department has placed the following orders on your desk. 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 in using

expressions representing the lengths of the sides.

6. 2 5 3 15x x x

Diagram:

Written description:

Algebraic descriptions

12

7. 2 4 6 24x x x

Diagram:



8. 2 9 2 18x x x

Diagram:



13

9. 2 5 5x x x

Diagram:



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.

10. 2 11 10x x 11.

2 7 10x x

12. 2 9 8x x 13.

2 6 8x x

14. 2 8 12x x 15.

2 7 12x x

16. 2 13 12x x

14

17. What relationships or patterns do you notice when you find the sides of the rectangles for a given area of

this type?

18. A customer called and asked for a rectangle with area given by: 2 7 9x x . The customer service

representative said that the shop couldn’t make that rectangle. Do you agree or disagree? How can you tell

if a rectangle can be constructed from a given area?

15

7.3 HW: Factor Fixin’

READY

Topic: Creating binomial quadratics

Multiply (use the distributive property, write the quadratics in standard form)

1. 4 7x x 2. 5 3 8x x 3. 3 3 2x x

4. The answers to problems 1-3 are quadratics that are represented in standard form. Which coefficient is

equal to zero (hint: standard form is: 2ax bx c )

Factor the following expressions (Write the expression as the product of two linear factors)

5. 2 4x x 6.

27 21x x 7. 212 60x x 8.

28 20x x

Multiply the two linear factors together

9. 9 9x x 10. 2 2x x 11. 6 5 6 5x x 12. 7 1 7 1x x

13. The answers to problems 9-12 are quadratics that are represented in standard form. Which coefficient is

equal to zero?

16

SET

Topic: Factoring trinomials

Find the sides of the rectangles that have each area described below. Write your answers in factored form [e.g.

rectangle with one side increased by one and the other side increased by five would be: 1 5x x ]

14. 2 14 45x x 15.

2 18 45x x 16. 2 46 45x x

17. 2 11 24x x 18.

2 10 24x x 19. 2 14 24x x

20. 2 12 36x x 21.

2 13 36x x 22. 2 20 36x x

23. 2 15 100x x 24.

2 20 100x x 25. 2 29 100x x

26. Look back at each factored expression in problems 14-25. Explain how it is possible that the coefficient (b)

of the middle term can be different numbers in problems when the “outside” coefficients (a) and (c) are the

same.

17

GO

Topic: Taking the square root of perfect squares

Only some of the expressions inside the radical sign are perfect squares. Identify which ones are perfect

squares and take the square root. Leave the ones that are not perfect squares under the radical sign. Do not

attempt to simplify them (hint: Check your answers by squaring them. You should be able to get what you

started with, if you are right.)

27. 2

17xyz 28. 2

3 7x 29. 2 6121a b

30. 2 8 16x x 31.

2 14 49x x 32. 2 14 49x x

33. 2 10 100x x 34.

2 20 100x x 35. 2 20 100x x

18

7.4 CW: The “x” Factor

Now that Optima’s Quilts is accepting orders for rectangular blocks,

their business in growing by leaps and bounds. Many customers want

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 2” more than the standard size.

19

4. One side is 2” less than the standard size and the other side is 3” more than the standard size.

5. One side is 2” more than the standard size and the other side is 3” less than the standard size.



8. An expression that has 3 terms in the form: 2ax 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?

20

9. One customer had an unusual request. She wanted a block that is extended 2 inches on one side and

decreased by 2 inches on the other. One of the employees 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.

10. The result of the unusual request made the employee 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 #8, and see if you can find a pattern in the result.

a. 1 1x x b. 3 1 3x x

11. What pattern did you notice? What is the result of x a x a ?

12. Some customers 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. 2 3x x b. 1 4x x

21

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

term in the trinomial expression 2ax bx c will be positive or negative.

14. Optima’s quilt shop has received 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. 2 9 18x x b.

2 3 18x x

c. 2 3 18x x d.

2 9 18x x

e. 2 5 4x x f.

2 3 4x x

g. 2 2 15x x

15. Write an explanation of how to factor a trinomial in the form: 2ax bx c

22

7.4 HW: The “x” Factor

READY

Topic: Exploring the density of the number line.

Find three numbers that are between the two given numbers.

1. 23 19

& 4 3

2. 9 3

& 4 2

3. 1 5

& 4 8

4. 3 & 5

5. 4 & 23 6. 39 19

& 4 2

7. 1 4

& 4 9

8. 13 & 14

SET

Topic: Factoring quadratics

The area of a rectangle is given in the form of a trinomial expression. Find the equivalent expression that

shows the lengths of the two sides of the rectangle.

9. 2 9 8x x 10.

2 6 8x x 11. 2 2 8x x 12.

2 7 8x x

13. 2 11 24x x 14.

2 14 24x x 15. 2 25 24x x 16.

2 10 24x x

23

17. 2 2 24x x 18.

2 5 24x x 19. 2 5 24x x 20.

2 10 25x x

21. 2 25x 22.

2 2 15x x 23. 2 10 75x x 24.

2 20 51x x

25. 2 14 32x x 26.

2 1x 27. 2 2 1x x 28.

2 12 45x x

24

GO

Topic: Graphing parabolas

Graph each parabola. Include the vertex and at least 3 accurate points on each side of the axis of symmetry.

Then describe the transformation in words.

29.

2f x x

30.

2 3g x x

31.

2

2h x x

32.

2

1 4b x x

25

7.5 Warm Up

Put the product of columns “a” and “c” in column “ac.” Find two numbers that multiply to the number in column

“ac” and add to the number in “b.” The first problem has been done for you.

a b c ac 1st number 2nd number

1. 2 11 15 30 5 6

2. 1 7 12

3. 1 11 24

4. 1 -8 7

5. 2 2 -4

6. 6 1 -2

7. 1 -5 6

8. 5 8 -4

9. 3 7 -10

10. 2 -11 9

26

7.5 CW: The “Wow” Factor

Optima’s Quilts sometimes gets orders for blocks that are multiples of a

given block. For instance, Optima 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. Optima has a lot of new orders. Use diagrams to help you find equivalent expressions for each of the

following

a. 5𝑥2 + 10𝑥

b. 3𝑥2 + 21𝑥 + 36

c. 2𝑥2 + 2𝑥 − 4

d. 2𝑥2 − 10𝑥 + 12

e. 3𝑥2 − 27

27

3. Because she is a great business manager, Optima offers her customers lots of options. One option is to

have rectangles that have side lengths that are more than one x. For instance, Optima made this cool

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

answer is correct.

4. Here we have some partial orders. 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: 22 7 3x x

Equivalent expression for area:

Side: (x+3)

b. Area: 25 8 3x x


Side: (x+1)

c. Area: 22 7 3x x


Side: (2x+1)

28

5. There’s one more twist on the kind of blocks that Optima makes. These are the trickiest of all because they

have more than one x in the length of both sides of the rectangle! Here’s an example. Write two equivalent

expressions for this block. Use the distributive property to verify that your answer is correct.

6. All right, let’s try the tricky ones. They may take a little messing around to get the factored expression to

match the given expression. Make sure you check your answers to be sure that you’ve got them right.

Factor each of the following:

a. 6𝑥2 + 7𝑥 + 2 b. 10𝑥2 + 17𝑥 + 3

c. 4𝑥2 − 8𝑥 + 3 d. 4𝑥2 + 4𝑥 − 3

29

7.5 HW: The “Wow” Factor

READY

Topic: Comparing arithmetic and geometric sequences

The first and fifth terms of each sequence are given. Fill in the missing numbers.

1.

Arithmetic 3 1875

Geometric 3 1875

2.

Arithmetic 3 1875

Geometric 3 1875

3.

Arithmetic 3 1875

Geometric 3 1875

30

SET

Topic: Writing an area model as a quadratic expression

Write two equivalent expressions for the area of each block. Let x be the side length of each of the large

squares.

4.

5.

6.

7. Problems 4-6 all contain the same number of squares measuring 2x and

21

a. What is different about the images?

b. How does this difference affect the quadratic expression that represents them?

c. Describe how the arrangement of the squares and rectangles affects the factored form

31

Topic: Factoring quadratic expressions when 1a

Factor the following quadratic expressions.

8. 24 7 2x x 9.

22 7 15x x 10. 26 7 3x x 11.

24 3x x

12. 24 19 5x x 13.

23 10 8x x 14. 26 2x x 15.

23 14 24x x

16. 22 9 10x x 17.

25 31 6x x 18. 25 7 6x x 19.

24 8 5x x

20. 23 75x 21.

23 7 2x x 22. 24 8 5x x 23.

22 1 6x x

32

GO

Topic: Find the equation of the line of symmetry of a parabola

Given the x-intercepts of a parabola, write the equation of the line of symmetry

24. x-intercepts: (-3, 0) and (3, 0) 25. x-intercepts: (-4, 0) and (16, 0)

26. x-intercepts: (-2, 0) and (5, 0) 27. x-intercepts: (-14, 0) and (-3, 0)

28. x-intercepts: (17, 0) and (33, 0) 29. x-intercepts: (-.75, 0) and (2.25, 0)

33

7.6 CW: Lining Up Quadratics

Graph each function and find the vertex, the y-intercept and the x-

intercepts. Be sure to properly write the intercepts as points

1. 1 3y x x

Line of symmetry:

Vertex:

x-intercepts:

y-intercept:

2. 2 2 6f x x x

Line of symmetry:

Vertex:

x-intercepts:

y-intercept:

34

3. 4g x x x

Line of symmetry:

Vertex:

x-intercepts:

y-intercept:

4. Based on these examples, how can you use a quadratic function in factored form to:

a. Find the line of symmetry of the parabola?

b. Find the vertex of the parabola?

c. Find the x-intercepts of the parabola?

d. Find they-intercept of the parabola?

e. Find the vertical stretch

35

5. Graph the two functions, f(x) and g(x), on the same graph

Linear function 1: 1 3f x x

Linear function 2: 3 1g x x

6. On the same graph as #5, graph the function

P x that is the product of the two linear functions.

What shape is created (hint: make a table)

7. Describe the relationship between x-intercepts of the linear functions and the x-intercepts of the function

P x . Why does this relationship exist?

8. Describe the relationship between the y-intercepts of the linear functions and the y-intercept of the function

P x . Why does this relationship exist?

36

9. Given the parabola to the right, sketch two lines that

could represent its linear factors

10. Write an equation for each of these two lines.

11. How did you use the x and y intercepts of the parabola to select the two lines?

12. Are these the only two lines that could represent the linear factors of the parabola? If so,

explain why. If not, describe the other possible lines.

13. Use your two lines to write the equation of the parabola. Is this the only possible equation of

the parabola?

37

7.6 HW: Lining up Quadratics

READY

Topic: Multiplying Binomials

Multiply the following binomials

1. 3 4 7 5x x 2. 9 2 6x x 3. 4 3 3 11x x

4. 7 3 7 3x x 5. 3 10 3 10x x 6. 11 5 11 5x x

7. 2

4 5x 8. 2

9x 9. 2

10 7x

10. Describe the relationship between the middle coefficient (b) and the outside coefficients (a and c)

38

SET

Topic: Factored Form of a Quadratic Function

Given the factored form of a quadratic function, identify the vertex, intercepts, and vertical stretch of the

parabola.

11. 4 2 6y x x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

12. 3 2 6y x x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

13. 5 7y x x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

14. 1

7 72

y x x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

15. 1

8 42

y x x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

16. 3

25 95

y x x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

17. 3

3 34

y x x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

18. 5 5y x x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

19. 2

10 103

y x x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

39

GO

Topic: Vertex Form of a Quadratic Equation

Given the vertex form of a quadratic function, identify the vertex, intercepts, and vertical stretch of the

parabola.

20. 2

2 4y x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

21. 2

3 6 3y x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

22. 2

2 1 8y x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

23. 2

4 2 64y x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

24. 2

3 2 48y x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

25. 2

6 1y x

a. Vertex:

b. x-int:

c. y-int:

d. stretch:

26. Did you notice that the parabolas in problems 11, 12, & 13 are the same as the ones in problems 23, 24, &

25 respectively? If you didn’t, go back and compare the answers in problems 11, 12, & 13 and problems

23, 24, & 25.

Prove that

a. 2

4 2 6 4 2 64x x x b. 2

3 2 6 3 2 48x x x

c. 2

5 7 6 1x x x

40

7.6C CW: Writing Quadratic Functions from Graphs

Use your notes to write a quadratic function (in vertex, intercept, or standard form) for each graph.

1.

Function:

What do you know?

vertex:

y – intercept:

x – intercepts:

Which function form(s) is the easiest to write

from the information you can find on the graph?

vertex form

intercept form

standard form

2.

Function:

What do you know?

vertex:

y – intercept:

x – intercepts:

Which function form(s) is the easiest to write from

the information you can find on the graph?

vertex form

intercept form

standard form

41

3.

Function:

What do you know?

vertex:

y – intercept:

x – intercepts:

Which function form(s) is the easiest to write from

the information you can find on the graph?

vertex form

intercept form

standard form

4.

Function:

What do you know?

vertex:

y – intercept:

x – intercepts:

Which function form(s) is the easiest

to write from the information you can

find on the graph?

vertex form

intercept form

standard form

42

7.7 CW: I’ve Got a Fill-in

For each problem below, you are given a piece of information that

tells you a lot. Use what you know about that information to fill in the

rest.

1. You get this: 2 12y x x

Fill in this:

Factored form of the equation

x-intercepts and y-intercept

2. You get this: 2 6 3y x x

Fill in this:

Vertex form of the equation

vertex and y-intercept

43

3. You get this:

Fill in this:

Factored form of the equation

Standard form of the equation

4. You get this: Fill in this:

Vertex form of the equation

Standard form of the equation

44


Fill in this:

Either form of the equation other than standard form

Vertex



Fill in this:


Vertex

y-intercept


Fill in this:


Vertex


45

7.7 HW: I’ve Got a Fill-in

READY

Topic: Identifying and interpreting parts of an equation

A golf-pro practices his swing by driving golf balls off the edge of a cliff into a lake. The height of the ball above

the lake (measured in meters) as a function of time (measured in seconds and represented by the variable t)

from the instant of impact with the golf club is:

258.8 19.6 4.9t t

The expressions below are equivalent:

standard form: 24.9 19.6 58.8t t

factored form: 4.9 6 2t t

vertex form: 2

4.9 2 78.4t

1. Which expression is the most useful for finding how many seconds it takes for the ball to hit the water?

Why?

2. Which expression is the most useful for finding the maximum height of the ball? Justify your answer.

3. If you wanted to know the height of the ball at exactly 3.5 seconds, which expression would help the most

to find the answer? Why?

4. If you wanted to know the height of the cliff above the lake, which expression would you use? Why?

46

SET

Topic: Finding multiple representations of a quadratic

One form of a quadratic function is given. Fill-in the missing forms.

5.

a. Standard Form b. Vertex Form c. Factored Form

5 3y x x

d. Table (include the vertex and 2 points on

either side of the vertex). Show the first and

second differences

𝒙 𝒚

e. Graph

6.

a. Standard Form b. Vertex Form

2

3 1 3y x

c. Factored Form



second differences

𝒙 𝒚

e. Graph

47

7.

a. Standard Form 2 10 25y x x

b. Vertex Form c. Factored Form



second differences

𝒙 𝒚

e. Graph

8.



either side of the vertex). Show the first

and second differences

𝒙 𝒚

e. Graph

48

9.




second differences

𝒙 𝒚

0 12

1 2

2 -4

3 -6

4 -4

5 2

6 12

e. Graph

GO

Topic: Factoring quadratics

Verify each factorization by multiplying the linear factors

10. 2 12 64 16 4x x x x 11. 2 64 8 8x x x

12. 2 20 64 16 4x x x x 13. 2 16 64 8 8x x x x

Factor the following quadratic expressions, if possible (some will not factor).

14. 2 5 6x x 15.

2 7 6x x 16. 2 5 36x x

49

17. 2 16 63m m 18.

22 3 1s 19. 2 7 2x x

20. 2 14 49x x 21.

2 9x 22. 2 11 3c cx

23. Which quadratic expression above could represent the area of a square? Explain.

24. Would any of the expressions above NOT be the side-lengths for a rectangle? Explain

50

7.8 CW: Throwing an Interception

The x-intercept(s) of the graph of a function 𝑓(𝑥) are often very important

because they are the solution to the equation 𝑓(𝑥) = 0. In previous tasks, we

learned how to find the x-intercepts of the function by factoring, which works

great for some functions, but not for others. In this task we are going to work

on a process to find the x-intercepts of any quadratic function that has them.

We’ll start by thinking about what we already know about a few specific

quadratic functions and then use what we know to generalize to all quadratic functions with x-intercepts.

1. Consider the graph of the function

21 16

8 4f x x x

a. Complete the table and graph the function.

𝒙 -8 -6 1 4 8

𝒇(𝒙)

b. What is the equation of the line of symmetry?

c. What is the vertex of the function?

d. What are the x-intercepts?

e. How far are the x-intercepts from the line of symmetry?

f. How far above the vertex are the x-intercepts?

g. What is the value of 𝑓(𝑥) at the x-intercepts?

51


2 6 4f x x x


𝒙 0 2 3 4 6

𝒇(𝒙)



d. What do you estimate the x-intercepts to be?

e. About how far are the x-intercepts from the line of symmetry?



52


2 5 2f x x x


𝒙 -4 -3 0 4 6

𝒇(𝒙)



d. What do you estimate the x-intercepts to be?

e. About how far are the x-intercepts from the line of symmetry?



53

7.8 HW: Throwing an Interception

READY Topic: converting units of measure

While working with areas it sometimes essential to convert between units of measure, for example changing

from square yards to square feet. Convert the areas below to the desired measure. (Hint: area is two

dimensional, for example 1 yd2 = 9 ft2 because 3 ft along each side of a square yard equals 9 square feet.)

1. 7 yd2 = ____________ ft2 2. 5 ft2 = ____________ in2 3. 1 mile2 = _______________ ft2

Set Topic: Transformations and Parabolas, Symmetry and Parabolas

4. Examine the differences and similarities between each

quadratic function

a. Graph each of the quadratic functions.

𝑓(𝑥) = 𝑥2

𝑔(𝑥) = 𝑥2 − 9

ℎ(𝑥) = (𝑥 + 2)2 − 9

b. How do the functions compare to each other?

c. What are the x-intercepts of 𝑔(𝑥)?

d. What are the coordinates of the points corresponding to the x-intercepts in g(x) in each of the other

functions?

𝑔(𝑥) (same answers from part c) (_____, 0) (_____, 0)

𝑓(𝑥)

ℎ(𝑥)

e. How do the coordinates compare to each other?

54

5. Examine the differences and similarities between each

quadratic function

a. Graph each of the quadratic functions.

𝑓(𝑥) = 𝑥2

𝑔(𝑥) = 𝑥2 − 4

ℎ(𝑥) = (𝑥 − 1)2 − 4

b. How do the functions compare to each other?

c. What are the x-intercepts of 𝑔(𝑥)?

d. What are the coordinates of the points corresponding to the x-intercepts in g(x) in each of the other

functions?

𝑔(𝑥) (same answers from part c) (_____, 0) (_____, 0)

𝑓(𝑥)

ℎ(𝑥)

e. How do the coordinates compare to each other?

55

GO Topic: Evaluating functions Use the given functions to find the missing values. (Check your work using a graph.)

6. 𝑓(𝑥) = 𝑥2 + 4𝑥 − 12

a. 𝑓(0) = b. 𝑓(2) = c. 𝑓(𝑥) = 0 d. 𝑓(𝑥) = 20

7. 𝑓(𝑥) = 𝑥2 − 6𝑥 + 9

a. 𝑓(0) = b. 𝑓(2) = c. 𝑓(𝑥) = 0 d. 𝑓(𝑥) = 20

8. 𝑓(𝑥) = (𝑥 − 2)2 − 3

a. 𝑓(0) = b. 𝑓(2) = c. 𝑓(𝑥) = 0 d. 𝑓(𝑥) = 20

9. 𝑓(𝑥) = −(𝑥 + 1)2 + 8

a. 𝑓(0) = b. 𝑓(2) = c. 𝑓(𝑥) = 0 d. 𝑓(𝑥) = 20

56

7.8B CW: Quadratic Formula

1. Factor each expression.

a. 𝑥2 + 2𝑥 + 1 b. 𝑥2 + 12𝑥 + 36

c. 𝑥2 − 4𝑥 + 4 d. 𝑥2 + 6𝑥 + 9

2. What patterns did you notice? The standard form of a quadratic equation is 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐; what is the

relationship between b & c in the problems above?

If the “c” term is not a then we can manipulate the equation and create

square factors to solve quadratic equations as long as we use the properties of equality.

Example:

Equation to solve 𝒙𝟐 + 𝟏𝟐𝒙 + 𝟏𝟏 = 𝟎 Can’t make a square factor

Move the “c” term to the other side (𝑥 + ? )(𝑥 + ? ) =

What two numbers (that are also the same) add

up to the “b” term? What is special about this

number?

(𝑥 + )(𝑥 + ) =

What number did we “add” to the left side of the

equation? What do we need to do on the right

side of the equation? What is special about this

number?

(𝑥 + )(𝑥 + ) =

Write the factor as a perfect square (𝑥 + )2 =

Take the square root of each side (what do you

need to remember?)

Solve for x (you should have two numbers)

57

3. Solve 𝑥2 + 6𝑥 − 7 = 0 using the same method.

4. Let’s try to solve 2𝑥2 + 3𝑥 − 7 = 0. It’s a little trickier since 𝑎 is not 1.

Equation to solve 𝟐𝒙𝟐 + 𝟑𝒙 − 𝟕 = 𝟎 Can’t make a square factor

Move the “c” term to the other side Can’t make a square factor

Divide both sides by “a” (𝑥 + ? )(𝑥 + ? ) =

What two numbers (that are also the same) add

up to the “b” term? What is special about this

number?

(𝑥 + )(𝑥 + ) =

What number did we “add” to the left side of the

equation? What do we need to do on the right

side of the equation? What is special about this

number?

(𝑥 + )(𝑥 + ) =

Write the factor as a perfect square (𝑥 + )2 =

Take the square root of each side (what do you

need to remember?)

Solve for x (you should have two numbers)

58

5. Try the same process with a quadratic equation in standard form: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0

6. Find the x-intercepts of the following quadratic functions.

a. 𝑓(𝑥) = 𝑥2 − 8𝑥 + 12

b. 𝑓(𝑥) = −2𝑥2 + 6𝑥 + 20

c. 𝑓(𝑥) = 3𝑥2 + 21𝑥 + 36

d. 𝑓(𝑥) = 𝑥2 + 2𝑥 + 4

59

7.8C CW: Quadratic Formula Practice

Use the quadratic formula to find the x-intercepts. Use the x-intercepts to find the line of symmetry and the

vertex. Write the function in vertex form and factored form.

1. 2 2 3f x x x

Line of symmetry: x-intercepts:

Vertex: Vertex form:

Factored form:

2. 2 14 25f x x x

Line of symmetry: x-intercepts:

Vertex: Vertex form:

Factored form:

60

Use the quadratic formula to find the solutions.

3. 𝑥2 + 𝑥 − 4 = 0 4. 6𝑥2 + 11𝑥 − 35 = 0

5. 𝑥2 − 4𝑥 − 8 = 0 6. 2𝑥2 + 9𝑥 = −7

61

7. 4𝑥2 + 8𝑥 + 7 = 4 8. 2𝑥2 − 3𝑥 − 5 = 0

9. 𝑥2 + 4𝑥 + 3 = 0 10. 2𝑥2 + 3𝑥 − 20 = 0

62

7.9 CW: Curbside Rivalry

Carlos and Clarita have a brilliant idea for how they will earn money this

summer. Since the community in which they live includes many high

schools, a couple of universities, and even some professional sports

teams, it seems that everyone has a favorite team they like to root for.

In Carlos’ and Clarita’s neighborhood these rivalries take on special

meaning, since many of the neighbors support different teams. They

have observed that their neighbors often display handmade posters and other items to make their support of

their favorite team known. The twins believe they can get people in the neighborhood to buy into their new

project: painting team logos on curbs or driveways.

For a small fee, Carlos and Clarita will paint the logo of a team on a neighbor’s curb, next to their house

number. For a larger fee, the twins will paint a mascot on the driveway. Carlos and Clarita have designed

stencils to make the painting easier and they have priced the cost of supplies. They have also surveyed

neighbors to get a sense of how many people in the community might be interested in purchasing their service.

Here is what they have decided, based on their research.

A curbside logo will require 48 in2 of paint

A driveway mascot will require 16 ft2 of paint

Surveys show the twins can sell 100 driveway mascots at a cost of $20, and they will sell 10 fewer

mascots for each additional $5 they charge

1. If a curbside logo is designed in the shape of a square, what will its dimensions be?

A square logo will not fit nicely on a curb, so Carlos and Clarita are experimenting with different types of

rectangles. They are using a software application that allows them to stretch or shrink their logo designs to fit

different rectangular dimensions.

2. Carlos likes the look of the logo when the rectangle in which it fits is 8 inches longer than it is wide. What

would the dimensions of the curbside logo need to be to fit in this type of rectangle? As part of your work,

write a quadratic equation that represents these requirements.

63

3. Clarita prefers the look of the logo when the rectangle in which it fits is 13 inches longer than it is wide.

What would the dimensions of the curbside logo need to be to fit in this type of rectangle? As part of your

work, write a quadratic equation that represents these requirements.

Your quadratic equations on the previous two problems probably started out looking like this: 𝑥(𝑥 + 𝑛) = 48

where 𝑛 represents the number of inches the rectangle is longer than it is wide. The expression on the left of

the equation could be multiplied out to get and equation of the form𝑥2 + 𝑛𝑥 = 48 If we subtract 48 from both

sides of this equation we get 𝑥2 + 𝑛𝑥 − 48 = 0. In this form, the expression on the left looks more like the

quadratic functions you have been working with in previous tasks, 𝑦 = 𝑥2 + 𝑛𝑥 − 48.

4. Consider Carlos’ quadratic equation where n = 8, so 𝑥2 + 8𝑥 − 48 = 0. Describe at least two different

strategies, other than guess-and-check, you could use to solve the equation.

5. After much disagreement, Carlos and Clarita agree to design the curbside logo to fit in a rectangle that is 6

inches longer than it is wide. What would the dimensions of the curbside logo need to be to fit in this type of

rectangle? As part of your work, write and solve a quadratic equation that represents these requirements.

64

6. What are the dimensions of a driveway mascot if it is designed to fit in a rectangle that is 6 feet longer than

it is wide (see the requirements for a driveway mascot given in the bulleted list above)? As part of your

work, write and solve a quadratic equation that represents these requirements.

7. What are the dimensions of a driveway mascot if it is designed to fit in a rectangle that is 10 feet longer

than it is wide (see the requirements for a driveway mascot given in the bulleted list above)? As part of

your work, write and solve a quadratic equation that represents these requirements.

65

Carlos and Clarita are also examining the results of their neighborhood survey, trying to determine how much

they should charge for a driveway mascot. Remember, this is what they have found from the survey: They can

sell 100 driveway mascots at a cost of $20, and they will sell 10 fewer mascots for each additional $5 they

charge.

8. Write an equation, make a table, and sketch a graph (on the same set of axes) for the price of the

driveway mascot for each $5 increment, x, in the price.

Equation Table

number of $5

increments, x

price of the

mascot m(x)

0 20

1

2

3

4

5

6

7

8

Sketch the Graph

9. Write an equation, make a table, and sketch a graph for the number of driveway mascots the twins can

sell for each $5 increment, x, in the price of the mascot. (Note: zero $5 increments is a price of $20)

Equation Table

number of $5

increments, x

number they

can sell, s(x)

0 100

1

2

3

4

5

6

7

8

Sketch the Graph

66

10. Write an equation, make a table, and sketch a graph for the revenue the twins will collect for each $5

increment in the price of the mascot.

What is revenue? How do

you find this?

Equation

Table

number of $5

increments, x

revenue, r(x)

0 2000

1

2

3

4

5

6

7

8

Sketch the Graph

11. The twins estimate that the cost of supplies will be $250 and they would like to make $2000 in profit from

selling driveway mascots. Therefore, they will need to collect $2250 in revenue. Find the prices they should

charge to collect at least $2250 in revenue.

12. What price should the twins charge for each mascot if they want to make the most profit?

67

7.9 HW: Curbside Rivalry

READY

Topic: Finding x-intercepts for linear equations

1. Find the x-intercept of each equation below. Write your answer as an ordered pair. Consider how the

format of the given equation either facilitates or inhibits your work.

a. 3 4 12x y b. 5 3y x c. 4 1 5y x

d. 4 1y x e. 2 7 6y x f. 5 2 10x y

2. Which of the linear equation formats above facilitates your work in finding x-intercepts? Why?

3. Using the same equations from question 1, find the y-intercepts. Write your answers as ordered pairs

a. 3 4 12x y b. 5 3y x c. 4 1 5y x

d. 4 1y x e. 2 7 6y x f. 5 2 10x y

4. Which of the formats above facilitate finding the y-intercept? Why?

68

SET

Topic: Solve Quadratic Equations, Connecting Quadratics with Area

For each of the given quadratic equations, (a) describe the rectangle the equation fits with. (b)

What constraints have been placed on the dimensions of the rectangle?

5. 2 7 170 0x x 6.

2 15 16 0x x

7. 2 2 35 0x x 8.

2 10 80 0x x

Solve the quadratic equations below

9. 2 7 170 0x x 10.

2 15 16 0x x

11. 2 2 35 0x x 12.

2 10 80 0x x

69

GO

Topic: Factoring Expressions

Write each of the expressions below in factored form.

13. 2 132x x 14.

2 5 36x x 15. 2 5 6x x

16. 2 13 42x x 17.

2 56x x 18. 2x x

19. 2 8 12x x 20.

2 10 25x x 21. 2 5x x

7.9B Warm Up

Rank the three quadratic equations on ease of solving (e.g. finding the x-intercepts). Explain why you arrived at

the rankings you did and find the x-intercept of the easiest one.

Easy: 2 5 4 0x x

Medium: 24 71 12 0x x

Hard: 2

2 25 0x

70

7.10 Warm Up

Find the quadratic equation from the given table:

𝒙 𝒇(𝒙)

1 1

2 4

3 11

4 22

71

7.10 CW: Writing Quadratic Functions w/ Matrices

Write an explicit quadratic function for each table of values.

1.

x f(x)

1 -1

2 -3

3 -9

4 -19

System of 3 equations

Matrix

a=

b=

c=

Quadratic Function:

check:

2.

x f(x)

-1 6

1 2

3 38

5 114


Matrix

a=

b=

c=

Quadratic Function:

check:

72

3.

x f(x)

2 12

4 66

6 152

8 270


Matrix

a=

b=

c=

Quadratic Function:

check:

4.

x f(x)

1 4

2 0

3 -2

4 -2


Matrix

a=

b=

c=

Quadratic Function:

check:

73

7.10 HW: Writing Quadratic Functions w/ Matrices

Create a matrix for the system of equations that can be used to find the solution.

1.

2 4 0

5 4 5 12

4 4 24

x y z

x y z

x y z

2.

2 5 15

4 12

6 4 12

x y z

x y z

x y z

We have learned several different methods for solving quadratic equations. Use the most efficient method for

each of the equations below. Check your solution(s).

Solve for x.

1. 2 3 4x x 2.

2 7 2 0x x

3. 2

4 8x 4. 24 6 1 0x x

74

5. 2 5 3x 6.

24 7 2 0x x

7. 2 3 1 6x x 8.

28 200x

9. 2 6 8 0x x 10.

2 10 25x x 25102 xx

75

Homework Checklist Section Pg Problems Checked?

76

Module 7 KUDOs: Quadratic Equations

Algebra 1 Honors I will be able to…

Essential Questions: How do I use algebraic expressions to model and solve problems? What are the limits of mathematical modeling? Know: (understand the meaning of these terms, memorize the formulas)

o Integer o rational number o radical o irrational number o expression

o coefficient o factor o term o quadratic formula o function

o roots, zeroes, solutions o vertex o line of symmetry o parabola

Understand: Big ideas B: beginning of the unit L: right after the lesson T: before the unit test

I understand... Not yet Got it Homework problems that

apply, text pages, …

Understand that polynomials are similar to integers and can be added, subtracted, multiplied, and divided

Quadratic functions have a distinct pattern of growth and can be interpreted as the product of two linear factors

Do B: beginning of the unit L: right after the lesson T: before the unit test

I can do the following... Not yet Got it Homework problems that

apply, text pages, questions I have

Factor quadratic expressions

Use the properties of exponents to simplify or compare expressions with rational exponents

Solve quadratic equations: by inspection (e.g., for x2 = 49), taking square roots, the quadratic formula, [completing the square], and factoring

Graph quadratic equations and identify important features (intercepts, maximum, minimum, line of symmetry, vertex, stretch, domain, range)

Write quadratic functions in standard form, vertex form, and intercept form and describe features highlighted by each form

Create the standard form of a quadratic equation using a matrix. (solve a system of 3 equations with 3 variables) [graphing calculator]

quadratic equations - wordpress.com · name: algebra 1 a learning cycle approach module 7 quadratic...

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