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VOLUME 2

ALGEBRA 1 ISBN: 979-8-89353-910-3

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SOUTH CAROLINA

Math Nation AGA was originally developed by Illustrative Mathematics®, and is copyright 2019 by Illustrative Mathematics. It is licensed under Creative Commons Attribution 4.0 International License (CCBY 4.0).

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The online curriculum resource platform oﬀered at "mathnation.com", including the videos and practice questions, as well as custom curriculum adaptations, are additions to the original Illustrative Mathematics content is copyright 2024 by Accelerate Learning, Inc.

The trademark "Math Nation" is owned by Accelerate Learning, Inc. All other trademarks and product names referred to in the Math Nation AGA curriculum are the property of their respective owners and used solely for educational purposes. Unless otherwise stated, Math Nation has no relationship with any of the companies or brands mentioned in the curriculum and does not endorse or have a preference for any of those companies or brands. This curriculum includes public domain images or openly licensed images that are copyright by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information.

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Table of Contents Unit 6: Exponential Functions........................................................ 1

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Lesson 1: Patterns of Growth.................................................. 3 Lesson 2: Representing Exponential Growth............................11 Lesson 3: Understanding Decay.............................................19 Lesson 4: Representing Exponential Decay..............................27

Lesson 5: Using Negative Exponents......................................35 Lesson 6: Exploring Exponential Functions...............................43

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Lesson 7: Interpreting Exponential Functions...........................51 Lesson 8: Looking at Rates of Change.....................................61

Lesson 9: Modeling Exponential Behavior................................69 Lesson 10: Reasoning about Exponential Graphs......................77

Lesson 11: Graphing Exponential Functions.............................87 Lesson 12: Solving One-Variable Exponential Equations............95 Lesson 13: Function Involving Percent Change....................... 103

Lesson 14: Interest Compounding at Different Intervals.......... 111 Lesson 15: Expressed in Different Ways................................ 123

Lesson 16: Which One Changes Faster?................................ 131 Lesson 17: Changes over Equal Intervals.............................. 141

Unit 7: Equivalent Expressions.................................................. 151 Lesson 1: Introducing Polynomials....................................... 153 Lesson 2: Adding and Subtracting Polynomials....................... 163

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Lesson 3: Multiplying Binomials with Algebra Tiles.................. 169 Lesson 4: Multiplying Binomials and Polynomials.................... 177 Lesson 5: Adding, Subtracting, and Multiplying Polynomials........... 185 Lesson 6: Translating between Rational Exponents and Radicals....................................................... 191

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Lesson 7: Rewriting Algebraic Expressions with Rational Exponents and Radicals........................................ 199 Lesson 8: Equivalent Radical Expressions.............................. 205 Lesson 9: Adding and Subtracting Radical Expressions............ 211 Lesson 10: Multiplying and Dividing Radical Expressions.......... 219

Lesson 11: Operations with Radical Expressions..................... 227 Unit 8: Quadratic Functions....................................................... 231

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Lesson 1: Understanding How Relationships Change............... 233 Lesson 2: Building Quadratic Functions from Geometric Patterns.............................................. 243

Lesson 3: Building Quadratic Functions to Describe Situations – Part 1............................................... 251

Lesson 4: Building Quadratic Functions to Describe Situations – Part 2............................................................. 261 Lesson 5: Standard Form and Factored Form......................... 273

Lesson 6: Graphs of Functions in Standard and Factored Forms.................................................................. 279 Lesson 7: Graphing the Standard Form – Part 1..................... 289

Lesson 8: Graphing the Standard Form – Part 2..................... 299 Lesson 9: Graphs That Represent Situations.......................... 307

Lesson 10: Vertex Form...................................................... 315 Lesson 11: Changing the Vertex.......................................... 323 Lesson 12: Identifying Key Features of Quadratic Functions............ 331

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Unit 9: Quadratic Equations....................................................... 341 Lesson 1: When and Why Do We Write Quadratic Equations?... 343 Lesson 2: Solving Quadratic Equations Using Reasoning ......... 353 Lesson 3: How Many Solutions?........................................... 363 Lesson 4: Rewriting Quadratic Expressions in Factored Form – Part 1.................................................................... 375

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Lesson 5: Rewriting Quadratic Expressions in Factored Form – Part 2....................................................... 385 Lesson 6: Solving Quadratic Equations by Using Factored Form................................................................... 395

Lesson 7: Rewriting Quadratic Expressions in Factored Form When ...................................................................... 403 Lesson 8: What Are Perfect Squares?.................................... 413

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Lesson 9: Completing the Square When a = 1 Using Algebra Tiles..................................................................... 421

Lesson 10: Completing the Square When a > 1 Using Algebra Tiles..................................................... 433 Lesson 11: Completing the Square Algebraically..................... 443

Lesson 12: Solving Quadratic Equations by Completing the Square........................................................ 451 Lesson 13: Quadratic Equations with Irrational Solutions......... 461

Lesson 14: The Quadratic Formula....................................... 469 Lesson 15: Applying Different Methods of Solving a Quadratic....................................................... 477

Lesson 16: Using Quadratic Expressions in Vertex Form to Solve Problems.............................................. 485 Lesson 17: Using Quadratic Equations to Model Situations and Solve Problems............................................ 493

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Unit 10: Sequences................................................................... 499 Lesson 1: A Towering Sequence........................................... 501 Lesson 2: Introducing Geometric Sequences.......................... 507 Lesson 3: Different Types of Sequences................................ 515 Lesson 4: Sequences Are Functions...................................... 523 Lesson 5: Representing Sequences....................................... 531

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Lesson 6: The th Term........................................................ 539 Lesson 7: Arithmetic Sequences and Linear Functions............. 547 Lesson 8: Geometric Sequence and Exponential Functions....... 557

Unit 11: Reasoning with Parent Functions................................. 563 Lesson 1: Classifying Parent Functions.................................. 565

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Lesson 2: Determining Which Function Type Best Models a Real-World Context........................................... 577 Lesson 3: Modeling Real-World Contexts............................... 585

Lesson 4: Transformations Using Horizontal and Vertical Translations............................................. 597 Lesson 5: Transformations by a Factor.................................. 609

Lesson 6: Types of Transformations...................................... 619 Lesson 7: Writing Equations of Transformed Functions............ 627

Lesson 8: Comparing Key Features of Functions..................... 637

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Unit 6: Exponential Functions

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Unit 6, Lesson 1: Patterns of Growth

Warm-Up: Splitting Bacteria

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There are some bacteria in a dish. Every hour, each bacterium splits into 3 bacteria.

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1. This diagram shows a bacterium in hour 0 and then hour 1. Draw what happens in hours 2 and 3.

2. How many bacteria are there in hours 2 and 3?

Exploration Activity: A Genie in a Bottle

You are walking along a beach and your toe hits something hard. You reach down, grab onto a handle, and pull out a bottle. It is sandy. You start to brush it off with your towel. Poof! A genie appears.

He tells you, “Thank you for freeing me from that bottle. I was getting claustrophobic. You can choose one of these purses as a reward.” • Purse A which contains $1,000 today. If you leave it alone, it will contain $1,200 tomorrow (by magic). The next day, it will have $1,400. This pattern of $200 additional dollars per day will continue. • Purse B which contains 1 penny today. Leave that penny in there, because tomorrow it will (magically) turn into 2 pennies. The next day, there will be 4 pennies. The amount in the purse will continue to double each day.

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1. How much money will be in each purse after a week? After two weeks?

2. The genie later added that he will let the money in each purse grow for three weeks. How much money will be in each purse then?

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3. Which purse contains more money after 30 days?

Exploration Activity: Graphing the Genie’s Offer

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Here are graphs showing how the amount of money in the purses change. Remember Purse A starts with $1,000 and grows by $200 each day. Purse B starts with $0.01 and doubles each day.

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1. Which graph shows the amount of money in Purse A? Which graph shows the amount of money in Purse B? Explain how you know.

2. Points 𝑃 and 𝑄 are labeled on the graph. Explain what they mean in terms of the genie’s offer.

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3. What are the coordinates of the vertical intercept for each graph? Explain how you know.

4. When does Purse B become a better choice than Purse A? Explain your reasoning.

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5. Knowing what you know now, which purse would you choose? Explain your reasoning.

Collaborative Activity: Flow and Followers

Here are verbal descriptions of two situations, followed by tables and expressions that could help to answer one of the questions in the situations. • Situation 1: A person has 80 followers on social media. The number of followers triples each year. How many followers will she have after 4 years?

• Situation 2: A tank contains 80 gallons (gal.) of water and is getting filled at rate of 3 gal. per minute (min.). How many gal. of water will be in the tank after 4 min.?

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Match each representation (a table or an expression) with one situation. Be prepared to explain how the table or expression answers the question.

𝒙

Situation

80 · 3 · 3 · 3 · 3

Situation 1 Situation 2

240

720

2,160

6,480

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Representation

80 + 3 + 3 + 3 + 3

80 + 4 · 3

𝒙

𝒚

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80 · 81

Situation 1 Situation 2

Lesson Summary

When a positive number is repeatedly doubled, it eventually becomes very large. The list shows what happens when doubling the number 0.001 begins.

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0.001 0.002 0.004 0.008 0.016

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To continue this process, it is convenient to use an exponent. For example, the last entry in the list, 0.016, is 0.001 doubled 4 times, or (0.001) ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2, which can be expressed as (0.001) ⋅ 24.

Even though this started with a very small number, 0.001, it doesn’t have to be doubled that many times before reaching a very large number. For example, if it is doubled 30 times, represented by (0.001) ⋅ 230, the result is greater than 1,000,000.

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Throughout this unit, many situations will be explored where quantities grow or decrease when the same factor is applied to them repeatedly. These are different from situations where quantities grow or decrease by equal differences. For example, tables representing 2 different situations are shown.

Pay (dollars)

Number of Errands 0 1 2 3

5 5 5

Day

People Who Have Heard the Rumor

Factor from Previous Number of People

0 1 2 3

1 5 25 125

5 5 5

• A student at a high school hears a rumor that a celebrity will be speaking at graduation. The table shows how the rumor is spreading over time in days.

Difference from Previous Pay

10 15 20 25

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• A student runs errands for a neighbor every week. The table shows the pay he may receive, in dollars, in any given week.

Once it is determined how the patterns change, they can be described mathematically.

The upcoming lessons will continue to explore how to describe and represent these patterns and then how to use the patterns to solve problems.

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Practice Problems 1. Which expression equals 27?

A. 2 + 2 + 2 + 2 + 2 + 2 + 2

B. 2 · 2 · 2 · 2 · 2 · 2 · 2

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C. 2 · 7

2. Evaluate the expression 3⋅5𝑥 when 𝑥 is 2.

D. 2 + 7

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3. The graph shows the yearly balance, in dollars, in an investment account.

a. What is the initial balance in the account?

b. Is the account growing by the same number of dollars each year? Explain how you know.

c. A second investment account starts with $2,000 and grows by $150 each year. Sketch the values of this account on the graph.

d. How does the growth of balances in the two account balances compare?

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4. A population of ants is 10,000 at the start of April. Since then, it triples each month. a. Complete the table.

Months since April

Number of Ants

0 1 2 3

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b. What do you notice about the population differences from month to month?

c. If there are 𝑛 ants one month, how many ants will there be a month later?

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5. A swimming pool contains 500 gal. of water. A hose is turned on, and it fills the pool at a rate of 24 gal. per min. Which expression represents the amount of water in the pool, in gal., after 8 min.? B. 500 + 24 + 8

C. 500 + 24 · 8

A. 500 · 24 · 8

D. 500 · 248

6. The population of a city is 100,000. It doubles each decade for 5 decades. Select all expressions that represent the population of the city after 5 decades.

□ 32,000

□ 320,000

□ 100,000 · 2 · 2 · 2 · 2 · 2 □ 100,000 · 5

□ 100,000 · 2

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Review Problems

B average

C average

D average

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a. What does the 21 in the table represent?

A average

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Participants are asked if they read books every night with another person when they were ages 2 to 5, as well as their grade average for all of their high school classes. The results are represented in the table.

Read books Did not read nightly books nightly

7. A study of 100 recent high school graduates investigates a link between their childhood reading habits and achievement in high school.

b. What does the 10 in the table represent?

8. The rule that defines function 𝑓 is 𝑓(𝑥) = 𝑥2 + 1. Complete the table. Then, sketch a graph of function 𝑓. 𝒇(𝒙)

𝒙

−4 −2

0 2 4

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Unit 6, Lesson 2: Representing Exponential Growth

Warm-Up: Exponent Rules Rewrite each expression as a power of 2. 25 ⋅ 2

210 ÷ 27

29 ÷ 2

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23 ⋅ 24

Exploration Activity: What Does 𝒙𝟎 Mean?

1. Complete the table. Take advantage of any patterns you notice.

3𝒙

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𝒙

2. Here are some equations. Find the solution to each equation using what you know about exponent rules. Be prepared to explain your reasoning.

a. 9? ⋅ 97 = 97

b. 9? = 912 12

3. What is the value of 50? What about 20?

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Guided Activity: Multiplying Microbes 1. In a biology lab, 500 bacteria reproduce by splitting. Every hour, on the hour, each bacterium splits into 2 bacteria.

Hour

Number of Bacteria

500

a. Complete the table for this situation.

b. Write an expression to show how to find the number of bacteria after each hour listed in the table.

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2 3 6 𝑡

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c. Write an equation relating 𝑛, the number of bacteria, to 𝑡, the number of hours.

d. Use your equation to find 𝑛 when 𝑡 is 0. What does this value of 𝑛 mean in this situation?

2. In a different biology lab, a population of single-cell parasites also reproduces hourly. An equation which gives the number of parasites, 𝑝, after 𝑡 hours is 𝑝 = 100 ⋅ 3𝑡. Explain what the numbers 100 and 3 mean in this situation.

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Collaborative Activity: Graphing the Microbes 1. Refer back to your work in the table of the previous task. Use that information and the given coordinate planes to graph the following. b. Graph (𝑡, 𝑝) when 𝑡 is 0, 1, 2, 3, and 4. (If you get stuck, you can create a table.)

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a. Graph (𝑡, 𝑛) when 𝑡 is 0, 1, 2, 3, and 4.

2. On the graph of 𝑛, where can you see each number that appears in the equation? 3. On the graph of 𝑝, where can you see each number that appears in the equation?

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Lesson Summary In relationships where the change is exponential, a quantity is repeatedly multiplied by the same amount. The multiplier is called the growth factor. In an exponential function, the output is multiplied by the same factor every time the input increases by one. This multiplier is called the growth factor.

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Suppose a population of cells starts at 500 and triples every day. The number of cells each day can be calculated as shown in the table. Number of Cells

500

Number of Days

1,500 (or 500 ⋅ 3)

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4,500 (or 500 ⋅ 3 ⋅ 3, or 500 ⋅ 32)

500 ⋅ 3𝑑

𝑑

13,500 (or 500 ⋅ 3 ⋅ 3 ⋅ 3, or 500 ⋅ 33)

The table shows that the number of cells, 𝑝, is changing exponentially, and that 𝑝 can be found by multiplying 500 by 3 as many times as the number of days, 𝑑, since the 500 cells were first observed. The growth factor is 3, and the equation 𝑝 = 500 ⋅ 3𝑑 models this situation.

The equation can be used to find the population on any day, including day 0, when the population was first measured. On day 0, the population is 500 ⋅ 30. Since 30 = 1, this is 500 ⋅ 1, or 500.

A graph of the daily cell population is shown. The point (0, 500) on the graph means that on day 0, the population starts at 500. Each point is 3 times higher on the graph than the previous point. •

•

(1, 1500) is 3 times higher than (0, 500).

(2, 4500) is 3 times higher than (1, 1500).

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Practice Problems 1. Which expression is equal to 40 ⋅ 42? A. 0 B. 1

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C. 16 D. 64

2. A bee population is measured each week and the results are plotted on the graph.

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a. What is the bee population when it is first measured?

b. Is the bee population growing by the same factor each week? Explain how you know.

c. What is an equation that models the bee population, 𝑏, 𝑤 weeks after it is first measured?

Decades since Bond Is Bought

a. Complete the table.

3. A bond is initially bought for $250. It doubles in value every decade.

b. How many decades does it take before the bond is worth more than $10,000? c. Write an equation relating 𝑣, the value of the bond, to 𝑑, the number of decades since the bond was bought.

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Dollar Value of Bond

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𝑦

4. A sea turtle population 𝑝 is modeled by the equation 𝑝 = 400 ⋅ �5� where 𝑦 is the 4 number of years since the population was first measured. a. How many turtles are in the population when it is first measured? Where do you see this in the equation?

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b. Is the population increasing or decreasing? How can you tell from the equation?

c. When will the turtle population reach 700? Explain how you know. Review Problems

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5. Select all expressions that are equivalent to 38.

□8

□ 33

□3 ⋅ 8 □ (3 )

4 2

□ (3 ⋅ 3)

□ 31

−8

6. Bank account A starts with $5,000 and grows by $1,000 each week. Bank account B starts with $1 and doubles each week. a. Which account has more money after one week? After two weeks?

b. Here is a graph showing the two account balances. Which graph corresponds to which situation? Explain how you know.

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c. Given a choice, which of the two accounts would you choose? Explain your reasoning.

7. Match each equation in the first list to an equation in the second list that has the same solution. a. 𝑦 = 2 𝑥 + 2

c. 𝑦 = 10 − 0.4𝑥 5

iii. − 2𝑥 + 5𝑦 = 10

d. 2𝑥 = 10 − 5𝑦

ii. − 2𝑥 − 5𝑦 = 10

b. 𝑥 = − 5 − 2.5𝑦

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i. 2𝑥 + 5𝑦 = 10

e. − 5𝑦 = 2𝑥 + 10 f. 𝑥 = 5 − 5 𝑦

8. Function 𝐹 is defined so that its output 𝐹(𝑡) is the number of followers on a social media account 𝑡 days after set up of the account. a. Explain the meaning of 𝐹(30) = 8,950 in this situation.

b. Explain the meaning of 𝐹(0) = 0.

c. Write a statement about function 𝐹 that represents the fact that there were 28,800 followers 110 days after the set up of the account. d. Explain the meaning of 𝑡 in the equation 𝐹(𝑡) = 100,000.

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Unit 6, Lesson 3: Understanding Decay

Warm-Up: Two Tables What do you notice? What do you wonder? Table B

𝒙

𝒚

9 2

27 4

81 8

𝒚

𝒙

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Table A

Collaborative Activity: What’s Left?

1. Here is one way to think about how much Diego has left after spending 1 of $100. 4 Explain each step.

Step 1: 100 − 1 ⋅ 100 4

Step 2: 100 �1 − 1 � 4

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Step 3: 100 ⋅ 3 4

Step 4: 3 ⋅ 100 4

2. A person makes $1,800 per month, but 1 of that amount goes to her rent. What two

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numbers can you multiply to find out how much she has after paying her rent?

3. Write an expression that only uses multiplication and that is equivalent to 𝑥 reduced

by 1 of 𝑥.

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Guided Activity: Value of a Vehicle

Every year after a new car is purchased, it loses 1 of its value. Let’s say that a new car 3 costs $18,000.

1. A buyer worries that the car will be worth nothing in three years. Do you agree? Explain your reasoning.

2. Write an expression to show how to find the value of the car for each year listed in the table.

3. Write an equation relating the value of the car in dollars, 𝑣, to the number of years, 𝑡.

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Year

Value of Car ($)

18,000

1 2 3 6 𝑡

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4. Use your equation to find 𝑣 when 𝑡 is 0. What does this value of 𝑣 mean in this situation? 5. A different car loses value at a different rate. The value of this different car in 𝑡

dollars, 𝑑, after 𝑡 years can be represented by the equation 𝑑 = 10,000 ⋅ �4 � . Explain 5

Lesson Summary

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what the numbers 10,000 and 4 mean in this situation. 5

Sometimes, a quantity grows by the same factor at regular intervals. For example, a population doubles every year. Other times, a quantity decreases by the same factor at regular intervals. For example, a car might lose 1 of its value every year. This lesson 3

introduced situations where the quantity decreases by the same factor at regular intervals.

Suppose a bacteria population starts at 100,000, and 1 of the population dies each day.

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• The population 1 day later is 100,000 − 1 ⋅100,000, which can be written as

4 1 100,000 �1 − �. The population after 1 day is 3 of 100,000, or 75,000. 4 4

• The population after 2 days is 3 ⋅ 75,000. 4

Additional values for the bacteria population are shown in the table.

Bacteria Population

100,000

Number of Days

1 2

75,000 (or 100,000 ⋅ 4 )

3 2

56,250 (or 100,000 ⋅ 3 ⋅ 3 , or 100,000 ⋅ � 4� ) 4

3 3

about 42,188 (or 100,000 ⋅ 3 ⋅ 3 ⋅ 3 , or 100,000 ⋅ � 4 � )

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In general, 𝑑 days after the bacteria population was 100,000, the population, 𝑝, is 𝑑

given by the equation 𝑝 = 100,000 ⋅ � 3 � , with a factor of 3 for each day. 4

Situations with quantities that decrease exponentially are described as having exponential decay. The multiplier, 3 in this case, is still called the growth factor, though people sometimes

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call it the decay factor instead.

Practice Problems

1. A new bicycle sells for $300. It is on sale for 1 off the regular price. Select all the 4

□ 300 ⋅ 14 □ 300 ⋅ 34

□ 300 − 14

□ 300 ⋅ �1 − 14 �

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expressions that represent the sale price of the bicycle in dollars?

□ 300 − 14 ⋅ 300

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2. A computer costs $800. It loses 1 of its value every year after it is purchased. 4

a. Complete the table to show the value of the computer at the listed times.

Time (years) 0 1 2

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b. Write an equation representing the value, 𝑣, of the computer, 𝑡 years after it is purchased.

Value of Computer ($)

c. Use your equation to find 𝑣 when 𝑡 is 5. What does this value of 𝑣 mean?

𝑡

3. A piece of paper is folded into thirds multiple times. The area, 𝐴, of the piece of 𝑛

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paper in square inches (sq. in.), after 𝑛 folds, is 𝐴 = 90 ⋅ � 1 � . 3

a. What is the value of 𝐴 when 𝑛 = 0? What does this mean in the situation?

b. How many folds are needed before the area is less than 1 sq. in.? c. The area of another piece of paper in sq. in. after 𝑛 folds, is given by 𝑛

𝐵 = 100 ⋅ � 1 � . What do the numbers 100 and 1 mean in this situation? 2

4. At the beginning of April, a colony of ants has a population of 5,000.

a. The colony decreases by 1 during April. Write an expression for the ant 5

population at the end of April.

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b. During May, the colony decreases again by 1 of its size. Write an expression for 5

the ant population at the end of May.

c. The colony continues to decrease by 1 of its size each month. Write an 5

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expression for the ant population after 6 months.

Review Problems

5. Lin has 13 mystery novels. Each month, she gets 2 more. Select all expressions that represent the total number of Lin’s mystery novels after 3 months.

□ 13 + 2 + 2 + 2

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□ 13 ⋅ 2 ⋅ 2 ⋅ 2 □ 13 ⋅ 8

□ 19

□ 13 + 6

6. An odometer is the part of a car’s dashboard that shows the number of miles a car has traveled in its lifetime. Before a road trip, a car odometer reads 15,000 miles (mi.). During the trip, the car travels 65 miles per hour (mph). a. Complete the table. b. What do you notice about the differences of the odometer readings each hour? c. If the odometer reads 𝑛 mi. at a particular hour, what will it read one hour later? 24 | Unit 6

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Duration of Trip (hours)

Odometer Reading (mi.)

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7. A function multiplies its input by 3 and then adds 7 to get its output. Use function 4 notation to represent this function.

8. A function is defined by the equation 𝑓(𝑥) = 2𝑥 − 5.

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a. What is 𝑓(0)?

b. What is 𝑓 � 1 �? 2

d. What is 𝑥 when 𝑓(𝑥) = 9?

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c. What is 𝑓(100)?

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Unit 6, Lesson 4: Representing Exponential Decay

Warm-Up: Two Other Tables Use the patterns you notice to complete the tables. Show your reasoning.

𝒚

2.5

17.5

160

2.5

𝒚

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Table B 𝒙

𝒙

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Table A

Collaborative Activity: The Algae Bloom

In order to control an algae bloom in a lake, scientists introduce some treatment products. Once the treatment begins, the area covered by

algae, 𝐴, in square yards (sq, yd.), is given by 𝑡

the equation 𝐴 = 240 ⋅ � 1 � . Time, 𝑡, is measured

in weeks.

1. In the equation, what does the 240 tell us about the algae? What does the 1 tell us? 3

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𝑡

2. Create a graph to represent 𝐴 = 240 ⋅ � 1 � when 𝑡 is 0, 1, 2, 3, and 4. Think carefully 3

about how you choose the scale for the axes. If you get stuck, consider creating a

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table of values.

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3. About how many sq. yd. will the algae cover after 2.5 weeks? Explain your reasoning.

Guided Activity: Insulin in the Body

A patient who is diabetic receives 100 micrograms of insulin. The graph shows the amount of insulin, in micrograms, remaining in his bloodstream over time, in minutes.

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1. Scientists have found that the amount of insulin in a patient’s body changes exponentially. How can you check if the graph supports the scientists’ claim?

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2. How much insulin broke down in the first minute? What fraction of the original insulin is that?

3. How much insulin broke down in the second minute? What fraction is that of the amount one minute earlier?

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4. What fraction of insulin remains in the bloodstream for each minute that passes? Explain your reasoning.

5. Complete the table to show the predicted amount of insulin 4 and 5 minutes after injection.

Time after injection (minutes)

100

72.9

Insulin in the bloodstream (micrograms)

6. Describe how you would find how many micrograms of insulin remain in his bloodstream after 10 minutes. After 𝑚 minutes?

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Lesson Summary A graph representing the amount of caffeine in a person’s body, measured in milligrams (mg), over a period of time, measured in hours (hr.), is shown. The amount of caffeine in the person’s body changes exponentially.

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• The graph includes the point (0, 200). This means that there were 200 mg of caffeine in the person’s body when it was initially measured.

• Between 6 and 7 hr. after the initial measurement, the amount of caffeine in the body fell below 100 mg.

• The point (1, 180) reveals there were 180 mg of caffeine 1 hr. later.

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Use the graph to determine what fraction of caffeine remains in the body each hr. Notice that 180 = 9 and 162 = 9 . As each hr. passes, the amount of caffeine that stays 200

180

in the body is multiplied by a factor of 9 .

If 𝑦 is the amount of caffeine, in mg, and 𝑡 is time, in hr., then this situation is modeled 𝑡

by the equation 𝑦 = 200 ⋅ � 9 � .

Practice Problems

𝑤

1. A population, 𝑝, of migrating butterflies satisfies the equation 𝑝 = 100,000 ⋅ � 4 � , 5 where 𝑤 is the number of weeks since they began their migration.

a. Complete the table with the population after different numbers of weeks. 𝒘 𝒑

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b. Graph the butterfly population. Think carefully about how to choose a scale for the axes.

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c. What is the vertical intercept of the graph? What does it tell you about the butterfly population?

d. About when does the butterfly population reach 50,000?

2. The graph shows the amount of a chemical in a water sample. It is decreasing exponentially.

Find the coordinates of the points labeled 𝐴, 𝐵, and 𝐶. Explain your reasoning.

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3. The graph shows the amount of a chemical in a water sample at different times after it was first measured.

Select all statements that are true.

□ The amount of the chemical in the water sample is decreasing exponentially.

□ The amount of the chemical in the water sample is not decreasing exponentially. decreasing exponentially.

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□ It is not possible to tell for certain whether or not the amount of the chemical is sample.

□ When it was first measured, there were 2,000 mg of the chemical in the water

□ After 4 hr., there were 100 mg of the chemical in the water.

4. The graph shows the amount of a chemical in a patient’s body at different times measured in hours since the levels were first checked.

Could the amount of this chemical in the patient be decaying exponentially? Explain how you know.

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Review Problems 5. The height of a plant in millimeters (mm) is 7. It doubles each week. Select all expressions that represent the height of the plant, in mm, after 4 weeks.

□7 + 4 ⋅ 2 □7 ⋅ 2

□2 + 7

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□7 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 □7 ⋅ 2 ⋅ 4

6. The number of people who have read a new book is 300 at the beginning of January. The number of people who have read the book doubles each month.

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Number of Months since January

Number of People Who Have Read the Book

a. Use this information to complete the table.

b. What do you notice about the difference in the number of people who have read the book from month to month?

0 1

2 3

c. What do you notice about the factor by which the number of people changes each month?

d. At the beginning of a month, 𝑛 people have read the book. How many people will have read the book at the beginning of the next month?

7. Solve each system of equations. �

𝑥 + y = 2

−3𝑥 − y = 5

� 2

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𝑥 + 2y = −13

𝑥 − 4y = 8

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Unit 6, Lesson 5: Using Negative Exponents

Warm-Up: Exponent Rules How would you rewrite each of the following as an equivalent expression with a single exponent? 24 ⋅ 2−1

24 ⋅ 2−3

Guided Activity: Coral in the Sea

24 ⋅ 2−4

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24 ⋅ 20

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A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters (cu. cm) and that its volume doubles each year.

1. Write an equation of the form 𝑦 = 𝑎 ⋅ 𝑏𝑡 representing the relationship, where 𝑡 is time in years since the coral was measured and 𝑦 is volume of coral in cu. cm.

2. Find the volume of the coral when 𝑡 is 5, 1, 0, −1, and −2.

3. What does it mean, in this situation, when 𝑡 is −2?

4. In a certain year, the volume of the coral is 37.5 cu. cm. Which year is this? Explain your reasoning.

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Exploration Activity: Windows of Graphs

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The volume, 𝑦, of coral in cu. cm is modeled by the equation 𝑦 = 1,200 ⋅ 2𝑥, where 𝑥 is the number of years since the coral was measured. Three students used graphing technology to graph the equation that represents the volume of coral as a function of time.

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For each graph:

1. Describe how well each graphing window does, or does not, show the behavior of the function.

2. For each graphing window you think does not show the behavior of the function well, describe how you would change it.

3. Make the change(s) you suggested and sketch the revised graph using graphing technology.

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Collaborative Activity: Measuring Meds A person took some medicine but does not remember how much. Concerned that she took too much, she has a blood test every hour for several hours. 1.

Time (hr.) Medicine (mg)

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b. Find the amounts of medicine in the patient’s body when 𝑡 is −1 and −3. Record them in the table.

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a. Time, 𝑡, is measured in hours (hr.) since the first blood test and amount of medicine in her body, 𝑚, is measured in milligrams (mg). What is the growth factor? That is, what is 𝑏 in an equation of the form 𝑚 = 𝑎 ⋅ 𝑏𝑡? What is 𝑎?

100

2. What do 𝑡 = 0 and 𝑡 = −3 mean in this context?

3. The medicine was taken when 𝑡 is −5. Assuming the person did not have any of the medication in her body beforehand, how much medicine did the patient take?

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ip t

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4. Plot the points whose coordinates are shown in the table. Make sure to draw and label tick marks on the axes.

5. Based on your graph, when do you think the patient will have:

a. 500 mg of medicine remaining in the body

b. no medicine remaining in the body

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Lesson Summary Equations are useful not only for representing relationships that change exponentially but also for answering questions about these situations. Suppose a bacteria population of 1,000,000 has been increasing by a factor of 2 every hr. What was the size of the population 5 hr. ago? How many hr. ago was the population less than 1,000?

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The population of bacteria can be calculated 1 hr. ago, 2 hr. ago, and so on. For example, if the population doubled each hr. and was 1,000,000 when first observed, then an hr. before that time, it must have been 500,000, and 2 hr. before then, it must have been 250,000, and so on.

Another way to reason through these questions is by representing the situation with an equation. If 𝑡 measures time in hr. since the population was 1,000,000, then the bacteria population can be described by the equation 𝑝 = 1,000,000 ⋅ 2𝑡.

ip t

The population is 1,000,000 when 𝑡 = 0, so 5 hr. earlier, 𝑡 = − 5. The population at that time can be calculated using the work shown.

1,000,000 ⋅ 2−5 = 1,000,000 ⋅ 15 2

1 = 1,000,000 ⋅ 32

= 31,250

Likewise, substituting − 10 for 𝑡 gives 1,000,000⋅2−10, or 1,000,000 ⋅ 110 , which is a little 2

less than 1,000. This means that 10 hr. before the initial measurement, the bacteria population was less than 1,000.

Note that for any given real-world context, it’s important to determine whether limitations on the model should exclude negative values of time. In the previous example, the calculation works as long as the population of bacteria was growing at that rate the entire time.

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Practice Problems 1. A forest fire has been burning for several days. The burned area, in acres, is given by the equation 𝑦 = (4,800) ⋅ 2𝑑, where 𝑑 is the number of days since the area of the fire was first measured. a. Complete the table.

−1

−2 −3

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b. Look at the value of 𝑦 = 4,800 ⋅ 2𝑑 when 𝑑 = −1. What does it tell you about the area burned in the fire? What about when 𝑑 = − 3?

Acres Burned since Fire Started

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Days since First Measurement

−5

c. How much area had the fire burned a week before it measured 4,800 acres? Explain your reasoning.

2. The value of a home in 2015 was $400,000. Its value has been doubling each decade.

a. If 𝑣 is the value of the home, in dollars, write an equation for 𝑣 in terms of 𝑑, the number of decades since 2015.

b. What is 𝑣 when 𝑑 = −1? What does this value mean?

c. What is 𝑣 when 𝑑 = −3? What does this value mean?

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𝑡

3. A fish population, 𝑝, can be represented by the equation 𝑝 = 800 ⋅ � 2 � where 𝑡 is time in years since the beginning of 2015. What was the fish population at the beginning of 2012? A. 100 B. 800

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C. 2,400

D. 6,400

𝑑

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4. The area, 𝐴, of a forest, in acres, is modeled by the equation 𝐴 = 5,000 ⋅ � 4 � where 𝑑 is the number of decades since the beginning of the year 1950.

a. Is the area of the forest increasing or decreasing with time? Explain how you know.

b. What was the area of the forest in 1950?

c. What was the area of the forest in 1940?

d. Was the area of the forest less than 1,000 acres in 1900? Explain how you know.

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Review Problems 5. A population of mosquitos 𝑝 is modeled by the equation 𝑝 = 1,000 ⋅ 2𝑤 where 𝑤 is the number of weeks after the population was first measured. a. Find and plot the mosquito population for 𝑤 = 0, 1, 2, 3, 4.

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c. Where on the graph can you see the 2 from the equation?

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b. Where on the graph do you see the 1,000 from the equation for 𝑝?

6. The number of copies of a book sold the year it was released was 600,000. Each year after that, the number of copies sold decreased by 1 .

a. Complete the table showing the number of copies of the book sold each year.

b. Write an equation representing the number of copies, 𝑐, sold 𝑦 years after the book was released.

c. Use your equation to find 𝑐 when 𝑦 = 6. What does this mean in terms of the book?

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Years since Published

Number of Copies Sold

0 1 2 3

𝑦

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Unit 6, Lesson 6: Exploring Exponential Functions

Warm-Up: Rainfall in Las Vegas

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Here is a graph of the accumulated rainfall in Las Vegas, Nevada, in the first 60 days of 2017.

Use the graph to support your answers to the following questions.

1. Is the accumulated amount of rainfall a function of time?

2. Is time a function of accumulated rainfall?

Collaborative Activity: Card Sort: Matching Descriptions to Graphs

Your teacher will give you a set of cards containing descriptions of situations and graphs. Match each situation with a graph that represents it. Record your matches and be prepared to explain your reasoning.

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Guided Activity: Functionally Speaking Here are some situations we have seen previously. For each situation: • Write a sentence of the form “______ is a function of ______.”

• Indicate which is the independent and which is the dependent variable.

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• Write an equation that represents the situation using function notation.

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1. In a biology lab, a population of 50 bacteria reproduce by splitting. Every hour, on the hour, each bacterium splits into two bacteria.

2. Every year after a new car is purchased, it loses 1 of its value. Let’s say that the 3 new car costs $18,000.

3. In order to control an algae bloom in a lake, scientists introduce some treatment products. The day they begin treatment, the area covered by algae is 240 square yards (sq. yd.). Each day since the treatment began, a third of the previous day’s area (in sq. yd.) remains covered by algae. Time 𝑡 is measured in days.

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Collaborative Activity: Deciding on Graphing Window The equation 𝑚 = 20 ⋅ (0.8)ℎ models the amount of medicine, 𝑚, in milligrams (mg), in a patient’s body as a function of hours, ℎ, after injection. 1. Without using a graphing tool, decide if the following horizontal and vertical boundaries are suitable for graphing this function. Explain your reasoning. −100 < 𝑚 < 1,000

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−10 < ℎ < 100

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2. Verify your answer by graphing the equation using graphing technology, and using the given graphing window. What do you see? Sketch or describe the graph.

3. If your graph in the previous question is unhelpful, modify the window settings so that the graph is more useful. Record the window settings here. Convince a partner why the horizontal and vertical boundaries that you set are better.

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Lesson Summary Situations that are characterized by exponential change can be seen as functions. In each situation, there is a quantity (an independent variable) that determines another quantity (a dependent variable). They are functions because any value of the independent variable corresponds to 1 and only 1 value of the dependent variable. Functions that describe exponential change are called exponential functions.

op y

An exponential function is a function with a constant percent rate of change.

For example, 𝑡 represents time, in hr., and 𝑝 is a bacteria population 𝑡 hr. after the bacteria population was measured. For each time 𝑡, there is only 1 value for the corresponding number of bacteria. Therefore, 𝑝 is a function of 𝑡 and is as written as 𝑝 = 𝑓(𝑡).

If there were 100,000 bacteria at the time it was initially measured, and the population

ip t

decreases so that 1 of it remains after each passing hour, function notation can be used 5

to model the bacteria population as shown.

𝑝(𝑡) = 100,000 ⋅ � 5 �

𝑡

Notice the expression in the form of 𝑎 ⋅ 𝑏𝑡 on the right side of the equation is the same as in previous equations used to represent situations characterized by exponential change, where 𝑎 represents the initial value and 𝑏 represents the growth factor.

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Practice Problems 1. A medicine is applied to a burn on a patient’s arm. The area of the burn in square centimeters (sq. cm) decreases exponentially and is shown in the graph.

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a. What fraction of the burn area remains each week?

b. Write an equation representing the area of the burn, 𝑎, after 𝑡 weeks.

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c. What is the area of the burn after 7 weeks? Round to three decimal places. 2. For an experiment, a scientist designs a can 20 centimeters (cm) in height that holds water. A tube is installed at the bottom of the can allowing water to drain out.

At the beginning of the experiment, the can is full. Every minute (min.) after the start of the experiment, 2 of the water is drained. 3

a. Explain why the height of the water in the can is a function of time.

b. The height, ℎ, in cm, is a function 𝑓 of time, 𝑡, in min. since the beginning of the experiment, ℎ = 𝑓(𝑡). Find an expression for 𝑓(𝑡). c. Find and record the values for 𝑓 when 𝑡 is 0, 1, 2, and 3.

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d. Find 𝑓(4). What does 𝑓(4) represent?

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e. Sketch a graph of 𝑓 by hand or use graphing technology.

f. What happens to the level of water in the can as time continues to elapse? How do you see this in the graph?

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3. The area covered by a city is 20 square miles (sq. mi.). The area grows by a factor of 1.1 each year since it was 20 sq. mi.

a. Explain why the area, 𝑎, covered by the city, in sq. mi., is a function of 𝑡, the number of years since its area was 20 sq. mi.

b. Write an equation for 𝑎 in terms of 𝑡.

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Review Problems 4. The graph shows an exponential relationship between 𝑥 and 𝑦.

b. What is the value of 𝑦 when 𝑥 = −1. Label this point on the graph.

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a. Write an equation representing this relationship.

c. What is the value of 𝑦 when 𝑥 = −2? Label this point on the graph.

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a. The area of a sheet of paper is 100 square inches (sq. in.). Write an equation that gives the area, 𝐴, of the sheet of paper, in sq. in., after being folded in half 𝑛 times.

b. The area of another sheet of paper is 200 sq. in. Write an equation that gives the area, 𝐵, of this sheet of paper, in sq. in., after being folded into thirds 𝑛 times.

c. Are the areas of the two sheets of paper ever the same after each being folded 𝑛 times? Explain how you know.

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6. Technology required. Use a graphing calculator to determine the equation of the line of best fit. Round numbers to 2 decimal places.

𝒙

12 22

15 21

20 14

24 10

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op y

𝒚

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Unit 6, Lesson 7: Interpreting Exponential Functions

Warm-Up: Equivalent or Not? Lin and Diego are discussing two expressions: 𝑥2 and 2𝑥.

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• Lin says, “I think the two expressions are equivalent.” • Diego says, “I think the two expressions are only equal for some values of 𝑥.”

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Do you agree with either of them? Explain or show your reasoning.

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Exploration Activity: Cost of Solar Cells The cost, in dollars, to produce 1 watt of solar energy is a function of the number of years since 1977, 𝑡.

ip t

op y

From 1977 to 1987, the cost could be modeled by an exponential function 𝑓. Here is the graph of the function.

1. What is the statement 𝑓(9) ≈ 6 saying about this situation?

2. What is 𝑓(4)? What about 𝑓(3.5)? What do these values represent in this context? 3. When 𝑓(𝑡) = 45, what is 𝑡? What does that value of 𝑡 represent in this context? 4. By what factor did the cost of solar cells change each year? (If you get stuck, consider creating a table.)

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Guided Activity: Paper Folding 1. The thickness, 𝑡, in millimeters (mm), of a folded sheet of paper after it is folded 𝑛 times is given by the equation 𝑡 = (0.05) ⋅ 2𝑛.

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a. What does the number 0.05 represent in the equation?

b. Use graphing technology to graph the equation 𝑡 = (0.05) ⋅ 2𝑛.

c. How many folds does it take before the folded sheet of paper is more than 1 mm thick? How many folds before it is more than 1 centimeter (cm) thick? Explain how you know.

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2. The area of a sheet of paper is 93.5 square inches (sq. in.).

a. Find the remaining, visible area of the sheet of paper after it is folded in half once, twice, and three times.

b. Write an equation expressing the visible area, 𝑎, of the sheet of paper in terms of the number of times it has been folded 𝑛. c. Use graphing technology to graph the equation.

d. In this context, can 𝑛 take negative values? Explain your reasoning. e. Can 𝑎 take negative values? Explain your reasoning.

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Collaborative Activity: Info Gap: Smartphone Sales Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. If your teacher gives you the problem card:

1. Silently read the information on your card.

1. Silently read your card and think about what information you need to answer the question.

3. Explain to your partner how you are using the information to solve the problem. 4. When you have enough information, share the problem card with your partner, and solve the problem independently.

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3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”

2. Ask your partner for the specific information that you need.

2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner.)

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If your teacher gives you the data card:

4. Read the problem card, and solve the problem independently.

5. Share the data card, and discuss your reasoning.

5. Read the data card, and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

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Lesson Summary Equations can be used to represent situations characterized by exponential change. For 9

𝑡

example, the equation 𝑐 = 100 ⋅ �10 � can be used to describe the amount of caffeine, 𝑐,

in a person’s body 𝑡 hours (hr.) after an initial measurement of 100 milligrams (mg). Notice that the amount of caffeine is a function of time, so another way to express 9

𝑡

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this relationship is 𝑐 = 𝑓(𝑡), where 𝑓 is the function given by 𝑓(𝑡) = 100 ⋅ �10 � . Use this

function to analyze the amount of caffeine.

729

For example, when 𝑡 = 3, the amount of caffeine in the body is 100 ⋅ �10 � , or 100 ⋅ 1,000 ,

which is 72.9. The statement 𝑓(3) = 72.9 means that 72.9 mg of caffeine are present in the body 3 hr. after the initial measurement.

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The function 𝑓 can be graphed to better understand what is happening. The point labeled 𝑃, for example, has coordinates of approximately (10, 35). This means it takes about 10 hr. after the initial measurement for the caffeine level to decrease to 35 mg.

Use the graph to determine viable values for the domain and range of the function. Because the body breaks down caffeine continuously over time, the domain of the function, the time in hr., can include nonwhole numbers. For example, the caffeine level can be determined when 𝑡 = 3.5. In this situation, negative values for the domain would represent the time before the initial measurement. For example, 𝑓 (−1) would represent the amount of caffeine in the person’s body 1 hr. before the initial measurement. The range of this function would not include negative values because a negative amount of caffeine does not make sense in this situation.

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Practice Problems 1. The number of people with the flu during an epidemic is a function, 𝑓, of the number 3 𝑑

of days, 𝑑, since the epidemic began. The equation 𝑓(𝑑) = 50 ⋅ � 2 � defines 𝑓.

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a. How many people had the flu at the beginning of the epidemic? Explain how you know.

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c. What does 𝑓(1) mean in this situation?

b. How quickly is the flu spreading? Explain how you can tell from the equation.

d. Does 𝑓(3.5) make sense in this situation?

2. The function, 𝑓, gives the dollar value of a bond 𝑡 years after the bond was purchased. The graph of 𝑓 is shown.

a. What is 𝑓(0)? What does it mean in this situation?

b. What is 𝑓(4.5)? What does it mean in this situation?

c. When is 𝑓(𝑡) = 1500? What does this mean in this situation?

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3. Technology required. A function 𝑓 gives the number of stray cats in a town 𝑡 years since the town started an animal control program. The program includes both sterilizing stray cats and finding homes to adopt them. An equation representing 𝑓 is 1 𝑡

𝑓(𝑡) = 243 � 3 � .

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a. What is the value of 𝑓(𝑡) when 𝑡 is 0? Explain what this value means in this situation.

b. What is the approximate value of 𝑓(𝑡) when 𝑡 is 2 ? Explain what this value means in this situation.

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c. What does the number 3 tell you about the stray cat population?

d. Use technology to graph 𝑓 for values of 𝑡 between 0 and 4. What graphing window allows you to see values of 𝑓(𝑡) that correspond to these values of 𝑡? 4. Function 𝑔 gives the amount of a chemical in a person’s body, in mg, 𝑡 hr. since 3 𝑡

the patient took the drug. The equation 𝑔(𝑡) = 600 ⋅ � 5 � defines this function. 3

a. What does the fraction 5 mean in this situation?

b. Sketch a graph of 𝑔.

c. What are the domain and range of 𝑔? Explain what they mean in this situation.

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5. The dollar value of a moped is a function of the number of years, 𝑡, since the moped 1 𝑡

was purchased. The function, 𝑓, is defined by the equation 𝑓(𝑡) = 2,500 ⋅ � 2 � .

What is the best choice of domain for the function 𝑓? A. −10 ≤ 𝑡 ≤ 10

B. −10 ≤ 𝑡 ≤ 0

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C. 0 ≤ 𝑡 ≤ 10

D. 0 ≤ 𝑡 ≤ 100

Review Problems

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6. A patient receives 1,000 mg of a medicine. Each hour, 1 of the medicine in the 5 patient’s body decays.

a. Complete the table with the amount of medicine in the patient’s body.

b. Write an equation representing the number of mg of the medicine, 𝑚, in the patient’s body ℎ hours after receiving the medicine.

c. Use your equation to find 𝑚 when ℎ = 10. What does this mean in terms of the medicine?

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Hours Since Receiving Medicine

Medicine Left in Body (mg)

0 1 2 3

ℎ

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7. The trees in a forest are suffering from a disease. The population of trees, 𝑝, in 3

𝑡

thousands, is modeled by the equation 𝑝 = 90 ⋅ � 4 � , where 𝑡 is the number of years since 2000. a. What was the tree population in 2001? What about in 1999?

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b. What does the number 4 in the equation for 𝑝 tell you about the population?

c. What is the last year when the population was more than 250,000? Explain how you know.

8. All of the students in a classroom list their birthdays.

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a. Is the birthdate, 𝑏, a function of the student, 𝑠?

b. Is the student, 𝑠, a function of the birthdate, 𝑏?

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Unit 6, Lesson 8: Looking at Rates of Change

Warm-Up: Falling Prices

1. Which expression best represents the average rate of change in solar cost between 1977 and 1987?

A. 𝑝(10) − 𝑝(0) B. 𝑝(10)

𝑝(10) D. 𝑝(0)

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𝑝(10) − 𝑝(0) 10 − 0

𝒑(𝒕)

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Let 𝑝 be the function that gives the cost 𝑝(𝑡), in dollars, of producing 1 watt of solar energy 𝑡 years after 1977. Here is a table showing the values of 𝑝 from 1977 to 1987. 𝒕

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33.75

18.98

2 4 6 7 8 9

25.31 14.24 10.68 8.01 6.01 4.51

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Guided Activity: Coffee Shops Here are a table and a graph that show the number of coffee shops worldwide that a company had in its first 10 years, between 1987 and 1997. The growth in the number of stores was roughly exponential. Number of Stores

1987

1988

1989

1990

1994 1995 1996 1997

165

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1993

116 272 425 677

1,015

1992

1991

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Year

1,412

1. Find the average rate of change for each period of time. Show your reasoning.

a. 1987 and 1990

b. 1987 and 1993

c. 1987 and 1997

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2. Make some observations about the rates of change you calculated. What do these average rates tell us about how the company was growing during this time period?

3. Use the graph to support your answers to these questions. How well do the average rates of change describe the growth of the company in:

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a. the first 3 years?

b. the first 6 years?

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c. the entire 10 years?

4. Let 𝑓 be the function so that 𝑓(𝑡) represents the number of stores 𝑡 years since

𝑓(20) − 𝑓(10) and say what it tells us about 20 − 10

1987. The value of 𝑓(20) is 15,011. Find

the change in the number of stores.

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Collaborative Activity: Revisiting Cost of Solar Cells

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Here is a graph of a function you saw in an earlier lesson. It represents the exponential function 𝑝, which models the cost 𝑝(𝑡), in dollars, of producing 1 watt of solar energy, from 1977 to 1988 where 𝑡 is years since 1977.

1. Clare said, “In the first five years, between 1977 and 1982, the cost fell by about $12 per year. But in the second five years, between 1983 and 1988, the cost fell only by about $2 a year.” Show that Clare is correct.

2. If the trend continues, will the average decrease in price be more or less than $2 per year between 1987 and 1992? Explain your reasoning.

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Lesson Summary When average rate of change is calculated for a linear function, no matter which interval is chosen, the value of the rate of change is the same. A constant rate of change is an important feature of linear functions. When a linear function is represented by a graph, the slope of the line is the rate of change of the function.

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Exponential functions also have important features. Exponential growth and exponential decay are both characterized by a constant quotient over equal intervals. But what does this mean for the value of the average rate of change for an exponential function over a specific interval?

Revisit an exponential function studied previously. Let 𝐴 be the function that models the area, 𝐴(𝑡), in square yards (sq. yd.), of algae covering a pond 𝑡 weeks after beginning treatment to control the algae bloom. The table shows about how many sq. yd. of algae remain during the first 5 weeks of treatment. • The average rate of change of 𝐴 from the start of treatment to

𝐴(2) − 𝐴(0) ≈ −107. 2−0

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week 2 is about −107 sq. yd. per week since

• The average rate of change of 𝐴 from week 2 to week 4, however, 𝐴(4) − 𝐴(2) ≈ −12. 4−2

is only about −12 sq. yd. per week since

𝒕

𝑨(𝒕)

0 2 3 4

240 27 9 3

These calculations show that 𝐴 is decreasing over both intervals, but the average rate of change is less in weeks 0 through 2 than in weeks 2 through 4, which is due to the effect of the decay factor. For an exponential growth function, the values for the average rate of change of each interval would be positive, with the second interval having a greater value than the first, due to the effect of a growth factor.

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Practice Problems 1. A store receives 2,000 decks of popular trading cards. The number of decks of cards is a function, 𝑑, of the number of days, 𝑡, since the shipment arrived. Here is a table showing some values of 𝑑.

𝒅(𝒕)

2,000

823

1,283

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Calculate the average rate of change for the following intervals:

𝒕

a. day 0 to day 5

338

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b. day 15 to day 20

528

2. A study was conducted to analyze the effects on deer population in a particular area. Let 𝑓 be an exponential function that gives the population of deer 𝑡 years after the study began. If 𝑓(𝑡) = 𝑎 ⋅ 𝑏𝑡 and the population is increasing, select all statements that must be true.

□ 𝑏 > 1 □ 𝑏 < 1

□ The average rate of change from year 0 to year 5 is less than the average rate of change from year 10 to year 15.

□ The average rate of change from year 0 to year 5 is greater than the average rate of change from year 10 to year 15.

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3. Function 𝑓 models the population, in thousands, of a city 𝑡 years after 1930.

The average rate of change of 𝑓 from 𝑡 = 0 to 𝑡 = 70 is approximately 14 thousand people per year.

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Is this value a good way to describe the population change of the city over that time period? Explain or show your reasoning.

4. The function, 𝑓, gives the number of copies a book has sold 𝑤 weeks after it was published. The equation 𝑓(𝑤) = 500 ⋅ 2𝑤 defines this function.

□ 0 ≤ 𝑤 ≤ 7

□ 5 ≤ 𝑤 ≤ 7

□ 0 ≤ 𝑤 ≤ 2

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Select all domains for which the average rate of change could be a good measure for the number of books sold.

□ 5 ≤ 𝑤 ≤ 10

□ 0 ≤ 𝑤 ≤ 10

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Review Problems 5. The graph shows a bacteria population decreasing exponentially over time. 2 ℎ

The equation 𝑝 = 100,000,000 ⋅ � 3 � gives the size of a

second population of bacteria, where ℎ is the number of hours since it was measured at 100 million.

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Which bacterial population decays more quickly? Explain how you know.

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6. Give a value for 𝑟 that would indicate that a line of best fit has a positive slope and models the data well.

7. The size of a district and the number of parks in it have a weak positive relationship.

Explain what it means to have a weak positive relationship in this context.

8. Here is a graph of Han’s distance from home as he drives.

Identify the intercepts of the graph and explain what they mean in terms of Han’s distance from home.

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Unit 6, Lesson 9: Modeling Exponential Behavior

Warm-Up: Wondering about Windows

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Here is a graph of a function 𝑓 defined by 𝑓(𝑥) = 400 ⋅ (0.2)𝑥.

1. Identify the approximate graphing window shown.

2. Suggest a new graphing window that would:

a. make the graph more informative or meaningful

b. make the graph less informative or meaningful

Be prepared to explain your reasoning.

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Guided Activity: Beholding Bounces Here are measurements for the maximum height in centimeters (cm) of a tennis ball after bouncing several times on a concrete surface.

Bounce Number

Height (cm)

1. Which is more appropriate for modeling the maximum height, ℎ, in cm, of the tennis ball after 𝑛 bounces: a linear function or an exponential function? Use data from the table to support your answer.

0 1

150

43 20

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2. Regulations say that a tennis ball, dropped on concrete, should rebound to a height between 53% and 58% of the height from which it is dropped. Does the tennis ball here meet this requirement? Explain your reasoning.

3. Write an equation that models the bounce height ℎ after 𝑛 bounces for this tennis ball.

4. About how many bounces will it take before the rebound height of the tennis ball is less than 1 cm? Explain your reasoning.

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Exploration Activity: Beholding More Bounces The table shows some heights of a ball after a certain number of bounces. Bounce Number

Height (cm)

op y

0 1

73.5

51.5 36

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1. Is this ball more or less bouncy than the tennis ball in the earlier task? Explain or show your reasoning.

2. From what height was the ball dropped? Explain or show your reasoning.

3. Write an equation that represents the bounce height of the ball, ℎ, in cm, after 𝑛 bounces.

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4. Which graph would more appropriately represent the equation for ℎ: Graph A or Graph B? Explain your reasoning.

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5. Will the 𝑛th bounce of this ball be lower than the 𝑛th bounce of the tennis ball? Explain your reasoning.

Collaborative Activity: Falling and Falling

The value of some cell phones changes exponentially after initial release. Here are graphs showing the depreciation of two phones 1, 2, and 3 years after they were released.

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1. Which phone is more expensive to buy when it is first released?

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2. How does the value of each phone change with every passing year?

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3. Which one is falling in value more quickly? Explain or show how you know.

4. If the phones continue to depreciate by the same factor each year, what will the value of each phone be 4 years after its initial release?

5. For each cell phone, write an equation that relates the value of the phone in dollars to the years since release, 𝑡. Use 𝑣 for the value of Phone A and 𝑤 for the value of Phone B.

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Lesson Summary Sometimes, data suggests an exponential relationship. For example, this table shows the bounce heights of a certain ball. The height decreases with each bounce. Bounce Height (cm)

op y

Bounce Number

To determine what fraction of the height remains after each bounce, divide 61

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3 consecutive values: 95 is about 0.642, 61 is about 0.639, and 39 is about 0.667.

All these quotients are close to 3 . This suggests that there is an exponential relationship between the number of bounces and the height of the bounce, and that the height is decreasing with

a factor of about 3 for each successive bounce. The height ℎ of the ball, in cm, after 𝑛 bounces can be modeled by the equation

(2 )

𝑛

ℎ = 142.5 ⋅ 3 .

A graph of the equation is shown.

This graph shows both the points from the data and the points generated by the equation, which can provide new insights. For example, the height from which the ball 2

was dropped is not given, but it can be determined. If 3 of the initial height is about 2

95 cm, then the initial height is about 142.5 cm because 95÷ 3 = 142.5. For a second example, the graph shows that it will take 7 bounces before the rebound height is less than 10 cm. 74 | Unit 6

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Practice Problems 1. Function ℎ describes the height of a ball, in inches (in.), after 𝑛 bounces and is defined by the equation ℎ(𝑛) = 120 ⋅

( 45 ) . 𝑛

b. Could ℎ(𝑛) be 150? Explain how you know.

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a. What is ℎ(3)? What does it represent in this situation?

c. Which ball loses its height more quickly, this ball or a tennis ball whose height

( )? 𝑛

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in inches after 𝑛 bounces is modeled by the function 𝑓 where 𝑓(𝑛) = 50 ⋅

5 9

d. How many bounces would it take before the ball bounces less than 12 in. from the surface?

A. 34

2. After its second bounce, a ball reached a height of 80 cm. The rebound factor for the ball was 0.7. From approximately what height, in cm, was the ball dropped?

B. 49

C. 115 D. 163

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3. Which equation is most appropriate for modeling this data?

101

124

158

195

244

A. 𝑦 = 64 ⋅(1.25)

𝑥

B. 𝑦 = 79 ⋅ (1.25)𝑥

op y

C. 𝑦 = 79 + 1.25𝑥

Number of Employees

Number of Customers

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4. The table shows the number of employees and number of active customer accounts for some different marketing companies.

D. 𝑦 = 64 + 22𝑥

Would a linear or exponential model for the relationship between number of employees and number of customers be more appropriate? Explain how you know.

Review Problem

5. A bank account has a balance of 1,000 dollars. It grows by a factor of 1.04 each year.

a. Explain why the balance, in dollars, is a function, 𝑓, of the number of years, 𝑡, since the account was opened.

b. Write an equation defining 𝑓.

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Unit 6, Lesson 10: Reasoning about Exponential Graphs

Warm-Up: Which One Doesn’t Belong: Four Functions 1. Which one doesn’t belong?

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A. 𝑓(𝑛) = 8 ⋅ 2𝑛

B. 𝑔(𝑛) = 2 ⋅ 8𝑛

𝑛

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D. 𝑗(𝑛) = 8 ⋅ � 2 �

C. ℎ(𝑛) = 8 + 2𝑛

Exploration Activity: Equations and Their Graphs

1. The functions 𝑓, 𝑔, ℎ, and 𝑘 each represent the amount of money in a bank account, in dollars, as a function of time 𝑥, in years. They are written in form 𝑚(𝑥) = 𝑎 ⋅ 𝑏𝑥. 𝑓(𝑥) = 50 ⋅ 2𝑥

𝑔(𝑥) = 50 ⋅ 3𝑥

ℎ(𝑥) = 50 ⋅ � 2 �

𝑥

𝑘(𝑥) = 50 ⋅ (0.5)𝑥

a. Use graphing technology to graph each function on the same coordinate plane.

b. Explain how changing the value of 𝑏 changes the graph.

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2. The equations defining functions 𝑝, 𝑞, and 𝑟 are shown. They are also written in the form 𝑚(𝑥) = 𝑎 ⋅ 𝑏𝑥. 𝑞(𝑥) = 40 ⋅4𝑥

𝑝(𝑥) = 10 ⋅ 4𝑥

𝑟(𝑥) = 100 ⋅ 4𝑥

a. Use graphing technology to graph each function.

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b. Explain how changing the value of 𝑎 changes the graph.

3. Complete the statement.

When the equation is written in the form 𝑦 = 𝑎(𝑏)𝑥, the value of 𝑏 is greater decay growth

and 1 when the equation is modeling exponential

decay. growth.

and is between 0

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than 1 when the equation is modeling exponential

Collaborative Activity: Graphs Representing Exponential Decay

Four graphs representing exponential decay functions are shown on the coordinate plane.

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1. Complete the table by writing which graph corresponds to each function. Be prepared to explain your reasoning. Graph 𝑥

𝑥

𝑛(𝑥) = 200 ⋅ � 2 � 𝑝(𝑥) = 200 ⋅ � 4 �

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𝑞(𝑥) = 200 ⋅ � 8 �

𝑚(𝑥) = 200 ⋅ � 4 �

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Function

𝑥

2. Function 𝑓 and 𝑔 are defined as 𝑓(𝑥) = 1,000 ⋅ �10 � and 𝑔(𝑥) = 1,000 ⋅ �10 � .

a. Which function is decaying more quickly? Explain your reasoning.

b. Use graphing technology to verify your response.

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Exploration Activity: Moldy Wall Here are graphs representing two functions, and descriptions of two functions. • Function 𝑓: The area of a wall that is covered by Mold A, in square inches (sq. in.), doubling every month.

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• Function 𝑔: The area of a wall that is covered by Mold B, in sq. in., tripling every month.

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1. Which graph represents each function? Label the graphs accordingly and explain your reasoning.

2. When the mold was first spotted and measured, was there more of Mold A or Mold B? Explain how you know.

3. What does the point (𝑝, 𝑞) tell us in this situation?

Lesson Summary

An exponential function can provide information about a graph that represents it.

For example, suppose the function 𝑞 represents a bacteria population 𝑡 hours (hr.) after it is first measured, and 𝑞(𝑡) = 5,000 ⋅ (1.5)𝑡. The number 5,000 is the bacteria population measured when 𝑡 is 0. The number 1.5 indicates that the bacteria population increases by a factor of 1.5 each hr. 80 | Unit 6

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A graph can help display how the starting population (5,000) and growth factor (1.5) influence the population. Suppose functions 𝑝 and 𝑟 represent 2 other bacteria populations and are given by 𝑝(𝑡) = 5,000 ⋅ 2𝑡 and 𝑟(𝑡) = 5,000 ⋅ (1.2)𝑡. The graphs of 𝑝, 𝑞, and 𝑟 are shown.

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All 3 graphs start at 5,000, but the graph of 𝑟 grows more slowly than the graph of 𝑞, but the graph of 𝑝 grows more quickly. This makes sense because a population that doubles every hr. is growing more quickly than one that increases by a factor of 1.5 each hr., and both grow more quickly than one that increases by a factor of 1.2 each hr.

Consider another example. Graphs representing 2 different exponential functions, labeled 𝑔 and ℎ, are shown. Each represents the area of algae, in square meters (sq. m), in a pond, 𝑥 days after certain fish were introduced. • Pond A had 40 sq. m of algae. Its area shrinks to

8 10 of the area on the previous day.

• Pond B had 50 sq. m of algae. Its area shrinks to 5 of the area on the previous day.

Which graph corresponds to which algae population? The graphs show that the 𝑦-intercept of 𝑔’s graph is greater than the 𝑦-intercept of ℎ’s graph. They also show that 𝑔 has a smaller growth factor than ℎ because as 𝑥 increases by the same amount, 𝑔 retains a smaller fraction of its value compared to ℎ. This suggests that 𝑔 corresponds to pond B, and ℎ corresponds to pond A.

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Practice Problems 1. Here are equations defining 3 exponential functions, 𝑓, 𝑔, and ℎ. 𝑔(𝑥) = 100 ⋅ (3.5)𝑥

𝑓(𝑥) = 100 ⋅ 3𝑥

ℎ(𝑥) = 100 ⋅ 4𝑥

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a. Which of these functions grows the most slowly? Which grows the most quickly? Explain how you know.

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b. The graphs of 𝑓, 𝑔, and ℎ are shown on the coordinate plane.

Complete the table by writing which graph corresponds to each function. Function

Graph

𝑓(𝑥) = 100 ⋅ 3𝑥

𝑔(𝑥) = 100 ⋅ (3.5)𝑥 ℎ(𝑥) = 100 ⋅ 4𝑥

c. Why do all 3 graphs share the same 𝑦-intercept?

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𝑥

2. The function 𝑓 is given by 𝑓(𝑥) = 160 ⋅ � 5 � and the 1

𝑥

function 𝑔 is given by 𝑔(𝑥) = 160 ⋅ � 5 � . The graph of

𝑓 is labeled A, and the graph of 𝑔 is labeled 𝐶.

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If 𝐵 is the graph of ℎ and ℎ is defined by ℎ(𝑥) = 𝑎 ⋅ 𝑏𝑥, what can you say about 𝑎 and 𝑏? Explain your reasoning. 3. Here is a graph of the function 𝑓 defined by 𝑓(𝑥) = 𝑎 ⋅ 𝑏𝑥. Select all possible values of 𝑏.

□0

□ 101

□ 1.3 □ 185

□1

□ 109

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□ 12

4. The function 𝑓 is defined by 𝑓(𝑥) = 50 ⋅ 3𝑥. The function 𝑔 is defined by 𝑔(𝑥) = 𝑎 ⋅ 𝑏𝑥.

Here are graphs of 𝑓 and 𝑔.

a. How does 𝑎 compare to 50? Explain how you know.

b. How does 𝑏 compare to 3? Explain how you know.

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Review Problems 5. Technology required. Start with a square with area 1 square unit (sq. unit), not shown. Subdivide it into 9 squares of equal area and remove the middle one to get the first figure shown.

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a. What is the area of the first figure shown?

b. Take the remaining 8 squares, subdivide each into 9 equal squares, and remove the middle one from each. What is the area of the figure now?

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c. Continue the process and find the area for stages 3 and 4.

d. Write an equation representing the area 𝐴 at stage 𝑛.

e. Use technology to graph your equation.

f. Use your graph to find the first stage when the area is first less than 2 sq. unit.

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6. The equation 𝑏 = 500 ⋅ (1.05)𝑡 gives the balance of a bank account 𝑡 years since the account was opened. The graph shows the annual account balance for 10 years.

a. What is the average annual rate of change for the bank account?

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b. Is the average rate of change a good measure of how the bank account varies? Explain your reasoning.

7. Technology required. The equation 𝑦 = 600,000 ⋅ (1.055)𝑡 represents the population of a country 𝑡 decades after the year 2000.

Use graphing technology to graph the equation. Then, set the graphing window so that you can simultaneously see points on the graph representing the population predicted by the model in 1980 and in the year 2020. What graphing window did you use?

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Unit 6, Lesson 11: Graphing Exponential Functions

Warm-Up: Rhind Papyrus

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Some of the earliest records of mathematical problem solving come from ancient Egypt in the form of papyri that were written between 1850 and 1600 BCE. One of these, the Rhind Mathematical Papyrus, is a collection of math problems written around 1650 BCE. The problems in the papyrus were used to manage food supplies for cities, among other things.

One of the problems asks this question: “In a certain village, there were 7 houses; each house had 7 cats; each cat caught 7 mice; each mouse would have (were it not for the cats) eaten 7 ears of spelt; each ear of spelt produced 7 hekats of grain at harvest. How many hekats of grain were saved by the presence of the cats?”

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1. Find the solution to the Rhind Papyrus problem.

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Guided Activity: Graphing an Exponential Equation When graphing an exponential function, knowing the key features and using a table of values can be helpful. 6

𝑥

1. Explain which key features can be determined from the function 𝐶(𝑥) = 5 � 5 � . Explanation

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Key Feature

Growth or decay Domain and range

Increasing/decreasing interval

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End behavior

𝑥

−2

1 2 3

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−1

𝑪(𝒙)

𝒙

2. Complete the table for 𝐶(𝑥) = 5 � 5 � . Then, graph the function.

-10

-5

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a. Discuss with your partner how the graph would change if the 𝑎-value of 5 was changed to − 5. 6

𝑥

b. Draw a sketch of 𝐶(𝑥) = −5 � 5 � .

-5

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-10

-5

-10

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-15

c. Discuss with your partner how the graph would change if the 𝑏-value was 6

changed from 5 to 6 .

𝑥

d. Draw a sketch of 𝐶(𝑥) = 5 � 6 � .

-5

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Collaborative Activity: Graphing Exponential Functions 1. Consider the function 𝑓(𝑥) = 3(0.68)𝑥.

a. Work with your partner to complete the statements about 𝑓(𝑥) = 3(0.68)𝑥.

increasing. decreasing.

the function

The left end behavior of the graph is

the right end behavior is

and

The 𝑦-intercept of the

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0.32 0.62 3.

𝑦 → 0 𝑦 → −∞.

𝑦 → 0 𝑦 → ∞

function is

between 0 and 1, greater than 1,

and 𝑏 is

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positive negative

Since 𝑎 is

b. Sketch the graph of 𝑓(𝑥) = 3(0.68)𝑥. y

5 4

3 2 1

-2

-1

-1 -2

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2. Consider the function 𝑓(𝑥) = − (1.4)𝑥.

a. Work with your partner to complete the statements for 𝑓(𝑥) = − (1.4)𝑥.

positive negative

increasing. decreasing.

the function

𝑦 → 0 𝑦 → ∞

The left end behavior of the graph is

𝑦 → 0 𝑦 → −∞.

right end behavior is

The 𝑦-intercept of the function

0.4. − 1. 1.04.

and the

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between 0 and 1, greater than 1,

and 𝑏 is

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Since 𝑎 is

b. Sketch the graph of 𝑓(𝑥) = − (1.4)𝑥. y

-1

-2

-1 -2 -3 -4 -5

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Collaborative Activity: Card Sort Work with your partner to complete the card sort. Fill in the table with the correct card labels that correspond to each graph. Card Sort

1.1

𝒚-intercept

Value of 𝒂 and 𝒃

Table

1.2

Function

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Graph

1.3

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1.4 1.5

1.6

Lesson Summary

An equation defining an exponential function has the form 𝑦 = 𝑎(𝑏)𝑥. In addition to the 𝑎-value representing the 𝑦-intercept of the graph, it also determines whether the graph is reflected over the 𝑥-axis. A reflection of the function over the 𝑥-axis occurs when 𝑎 < 0. The value of 𝑎 in conjunction with the value of 𝑏 determines the end behavior of an exponential function. The graphs of 1

𝑥

𝑓(𝑥) = 3(2)𝑥, 𝑔(𝑥) = 3� 2 � , and ℎ(𝑥) = −3 � 2 �

𝑥

are shown. Notice how the change in 𝑏 from 2 1

to 2 and the change in 𝑎 from 3 to − 3 transform

the end behavior of each function.

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To sketch the graph of an exponential function, the equation that represents the function, along with knowledge of how the values of 𝑎 and 𝑏 affect the shape and end behavior of the function, can be used to create a table of values. All these representations can be combined to identify key features and answer other questions about exponential functions.

Practice Problems

𝒚

𝒙

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1. Graph the exponential equation 𝑦 = 45(0.95)𝑥. If necessary, complete a table of values.

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2. Determine all the key features of the graph of 𝑦 = 45(0.95)𝑥.

𝑥-intercept(s)

Key Features Increasing interval(s)

𝑦-intercept

Decreasing interval(s)

Domain

Left end behavior

Range

Right end behavior

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As 𝑥→_____, 𝑦→_____.

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Review Problem

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3. The graph shows a population of butterflies, 𝑡 weeks since their migration began.

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a. How many butterflies were in the population when they started the migration? Explain how you know.

b. How many butterflies were in the population after 1 week? What about after 2 weeks?

c. Write an equation for the population, 𝑞, after 𝑡 weeks.

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Unit 6, Lesson 12: Solving One-Variable Exponential Equations

Warm-Up: Visual Patterns

Step 2

Step 3

Step 4

Step 1

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1. The first 4 steps in a pattern are shown.

a. Describe how the pattern changes in each step.

b. How many purple circles will be in step 5?

c. Draw step 10 of the pattern.

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Guided Activity: Finding Solutions of Exponential Equations 1. Use graphing technology to graph the functions 𝑓(𝑥) = 4 and 𝑔(𝑥) = 1 (10)𝑥. 25

a. Sketch the graphs of the functions. Label the point of intersection.

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b. Discuss with your partner what the point of intersection represents.

2. Discuss with your partner how to solve 1 (10)𝑥 = 4 without graphing technology but, 25

instead, using algebra. Summarize your discussion.

𝑥

3. Solve 8 � 2 � = 27 without a graphing tool.

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4. Consider the equation 4𝑥 = 2(𝑥 + 1).

a. Use graphing technology to solve 4𝑥 = 2(𝑥 + 1). Sketch the graph, and label the point of intersection.

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b. Verify that the 𝑥-value of the point of intersection is a solution to 4𝑥 = 2(𝑥 + 1). Show your work.

c. Complete the steps for solving the original equation algebraically. Description

4𝑥 = 2(𝑥 + 1)

Given equation

Equivalent Equation

(22)𝑥 = 2(𝑥 + 1)

Rewrite 1 or both expressions so that both have the same base. Apply the power of a power exponent law.

If both expressions have the same base, set the exponents equal to each other to create a new equation. Solve for 𝑥.

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Collaborative Activity: Solving Exponential Equations 1

𝑥

1. The graphs of 𝑏(𝑥) = � 3 � and 𝑤(𝑥) = 3𝑥 + 2 are shown on the coordinate grid.

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a. What is the solution to the equation 𝑏(𝑥) = 𝑤(𝑥)? 1

𝑥

b. Rewrite the equation � 3 � = 3𝑥 + 2 with a common base.

c. Write a new equation that can be used to solve for 𝑥.

d. Verify algebraically that your solutions to parts A and B satisfy the 1

𝑥

equation � 3 � = 3𝑥 + 2.

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2. Use graphing technology to graph 𝑝(𝑥) = 5 ∙ 2(3𝑥 + 5) and 𝑟(𝑥) = 20.

a. Sketch the graphs of 𝑝(𝑥) and 𝑟(𝑥) on the same coordinate plane.

ip t

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b. Rewrite 20 = 5 ∙ 2(3𝑥 + 5) with a common base.

c. Write a new equation that can be used to solve for 𝑥.

d. Verify algebraically that the solution satisfies the equation 20 = 5 ∙ 2(3𝑥 + 5).

3. Solve 3 ∙ 4(𝑥 + 6) = 768 without graphing technology.

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Lesson Summary To solve a one-variable exponential equation without using a graphing tool, determine if the expressions on each side of the equation can be rewritten as powers of the same base. Then, use the laws of exponents to simplify the exponents. Because the bases are the same, the exponents are equivalent. Use this fact to write a new equation where the exponents are set equal to each other. Solve the equation and determine the value of the variable by using the properties of operations and properties of equality.

op y

An example is shown. Solve 27𝑥 = 243𝑥 + 1 without technology. 243𝑥 + 1

33𝑥 =

35𝑥 + 5

Power of a power property

Subtraction property of equality

𝑥 =

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− 2𝑥 =

When the same base is raised to an exponent, the exponents are equal.

5𝑥 + 5 5

− 2

3𝑥 =

Rewrite both sides as a power of the same base.

(35)𝑥 + 1

Division property of equality

(33)𝑥 =

27𝑥 =

The solution can be verified using graphing technology or substituting the result into the original equation.

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Practice Problems 1. Use graphing technology to graph 𝑞(𝑥) = 81 and 𝑘(𝑥) = 33𝑥 − 1. a. Sketch the graphs of 𝑞(𝑥) and 𝑘(𝑥) on the same coordinate plane. Label the solution of 𝑞(𝑥) = 𝑘(𝑥).

op y

ip t

b. Rewrite 81 = 3(3𝑥 − 1) with a common base.

c. Write and solve a new equation to determine the value of 𝑥 algebraically.

2. Solve 64 = 4 (2)𝑥 + 3 without a calculator.

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Review Problems 3. Here are the equations that define three functions. 𝑓(𝑥) = 4𝑥 − 5

𝑔(𝑥) = 4(𝑥 − 5)

𝑥

ℎ(𝑥) = 4 − 5

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a. Which function value is the largest: 𝑓(100), 𝑔(100), or ℎ(100)?

1 1 1 �, 𝑔� �, or ℎ� �? 100 100 100

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c. Which function value is the largest: 𝑓�

b. Which function value is the largest: 𝑓(−100), 𝑔(−100), or ℎ(−100)?

4. Technology required. A moth population, 𝑝, is modeled by the equation 1

𝑤

measured.

𝑝 = 500,000 ⋅ � 2 � , where 𝑤 is the number of weeks since the population was first

a. What was the moth population when it was first measured?

b. What was the moth population after 1 week? What about 1.5 weeks?

c. Use technology to graph the population and find out when it falls below 10,000.

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Unit 6, Lesson 13: Function Involving Percent Change

Warm-Up: Expressing Percent Increase and Decrease 1. There is an 8% tax on a car priced at $12,000.

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Linda and Jono wrote different expressions to determine the total cost of the car. Linda’s Expression

Jono’s Expression

12,000 + 12,000(0.08)

12,000(1.08)

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Explain whose expression is correct, if either.

2. Complete the table so that each description has 2 different expressions that can be used to answer the question asked. The second expression should use only multiplication. Be prepared to explain how the 2 expressions are equivalent.

Description

A one-night stay at a hotel in Anaheim, California, costs $160. Hotel room occupancy tax is 15%. What is the total cost of a one-night stay?

Teachers receive a 30% educators’ discount at a museum. An adult ticket costs $24. How much would a teacher pay for admission to the museum?

Expression 1

Expression 2 (using only multiplication)

160 + (0.15) ⋅ 160

(0.7) ⋅ 24

After a major hurricane, 46% of the 90,500 households on an island lost their access to electricity. How many households still have electricity?

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Guided Activity: Owing Interests To get a new computer, a recent college graduate obtains a loan of $450. She agrees to pay 18% annual interest, which will apply to any money she owes. She makes no payments during the first year.

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1. How much will she owe at the end of one year? Show your reasoning.

2. Assuming she continues to make no payments to the lender, how much will she owe at the end of two years? Three years?

3. To find the amount owed at the end of the third year, a student started by writing:

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[Year 3 Amount] = [Year 2 Amount] + [Year 2 Amount] ⋅ (0.18) and ended with = 450 ⋅ (1.18) ⋅ (1.18) ⋅ (1.18).

Does her final expression correctly reflect the amount owed at the end of the third year? Explain or show your reasoning.

4. Write an expression for the amount she owes at the end of 𝑥 years without payment.

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Exploration Activity: Comparing Loans Suppose three people each have taken loans of $1,000 but they each pay different annual interest rates. 1. For each loan, write an expression, using only multiplication, for the amount owed at the end of each year if no payments are made.

Loan A 𝟭𝟮% %

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10 𝑥

Loan C 𝟯𝟬.𝟲% %

Loan B 𝟮𝟰% %

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Years without Payment

2. Use graphing technology to plot the graphs of the account balances.

3. Based on your graph, about how many years would it take for the original unpaid balance of each loan to double?

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Collaborative Activity: Comparing Average Rates of Change The functions 𝑎, 𝑏, and 𝑐 represent the amount owed (in dollars) for Loans A, B, and C, respectively; the input for the functions is 𝑡, the number of years without payments. 1. For each loan, find the average rate of change per year between:

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a. the start of the loan and the end of the second year

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b. the end of the tenth year and the end of the twelfth year

2. How do the average rates of change for the three loans compare in each of the two-year intervals?

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Lesson Summary This lesson introduces another form of an exponential function, 𝑦 = 𝑎(1±𝑟)𝑥, where 𝑟 is the constant percent rate of change. In the form 𝑦 = 𝑎(𝑏)𝑥 that you have been using throughout the unit, 𝑏 = (1 + 𝑟) or 𝑏 = (1 − 𝑟), depending on whether it is exponential growth or exponential decay.

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When money is borrowed from a lender, the lender usually charges interest, a percentage of the borrowed amount as payment for allowing use of the money. The interest is usually calculated at a regular interval of time (monthly, yearly, etc.).

Suppose a loan is taken out for $500, and the interest rate is 15%, calculated at the end of each year. If no other purchases or payments are made, the amount owed after 1 year would be 500 + (0.15) ⋅ 500, or 500 ⋅ (1 + 0.15). If no payments or other purchases are made in the second year, the amount owed would increase by another 15%. The table shows the calculation of the amount owed for the first 3 years.

500 ⋅ (1 + 0.15) (1 + 0.15), or 500 ⋅ (1 + 0.15)2

500 ⋅ (1 + 0.15)

500 ⋅ (1 + 0.15) (1 + 0.15) (1 + 0.15), or 500 ⋅ (1 + 0.15)3

Amount Owed (dollars)

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Time (years)

The pattern in the table continues. Each additional year means multiplication by another factor of (1 + 0.15). If no further purchases or payments are made, after 𝑡 years, the debt in dollars is given by the expression 500 ⋅ (1 + 0.15)𝑡. To determine the amount owed after 𝑡 years, use the equation 𝑦 = 500(1.5)𝑡.

Because exponential functions eventually grow very quickly, leaving a debt unpaid can be very costly.

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Practice Problems 1. In 2011, the population of deer in a forest was 650.

a. In 2012, the population increases by 15%. Write an expression, using only multiplication, that represents the deer population in 2012.

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b. In 2013, the population increases again by 15%. Write an expression that represents the deer population in 2013.

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c. If the deer population continues to increase by 15% each year, write an expression that represents the deer population 𝑡 years after 2011.

2. Mai and Elena are shopping for back-to-school clothes. They found a skirt that originally cost $30 on a 15% off sale rack. Today, the store is offering an additional 15% off. To find the new price of the skirt, in dollars, Mai says they need to calculate 30 ⋅ 0.85 ⋅ 0.85. Elena says they can just multiply 30 ⋅ 0.70.

a. How much will the skirt cost using Mai’s method?

b. How much will the skirt cost using Elena’s method?

c. Explain why the expressions used by Mai and Elena give different prices for the skirt. Which method is correct?

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3. Technology required. One $1,000 loan charges 5% interest at the end of each year, 𝑏, while a second loan charges 8% interest at the end of each year, 𝑐. Number of years 1

Loan balance with % interest 𝟱%

Loan balance with 𝟴% % interest

op y

3 𝑡

a. Complete the table with the balances for each loan. Assume that no payments are made and that the interest applies to the entire loan balance, including any previous interest charges.

ip t

b. Which loan balance grows more quickly? How will this be visible in the graphs of the two loan balances 𝑏 and 𝑐 as functions of the number of years 𝑡?

c. Use technology to create graphs representing 𝑏 and 𝑐 over time. The graph should show the starting balance of each loan as well as the amount of the loan after 15 years. Write down the graphing window needed to show these points. 1

4. Lin opened a savings account that pays 5 4 % interest annually and deposited $5,000. If she makes no deposits and no withdrawals for 3 years, how much money will be in her account?

5. A person loans his friend $500. They agree to an annual interest rate of 5%. Write an expression for computing the amount owed on the loan, in dollars, after 𝑡 years if no payments are made.

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Review Problems 6. Select all situations that are accurately described by the expression 15 ⋅ 35.

□ A population of bacteria begins at 15,000. The population triples each hour (hr.). How many bacteria are there after 5 hr.?

□ A population of bacteria begins at 15,000. The population triples each hr. How many thousand bacteria are there after 5 hr.?

op y

□ A population of bacteria begins at 15,000. The population quintuples each hour. How many thousand bacteria are there after 3 hr.?

□ A bank account balance is $15. The account balance triples each year. What is the bank account balance, in dollars, after 5 years?

□ A bank account balance is $15,000. It grows by $3,000 each year. What is the bank

ip t

account balance, in thousands of dollars, after 5 years?

7. Here are graphs of two exponential functions, 𝑓 and 𝑔. 2

𝑥

A. 3 3

B. 4

the value of 𝑏?

If 𝑓(𝑥) = 100 ⋅ � 3 � and 𝑔(𝑥) = 100 ⋅ 𝑏𝑥, what could be

C. 1

D. 2

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Unit 6, Lesson 14: Interest Compounding at Different Intervals

Warm-Up: Resizing Images

op y

Andre and Mai need to enlarge two images for a group project. The two images are the same size. Andre makes a scaled copy of his image, increasing the lengths by 10%. It was still a little too small, so he increases the lengths by 10% again.

Mai says, “If I scale my image and increase the lengths by 20%, our images will be exactly the same size.”

ip t

Do you agree with Mai? Explain or show your reasoning.

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Guided Activity: Contemplating Credit Cards A credit card company lists a nominal APR (annual percentage rate) of 24% but compounds interest monthly, so it calculates 2% per month.

op y

Suppose a cardholder made $1,000 worth of purchases using his credit card and made no payments or other purchases. Assume the credit card company does not charge any additional fees other than the interest.

ip t

1. Write expressions for the balance on the card after 1 month, 2 months, 6 months, and 1 year.

2. Write an expression for the balance on the card, in dollars, after 𝑚 months without payment.

3. How much does the cardholder owe after 1 year without payment? What is the effective APR of this credit card?

4. Write an expression for the balance on the card, in dollars, after 𝑡 years without payment. Be prepared to explain your expression.

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Exploration Activity: Which One Would You Choose? Suppose you have $500 to invest and can choose between two investment options. • Option 1: every 3 months 3% interest is applied to the balance

op y

• Option 2: every 4 months 4% interest is applied to the balance

ip t

Which option would you choose? Build a mathematical model for each investment option and use them to support your investment decision. Remember to state your assumptions about the situation.

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Collaborative Activity: Changes over the Years 1. The function 𝑓 defined by 𝑓(𝑥) = 15 ⋅ (1.07)𝑥 models the cost of tuition, in thousands of dollars, at a local college 𝑥 years since 2017.

op y

a. What is the cost of tuition at the college in 2017?

b. At what annual percentage rate does the tuition grow?

ip t

c. Assume that before 2017 the tuition had also been growing at the same rate as after 2017. What was the tuition in 2000? Show your reasoning.

d. What was the tuition in 2010?

e. What will the tuition be when you graduate from high school?

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2. Between 2000 and 2010 the tuition nearly doubled.

op y

a. By what factor will the tuition grow between 2017 and 2027? Show your reasoning.

ip t

b. Choose another 10-year period and find the factor by which the tuition grows. Show your reasoning.

c. What can you say about how the tuition changes over any 10-year period (assuming the function 𝑓 continues to be an accurate model)? Explain or show how you know that this will always be the case.

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Lesson Summary

1 2

Account Balance ($ $) 300 ⋅ (1.02)

300 ⋅ (1.02)2

(1.02)12 ≈ 1.2682, so the account will grow by about 26.82% in 1 year. This rate is called the effective interest rate. It reflects how the account balance actually changes after 1 year.

300 ⋅ (1.02)3

300 ⋅ (1.02)12

•

Number of Months

op y

Applying a percent increase on an amount that has had a prior percent increase is called compounding. Compounding happens when interest is calculated on money in a bank account or on a loan. An account that earns 2% interest every month does not actually earn 24% a year. For example, suppose a savings account has $300, and no other deposits or withdrawals are made. The account balances after some months are shown in the table.

ip t

• The 24% is called the nominal interest rate. It is the stated or published rate and is usually used to determine the monthly, weekly, or daily rates.

In many situations, interest is calculated more frequently than once a year. How often the interest is compounded, calculated, and added to the previous amount affects the overall amount of interest earned or owed over time.

Suppose a bank account has a balance of $1,000 and a nominal annual interest rate of 6% per year. No additional deposits or withdrawals are made.

If the bank compounds interest annually, the account will have 1 interest calculation in 1 year at a 6% rate. If it compounds interest every 6 months, the account will see 2 interest calculations in 1 year at a 3% rate each time (because 6÷2 = 3). If it is compounded every 3 months, there will be 4 calculations at 1.5% each time, and so on. The table shows the nominal interest rates used for different compounding intervals and the corresponding expressions for the account balance in 1 year.

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Compounding Frequency per Year

Nominal Interest Rate

Account Balance in 1 Year

1 time

1,000 ⋅ (1 + 0.06)

2 times

1,000 ⋅ (1 + 0.03)2

4 times

1.5%

Annually (12 months) Semi-annually (6 months) Quarterly (3 months) Monthly

op y

Compounding Interval

12 times

0.5%

□ Annually: 1,000 ⋅ (1.06) = $1,060

1,000 ⋅ (1 + 0.005)12

The account balances for each compounding interval are shown.

1,000 ⋅ (1 + 0.015)4

□ Semi-annually: 1,000 ⋅ (1.03) = $1,060.90

ip t

□ Quarterly: 1,000 ⋅ (1.015) ≈ $1,061.36 4

□ Monthly: 1,000 ⋅ (1.005) ≈ $1,061.68 12

Notice that the more frequently interest is calculated, the greater the balance becomes.

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Practice Problems 1. The population of a city in 2010 is 50,000, and it grows by 5% each year after.

a. Write a function 𝑓 which models the population of the city 𝑡 years after 2010.

op y

b. What is the population of the city in 2017?

c. What will the population of the city be in 2020? What about in 2030?

ip t

d. By what factor does the population grow between 2010 and 2020? What about between 2020 and 2030?

2. A person charges $100 to a credit card with a 24% nominal annual interest rate.

a. annually

Assuming no other charges or payments are made, find the balance on the card, in dollars, after 1 year if interest is calculated:

b. every 6 months c. every 3 months d. monthly e. daily

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3. A couple has $5,000 to invest and has to choose between three investment options. 1

• Option A: 2 4 % interest applied each quarter

• Option B: 3% interest applied every 4 months 1

• Option C: 4 2 % interest applied twice each year

op y

If they plan on no deposits and no withdrawals for 5 years, which option will give them the largest balance after 5 years? Use a mathematical model for each option to explain your choice.

ip t

4. Elena says that 6% interest applied semi-annually is the same as 1% interest applied every month: she reasons they are the same because they are both a 12% nominal annual interest rate.

a. Is Elena correct that these two situations both offer a 12% nominal annual interest rate?

b. Is Elena correct that the two situations pay the same amount of interest?

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5. A bank pays 8% nominal annual interest, compounded at the end of each month. An account starts with $600, and no further withdrawals or deposits are made. a. What is the monthly interest rate?

op y

b. Write an expression for the account balance, in dollars, after one year.

ip t

c. What is the effective annual interest rate?

Review Problems

d. Write an expression for the account balance, in dollars, after 𝑡 years.

6. At the end of each year, 10% interest is charged on a $500 loan. The interest applies to any unpaid balance on the loan, including previous interest.

Select all the expressions that represent the loan balance after two years if no payments are made.

□ 500 + 2 ⋅ (0.1) ⋅ 500 □ 500 ⋅ (1.1) ⋅ (1.1)

□ 500 + (0.1) + (0.1) □ 500 ⋅ (1.1)

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op y

7. Here is a graph of the function 𝑓 given by 𝑓(𝑥) = 100 ⋅ 2𝑥.

Suppose 𝑔 is the function given by 𝑔(𝑥) = 50 ⋅ (1.5)𝑥.

ip t

Will the graph of 𝑔 meet the graph of 𝑓 for any positive value of 𝑥? Explain how you know.

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Unit 6, Lesson 15: Expressed in Different Ways

Warm-Up: Equal Expressions

op y

Decide if each expression is equal to (1.21)100. ((1.21)10)10

((1.21)50)50

(1.1)200

ip t

((1.1)2)100

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Guided Activity: Population Projections 1. From 1790 to 1860, the United States population, in thousands, is modeled by the equation 𝑃 = 4,000 ⋅ (1.031)𝑡, where 𝑡 is the number of years since 1790.

op y

a. About how many people were living in the U.S. in 1790? What about in 1860? Show your reasoning.

b. What is the approximate annual percent increase predicted by the model?

ip t

c. What does the model predict for the population in 2017? Is it accurate? Explain.

a. What percent increase does the model predict each decade? Explain.

b. Suppose 𝑑 represents the number of decades since 1790. Write an equation for 𝑃 in terms of 𝑑 modeling the population in the US (in thousands).

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3. a. What percent increase does the model predict each century? Explain.

op y

b. Suppose 𝑐 represents the number of centuries since 1790. Write an equation for 𝑃 in terms of 𝑐 modeling the population in the United States (in thousands).

Collaborative Activity: Interest Calculations

1. Three expressions and 3 descriptions are shown. In each case, $1,000 has been put in an interest-bearing bank account. No withdrawals or other deposits (aside from the earned interest) are made for 6 years.

ip t

a. Match each expression with the description that represents the same amount of interest. More than 1 expression matches 1 of the descriptions. One description has no match.

Expression 0.07

a. 1,000 ∙ �1 + 12 �

0.07 12 2 �

b. 1,000 ∙ �1 +

7% annual interest compounded semi-annually 7% annual interest compounded monthly 7% annual interest compounded every 2 months

0.07 12 c. 1,000 ∙ ��1 + 12 � �

Description

b. Write an expression for the description that does not have a match.

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Lesson Summary Expressions can be written in different ways to highlight various aspects of a situation or to understand it better. A growth rate in an exponential function represents the constant percent rate of change.

op y

In an exponential function, the growth rate is the fraction or percentage of the output that gets added every time the input is increased by one. As always, in percent change situations, it is important to know if the context describes a change that is an increase or decrease. Some examples are listed.

• A population is increasing by 20% each year. The growth rate is 20%, so after 1 year, 0.2 times the population at the beginning of that year has been added. If the initial population is 𝑝, the new population is 𝑝 + 0.2𝑝, which equals (1 + 0.2)𝑝 or 1.2𝑝.

ip t

• A population is decreasing by 20% each year. The growth rate is − 20%, so after 1 year, 0.2 times the population at the beginning of that year has been lost. If the initial population is 𝑝, the new population is 𝑝 − 0.2𝑝, which equals (1 − 0.2)𝑝 or 0.8𝑝.

Suppose the area, 𝑎, covered by a forest is currently 50 square miles (sq. mi.), and it is growing by 0.2% each year. If 𝑡 represents time from now, in years, the area of the forest can be expressed as 𝑎 = 50 ⋅ (1 + 0.002)𝑡 or 𝑎 = 50 ⋅ (1.002)𝑡.

In this situation, the growth rate is 0.002, and the growth factor is 1.002. However, 0.002 is such a small number that it may be difficult to tell from this function how quickly the forest is growing. It might be more meaningful to measure the growth every decade or every century.

• There are 10 years in a decade, so to find the growth rate in decades, use the expression (1.002)10, which is approximately 1.02. This means a growth rate of about 2% per decade. Using 𝑑 for time from now in decades, the area of the forest can be expressed as 𝑎 = 50 ⋅ ((1 + 0.002)10)𝑑 or 𝑎 ≈ 50 ⋅ (1.02)𝑑.

• If time is measured in centuries, the growth rate is about 22% per century because 1.002100 ≈ 1.22. Using 𝑐 for time from now in centuries, the equation for area now becomes 𝑎 = 50 ⋅ ((1 + 0.002)100)𝑐 or 𝑎 ≈ 50 ⋅ (1.22)𝑐.

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Practice Problems 1. For each growth rate, find the associated growth factor. e. 0.04% increase

b. 30% decrease

f. 0.04% decrease

c. 2% increase

g. 100% increase

op y

a. 30% increase

d. 2% decrease

ip t

2. In 1990, the population 𝑝 of India was about 870.5 million people. By 1995, there were about 960.9 million people. The equation 𝑝 = 870.5 ⋅ (1.021)𝑡 approximates the number of people, in millions, in terms of the number of years 𝑡 since 1990.

a. By what factor does the number of people grow in one year?

b. If 𝑑 is time in decades, write an equation expressing the number of people in millions, 𝑝, in terms of decades, 𝑑, since 1990.

c. Use the model 𝑝 = 870.5 ⋅ (1.021)𝑡 to predict the number of people in India in 2015. d. The 2015, the population of India was 1,311 million. How does this compare with the predicted number?

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3. An investor paid $156,000 for a condominium in Texas in 2008. The value of the homes in the neighborhood have been appreciating by about 12% annually.

Select all the expressions that could be used to calculate the value of the house, in dollars, after 𝑡 years.

□ 156,000 ⋅ (0.12)

𝑡

□ 156,000 ⋅ (1.12)

𝑡

□ 156,000 ⋅ (1 + 0.12) □ 156,000 ⋅ (1 − 0.12)

op y

𝑡 𝑡 𝑡

□ 156,000 ⋅ �1 + 0.12 12 �

ip t

4. A credit card has a nominal annual interest rate of 18%, and interest is compounded monthly. The cardholder uses the card to make a $30 purchase.

a. 30 (1 + 18)5

b. 30(1 + 0.18)5

Which expression represents the balance on the card after 5 years, in dollars, assuming no further charges or payments are made?

0.18 5

c. 30 �1 + 12 �

0.18 5·12

d. 30�1 + 12 �

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5. The expression 1,500 ⋅ (1.085)3 represents an account balance in dollars after three years with an initial deposit of $1,500. The account pays 8.5% interest, compounded annually for three years.

op y

a. Explain how the expression would change if the bank had compounded the interest quarterly for the three years.

b. Write a new expression to represent the account balance, in dollars, if interest is compounded quarterly.

Review Problems

ip t

6. The function, 𝑓, defined by 𝑓(𝑡) = 1,000 ⋅ (1.07)𝑡, represents the amount of money in a bank account 𝑡 years after it was opened.

a. How much money was in the account when it was opened?

b. Sketch a graph of 𝑓.

c. When does the account value reach $2,000?

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7. The graph shows the number of patients with an infectious disease over a period of 15 weeks.

op y

a. Give an example of a domain for which the average rate of change is a good measure of how the function changes.

ip t

b. Give an example of a domain for which the average rate of change is not a good measure of how the function changes.

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Unit 6, Lesson 16: Which One Changes Faster?

Warm-Up: Graph of Which Functions?

op y

Here is a graph.

𝑦 = 120 ⋅ (1.03)𝑥

𝑦 = 120 + (3.7) ⋅ 𝑥

ip t

1. Which equation do you think the graph represents? Use the graph to support your reasoning.

2. What information might help you decide more easily whether the graph represents a linear or an exponential function?

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Exploration Activity: Simple and Compound Interests A family has $1,000 to invest and is considering two options: investing in government bonds that offer 2% simple interest, or investing in a savings account at a bank, which charges a $20 fee to open an account and pays 2% compound interest. Both options pay interest annually.

op y

Here are two tables showing what they would earn in the first couple of years if they do not invest additional amounts or withdraw any money. Bond

Savings Account

$1,000

$1,040

$1,020

0 1 2

Amount (dollars) $980

$999.60

$1,019.59

Years of Investment

Amount (dollars)

ip t

Years of Investment

1. Bonds: How does the investment grow with simple interest?

2. Savings account: How are the amounts $999.60 and $1,019.59 calculated?

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3. For each option, write an equation to represent the relationship between the amount of money and the number of years of investment.

op y

4. Which investment option should the family choose? Use your equations or calculations to support your answer.

ip t

5. Use graphing technology to graph the two investment options and show how the money grows in each.

Collaborative Activity: Reaching 2,000

1. Complete the table of values for the functions 𝑓(𝑥) = 2𝑥 and 𝑔(𝑥) = (1.01)𝑥.

𝒙 1

𝒇�𝒙�

𝒈�𝒙�

100 500

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2. Based on the table of values, which function do you think grows faster? Explain your reasoning.

Lesson Summary

ip t

op y

3. Which function do you think will reach a value of 2,000 first? Show your reasoning. If you get stuck, consider increasing 𝑥 by 100 a few times and record the function values in the table.

The top prize from a game show offers 2 options. The first option is a cash gift of $10,000 plus another $1,000 per day for the next 7days. The second option is a cash gift of 1 cent (or $0.01) that grows tenfold each day for 7 days. Which option would you choose?

• In the first option, the amount of money increases by the same amount, $1,000, each day, so it can be represented with a linear function. • In the second option, the money grows by multiples of 10, so it can be represented with an exponential function.

Let 𝑓 represent the amount of money 𝑥 days after winning with the first option, and let 𝑔 represent the amount of money 𝑥 days after winning with the second option.

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Option 1: 𝑓(𝑥) = 10,000 + 1,000𝑥

Option 2: 𝑔(𝑥) = (0.01) ⋅ 10𝑥

...

𝑓(1) = 11,000 𝑓(2) = 12,000 𝑓(3) = 13,000

𝑔(1) = 0.1 𝑔(2) = 1 𝑔(3) = 10

𝑔(6) = 10,000 𝑔(7) = 100,000

op y

𝑓(6) = 16,000 𝑓(7) = 17,000

For the first few days, the second option trails far behind the first. Because of the repeated multiplication by 10, however, it surges past the amount in the first option after 7 days.

ip t

What if the factor of growth is much smaller than 10? Suppose there is a third option, represented by a function ℎ. The starting amount is still $0.01, and it grows by a factor of 1.5 each day. The graph of the function ℎ(𝑥) = (0.01) ⋅ (1.5)𝑥 shows that it takes many more days before it starts to grow rapidly. But given time to continue growing, the amount in this exponential option will eventually also outpace that in the linear option.

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Practice Problems 1. Functions 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, and 𝑓 are given below. Classify each function as linear, exponential, or neither. d. 𝑑(𝑥) = 9 + 3𝑥

b. 𝑏(𝑥) = 3𝑥

e. 𝑒(𝑥) = 9 ⋅ 3𝑥

c. 𝑐(𝑥) = 𝑥3

f. 𝑓(𝑥) = 9 ⋅ 3𝑥

ip t

op y

a. 𝑎(𝑥) = 3𝑥

𝑏(𝑥) = 3𝑥 + 5

𝑐(𝑥) = 𝑥 + 4

𝑑(𝑥) = 1 + 4𝑥

𝑎(𝑥) = 5𝑥 + 3

2. Here are 4 equations defining 4 different functions, 𝑎, 𝑏, 𝑐, and 𝑑. List them in order of increasing rate of change. That is, start with the one that grows the slowest and end with the one that grows the quickest.

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3. Technology required. Function 𝑓 is defined by 𝑓(𝑥) = 3𝑥 + 5 and function 𝑔 is defined by 𝑔(𝑥) = (1.1)𝑥. a. Complete the table with values of 𝑓(𝑥) and 𝑔(𝑥). When necessary, round to 2 decimal places.

𝒙

𝒇�𝒙�

𝒈�𝒙�

5 10

op y

b. Which function do you think grows faster? Explain your reasoning.

ip t

c. Use technology to create graphs representing 𝑓 and 𝑔. What graphing window do you have to use to see the value of 𝑥 where 𝑔 becomes greater than 𝑓 for that 𝑥?

4. Functions 𝑚 and 𝑛 are given by 𝑚(𝑥) = (1.05)𝑥 and 𝑛(𝑥) = 5 𝑥. As 𝑥 increases from 0:

a. Which function reaches 30 first?

b. Which function reaches 100 first?

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5. The functions 𝑓 and 𝑔 are defined by 𝑓(𝑥) = 8𝑥 + 33 and 𝑔(𝑥) = 2 ⋅ (1.2)𝑥.

a. Which function eventually grows faster, 𝑓 or 𝑔? Explain how you know.

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b. Explain why the graphs of 𝑓 and 𝑔 meet for a positive value of 𝑥.

Review Problems

6. A line segment of length 𝑙 is scaled by a factor of 1.5 to produce a segment with length 𝑚. The new segment is then scaled by a factor of 1.5 to give a segment of length 𝑛.

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What scale factor takes the segment of length 𝑙 to the segment of length 𝑛? Explain your reasoning.

7. A couple needs to get a loan of $5,000 and has to choose between three options. • Option A: 2 1 % applied quarterly 4

• Option B: 3% applied every 4 months

• Option C: 4 1 % applied semi-annually 2

If they make no payments for 5 years, which option will give them the least amount owed after 5 years? Use a mathematical model for each option to explain your choice.

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8. Here are graphs of five absolute value functions. Match the graph and equation that represent the same function.

Function

Graph 1

a. 𝑓(𝑥) = |𝑥|

Graph

Graph 2

d. 𝑓(𝑥) = |𝑥 + 5|

Graph 4

b. 𝑓(𝑥) = |𝑥 − 5|

c. 𝑓(𝑥) = |𝑥| − 5

e. 𝑓(𝑥) = |𝑥| + 5

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Graph 3

Graph 5

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Unit 6, Lesson 17: Changes over Equal Intervals

Warm-Up: Writing Equivalent Expressions For each given expression, write an equivalent expression with as few terms as possible. 3. 95 ⋅ 92 ⋅ 9𝑥

2. [4(𝑛 + 1) + 10] − 4(𝑛 + 1)

4. 2𝑛

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24𝑛

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1. 7𝑝 − 3 + 2(𝑝 + 1)

Exploration Activity: Outputs of a Linear Function Here is a graph of 𝑦 = 𝑓(𝑥) where 𝑓(𝑥) = 2𝑥 + 5.

1. How do the values of 𝑓 change whenever 𝑥 increases by 1, for instance, when it increases from 1 to 2, or from 19 to 20? Be prepared to explain or show how you know.

2. Here is an expression we can use to find the difference in the values of 𝑓 when the input changes from 𝑥 to 𝑥 + 1. [2(𝑥 + 1) + 5] − [2𝑥 + 5]

Does this expression have the same value as what you found in the previous questions? Show your reasoning.

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3. a. How do the values of 𝑓 change whenever 𝑥 increases by 4? Explain or show how you know.

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b. Write an expression that shows the change in the values of 𝑓 when the input value changes from 𝑥 to 𝑥 + 4.

c. Show or explain how that expression has a value of 8.

Collaborative Activity: Outputs of an Exponential Function

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Here is a table that shows some input and output values of an exponential function 𝑔. The equation 𝑔(𝑥) = 3𝑥 defines the function.

1. How does 𝑔(𝑥) change every time 𝑥 increases by 1? Show or explain your reasoning.

2. Choose two new input values that are consecutive whole numbers and find their output values. Record them in the table. How do the output values change for those two input values?

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𝒈(𝒙)

243

2,187

6 8

3. Complete the table with the output when the input is 𝑥 and when it is 𝑥 + 1. 142 | Unit 6

𝒙

729

6,561

𝑥

𝑥 + 1

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4. Look at the change in output values as the 𝑥 increases by 1. Does it still agree with your findings earlier? Show your reasoning.

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Pause here for a class discussion. Then, work with your group on the next few questions.

5. Choose two 𝑥-values where one is 3 more than the other (for example, 1 and 4). How do the output values of 𝑔 change as 𝑥 increases by 3? (Each group member should choose a different pair of numbers and study the outputs.)

𝒙 𝑥

𝒈(𝒙)

𝑥 + 3

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6. Complete this table with the output when the input is 𝑥 and when it is 𝑥 + 3. Look at the change in output values as 𝑥 increases by 3. Does it agree with your group’s findings in the previous question? Show your reasoning.

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Guided Activity: Which Function? 1. Complete the statements. factors

Linear functions change by equal

constant percent rate

per unit interval. factors

Exponential functions change by equal

per unit interval.

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constant percent rate

over equal intervals or at a differences

constant rate

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constant rate

over equal intervals or at a differences

Determine which real-world situations can be modeled with a linear function and which can be modeled with an exponential function. Then, select the justification for your choice.

2. The share of US households that used a smartphone in 2017 was 73%. Each year since 2017, the share of US households that used a smartphone increased 4%. A table of values representing this situation is shown. 0

Share of US Households that Used a Smartphone (%)

Number of Years since 2017

Type of Function

Justification

Linear function

The function grows by equal factors over equal intervals.

Exponential function

The function grows by equal differences over equal intervals.

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3. In 1950, the life expectancy of a female born in the US was 71.3. Since 1950, the life expectancy of a female born in the US has increased 1.8 years for each decade. Type of Function

Justification

Linear function

The function grows by equal factors over equal intervals.

Exponential function

The function grows by equal differences over equal intervals.

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4. In 2011, the share of the US population that was exposed to air pollution of concentrations of 10 micrograms per cubic meter was 46.5%. Each subsequent year, the rate of decay of the share of the US population exposed to air pollution of concentrations of 10 micrograms per cubic meter was 32%. Type of Function

Justification

The function grows by equal factors over equal intervals.

Exponential function

The function grows by equal differences over equal intervals.

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Linear function

5. A research scientist found that under a controlled environment, the area, in square inches (sq. in.), of algae on a pond changed by a factor of 4 each week. 2

144

576

Area (sq. in.)

Week of the Study

Type of Function

Linear function

The function grows by equal factors over equal intervals. The function grows by equal differences over equal intervals.

Exponential function

Justification

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Lesson Summary

Take the linear function 𝑓 defined by 𝑓(𝑥) = 5𝑥 + 3. The graph of this function has a slope of 5. That means that each time 𝑥 increases by 1, 𝑓(𝑥) increases by 5. For example, the points (7, 38) and (8, 43) are both on the graph. When 𝑥 increases by 1 (from 7 to 8), 𝑦 increases by 5 (because 43 − 38 = 5).

= = =

5(𝑥 + 1) + 3 − (5𝑥 + 3) 5𝑥 + 5 + 3 − 5𝑥 − 3 5

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𝑓(𝑥 + 1) − 𝑓(𝑥)

Algebraically, it can be proven that this is always true, regardless of the value of 𝑥. The value of 𝑓 when 𝑥 increases by 1, or 𝑓(𝑥 + 1), is 5(𝑥 + 1) + 3. Subtracting 𝑓(𝑥) from 𝑓(𝑥 + 1) gives the result shown.

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Linear and exponential functions each behave in a particular way every time their input value increases by the same amount.

This shows that whenever 𝑥 increases by 1, the difference in the output is always 5. When 𝑥 increases by an amount other than 1, the output always increases by the same amount if the function is linear.

Now consider the exponential function 𝑔 defined by 𝑔(𝑥) = 2𝑥. The graph of 𝑔 shows that each time 𝑥 increases by 1, the value of 𝑔(𝑥) doubles.

Algebraically, it can be proven that this is always true, regardless of what value 𝑥 takes. The value of 𝑔 when 𝑥 increases by 1, or 𝑔(𝑥 + 1), is 2𝑥+1. Dividing 𝑔(𝑥 + 1) by 𝑔(𝑥) gives the results shown. 𝑔(𝑥 + 1) 𝑔(𝑥)

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2(𝑥 + 1) 2𝑥

= 2𝑥+1−𝑥 = 21 = 2

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This means that whenever 𝑥 increases by 1, the value of 𝑔 always increases by a multiple of 2. When 𝑥 increases by an amount other than 1, the output always increases by the same factor if the function is exponential. A linear function always increases (or decreases) by the same amount over equal intervals. An exponential function always increases (or decreases) by equal factors over equal intervals.

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Practice Problems

1. Whenever the input of a function 𝑓 increases by 1, the output increases by 5. Which of these equations could define 𝑓?

A. 𝑓(𝑥) = 3𝑥 + 5 B. 𝑓(𝑥) = 5𝑥 + 3

D. 𝑓(𝑥) = 𝑥5

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C. 𝑓(𝑥) = 5𝑥

2. The function 𝑓 is defined by 𝑓(𝑥) = 2𝑥. Which of the following statements is true about the values of 𝑓? Select all that apply.

□ When the input 𝑥 increases by 1, the value of 𝑓 increases by 2.

□ When the input 𝑥 increases by 1, the value of 𝑓 increases by a factor of 2. □ When the input 𝑥 increases by 3, the value of 𝑓 increases by 8.

□ When the input 𝑥 increases by 3, the value of 𝑓 increases by a factor of 8. □ When the input 𝑥 increases by 4, the value of 𝑓 increases by a factor of 4.

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3. The two lines on the coordinate plane are graphs of functions 𝑓 and 𝑔. a. Use the graph to explain why the value of 𝑓 increases by 2 each time the input 𝑥 increases by 1.

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b. Use the graph to explain why the value of 𝑔 increases by 2 each time the input 𝑥 increases by 1. 4. The function ℎ is given by ℎ(𝑥) = 5𝑥. ℎ(𝑥 + 2) ℎ(𝑥) .

a. Find the quotient

ℎ(𝑥 + 3) . ℎ(𝑥)

c. Find the quotient

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b. What does this tell you about how the value of ℎ changes when the input is increased by 2?

d. What does this tell you about how the value of ℎ changes when the input is increased by 3?

5. The table shows the share of US households whose home had air-conditioning for the years 1966 to 1971. Number of Years since 1966

Share of US Households that Had Air-Conditioning (%)

21.6

25.92

31.1

37.3

44.79

Determine whether the data changes at a constant rate per unit interval relative to another or whether it grows or decays by a constant percent rate per unit interval relative to another. 148 | Unit 6

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Review Problems 6. For each of the functions 𝑓, 𝑔, ℎ, 𝑝, and 𝑞, the domain is 0 ≤ 𝑥 ≤ 100. For which functions is the average rate of change a good measure of how the function changes for this domain? Select all that apply.

□ 𝑓(𝑥) = 𝑥 + 2 □ 𝑔(𝑥) = 2

𝑥

□ 𝑝(𝑥) = 50,000 ⋅ 3

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□ ℎ(𝑥) = 111𝑥 − 23

𝑥

□ 𝑞(𝑥) = 87.5

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7. The average price of a gallon of regular gasoline in 2016 was $2.14. In 2017, the average price was $2.42 a gallon, an increase of 13%.

At that rate, what will the average price of gasoline be in 2020?

8. A credit card charges a 14% annual nominal interest rate and has a balance of $500.

If no payments are made and interest is compounded quarterly, which expression could be used to calculate the account balance, in dollars, in 3 years? A. 500 ⋅ (1 + 0.14)3 0.14 3

B. 500 ⋅ �1 + 4 �

0.14 12

C. 500 ⋅ �1 + 4 �

0.14 48

D. 500 ⋅ �1 + 4 �

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Unit 7: Equivalent Expressions

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Unit 7, Lesson 1: Introducing Polynomials

Warm-Up: Which One Doesn’t Belong? 1. Julius wrote a list of expressions using variables, numbers, and operators.

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a. Circle the expression that does not belong.

(𝑐 + 1)( − 𝑐)

2( − 3 − 𝑥)

− 5(2 ∙ 𝑛)

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ℎ(4∙ℎ)

b. Explain why the expression does not belong.

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Exploration Activity: Defining a Polynomial

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1. The charts show several examples and non-examples of polynomials.

Create a list of identifying features of polynomials and features that polynomials cannot have. Polynomials cannot have . . .

Polynomials can have . . .

2. Compare your list with your partner’s. Add any additional features you found after your discussion. 3. With your partner, write a definition for polynomial.

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Some polynomials are described with specific names based on the number of terms. 4. Four prefix options are given.

Prefixes

mono-

poly-

bi-

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tri-

a. Match the prefixes to the types of polynomials in the table, based on the given description and examples.

Polynomials

_____nomials

______nomials

(one term)

(two terms)

(three terms)

−7.8𝑡𝑤 + 2.04𝑤2

−7.8𝑡2 + 2.04𝑤 − 𝑡

−8 𝑦2 + 14

−8 𝑦2 − 12𝑚 + 14

−7.8𝑡𝑤

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______mials

b. In the last row of the table, create your own polynomial that fits each category.

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Collaborative Activity: Polynomials in Standard Form The degree of a monomial is the sum of the exponents of the variables in the monomial. 1. Identify the degree of each monomial.

Degree

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Monomial

− 4𝑥2

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− 7 𝑟

1.6𝑝𝑐

The degree of a polynomial is the greatest degree of its monomials.

1. What is the degree of 7𝑎 − 38𝑔2?

The leading coefficient is the coefficient of the first term of a polynomial whose terms are written in descending order from largest degree to smallest degree.

2. What is the leading coefficient of 7𝑎 − 38𝑔2?

3. Refer to the Lesson Summary for a description of the standard form of a polynomial. Rewrite the polynomial 7𝑎 − 38𝑔2 in standard form. 156 | Unit 7

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Guided Activity: Distributing a Monomial to a Polynomial The distributive property of multiplication over addition applies to polynomials. 1. An example of distributing a monomial to a polynomial is shown. a. Discuss with your partner what operation the arrows are demonstrating in the example.

(

)

1 m³ + 2m – 8 4

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–3b²

b. In the resulting product, why are the first 2 terms negative while the third term is positive?

– 34 b²m³ – 6b²m + 24b²

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2. Find each product by distributing the monomial to the polynomial. Then, complete the table for the resulting product.

(𝟏 + 𝟑𝒚 − 𝟐𝒚𝟐) (− 𝟔𝒚𝟐)

Product in standard form

Leading coefficient

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Degree of polynomial

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𝟑 𝟐 𝟏 𝟐 𝟐 𝒂 �𝟖𝒂 − 𝟗𝒃 + 𝟑 𝒃�

Product in standard form

Degree of polynomial

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Leading coefficient

− 𝟎.𝟐𝟓𝒃𝒄(𝟏.𝟓𝒄 + 𝟔𝒃)

Product in standard form

Leading coefficient

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Degree of polynomial

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Lesson Summary The word polynomial is made up of a prefix, poly-, which means “many,” and a suffix, - nomial, which means “name,” or in mathematics, “term.” A polynomial is the sum or difference of terms that have variables raised to non-negative integer powers and that have coefficients, which may be real or complex.

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As explored in the lesson, some polynomials have special names, such as monomial, binomial, and trinomial.

A monomial is a polynomial with one term.

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Polynomials are often written in standard form, such that the term with the highest degree is written first, and the other terms follow in descending order by degree. The variables in each term should be written in alphabetical order. In cases where terms are the same degree, arrange terms by alphabetical order of the variables.

For example, the polynomial 8𝑥𝑔 − 7 − 5𝑥𝑔3 + 10𝑔4 is rewritten in standard form as shown. 10𝑔4 − 5𝑔3𝑥 + 8𝑔𝑥 − 7

To multiply a monomial by a polynomial, apply the distributive property of multiplication over addition. Recall that the distributive property “distributes” a factor to all terms within a set of parentheses.

An example of distributing a monomial to a polynomial is shown.

–3b²

(

)

1 m² + 2m – 8 4

– 34 b²m² – 6b²m + 24b² The monomial − 3𝑏2 is multiplied by each term of the polynomial. In this case, the resulting polynomial is already in standard form. © Accelerate Learning Inc. - All Rights Reserved

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Practice Problems 1. Three polynomials are given in the table. a. Determine whether each polynomial is written in standard form. If it is not, rewrite it in standard form.

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b. Then, in the second column, identify the features of each polynomial.

Features of the Polynomial

Polynomial

Number of terms: ______

− 52𝑎 + 3𝑎𝑏 + 21𝑏2

Degree of the polynomial: ______ Leading term: ________

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Leading coefficient: ________ Number of terms: ______

− 9𝑥𝑦 + 𝑦

Degree of the polynomial: ______

Leading term: ________

5𝑛 + 13𝑚 − 1

Leading coefficient: ________ Number of terms: ______ Degree of the polynomial: ______

Leading term: ________ Leading coefficient: ________

2. Find the product, and write the answer in standard form. − 4𝑥(3𝑥2 + 7 − 1 𝑥) 2

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3. Match each polynomial in the left column to a description or descriptive feature in the right column. Mark each match with an arrow.

𝑥3 + 4𝑥2 − 5𝑥 + 9

Second-degree polynomial

3𝑥2 − 9𝑥3 + 4𝑥4

First-degree polynomial

𝑥 − 9𝑥 + 2𝑦

4 terms

Constant term of − 2

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45𝑎𝑏

Leading coefficient of 3

3𝑥3 + 7𝑥2 − 11

5 terms

7𝑎2 + 𝑎𝑐 − 9𝑐3 − 𝑐 + 1

Equivalent to 4𝑥4 − 9𝑥3 + 3𝑥2

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𝑥𝑦2 − 2

Review Problem

4. A piece of paper is 0.004 inches (in.) thick.

a. Explain why the thickness in inches, 𝑡, is a function of the number of times the paper is folded, 𝑛.

b. Using function notation, represent the relationship between 𝑡 and 𝑛. That is, find a function 𝑓 so that 𝑡 = 𝑓(𝑛).

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Unit 7, Lesson 2: Adding and Subtracting Polynomials

Warm-Up: Degree and Leading Coefficient Hani and Scott were asked to determine the leading coefficient and degree of the polynomial 3𝑎 − 5𝑓 + 13𝑎𝑔. Their answers are shown. Degree

Leading Coefficient

Scott

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Hani

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Whose answers do you agree with, if either? Explain your thinking.

Guided Activity: Adding Polynomials

1. Discuss with your partner why 5𝑣2 and 5𝑣 are not like terms.

2. Draw an arrow to match the like terms between column A to column B. Then, check your matches against your partner’s. Column B

4𝑥𝑦

9𝑎𝑏

11 𝑥2 4

0.2𝑎2

Column A

3𝑏𝑎

𝑦

4𝑦

−2.9𝑏2

−6 7 𝑎2

−3𝑥2

−1.6𝑏2 8

−7𝑦𝑥

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3. Ziare and Colby used different strategies to add the polynomials (2𝑡2 + 6𝑡 + 5) and ( − 7𝑡2 + 1 − 4𝑡).

a. Work with your partner to describe what each student did, and identify a helpful aspect of their strategy. (𝟐𝒕𝟐 + 𝟔𝒕 + 𝟓) + ( − 𝟕𝒕𝟐 + 𝟏 − 𝟒𝒕)

(2t ² + 6t +5) + (–7t ² + 1 – 4t)

–5t ² + 2t + 6

− 5𝑡2

+2𝑡

−7𝑡2

−4𝑡

What was helpful?

+6𝑡

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Description of the Strategy

2𝑡2

Colby’s Strategy

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Ziare’s Strategy

b. Find the sum of (19ℎ2 − 2ℎ + 25𝑘) + (8ℎ − 14𝑘 + ℎ2) using both Ziare’s and Colby’s strategies.

(𝟏𝟗𝒉𝟐 − 𝟐𝒉 + 𝟐𝟓𝒌) + (𝟖𝒉 − 𝟏𝟒𝒌 + 𝒉𝟐)

Ziare’s Strategy

Colby’s Strategy

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4. Find the sum using your preferred method. Write the sum in standard form.

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( − 3𝑝𝑘 − 2𝑘2 + 1.4𝑝) + (2.7𝑝𝑘 − 4.6𝑝 + 5.8𝑘2)

Collaborative Activity: Subtracting Polynomials

The process for subtracting polynomials is similar to the process for adding polynomials.

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1. With your partner, find the difference of (8𝑥2 + 4𝑥 − 7) and (11 − 𝑥 + 5𝑥2) by completing Ziare’s and Colby’s strategies.

(𝟖𝒙𝟐 + 𝟒𝒙 − 𝟕) − (𝟏𝟏 − 𝒙 + 𝟓𝒙𝟐)

Colby’s Strategy

Ziare’s Strategy

(8x2 + 4x − 7) − (11 − x + 5x2 )

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8𝑥2

− (5𝑥2)

+4𝑥

− (−𝑥)

−7

−(11)

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2. Oliver noticed another strategy that can work for subtraction. He said, “I know the subtraction sign, or negative, can also mean ‘opposite,’ so I read the problem as (8𝑥2 + 4𝑥 − 7) plus the opposite of (11 − 𝑥 + 5𝑥2).”

He rewrote the problem to model his thinking.

(8𝑥2 + 4𝑥 – 7) + (−11 + 𝑥 − 5𝑥2)

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a. Discuss Oliver’s strategy with your partner. Explain why you agree or disagree with Oliver’s thinking.

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b. Why might it be helpful to rewrite subtraction of polynomials as addition of polynomials?

3. Find the difference of (5𝑚2 − 4) and (8𝑚 + 4 + 3𝑚2). Write the resulting polynomial in standard form.

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Lesson Summary Like terms are terms whose variables and their corresponding exponents are the same. When adding and subtracting polynomials, only like terms can be combined. The following are examples of like terms. Example 2

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Example 1

−𝑏2, 81𝑏2, and 1 𝑏2

3.7𝑚𝑛 and −12𝑛𝑚

When combining like terms in a polynomial, add or subtract the coefficients until the terms cannot be combined any further. Then, rearrange the resulting polynomial into standard form, if necessary. Two examples are shown.

5𝑛 + 3.7𝑚𝑛 + (−12𝑛𝑚)

81𝑏2 − 𝑏2 − 5 𝑐2 + 1 𝑏2

5𝑛 − 8.3𝑚𝑛

Example 2

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Example 1

−8.3𝑚𝑛 + 5𝑛

2 2 5 1 2 80𝑏 − 𝑐 + 𝑏2 2 2 1 5 2 80 𝑏 − 𝑐2 2 2 2

Practice Problems

1. Subtract (14𝑚2 + 2𝑚𝑛 – 9) − (5𝑚2 − 6𝑚𝑛 + 17). Write the result in standard form.

2. Subtract (𝑏2 + 13𝑏 − 4𝑎) − (7𝑏 + 𝑎 + 1 𝑏2). Write the result in standard form. 3

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3. Add (−9𝑑 + 5𝑑2 + 3) + (5𝑑 − 𝑑2 + 13). Write the result in standard form.

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4. Add 3𝑥𝑦 + (𝑥2 − 3𝑦𝑥). Write the result in standard form.

Review Problem

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1. Here are the graphs of three equations: 𝑦 = 50 ⋅ (1.5)𝑥, 𝑦 = 50 ⋅ 2𝑥, and 𝑦 = 50 ⋅ (2.5)𝑥.

Which equation matches each graph? Explain how you know.

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Unit 7, Lesson 3: Multiplying Binomials with Algebra Tiles

Warm-Up: Math Puzzle

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A math puzzle is shown with a value missing.

1. Find the missing number in the puzzle.

2. Describe the pattern used to find the missing number.

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Exploration Activity: Multiplying with the Area Model Work with your partner to complete the following. 1. Three multiplication problems are set up, each using an area model. a. Complete the area models to find each product.

−5

Model B’s product

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Model A’s product

Model C

Model B

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Model A

_________

−2

Model C’s product _________

_________

b. Match each area model to the multiplication expression it represents.

Multiplication Expression

(15)(27) (15)(25) (15)(18)

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c. Complete the statements. In the area models, each rectangle represents part of the of each

factor product

Parts

are multiplied to find 4 partial products. The product adding multiplying

the partial products.

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of the 2 factors is found by

Guided Activity: Multiplying Binomials Similar to the area model, algebra tiles can also be used to model the product of two binomials.

–1 –1

x²

–x –x

1 1 1

–1

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1. Ty wants to find the product of (𝑥 + 5) and (𝑥 – 2). He uses algebra tiles to model multiplying (𝑥 + 5)(𝑥 – 2).

The factors are placed along the width and the height of a rectangular area.

1 1 1 1

a. Discuss with your partner how each factor is represented using algebra tiles. Write a summary of your discussion.

b. Complete Ty’s model to find the product of (𝑥 + 5) and (𝑥 – 2) by drawing and labeling the missing tiles on the partially completed area model.

–1

1 1

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c. Determine how many of each tile are represented in the product, and write the numbers in the table. 𝒙𝟐 tiles

+ 𝒙 tiles

− 𝒙 tiles

+ 𝟏 tiles

− 𝟏 tiles

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d. Complete the expression that represents the product of (𝑥 + 5) and (𝑥 − 2) using the algebra tiles. (𝑥 + 5)(𝑥 – 2) = _____𝑥2 + _____𝑥 − ______

2. Monica is using algebra tiles to find the product of (2ℎ − 3)2.

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a. How does the algebra tiles model represent (2ℎ − 3)2?

–1 –1 –1

b. Complete Monica’s model.

c. What shape did the product of (2ℎ − 3) form with the algebra tiles?

–1 –1

–1

Remember, when a value is “squared,” the value is being multiplied by itself.

d. Complete the table by writing how many of each tile are in the model of the product of (2ℎ − 3)2. 𝒉𝟐 tiles

+ 𝒉 tiles

− 𝒉 tiles

+ 𝟏 tiles

− 𝟏 tiles

e. Write the algebraic expression that represents the product of (2ℎ − 3)2.

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Collaborative Activity: Multiplying Binomials Work with your partner to complete the following.

a. Complete the statement.

-b²

-b

width height

of Annalee’s model,

The factor (𝑏 + 2) is represented along the

-b

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1. Annalee used algebra tiles to find the product of (𝑏 + 2)(3 − 2𝑏). Her model is shown.

-b

width height

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while the factor (3 − 2𝑏) is represented along the

b. Annalee wrote that the product is 2𝑏2 − 7𝑏 + 6. Describe the error(s) Annalee made.

c. Complete the statement.

The product of (𝑏 + 2)(3 − 2𝑏) is _______________.

2. Use algebra tiles to find the product of (1 – 𝑥) (4𝑥 – 1).

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Lesson Summary Algebra tiles can be used to model the product of 2 binomials. Use 𝑥, 𝑥2, and unit tiles to represent the terms of each factor along the width and height of the area model. Multiply each term along the width by each term along the height to find each partial product within the area model. The tiles inside the area model represent the product. Simplify the product by combining like terms.

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An example of how to multiply (2𝑔 – 4)( − 𝑔 – 2) using algebra tiles is shown.

-1 -1 -1 -1

-g

-g²

-1

-g

-1

-g

g 1

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The tiles inside the area model represent the polynomial −2𝑔2 − 4𝑔 + 4𝑔 + 8, which simplifies to −2𝑔2 + 8. Therefore, (2𝑔 – 4)( − 𝑔 – 2) = − 2𝑔2 + 8.

Practice Problems

1. Use algebra tiles to find the product of (2𝑥 – 1)(𝑥 + 6).

(2𝑥 – 1)(𝑥 + 6) = ____________________________

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2. Use algebra tiles to find the product of (𝑥 – 4)(𝑥 + 4).

(𝑥 – 4)(𝑥 + 4) = _________________

Review Problems

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3. Lin says that a snack machine is like a function because it outputs an item for each code input. Explain why Lin is correct.

4. At a gas station, a gallon of gasoline costs $3.50. The relationship between the dollar cost of gasoline and the gallons purchased can be described with a function.

a. Identify the input variable and the output variable in this function.

b. Describe the function with a sentence of the form “________ is a function of _______.”

c. Identify an input-output pair of the function, and explain its meaning in this situation.

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Unit 7, Lesson 4: Multiplying Binomials and Polynomials

Warm-Up: Notice and Wonder

1970

1980

1990

2000

1960

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1. The dot plot shows the number of hurricanes that made landfall in the continental United States from 1959–2017.

2010

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Number of Hurricanes 1959–2017

a. What do you notice when looking at the dot plot?

b. What do you wonder about the data presented?

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Guided Activity: Multiplying Binomials An algebra tiles model and an area model of (𝑥 + 2)(𝑥 + 3) are shown. The orange box in both models shows that the 𝑥 from the first factor, (𝑥 + 2), was distributed to the (𝑥 + 3).

x²

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The pink boxes show the 2 from (𝑥 + 2) being distributed to the (𝑥 + 3).

x²

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Distributing the 𝑥 and the 2 from (𝑥 + 2) to (𝑥 + 3) can be algebraically represented as 𝑥(𝑥 + 3) + 2(𝑥 + 3).

1. Apply the distributive property. Then, combine like terms. 𝑥(𝑥 + 3) + 2(𝑥 + 3)

2. Discuss with your partner how the algebra tile model or area model relates to applying the distributive property. Brayden and Jackson were asked to find the product of (𝑤 – 8)(𝑤 + 5) using different methods. 178 | Unit 7

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3. Complete each student’s work using the method they were assigned. Brayden’s

Jackson’s

Distributive Property

Area Model

w 𝑤(𝑤 + 5) − 8(𝑤 + 5)

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–8

4. Multiply (𝑚2 + 2𝑘)(5𝑚 − 6𝑘) using the distributive property.

5. Find the product (2𝑎 + 3𝑏)2 using an area model.

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Collaborative Activity: Multiplying Polynomials The same methods can be extended to multiply any polynomial. 1. Work with your partner to complete the following.

y²

+7x

–26

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–1

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a. Multiply (𝑦 – 1)(𝑦2 + 7𝑥 – 26) using the area model.

(𝑦 – 1)1(𝑦2 + 7𝑥 – 26) = ________________________

b. Multiply (1.5𝑥 – 5)(4𝑥2 − 7𝑥 + 3) using the distributive property.

1.5𝑥(4𝑥2 − 7𝑥 + 3) − 5(4𝑥2 − 7𝑥 + 3)

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2. Decide who will be partner A and who will be partner B. a. Find the product of (2𝑛2 − 𝑛 – 14)(3𝑛 + 2) using the method indicated. Partner A

Partner B

Distributive Property

Area Model

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(2𝑛2 − 𝑛 – 14)(3𝑛 + 2)

b. Compare your answers with your partner’s and resolve any differences.

c. Discuss any connections you notice between the methods used to find the product.

d. Complete the statement.

When multiplying polynomials, I prefer to use the

area model distributive property

because . . .

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Lesson Summary Use the distributive property to multiply polynomials. Each term of the first polynomial must be multiplied by each term of the other polynomial. Then, combine like terms, if possible, to simplify the resulting product. 2

� 1 𝑥 – 3�� 𝑥 − 4𝑥2 + 8 � 2

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An example of multiplying � 1 𝑥 – 3�� 𝑥 − 4𝑥2 + 8 � using the distributive property is shown. 1 𝑥� 𝑥 − 4𝑥2 + 8 � − 3� 𝑥 − 4𝑥2 + 8 � 2 5 5 1 𝑥2 − 2𝑥3 + 4 𝑥 − 3𝑥 + 12𝑥2 − 24 2 5 5 2

−2𝑥3 + 25 𝑥2 − 11 𝑥 − 24 5

Practice Problems

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If needed, algebra tiles or an area model can be extended to a 2-by-3 or 3-by-3 rectangular area when multiplying polynomials.

1. Find the product of � 2𝑥 − 1 �� 𝑥 + 5 �. 4

2. Find the product of � 2 + 𝑎�( − 5𝑎 + 2𝑎2 − 8).

3. Find the product of (7𝑟 − 0.18)( − 6 + 𝑟2 + 4𝑟).

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Review Problems 4. From 2005 to 2015, a population of 𝑃 lions is modeled by the equation 𝑃 = 1,500 ⋅ (0.98)𝑡, where 𝑡 is the number of years since 2005. a. About how many lions were there in 2005?

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b. Describe what is happening to the population of lions over this decade.

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c. About how many lions are there in 2015? Show your reasoning.

5. The scatter plot shows the rent prices for apartments in a large city over ten years.

a. The best fit line is given by the equation 𝑦 = 134.02𝑥 + 655.40, where 𝑦 represents the rent price in dollars, and 𝑥 the time in years. Use it to estimate the rent price after 8 years. Show your reasoning.

b. Use the best fit line to estimate the number of years it will take the rent price to equal $2,500s. Show your reasoning.

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Unit 7, Lesson 5: Adding, Subtracting, and Multiplying Polynomials

Warm-Up: Order of Operations Evaluate each expression. Show your work.

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1. 8 − 2 − ( − 1 + 3)2 3 5

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2. 1 (28 − 6) − 2 3 2 4

Collaborative Activity: Adding and Subtracting Polynomials with Distribution

When performing operations on polynomials, the order of operations applies.

Work with your partner to complete the following. 1. The order of operations can be applied to simplify the polynomial 8𝑥 + 3(11𝑥 − 2).

a. Rewrite the expression in each step using the description.

Step 1

𝟖𝒙 + 𝟑(𝟏𝟏𝒙 − 𝟐)

Step 2

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Description Distribute 3 to (11𝑥 − 2). Combine like terms.

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b. Complete the statement. Addition and subtraction are Multiplication is addition and subtraction multiplication

performed before

in the order of operations.

Description

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𝟏𝟎 �𝟐 𝒎 − 𝟑𝒑𝟐� + 𝟐𝒎 − (𝟕𝒑𝟐 + 𝟔) 𝟓

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2. Use the order of operations to simplify the polynomial, and describe each step you perform.

3. Complete the statement. When adding and subtracting polynomials that include distribution, first

distribute monomials, add or subtract like terms,

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and then

distribute monomials. add or subtract like terms.

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Collaborative Activity: Polynomial Scavenger Hunt 1. With your partner, complete the polynomial scavenger hunt.

• Start at 1 of the Polynomial Cards posted. • Draw (or describe) your card’s emoji, and write the polynomial expression on the recording sheet.

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• Simplify the expression by applying the order of operations.

• Use your simplified answer to find your next card posted somewhere in the room. • Record the emoji on that Polynomial Card and your next expression to simplify.

• Repeat until you have completed 5 problems.

Polynomial Emoji Scavenger Hunt Recording Sheet Polynomial Expression

Answer:

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Card Emoji

Answer:

The types of expressions you and your partner simplified in the emoji scavenger hunt were all polynomial expressions.

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2. Review your answers. Discuss with your partner what the results have in common. 3. Complete the statements. When adding polynomials, the result is

always not always

a polynomial.

always not always

When multiplying polynomials, the result is

always not always

a polynomial.

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When subtracting polynomials, the result is

a polynomial.

4. Select all of the operations under which polynomials are closed.

□ addition

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□ subtraction □ multiplication

□ division

Lesson Summary

When performing operations on polynomials, the order of operations applies just as it does when evaluating other numerical or algebraic expressions that include multiple operations.

• Multiplication, which includes distribution, is performed before any addition and sub-

traction. Distribute any monomials to the terms in parentheses.

• After multiplying, combine like terms by adding or subtracting.

In mathematics, a system is said to be closed under certain operations when the result of the operation will produce a number, term, or expression from the same system. For example, the system of integers is closed under the operations of addition, subtraction, and multiplication because the result of any of these operations on integers will always result in another integer.

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Polynomials are also closed under the operations of addition, subtraction, and multiplication. Polynomials are not closed under division because division of some 9𝑥2 + 𝑥2 polynomials results in answers from other systems. For example, results in 10, 3𝑥2 3 which is rational number and not a polynomial. Division of polynomials will be explored in future courses.

Practice Problems

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1. Simplify 7 (𝑛2 + 6𝑘) − 𝑘(3𝑘2 + 11) + 2𝑛2. 3

2. Simplify 2.7 + 5(1.2𝑛𝑔 − 0.3𝑛) − 𝑛(9𝑔 − 𝑛).

Review Problems

3. A major retailer has a staff of 6,400 employees for the holidays. After the holidays, they will decrease their staff by 30%. How many employees will they have after the holidays?

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4. A bank account pays 0.5% monthly interest.

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a. If $500 is put in the account, what will the balance be at the end of one year, assuming no additional deposits or withdrawals are made?

b. What is the effective annual interest rate?

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c. Is the effective annual interest rate more or less than 6% (the nominal interest rate)?

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Unit 7, Lesson 6: Translating between Rational Exponents and Radicals

Warm-Up: Reviewing Laws of Exponents

Exponent Law

82 ∙ 84

Product of powers

(75)3

Power of a power

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Product of powers

Power of a power

212 210

Quotient of powers

Quotient of powers Power of a product

Power of a quotient

(3 ∙ 7)5

�7�

�5�

−2

Equivalent Expression

Expression

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Complete the table of examples of each exponent law by writing the missing expression.

Power of a product

Power of a quotient Negative exponents

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107

432

135 ∙ 25

63 253

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Product of powers

113 ∙ 11 − 5

1 45

Negative exponents

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Exploration Activity: Relating Rational Exponents to Radical Expressions

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1. Use the product of powers law, 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚 + 𝑛, to evaluate the expression 5 3 ∙5 3 ∙5 3 .

2. Complete the equation with a whole number.

─ ∙ ∛5 ─∙ ∛5 ─ = _______ ∛5

3. Complete the statement.

1 1 1 ─ ∙ ∛5 ─∙ ∛5 ─ = _____, 5 13 is equivalent to _____. Because 5 3 ∙5 3 ∙5 3 = _____ and ∛5

𝑚 𝑚 𝑛 ─ 𝑛 ─ Rational exponents are defined as 𝑎 𝑛 = (√ 𝑎 )𝑚, and conversely, (√ 𝑎 )𝑚 = 𝑎 𝑛 , where 𝑎>0.

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4. Kalini’s teacher asked her to rewrite �(225) 2 � as an equivalent exponential expression of a base raised to a single power.

a. Using the laws of exponents, write an equivalent exponential expression 1

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for �(225) 2 � .

b. Rewrite your answer to part A as an equivalent radical expression.

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c. Evaluate �(225) 2 � .

5. Rewrite each expression using the laws of exponents. Then, rewrite the expression using a radical. Equivalent Exponential Expression

Given Expression 1

Equivalent Radical Expression

65 3 ∙ 65 3 7

23 3 2

23 3

−

12512 ∙ 125 12

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Guided Activity: Writing Equivalent Expressions 1. Use the power of a power law of exponents to write an equivalent exponential expression for the following 2 expressions. 𝑏

�𝑎 𝑐 � = 𝑎(________ ∙ ________) = _________

b. Expression 2:

�𝑎𝑏 � 𝑐 = 𝑎(_______ ∙ ________) = _________

𝑏

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a. Expression 1:

2. Explain why the expressions �𝑎 𝑐 � and �𝑎𝑏 � 𝑐 are equivalent.

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3. Two students evaluated the same expression using different methods. Jerry

─ )3 (√─ 81

(9)3

�(81) 2 �

Antonio

729

3 1

81 2 ∙ 81 2 ∙ 81 2

─ ∙ √─ ─ ∙√─ ─ 81 81 81 √─ 9∙9∙9 729

Explain the difference between the 2 methods.

�(81) 2 �

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Collaborative Activity: Writing Equivalent Expressions 1. Complete the table by writing equivalent expressions in each row following the form indicated.

(100) 2

�(100) 2 �

�𝒂 �

𝟏

𝒄 ─ 𝒂𝒃 √─

𝒄 ─ 𝒃 �√ 𝒂�

𝒃 𝒄

─� �∛─ 60

−3 2

8─ √─

�𝒂 �

𝒃

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𝟏 𝒄

𝒃

�𝒂� 𝒄

1 8─ √─ 3

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�(27) � 3

−3

─ )2. 2. Write 3 equivalent expressions for (√─ 17

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Lesson Summary An expression containing a rational exponent can be rewritten as a radical expression. The related parts of a radical and a rational exponent are shown.

√a m radicand

power base

base

m an

exponent root

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index

When rewriting an expression with a rational exponent in fraction form as a radical expression, the exponent’s denominator becomes the radical’s index, and the exponent’s numerator becomes the radicand’s power.

ip t

By reversing the process of converting rational exponents to radicals, radicals can be translated to equivalent expressions with rational exponents. Then, the laws of exponents can be applied to simplify the expression if necessary. An example is shown. ─── = ((7)2) 13 = (7) 23 ∛(7) 2

This course explores indices of 2, square roots, and 3, cube roots. Future courses will work with other indices.

The square root of a positive number 𝑛 is the positive number whose square is 𝑛.

The cube root of a number 𝑛 is the positive number whose cube is 𝑛.

Every square root has both a positive and a negative value. For example, the value 9 has 2 square roots, 3 and − 3, because 9 = 3∙3 and 9 = (− 3)∙(− 3). Unless otherwise specified, it is typical to work with the principal square root of a number when writing equivalent expressions. The principal square root is the positive square root of a positive real number.

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Practice Problems ─ )5 using a rational exponent. 1. Rewrite (∛─ 72 3

op y

2. Rewrite 5 2 using a radical. ─ )4 using a rational exponent. 3. Rewrite (∛─ 20 1

ip t

Review Problems

4. Rewrite 6− 2 using a radical.

5. The real estate tax rate in 2018 in a small rural county increased by 1 %. 4 In 2017, a family paid $1,200.

Which expression represents the real estate tax, in dollars, that the family will pay in 2018? A. 1,200 + 1,200 ⋅� 1 �

B. 1,200 ⋅ (1.25)

C. 1,200 ⋅ (1.025)

D. 1,200 ⋅ (1.0025)

6. Automobiles start losing value, or depreciating, as soon as they leave the car dealership. Five years ago, a family purchased a new car that cost $16,490. If the car lost 13% of its value each year, what is the value of the car now?

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7. The number of trees in a rainforest decreases each month by 0.5%. The forest currently has 2.5 billion trees.

ip t

op y

Write an expression to represent how many trees will be left in 10 years. Then, evaluate the expression.

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Unit 7, Lesson 7: Rewriting Algebraic Expressions with Rational Exponents and Radicals

Warm-Up: Identifying Equivalent Expressions Three lists of numbers are shown. List B

8∙8∙8

4 ∙ 16 ∙ 2

6∙4

2 ∙ 16

�2�

1 1 1 4 ∙ 4 ∙4

1 32

512

22 ∙ 23

128

2 ∙ 43

4+4+4+4+4+4

ip t

List C

op y

List A

1. Choose an expression from list A, and match it with an equivalent expression from list B and another from list C.

2. Explain to your partner how you know the expressions you matched are equivalent.

3. Listen to your partner explain the match they found. If you disagree, discuss your thinking, and work to reach an agreement.

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Collaborative Activity: Card Sort: Matching Equivalent Expressions Your teacher will give you 20 cards. Match the cards into groups of 4 equivalent expressions. Record your matches in the table. Cards with Equivalent Expressions

op y

Group Group 1

Group 2

ip t

Group 3

Group 5

Group 4

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Guided Activity: Converting Radicals to Rational Exponents 1. Convert each exponential expression into an equivalent radical expression. Exponential Expression

Equivalent Radical Expression

op y

2(8𝑥) 3 1

(8𝑦𝑧2) 3

1 ─ because �4 13 �3 = 4. 2. The exponential expression 4 3 is equivalent to ∛4

ip t

Fill in the blanks to reverse the process and generate an equivalent expression ─ �2. using rational exponents for �∛4 _______

─ �2 = � 1 � �∛4 4

3. Rewrite the expression (∛── 10─ 𝑥5 )10 using rational exponents.

4. Discuss with your partner how to rewrite 2�

5. Rewrite the expression 2�

3𝑥2 using rational exponents. 8𝑧15

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Collaborative Activity: Equivalent Expressions 1

1. Select all the expressions that are equivalent to (25𝑚4𝑛6) 2 .

□ √25𝑚 𝑛

4 6

□ √25𝑚 𝑛 2

4 6

□ 5𝑚 𝑛

op y

2 3

□ �𝑚25𝑛

4 6

□ 25𝑚1 𝑛

4 6

ip t

2. Rewrite 3(15𝑤2) 3 as an equivalent radical expression.

a. ∛125𝑥12𝑦9

3. Rewrite each expressions as an equivalent expression using rational exponents.

100𝑢7

b. �49𝑢3

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Lesson Summary The process for rewriting an algebraic expression containing a rational exponent into a radical expression is the same as for numerical expressions. Revisit the related parts of a radical and rational exponent.

√a m radicand

power base

base

m an

exponent root

op y

index

When rewriting an algebraic expression with a rational exponent as a radical, the exponent’s denominator becomes the radical’s index, and the exponent’s numerator becomes the radical’s power.

ip t

By reversing the process of converting rational exponents to radicals, radicals can be converted to equivalent expressions with rational exponents. Then, the laws of exponents can be applied to simplify the expression if necessary. An example is shown. 1

∛(𝑦)2

−

= ((𝑦)−2) 3 = (𝑦) 3

Practice Problems

1 2

1. Which expression is equivalent to 23 ? A. 1

1 ─ 23 √─

M B.

C. 11.5 ──

D. √23

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2. Write 2 expressions that are equivalent to (81𝑥8𝑦4) 2 using the laws of exponents.

10𝑏2𝑐2 𝑐√𝑏

using the laws of exponents.

op y

3. Write 2 expressions that are equivalent to

4. Complete the table by rewriting each given expression in its equivalent form.

2 ∛(27𝑥ℎ)2

Review Problem

ip t

𝑡9 ∙ 𝑡9

Radical Expression

Exponential Expression

5. The function 𝑓 represents the amount of a medicine, in milligrams (mg), in a person’s body 𝑡 hours (hr.) after taking the medicine. Here is a graph of 𝑓.

a. How many mg of the medicine did the person take?

b. Write an equation that defines 𝑓. c. After 7 hr., how many mg of medicine remain in the person’s body?

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Unit 7, Lesson 8: Equivalent Radical Expressions

Warm-Up: Rewriting Expressions 11

op y

1. Rewrite �∛30𝑦8�12 using rational exponents.

ip t

2. Rewrite (4ℎ2) 2 as a radical expression.

Guided Activity: Equivalent Expressions 3 2

3 2

1. Consider the expression 𝑟 ∙ 𝑟 , where 𝑟 is a positive real number.

a. Rewrite each term using radicals.

b. Rewrite the expression as a single radical.

c. Discuss with your partner which laws of exponents you used to write the expression as a single radical. Summarize your discussion.

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d. Generate an equivalent expression by rewriting the radical expression that is the answer to part B as an expression with a rational exponent.

11 7

2 21

2. Consider the expression 𝑑 ∙ 𝑑 , where 𝑑 is a positive real number. Complete the following with your partner. 11

2 21

11 7

op y

a. Write the fractional exponents in the terms 𝑑 7 and 𝑑21 as equivalent fractions with the same denominator.

ip t

b. Write the expression 𝑑 ∙ 𝑑 with 1 base raised to a rational exponent.

c. Rewrite the expression that is the answer to part B as a single radical expression.

3. How were the expressions in questions 1 and 2 the same? How were they different?

𝑚

𝑥

4. Consider the expression 𝑎 𝑛 ∙ 𝑎 𝑦 .

With your partner, discuss how to rewrite the terms as equivalent exponential expressions, where the exponents have the same denominator. Summarize your discussion.

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𝑚

𝑥

5. The table in the Lesson Summary shows the process to rewrite 𝑎 𝑛 ∙ 𝑎 𝑦 as an 5

𝑛𝑦

equivalent expression of √𝑎𝑚𝑦 + 𝑛𝑥 . Apply this process to rewrite 𝑥 6 ∙ 𝑥 2 as a radical.

√432𝑓6𝑔4ℎ3𝑝2

op y

6. A radical expression is shown.

a. Discuss with your partner which factors in the radical expression are perfect squares. List them.

ip t

b. Use the laws of exponents to rewrite the given expression as 3 different equivalent expressions.

c. Compare your equivalent expressions with your partner. Discuss any differences, and determine if those different expressions are equivalent. Then, add any other equivalent expressions to your list.

7. Consider the expression ∛2 ∙ √4 .

a. The terms have different indices. Rewrite each radical with rational exponents.

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b. With your partner, determine the next steps, and then find at least 2 additional equivalent expressions to determine the product of ∛2 ∙ √4 .

Collaborative Activity: Generating Equivalent Expressions

op y

Use the laws of exponents to generate at least two equivalent expressions where one expression is in radical form and one is in rational exponent form. 5

3. (𝑑𝑦) 3

ip t

2. �𝑑 2 �

1. 𝑔 6 ∙ 𝑔 2

Use the laws of exponents to generate at least two equivalent expressions where one expression is in radical form and one is in rational exponent form. 2

4. √4𝑣6 ∙ ∛𝑣 5. 4𝑏 ∙ ∛𝑏2

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Lesson Summary 𝑚

𝑥

To use the laws of exponents to generate equivalent expressions of 𝑎 𝑛 ∙ 𝑎 𝑦 , apply the processes described and shown in the table. Description

Equivalent Expression

Rewrite the exponents to have the common denominator of 𝑛𝑦.

𝑚𝑦 𝑛𝑥 𝑎� 𝑛𝑦 �∙ 𝑎� 𝑛𝑦 �

𝑛𝑦

Rewrite each term as a radical.

𝑛𝑦

op y

√𝑎𝑚𝑦 ∙ √𝑎𝑛𝑥 𝑛𝑦

Rewrite as a product with the same index.

√𝑎𝑚𝑦 ∙ 𝑎𝑛𝑥

𝑛𝑦

Apply the product of powers law.

√𝑎𝑚𝑦 + 𝑛𝑥

𝑚𝑦 + 𝑛𝑥 � 𝑎� 𝑛𝑦

Rewrite as the sum of 2 fractional exponents.

𝑚𝑦 𝑛𝑥 + 𝑎� 𝑛𝑦 𝑛𝑦 �

𝑎� 𝑛

ip t

Simplify each fractional exponent.

Rewrite as a fractional exponent.

−

𝑥 𝑦

�

An example is shown.

𝑚

𝑥 3 ∙ 4𝑥2 = 𝑥 3 ∙ 4𝑥 3 = 4𝑥 3 = 4∛𝑥5

When given a radical expression, the process can be reversed to create an equivalent expression with rational exponents. An example is shown.

∛5𝑏6 ∙ √𝑏 = ∛5 ∙ 𝑏6 ∙ √𝑏 = ∛5 ∙ ∛𝑏6 ∙ √𝑏 = 5 3 ∙ 𝑏 3 ∙ 𝑏 2 = 5 3 ∙ 𝑏 6 ∙ 𝑏 6 = 5 3 ∙ 𝑏 6 = 5 3 ∙ 𝑏 2

Note that there are many ways to rewrite equivalent expressions involving radicals and rational exponents. Different laws or a different order of rewriting expressions may be applied but, if done correctly, will still end in the same simplified expression.

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Practice Problems 2

1. Find an equivalent expression to √𝑥𝑦 ∙ ∛2𝑥𝑦.

op y

2. Rewrite ∛96𝑎3𝑏6𝑐15 as an equivalent exponential expression.

ip t

expression.

3. Rewrite �3𝑡 4 � as an equivalent exponential expression and as an equivalent radical

Review Problem

𝑑

𝑓(𝑑) = 550 ⋅ � 2 � .

4. The number of grams of a chemical in a pond is a function of the number of days, 𝑑, since the chemical was first introduced. The function, 𝑓, is defined by

a. What is the average rate of change between day 0 and day 7?

b. Is the average rate of change a good measure for how the amount of the chemical in the pond has changed over the week? Explain your reasoning.

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Unit 7, Lesson 9: Adding and Subtracting Radical Expressions

Warm-Up: Constraints Sometimes a question gives a constraint.

op y

Suppose a question began, “Given that 𝑥 is a negative integer . . .” What are some possible values that could be substituted in for 𝑥? When choosing values to substitute, always try at least 2 different values. When possible, use − 1, 0, or 1 as one of those values.

1. Given that 𝑥 is a negative integer, select all the values that would result in a positive value.

□ 3𝑥 □ 3𝑥

ip t

−1

□ −3𝑥

−1 −2

□ (3𝑥)

−2

□ (3𝑥)

−1

□ −3𝑥

−2

Exploration Activity: Adding and Subtracting Expressions

1. Find the sum of 2𝑥 + 3𝑦 − 4 + 5𝑦 − 𝑥.

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2. Miren and Riley are finding ─ + 3�𝑥 ─ + 6∛𝑥 ─. the sum of 5�𝑥 Their work is shown.

Miren

Riley

─ ─ + 6∛𝑥 ─ 5�𝑥 + 3�𝑥

Explain why Miren’s and Riley’s answers are equivalent.

1 2

1 3

5𝑥 + 6𝑥 + 3𝑥 1

1 2

op y

8𝑥2 + 6𝑥3

─ + 3�𝑥 ─ + 6∛𝑥 ─ 5�𝑥 ─ ─ + 6∛𝑥 8�𝑥

3. How are like terms in a polynomial such as 2𝑥 + 3𝑦 − 4 + 5𝑦 − 𝑥 similar to like terms ─ + 3�𝑥 ─ + 6∛𝑥 ─? in a radical expression such as 5�𝑥

Felipe

ip t

─2, but they each got a different result. ─2 − 7∛𝑥 4. Felipe and Adele were subtracting 4∛𝑥 Their work is shown.

─2 = − 3∛𝑥 ─2 ─2 − 7∛𝑥 4∛𝑥

Adele

─2 using rational exponents. ─2 − 7∛𝑥 a. Rewrite 4∛𝑥

6 ─4 ─ 2 − 7∛𝑥 ─2 = − 3�𝑥 4∛𝑥

─2 − 7∛𝑥 ─2 is correct. b. Use your answer to part A to explain whose result for 4∛𝑥

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Guided Activity: Adding and Subtracting Radical Expressions Similar to like terms, like radicals can be combined through addition and subtraction. Often, to know whether a radical has the same radicand, the radicand must first be rewritten in simplest form. 1. Determine which pairs of radicals are like radicals by first rewriting the radicands in simplest form. ─ 4�𝑥5 , −9𝑥2�𝑥

op y

─ −11�𝑥7𝑦8, 4𝑥3𝑦4�𝑥

─ 5�𝑥6𝑦 , 8𝑥3�𝑦

─ −2𝑥𝑦2 �𝑥5𝑦2, 𝑥2𝑦3�𝑥

6�𝑥3𝑦9, 𝑥2𝑦4 �𝑥𝑦

ip t

─ 4𝑥�𝑥7 , −𝑥3�𝑥

6 ─3 2. Discuss with your partner whether �16𝑥 and �𝑥 are like terms. Summarize your discussion.

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3. Find the sum or difference of each expression. Show your work. Assume all variables are greater than 0.

op y

a. −10𝑏 �6𝑏 − �24𝑏3

c. 3�12𝑓2 − 2𝑓�48 + 4� 27𝑓2

ip t

b. ∛16𝑎7 − ∛54𝑎4

d. 2∛16𝑟5𝑡 + 𝑟 ∛2𝑟2𝑡 − ∛54𝑟5𝑡

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Collaborative Activity: Equivalent Expressions With your partner, determine who will be partner A and who will be partner B. Then, complete the following. • Add and/or subtract each of the expressions in your assigned column.

op y

• After you finish the 3 problems, compare your answers with your partner. • If your answers for each row are not equivalent, work together to determine the correct answers for the row. • Assume all variables are greater than 0.

Partner B

Partner A

ip t

5 � 𝑝3 + 3 �𝑝3

(500𝑛5)2 − 𝑛(20𝑛3)2

7𝑦 ∛ 𝑥5 + 5 ∛𝑥5𝑦3

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2𝑝2 + 6𝑝2

𝑛 �45𝑛3 + � 125𝑛5

3(𝑥5𝑦3)3 + 9𝑦(𝑥5)3

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Lesson Summary When adding and subtracting polynomial expressions, only like terms can be combined. Adding and subtracting radical expressions is similar because only like radicals can be combined.

For example, consider the difference 3 �32𝑡3 − 10 �8𝑡3.

op y

Like radicals are radical terms that have the same index and the same radicand. In other words, when the radical is rewritten using a rational exponent, the rational exponents are the same. Often, to know if a radical has the same radicand, the radicals must first be rewritten in simplest form.

• Before it can be determined whether the radicals in the expression 3� 32𝑡3 − 10 � 8𝑡3 can be combined, they need to be rewritten in simplest radical form, as shown. 3 �32𝑡3 − 10 �8𝑡3 = 3 �16 ∙ 2 ∙ 𝑡3 − 10 �4 ∙2 ∙ 𝑡3 = 12𝑡 �2𝑡 − 20𝑡 �2𝑡

12𝑡 � 2𝑡 − 20𝑡 �2𝑡 = − 8𝑡 �2𝑡

�

ip t

• The resulting radicals have the same index and radicand, � 2𝑡. The like radical terms can then be combined.

Therefore, 3 �32𝑡 − 10 8𝑡3 = − 8𝑡 � 2𝑡. 3

Expressions can be added or subtracted in either radical form or rational exponent form. Consider translating between forms if it helps to simplify the expression. If needed, the simplified expression can then be rewritten in the form requested.

An example of how to find the sum 9 ∛𝑥2 + ∛125𝑥2 is shown in both forms.

Rational Exponent Form

─2 +∛125𝑥2 9∛𝑥

9(𝑥)3 + (125)3 (𝑥)3

─2 14∛𝑥

14(𝑥)3

Radical Form

─2 + 5∛𝑥 ─2 9∛𝑥

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9(𝑥)3 + 5(𝑥)3 2

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Practice Problems 1

op y

1. Simplify 2(𝑥)3 + 3(4𝑥)2 .

ip t

2. Simplify 2 �8𝑥2 − 5𝑥 �32 + 5 �18𝑥2.

3. Simplify 2 ∛40𝑥3 − 5𝑥 ∛5 + ∛135𝑥3

Review Problems

4. Simplify ∛ 81𝑘5𝑚6.

5. Simplify �162𝑔7ℎ4.

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op y C ip t cr us an M 9798888261170_Math Nation_Illustrative Math_Algebra 1 Volume 2_Version2_652 pages.indb 218

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Unit 7, Lesson 10: Multiplying and Dividing Radical Expressions

Warm-Up: Simplifying Radical Expressions

op y

Simplify each expression. 2. �600𝑗2𝑘5 − 𝑘 � 20𝑗2𝑘3

ip t

1. 9ℎ ∛10𝑔7 + 2 ∛10𝑔7ℎ3

Exploration Activity: Multiplying Using the Laws of Exponents

1. Multiply (6𝑥2𝑦4𝑧)(8𝑥7𝑦3𝑧5).

2. Multiply �− 11𝑥3 𝑦3 ��2𝑥3 𝑦3 �.

3. Discuss with your partner why −11 ∛𝑥2𝑦 ∙ 2 ∛𝑥4𝑦 is the same as multiplying 1

�−11𝑥3 𝑦3 ��2𝑥3 𝑦3 �.

4. Use a calculator to complete the table for each value of 𝑥.

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𝟱

Value of � 𝒙 ∙ ∛𝒙

𝒙

Value of 𝒙𝟲

729

op y

4,096

─ is possible. Summarize ─ ∙ ∛𝑥 5. Discuss with your partner whether or not multiplying �𝑥 your discussion. Use the information from the table to help justify your answer.

ip t

6. Esther and Bahman were asked to explain how they would multiply �𝑥𝑦 ∙ �𝑥𝑦 Their explanations are shown. Assume the domain of all variables is greater than or equal to 0.

Esther’s Explanation

I would first rewrite the radical expressions using rational exponents and then use the product of powers law of exponents to multiply the like bases.

� 𝑥𝑦 ∙� 𝑥𝑦 = 𝑥2 ∙ 𝑥2 ∙ 𝑦2 ∙ 𝑦2 = 𝑥2

1 2

∙ 𝑦2 + 2 = 𝑥2 ∙ 𝑦2 = 𝑥1 ∙ 𝑦1 = 𝑥𝑦

I would use the product law of radicals to multiply the radical expressions and then use the law again to rewrite the square root.

Bahman’s Explanation

�𝑥𝑦 ∙ �𝑥𝑦 = �𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 = � 𝑥2 ∙ 𝑦2 = 𝑥𝑦

a. Both students’ methods resulted in the same expression, 𝑥𝑦. Discuss each method with your partner.

b. Identify which method you would use to multiply �𝑎3𝑐 ∙ �𝑎3𝑐. Then, use the method to find the product.

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Collaborative Activity: Multiplying Radical Expressions To multiply radical expressions with the same index, multiply the coefficients together and then the variables. Multiply each expression, and simplify. Assume the domain of all variables is greater than or equal to 0.

op y

1. 2 ∛6𝑥4𝑦 ∙ 3 ∛9𝑥5𝑦2

3. 3𝑚 �24𝑚5𝑛6 ∙ 4𝑛 � 2𝑚10𝑛7

ip t

2. 5𝑝 � 25𝑝7𝑞5 ∙ 𝑞 � 50𝑝3𝑞2

4. 2 ∛ 12𝑔6ℎ4 ∙ 3𝑔 ∛ 36𝑔ℎ

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Guided Activity: Dividing Radical Expressions ─ �𝑦 Ariana used the quotient of powers law of exponents to simplify ─ . Her work is �𝑦 shown. 1

─ 1 1 𝑦2 �𝑦 = = 𝑦 2 − 2 = 𝑦0 = 1 ─ 1 �𝑦 𝑦2

� 𝑥3𝑓

Description

147𝑔𝑡8

�3𝑔3𝑡4

, where 𝑔>0 and 𝑡>0.

ip t

1. Complete the table to simplify

3 �𝑥3𝑓 �𝑥 𝑓 =1=1 = �𝑥3𝑓 𝑥3𝑓

op y

She then used the quotient of radicals law to rewrite the expression 3 . Her work is �𝑥 𝑓 shown.

Step

Rewrite using the quotient property.

Remove the common factors.

Simplify the radical.

2. Simplify ∛ 32𝑏2 ÷ ∛4𝑏5, where 𝑏≠0.

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3. With your partner, discuss why there is a domain restriction of 𝑏≠0.

∛54𝑎3𝑏5

∛16𝑎2𝑏2

op y

4. Rewrite radical expressions in equivalent forms, and use the quotient law to eliminate radicals where possible. List the domain restrictions.

cr us

�8𝑤3𝑧 �2𝑤𝑧3

ip t

d. 4 ∛ 15ℎ7𝑘9÷6 ∛6ℎ3

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Lesson Summary When multiplying radicals with the same index, the coefficients are multiplied together and then the variables. This is true for the terms outside the radical as well as for the terms of the radicand. The steps to multiply 2𝑦 �4𝑥 · 𝑥 �3𝑦4 · �5𝑥11 · 3 � 7𝑦 are shown as an example.

op y

Given expression

2𝑦 �4𝑥 · 𝑥 �3𝑦4 · � 5𝑥11 · 3 � 7𝑦

Multiply coefficients, and multiply radicands.

6𝑥𝑦 �420𝑥12𝑦5

12𝑥7𝑦3 �105𝑦

ip t

Simplify.

Take the square root of perfect square factors of the radicand.

6 ∙ 2 ∙ 𝑥 ∙ 𝑥6 ∙ 𝑦 ∙ 𝑦2 �105𝑦

4∛3𝑡2𝑥7

∛ 24𝑡8𝑥2

are shown as an example.

The steps to divide

When dividing radicals with the same index, the coefficients are divided and then the variables. This is true for the terms outside the radical as well as for the terms of the radicand.

4∛3𝑡2𝑥7

∛ 24𝑡8𝑥2

4∙𝑥2 ∛3𝑡2𝑥

2∙𝑡 ∛3𝑡 𝑥

2 2

2𝑥2 𝑡2 ∛𝑥

Given expression

Take the cube root of perfect cube factors in radicands.

Divide coefficients, and divide variables.

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Practice Problems 1. Declan and Munroe were asked to show their work for multiplying �𝑎3𝑏2 ∙4� 2𝑎5𝑏6.

Declan’s Work

a. Whose work is correct? Explain why.

�𝑎3𝑏2 ∙4 �2𝑎5𝑏6 1 ∙ 4 �2𝑎8𝑏8 4 � 2𝑎8𝑏8

op y

b. Finish simplifying the correct expression.

�𝑎3𝑏2 ∙ 4 � 2𝑎5𝑏6

Munroe’s Work

1 ∙ 4 �2𝑎15𝑏12

ip t

4 � 2𝑎15𝑏12

2. Harley and Leva were asked to show their work for dividing ∛192𝑢12𝑣10 ÷ ∛ 3𝑢3𝑣2.

∛192𝑢12𝑣10

Leva’s Work

∛192𝑢12𝑣10

∛3𝑢 𝑣

3 2

∛3𝑢3𝑣2

= ∛64𝑢4𝑣5 = ∛ 64𝑢9𝑣8

a. Whose work is correct? Explain why.

Harley’s Work

b. Finish simplifying the correct expression.

c. What are the restrictions on the domain?

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Review Problem 3. Ten students guessed the number of cubes in a jar that contains 202 cubes. Their names and guesses are listed in the table.

Student

Guess

Andre

205

Clare

190

Diego

197

Elena

200

Han

220

Jada

210

Kiran

202

Lin

203

Mai

199

Noah

185

ip t

op y

Create a scatter plot with the guesses as the horizontal values and the absolute guessing errors as the vertical values.

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Unit 7, Lesson 11: Operations with Radical Expressions

Warm-Up: Riddle Me This

op y

1. If 6666 = 4, 8888 = 8, 1618 = 6, 1313 = 0, then 1616 = _______?

2. Explain your thinking.

ip t

Collaborative Activity: Operations on Radical Expressions

1. Aaliyah simplified the following radical expression, but she made a mistake in her process. Her work is shown.

(∛ 192𝑟8𝑐9 − 𝑟 ∛648𝑟5𝑐9 )

(4𝑟2𝑐3 ∛3𝑟2 − 6𝑟2𝑐3 ∛3𝑟2) ∛2𝑟

∛2𝑟

− 2𝑟2𝑐3 ∛3𝑟2 ∛2𝑟

a. With your partner, discuss Aaliyah’s mistake.

= − 𝑟𝑐3 ∛3𝑟2

b. Fix the error, and find the correct simplification.

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Collaborative Activity: Riddled by Operations Involving Radical Expressions 1. Simplify each expression. Include domain restrictions, if necessary. 9𝑥 � 8𝑥 + 8 �18𝑥3 − �72𝑥3

op y

T �18𝑥7𝑦2

∛𝑥 � ∛𝑥2 − 73 ∛𝑥5 + 2 ∛𝑥8�

� 48𝑥2𝑦7

ip t

9𝑥 �𝑥4𝑦 − 4𝑥3�𝑦 + 5 �81𝑥2 − 3𝑥

cr us

∛ 64𝑥−2𝑧10 ∙ ∛8𝑥12𝑧−4

N 3 ∛3𝑥5𝑦4 − 𝑥 ∛375𝑥2𝑦4

�32𝑥3𝑦2 − 𝑥𝑦 �18𝑥

─(3�𝑥 ─ − 5) �𝑥

∛6𝑥4𝑦5 ∙ 2∛36𝑥𝑦2

5𝑥 ∛24𝑥2 + 2 ∛81𝑥5 − 3𝑥 √3𝑥2

5 ∛𝑥 + �36𝑥

3 7 −5 5 � 16𝑥 𝑦 𝑧 54𝑥−2𝑦7𝑧−7

∛24𝑥8𝑧6 − 𝑥 ∛3𝑥5𝑧6 + 𝑧 ∛3𝑥8𝑧3

7 � 𝑥(2� 𝑥𝑦 + �𝑦 )

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2. Fill in the letter that corresponds to each expression in order to solve the riddle.

What did the ocean say to the iceberg? _____

_____

_____,

𝑥𝑦 �2𝑥

5 ∛𝑥 + 6 �𝑥

36𝑥 � 2𝑥

2𝑥3 − 7𝑥2 + 𝑥

𝑥2 � 3𝑥

𝑥𝑦 �2𝑥

2𝑥3 𝑧4 3𝑦4

𝑥2 � 3𝑥

36𝑥 �2𝑥

16𝑥 ∛3𝑥2 − 3𝑥2�3

_____ 14𝑥 � 𝑦 + 7 � 𝑥𝑦

_____

12𝑥𝑦2∛ 𝑥2𝑦

_____

− 2𝑥𝑦 ∛3𝑥2𝑦

5𝑥3� 𝑦 + 42𝑥

36𝑥 � 2𝑥

3𝑥 − 5 � 𝑥

8𝑥3𝑧2 ∛𝑥

_____

_____.

Lesson Summary

2𝑥2𝑧2 ∛3𝑥2

_____

op y

2𝑦2 �2𝑦 _____

_____

ip t

_____

2𝑦2 �2𝑦 _____

To simplify algebraic expressions with radicals, remember the following key ideas.

• When adding and subtracting radical expressions, make sure that radical terms are in simplest form first to determine if they can be combined.

• When multiplying and dividing radical expressions, make sure the radical terms have the same index before multiplying or dividing the coefficients and the radicands.

• A radical expression can be rewritten into an equivalent expression with rational exponents to help simplify the expression. Apply the laws of exponents as needed. In this course, restrictions on the domain when simplifying algebraic expressions with radicals are limited to situations in which the values of the variables cannot be 0. This occurs when dividing radicals with variables in the divisor. Since division by 0 is undefined, this caveat, in the form of a domain restriction, should be included when simplifying radical expressions. Future courses will reveal more about why division by 0 is undefined, as well as other restrictions for the domain when simplifying radical expressions.

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Practice Problems 1. Travis simplified the radical expression − 𝑝 ��12𝑟5𝑝4 + �3𝑟𝑝2�, but he made an error. His work is shown. − 𝑝 � �12𝑟5𝑝4 + �3𝑟𝑝2� = − 𝑝�2𝑟2𝑝2 �3𝑟 + 𝑝 �3𝑟� = − 2𝑟2𝑝3 �3𝑟 − 𝑝2 �3𝑟 = − 2𝑟2𝑝 �3𝑟

ip t

b. Correctly simplify − 𝑝� �12𝑟5𝑝4 + �3𝑟𝑝2�.

op y

a. Describe the error Travis made in his work.

Review Problem

2. Simplify each expression.

b. ∛ 81𝑘5𝑚6

a. �54𝑥3𝑦8

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op y

ip t

Unit 8: Quadratic Functions

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Unit 8, Lesson 1: Understanding How Relationships Change

Warm-Up: Squares in a Figure

a. Expression A: 6 ⋅ 8 − 2 ⋅ 3

ip t

op y

1. How does each expression represent the number of small squares in the figure?

b. Expression B: 4 ⋅ 8 + 2 ⋅ 5

c. Expression C: 8 + 8 + 8 + 8 + 5 + 5

d. Expression D: 5 ⋅ 6 + 3 ⋅ 4

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Exploration Activity: Plotting the Measurements of a Garden Nadine has 50 meters (m) of fencing to enclose a rectangular garden in her backyard. Length (m)

Width (m)

12.5 20 24

12.5 7 5

Area (sq. m)

op y

1. The table lists some possible lengths and widths of the garden if Nadine uses all her fencing. Complete the table by determining the area of the garden, in square meters (sq. m), for each pair of dimensions.

ip t

2. Plot some values for the length and area of the garden on the coordinate plane.

3. What do you notice about the plotted points?

4. The points (3, 66) and (22, 66) each represent the length and area of the garden.

a. Plot these 2 points on the coordinate plane.

b. What do these points mean in this situation?

5. Explain whether the point (1, 25) could represent the length and area of the garden.

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Exploration Activity: Patterns of Dots 1. Study the 2 patterns of dots. Pattern 2

op y

Pattern 1

a. How are the number of dots in each pattern changing?

ip t

b. How would you find the number of dots in the 5th step in each pattern?

2. Complete the table with the number of dots in each pattern.

Step

Number of Dots in Pattern 1

Number of Dots in Pattern 2

0 1 2 3 4 5

10 𝑛

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3. Plot the number of dots at each step number. Pattern 2

op y

Pattern 1

ip t

4. Explain why the graphs of the 2 patterns look the way they do.

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Collaborative Activity: Expressing a Growth Pattern

op y

Here is a pattern of squares.

1. Is the number of small squares growing linearly? Explain how you know.

ip t

2. Complete the table.

3. Is the number of small squares growing exponentially? Explain how you know.

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Step

Number of Small Squares

1 2 3 4 5

10 12 𝑛

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Lesson Summary This lesson explored quantities that change in a particular way, but the change is neither linear nor exponential. A pattern of shapes and a table displaying the relationship between the step number and the number of small squares are shown. Total Number of Small Squares

op y

Step

𝑛2 + 1

𝑛

The number of small squares increases by 3 and then by 5, so the growth is not linear. It is also not exponential because it is not changing by the same factor each time. From 5

ip t

step 1 to step 2, the number of small squares grows by a factor of 2 , but from step 2 to step 3, it grows by a factor of 2.

The diagram shows that in step 2, there is a 2-by-2 square plus 1 small square added on top. Likewise, in step 3, there is a 3-by-3 square plus 1 small square added. It stands to reason that the 𝑛th step is an 𝑛-by-𝑛 arrangement of small squares with an additional small square on top, giving the expression 𝑛2 + 1 for the number of small squares.

The relationship between the step number and the number of small squares is a quadratic relationship because it is given by the expression 𝑛2 + 1, which is an example of a quadratic expression. A quadratic expression is a polynomial expression that contains a term of degree 2, but no term of higher degree.

Quadratic expressions will be explored more thoroughly in upcoming lessons.

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Practice Problems 1. Here are some lengths and widths of a rectangle whose perimeter is 20 m. Length (m)

Width (m)

3 5 7

Area (sq. m)

op y

a. Complete the table. What do you notice about the areas?

ip t

9 b. Without calculating, predict whether the area of the rectangle will be greater or less than 25 sq. m if the length is 5.25 m.

c. On the coordinate plane, plot the points for length and area from your table. Do the values change in a linear way? Do they change in an exponential way?

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2. Which statement best describes the relationship between a rectangle’s side length and area as represented by the graph? A. As the side length increases by 1, the area increases and then decreases by an equal amount.

op y

B. As the side length increases by 1, the area increases and then decreases by an equal factor.

C. As the side length increases by 1, the area does not increase or decrease by an equal amount.

ip t

D. As the side length increases by 1, the area does not change.

3. Here are descriptions for how 2 dot patterns are growing. • Pattern A: Step 2 has 10 dots. It grows by 3 dots at each additional step.

• Pattern B: The total number of dots can be expressed by 2𝑛2 + 1, where 𝑛 is the step number.

For each pattern, draw a diagram of Step 0 to Step 3.

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4. Each expression represents the total number of dots in a pattern where 𝑛 represents the step number. Select all the expressions that represent a quadratic relationship between the step number and the total number of dots. (If you get stuck, consider sketching the first few steps of each pattern as described by the expression.)

□𝑛

op y

□ 2𝑛

□ 𝑛 ⋅ 𝑛

□ 𝑛 + 𝑛

□ 𝑛 + 2

ip t

□ 𝑛 ÷ 2

Review Problems

5. The value of a phone when it was purchased was $500. It loses 5 of its value a year.

a. What is the value of the phone after 1 year? What about after 2 years? 3 years? 1

b. Tyler says that the value of the phone decreases by $100 each year since 5 of 500 is 100. Do you agree with Tyler? Explain your reasoning.

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6. A bacteria population is 10,000 when it is first measured and then doubles each day. Time (days)

Population (thousands)

op y

ip t

𝑑

a. Use this information to complete the table.

b. Which is the first day, after the population was originally measured, that the bacteria population is more than 1,000,000?

c. Write an equation relating 𝑝, the bacteria population, to 𝑑, the number of days since it was first measured.

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Unit 8, Lesson 2: Building Quadratic Functions from Geometric Patterns

Warm-Up: Quadratic Expressions and Area

Figure B

Figure C

Figure A

op y

Figure A is a large square. Figure B is a large square with a smaller square removed. Figure C is composed of two large squares with one smaller square added.

ip t

Area of A

Area of B

Area of C

Side Length of Large Square

Write an expression to represent the area of each shaded figure when the side length of the large square is as shown in the first column.

𝑥

4𝑥

(𝑥 + 3)

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op y

Exploration Activity: Expanding Squares

1. If the pattern continues, what will we see in Step 5 and Step 18?

ip t

a. Sketch or describe the figure in each of these steps.

b. How many small squares are in each of these steps? Explain how you know.

2. Write an equation to represent the relationship between the step number, 𝑛, and the number of squares, 𝑦. Be prepared to explain how each part of your equation relates to the pattern. (If you get stuck, try making a table.)

3. Sketch the first 3 steps of a pattern that can be represented by the equation 𝑦 = 𝑛2 −1 .

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op y

Collaborative Activity: Growing Steps

ip t

1. Sketch the next step in the pattern.

2. Kiran says that the pattern is growing linearly because as the step number goes up by 1, the number of rows and the number of columns also increase by 1. Do you agree? Explain your reasoning.

3. To represent the number of squares after 𝑛 steps, Diego and Jada wrote different equations. Diego wrote the equation 𝑓(𝑛) = 𝑛(𝑛 + 2). Jada wrote the equation 𝑓(𝑛) = 𝑛2 + 2𝑛. Are either Diego or Jada correct? Explain your reasoning.

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Lesson Summary

op y

Sometimes, a quadratic relationship can be expressed without using a squared term. Consider this pattern of squares for example.

The first 3 steps show that both the length and the width of the rectangle increase by 1 at each step. Step 1 is a 1-by-2 rectangle, step 2 is a 2-by-3 rectangle, and step 3 is a 3-by-4 rectangle. This suggests that step 𝑛 is a rectangle with side lengths 𝑛 and 𝑛 + 1, so the number of squares at step 𝑛 is 𝑛(𝑛 + 1).

ip t

This expression may not look like a quadratic expression because it has no squared term, but applying the distributive property shows 𝑛(𝑛 + 1) is equivalent to 𝑛2 + 𝑛.

It can be demonstrated that these expressions are equivalent by breaking each rectangle into an 𝑛-by-𝑛 square to represent the 𝑛2 in the expression and an 𝑛-by-1 rectangle to represent the 𝑛 in the expression.

The relationship between the step number and the number of squares can be described by a quadratic function, 𝑓, whose input is 𝑛 and whose output is the number of squares at step 𝑛. Define 𝑓 as 𝑓(𝑛) = 𝑛(𝑛 + 1) or 𝑓(𝑛) = (𝑛)2 + 𝑛. A quadratic function is a polynomial function with degree of 2.

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Practice Problems

op y

1. Sketch or describe the figure in Step 4 and Step 15.

a. How many small squares will there be in each of these steps?

ip t

b. Write an equation to represent the relationship between the step number, 𝑛, and the number of small squares, 𝑦, in each step.

c. Explain how your equation relates to the pattern.

2. Which expression represents the relationship between the step number 𝑛 and the total number of small squares in the pattern?

A. 𝑛2 + 1

B. 𝑛2 − 1

C. 𝑛2 − 𝑛 D. 𝑛2 + 𝑛

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3. Each figure is composed of large squares and small squares. The side length of the large square is 𝑥. Write an expression for the area of the shaded part of each figure. Figure B

op y

Figure A

Review Problems

4. Here are a few pairs of positive numbers whose difference is 5.

ip t

a. Find the product of each pair of numbers. Then, plot some points to show the relationship between the first number and the product.

7 8

First Second Product Number Number

10 12

b. Is the relationship between the first number and the product exponential? Explain how you know.

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5. Here is a pattern of dots.

Total Number of Dots

op y

Step 0 2

ip t

a. Complete the table.

b. How many dots will there be in Step 10?

c. How many dots will there be in Step 𝑛?

6. A stadium can seat 63,026 people. For each game, the amount of money that the organization brings in through ticket sales is a function of the number of people, 𝑛, in attendance.

If each ticket costs $30.00, find the domain and range of this function.

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Unit 8, Lesson 3: Building Quadratic Functions to Describe Situations – Part 1

Warm-Up: Sky Bound

op y

A cannon is 10 feet (ft.) off the ground. It launches a cannonball straight up with a velocity of 406 ft. per second (ft./sec.).

Imagine that there is no gravity and that the cannonball continues to travel upward with the same velocity.

𝑡

Distance above Ground (ft.)

ip t

Time (sec.)

1. Complete the table with the heights of the cannonball at different times.

2. Write an equation to model the distance, in ft., 𝑑, of the ball 𝑡 sec. after it was fired from the cannon if there was no gravity.

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Exploration Activity: Falling from the Sky A rock is dropped from the top floor of a 500-foot-tall building. A camera captures the distance the rock traveled, in ft., after each sec.

16 = 16 ∙ 1 64 = 16 ∙ 4 144 = 16 ∙ 9 256 = 16 ∙ 16 400 = 16 ∙ 25

ip t

She wrote down:

3. Jada noticed that the distances fallen are all multiples of 16.

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1. How far will the rock have fallen after 6 sec.? Show your reasoning.

Then, she noticed that 1, 4, 9, 16, and 25 are 12, 22, 32, 42 and 52.

a. Use Jada’s observations to predict the distance fallen after 7 sec. (Assume the building is tall enough that an object dropped from the top of it will continue falling for at least 7 sec.) Show your reasoning.

b. Write an equation for the function, with 𝑑 representing the distance dropped after 𝑡 sec.

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Exploration Activity: Galileo and Gravity Galileo Galilei, an Italian scientist, and other medieval scholars studied the motion of free-falling objects. The law they discovered can be expressed by the equation 𝑑 = 16 ⋅ 𝑡2, which gives the distance fallen, in ft., 𝑑, as a function of time, 𝑡, in sec. An object is dropped from a height of 576 ft.

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1. How far does it fall in 0.5 sec.?

2. To find out where the object is after the first few seconds after it was dropped, Elena and Diego created different tables. Elena’s Table

Diego’s Table

Distance Fallen (ft.)

Time (sec.)

Distance Fallen (ft.)

576

512

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Time (sec.)

𝑡

1 3

560

4 𝑡

a. How are the two tables alike? How are they different?

b. Complete Elena’s and Diego’s tables. Be prepared to explain your reasoning.

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Guided Activity: Graphing a Cannonball The function defined by 𝑑 = 50 + 312𝑡 − 16𝑡2 gives the height, in ft., of a cannonball 𝑡 sec. after the ball leaves the cannon.

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1. What do the terms 50, 312𝑡, and −16𝑡2 tell us about the cannonball?

3. Observe the graph and:

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2. Use graphing technology to graph the function. Adjust the graphing window to the following boundaries: 0 < 𝑥 < 25 and 0 < 𝑦 < 2,000.

a. Describe the shape of the graph. What does it tell us about the movement of the cannonball?

b. Estimate the maximum height the ball reaches. When does this happen?

c. Estimate when the ball hits the ground.

4. What domain is appropriate for this function? Explain your reasoning.

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Lesson Summary The distance traveled by a falling object in a given amount of time is an example of a quadratic function. Galileo is said to have dropped balls of different masses from the Leaning Tower of Pisa, which is about 190 ft. tall, to show that they travel the same distance in the same amount of time. In fact, the equation 𝑑 = 16𝑡2 models the distance 𝑑, in ft., that a cannonball falls after 𝑡 sec., no matter what its mass.

Distance Fallen (ft.)

144

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Time (sec.)

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Because 16 ⋅ 42 = 256 and the tower is only 190 ft. tall, the cannonball hits the ground before 4 sec. A table and graph showing how far the cannonball has fallen over the first few seconds are shown.

Notice that the distance fallen increases each second. The average rate of change increases each second, which means that the cannonball speeds up over time. This comes from the influence of gravity, which is represented by the quadratic expression 16𝑡2. It is the exponent 2 in that expression that makes it increase by larger and larger amounts. Objects that are launched upward and then come back down because of gravity also fit a quadratic model.

Suppose an object is thrown upward from a height of 5 ft. with a velocity of 60 ft./sec. Its height, ℎ(𝑡), in ft., after 𝑡 sec. is modeled by the function ℎ(𝑡) = 5 + 60𝑡 − 16(𝑡)2.

• The linear expression 5 + 60𝑡 represents the height the object would have at time 𝑡 if there were no gravity. The object would keep going up at the same speed at which it was thrown. The graph would be a line with a slope of 60, which relates to the constant speed of 60 ft./sec. • The expression −16𝑡2 represents the effect of gravity, which eventually causes the object to slow down, stop, and start falling.

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The graph of ℎ is shown. Notice how it intersects the vertical axis at 5, which means the object was thrown into the air from 5 ft. off the ground. The graph indicates that the object reaches its peak height of about 60 ft. after a little less than 2 sec. That peak is the point on the graph where the function reaches a maximum value. The point on a quadratic function where the curve changes direction is called the vertex of the graph.

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The vertex of the graph of a quadratic function is the point where the graph changes from increasing to decreasing or vice versa. The graph representing any quadratic function is a special kind of “U” shape called a parabola. Every parabola has a vertex because there is a point where it changes direction from increasing to decreasing or the other way around.

The object hits the ground a little before 4 sec. That time corresponds to the horizontal intercept of the graph. This is called a zero of the function.

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A zero of a function is an input that yields an output of zero. In other words, if 𝑓(𝒶) = 0 then 𝒶 is a zero of 𝑓.

A zero of the function ℎ is approximately 3.8 because ℎ(3.8) ≈ 0. In this situation, input values less than 0 sec. or more than about 3.8 sec. would not be meaningful, so an appropriate domain for this function would include all values of 𝑡 between 0 and about 3.8.

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Practice Problems 1. A baseball travels 𝑑 meters (m) 𝑡 sec. after being dropped from the top of a building. The distance traveled by the baseball can be modeled by the equation 𝑑 = 5𝑡2. a. Complete the table and plot the data on the coordinate plane.

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Time (sec.) Distance (m) 0

0.5 1

1.5

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b. Is the baseball traveling at a constant speed? Explain how you know.

2. The height of a diver above the water is given by ℎ(𝑡) = − 5𝑡2 + 10𝑡 + 3, where 𝑡 is time measured in sec. and ℎ(𝑡) is measured in m. Select all statements that are true about the situation.

□ The diver begins 5 m above the water.

□ The diver begins 3 m above the water.

□ The function has 1 zero that makes sense in this situation.

□ The function has 2 zeros that make sense in this situation. □ The graph that represents ℎ starts at the origin and curves upward. □ The diver begins at the same height as the water level.

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3. The height of a baseball, in ft., is modeled by the function ℎ given by the equation ℎ(𝑡) = 3 + 60𝑡 − 16𝑡2. The graph of the function is shown. a. About when does the baseball reach its maximum height?

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b. About how high is the maximum height of the baseball?

c. About when does the ball hit the ground?

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4. Each expression represents an object’s distance from the ground in meters as a function of time, 𝑡, in seconds. Object B: −5𝑡2 + 50𝑡 + 25

Object A: −5𝑡2 + 25𝑡 + 50

a. Which object was launched with the greatest vertical speed?

b. Which object was launched from the greatest height?

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Review Problems 5. Select all of the expressions that give the number of small squares in step 𝑛.

□ 2𝑛 □𝑛

□ 𝑛 + 1

□𝑛 + 1

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□ 𝑛(𝑛 + 1) □𝑛 + 𝑛 2

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□ 𝑛 + 𝑛 + 1

6. The graph shows how much insulin, in micrograms (mcg), is in a patient’s body after receiving an injection. Assume the amount of insulin continues to decay exponentially.

a. Write an equation giving the number of mcg of insulin, 𝑚, in the patient’s body ℎ hours (hr.) after receiving the injection.

b. After 3 hr., will the patient still have at least 10 mcg of insulin in their body? Explain how you know.

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Unit 8, Lesson 4: Building Quadratic Functions to Describe Situations – Part 2

Warm-Up: Graphs of Four Functions

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Which one doesn’t belong?

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Guided Activity: What Price to Charge? A company that sells movies online is deciding how much to charge customers to download a new movie. Based on data from previous sales, the company predicts that if they charge 𝑥 dollars for each download, then the number of downloads, in thousands, is 18 − 𝑥.

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1. Complete the table to show the predicted number of downloads at each listed price. Then, find the revenue at each price. The first row has been completed for you. Number of Downloads (thousands)

Revenue (thousands of dollars)

𝑥

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Price (dollars per download)

2. Is the relationship between the price of the movie and the revenue (in thousands of dollars) quadratic? Explain how you know.

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3. Plot the points that represent the revenue, 𝑟, as a function of the price of one download in dollars, 𝑥.

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4. What price would you recommend the company charge for a new movie? Explain your reasoning.

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Collaborative Activity: Domain, Vertex, and Zeros Here are 4 sets of descriptions and equations that represent some familiar quadratic functions. The graphs show what a graphing technology may produce when the equations are graphed. For each function:

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• Describe a domain that is appropriate for the situation. Think about any upper or lower limits for the input, as well as whether all numbers make sense as the input. Then, describe how the graph should be modified to show the domain that makes sense. • Identify or estimate the vertex on the graph. Describe what it means in the situation.

a. Domain:

b. Vertex:

(25 − 2𝑥) 2

𝐴 (𝑥) = 𝑥 ∙

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1. The area of a rectangle with a perimeter of 25 meters (m) and a side length 𝑥,

• Identify or estimate the zeros of the function. Describe what it means in the situation.

c. Zeros:

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2. The number of squares as a function of step number 𝑛, 𝑓(𝑛) = 𝑛2 + 4 a. Domain:

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b. Vertex:

c. Zeros:

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3. The distance, in feet (ft.), that an object has fallen 𝑡 seconds (sec.) after being dropped, 𝑔(𝑡) = 16𝑡2

c. Zeros:

b. Vertex:

a. Domain:

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4. The height, in ft., of an object 𝑡 sec. after being dropped, ℎ(𝑡) = 576 − 16(𝑡)2 a. Domain:

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b. Vertex:

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c. Zeros:

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Lesson Summary Quadratic functions often come up when studying revenue. Revenue is the money collected when someone sells something. When raffle tickets are being sold, sellers need to determine how much to charge for each ticket. Typically, when the price of the tickets is higher, fewer tickets will be sold.

ip t

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Assume that with a price of 𝑑 dollars, it is possible to sell 600 − 75𝑑 tickets. The revenue can be found by multiplying the price by the number of tickets expected to be sold. A function that models the revenue, 𝑟, collected is 𝑟(𝑑) = 𝑑(600 − 75𝑑). A graph that represents this function is shown.

It makes sense that the revenue goes down after a certain point because if the price is too high, nobody will buy a ticket. The graph shows the greatest revenue, $1,200, comes from selling the tickets for $4 each.

The graph also shows the domain of function 𝑟 is between 0 and 8, or 0 ≤ 𝑑 ≤ 8. This makes sense because the cost of the tickets can’t be negative, and if the price were more than $8, then the model would not work because the revenue collected cannot be negative. A negative revenue would mean the number of tickets sold was negative, which is not possible.

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Practice Problems 1. Based on past musical productions, a theater predicts selling 400 − 8𝑝 tickets when each ticket is sold at 𝑝 dollars. a. Complete the table to find out how many tickets the theater expects to sell and what revenues it expects to receive at the given ticket prices. Ticket Price (dollars) 5

15 20 25

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𝑝

45 50

Revenue (dollars)

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Number of Tickets Sold

b. For which ticket prices will the theater earn no revenue? Explain how you know.

c. At what ticket prices should the theater sell the tickets if it must earn at least $3,200 in revenue to break even (to not lose money) on the musical production? Explain how you know.

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2. A company sells running shoes. If the price of a pair of shoes in dollars is 𝑝, the company estimates that it will sell 50,000 − 400𝑝 pairs of shoes. Write an expression that represents the revenue in dollars from selling running shoes if a pair of shoes is priced at 𝑝 dollars.

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3. The function 𝑓 represents the revenue in dollars the school can expect to receive if it sells 220 − 12𝑥 coffee mugs for 𝑥 dollars each.

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Here is the graph of 𝑓.

Select all the statements that describe this situation.

□ At $2 per coffee mug, the revenue will be $196.

□ The school expects to sell 160 mugs if the price is $5 each.

□ The school will lose money if it sells the mugs for more than $10 each. □ The school will earn about $1,000 if it sells the mugs for $10 each.

□ The revenue will be more than $700 if the price is between $4 and $14.

□ The expected revenue will increase if the price per mug is greater than $10.

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Review Problems 4. Technology required. A small marshmallow is launched straight up in the air with a slingshot. The function ℎ, given by the equation ℎ(𝑡) = 5 + 20𝑡 − 5𝑡2, describes the height of the marshmallow in meters (m) as a function of time, 𝑡, in seconds (sec.) since it was launched.

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This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

a. Use graphing technology to graph the function ℎ.

b. About when does the marshmallow reach its maximum height?

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c. About how long does it take before the marshmallow hits the ground?

d. What domain makes sense for the function ℎ in this situation?

5. A bacteria population, 𝑝, is modeled by the equation 𝑝 = 100,000 ∙ 2𝑑, where 𝑑 is the number of days since the population was first measured. Select all statements that are true in this situation.

□ 100,000 ∙ 2 represents the bacteria population 2 days before it was first mea-

sured.

-2

□ The bacteria population 3 days before it was first measured was 800,000.

□ The population was more than 1,000 one week before it was first measured.

□ The population was more than 1,000,000 one week after it was first measured. □ The bacteria population 4 days before it was first measured was 6,250. 270 | Unit 8

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C. Graph C

B. Graph B

A. Graph A

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6. A rock is dropped from a bridge over a river. Which graph could represent the distance fallen, in feet, as a function of time in seconds?

D. Graph D

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Unit 8, Lesson 5: Standard Form and Factored Form

Warm-Up: Equivalent Quadratic Expressions

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Applying the distributive property to multiply out the factors of, or expand, 4(𝑥 + 2) gives us 4𝑥 + 8, so we know the two expressions are equivalent. We can use a rectangle with side lengths (𝑥 + 2) and 4 to illustrate the multiplication.

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1. Draw a diagram to show that 𝑛(2𝑛 + 5) and 2𝑛2 + 5𝑛 are equivalent expressions.

b. 𝑝 (4𝑝 + 9)

a. 6 � 1 𝑛 + 2�

2. For each expression, use the distributive property to write an equivalent expression. If you get stuck, consider drawing a diagram.

c. 5𝑟 �𝑟 + 3 �

d. (0.5𝑤 + 7)𝑤

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Collaborative Activity: Using Diagrams to Find Equivalent Quadratic Expressions

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1. A diagram of a rectangle with side lengths 𝑥 + 1 and 𝑥 + 3 is shown. Use this diagram to show that (𝑥 + 1)(𝑥 + 3) and 𝑥2 + 4𝑥 + 3 are equivalent expressions.

2. Show that (𝑥 − 1)(𝑥 − 1) and 𝑥2 − 2𝑥 + 1 are equivalent expressions by drawing a diagram or applying the distributive property. Show your reasoning.

3. Draw diagrams to help you write an equivalent expression for each of the following.

c. (𝑥 + 1)(𝑥 − 1)

b. (𝑥 + 𝑚)(𝑥 + 𝑛)

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a. 2𝑥 (𝑥 + 4)

d. (𝑥 − 2)2

4. Write an equivalent expression for each expression without drawing a diagram. a. (𝑥 + 2)(𝑥 + 6) b. (𝑥 − 2)(𝑥 + 3) 274 | Unit 8

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Guided Activity: What Is the Standard Form? What Is the Factored Form? The quadratic expression 𝑥2 + 4𝑥 + 3 is written in standard form.

Here are some other quadratic expressions. The expressions on the left are written in standard form and the expressions on the right are not. Not written in standard form:

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Written in standard form: 𝑥2 − 1

(2𝑥 + 3)𝑥

𝑥2 + 9𝑥

(𝑥 + 1)(𝑥 − 1)

1 𝑥2 2

3(𝑥 − 2)2 + 1

− 4(𝑥2 + 𝑥) + 7

4𝑥2 − 2𝑥 + 5 − 𝑥2 + 1

(𝑥 + 8)(− 𝑥 + 5)

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− 3𝑥2 − 𝑥 + 6

1. What are some characteristics of expressions in standard form?

2. (𝑥 + 1)(𝑥 − 1) and (2𝑥 + 3)𝑥 in the right column are quadratic expressions written in factored form. Why do you think that form is called factored form?

Lesson Summary

A quadratic function can often be defined by many different but equivalent expressions. For example, in the previous lesson, the predicted revenue, in thousands of dollars, from selling a downloadable movie at 𝑥 dollars was expressed with 𝑥(18 − 𝑥), which can also be written as 18𝑥 − 𝑥2.

Sometimes, a quadratic expression is a product of 2 factors that are each a linear expression, such as (𝑥 + 2)(𝑥 + 3). An equivalent expression can be written by thinking about each factor, (𝑥 + 2) and (𝑥 + 3), as the side lengths of a rectangle, and decomposing each side length into a variable expression and a number. The function 𝑓 can be defined by the equivalent expression (𝑥 + 2)(𝑥 + 3). © Accelerate Learning Inc. - All Rights Reserved

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The quadratic expression (𝑥 + 2)(𝑥 + 3) is in factored form. A quadratic expression that is written as the product of a constant times two linear factors is said to be in factored form.

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Applying the distributive property to multiply (𝑥 + 2) and (𝑥 + 3) results in the area of the rectangle. Adding the areas of the 4 sub-rectangles also gives the area of the rectangle. This means that (𝑥 + 2)(𝑥 + 3) is equivalent to 𝑥2 + 2𝑥 + 3𝑥 + 6, or 𝑥2 + 5𝑥 + 6. Therefore, the function 𝑓 can also be defined by 𝑥2 + 5𝑥 + 6. This form, the sum of a multiple of 𝑥2 and a linear expression, 5𝑥 + 6, is called standard form.

The standard form of a quadratic expression in x is ax2 + bx + c, where a, b, and c are constants, and a is not 0.

Practice Problems

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The 𝑎 is referred to as the coefficient of the squared term, 𝑥2, 𝑏 as the coefficient of the linear term, 𝑥, and 𝑐 as the constant term.

1. Explain why 3𝑥2 can be said to be in both standard form and factored form.

Factored Form

Standard Form

(𝑥 + 2)(𝑥 + 6)

(2𝑥 + 8)(𝑥 + 2)

𝑥2 + 12𝑥 + 32

2𝑥2 + 10𝑥 + 12

2. Match each quadratic expression that is written in factored form with an equivalent expression that is written in standard form.

(𝑥 + 8)(𝑥 + 4)

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2𝑥2 + 12𝑥 + 16 𝑥2 + 8𝑥 + 12

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3. Tyler drew a diagram to expand (𝑥 + 5)(2𝑥 + 3). a. Explain Tyler’s mistake.

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b. What is the correct expanded form of (𝑥 + 5)(2𝑥 + 3)?

4. Write each quadratic expression in standard form. Draw a diagram if needed.

a. (𝑥 + 4)(𝑥 − 1)

Review Problems

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b. (2𝑥 − 1)(3𝑥 − 1)

5. Equations defining functions 𝑎, 𝑏, 𝑐, 𝑑, and 𝑓 are shown here.

Select all the equations that represent exponential functions.

□ 𝑎(𝑥) = 2 ∙ 𝑥 3

□ 𝑏(𝑡) = � 23 �

𝑡

□ 𝑐(𝑚) = 15 ∙ 2 □ 𝑑(𝑥) = 3𝑥

𝑚

□ 𝑓(𝑡) = 3 ⋅ 2

𝑡

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6. A football player throws a football. The function ℎ given by ℎ(𝑡) = 6 + 75𝑡 − 16𝑡2 describes the football’s height in feet (ft.), 𝑡 sec. after it is thrown. Select all the statements that are true about this situation.

□ The football is thrown from ground level. □ The football is thrown from 6 ft. off the ground.

□ In the function, − 16𝑡 represents the effect of gravity.

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□ The outputs of ℎ decrease then increase in value.

□ The function ℎ has 2 zeros that make sense in this situation.

□ The vertex of the graph of ℎ gives the maximum height of the football.

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7. Each day after an outbreak of a new strain of the flu virus, a public health scientist receives a report of the number of new cases of the flu reported by area hospitals. 2

Time since Outbreak (days)

Number of New Cases of the Flu

105

Would a linear or exponential model be more appropriate for this data? Explain how you know.

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Unit 8, Lesson 6: Graphs of Functions in Standard and Factored Forms

Warm-Up: Comparing Two Graphs Consider two functions defined by 𝑓(𝑥) = 𝑥(𝑥 + 4) and 𝑔(𝑥) = 𝑥(𝑥 − 4).

−5

𝒇(𝒙) 5

𝑥-intercepts:

−2 −1

Vertex: −4

−1

−3

𝑥-intercepts:

Vertex:

−2

−3

𝒈(𝒙)

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−4

𝒙

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1. Complete the table of values for each function. Then, determine the 𝑥-intercepts and vertex of each graph.

−3

4 5

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2. Plot the points from the tables on the same coordinate plane. (Consider using different colors or markings for each set of points so you can tell them apart.)

12 10 8 6

Then, make a couple of observations about how the two graphs compare.

4 2 -6 -5 -4 -3 -2 -1 0 -2

x 1

-4

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-6 -8

Exploration Activity: Relating Expressions and Their Graphs

Here are pairs of expressions in standard and factored forms. Each pair of expressions define the same quadratic function, which can be represented with the given graph.

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1. Identify the 𝑥 -intercepts and the 𝑦 -intercept of each graph.

(𝑥 + 3)(𝑥 + 1)

𝑥2 + 4𝑥 + 3

Function 𝑓

𝑥-intercepts: 𝑦-intercept:

Function 𝑔

𝑥-intercepts:

(𝑥 − 4)(𝑥 − 1)

𝑦-intercept:

𝑥2 − 5𝑥 + 4

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Function ℎ

𝑥-intercepts:

(𝑥 − 3)(𝑥 + 3)

𝑦-intercept:

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𝑥2 − 9

Function 𝑖

𝑥-intercepts:

𝑥2 − 5𝑥

ip t

𝑥(𝑥 − 5)

𝑥(5 − 𝑥)

5𝑥 − 𝑥2

Function 𝑗

𝑦-intercept:

𝑥-intercepts: 𝑦-intercept:

𝑥-intercepts:

(𝑥 + 2)(𝑥 + 2)

𝑦-intercept:

Function 𝑘 𝑥2 + 4𝑥 + 4

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2. What do you notice about the 𝑥-intercepts, the 𝑦-intercept, and the numbers in the expressions defining each function? Make a couple of observations.

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3. Here is an expression that models function 𝑝, another quadratic function: (𝑥 − 9)(𝑥 − 1). Predict the 𝑥-intercepts and the 𝑦-intercept of the graph that represent this function.

Collaborative Activity: What Do We Need to Sketch a Graph?

𝑔(𝑥) = 2𝑥(𝑥 − 3)

𝒙-coordinate of the Vertex

𝑓(𝑥) = (𝑥 + 3)(𝑥 − 5)

𝒙-intercepts

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Equation

1. The functions 𝑓, 𝑔, and ℎ are given. Predict the 𝑥-intercepts and the 𝑥-coordinate of the vertex of each function.

ℎ(𝑥) = (𝑥 + 4)(4 − 𝑥)

2. Use graphing technology to graph the functions 𝑓, 𝑔, and ℎ. Use the graphs to check your predictions.

3. Sketch a graph that represents the expression (𝑥 − 7)(𝑥 + 11) and that shows the 𝑥-intercepts and the vertex. Think about how to find the 𝑦- coordinate of the vertex. Be prepared to explain your reasoning.

100

90 80 70 60 50 40 30 20 10 -14 -12 -10 -8 -6 -4 -2 0 -10

x 2

10 12 14

-20 -30 -40 -50 -60 -70 -80 -90 -100

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Lesson Summary Different forms of quadratic functions reveal key information about the function’s graph.

y 6 4 2

-2 -1 0 -2

• When a function is expressed in factored form, it reveals the 𝑥-intercepts of its graph. Consider the function 𝑓 given by 𝑓(𝑥) = (𝑥 − 4)(𝑥 − 1). The graph of 𝑦 = 𝑓(𝑥) shows that the 𝑥-intercepts of the graph are (1, 0) and (4, 0). Notice that 1 and 4 also appear in 𝑓(𝑥) = (𝑥 − 4)(𝑥 − 1), and they are subtracted from 𝑥.

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• When a quadratic function is expressed in standard form, the 𝑦-intercept of the graph representing the function can be determined easily. For example, the graph representing 𝑦 = 𝑥2 − 5𝑥 + 7 has a 𝑦-intercept of (0, 7). This makes sense because the 𝑦-coordinate is the 𝑦-value when 𝑥 is 0. Evaluating the expression at 𝑥 = 0 gives 𝑦 = 02 − 5(0) + 7, which equals 7.

4 2

ip t

-2 -1 0 -2

x 1

-4

The function 𝑔, given by 𝑔(𝑥) =( 𝑥 + 1)(𝑥 − 3), is written in factored form. The graph of 𝑔 is shown.

It shows 2 𝑥-intercepts at 𝑥 = − 1 and 𝑥 = 3.

Use − 1 and 3 as inputs into 𝑔. What are the outputs?

𝑔( − 1) = (− 1 + 1)( − 1 − 3) = (0) (− 4) = 0 𝑔(3) = (3 + 1)(3 − 3) = (4)(0) = 0

Because the inputs − 1 and 3 produce an output of 0, they are the zeros of the function. And because both 𝑥-values have 0 for their 𝑦-value, they also give us the 𝑥-intercepts of the graph. So, the zeros of a function have the same values as the 𝑥-coordinates of the 𝑥-intercepts of the graph of the function.

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The factored form can also help identify the vertex of the graph, which is the point where the function reaches its minimum or maximum value. Notice that the 𝑥-coordinate of the vertex is 1 and that 1 is halfway between − 1 and 3. Once the 𝑥-coordinate of the vertex is known, the 𝑦-coordinate can be found by evaluating the function for that value. 𝑔(1) = (1 + 1)(1 − 3) = 2( − 2) = − 4, so the vertex is at (1, − 4).

op y

When a quadratic function is in standard form, the 𝑦-intercept is clear. Its 𝑦-coordinate is the constant term, 𝑐, in 𝑎𝑥2 + 𝑏𝑥 + 𝑐. To find the 𝑦-intercept from factored form, evaluate the function at 𝑥 = 0 because the 𝑦-intercept is the point where the graph has an input value of 0. 𝑔(0) = (0 + 1)(0 − 3) = (1)( − 3) = − 3.

Practice Problems

1. A quadratic function 𝑓 is defined by 𝑓(𝑥) = (𝑥 − 7)(𝑥 + 3).

ip t

a. Without graphing, identify the 𝑥-intercepts of the graph of 𝑓. Explain how you know.

b. Expand (𝑥 − 7)(𝑥 + 3) and use the expanded form to identify the 𝑦-intercept of the graph of 𝑓.

2. Noah said that if we graph 𝑦 = (𝑥 − 1)(𝑥 + 6), the 𝑥-intercepts will be at (1, 0) and ( − 6, 0). Explain how you can determine, without graphing, whether Noah is correct. 3. Select all true statements about the graph that represents 𝑦 = 2𝑥(𝑥 − 11).

□ Its 𝑥-intercepts are at ( − 2,0) and (11, 0). □ Its 𝑥-intercepts are at (0, 0) and (11, 0).

□ Its 𝑥-intercepts are at (2, 0) and ( − 11, 0). □ It has only one 𝑥-intercept.

□ The 𝑥-coordinate of its vertex is − 4.5.

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□ The 𝑥-coordinate of its vertex is 11.

□ The 𝑥-coordinate of its vertex is 4.5. □ The 𝑥-coordinate of its vertex is 5.5.

Equation 𝑦 = 𝑥(𝑥 − 2)

𝑦 = (𝑥 − 4)(𝑥 + 5)

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𝑦 = − 5𝑥(3 − 𝑥)

𝒙-coordinate of the Vertex

𝒙-intercepts

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4. Determine the 𝑥-intercepts and the 𝑥-coordinate of the vertex of the graph that represents each equation.

5. Which one is the graph of the equation 𝑦 = (𝑥 − 3)(𝑥 + 5)?

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A. Graph A B. Graph B C. Graph C D. Graph D

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6. a. What are the 𝑥-intercepts of the graph of 𝑦 = (𝑥 − 2)(𝑥 − 4)? 10 9 8 7

b. Find the coordinates of another point on the graph. Show your reasoning.

6 5

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c. Sketch a graph of the equation 𝑦 = (𝑥 − 2)(𝑥 − 4).

3 2

-3 -2 -1 0 -1

x 1

-2 -3

Review Problems

7. A company sells a video game. If the price of the game in dollars is 𝑝 the company estimates that it will sell 20,000 − 500𝑝 games.

Which expression represents the revenue in dollars from selling games if the game is priced at 𝑝 dollars? A. (20,000 − 500𝑝) + 𝑝 B. (20,000 − 500𝑝) − 𝑝 C.

20,000 − 500𝑝 𝑝

D. (20,000 − 500𝑝) ⋅ 𝑝

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8. Here are graphs of two functions 𝑓 and 𝑔. An equation defining 𝑓 is 𝑓(𝑥) = 100 ⋅ 2𝑥.

Which of these could be an equation defining the function 𝑔? A. 𝑔(𝑥) = 25 ⋅ 3𝑥

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B. 𝑔(𝑥) = 50 ⋅ (1.5)𝑥 C. 𝑔(𝑥) = 100 ⋅ 3𝑥

D. 𝑔(𝑥) = 200 ⋅ (1.5)𝑥

ip t

9. Here are graphs of two exponential functions, 𝑓 and 𝑔.

The function 𝑓 is given by 𝑓(𝑥) = 100 ⋅ 2𝑥 while 𝑔 is given by 𝑔(𝑥) = 𝑎 ⋅ 𝑏𝑥.

Based on the graphs of the functions, what can you conclude about 𝑎 and 𝑏?

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Unit 8, Lesson 7: Graphing the Standard Form – Part 1

Warm-Up: Matching Graphs to Linear Equations

ip t

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Graphs A, B, and C represent 3 linear equations: 𝑦 = 2𝑥 + 4, 𝑦 = 3 − 𝑥, and 𝑦 = 3𝑥 − 2. Which graph corresponds to which equation? Explain your reasoning.

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Exploration Activity: Quadratic Graphs Galore Using graphing technology, graph 𝑦 = 𝑥2, and then experiment with each of the following changes to the function. Record your observations (include sketches, if helpful).

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1. Adding different constant terms to 𝑥2 (for example: 𝑥2 + 5, 𝑥2 + 10, 𝑥2 − 3, etc.)

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2. Multiplying 𝑥2 by different positive coefficients greater than 1 (for example: 3𝑥2, 7.5𝑥2, etc.)

3. Multiplying 𝑥2 by different negative coefficients less than or equal to −1 (for example: −𝑥2, −4𝑥2, etc.)

4. Multiplying 𝑥2 by different coefficients between −1 and 1 (for example: 2 𝑥2, 0.25𝑥2, etc.)

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Guided Activity: What Do These Tables Reveal? 1. a. Complete the table with values of 𝑥2 + 10 and 𝑥2 − 3 at different values of 𝑥. (You may also use a spreadsheet tool, if available.)

𝒙𝟐

𝒙𝟐 + 𝟏𝟎

−3 9

−2 4

𝒙𝟐 − 𝟑

−1

op y

𝒙

ip t

b. Earlier, you observed the effects on the graph of adding or subtracting a constant term from 𝑥2. Study the values in the table. Use them to explain why the graphs changed the way they did when a constant term is added or subtracted. 2.

a. Complete the table with values of 2𝑥2, 2 𝑥2, and −2𝑥2 at different values of 𝑥.

(You may also use a spreadsheet tool, if available.) 𝒙

𝒙𝟐

−3

𝟐𝒙𝟐

−2 4

−1 1

𝟏 𝟐 𝒙 𝟐

−𝟐𝒙𝟐

b. You also observed the effects on the graph of multiplying 𝑥2 by different coefficients. Study the values in the table. Use them to explain why the graphs changed they way they did when 𝑥2 is multiplied by a number greater than 1, by a negative number less than or equal to − 1, and by numbers between −1 and 1.

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Collaborative Activity: Representations of Quadratic Functions Your teacher will give your group a set of cards. Each card contains a graph or an equation.

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• Take turns with your partner to sort the cards into sets so that each set contains two equations and a graph that all represent the same quadratic function. • For each set of cards that you put together, explain to your partner how you know they belong together. • For each set that your partner puts together, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

ip t

• Once all the cards are sorted and discussed, record the equivalent equations, sketch the corresponding graph, and write a brief note or explanation about why the representations were grouped together.

Explanation:

Factored form:

Standard form:

Explanation:

Standard form:

Factored form:

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Standard form:

Explanation:

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Factored form:

Explanation:

Standard form:

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Factored form:

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Lesson Summary Remember that the graph representing any quadratic function is a shape called a parabola. It is commonly said that a parabola “opens upward” when the vertex is the lowest point on the graph and “opens downward” when the vertex is the highest point on the graph. Each coefficient in a quadratic expression written in standard form, 𝑎𝑥2 + 𝑏𝑥 + 𝑐, tells something important about the graph that represents it.

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The graph of 𝑦 = 𝑥2 is a parabola opening upward with a vertex at (0, 0). Adding a constant term 5 results in 𝑦 = 𝑥2 + 5 and raises the graph by 5 units. Subtracting 4 from 𝑥2 results in 𝑦 = 𝑥2 − 4 and moves the graph down 4 units.

𝒙

−3

−2

−1

𝒙𝟐 + 𝟓

𝒙𝟐 − 𝟒

−3

−4

−3

𝒙𝟐

ip t

A table of values can be used to show that adding 5 to 𝑥2 increases all the output values of 𝑦 = 𝑥2 by 5, which explains why the graph moves up 5 units. Subtracting 4 from 𝑥2 decreases all the output values of 𝑦 = 𝑥2 by 4, which explains why the graph shifts down 4 units.

14 5

In general, the constant term of a quadratic expression in standard form influences the vertical position of the graph. For an expression with no constant term, such as 𝑥2 or 𝑥2 + 9𝑥, the constant term is 0, so the 𝑦-intercept of the graph is on the 𝑥-axis. It’s not shifted up or down relative to the 𝑥-axis.

The coefficient of the squared term in a quadratic function also tells something about its graph. The coefficient of the squared term in 𝑦 = 𝑥2 is 1. Its graph is a parabola that opens upward. • Multiplying 𝑥2 by a number greater than 1 makes the graph steeper, so the parabola is narrower than that representing 𝑥2.

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• Multiplying 𝑥2 by a number less than 0 makes the parabola open downward. These impacts can also be observed using a table of values.

𝒙

−3

−2

−1

𝟐𝒙𝟐

𝒙𝟐

−𝟐𝒙𝟐

−18

−8

−2

−8

−18

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Comparing the output values of 2𝑥2 and −2𝑥2 shows they are opposites, which suggests that 1 graph is a reflection of the other across the 𝑥-axis.

Practice Problems

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1. Here are four graphs. Match each graph with a quadratic equation that it represents.

Graph

Equation

Graph A

𝑦 = 𝑥2

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𝑦 = 𝑥2 + 5 𝑦 = 𝑥2 + 7

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2. The two equations 𝑦 = (𝑥 + 2)(𝑥 + 3) and 𝑦 = 𝑥2 + 5𝑥 + 6 are equivalent. a. Which equation helps find the 𝑥-intercepts most efficiently?

op y

b. Which equation helps find the 𝑦-intercept most efficiently?

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3. Here is a graph that represents 𝑦 = 𝑥2.

a. 𝑦 = 𝑥2 − 4

On the same coordinate plane, sketch and label the graph that represents each equation.

b. 𝑦 = − 𝑥2 + 5

4. Select all equations whose graphs have a 𝑦-intercept with a positive 𝑦-coordinate.

□ 𝑦 = 𝑥 + 3𝑥 − 2

□ 𝑦 = 𝑥 − 10𝑥 2

□ 𝑦 = (𝑥 − 1)

□ 𝑦 = 5𝑥 − 3𝑥 − 5 2

□ 𝑦 = (𝑥 + 1)(𝑥 + 2)

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Review Problems 5. Here is a graph of the function 𝑔 given by 𝑔(𝑥) = 𝑎 ⋅ 𝑏𝑥.

What can you say about the value of 𝑏? Explain how you know.

op y

a. What are the 𝑥-intercepts of the graph that represents 𝑦 = (𝑥 + 1)(𝑥 + 5)? Explain how you know.

ip t

b. What is the 𝑥-coordinate of the vertex of the graph that represents 𝑦 = (𝑥 + 1)(𝑥 + 5)? Explain how you know.

c. Find the 𝑦-coordinate of the vertex. Show your reasoning.

d. Sketch a graph of 𝑦 = (𝑥 + 1)(𝑥 + 5).

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7. Determine the 𝑥-intercepts, the vertex, and the 𝑦-intercept of the graph of each equation. Equation

𝒙-intercepts

Vertex

𝒚-intercept

op y

𝑦 = (𝑥 − 5)(𝑥 − 3)

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𝑦 = 2𝑥(8 − 𝑥)

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Unit 8, Lesson 8: Graphing the Standard Form – Part 2

Warm-Up: Equivalent Expressions 1. Complete each row with an equivalent expression in standard form or factored form. Factored Form

op y

Standard Form 𝑥2

𝑥2 − 18𝑥

𝑥(𝑥 + 9)

ip t

− 𝑥2 + 10𝑥

𝑥(6 − 𝑥)

− 𝑥(𝑥 + 2.75)

2. What do the quadratic expressions in each column have in common (besides the fact that everything in the left column is in standard form and everything in the other column is in factored form)? Be prepared to share your observations.

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Exploration Activity: What about the Linear Term? 1. Using graphing technology: a. Graph 𝑦 = 𝑥2, and then experiment with adding different linear terms (for example, 𝑥2 + 4𝑥, 𝑥2 + 20𝑥, 𝑥2 − 50𝑥). Record your observations.

op y

b. Graph 𝑦 = 𝑥2, and then experiment with adding different linear terms. Record your observations.

Equation

𝑦 = 𝑥2 + 50𝑥

𝑦 = 𝑥2 − 36𝑥

𝑦 = 𝑥2 + 6𝑥

𝑦 = 𝑥2 − 10𝑥

𝒙-coordinate of Vertex

ip t

𝒙-intercepts

2. Use your observations to help you complete the table without graphing the equations.

3. Some quadratic expressions have no linear terms. Find the 𝑥-intercepts and the 𝑥-coordinate of the vertex of the graph representing each equation. (Note it is possible for the graph to not intersect the 𝑥-axis.) If you get stuck, try graphing the equations.

a. 𝑦 = 𝑥2 − 25

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b. 𝑦 = 𝑥2 + 16

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Collaborative Activity: Writing Equations to Match Graphs Use graphing technology to graph a function that matches each graph. Make sure your graph goes through all 3 points shown. Record the equation you use for each graph. B

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op y

Equation: D

Equation: C

Equation:

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Equation:

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Equation:

Equation: J

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op y

Equation:

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Equation:

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Lesson Summary A quadratic function written in standard form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 reveals features about the graph that represents it. The coefficient 𝑎 indicates whether the graph of the function opens upward or downward, and it also gives information about whether the parabola is narrow or wide. The constant term 𝑐 provides information about the graph’s vertical position.

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Recall that the graph representing 𝑦 = 𝑥2 is an upward-opening parabola with a vertex at (0, 0). The vertex is also the 𝑥-intercept and the 𝑦-intercept.

Adding a 6 to the squared term, 𝑦 = 𝑥2 + 6, shifts the graph up so the vertex is at (0, 6). The vertex is the 𝑦-intercept, and the graph is centered on the 𝑦-axis. What can the linear term, 𝑏𝑥, tell us about the graph representing a quadratic function?

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The linear term has a somewhat mysterious effect on the graph of a quadratic function. The graph seems to shift both horizontally and vertically. When 𝑏𝑥 is added to 𝑥2, where 𝑏 is not 0, the graph of 𝑦 = 𝑥2 + 𝑏𝑥 is no longer centered on the 𝑦-axis.

• When 6𝑥 is added to the squared term, the result is 𝑦 = 𝑥2 + 6𝑥. Writing 𝑥2 + 6𝑥 in factored form as 𝑥(𝑥 + 6) gives the zeros of the function, 0 and − 6. Adding the term 6𝑥 shifts the graph down and to the left, and the 𝑥-intercepts are now ( − 6, 0) and (0, 0). The vertex is no longer the 𝑦-intercept, and the graph is no longer centered on the 𝑦-axis.

• Suppose − 6𝑥 is added to 𝑥2. The expression 𝑥2 − 6𝑥 can be rewritten as 𝑥(𝑥 − 6), which reveals the zeros, 0 and 6. Adding a negative linear term to a squared term shifts the graph to the right and down. The 𝑥-intercepts are now (0, 0) and (6, 0). The vertex is no longer the 𝑦-intercept and the graph is not centered on the 𝑦-axis.

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Practice Problems

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op y

1. Here are four graphs. Match each graph with the quadratic equation that it represents.

Graph

Graph A

Graph B

Graph C Graph D

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Equation 𝑦 = 𝑥2 + 𝑥

𝑦 = − 𝑥2 + 2 𝑦 = 𝑥2 − 𝑥

𝑦 = 𝑥2 + 3𝑥

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2. Complete the table without graphing the equations. Equation

𝒙-coordinate of Vertex

𝒙-intercepts

𝑦 = 𝑥2 + 12𝑥 𝑦 = 𝑥2 − 3𝑥

𝑦 = 𝑥2 + 16𝑥

3. Here is a graph that represents 𝑦 = 𝑥2.

op y

𝑦 = −𝑥2 − 24𝑥

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a. Describe what would happen to the graph if the original equation were changed to 𝑦 = 𝑥2 − 6𝑥. Predict the 𝑥- and 𝑦-intercepts of the graph and the quadrant where the vertex is located.

b. Sketch the graph of the equation 𝑦 = 𝑥2 − 6𝑥 on the same coordinate plane as 𝑦 = 𝑥2.

4. Select all equations whose graph opens upward.

□ 𝑦 = − 𝑥 + 9𝑥 2

□ 𝑦 = 10𝑥 − 5𝑥 □ 𝑦 = (2𝑥 − 1)

□ 𝑦 = (1 − 𝑥)(2 + 𝑥) □ 𝑦 = 𝑥 − 8𝑥 − 7 2

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Equation:

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5. Technology required. Write an equation for a function that can be represented by each given graph. Then, use graphing technology to check each equation you wrote.

Review Problems

(𝑥 + 3)(𝑥 + 4)

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6. Match each quadratic expression that is written as a product with an equivalent expression that is expanded. 𝑥2 + 10𝑥 + 21

(𝑥 + 3)(𝑥 + 7)

3𝑥2 + 13𝑥 + 12

(𝑥 + 7)(3𝑥 + 1)

𝑥2 + 7𝑥 + 12

(3𝑥 + 4)(𝑥 + 3)

3𝑥2 + 22𝑥 + 7

7. A bank loans $4,000 to a customer at a 9 2 % annual interest rate.

Write an expression to represent how much the customer will owe, in dollars, after 5 years without payment.

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Unit 8, Lesson 9: Graphs That Represent Situations

Warm-Up: A Jumping Frog

1. Find ℎ(0) and ℎ(0.8). What do these values mean in terms of the frog’s jump?

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The height, in inches (in.), of a frog’s jump is modeled by the equation ℎ(𝑡) = 60𝑡 − 75𝑡2, where the time, 𝑡, after it jumped is measured in seconds (sec.).

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2. How much time after it jumped did the frog reach the maximum height? Explain how you know.

Exploration Activity: A Catapulted Pumpkin

The equation ℎ = 2 + 23.7𝑡 − 4.9𝑡2 represents the height of a pumpkin that is catapulted up in the air as a function of time, 𝑡, in sec. The height is measured in meters (m) above ground. The pumpkin is shot up at a vertical velocity of 23.7 m per sec.

1. Without writing anything down, consider these questions:

a. What do you think the 2 in the equation tells us in this situation? What about the −4.9𝑡2?

b. If we graph the equation, will the graph open upward or downward? Why? c. Where do you think the vertical intercept would be?

d. What about the horizontal intercepts?

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2. Graph the equation using graphing technology.

3. Identify the vertical and horizontal intercepts, and the vertex of the graph. Explain what each point means in this situation.

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Guided Activity: Flight of Two Baseballs

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Here is a graph that represents the height of a baseball, ℎ, in feet (ft.), as a function of time, 𝑡, in sec., after it was hit by Player A.

The function 𝑔, defined by 𝑔(𝑡) = (− 16𝑡 − 1)(𝑡 − 4), also represents the height in ft. of a baseball 𝑡 sec. after it was hit by Player B. Without graphing function 𝑔, answer the following questions and explain or show how you know.

1. Which player’s baseball stayed in flight longer?

2. Which player’s baseball reached a greater maximum height?

3. How can you find the height at which each baseball was hit?

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Collaborative Activity: Info Gap: Rocket Math Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

2. Ask your partner for the specific information that you need.

3. Explain to your partner how you are using the information to solve the problem.

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3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”

1. Silently read your card and think about what information you need to answer the question.

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1. Silently read the information on your card.

If your teacher gives you the problem card:

If your teacher gives you the data card:

5. Share the data card, and discuss your reasoning.

5. Read the data card, and discuss your reasoning.

4. Read the problem card, and solve the problem independently.

4. When you have enough information, share the problem card with your partner, and solve the problem independently.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Lesson Summary

A tennis ball is hit straight up in the air, and its height, in ft. above the ground, is modeled by the equation 𝑓(𝑡) = 4 + 12𝑡 − 16𝑡2, where 𝑡 represents the time, in sec., after the ball is hit. A graph representing the function is shown, from the time the tennis ball is hit until the time it reaches the ground. © Accelerate Learning Inc. - All Rights Reserved

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The graph shows the following information: • The 4 in the equation means the graph of the function intersects the vertical axis at 4. It shows that the tennis ball is 4 ft. off the ground, at 𝑡 = 0, when it is hit.

• The horizontal intercept is (1, 0). This means that the tennis ball reaches the ground 1 sec. after it is hit.

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• The vertex of the graph is at approximately (0.4, 6.3). This means that about 0.4 sec. after the ball is hit, it reaches a maximum height of about 6.3 ft. above the ground.

The equation can be rewritten in factored form as 𝑓(𝑡) = (−16𝑡 − 4)(𝑡 − 1). From this form, it can be determined that the zeros of the function are 𝑡 = 1 and 𝑡 = − 1 . The

Practice Problems

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was hit is irrelevant in this context.

4 1 negative zero, − , is not meaningful in this situation because the time before the ball 4

1. The height, in ft., of a thrown football is modeled by the equation 𝑓(𝑡) = 6 + 30𝑡 − 16𝑡2, where time 𝑡 is measured in sec.

a. What does the constant 6 mean in this situation?

b. What does the 30𝑡 mean in this situation?

c. How do you think the squared term − 16𝑡2 affects the value of the function 𝑓? What does this term reveal about the situation?

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2. Here are graphs of functions 𝑓 and 𝑔.

Each represents the height of an object being launched into the air as a function of time. a. Which object was launched from a higher point?

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b. Which object reached a higher point?

c. Which object was launched with the higher upward velocity?

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d. Which object landed last?

3. Technology required. The function ℎ given by ℎ(𝑡) = (1 − 𝑡)(8 + 16𝑡) models the height of a ball, in ft., 𝑡 sec. after it was thrown.

a. Find the zeros of the function. Show or explain your reasoning.

b. What do the zeros tell us in this situation? Are both zeros meaningful?

c. From what height is the ball thrown? Explain your reasoning.

d. About when does the ball reach its highest point, and about how high does the ball go? Show or explain your reasoning.

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4. The height, in ft., of an arrow is modeled by the equation ℎ(𝑡) = (1 + 2𝑡)(18 − 8𝑡), where 𝑡 is sec. after the arrow is shot. a. When does the arrow hit the ground? Explain or show your reasoning.

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b. From what height is the arrow shot? Explain or show your reasoning.

5. Two objects are launched into the air.

• The height, in ft., of Object A is given by the equation 𝑓(𝑡) = 4 + 32𝑡 − 16𝑡2. In both functions, 𝑡 is sec. after launch.

• The height, in ft., of the Object B is given by the equation 𝑔(𝑡) = 2.5 + 40𝑡 − 16𝑡2.

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a. Which object was launched from a greater height? Explain how you know.

b. Which object was launched with a greater upward velocity? Explain how you know.

Review Problems

a. Predict the 𝑥- and 𝑦-intercepts of the graph of the quadratic function defined by the expression (𝑥 + 6)(𝑥 − 6). Explain how you made your predictions. b. Technology required. Check your predictions by graphing 𝑦 = (𝑥 + 6)(𝑥 − 6).

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7. Technology required. A student needs to get a loan of $12,000 for the first year of college. Bank A has an annual interest rate of 5.75%, Bank B has an annual interest rate of 7.81%, and Bank C has an annual rate of 4.45%. a. If we graph the amount owed for each loan as a function of years without payment, predict what the three graphs would look like. Describe or sketch your prediction.

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b. Use graphing technology to plot the graph of each loan balance.

c. Based on your graph, how much would the student owe for each loan when they graduate from college in four years?

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d. Based on your graph, if no payments are made, how much would the student owe for each loan after 10 years?

8. Technology required. The functions 𝑓 and 𝑔 are given by 𝑓(𝑥) = 13𝑥 + 6 and 𝑔(𝑥) = 0.1 ⋅ (1.4)𝑥.

a. Which function eventually grows faster, 𝑓 or 𝑔? Explain how you know.

b. Use graphing technology to decide when the graphs of 𝑓 and 𝑔 meet.

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Unit 8, Lesson 10: Vertex Form

Warm-Up: Equivalent Expressions

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Two sets of equations for quadratic functions are shown. In each set, the expressions that define the output are equivalent. Set 1

Set 2

𝑝(𝑥) = − 𝑥2 + 6𝑥 − 5

𝑔(𝑥) = 𝑥(𝑥 + 4)

𝑞(𝑥) = (5 − 𝑥)(𝑥 − 1)

𝑟(𝑥) = − 1(𝑥 − 3)2 + 4

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ℎ(𝑥) = (𝑥 + 2)2 − 4

𝑓(𝑥) = 𝑥2 + 4𝑥

1. Discuss with a partner how functions 𝑓 and 𝑔 or functions 𝑝 and 𝑞 are equivalent.

2. Show that the expressions defining ℎ and 𝑓 are equivalent.

3. Show that the expressions defining 𝑟 and 𝑝 are equivalent.

Exploration Activity: Playing with Parameters

1. In the Warm-Up, the expressions that define ℎ and 𝑟 are written in vertex form. ℎ(𝑥) = (𝑥 + 2)2 − 4

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𝑟(𝑥) = − 1(𝑥 − 3)2 + 4

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The graphs representing the quadratic functions ℎ and 𝑟 are shown. Function 𝒓

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Function 𝒉

Discuss with your partner why you think expressions such as those defining ℎ and 𝑟 are said to be written in vertex form. Summarize your discussion.

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2. Using graphing technology, graph 𝑦 = 𝑥2. Then, add different numbers to 𝑥 before it is squared (for example, 𝑦 = (𝑥 + 4)2, 𝑦 = (𝑥 − 3)2), and observe how the graph changes. Record your observations.

3. Graph 𝑦 = (𝑥 − 1)2. Then, experiment with each of the following changes to the function, and see how they affect the graph and the vertex.

a. Adding different constant terms to (𝑥 − 1)2 (for example, (𝑥 − 1)2 + 5, (𝑥 − 1)2 − 9)

b. Multiplying (𝑥 − 1)2 by different coefficients (for example, 𝑦 = 3(𝑥 − 1)2, 𝑦 = −2(𝑥 − 1)2)

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4. For rows 1–6, without graphing, predict the coordinates of the vertex of the graph of each quadratic function, and predict whether the graph opens up or down. Leave the last row blank until the next question. Coordinates of Vertex

Equations 𝑦 = (𝑥 + 10)2

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𝑦 = (𝑥 − 4)2 + 8

𝑦 = − (𝑥 − 4)2 + 8

𝑦 = 𝑥2 − 7

𝑦 = 1 (𝑥 + 3)2 − 5

𝑦 = 𝑎(𝑥 + 𝑚)2 + 𝑛

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𝑦 = − (𝑥 + 100)2 + 50

Graph Opens Up or Down?

5. Use graphing technology to check your predictions. If they are incorrect, revise them. Then, complete the last row of the table.

Guided Activity: Sharing a Vertex

Here are two equations that define quadratic functions. 𝑝(𝑥) = −(𝑥 − 4)2 + 10

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𝑞(𝑥) = 1 (𝑥 − 4)2 + 10 2

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1. The graph of 𝑝 passes through (0, −6) and (4, 10), as shown on the coordinate plane.

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Find the coordinates of another point on the graph of 𝑝. Explain or show your reasoning. Then, use the points to sketch and label the graph.

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2. On the same coordinate plane, identify the vertex and two other points that are on the graph of 𝑞. Explain or show your reasoning. Sketch and label the graph of 𝑞.

3. Priya says, “Once I know the vertex is (4, 10), I can find out, without graphing, whether the vertex is the maximum or the minimum of function 𝑝. I would just compare the coordinates of the vertex with the coordinates of a point on either side of it.”

Complete the table and then explain how Priya might have reasoned about whether the vertex is the minimum or maximum.

𝑥

𝒑(𝑥)

Lesson Summary

Sometimes, the expressions that define quadratic functions are written in vertex form. The vertex form of a quadratic expression in 𝑥 is 𝑎(𝑥 − h)2 + 𝑘, where 𝑎, ℎ, and 𝑘 are constants and 𝑎 is not 0.

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The vertex form is especially helpful for finding the vertex of the graph of a quadratic function. For the function 𝑝, given by 𝑝(𝑥) = (𝑥 − 3)2 + 1, the expression (𝑥 − 3)2 reveals that the vertex has an 𝑥-coordinate of 3, and the constant term 1 reveals its 𝑦-coordinate. The vertex of function 𝑝 is at (3, 1).

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Also notice that when the squared expression (𝑥 − 3)2 has a positive coefficient, the graph opens upward. This means that the vertex (3, 1) represents the minimum function value.

But why does the function 𝑝 take on its minimum value when 𝑥 is 3?

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One explanation is that when 𝑥 = 3, the squared term (𝑥 − 3)2 equals 0 because (3 − 3)2 = 02 = 0. When 𝑥 is any other value besides 3, the squared term (𝑥 − 3)2 is a positive number greater than 0. This means that the output when 𝑥 ≠ 3 will always be greater than the output when 𝑥 = 3, so the function 𝑝 has a minimum value at 𝑥 = 3.

𝒙

This table shows some values of the function for some values of 𝑥. Notice that the output is the lowest when 𝑥 = 3, and it increases both as 𝑥 increases and as it decreases.

(𝒙 − 𝟯)𝟮 � 𝟭

The squared term sometimes has a negative coefficient; for instance, (ℎ)𝑥 = −2(𝑥 + 4)2. The 𝑥- value that makes (𝑥 + 4)2 equal 0 is −4 because (− 4 + 4)2 = 02 = 0. Any other 𝑥-value makes (𝑥 + 4)2 greater than 0. But when (𝑥 + 4)2 is multiplied by a negative number, − 2, the resulting expression, − 2(𝑥 + 4)2, ends up being negative. This means that the output when 𝑥 ≠ −4 will always be less than the output when 𝑥 = −4, so the function ℎ has its maximum value when 𝑥 = −4.

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Remember the 𝑦-coordinate of the 𝑦-intercept is the value of the function when 𝑥 = 0. If 𝑔 is defined by 𝑔(𝑥) = (𝑥 + 1)2 − 5, then the 𝑦-intercept is (0, − 4) because 𝑔(0) = (0 + 1)2 − 5 = − 4. Its vertex is at (−1, −5). Another point on the graph with the same 𝑦-coordinate is located the same horizontal distance from the vertex but on the other side.

Practice Problems

1. Select all of the quadratic expressions in vertex form.

□ (𝑥 − 2) + 1 2

□𝑥 −4

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□ 𝑥(𝑥 + 1) □ (𝑥 + 3)

□ (𝑥 − 4) + 6

2. Here are two equations. One defines function 𝑚 and the other defines function 𝑝. 𝑚(𝑥) = 𝑥(𝑥 + 6)

𝑝(𝑥) = (𝑥 + 3)2 − 9

a. Show that the expressions defining 𝑚 and 𝑝 are equivalent.

b. What is the vertex of the graph of 𝑚? Explain how you know. c. What are the 𝑥-intercepts of the graph of 𝑝? Explain how you know.

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3. Andre thinks the vertex of the graph of the equation 𝑦 = (𝑥 + 2)2 − 3 is (2, − 3). Lin thinks the vertex is ( − 2, 3). Do you agree with either of them? Explain your reasoning. 4. a. Where is the vertex of the graph that represents 𝑦 = (𝑥 − 2)2 − 8?

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c. Identify one other point on the graph of the equation. Explain or show how you know.

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b. Where is the 𝑦-intercept? Explain how you know.

d. Sketch a graph that represents the equation.

5. The function 𝑣 is defined by 𝑣(𝑥) = 1 (𝑥 + 5)2 − 7. 2

Without graphing, determine if the vertex of the graph representing 𝑣 shows the minimum or maximum value of the function. Explain how you know.

6. Match each graph to an equation that represents it.

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Graph

Equation

Graph A

𝑦 = − 2(𝑥 − 6)2 − 5

Graph B

𝑦 = (𝑥 − 6)2 − 5

Graph C

𝑦 = 6(𝑥 − 6)2 − 5

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𝑦 = − 1 (𝑥 − 6)2 − 5

Graph D

Review Problem

7. Technology required. Two objects are launched into the air.

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• The height, in feet (ft.), of Object A is given by the equation 𝑓(𝑡) = 4 + 32𝑡 − 16𝑡2.

• The height, in ft., of the Object B is given by the equation 𝑔(𝑡) = 2.5 + 40𝑡 − 16𝑡2.

In both functions, 𝑡 is seconds (sec.) after launch.

Use technology to graph each function in the same graphing window.

a. What is the maximum height of each object?

b. Which object hits the ground first? Explain how you know.

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Unit 8, Lesson 11: Changing the Vertex

Warm-Up: Graphs of Two Functions

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Here are graphs representing the functions 𝑓 and 𝑔, given by 𝑓(𝑥) = 𝑥(𝑥 + 6) and 𝑔(𝑥) = 𝑥(𝑥 + 6) + 4.

1. Which graph represents each function? Explain how you know.

2. Where does the graph of 𝑓 meet the 𝑥-axis? Explain how you know.

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Collaborative Activity: Shifting the Graph 1. How would you change the equation 𝑦 = 𝑥2 so that the vertex of the graph of the new equation is located at the following coordinates and the graph opens as described?

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a. (0, 11), opens up b. (7, 11), opens up

c. (7, −3), opens down

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2. Use graphing technology to verify your predictions. Adjust your equations if necessary.

3. Kiran graphed the equation 𝑦 = 𝑥2 + 1 and noticed that the vertex is at (0, 1). He changed the equation to 𝑦 = (𝑥 − 3)2 + 1 and saw that the graph shifted 3 units to the right and the vertex is now at (3, 1).

Next, he graphed the equation 𝑦 = 𝑥2 + 2𝑥 + 1, observed that the vertex is at (−1, 0). Kiran thought, “If I change the squared term 𝑥2 to (𝑥 − 5)2, the graph will move 5 units to the right and the vertex will be at (4, 0).”

Do you agree with Kiran? Explain or show your reasoning.

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Exploration Activity: A Peanut Jumping over a Wall This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

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Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to show the path of the main character, an animated peanut, jumping over a wall.

Mai uses the equation 𝑦 = −0.1(𝑥 − ℎ)2 + 𝑘 to represent the path of the jump, where 𝑦 represents the height of the peanut as a function of the horizontal distance it travels 𝑥. On the screen, the base of the wall is located at (22, 0), with the top of the wall at (22, 4.5).

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The dashed curve in the picture shows the graph of one equation Mai tried, where the peanut fails to make it over the wall.

1. What are the values of 𝑘 and ℎ in this equation?

2. This applet is programmed to use Mai’s equation. Experiment with the applet to find an equation to get the peanut over the wall and keep it on the screen. Be prepared to explain your reasoning.

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Guided Activity: Smiley Face

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Do you see 2 “eyes” and a smiling “mouth” on the graph? The 3 arcs on the graph all represent quadratic functions that were initially defined by 𝑦 = 𝑥2, but whose equations were later modified.

1. Write equations to represent each curve in the smiley face.

2. What domain is used for each function to create this graph?

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Lesson Summary The graphs of 𝑦 = 𝑥2, 𝑦 = 𝑥2 + 12, and 𝑦 = (𝑥 + 3)2 all have the same shape, but their locations are different. The graph that represents 𝑦 = 𝑥2 has its vertex at (0, 0).

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Notice that adding 12 to 𝑥2 raises the graph by 12 units, so the vertex of that graph is at (0, 12). Replacing 𝑥2 with (𝑥 + 3)2 shifts the graph 3 units to the left, so the vertex is at (−3, 0).

A graph can also be shifted both horizontally and vertically.

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The graph that represents 𝑦 = (𝑥 + 3)2 + 12 will look like the graph of 𝑦 = 𝑥2, but it will be shifted 12 units up and 3 units to the left. Its vertex is at (−3, 12).

The graph representing the equation 𝑦 = −(𝑥 + 3)2 + 12 has the same vertex, at (−3, 12), but because the squared term (𝑥 + 3)2 is multiplied by a negative number, the graph is reflected horizontally over the line 𝑦 = 3, so that it opens downward.

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Practice Problems 1. Here is the graph of quadratic function 𝑓.

• Andre uses the expression (𝑥 − 5)2 + 7 to define 𝑓. • Noah uses the expression (𝑥 + 5)2 − 7 to define 𝑓.

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2. Here are the graphs of 𝑦 = 𝑥2, 𝑦 = 𝑥2 − 5, and 𝑦 = (𝑥 + 2)2 − 8.

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Do you agree with either of them? Explain your reasoning.

a. How do the 3 graphs compare?

b. Compare the graphs of 𝑦 = 𝑥2 and 𝑦 = 𝑥2 − 5. What role does the −5 play in the comparison?

c. Compare the graphs of 𝑦 = 𝑥2 and 𝑦 = (𝑥 + 2)2 − 8. What role does the +2 play in the comparison?

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3. Select all the equations with a graph whose vertex has both a positive 𝑥- and a positive 𝑦-coordinate.

□ 𝑦 = 𝑥

□ 𝑦 = (𝑥 − 1)

□ 𝑦 = (𝑥 − 3) + 2 2

□ 𝑦 = 2(𝑥 − 4) − 5

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□ 𝑦 = 0.5(𝑥 + 2) + 6 2

□ 𝑦 = −(𝑥 − 4) + 3 2

□ 𝑦 = −2(𝑥 − 3) + 1

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Review Problems

4. The height, in feet (ft.), of a soccer ball is modeled by the equation 𝑔(𝑡) = 2 + 50𝑡 − 16𝑡2, where time 𝑡 is measured in seconds (sec.) after it was kicked.

a. How far above the ground was the ball when kicked?

b. What was the initial upward velocity of the ball?

c. Why is the coefficient of the squared term negative?

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5. a. What is the vertex of the graph of the function 𝑓 defined by 𝑓(𝑥) = − (𝑥 − 3)2 + 6? b. Identify the 𝑦-intercept and one other point on the graph of this function.

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c. Sketch the graph of 𝑓.

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6. Technology required. Function ℎ is defined by ℎ(𝑥) = 5𝑥 + 7 and function 𝑘 is defined by 𝑘(𝑥) = (1.005)𝑥. 𝒙

a. Complete the table with values of ℎ(𝑥) and 𝑘(𝑥). When necessary, round to 2 decimal places.

𝒌(𝒙)

10 50

100

b. Which function do you think eventually grows faster? Explain your reasoning.

𝒉(𝒙)

c. Use graphing technology to verify your answer to the previous question.

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Unit 8, Lesson 12: Identifying Key Features of Quadratic Functions

Warm-Up: Which Form to Use?

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Expressions in different forms can be used to define the same function. Here are three ways to define a function 𝑓. 𝑓(𝑥) = 𝑥2 − 4𝑥 + 3

(standard form)

𝑓(𝑥) = (𝑥 − 2)2 − 1

(vertex form)

(factored form)

𝑓(𝑥) = (𝑥 − 3)(𝑥 − 1)

b. the vertex

a. the 𝑥-intercepts

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1. Which form would you use if you want to find the following features of the graph of 𝑓? Be prepared to explain your reasoning.

c. the 𝑦-intercept

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Exploration Activity: Exploring Additional Key Features of Quadratic Functions 1. Work with your partner to describe the red and purple parabolas. Note any similarities or differences.

5 y-intercept vertex

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vertex

x-intercept

-5

x-intercept

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-5

2. On the graph, circle where the red parabola has negative 𝑦-values.

3. Draw a square on the graph where the red parabola has positive 𝑦-values.

4. Discuss with your partner what key feature can be helpful in determining where a quadratic function has positive 𝑦-values and negative 𝑦-values.

5. Discuss with your partner why the purple parabola only has positive 𝑦-values. Summarize your discussion.

6. Every parabola has a line of symmetry. Draw the lines of symmetry for the red and purple parabolas on the graph. 7. What do you notice about the 2 lines of symmetry?

8. Complete the statements about the end behavior of each parabola. a. For the red parabola, as the 𝑥-values get infinitely smaller, the ends of the graph of the red parabola approach

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−∞. ∞.

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b. For the red parabola, as the 𝑥-values get infinitely larger, the ends of the graph of the red parabola approach

−∞. ∞.

c. For the purple parabola, as the 𝑥-values get infinitely smaller,

−∞. ∞.

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the ends of the graph of the purple parabola approach

d. For the purple parabola, as the 𝑥-values get infinitely larger,

the ends of the graph of the purple parabola approach

−∞. ∞.

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9. How does the fact that a parabola opens down help to determine the end behavior?

10. Complete the statements about the domain and range of each function.

−∞ � 𝑥 � ∞. −∞ � 𝑥 � 2. 2 � 𝑥 � ∞.

a. For the red parabola, the domain is

b. For the red parabola, the range is

−∞ � 𝑦 � ∞. −∞ � 𝑦 � 2. 2 � 𝑦 � ∞.

c. For the purple parabola, the domain is

d. For the purple parabola, the range is

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−∞ � 𝑥 � ∞. −∞ � 𝑥 � 2. 2 � 𝑥 � ∞. −∞ � 𝑦 � ∞. −∞ � 𝑦 � 2. 2 � 𝑦 � ∞.

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Guided Activity: Using Key Features of Quadratics to Graph 1. Work with your partner to complete the following steps for the equation 𝑦 = −1(𝑥 − 2)2 + 3.

• Determine all of the key features possible using the equation and table of values.

Equation

Table of Values 𝒙

𝒚

Vertex

−1

Key Features

−1

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Decreasing interval(s)

Positive interval(s)

Negative interval(s) Left end behavior

As 𝑥 →_____, 𝑦 →_____.

Right end behavior

Opens

−6

Range

Increasing interval(s)

𝑥-intercept(s) Domain

−1

𝑦 = −1(𝑥 − 2)2 + 3

𝑦-intercept

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• Use the key features to graph the quadratic.

As 𝑥 →_____, 𝑦 →_____.

-5

Graph

-5

-10

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Lesson Summary Key features of quadratic functions can be determined from various representations of the function, including equations, graphs, and tables. The given representations can be used to create any other representations needed until all key features in question are determined.

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The graph of 𝑦 = −1(𝑥 + 1)2 + 9 is shown, along with 5 equivalent forms of the equation. The key features of the graph are color-coded in the graph, key features table, and forms of the equation shown. Key Features

Graph

(0, 8)

-10

𝑦-intercept

(0, 8)

𝑥-intercept Vertex

Axis of symmetry

(2, 0)

(–4, 0)

(−4, 0)

𝑥-intercept

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(–1, 9)

Increasing interval

-10

Decreasing interval Positive interval Negative intervals

𝑦 = −1(𝑥 − (−1)2 + 9

Domain

𝑦 = −1(𝑥 − (−4))(𝑥 − 2)

Left end behavior

Equations

𝑦 = −1(𝑥 + 1)2 + 9

𝑦 = −1(𝑥 + 4)(𝑥 − 2) 𝑦 = −1𝑥2 − 2𝑥 + 8

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Range

(2, 0)

(−1, 9) 𝑥 = −1

𝑥 < −1 𝑥 > −1

−4 < 𝑥 < 2

𝑥 < −4 or 𝑥 > 2 𝑅

𝑦 ≤ 9

Right end behavior

𝑥 → −∞, 𝑦 → −∞

Opens

Down

𝑥 → ∞, 𝑦 → −∞

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Practice Problems 1. Three different equations for the same quadratic function are given. Vertex Form

Factored Form

𝑦 = 2𝑥2 + 8𝑥 − 24

𝑦 = 2(𝑥 + 2)2 − 32

𝑦 = 2(𝑥 + 6)(𝑥 − 2)

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Standard Form

Use the equations to determine the key features of the function. If none of the equations can be used to determine a key feature, write “no equation.” Form Used

Key Feature

Name of Key Feature

Factored form

Standard form

𝑥-intercepts

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Vertex form

(_______ , _______)

Factored form

Standard form

𝑦-intercept

Vertex form

(_______ , _______)

Factored form

Vertex

Standard form

Vertex form

Axis of symmetry

(_______ , _______)

Factored form Standard form Vertex form

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𝑥 = ________

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Name of Key Feature

Form Used

Key Feature

Factored form Increasing interval

Standard form

______ < 𝑥 < ______

Vertex form Factored form Standard form Vertex form

Factored form Positive interval

Standard form

_______ < 𝑥 < ______

Vertex form

______ < 𝑥 < ______

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Decreasing interval

______ < 𝑥 < ______

Factored form

Standard form

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Negative interval

Vertex form

______ < 𝑥 < ______

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2. Determine all of the key features possible using the equation of the quadratic 𝑦 = 4(𝑥 + 3)(𝑥 − 1). Then, use the key features to graph the quadratic. If necessary, complete the table of values. Key Features Increasing interval(s)

𝑥-intercept(s)

Decreasing interval(s)

Domain

Positive interval(s)

Range

Negative interval(s)

Vertex

Left end behavior

Opens

Right end behavior

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𝑦-intercept

As 𝑥 →_____, 𝑦 →_____.

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Graph y

20 15

As 𝑥 →_____, 𝑦 →_____.

Table of Values

𝒙

𝒚

-10

-5

0 -5

-15

-10

-15 -20 -25

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Review Problems 3. For each function, write the coordinates of the vertex of its graph and tell whether the graph opens up or down. Function

Coordinates of Vertex

Graph Opens Up or Down?

𝑓(𝑥) = (𝑥 − 4)2 − 5

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𝑔(𝑥) = −𝑥2 + 5

4. Here is a graph that represents 𝑦 = 𝑥2.

ℎ(𝑥) = 2(𝑥 + 1)2 − 4

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a. Describe what would happen to the graph if the original equation were modified as follows:

ii. 𝑦 = 3𝑥2

i. 𝑦 = −𝑥2

iii. 𝑦 = 𝑥2 + 6

b. Sketch the graph of the equation 𝑦 = −3𝑥2 + 6 on the same coordinate plane as 𝑦 = 𝑥2.

5. Noah is going to put $2,000 in a savings account. He plans on putting the money in an account and leaving it there for 5 years. He can put the money in an account that pays 1% interest monthly, an account that pays 6% interest every six months, or an account that pays 12% interest annually. Which account will give him the most money in his account at the end of the 5 years?

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Unit 9: Quadratic Equations

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Unit 9, Lesson 1: When and Why Do We Write Quadratic Equations?

Warm-Up: What Goes Up Must Come Down

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A mechanical device is used to launch a potato vertically into the air. The potato is launched from a platform 20 feet (ft.) above the ground, with an initial vertical velocity of 92 ft. per second (sec.).

The function ℎ(𝑡) = − 16𝑡2 + 92𝑡 + 20 models the height of the potato over the ground, in ft., 𝑡 sec. after launch.

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Here is the graph representing the function.

For each question, be prepared to explain your reasoning.

1. What is the height of the potato 1 sec. after launch?

2. Eight sec. after launch, will the potato still be in the air?

3. Will the potato reach 120 ft.? If so, when will it happen? 4. When will the potato hit the ground?

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Exploration Activity: A Framed Picture A framed picture has a total area 𝑦, in square inches (sq. in.). The thickness of the frame is represented by 𝑥, in inches (in.). The equation 𝑦 = (8 + 2𝑥)(10 + 2𝑥) relates these 2 variables.

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1. What are the length and width of the picture without the frame?

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2. What would a solution to the equation 100 = (8 + 2𝑥)(10 + 2𝑥) mean in this situation?

Guided Activity: The Flying Potato Again

In the Warm-Up, you saw an equation that defines the height of a potato as a function of time after it was launched from a mechanical device. A different function modeling the height of a potato, in ft., 𝑡 sec. after being fired from a different device is shown. 𝑓(𝑡) = −16𝑡2 + 80𝑡 + 64

1. Which equation would we solve to find the time at which the potato hits the ground?

2. Use any method except graphing to find a solution to this equation.

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Collaborative Activity: Revenue from Ticket Sales The expressions 𝑝(200 − 5𝑝) and −5𝑝2 + 200𝑝 define the same function. The function models the revenue a school would earn from selling raffle tickets at 𝑝 dollars each.

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1. At what price or prices would the school collect $0 revenue from raffle sales? Explain or show your reasoning.

Lesson Summary

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2. The school staff noticed that there are two ticket prices that would both result in a revenue of $500. How would you find out what those two prices are?

The height of a softball, in ft., 𝑡 sec. after someone throws it straight up, can be defined by 𝑓(𝑡) = −16𝑡2 + 32𝑡 + 5. The input of function 𝑓 is time, and the output is height. The output of this function can be determined at any given input. For example,

• At the beginning of the softball’s journey, when 𝑡 = 0, its height is given by 𝑓(0). • Two seconds later, when 𝑡 = 2, its height is given by 𝑓(2).

The values of 𝑓(0) and 𝑓(2) can be found using a graph or by evaluating the expression −16𝑡2 + 32𝑡 + 5 at those values of 𝑡. When the output of the function is known, the inputs can be determined. For example, • When does the softball hit the ground? Answering this question means finding the values of 𝑡 that make 𝑓(𝑡) = 0 or solving the equation −16𝑡2 + 32𝑡 + 5 = 0.

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One way to solve these equations is by graphing, as explored in the previous unit. This unit will explore how to solve quadratic equations algebraically. Sometimes, the form of the equation can help make solutions easier to determine. The height of a potato that is launched from a mechanical device can be modeled by a function, 𝑔, where 𝑥 represents time, in sec. Two expressions that are equivalent, and both define function 𝑔, are shown. −16(𝑥 − 6)(𝑥 + 1)

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−16𝑥2 + 80𝑥 + 96

Notice that 1 expression is in standard form, and the other is in factored form.

The potato will hit the ground when the value of the function is 0, so either expression can be set equal to 0 to determine when the potato hit the ground. −16𝑥2 + 80𝑥 + 96 = 0

−16(𝑥 − 6)(𝑥 + 1) = 0

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Solving −16𝑥2 + 80𝑥 + 96 = 0 is an option that will be explored later in the unit.

However, the solutions to −16(𝑥 − 6)(𝑥 + 1) = 0 can be determined using concepts learned in the prior unit.

Recall that the zeros of a quadratic function can be identified when the expression defining the function is in factored form. The solutions to −16(𝑥 − 6)(𝑥 + 1) = 0 are the zeros of the function, so this form may be more helpful in this case. • If 𝑥 is 6, then the value of 𝑥 − 6 is 0, so the entire expression has a value of 0.

• If 𝑥 is −1, then the value of 𝑥 + 1 is 0, so the entire expression also has a value of 0.

This means that 6 and −1 are solutions to the equation and that the potato hits the ground after 6 sec. A negative value of time is not viable in this context, so the solution of −1 is excluded.

These equations are quadratic equations. In general, a quadratic equation is an equation that can be expressed as 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 ≠ 0.

In upcoming lessons, rewriting quadratic equations into forms that make the solutions easier to see will be explored.

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Practice Problems 1. A square picture has a frame that is 3 in. thick all the way around. The total side length of the picture and frame is 𝑥 in.

Which expression represents the area of the square picture, without the frame? If you get stuck, try sketching a diagram.

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A. (2𝑥 + 3)(2𝑥 + 3) B. (𝑥 + 6)(𝑥 + 6)

C. (2𝑥 − 3)(2𝑥 − 3)

D. (𝑥 − 6)(𝑥 − 6)

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2. A garden designer designed a square decorative pool. The pool is surrounded by a walkway. On two opposite sides of the pool, the walkway is 8 ft. On the other two opposite sides, the walkway is 10 ft.

Here is a diagram of the design.

The final design for the pool and walkway covers a total area of 1,440 square feet (sq. ft.). a. The side length of the square pool is 𝑥. Write an expression that represents:

i. the total length of the rectangle (including the pool and walkway)

ii. the total width of the rectangle (including the pool and walkway)

iii. the total area of the pool and walkway

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b. Write an equation of the form: your expression = 1,440. What does a solution to the equation mean in this situation?

3. Select all values of that are solutions to the equation (𝑥 − 5)(7𝑥 − 21) = 0.

□ −7

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□ −5 □ −3 □0

□3 □5

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□7

4. The expressions 30𝑥2 − 105𝑥 − 60 and (5𝑥 − 20)(6𝑥 + 3) define the same function, 𝑓.

a. Which expression makes it easier to find 𝑓(0)? Explain your reasoning.

b. Find 𝑓(0).

c. Which expression makes it easier to find the values of 𝑥 that make the equation 𝑓(𝑥) = 0 true? Explain or show your reasoning.

d. Find the values of 𝑥 that make 𝑓(𝑥) = 0.

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5. A band is traveling to a new city to perform a concert. The revenue from their ticket sales is a function of the ticket price, 𝑥, and can be modeled with (𝑥 − 6)(250 − 5𝑥).

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What are the ticket prices at which the band would make no money at all?

6. Two students built a small rocket from a kit and attached an altimeter (a device for recording altitude or height) to the rocket. They record the height of the rocket over time since it is launched in the table, based on the data from the altimeter.

Height (m)

110.25

236.25

252

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Time (sec.)

110.25

Function ℎ gives the height in meters (m) as a function of time in sec., 𝑡.

a. What is the value of ℎ(3)?

b. What value of 𝑡 gives ℎ(𝑡) = 252?

c. Explain why ℎ(0) = ℎ(8).

d. Based on the data, which equation about the function could be true: ℎ(2) = 189 or ℎ(189) = 2? Explain your reasoning.

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Review Problems 7. The revenue from a youth league baseball game depends on the price per ticket, 𝑥.

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Here is a graph that represents the revenue function, 𝑅.

□ 𝑅(600) is a little less than 5.

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□ 𝑅(5) is a little more than 600.

Select all the true statements.

□ The maximum possible ticket price is $15.

□ The maximum possible revenue is about $1,125.

□ If tickets cost $10, the predicted revenue is $1,000.

□ If tickets cost $20, the predicted revenue is $1,000.

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8. A random sample of people are asked to give a taste score—either “low” or “high”— to two different types of ice cream. The two types of ice cream have identical formulas, except they differ in the percentage of sugar in the ice cream. What values could be used to complete the table so that it suggests there is an association between taste score and percentage of sugar? Explain your reasoning.

Low Taste Score

126

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High Taste Score

239

𝟭𝟱% % Sugar

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𝟭𝟮% % Sugar

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Unit 9, Lesson 2: Solving Quadratic Equations Using Reasoning

Warm-Up: How Many Solutions?

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How many solutions does each equation have? What are the solution(s)? Be prepared to explain how you know. 1. 𝑥2 = 9

2. 𝑥2 = 0

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3. 𝑥2 − 1 = 3

4. 2𝑥2 = 50

5. (𝑥 + 1)(𝑥 + 1) = 0

6. 𝑥(𝑥 − 6) = 0

7. (𝑥 − 1)(𝑥 − 1) = 4

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Guided Activity: Finding Pairs of Solutions Each of these equations has two solutions. What are they? Explain or show your reasoning.

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1. 𝑛2 + 4 = 404

2. 432 = 3𝑛2

4. (𝑛 − 5)2 − 30 = 70

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3. 144 = (𝑛 + 1)2

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Exploration Activity: Take the Zero Product Property out for a Spin For each equation, find its solution or solutions. Be prepared to explain your reasoning.

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1. 𝑥 − 3 = 0

2. 𝑥 + 11 = 0

4. 𝑥(2𝑥 + 11) = 0

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3. 2𝑥 + 11 = 0

5. (𝑥 − 3)(𝑥 + 11) = 0

6. (𝑥 − 3)(2𝑥 + 11) = 0

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Collaborative Activity: Revisiting a Projectile We have seen quadratic functions modeling the height of a projectile as a function of time. Here are two ways to define the same function that approximates the height of a projectile, in meters (m), 𝑡 seconds (sec.) after launch. ℎ(𝑡) = (−5𝑡 − 3)(𝑡 − 6)

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ℎ(𝑡) = −5𝑡2 + 27𝑡 + 18

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1. Which way of defining the function allows us to use the zero product property to find out when the height of the object is 0 m?

2. Without graphing, determine at what time the height of the object is 0 m. Show your reasoning.

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Lesson Summary Some quadratic equations can be solved by applying the properties of equality and by reasoning about which values of the variable would make the equation true. For example, the steps in a solution method for 3(𝑥 + 1)2 − 75 = 0 are shown. Description

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Step

Original equation

3(𝑥 + 1)2 − 75 = 0

Apply the addition property of equality.

3(𝑥 + 1)2 = 75

Apply the division property of equality.

𝑥 + 1 = 5 and 𝑥 + 1 = −5

Write 2 new equations.

Apply the subtraction property of equality.

𝑥 = 4 and 𝑥 = −6

Reason, “What can be squared to get 25?” Those values are 5 and −5.

ip t

(_____)2 = 25

(𝑥 + 1)2 = 25

This means that both 𝑥 = 4 and 𝑥 = −6 make the equation true and are, therefore, solutions to the equation.

Another strategy for solving quadratic equations is applying the zero product property.

The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.

In other words, if 𝑎⋅𝑏 = 0, then either 𝑎 = 0 or 𝑏 = 0. This property is handy when an equation that needs to be solved states that the product of 2 factors is 0.

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• The equation 𝑚(𝑚 + 9) = 0 says that the product of 𝑚 and (𝑚 + 9) is 0. For this to be true, either 𝑚 = 0 or 𝑚 + 9 = 0, so both 0 and −9 are solutions. • Consider another example using the equation (𝑢 − 2.345)(14𝑢 + 2) = 0.

For 𝑢 − 2.345 = 0 to be true, 𝑢 would have to be 2.345. For 14𝑢 + 2 = 0 or 14𝑢 = −2 to

be true, 𝑢 would have to be − 2 or − 1 . Therefore, the solutions are 2.345 and − 1 . 14

Practice Problems

ip t

1. Solve (𝑥 − 1)2 = 16. Explain or show your reasoning.

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In general, when a quadratic expression in factored form is on 1 side of an equation and 0 is on the other side, the zero product property can be used to find its solutions.

2. Here is one way to solve the equation 5 𝑦2 = 5. Explain what is done in each step. 9

Step 1

5𝑦2 = 45

Original equation

5 2 𝑦 =5 9

𝑦2 = 9

𝑦 = 3 or 𝑦 = −3

Step 2 Step 3

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3. Diego and Jada are working together to solve the quadratic equation (𝑥 − 2)2 = 100. Diego solves the equation by dividing each side of the equation by 2 and then adding 2 to each side. He writes: (𝑥 − 2) = 50 𝑥 = 52

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Jada asks Diego why he divides each side by 2 and he says, “I want to find a number that equals 100 when multiplied by itself. That number is half of 100.”

a. What mistake is Diego making?

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b. If you were Jada, what could you say to Diego to help him realize his mistake?

4. If the equation (𝑥 + 10)𝑥 = 0 is true, which statement is also true according to the zero product property? A. only 𝑥 = 0

B. either 𝑥 = 0 or 𝑥 + 10 = 0

C. either 𝑥2 = 0 or 10𝑥 = 0

D. only 𝑥 + 10 = 0

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5. What are the solutions to the equation (10 − 𝑥)(3𝑥 − 9) = 0? A. −10 and 3

B. −10 and 9 C. 10 and 3

op y

D. 10 and 9

6. Solve each equation.

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a. (𝑥 − 6)(𝑥 + 5) = 0

b. (𝑥 − 3)( 2 𝑥 − 6) = 0

c. (−3𝑥 − 15)(𝑥 + 7) = 0

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Review Problems 7. As part of a publicity stunt (an event designed to draw attention), a TV host drops a watermelon from the top of a tall building. The height of the watermelon 𝑡 seconds (sec.) after it is dropped is given by the function ℎ(𝑡) = 850 − 16𝑡2, where ℎ is in feet (ft.).

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a. Find ℎ(4). Explain what this value means in this situation.

b. Find ℎ(0). What does this value tell us about the situation?

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c. Is the watermelon still in the air 8 sec. after it is dropped? Explain how you know.

8. Consider the expressions (𝑥 − 4)(3𝑥 − 6) and 3𝑥2 − 18𝑥 + 24.

Show that the two expressions define the same function.

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Unit 9, Lesson 3: How Many Solutions?

Warm-Up: Math Talk: Four Equations 1. Decide whether each statement is true or false.

ip t

b. A solution to 𝑥2 + 25 = 0 is −5.

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a. 3 is the only solution to 𝑥2 − 9 = 0.

c. 𝑥(𝑥 − 7) = 0 has two solutions.

d. 5 and −7 are the solutions to (𝑥 − 5)(𝑥 + 7) = 12.

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Exploration Activity: Solving by Graphing Han is solving three equations by graphing. (𝑥 − 5)(𝑥 − 3) = 0

(𝑥 − 5)(𝑥 − 3) = −4

op y

(𝑥 − 5)(𝑥 − 3) = −1 1. To solve the first equation, (𝑥 − 5)(𝑥 − 3) = 0, he graphed 𝑦 = (𝑥 − 5)(𝑥 − 3) and then looked for the 𝑥-intercepts of the graph.

b. What are the solutions?

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a. Explain why the 𝑥-intercepts can be used to solve (𝑥 − 5)(𝑥 − 3) = 0.

2. To solve the second equation, Han rewrote it as (𝑥 − 5)(𝑥 − 3) + 1 = 0. He then graphed 𝑦 = (𝑥 − 5)(𝑥 − 3) + 1.

Use graphing technology to graph 𝑦 = (𝑥 − 5)(𝑥 − 3) + 1. Then, use the graph to solve the equation. Be prepared to explain how you use the graph for solving.

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op y

3. Solve the third equation using Han’s strategy.

4. Think about the strategy you used and the solutions you found.

a. Why might it be helpful to rearrange each equation to equal 0 on one side and then graph the expression on the non-zero side?

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b. How many solutions do each of the three equations have?

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Guided Activity: Finding All the Solutions Solve each equation. Be prepared to explain or show your reasoning.

op y

1. 𝑥2 = 121

2. 𝑥2 − 31 = 5

5. (𝑥 + 1)2 = −4

4. (𝑥 + 3)(𝑥 − 1) = 5

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3. (𝑥 − 4)(𝑥 − 4) = 0

6. (𝑥 − 4)(𝑥 − 1) = 990

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Collaborative Activity: Analyzing Errors in Equation Solving 1. Consider (𝑥 − 5)(𝑥 + 1) = 7. Priya reasons that if this is true, then either 𝑥 − 5 = 7 or 𝑥 + 1 = 7. So, the solutions to the original equation are 12 and 6.

ip t

op y

Do you agree? If not, where was the mistake in Priya’s reasoning?

2. Consider 𝑥2 − 10𝑥 = 0. Diego says to solve we can just divide each side by 𝑥 to get 𝑥 − 10 = 0, so the solution is 10. Mai says, “I wrote the expression on the left in factored form, which gives 𝑥(𝑥 − 10) = 0, and ended up with two solutions: 0 and 10.”

Do you agree with either strategy? Explain your reasoning.

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Lesson Summary Quadratic equations can have 2, 1, or 0 real solutions. To determine how many solutions a quadratic equation has and what the solutions are, rearrange the equation so it is set equal to 0. Then, graph the function that the expression defines, and determine its zeros. Some examples are shown. Example 1: 𝒙𝟮 � 𝟱𝒙

op y

First, subtract 5𝑥 from each side, and rewrite the equation as 𝑥2 − 5𝑥 = 0. Think of solving this equation as finding the zeros of a function defined by 𝑥2 − 5𝑥.

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The graph shows the 𝑥-intercepts are (0, 0) and (5, 0), so 𝑥2 − 5𝑥 equals 0 when 𝑥 = 0 and when 𝑥 = 5.

If the output of this function is 𝑦, graph 𝑦 = 𝑥2 − 5𝑥, and identify where the graph intersects the 𝑥-axis, where the 𝑦-coordinate is 0.

The graph readily shows that there are 2 solutions to the equation.

Note that the equation 𝑥2 = 5𝑥 can be solved without graphing, but be careful not to divide both sides by 𝑥. Doing so will result in 𝑥 = 5 but will show no trace of the other solution, 𝑥 = 0. Even though dividing both sides by the same value is usually acceptable for solving equations, we avoid dividing by the same variable because it may eliminate a solution.

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Example 2: �𝒙 − 𝟔��𝒙 − 𝟒� � − 𝟏

Rewrite the equation as (𝑥 − 6)(𝑥 − 4) + 1 = 0, and consider it to represent a function defined by (𝑥 − 6)(𝑥 − 4) + 1 and whose output, 𝑦, is 0.

The graph shows 1 𝑥-intercept at (5, 0). This means the function defined by (𝑥 − 6)(𝑥 − 4) + 1 has only 1 zero.

op y

Graph 𝑦 = (𝑥 − 6)(𝑥 − 4) + 1, and identify the 𝑥-intercepts.

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Example 3: �𝒙 − 𝟑��𝒙 − 𝟑� � − 𝟒

It also means that the equation (𝑥 − 6)(𝑥 − 4) + 1 = 0 is true only when 𝑥 = 5. The value 5 is the only solution to the equation.

Rearranging the equation gives (𝑥 − 3)(𝑥 − 3) + 4 = 0.

Graph 𝑦 = (𝑥 − 3)(𝑥 − 3) + 4, and find the 𝑥-intercepts.

The graph does not intersect the 𝑥-axis, so there are no 𝑥-intercepts.

This means there are no 𝑥-values that can make the expression (𝑥 − 3)(𝑥 − 3) + 4 equal 0, so the function defined by 𝑦 = (𝑥 − 3)(𝑥 − 3) + 4 has no zeros. Therefore, the equation (𝑥 − 3)(𝑥 − 3) = −4 has no real solutions.

Reasoning can be used to determine there are no real solutions, even without graphing. (𝑥 − 3)(𝑥 − 3) = −4 is (𝑥 − 3)2 = −4. Because no number can be squared to get a negative value, the equation has no real solutions. Future courses will explore nonreal solutions.

Keep in mind that graphing is not always reliable for showing precise solutions. The 𝑥-intercepts of a graph are not always integer values. Although graphs can provide an idea of how many real solutions there are and what the values may be (at least approximately), algebraic solution methods should be used to determine exact solutions. © Accelerate Learning Inc. - All Rights Reserved

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Practice Problems 1. Rewrite each equation so that the expression on one side could be graphed and the 𝑥-intercepts of the graph would show the solutions to the equation.

op y

a. 3𝑥2 = 81

b. (𝑥 − 1)(𝑥 + 1) − 9 = 5𝑥

d. 6𝑥(𝑥 − 8) = 29

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c. 𝑥2 − 9𝑥 + 10 = 32

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a. Here are equations that define quadratic functions 𝑓, 𝑔 and ℎ. Sketch a graph, by hand or using technology, that represents each equation.

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b. Determine how many solutions each 𝑓(𝑥) = 0, 𝑔(𝑥) = 0, and ℎ(𝑥) = 0 have. Explain how you know.

3. Mai is solving the equation (𝑥 − 5)2 = 0. She writes that the solutions are 𝑥 = 5 and 𝑥 = −5. Han looks at her work and disagrees. He says that only 𝑥 = 5 is a solution. Who do you agree with? Explain your reasoning.

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4. Decide if each equation has 0, 1, or 2 real solutions and explain how you know. a. 𝑥2 − 144 = 0

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b. 𝑥2 + 144 = 0 c. 𝑥(𝑥 − 5) = 0

d. (𝑥 − 8)2 = 0

Review Problems

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e. (𝑥 + 3 )(𝑥 + 7) = 0

5. The graph shows the number of square meters, 𝐴, covered by algae in a lake 𝑤 weeks after it was first measured. In a second lake, the number of square meters, 𝐵, covered by algae is defined by the equation 2

𝐵 = 975 ⋅ ( 5 )𝑤, where 𝑤 is the number of weeks since

it was first measured.

For which algae population is the area decreasing more rapidly? Explain how you know.

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6. If the equation (𝑥 − 4)(𝑥 + 6) = 0 is true, which is also true according to the zero product property? A. only 𝑥 − 4 = 0

B. only 𝑥 + 6 = 0

D. 𝑥 = −4 or 𝑥 = 6

op y

C. 𝑥 − 4 = 0 or 𝑥 + 6 = 0

7. To solve the quadratic equation 3(𝑥 − 4)2 = 27, Andre and Clare wrote the following: Clare

3(𝑥 − 4)2 = 27

Andre

𝑥2 − 16 = 9

𝑥 = 7

𝑥2 = 25

𝑥 − 4 = 3

𝑥2 − 42 = 9

(𝑥 − 4)2 = 9

ip t

(𝑥 − 4)2 = 9

𝑥 = 5 or 𝑥 = −5

a. Identify the mistake each student made.

b. Solve the equation and show your reasoning.

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Unit 9, Lesson 4: Rewriting Quadratic Expressions in Factored Form – Part 1

Warm-Up: Puzzles of Rectangles

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Here are two puzzles that involve side lengths and areas of rectangles. Can you find the missing area in Figure A and the missing length in Figure B? Be prepared to explain your reasoning.

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Exploration Activity: Using Diagrams and Reasoning to Understand Equivalent Expressions 1. Use a diagram to show that each pair of expressions is equivalent. b. (𝑥 + 2)(𝑥 + 4) and 𝑥2 + 6𝑥 + 8𝑥

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a. 𝑥(𝑥 + 3) and 𝑥2 + 3𝑥

d. (𝑥 + −6) and 𝑥2 − 6𝑥

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c. (𝑥 + −5)(𝑥 + −1) and 𝑥2 − 6𝑥 + 5

f. (𝑥 − 1)(𝑥 − 7) and 𝑥2 − 8𝑥 + 7

e. (𝑥 + 4)(𝑥 + 10) and 𝑥2 + 14𝑥 + 40

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op y

2. Observe the pairs of expressions that involve the product of two sums or two differences from problems 1 and 2. How is each expression in factored form related to the equivalent expression in standard form?

3. Abby uses a different method of factoring. She rewrote 𝑘2 + 10𝑘 + 24 as 𝑘2 + 6𝑘 + 4𝑘 + 24, and then paired the terms as shown. (𝑘2 + 6𝑘) + (4𝑘 + 24)

ip t

Work with your partner to finish factoring using Abby’s method, known as factoring by grouping, through the following steps.

(𝒌2 + 𝟔𝒌) + (𝟒𝒌 + 𝟐𝟒)

Factor out the greatest common factor from each pair of terms.

(𝒌 + 𝟔)(𝒌 + ________)

Rewrite the sum using the common factor, (𝑘 + 6).

𝒌(𝒌 + 𝟔) + _______(𝒌 + 𝟔)

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Collaborative Activity: Let’s Rewrite Some Expressions Each row in the table contains a pair of equivalent expressions. Complete the table with the missing expressions. If you get stuck, consider drawing a diagram. Standard Form

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Factored Form 𝑥(𝑥 + 7)

𝑥2 − 8𝑥

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(𝑥 + 6)(𝑥 + 2)

𝑥2 + 9𝑥

(𝑥 − 6)(𝑥 − 2)

𝑥2 + 13𝑥 + 12

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𝑥2 − 7𝑥 + 12 𝑥2 + 6𝑥 + 9

𝑥2 + 10𝑥 + 9 𝑥2 − 10𝑥 + 9 𝑥2 − 6𝑥 + 9

𝑥2 + (𝑚 + 𝑛)𝑥 + 𝑚𝑛

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Lesson Summary Previous lessons explored how to expand a quadratic expression given in factored form, converting it to standard form by applying the distributive property. For example, to expand (𝑥 + 4)(𝑥 + 5), apply the distributive property to multiply 𝑥 by (𝑥 + 5) and 4 by (𝑥 + 5). Then, apply the property again to multiply 𝑥 by 𝑥 and 𝑥 by 5, and multiply 4 by 𝑥 and 4 by 5.

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To keep track of all the products, a diagram can be used. The diagram shown illustrates that (𝑥 + 4)(𝑥 + 5) is equivalent to 𝑥2 + 5𝑥 + 4𝑥 + 4 ⋅ 5, or in standard form, 𝑥2 + 9𝑥 + 20.

• The linear term, 9𝑥, has a coefficient of 9, which is the sum of 5 and 4.

𝑥2

5𝑥

4𝑥

4⋅5

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• The constant term, 20, is the product of 5 and 4.

𝑥

These relationships can be used to reason in the other direction, that is, to start with an expression in standard form and rewrite it in factored form. For example, rewrite 𝑥2 − 11𝑥 + 24 in factored form.

• Start by creating a diagram and writing in the terms 𝑥2 and 24, as shown. Then, think of 2 numbers that multiply to make 24 and add up to −11.

𝑥

𝑥2

• The numbers −8 and −3 meet these conditions.

The product of −8 and −3 is 24.

The sum of − 8 and − 3 is − 11.

• Use the terms in the diagram to determine the factors.

Alternatively, the numbers − 8 and − 3 can be used to break apart the 𝑏-term into 2 terms, 𝑥2 − 8𝑥 − 3𝑥 + 24. This expression can then be used to factor the expression by grouping. © Accelerate Learning Inc. - All Rights Reserved

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𝑥

−3

𝑥

𝑥2

− 3𝑥

−8

− 8𝑥 24

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(𝑥2 − 8𝑥) + (−3𝑥 + 24) 𝑥(𝑥 − 8) − 3(𝑥 − 8) (𝑥 − 8)(𝑥 − 3)

Group the terms into pairs. Factor each pair to reveal 1 common binomial factor. Factor out the common binomial factor, creating the other binomial factor.

op y

Either method can be used to rewrite 𝑥2 − 11𝑥 + 24 into factored form as (𝑥 − 8)(𝑥 − 3).

Practice Problems

a. Factor 𝑥2 − 8𝑥 + 12 using a diagram.

1. Consider the trinomial 𝑥2 − 8𝑥 + 12.

ip t

b. Factor 𝑥2 − 8𝑥 + 12 using factoring by grouping.

2. Find two numbers that satisfy the requirements. If you get stuck, try listing all the factors of the first number.

a. Find two numbers that multiply to 17 and add to 18.

b. Find two numbers that multiply to 20 and add to 9.

c. Find two numbers that multiply to 11 and add to −12.

d. Find two numbers that multiply to 36 and add to −20. 380 | Unit 9

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3. Use the diagram to show that: a. (𝑥 + 4)(𝑥 + 2) is equivalent to 𝑥2 + 6𝑥 + 8. 2

4 b. (𝑥 − 10)(𝑥 − 3) is equivalent to 𝑥2 − 13𝑥 + 30.

− 10

𝑥

op y

𝑥

ip t

−3

□ 𝑥 + (−5)

□ 𝑥 − (−5)

4. Select all expressions that are equivalent to 𝑥 − 5.

□ −5 + 𝑥

□ −5 − 𝑥 □5 − 𝑥

□ −5 − (−𝑥) □5 + 𝑥

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5. Here are pairs of equivalent expressions, one in standard form and the other in factored form. Find the missing numbers. a. 𝑥2 + ________ 𝑥 + ________ and (𝑥 − 9)(𝑥 − 3) b. 𝑥2 + 12𝑥 + 32 and (𝑥 + 4)(𝑥 + ________)

c. 𝑥2 − 12𝑥 + 35 and (𝑥 − 5)(𝑥 + ________)

op y

d. 𝑥2 − 9𝑥 + 20 and (𝑥 − 4)(𝑥 + ________) Review Problems

6. Find all the values for the variable that make each equation true.

c. (2𝑥 + 4)(𝑥 − 4) = 0

b. (7𝑥 + 14)(7𝑥 + 14) = 0

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a. 𝑏(𝑏 − 4.5) = 0

d. (−2 + 𝑢)(3 − 𝑢) = 0

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7. Lin charges $5.50 per hour (hr.) to babysit. The amount of money earned, in dollars, is a function of the number of hr. that she babysits. Which of the following inputs is impossible for this function? A. −1 B. 2

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C. 5

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D. 8

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Unit 9, Lesson 5: Rewriting Quadratic Expressions in Factored Form – Part 2

Warm-Up: Sums and Products

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1. The product of the integers 2 and −6 is −12. List all the other pairs of integers whose product is −12.

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2. Of the pairs of factors you found, list all pairs that have a positive sum. Explain why they all have a positive sum.

3. Of the pairs of factors you found, list all pairs that have a negative sum. Explain why they all have a negative sum.

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Collaborative Activity: Factors of 𝟭𝟬𝟬 and −𝟭𝟬𝟬

1. Consider the expression 𝑥2 + 𝑏𝑥 + 100.

Complete the first table with all pairs of factors of 100 that would give positive values of 𝑏, and the second table with factors that would give negative values of 𝑏.

Factor 2

𝒃 (Positive)

Factor 1

Negative value of 𝑏 Factor 2

𝒃 (Negative)

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Factor 1

Positive value of 𝑏

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For each pair, state the 𝑏-value they produce. (Use as many rows as needed.)

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2. Consider the expression 𝑥2 + 𝑏𝑥 − 100.

Complete the first table with all pairs of factors of −100 that would result in positive values of 𝑏, the second table with factors that would result in negative values of 𝑏, and the third table with factors that would result in a zero value of 𝑏.

For each pair of factors, state the 𝑏-value they produce. (Use as many rows as there are pairs of factors. You may not need all the rows.)

op y

Factor 2

Negative value of 𝑏

Factor 1

𝒃 (Positive)

Factor 2

𝒃 (Negative)

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Factor 1

Positive value of 𝑏

us Factor 2

𝒃 (Positive)

Factor 1

Zero value of 𝑏

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3. Write each expression in factored form. b. 𝑥2 + 15𝑥 − 100

c. 𝑥2 − 15𝑥 − 100

d. 𝑥2 − 15𝑥 − 100

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a. 𝑥2 − 25𝑥 + 100

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Exploration Activity: Can Products Be Written as Differences?

1. Clare claims that (10 + 3)(10 − 3) is equivalent to 102 − 32 and (20 + 1)(20 − 1) is equivalent to 202 − 12. Do you agree? Show your reasoning.

a. Use your observations from the first question and evaluate (100 + 5)(100 − 5). Show your reasoning.

b. Check your answer by computing 105 ⋅ 95.

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3. Is (𝑥 + 4)(𝑥 − 4) equivalent to 𝑥2 − 42? Support your answer:

With a diagram: Without a diagram:

𝑥

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−4

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4. Is (𝑥 + 4)2 equivalent to 𝑥2 + 42? Support your answer, either with or without a diagram.

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Collaborative Activity: What if There Is No Linear Term? Each row has a pair of equivalent expressions. Complete the table.

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If you get stuck, consider drawing a diagram. (Heads up: one of them is impossible.)

(𝑥 − 10)(𝑥 + 10) (2𝑥 + 1)(2𝑥 − 1)

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(4 − 𝑥)(4 + 𝑥)

Standard Form

Factored Form

𝑥2 − 81

(𝑐 + 25 )(𝑐 − 25 ) (𝑥 + 5)(𝑥 + 5)

49 − 𝑦2

9𝑧2 − 16

25𝑡2 − 81

49 − 𝑑2 16

𝑥2 − 6

𝑥2 + 100

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Lesson Summary To rewrite a quadratic expression given in standard form to factored form, the process is the same whether the constant term is a positive or a negative value. For example, to rewrite 𝑥2 + 6𝑥 − 7 in factored form, find 2 numbers that meet the following criteria: • Multiply to make −7. The candidates are 7 and −1 or −7 and 1.

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• Add up to 6. Only 7 and −1 from the list of candidates add up to 6.

Therefore, the factored form of 𝑥2 + 6𝑥 − 7 is (𝑥 + 7)(𝑥 − 1).

Sometimes expressions in standard form don’t have a linear term. Can they still be written in factored form?

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Take 𝑥2 − 9 as an example. To help write it in factored form, think of it as having a linear term with a coefficient of 0, or 𝑥2 + 0𝑥 − 9. The expression 𝑥2 − 0𝑥 − 9 is equivalent to 𝑥2 − 9 because 0 times any number is 0, so 0𝑥 is 0.

Find 2 numbers that multiply to make −9 and add up to 0. The numbers 3 and −3 meet both requirements, so the factored form is (𝑥 + 3)(𝑥 − 3).

To check that this expression is indeed equivalent to 𝑥2 − 9, expand the factored expression by applying the distributive property: (𝑥 + 3)(𝑥 − 3) = 𝑥2 − 3𝑥 + 3𝑥 + (−9). Adding −3𝑥 and 3𝑥 gives 0, so the expanded expression is 𝑥2 − 9.

In general, a quadratic expression that is a difference of 2 squares, 𝑎2 − 𝑏2, can be rewritten as (𝑎 + 𝑏)(𝑎 − 𝑏).

Consider another example, 49 − 16𝑦2. This expression can be written 72 − (4𝑦)2, which is a difference of 2 squares. An equivalent expression in factored form is (7 + 4𝑦)(7 − 4𝑦).

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What about 𝑥2 + 9? Can it be written in factored form? Think about this expression as 𝑥2 + 0𝑥 + 9. Is it possible to find 2 numbers that multiply to make 9 but add up to 0? •

9 and 1, sum: 10

•

3 and 3, sum: 6

•

−9 and −1, sum: −10 −3 and −3, sum: −6

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•

For 2 numbers to add up to 0, they need to be opposites (a negative and a positive), but a pair of opposites cannot multiply to make positive 9 because multiplying a negative number and a positive number always gives a negative product. Because there are no numbers that multiply to make 9 that also add up to 0, it is not possible to write 𝑥2 + 9 in factored form using real numbers.

1. Find two numbers that . . .

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Practice Problems

a. multiply to − 40 and add to − 6.

b. multiply to − 40 and add to 6.

c. multiply to − 36 and add to 9. d. multiply to − 36 and add to − 5.

If you get stuck, try listing all the factors of the first number.

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2. Write a + or a − sign in each box so the expressions on each side of the equal sign are equivalent. a. (𝑥 30 18)(𝑥 30 3) = 𝑥2 − 15𝑥 − 54

b. (𝑥 30 18)(𝑥 30 3) = 𝑥2 + 21𝑥 + 54 c. (𝑥 30 18)(𝑥 30 3) = 𝑥2 + 15𝑥 − 54

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d. (𝑥 30 18)(𝑥 30 3) = 𝑥2 − 21𝑥 + 54

3. Rewrite each expression in factored form. If you get stuck, try drawing a diagram. b. 𝑥2 + 3𝑥 − 28

a. 𝑥2 − 3𝑥 − 28

d. 𝑥2 − 28𝑥 − 60

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c. 𝑥2 + 12𝑥 − 28

4. Match each quadratic expression given in factored form with an equivalent expression in standard form. One expression in standard form has no match. Factored Form

121 − 𝑥2

a. (𝑦 + 𝑥)(𝑦 − 𝑥)

Standard Form

𝑥2 + 2𝑥𝑦 − 𝑦2

d. (𝑥 − 𝑦)(𝑥 − 𝑦)

𝑥2 − 2𝑥𝑦 + 𝑦2

b. (11 + 𝑥)(11 − 𝑥)

c. (𝑥 − 11)(𝑥 + 11)

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𝑦2 − 𝑥2

𝑥2 − 121

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Review Problems 5. Which equation has exactly one solution? A. 𝑥2 = − 4

B. (𝑥 + 5)2 = 0

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C. (𝑥 + 5)(𝑥 − 5) = 0 D. (𝑥 + 5)2 = 36

6. Two bacteria populations are measured at the same time. One bacteria population,

𝑝, is modeled by the equation 𝑝 = 250,000 ⋅ ( 1 ) , where 𝑑 is the number of days since 𝑑

it was first measured. The second bacteria population, 𝑞, is modeled by the equation

𝑞 = 500,000 ⋅( 1 ) . 𝑑

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Which statement is true about the two populations?

A. The second population will always be larger than the first. B. Both populations are increasing.

C. The second bacteria population decreases more rapidly than the first.

D. When initially measured, the first population is larger than the second.

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Unit 9, Lesson 6: Solving Quadratic Equations by Using Factored Form

Warm-Up:Why Would You Do That?

1. Choose a whole number between 0 and 10.

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Let’s try to find at least one solution to 𝑥2 − 2𝑥 − 35 = 0.

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2. Evaluate the expression 𝑥2 − 2𝑥 − 35, using your number for 𝑥.

3. If your number doesn’t give a value of 0, look for someone in your class who may have chosen a number that does make the expression equal 0. Which number is it?

4. There is another number that would make the expression 𝑥2 − 2𝑥 − 35 equal 0. Can you find it?

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Collaborative Activity: Let’s Solve Some Equations! 1. To solve the equation 𝑛2 − 2𝑛 = 99, Tyler wrote out the following steps. Analyze Tyler’s work. Write down what Tyler did in each step. 𝑛2 − 2𝑛 = 99

Original equation

(𝑛 − 11)(𝑛 + 9) = 0

Step 2

Step 4

Step 1

𝑛 − 11 = 0 𝑛 = 11

𝑛 + 9 = 0

𝑛 = − 9

Step 3

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𝑛2 − 2𝑛 − 99 = 0

2. Solve each equation by rewriting it in factored form and using the zero product property. Show your reasoning.

b. 𝑥2 − 8𝑥 + 12 = 5

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a. 𝑥2 + 8𝑥 + 15 = 0

c. 𝑥2 − 10𝑥 − 11 = 0

d. 49 − 𝑥2 = 0

e. (𝑥 + 4)(𝑥 + 5) − 30 = 0

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Collaborative Activity: Revisiting Quadratic Equations with Only One Solution

op y

1. The other day, we saw that a quadratic equation can have 0, 1, or 2 real solutions. Sketch graphs that represent three quadratic functions: one that has no zeros, one with 1 zero, and one with 2 zeros.

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2. Use graphing technology to graph the function defined by 𝑓(𝑥) = 𝑥2 − 2𝑥 + 1. What do you notice about the 𝑥-intercepts of the graph? What do the 𝑥-intercepts reveal about the function?

3. Solve 𝑥2 − 2𝑥 + 1 = 0 by using the factored form and zero product property. Show your reasoning. What solutions do you get?

4. Write an equation to represent another quadratic function that you think will only have one zero. Graph it to check your prediction.

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Lesson Summary In recent lessons, strategies for transforming expressions from standard form to factored form were explored. A quadratic expression in factored form makes it simpler to find values of the variable that make the expression equal zero. The equation 𝑥(𝑥 + 4) = 0 means that the product of 𝑥 and 𝑥 + 4 is 0. By the zero product property, this means that either 𝑥 = 0 or 𝑥 + 4 = 0. Therefore, 0 and −4 are solutions.

Original equation

𝑛2 − 4𝑛 = 140

Subtract 140 from each side to set 1 side equal to 0.

𝑛2 − 4𝑛 − 140 = 0

Rewrite the expression in factored form.

𝑛 − 14 = 0 or 𝑛 + 10 = 0

Apply the zero product property.

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(𝑛 − 14)(𝑛 + 10) = 0 𝑛 = 14 or 𝑛 = −10

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Together, these 2 skills, writing quadratic expressions in factored form and using the zero product property when a factored expression is set equal to 0, can be used to solve quadratic equations given in other forms. An example is shown.

Solve each equation.

When a quadratic equation is written as a quadratic expression set equal to 0, the number of real solutions the equation has is revealed.

In the previous example, it was not obvious how many solutions there would be when the equation was 𝑛2 − 4𝑛 − 140 = 0. Rewriting the equation as (𝑛 − 14)(𝑛 + 10) = 0 reveals there are 2 values that could make the equation true, 𝑛 = 14 and 𝑛 = −10. How many solutions does the equation 𝑥2 − 20𝑥 + 100 = 0 have?

• Rewrite it in factored form.

(𝑥 − 10)(𝑥 − 10) = 0

• The 2 factors are identical, which means that there is only 1 value of 𝑥 that makes the expression (𝑥 − 10)(𝑥 − 10) equal 0. • The equation has only 1 solution, 𝑥 = 10. 398 | Unit 9

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Practice Problems 1. Find all the solutions to each equation. a. 𝑥(𝑥 − 1) = 0

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b. (5 − 𝑥)(5 + 𝑥) = 0

c. (2𝑥 + 1)(𝑥 + 8) = 0

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d. (3𝑥 − 3)(3𝑥 − 3) = 0

e. (7 − 𝑥)(𝑥 + 4) = 0

2. Rewrite each equation in factored form and solve using the zero product property.

a. 𝑑2 − 7𝑑 + 6 = 0

c. 𝑢2 + 7𝑢 − 60 = 0

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b. 𝑥2 + 18𝑥 + 81 = 0

d. 𝑥2 + 0.2𝑥 + 0.01 = 0

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3. Here is how Elena solves the quadratic equation 𝑥2 − 3𝑥 − 18 = 0. 𝑥2 − 3𝑥 − 18 = 0

(𝑥 − 3)(𝑥 + 6) = 0

𝑥 = 3

𝑥 + 6 = 0

𝑥 = −6

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𝑥 − 3 = 0

Is her work correct? If you think there is an error, explain the error and correct it.

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Otherwise, check her solutions by substituting them into the original equation and showing that the equation remains true.

4. Jada is working on solving a quadratic equation, as shown here.

𝑝2 − 5𝑝 = 0

𝑝(𝑝 − 5) = 0 𝑝 − 5 = 0 𝑝 = 5

She thinks that her solution is correct because substituting 5 for 𝑝 in the original expression 𝑝2 − 5𝑝 gives 52 − 5(5), which is 25 − 25 or 0. Explain the mistake that Jada made and show the correct solutions.

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Review Problems 5. Which expression is equivalent to 𝑥2 − 7𝑥 + 12? A. (𝑥 + 3)(𝑥 + 4) B. (𝑥 − 3)(𝑥 − 4)

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C. (𝑥 + 2)(𝑥 + 6) D. (𝑥 − 2)(𝑥 − 6)

6. These quadratic expressions are given in standard form. Rewrite each expression in factored form. If you get stuck, try drawing a diagram.

b. 𝑥2 − 7𝑥 + 6

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a. 𝑥2 + 7𝑥 + 6

c. 𝑥2 − 5𝑥 + 6

d. 𝑥2 + 5𝑥 + 6

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7. Select all the functions whose output values will eventually overtake the output values of function 𝑓 defined by 𝑓(𝑥) = 25𝑥2.

□ 𝑔(𝑥) = 5(2)

𝑥

□ ℎ(𝑥) = 5

𝑥

□ 𝑗(𝑥) = 𝑥 + 5 □ 𝑘(𝑥) = �52 �

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𝑥

□ 𝑚(𝑥) = 5 + 2

𝑥

□ 𝑛(𝑥) = 2𝑥 + 5

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Unit 9, Lesson 7: Rewriting Quadratic Expressions in Factored Form When 𝒂≠𝟭 Warm-Up: Which One Doesn’t Belong: Quadratic Expressions 1. Which one doesn’t belong?

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A. (𝑥 + 4)(𝑥 − 3)

B. 3𝑥2 − 8𝑥 + 5 C. 𝑥2 − 25

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D. 𝑥2 + 2𝑥 + 3

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Guided Activity: Factoring Quadratic Expressions Where 𝒂≠𝟏

1. Use the algebra tiles to determine the 2 binomials that are factors of 3𝑥2 + 17𝑥 + 10.

x²

1 1

op y

x²

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x²

2. Discuss with your partner any differences you notice between the binomial factors in question 1 and the factors of a quadratic expression where 𝑎 = 1. Summarize your discussion.

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3. The steps are shown for multiplying (3𝑥 + 2)(𝑥 − 5) using the distributive property. Step

Result

(𝟑𝒙 + 𝟐)(𝒙 − 𝟓)

Distribute 3𝑥 to − 5

(3x + 2) (x – 5)

𝟐𝒙

(3x + 2) (x – 5)

− 𝟏𝟎

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Distribute 2 to − 5

− 𝟏𝟓𝒙

(3x + 2) (x – 5)

Distribute 2 to 𝑥

𝟑𝒙𝟐

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Distribute 3𝑥 to 𝑥

a. Based on the results in the table, what is the product of (3𝑥 + 2)(𝑥 − 5)?

b. The algebra tiles diagram models the result of (3𝑥 + 2)(𝑥 − 5).

Draw a rectangle around − 𝟏𝟓𝒙 in the algebra tiles diagram for 3𝑥2 − 13𝑥 − 10.

c. Discuss with your partner how the algebra tiles diagram demonstrates the distribution of 3𝑥 to − 5. Summarize your discussion.

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x²

–x –x –x –x –x

x²

–x –x –x –x –x

x²

–x –x –x –x –x

–1 –1 –1 –1 –1

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4. Compare the algebra tiles diagram for 3𝑥2 + 17𝑥 + 10 to the algebra tiles diagram for 3𝑥2 − 13𝑥 − 10. Describe the differences.

Factoring quadratic expressions of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 where 𝑎 = 1 results in the factored form (𝑥 + 𝑔)(𝑥 + ℎ), where 𝑔∙ℎ = 𝑐 and 𝑔 + ℎ = 𝑏.

3𝑥2 + 17𝑥 + 10

Factored form

Similarities

Differences

(3𝑥 + 2)(𝑥 − 5)

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(3𝑥 + 2)(𝑥 + 5)

3𝑥2 − 13𝑥 − 10

Trinomial

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5. Use the examples in the table to explore factoring trinomials of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 where 𝑎≠1. Summarize the similarities and differences of the factored forms where 𝑎 = 1 and where 𝑎≠1.

When factoring quadratic expressions of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐, the coefficient, 𝑏, is the sum of factors of the product of 𝑎 and 𝑐.

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6. In 3𝑥2 + 17𝑥 + 10 and 3𝑥2 − 13𝑥 − 10, the products of 𝑎 and 𝑐 are 30 and −30, respectively. a. List all the pairs of factors for 30 and all the pairs of factors for −30. Value

Pairs of Factors

op y

30 −30

b. Circle the factor pair of 30 whose sum is 17.

c. Circle the factor pair of −30 whose sum is −13.

𝟑𝒙𝟐 + 𝟏𝟕𝒙 + 𝟏𝟎

ip t

d. The factor pairs can be used to factor by grouping. For each trinomial expression, complete the steps for factoring by grouping. 𝟑𝒙𝟐 − 𝟏𝟑𝒙 − 𝟏𝟎

3𝑥2 + 2𝑥 − 15𝑥 − 10

3𝑥(𝑥 + _______) + 2(𝑥 + _______)

3𝑥(𝑥 − _______) + 2(𝑥 − _______)

(3𝑥 + 2)(𝑥 + 5)

(3𝑥 + 2)(𝑥 − 5)

3𝑥2 + 2𝑥 + 15𝑥 + 10

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Collaborative Activity: Factoring Quadratic Expressions Where 𝒂≠𝟏

op y

1. Factor 5𝑥2 + 6𝑥 − 8 by grouping.

2. Consider the trinomial 3𝑥2 − 13𝑥 − 10.

a. Complete the table by choosing the appropriate values.

−𝟏

𝟓

−𝟓

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𝟏 𝟐

𝟑𝒙

𝒙

𝟏 𝟐 𝟓

𝟏𝟎

−𝟏 −𝟐 −𝟓

−𝟏𝟎

3. 3𝑥2

3𝑥 6𝑥

15𝑥 30𝑥

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𝟏𝟎 1𝑥 2𝑥 5𝑥

−3𝑥 −6𝑥

−15𝑥

10𝑥

−𝟐

−𝟏𝟎 −1𝑥 −2𝑥 −5𝑥

−10𝑥

4. −10

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b. Use the values to complete the area model. ________

𝟑𝒙 𝒙

3𝑥2

________

3. Factor 5𝑥2 − 3𝑥 − 8 using the area model.

________

ip t

________

Lesson Summary

________

op y

−10

Only some quadratic equations in the form of 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 can be solved by rewriting the quadratic expression into factored form and using the zero product property. In some cases, finding the right factors of the quadratic expression can be more challenging.

For example, what is the factored form of 6𝑥2 + 11𝑥 − 35?

Modeling and diagrams can help in some cases, but often some guessing and checking must be done before finding the equivalent expression that would help solve the equation 6𝑥2 + 11𝑥 − 35 = 0. Streamline the guessing and checking by listing all the factor pairs of 𝑎∙𝑐 and then determining which pair has a sum of 𝑏.

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Once the right factors are found, proceed to solving the quadratic equation using the zero product property. An example is shown. 6𝑥2 + 11𝑥 − 35 = 0 (3𝑥 − 5)(2𝑥 + 7) = 0 3𝑥 − 5 = 0 or 2𝑥 + 7 = 0 𝑥 = 5 or 𝑥 = − 7 3

op y

Many quadratic expressions cannot be written in factored form. Rewriting quadratic expressions in factored form and using the zero product property is only 1 method, with limitations, for solving quadratic equations. The next several lessons will explore other ways to solve quadratic equations that work for any equation.

Practice Problems

1. To write 11𝑥2 + 17𝑥 − 10 in factored form, Diego first listed pairs of factors of −10.

(____ + 10)(____ + −1)

(____ + 1)(____ + −10)

(____ + 2)(____ + −5)

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(____ + 5)(____ + −2)

a. Use what Diego started to complete the rewriting.

b. How did you know you’ve found the right pair of expressions? What did you look for when trying out different possibilities?

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2. To rewrite 4𝑥2 − 12𝑥 − 7 in factored form, Jada listed some pairs of factors of 4𝑥2: (2𝑥 + ____)(2𝑥 + ____)

(4𝑥 + ____)(1𝑥 + ____)

Use what Jada started to rewrite 4𝑥2 − 12𝑥 − 7 in factored form.

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3. Rewrite each quadratic expression in factored form. Then, use the zero product property to solve the equation.

a. 7𝑥2 − 22𝑥 + 3 = 0

c. 9𝑥2 − 25 = 0

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b. 4𝑥2 + 𝑥 − 5 = 0

4. Han is solving the equation 5𝑥2 + 13𝑥 − 6 = 0.

Here is his work:

5𝑥2 − 13𝑥 − 6 = 0 (5𝑥 − 2)(𝑥 + 3) = 0 𝑥 = 2 or 𝑥 = −3

Describe Han’s mistake. Then, find the correct solutions to the equation.

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Review Problems 5. A picture is 10 inches (in.) wide by 15 in. long. The area of the picture, including a frame that is 𝑥 in. thick, can be modeled by the function 𝐴(𝑥) = (2𝑥 + 10)(2𝑥 + 15).

a. Use function notation to write a statement that means: the area of the picture, including a frame that is 2 in. thick, is 266 sq. in.

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b. What is the total area if the picture has a frame that is 4 in. thick?

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6. Which equation shows a next step in solving 9(𝑥 − 1)2 = 36

A. 3(𝑥 − 1) = 6

B. (𝑥 − 1)2 = 4

A. 9(𝑥 − 1) = 6 or 9(𝑥 − 1) = −6

C. (9𝑥 − 9)2 = 36

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Unit 9, Lesson 8: What Are Perfect Squares?

Warm-Up: The Thing We Are Squaring In each equation, what expression could be substituted for 𝑎 so the equation is true for all values of 𝑥?

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1. 𝑥2 = 𝑎2

2. (3𝑥)2 = 𝑎2

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3. 𝑎2 = 7𝑥 ⋅ 7𝑥

5. 𝑎2 = 1 𝑥2

4. 25𝑥2 = 𝑎2

6. 𝑎2 = (𝑥 + 1)2

7. (2𝑥 − 9)(2𝑥 − 9) = 𝑎2

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Exploration Activity: Perfect Squares in Different Forms 1. Each expression is written as the product of factors. Write an equivalent expression in standard form.

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a. (3𝑥)2

b. 7𝑥 ⋅ 7𝑥

cr us

d. (𝑥 + 1)2

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c. (𝑥 + 4)(𝑥 + 4)

e. (𝑥 − 7)2

f. (𝑥 + 𝑛)2

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2. Why do you think the following expressions can be described as perfect squares? 𝑥2 + 6𝑥 + 9

𝑥2 + 1 𝑥 + 1

𝑥2 − 16𝑥 + 64

Collaborative Activity: Two Methods

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Han and Jada solved the same equation with different methods. Here they are: Jada’s Method

Han’s Method

(𝑥 − 6)2 = 25 𝑥 − 6 = 5 or 𝑥 − 6 = − 5 𝑥 = 11 or 𝑥 = 1

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(𝑥 − 6)2 = 25 (𝑥 − 6)(𝑥 − 6) = 25 𝑥2 − 12𝑥 + 36 = 25 𝑥2 − 12𝑥 + 11 = 0 (𝑥 − 11)(𝑥 − 1) = 0 𝑥 = 11 or 𝑥 = 1

Work with a partner to solve these equations. For each equation, one partner solves with Han’s method, and the other partner solves with Jada’s method. Make sure both partners get the same solutions to the same equation. If not, work together to find your mistakes. (𝑥 + 4)2 = 9

(𝑦 − 5)2 = 49

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(𝑣 − 0.1)2 = 0.36

(𝑧 + 1 )2 = 4 3

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Lesson Summary This lesson explored perfect squares.

A perfect square is an expression that is something times itself.

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Some examples of perfect squares are shown. •

49, because 49 is 7⋅7 or 72

•

(𝑥 + 5)2, because it is equivalent to (𝑥 + 5)(𝑥 + 5)

•

81𝑎2, because it is equivalent to (9𝑎) ⋅ (9𝑎) or (9𝑎)2

𝑥2 − 12𝑥 + 36, because it is equivalent to (𝑥 − 6)2 or (𝑥 − 6)(𝑥 − 6)

•

When expressions that are perfect squares are written in factored form and standard form, there is a predictable pattern. (𝑥 + 5)(𝑥 + 5) is equivalent to 𝑥2 + 10𝑥 + 25.

•

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•

(𝑥 − 6)2 is equivalent to 𝑥2 − 12𝑥 + 36. (𝑥 − 9)2 is equivalent to 𝑥2 − 18𝑥 + 81.

In general, (𝑥 + 𝑛)2 is equivalent to 𝑥2 + (2𝑛)𝑥 + 𝑛2.

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Quadratic equations in which a perfect square equals a perfect square can be solved by taking the square root of both sides of the equation. An example is shown.

𝑥2 − 18𝑥 + 81 = 25 (𝑥 − 9)(𝑥 − 9) = 25 (𝑥 − 9)2 = 25 �(𝑥 − 9)2 = �25 𝑥 − 9 = 5 or 𝑥 − 9 = − 5 𝑥 = 14 or 𝑥 = 4

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Recall that taking the square root produces 2 solutions, where 1 is positive and 1 is negative. In this case, �25 is both 5 and − 5, generating 2 equations to solve to determine the solutions to the original equation.

Practice Problems

1. Select all the expressions that are perfect squares.

□ (𝑥 + 5)(𝑥 + 5)

□ ( − 9 + 𝑐)(𝑐 − 9)

□ (𝑦 − 10)(10 − 𝑦)

□ (𝑎 + 3)(3 + 𝑎)

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□ (2𝑥 − 1)(2𝑥 + 1)

□ (4 − 3𝑥)(3 − 4𝑥) □ (𝑎 + 𝑏)(𝑏 + 𝑎)

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2. Solve each equation. a. (𝑥 − 1)2 = 4

b. (𝑥 + 5)2 = 81

d. (𝑥 + 11)2 = 121

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c. (𝑥 − 2)2 = 0

e. (𝑥 − 7)2 = 64

3. To solve the equation (𝑥 + 3)2 = 4, Han first expanded the squared expression. Here is his incomplete work: (𝑥 + 3)2 = 4

(𝑥 + 3)(𝑥 + 3) = 4

𝑥2 + 3𝑥 + 3𝑥 + 9 = 4

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𝑥2 + 6𝑥 + 9 = 4

a. Complete Han’s work and solve the equation.

Use Jada’s reasoning to solve the equation.

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b. Jada saw the equation (𝑥 + 3)2 = 4 and thought, “There are two numbers, 2 and − 2, that equal 4 when squared. This means 𝑥 + 3 is either 2 or it is − 2. I can find the values of 𝑥 from there.”

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c. Can Jada use her reasoning to solve (𝑥 + 3)(𝑥 − 3) = 5? Explain your reasoning.

Review Problem

4. A jar full of marbles is displayed. The following table shows the guesses for 10 people. The actual number of marbles in the jar is 145. Calculate the absolute guessing error for all 10 guesses.

Guess

Absolute Guessing Error

190

150

125

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133

167

160

148

200

170

115

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Unit 9, Lesson 9: Completing the Square When a = 1 Using Algebra Tiles

Warm-Up: Perfect or Imperfect? Select all expressions that are perfect squares. Explain how you know.

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□ (𝑥 + 5)(5 + 𝑥)

□ (𝑥 + 5)(𝑥 − 5) □ (𝑥 − 3) 2

□ 𝑥 − 3

□ 𝑥 + 8𝑥 + 16 2

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□ 𝑥 + 10𝑥 + 20

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Guided Activity: Completing the Square Using Algebra Tiles The algebra tiles shown represent the expression 𝑥2 − 6𝑥 + 11.

–x –x –x –x –x –x

1 1

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x²

1 1

The algebra tiles are arranged onto a product mat to create a square. There are 2 single unit tiles left over. The factors (𝑥 − 3) and (𝑥 − 3) are located above the top bar and on the left bar.

–1 –1 –1

x²

–x –x –x

–1

–x

1 1

–1

–x

1 1

–1

–x

1 1

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1. Use the algebra tiles on the product mat to complete the chart.

Number of 𝒙2 Tiles

Number of 𝒙 Tiles

Length of the Square

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Area of the Square (length2)

1 1

Unit Tiles Left Over

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2. Complete the statements. The algebra tiles show an arrangement where part of the trinomial 𝑥2 − 6𝑥 + 11 is made into a perfect square. The product mat shows that the area of the square is _______________________ with _____ 1 unit tiles left over. Therefore, 𝑥2 − 6𝑥 + 11 is equivalent to (_________________)2 + _____. This process is known as completing the square.

–x

–1 –1 –1

–x

–1 –1

–x

–1 –1

–x

–1 –1

x²

–x

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3. The algebra tiles shown represent the expression 𝑥2 − 12𝑥 − 9.

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a. The algebra tiles are arranged onto a product mat. Work with your partner to determine how many positive 1 unit tiles should be added to complete the square.

–x –x –x –x –x –x

x²

–1 –1 –1 –1 –1 –1 –1 –1 –1

–x –x

–x

–x –x –x

Since there are only 9 negative unit tiles, there are no positive unit tiles to place on the product mat. To add the unit squares needed, they will need to be “borrowed” from somewhere. © Accelerate Learning Inc. - All Rights Reserved

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b. To complete the perfect square trinomial, ______ positive unit tiles must be added. Adding only the positive unit tiles will change the value of the original expression, so instead, for every positive unit tile added, a negative unit tile also must be added to the model. For each pair of positive and negative unit tiles added, the value actually being added to the expression is 0.

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1 –1 = 0

c. Complete the product mat by drawing the positive unit tiles needed. Then, below the original 9 negative unit tiles, draw the negative unit tiles that must be added so that the sum of the tiles being added is 0.

Length of the Square

Number of 𝒙 Tiles

Area of the Square (length2)

Unit Tiles Left Over

Number of 𝒙2 Tiles

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d. Use the algebra tiles on the completed product mat to complete the chart.

e. On the product mat, draw the length of each side of the square to show each factor.

f. Complete the statement. The product mat shows that the area of the square is ______________ with ____ tiles left over. Therefore, 𝑥2 − 12𝑥 − 9 is equivalent to (____________)2 + ____.

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4. The algebra tiles on the product mat represent the expression 𝑥2 + 8𝑥 + 5. a. How many positive unit tiles are needed to complete the square?

1 1

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b. Add the positive unit tiles to complete the square and the same number of negative unit tiles to the side of the product mat. Show the addition of these tiles by drawing them.

x²

x x

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d. Complete the statement.

c. On the product mat, draw the length of each side of the square to show each factor.

The expression 𝑥2 + 8𝑥 + 5 is equivalent to (______________)2 + _____.

5. The algebra tiles on the product mat represent the expression 𝑥2 + 16𝑥 + 55.

a. On the product mat, complete the square, and draw the length of each side of the square to show each factor.

b. Complete the statement.

The expression 𝑥 + 16𝑥 + 55 is equivalent to (______________)2 + _____. 2

x²

1 1

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Collaborative Activity: Conjectures about Completing the Square Work with your partner to answer the questions.

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1. What do you notice about the number of 𝑥-tiles along the top of each model and the number of 𝑥-tiles along the left side of each model?

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2. What is the relationship between the coefficient of the 𝑥-term of the quadratic in standard form and the number of unit tiles along the left side of the model?

3. The table shows the 4 quadratic expressions from the Guided Activity. Quadratic Expression C

Quadratic Expression D

𝑥2 − 6𝑥 + 11

𝑥2 − 12𝑥 − 9

𝑥2 + 8𝑥 + 5

𝑥2 + 16𝑥 + 55

(𝑥 − 3)2 + 2

(𝑥 − 6)2 − 45

(𝑥 + 4)2 − 11

(𝑥 + 8)2 − 9

Vertex Form

Standard Form

Quadratic Expression B

Quadratic Expression A

a. Make a conjecture about how 𝑏, the coefficient of the 𝑥-term of the quadratic expression in standard form, 𝑥2 + 𝑏𝑥 + 𝑐, is related to the ℎ-value of the equivalent quadratic expression in vertex form, (𝑥 − ℎ)2 + 𝑘.

b. Make a conjecture about how to determine the 𝑘-value in vertex form of a quadratic expression from standard form.

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4. Use your conjectures to determine both the ℎ-value and 𝑘-value of the quadratic in vertex form (𝑥 − ℎ)2 + 𝑘 for each equation given in standard form. Standard Form

𝑥2 − 14𝑥 − 2

𝑥2 + 18𝑥 + 16

𝑥2 − 48𝑥 − 27

𝑥2 + 100𝑥 + 86

𝒌-value

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𝒉-value

5. With a different partner, review your conjectures about completing the square. Work through any differences, and revise the conjectures and answers as necessary.

Lesson Summary

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6. Verify any revisions with your original partner.

Rewriting a quadratic expression as a perfect square can be helpful when trying to solve a quadratic equation.

To rewrite the expression 𝑥2 + 6𝑥 + 10 as a perfect square, algebra tiles or area models can be used while learning how to complete the square. Completing the square is the method used to rewrite a quadratic expression, equation, or function from standard form to vertex form.

Recall that the vertex form of a quadratic expression is (𝑥 − ℎ)2 + 𝑘, where (ℎ,𝑘) is the vertex of the graph of the function.

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The tiles on the product mat represent 𝑥2 + 6𝑥 + 10. 1

𝑥

𝑥2

𝑥

1 1

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𝑥

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After completing the square, there is 1 unit tile left over. The factors (𝑥 + 3) and (𝑥 + 3) are located along the top and left edges of the product mat. This shows the expression 𝑥2 + 6𝑥 + 10 is equivalent to (𝑥 + 3)2 + 1.

Closely studying the examples in this lesson reveals a relationship between the value of 𝑏 in standard form and the value of ℎ in vertex form of quadratic expressions. The value of ℎ is always half of the opposite value of 𝑏. The value of 𝑘 also has a relationship to 𝑏. The value of 𝑘 is always half of 𝑏2, subtracted from 𝑐. Understanding this relationship helps to more efficiently apply the completing the square strategy when algebra tiles are not available or are impractical for solving problems.

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Practice Problems 1. Use the diagram to complete the square for the expression 𝑥2 + 10𝑥 + 8. Then, rewrite the expression in vertex form. 1

𝑥

𝑥2

𝑥

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𝑥

2. Determine both the ℎ-value and 𝑘-value of the quadratic expression in vertex form (𝑥 − ℎ)2 + 𝑘 for each equation given in standard form. 𝑥2 − 6𝑥 − 7

Standard Form

𝑥2 − 2𝑥 − 8

𝑥2 + 4𝑥 + 8

𝑥2 − 18𝑥 + 81

𝒉-value 𝒌-value

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3. Match each expression in standard form to an equivalent expression in vertex form. Standard Form

Vertex Form

𝑥2 + 8𝑥 − 2

(𝑥 − 10)2 − 54

𝑥2 − 20𝑥 + 46

(𝑥 − 10)2 − 91

(𝑥 + 5)2 − 12

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𝑥2 + 10𝑥 + 13 𝑥2 + 2𝑥

(𝑥 + 4)2 − 18

𝑥2 + 10𝑥 + 16

(𝑥 + 1)2 − 1

(𝑥 + 5)2 − 9

𝑥2 − 20𝑥 + 9

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Review Problems

a. (𝑥 + 3)(𝑥 − 3)

4. Rewrite each expression in standard form.

b. (7 + 𝑥)(𝑥 − 7)

c. (2𝑥 − 5)(2𝑥 + 5) d. �𝑥 +

1 1 ��𝑥 − � 8 8

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5. A basketball is dropped from the roof of a building and its height in feet (ft.) is modeled by the function ℎ. Here is a graph representing ℎ.

Select all the true statements about this situation.

□ When 𝑡 = 0 the height is 0 ft.

□ The expression that defines ℎ is linear.

□ The expression that defines ℎ is quadratic.

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□ The basketball falls at a constant speed.

□ When 𝑡 = 0 the ball is about 50 ft. above the ground.

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□ The basketball lands on the ground about 1.75 seconds (sec.) after it is dropped.

6. A group of students are guessing the number of paper clips in a small box.

The guesses and the guessing errors are plotted on a coordinate plane.

What is the actual number of paper clips in the box?

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Unit 9, Lesson 10: Completing the Square When a > 1 Using Algebra Tiles

Warm-Up: Equations with Fractions 1. Solve each equation mentally. b. ( 3 )2 = 𝑥

3 + 𝑥 = 9 5 5

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a. 𝑥 + 𝑥 = 1

d. 25 + 𝑥 = 4

Exploration Activity: Completing the Square When a > 1

𝑥

𝑥2

𝑥

𝑥2

1 𝑥

1 1

𝑥

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1. The diagram shows 3 perfect square trinomials.

𝑥

𝑥2

𝑥

Work with your partner to complete each algebraic representation of the diagram.

𝑥

Three Perfect Square Trinomials

Three Binomials Squared

All Like Terms Added

3(𝑥2 + _____𝑥 + _______)

3(𝑥 + ______)2

_____𝑥2 + _____𝑥 + ______

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2. The expression 3(𝑥2 + 6𝑥) + 11 is represented in the diagram using 3 product mats.

1 1 1

𝑥 𝑥 𝑥

1 1 1

𝑥2

𝑥

𝑥 𝑥 𝑥

𝑥2

1 1 1

𝑥

𝑥2

𝑥

1 1

𝑥 𝑥

1 𝑥

a. Discuss the representation with your partner.

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𝑥

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b. Name the property of operations that is applied to rewrite 3𝑥2 + 18𝑥 + 11 as 3(𝑥2 + 6𝑥) + 11.

c. How many 1-unit tiles should be added to complete all of the squares?

By adding those unit tiles, the diagram now shows 3(𝑥 + 3)2. As a reminder, if 1 unit tiles are added to the model, then the same number of −1 unit tiles must be added to the model so that its value doesn’t change. Algebraically, this can be represented by subtracting that number of units.

d. Complete the equivalent expression. 3(𝑥 + 3)2 − _______

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3. The diagram of algebra tiles shown represents the expression 2𝑥2 − 16𝑥 − 5.

−1 −1 −1 −1

𝑥2

−𝑥

−1 −𝑥

−𝑥

𝑥

𝑥2

−1

−𝑥

−1

−𝑥

−1

−𝑥

−1

−𝑥

−1

−𝑥

−1 −𝑥

−1

−1 −1

−𝑥

−1 −1

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𝑥

−1

a. Rewrite the expression 2𝑥2 − 16𝑥 − 5 to show the representation of the 2 separate squares by factoring 2 out from the first 2 terms.

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2(𝑥2 _____ _________) − 5

b. How many 1 unit tiles should be added to complete one of the squares?

c. Add the number of unit tiles for one square to the expression inside the parentheses. 2(𝑥2 − 8𝑥 + _______) − 5

d. Describe how the coefficient of 𝑥 is related to the number of unit tiles added for one square.

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e. Discuss with your partner how the 2 outside of the parentheses is related to the diagram.

2(x ² – 8x +

)–5

f. What is the total number of unit tiles added to complete the squares?

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g. Discuss with your partner how multiplying the 2 outside of the parentheses to the number of unit tiles added for one square also gives the total number of unit tiles added.

2(x ² – 8x +

)–5

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h. Complete the expression to show the subtraction of unit tiles needed so the new expression remains equivalent to the original expression, 2𝑥2 − 16𝑥 − 5.

2(𝑥2 − 8𝑥 + _______) − 5 − ________

2(𝑥________)2__________

i. Use the completed squares and the added tiles to rewrite the expression 2𝑥2 − 16𝑥 − 5.

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Guided Activity: Completing the Square When a > 1 The diagram of algebra tiles shown represents the expression 4𝑥2 − 48𝑥 − 3.

−1

−𝑥

−1

−𝑥

−1

𝑥

−1

−𝑥

𝑥

−1

−𝑥

−1 −1

−𝑥

−1

−𝑥

−1

−𝑥

−1

𝑥

−1

−𝑥

𝑥2

−𝑥

−1 −𝑥

−1

−𝑥

𝑥

−1

−𝑥 −𝑥

−1 −1

𝑥2

−𝑥

−1 −𝑥

−𝑥 −𝑥 −𝑥 −𝑥

−𝑥

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−1

𝑥2

−𝑥

−1 −1

𝑥

−1 −1

−𝑥

−1

𝑥

−1 −1

−1

−𝑥

−1

−𝑥

−1

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−1

−𝑥

−1

−𝑥

−1

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𝑥2

−1

𝑥

−1 −𝑥

−𝑥 −𝑥 −𝑥 −𝑥

−1

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1. Fill in the table to complete the square algebraically using the process described. Refer to the model of the expression 4𝑥2 − 48𝑥 − 3 as needed. Process

Expression

Start with the original expression.

op y

4𝑥2 − 48𝑥 − 3

Factor the first 2 terms to reflect the algebra tiles in each square. The number factored out is also the number of squares.

______(𝑥2 − ______𝑥 + _______) − 3

ip t

Multiply the number of squares by the number of unit tiles added to each square. Subtract the total number of unit tiles added so the expression remains equivalent.

Determine the number of unit tiles that should be added to complete each square.

______(𝑥2 − ______𝑥) − 3

Rewrite the perfect square trinomial represented in the square as a binomial squared.

______(𝑥 − _______)2 − 3 + _______ ______(𝑥 − _______)2 + _______

Combine like terms.

______(𝑥2 − ______𝑥 + _______) − 3 + _______

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Lesson Summary The process of completing the square can be used to rewrite quadratic expressions given in standard form as an equivalent expression in vertex form. When possible, algebra tiles can be used to model completing the square. The algebra tiles used in this lesson help make sense of the algebraic process of completing the square when 𝑎 > 1.

op y

With algebra tiles, the value of 𝑎 indicates how many squares need to be built. As the value of 𝑎 gets larger, it gets less practical to physically use algebra tiles to complete the square. However, knowledge of how algebra tiles work to complete the square can help to visualize the algebraic process.

An example of the process of completing the square to rewrite 16𝑥2 + 96𝑥 − 11 is shown and explained in the table. Explanation

Equivalent Expression

Start with the original expression in standard form.

Determine the number of 𝑥 tiles that go across the top and left side of each square (half of 𝑏).

16(𝑥2 + 6𝑥 + 9) − 11

Factor out the number of squares that would need to be built (the common factor) from the first 2 terms.

16(𝑥2 + 6𝑥) − 11

ip t

16𝑥2 + 96𝑥 − 11

16(𝑥2 + 6𝑥 + 9) − 11 − 144 16(𝑥 + 3)2 − 11 − 144 16(𝑥 + 3)2 − 155

Add the square of that number to indicate the number of 1-unit tiles in 1 complete square (the perfect square trinomial).

Distribute 16 to 9. This represents the total amount of 1-unit tiles needed to complete all the squares.

Subtract that value to keep the expression equivalent.

Rewrite the perfect square trinomial as a binomial squared. Combine like terms.

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Practice Problems 1. The diagram of algebra tiles shown represents the expression 2𝑥2 − 8𝑥 + 13.

−1

−𝑥

𝑥2

−𝑥

𝑥

−1

𝑥

−1

−𝑥

𝑥2

−𝑥

−𝑥 1

−𝑥

−1

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𝑥

b. 2(𝑥________)2__________

a. 2(____𝑥2 − _____𝑥 + _____) + _____

ip t

Use the diagram to complete each expression equivalent to 2𝑥2 − 8𝑥 + 13.

2. For each expression, find the missing value that makes the expression in parentheses a perfect square trinomial. Then, rewrite the entire expression as an equivalent expression in vertex form. Use algebra tiles to help you, if needed.

a. (2𝑥2 + 16𝑥 + _______ )

b. (6𝑥2 − 24𝑥 + _______ ) + 3

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Review Problem 3. Add the number that would make the expression a perfect square. Next, write an equivalent expression in factored form.

op y

a. 𝑥2 − 6𝑥

b. 𝑥2 + 2𝑥

e. 𝑥2 + 24𝑥

d. 𝑥2 − 4𝑥

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c. 𝑥2 + 14𝑥

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Unit 9, Lesson 11: Completing the Square Algebraically

Warm-Up: Perfect Squares in Two Forms Elena says, “(𝑥 + 3)2 can be expanded into 𝑥2 + 6𝑥 + 9. Likewise, (2𝑥 + 3)2 can be expanded into 4𝑥2 + 6𝑥 + 9.”

op y

Find an error in Elena’s statement and correct the error. Show your reasoning.

Collaborative Activity: Completing the Square Algebraically For each expression, the first step to completing the square is shown.

𝑥2 − 24𝑥 + 15

4𝑥2 − 24𝑥 + 15

(𝑥2 − 24𝑥) + 15

4(𝑥2 − 6𝑥) + 15

6(𝑥2 − 4𝑥) + 15

Expression C

6𝑥2 − 24𝑥 + 15

Expression B

ip t

Expression A

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1. Discuss with your partner how to determine what value to add to the expression inside the parentheses so that it results in a perfect square trinomial. Use the algebra tiles diagram provided if sketching a model will help you.

2. Use your process to complete each perfect square trinomial.

Expression A Expression B

Expression C

6(𝑥2 − 4𝑥 + ________) + 15

(𝑥2 − 24𝑥 + ________) + 15 4(𝑥2 − 6𝑥 + ________) + 15

3. In Expression A, the value that was added to create the perfect square trinomial is being multiplied by ______, the coefficient of the perfect square trinomial. Therefore, the value that should be subtracted from 15 to maintain equivalence is _______. 4. In Expression B, the value that was added to create the perfect square trinomial is being multiplied by ______, so the value that should be subtracted from 15 to maintain equivalence is _______.

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5. In Expression C, the value that was added to create the perfect square trinomial is being multiplied by ______, the coefficient of the perfect square trinomial. Therefore, the value that should be subtracted from 15 to maintain equivalence is _______. Expression A Expression B

6(𝑥 − ________)2 + 15 _____ ________ (𝑥 − ________)2 + 15 _____ ________

op y

6. Factor each perfect square trinomial, and rewrite it as a binomial squared. Also, enter the value determined from the steps described above, including the appropriate sign, + or − .

Expression C

4(𝑥 − ________)2 + 15 _____ _______

ip t

Rewriting the resulting perfect square trinomial as a binomial squared is the next step in the process, as it represents the factored form of a perfect square trinomial. The algebra tiles are a visual representation of this perfect square.

1 1

𝑥2 + 8𝑥 + 16 = (𝑥 + 4)2

x²

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7. While learning the process of completing the square with algebra tiles, the leading coefficients were all positive. The table shown gives a quadratic equation in standard form that has negative coefficients. Complete the table to complete the square without using algebra tiles. Equation

Original equation

𝑦 = −3𝑥2 − 24𝑥 + 11

op y

Process

a. Factor out the common factor of the first two terms.

𝑦 = _____(𝑥2 + ______𝑥) + 11

b. Discuss with your partner why there is a plus sign instead of a minus sign in the parentheses.

ip t

c. Complete the statement.

(𝑥2 + ______𝑥)

There is a plus sign instead of a minus sign in the parentheses because . . .

d. Determine the value that makes a perfect square trinomial in the parentheses.

e. Then, to maintain equivalence, multiply the factor by the value. Add the opposite of that value.

𝑦 = ____(𝑥2 + ______𝑥 + _______) + 11 𝑦 = ____(𝑥2 + _____𝑥 + ______) + 11 + ______

f. Discuss with your partner why, after finding the product of the factor and the value added to make a perfect square trinomial, the opposite of this product is added to the expression. g. Rewrite the perfect square trinomial represented in the parentheses as a binomial squared. h. Combine like terms.

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Guided Activity: Completing the Square Algebraically 1. Consider the quadratic equation 𝑦 = 𝑥2 + 22𝑥 − 53.

op y

a. Rewrite the equation in vertex form by completing the square.

ip t

b. Use both forms of the equation to determine key features of the quadratic. It may not be possible to determine all key features from the given form and the rewritten form.

Key Features

𝑥-intercept(s)

Decreasing interval(s) Positive interval(s)

Domain

Negative interval(s)

Range

Opens up/down

Vertex

Axis of symmetry

𝑦-intercept

Increasing interval(s)

Left end behavior

As 𝑥 →_____, 𝑦 →_____.

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Right end behavior

As 𝑥 →_____, 𝑦 →_____.

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Lesson Summary The process of completing the square is used to rewrite quadratic expressions, equations, or functions from standard form, 𝑎𝑥2 + 𝑏𝑥 + 𝑐, to vertex form, 𝑎(𝑥 − ℎ)2 + 𝑘.

This process is helpful when trying to find key features of quadratic equations that are not easy to determine using standard form.

op y

An example of completing the square to rewrite 𝑦 = 16𝑥2 + 40𝑥 − 11 is shown and explained. Explanation

𝑦 = 16𝑥2 + 48𝑥 − 11

Start with the original equation in standard form.

Expression

Factor out the common factor from the coefficients of the first 2 terms.

𝑦 = 16�𝑥2 + 3𝑥� − 11

ip t

Determine half of 𝑏. Add the square of that value to complete the perfect square trinomial in parentheses. Distribute 16 to 9 . Subtract that value to keep 4 the expression equivalent.

𝑦 = 16�𝑥2 + 3𝑥 + 9 � − 11 4 𝑦 = 16�𝑥2 + 3𝑥 + 9 � − 11 − 144 4

𝑦 = 16�𝑥 + 3 � − 11 − 36 2

𝑦 = 16�𝑥 + 3 � − 47 2

Rewrite the perfect square trinomial as a binomial squared.

Combine like terms. 2

The equation 𝑦 = 16𝑥2 + 40𝑥 − 11 rewritten in vertex form is 𝑦 = 16�𝑥 + 3 � − 47. The

graph of the function is a parabola with its vertex at �− 3 , −47�. 2

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Practice Problems 1. Consider the quadratic function 𝑓(𝑥) = −8𝑥2 + 80𝑥 − 71.

op y

a. Rewrite the function in vertex form by completing the square.

ip t

b. Use both forms of the function to determine key features of the quadratic. It may not be possible to determine all key features from the given form and the rewritten form. Key Features

Decreasing interval(s)

𝑥-intercept(s)

Increasing interval(s)

𝑦-intercept

Positive interval(s) Negative interval(s)

Range

Opens up/down

Vertex

Axis of symmetry

Domain

Left end behavior

As 𝑥 →_____, 𝑦 →_____.

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Right end behavior

As 𝑥 →_____, 𝑦 →_____.

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Review Problems 2. Solve each equation. 𝑥2 + 11𝑥 + 27 = 3

(𝑦 + 2)(𝑦 + 6) = −3

op y

𝑝2 + 10 = 7𝑝

ip t

3. Which function could represent the height in meters of an object thrown upwards from a height of 25 meters (m) above the ground 𝑡 seconds (sec.) after being launched?

B. 𝑓(𝑡) = − 5𝑡2 + 25

C. 𝑓(𝑡) = − 5𝑡2 + 25𝑡 + 50

D. 𝑓(𝑡) = − 5𝑡2 + 50𝑡 + 25

A. 𝑓(𝑡) = − 5𝑡2

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Unit 9, Lesson 12: Solving Quadratic Equations by Completing the Square

Warm-Up: Three Expressions, One Function

𝑥2 + 6𝑥 + 8

(𝑥 + 2)(𝑥 + 4)

op y

These expressions each define the same function.

(𝑥 + 3)2 − 1

Without graphing or doing any calculations, determine where the following features would be on a graph that represents the function.

2. the 𝑥-intercepts

ip t

1. the vertex

3. the 𝑦-intercept

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Guided Activity: Solving Quadratic Equations by Completing the Square 1. The expression 𝑥2 + 8𝑥 + 3 is represented using algebra tiles.

ip t

op y

Write an expression in vertex form.

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2. The equations 𝑦 = 𝑥2 + 8𝑥 + 3 and 𝑦 = −4 are shown on the coordinate plane. Where does the line 𝑦 = −4 intersect with the quadratic 𝑦 = 𝑥2 + 8𝑥 + 3?

3 2 1 -10

-8

-6

-4

-2

0 -1

x 2

-2 -3 -4 -5 -6 -7

op y

-8 -9

-10

-11

-12 -13 -14

The points of intersection show the solutions to 𝑥2 + 8𝑥 + 3 = −4.

ip t

3. Daniella tried to solve 𝑥2 + 8𝑥 + 3 = −4 by using the square root method. Her steps are shown. 𝑥2 + 8𝑥 + 3 = −4

𝑥2 + 8𝑥 = −7

𝑥2 = −7 − 8𝑥

Daniella got stuck on what to do next. Discuss with your partner why Daniella cannot solve 𝑥2 = −7 − 8𝑥 using square roots.

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4. One way of solving 𝑥2 + 8𝑥 + 3 = −4 is by completing the square. a. Complete the steps in the table. Process

Equation

Start with the original equation.

𝑥2 + 8𝑥 + 3 = −4

op y

Add 4 to both sides of the equation to set the equation to 0.

𝑥2 + 8𝑥 + 3 + 4 = −4 + 4 𝑥2 + 8𝑥 + _______ = _______ (𝑥2 + 8𝑥) + 7 = 0

Complete the square.

(𝑥2 + 8𝑥 + _______) + 7 − _______ = 0

ip t

(𝑥 + 4)2 − 9 = 0 (𝑥 + 4)2 − 9 = 0

(𝑥 + 4)2 − 9 + ______ = 0 + ______

Add 9 to both sides of the equation to isolate the quantity being squared.

Take the square root of both sides of the equation.

Solve for 𝑥.

�

(𝑥 + 4)2 = 9

=�

(𝑥 + 4) = _____

𝑥 + 4 = 3 or 𝑥 + 4 = −3

𝑥 = ________ or 𝑥 = ________

b. Check your solutions with the intersections found using the graph.

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op y

5. Solve −3(𝑥 − 5)2 + 6 = −21.

ip t

6. Solve −4𝑥2 − 16𝑥 − 11 = −3 by completing the square.

Collaborative Activity: Solving Quadratic Equations by Completing the Square

Work with your partner to complete the following.

𝑥2 − 12𝑥 + 7 = −17

1. Solve by completing the square. Show your work.

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2. The function 𝑓(𝑥) = −3𝑥2 + 54𝑥 + 83 models the sales of music cassette tapes for the years 1980 to 2000, in tens of millions of dollars, where 𝑥 is the number of years since 1980. a. A step is shown in the process of rewriting the function 𝑓(𝑥) = − 3𝑥2 + 54𝑥 + 83. Complete the step.

op y

𝑓(𝑥) = −3(𝑥 ____ _______)2 + 83 ___ ________ b. The function, 𝑓, models the sales in tens of millions of dollars. How many tens of millions of dollars is $40,000,000?

ip t

c. Solve −3𝑥2 + 54𝑥 + 83 = 4 by completing the square.

d. Determine which solution is not viable.

e. Justify why that solution is not viable.

f. In what year were the sales of cassette tapes $40,000,000?

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Lesson Summary One way to solve a quadratic equation is by completing the square. This method can be helpful when factoring is not practical. The steps to solve 25𝑥2 − 50𝑥 = − 8 by completing the square are shown. Equivalent Equations

25(𝑥2 − 2𝑥) + 9 = 0

25(𝑥2 − 2𝑥 + 1) + 9 − 25 = 0 25(𝑥 − 1)2 − 16 = 0 25(𝑥 − 1)2 − 16 = 0

Complete the square.

25(𝑥 − 1)2 − 16 + 16 = 0 + 16

25𝑥2 − 50𝑥 + 9 = 0

Add 9 to both sides of the equation to set the equation equal to 0.

ip t

25𝑥2 − 50𝑥 + 9 = −9 + 9

op y

Start with the original equation.

25𝑥2 − 50𝑥 = −9

Add 16 to both sides of the equation, and then divide by 25 to isolate the quantity being squared.

25(𝑥 − 1)2 = 16

Process

25(𝑥−1)2 16 = 25 25

(𝑥−1)2 = 16 25

�(𝑥−1)2 = �16 25

x−1=± 4 5

𝑥 − 1 = 4 or 𝑥 − 1 = − 4 5

𝑥 = 9 or 𝑥 = 1 5

Take the square root of both sides of the equation.

Solve for 𝑥.

The solutions to 25𝑥2 − 50𝑥 = − 9 are 𝑥 = 9 and 𝑥 = 1

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Practice Problems

op y

1. Solve −𝑥2 + 10𝑥 + 8 = −6 by completing the square. Show your work.

ip t

2. Solve 6𝑥2 − 48𝑥 − 2 = 4 by completing the square. Show your work.

3. Solve 4𝑥2 + 4𝑥 = 3 by completing the square. Show your work.

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Review Problem

ip t

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4. For each function 𝑓, decide if the equation 𝑓(𝑥) = 0 has 0, 1, or 2 real solutions. Explain how you know.

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Unit 9, Lesson 13: Quadratic Equations with Irrational Solutions

Warm-Up: Roots of Squares

op y

Here are some squares whose vertices are on a grid.

Side length (units)

Area (square units)

Square

ip t

Find the area and the side length of each square.

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Collaborative Activity: Solutions Written as Square Roots Solve each equation. Use the ± notation when appropriate. 2. (𝑥 − 6)2 = 0

3. 𝑥2 + 9 = 0

4. 𝑥2 = 18

6. (𝑥 + 1)2 = 18

5. 𝑥2 + 1 = 18

ip t

op y

1. 𝑥2 − 13 = −12

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Guided Activity: Finding Irrational Solutions by Completing the Square Here is an example of an equation being solved by graphing and by completing the square.

𝑥2 + 6𝑥 + 7 = 0

op y

𝑥2 + 6𝑥 + 9 = 2

(𝑥 + 3)2 = 2

𝑥 + 3 = ±�2

𝑥 = −3 ± �2

Verify: �2 is approximately 1.414. So − 3 + �2 ≈ −1.586 and −3 − �2 ≈ − 4.414.

2. 𝑥2 − 10𝑥 + 18 = 0

1. 𝑥2 + 4𝑥 + 1 = 0

ip t

For each equation, find the exact solutions by completing the square and the approximate solutions by graphing. Then, verify that the solutions found using the two methods are close. If you get stuck, study the example.

3. 𝑥2 + 5𝑥 + 1 = 0

4. 𝑥2 + 8 𝑥 + 14 = 0 3

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Lesson Summary When solving quadratic equations, it is important to remember the following. • Any positive number has 2 square roots, 1 positive and 1 negative, because there are 2 numbers that can be squared to make that number. For example, 62 and (−62) both equal 36, so 6 and − 6 are both square roots of 36.

op y

• The square root symbol (� ) can be used to express the positive square root of a number, known as the principal square root. For example, the square root of 36 is 6, but it can also be written as �36 because �36 ⋅ �36 = 36. • To express the negative square root of a number like 36, write − 6 or − �36.

• When a number is not a perfect square, such 40, express its square roots by writing �40 and −�40 . This can also be expressed as ±�40 .

ip t

How are the solutions to an equation like (𝑥 + 4)2 = 11 written? This equation is saying, “a value squared is 11.” To make the equation true, that value must be either �11 or − �11 . This can be expressed as shown. or

𝑥 + 4 = − �11

𝑥 + 4 = �11

𝑥 = − 4 +�11

𝑥 = − 4 − �11

A more compact way to write the 2 solutions to the equation is 𝑥 = − 4 ± �11 .

A calculator can be used to compute the approximate values of both expressions. − 4 + �11 ≈ − 0.683

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or − 4 − �11 ≈ − 7.317

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op y

The solutions can also be approximated by graphing. The equation (𝑥 + 4)2 = 11 is equivalent to (𝑥 + 4)2 − 11 = 0, so graph the function 𝑦 = (𝑥 + 4)2 − 11, and find its zeros by locating the 𝑥-intercepts of the graph.

Practice Problems

1. Solve each equation and write the solutions using ± notation. b. 𝑥2 = 5

cr us

c. 4𝑥2 = 28

ip t

a. 𝑥2 = 144

e. 2𝑥2 = 22

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d. 𝑥2 = 25 4

f. 7𝑥2 = 16

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2. Match each expression to an equivalent expression.

Expression

Equivalent Expression

4±1

− 17 and 5

4 + �2 and 4 − �2

10 ± �4

8 and 12

op y

−6 ±11

3 and 5

4 ±�10

4 + �10 and 4 − �10

�16 ±�2

ip t

a. Is �4 a positive or negative number? Explain your reasoning.

b. Is �5 a positive or negative number? Explain your reasoning.

c. Explain the difference between �9 and the solutions to 𝑥2 = 9.

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4. Technology required. For each equation, find the exact solutions by completing the square and the approximate solutions by graphing. Then, verify that the solutions found using the two methods are close. b. 𝑥2 − 4𝑥 − 11 = 0

op y

a. 𝑥2 + 10𝑥 + 8 = 0

Review Problems

5. Jada is working on solving a quadratic equation, as shown here.

ip t

𝑝2 − 5𝑝 = 0

𝑝 − 5 = 0

𝑝 = 5

𝑝(𝑝 − 5) = 0

She thinks that her solution is correct because substituting 5 for 𝑝 in the original expression 𝑝2 − 5𝑝 gives 52 − 5(5), which is 25 − 25 or 0.

Explain the mistake that Jada made and show the correct solutions.

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6. Which expression in factored form is equivalent to 30𝑥2 + 31𝑥 + 5? A. (6𝑥 + 5)(5𝑥 + 1) B. (5𝑥 + 5)(6𝑥 + 1)

C. (10𝑥 + 5)(3𝑥 + 1)

op y

D. (30𝑥 + 5)(𝑥 + 1)

7. Two rocks are launched straight up in the air. The height of Rock A is given by the function f, where 𝑓(𝑡) = 4 + 30𝑡 − 16𝑡2. The height of Rock B is given by 𝑔, where 𝑔(𝑡) = 5 + 20𝑡 − 16𝑡2. In both functions, 𝑡 is time measured in seconds (sec.) after the rocks are launched and height is measured in feet (ft.) above the ground.

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a. Which rock is launched from a higher point?

b. Which rock is launched with a greater velocity?

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Unit 9, Lesson 14: The Quadratic Formula

Warm-Up: Evaluate It Each expression represents two numbers. Evaluate the expressions and find the two numbers.

−18±�36 2∙3

3. ± �± (−5)2 − 4 ∙ 4 ∙1

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8±2 5

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1. 1±�49

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Exploration Activity: Easing into the Quadratic Formula

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1. Rewrite 𝑥 = − 5±�81 as 2 separate values.

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2. Determine the 2 different values of 𝑥 = − 𝑏 ± �𝑏2 − 4𝑎𝑐 given 𝑎 = −2, 𝑏 = −15, and 𝑐 = − 6. Leave in square root form.

3. Determine the 2 different values of 𝑥 = − 𝑏 ± �𝑏2 − 4𝑎𝑐 given 𝑎 = 1, 𝑏 = 4, and 𝑐 = 8. Leave in square root form.

4. Discuss with your partner whether the solutions found in question 3 are real number solutions. Summarize your discussion.

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Guided Activity: Meet the Quadratic Formula Here is a formula called the quadratic formula. 𝑥 =

−𝑏 ± �𝑏2 − 4𝑎𝑐 2𝑎

op y

The formula can be used to find the solutions to any quadratic equation in the form of 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏, and 𝑐 are numbers and 𝑎 is not 0.

𝑥 =

−(−8) �± (−8)2 − 4(1)(15) 2(1) 8±�64 − 60 2 8±�4 2

ip t

𝑥 =

Quadratic formula

Substitute the values of 𝑎, 𝑏, and 𝑐. Evaluate each part of the expression.

𝑥 =

−𝑏 ± �𝑏2 − 4𝑎𝑐 2𝑎

𝑥 =

This example shows how it is used to solve 𝑥2 − 8𝑥 + 15 = 0, in which 𝑎 = 1, 𝑏 = − 8, and 𝑐 = 15.

8±2

𝑥 = 2 10

𝑥 = 2

or 𝑥 = 2

𝑥 = 5 or 𝑥 = 3

Here are some quadratic equations and their solutions. Use the quadratic formula to show that the solutions are correct. 1. 𝑥2 + 4𝑥 − 5 = 0. The solutions are 𝑥 = −5 and 𝑥 = 1. © Accelerate Learning Inc. - All Rights Reserved

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2. 𝑥2 + 7𝑥 + 12 = 0. The solutions are 𝑥 = −3 and 𝑥 = −4.

�28

�20 . 2

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4. 𝑥2 − 8𝑥 + 11 = 0. The solutions are 𝑥 = 4 ±

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3. 𝑥2 + 10𝑥 + 18 = 0. The solutions are 𝑥 = −5 ± 2 .

5. 9𝑥2 − 6𝑥 + 1 = 0. The solution is 𝑥 = 1 .

6. 6𝑥2 + 9𝑥 − 15 = 0. The solutions are 𝑥 = − 5 and 𝑥 = 1.

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Lesson Summary Previous lessons showed how to solve quadratic equations by graphing, factoring, and completing the square.

op y

Some equations can be solved quickly and exactly with 1 of these methods, but many cannot. For example, consider the equation 5𝑥2 − 3𝑥 − 1 = 0. The expression on the left cannot be rewritten in factored form with rational coefficients. Because the coefficient of the squared term is not a perfect square and the coefficient of the linear term is an odd number, completing the square would be inconvenient and would result in a perfect square with fractions.

The quadratic formula can be used to find the solutions to any quadratic equation, including those that are tricky to solve with other methods or those that have nonreal solutions.

−𝑏 ± �𝑏2 − 4𝑎𝑐

ip t

The quadratic formula is 𝑥 = , which gives the 2𝑎 solutions of the quadratic equation, 𝑎𝑥2 − 𝑏𝑥 + 𝑐 = 0, where 𝑎 is not 0.

Nonreal solutions occur when there is a negative number under the radical because a negative number does not have a real number square root. Nonreal solutions will be explored in future courses.

𝑥

2𝑎

= −(−3) �± (−3)2 −4(5)(−1) 2(5)

= 3±�9 − 60 10

Quadratic formula

Substitute the values of 𝑎, 𝑏, and 𝑐.

Evaluate each part of the expression.

= 3±�29 10

𝑥

= −𝑏 ± �𝑏2 − 4𝑎𝑐

𝑥

For the equation 5𝑥2 − 3𝑥 − 1 = 0, 𝑎 = 5, 𝑏 = − 3, and 𝑐 = − 1.

The quadratic formula can be used to solve simpler equations like 𝑥2 − 9𝑥 + 8 = 0, but it may not be the most efficient way. If the quadratic expression can be easily rewritten in factored form or made into a perfect square, these methods may be preferable. For example, rewriting 𝑥2 − 9𝑥 + 8 = 0 as (𝑥 − 1)(𝑥 − 8) = 0 immediately reveals that the solutions are 1 and 8. © Accelerate Learning Inc. - All Rights Reserved

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Practice Problems 1. For each equation, identify the values of 𝑎, 𝑏, and 𝑐 that you would substitute into the quadratic formula to solve the equation. a. 3𝑥2 + 8𝑥 + 4 = 0

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b. 2𝑥2 − 5𝑥 + 2 = 0

c. −9𝑥2 + 13𝑥 − 1 = 0

e. − 𝑥2 + 16𝑥 + 64 = 0

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d. 𝑥2 + 𝑥 − 11 = 0

2. Use the quadratic formula to show that the given solutions are correct.

a. 𝑥2 + 9𝑥 + 20 = 0. The solutions are 𝑥 = −4 and 𝑥 = −5.

b. 𝑥2 − 10𝑥 + 21 = 0. The solutions are 𝑥 = 3 and 𝑥 = 7. 5

�13

c. 3𝑥2 − 5𝑥 + 1 = 0. The solutions are 𝑥 = 6 ± 6 .

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3. Identify the values of 𝑎, 𝑏, and 𝑐 that you would substitute into the quadratic formula to solve the equation. a. 𝑥2 + 9𝑥 + 18 = 0

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b. 4𝑥2 − 3𝑥 + 11 = 0 c. 81 − 𝑥 + 5𝑥2 = 0

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d. 4 𝑥2 + 3𝑥 = 1

f. 7𝑥 + 14𝑥2 = 42

Review Problems

e. 121 = 𝑥2

4. Select all the equations that are equivalent to 81𝑥2 + 180𝑥 − 200 = 100.

□ 81𝑥 + 180𝑥 − 100 = 0

□ 81𝑥 + 180𝑥 + 100 = 200

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□ 81𝑥 + 180𝑥 + 100 = 400 2

□ (9𝑥 + 10) = 400 2

□ (9𝑥 + 10) = 0 2

□ (9𝑥 − 10) = 10 2

□ (9𝑥 − 10) = 20

op y

5. Technology required. Two objects are launched upward. Each function gives the distance from the ground, in meters, as a function of time, 𝑡, in seconds (sec.). Object B: 𝑔(𝑡) = 30 + 10𝑡 − 5𝑡2

Object A: 𝑓(𝑡) = 25 + 20𝑡 − 5𝑡2

Use graphing technology to graph each function.

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a. Which object reaches the ground first? Explain how you know.

b. What is the maximum height of each object?

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Unit 9, Lesson 15: Applying Different Methods of Solving a Quadratic

Warm-Up: No Solutions for You!

(𝑥 + 3)2 + 9 = 0 (𝑥 + 3)2 = −9

── 𝑥 + 3 = ± �−9

op y

Here is an example of someone solving a quadratic equation that has no solutions:

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1. Study the example. At what point did you realize the equation had no real solutions?

2. Explain how you know the equation 49 + 𝑥2 = 0 has no real solutions.

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Collaborative Activity: Applying Different Methods of Solving a Quadratic

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2. Solve 9𝑥2 + 36𝑥 + 36 = 0 by factoring.

op y

1. Use the quadratic formula to solve 9𝑥2 + 36𝑥 + 36 = 0.

3. Solve 9(𝑥 + 2)2 = 0.

4. Check your work with your partner, and resolve any differences if necessary.

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5. The graph of 𝑦 = 9(𝑥 + 2)2 is shown.

Work with your partner to write an explanation that connects the graph of 𝑦 = 9(𝑥 + 2)2 to the solutions for 9𝑥2 + 36𝑥 + 36 = 0 and 9(𝑥 + 2)2 = 0.

16 14 12 10 8

op y

6 4 2

-2

(–2, 0)

-1

0 -2

-3

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6. Work with your partner to complete the statements using the word bank provided. different factored factored form same

standard form vertex form 𝑥-intercept(s)

-intercepts zero product property

The 3 methods for solving quadratic equations in questions 1–3 used different forms of a quadratic. Each method results in the __________ solution(s). To solve by factoring, the quadratic is factored, and then the _____________________________ is applied to find the solutions. To solve by completing the square, the quadratic is rewritten to ____________________________. Then, the equation is solved for 𝑥. The quadratic formula is best used when the quadratic cannot be _________________ or if completing the square is difficult. The solution(s) are the ___________________ of the parabola. These are also known as the zeros of the graph, or roots of the quadratic equation.

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Guided Activity: Choosing a Method 1. The equation 𝑦 = −16𝑥2 + 128𝑥 + 1 models the height, in feet (ft.), of a toy rocket 𝑥 seconds (sec.) after it is launched from 1 ft. above the ground. a. What is the height of the rocket when it is on the ground?

−319

113

241

113

−319

ip t

Height (ft.)

−2

Time (sec.)

op y

b. Use the table to determine when the rocket reaches its maximum height.

c. The equation −16𝑥2 + 128𝑥 + 1 = 221 can be used to determine the time that the rocket was at 221 ft. Rewrite the equation by filling in the blanks.

_________𝑥2 + _________𝑥 + __________ = 0

d. Consider the 3 methods of solving quadratics, and choose 1 method to solve the equation. Completing the square

Factoring

Quadratic formula

Explain your choice.

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e. Ask your partner which method they selected. Record their choice, and write your summary of their reason. You are the only person who should write in the first 2 columns of the table. Have the person initial next to your summary, indicating that your summary is correct. Method

My Summary of Their Reason

Initials

Completing the square

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Factoring

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f. Solve −16𝑥2 + 128𝑥 + 1 = 221.

Quadratic formula

g. Complete the statement about the solution to −16𝑥2 + 128𝑥 + 1 = 221.

At _____ sec. and _____ sec., the toy rocket is at a height of 221 ft.

2. The function 𝑀(𝑥) = 5𝑥2 − 160𝑥 + 6,179 models the number of miles, in millions, that US passengers traveled on a train for the period 1970 to 2000, where 𝑥 is the number of years since 1970.

a. Use the equation 5𝑥2 − 160𝑥 + 6,179 = 8,279 to determine in what year US passengers traveled 8,279 million miles.

b. Interpret the solutions, and determine if the solutions are viable.

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Lesson Summary Quadratic equations, especially ones that represent situations, cannot always be neatly put into factored form or easily solved by finding square roots. Completing the square is a workable strategy, but for some equations, it may involve many cumbersome steps. Graphing is also a handy way to solve the equations, but it is not always precise. The quadratic formula can be used to solve any quadratic equation. It is not always the most convenient, but it does solve quadratic equations precisely.

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For example, function ℎ models the height of an object, in meters (m), 𝑡 sec. after it is launched into the air. It is defined by ℎ(𝑡) = −5𝑡2 + 25𝑡.

To know how much time it would take the object to reach 15 m, solve the equation 15 = −5𝑡2 + 25𝑡. Determining which method to use of those available is a personal choice.

• Rewriting it in standard form gives −5𝑡2 + 25𝑡 − 15 = 0. The expression on the left side of the equation cannot be written in factored form, however.

ip t

• Completing the square is not convenient either because the coefficient of the squared term is not a perfect square and the coefficient the linear term is an odd number. Use the quadratic formula, using 𝑎 = −5, 𝑏 = 25, and 𝑐 = −15. −10

The expression −25±�325 represents the 2 exact solutions of the equation. A calculator can give approximate

solutions, or by reasoning that �325 ≈ 18.

The solutions reveal that there are 2 times after the launch when the object is at a height of 15 m, at about 0.7 sec. as the object is going up and 4.3 sec. as it comes back down.

𝑡 =

−4𝑎𝑐 𝑡 = �𝑏 2𝑎 2

−25±�252−4 (−5) (−15) 2(−5)

𝑡 = −25±�325 −10

The quadratic formula has many parts in it. A small error in any part can lead to incorrect solutions. Use care when working with the quadratic formula to solve problems involving quadratic functions.

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Be aware of these common errors: • Using the wrong values for 𝑎, 𝑏, and 𝑐 in the formula

• Forgetting to multiply 2 by 𝑎 for the denominator in the formula

• Forgetting that squaring a negative number produces a positive number • Forgetting that a negative number times a positive number is a negative number

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• Making calculation errors or not following the properties of algebra

To make sure solutions are correct, substitute the solutions back into the original equation and see whether each solution keeps the equation true. The equation can also be graphed to find its 𝑥-intercepts to see whether solutions are accurate (or close to accurate).

ip t

Practice Problems

1. The function 𝑓(𝑥) = −3𝑥2 + 54𝑥 + 83 models the sales of music cassette tapes for the years 1980 to 2000, in tens of millions of dollars, where 𝑥 is the number of years since 1980.

a. Use the equation −3𝑥2 + 54𝑥 + 83 = 0 to determine when the sales of music cassette tapes are $0.

b. Interpret the solutions, and determine if the solutions are viable.

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2. A tennis ball is hit straight up in the air, and its height, in ft. above the ground, is modeled by the equation 𝑓(𝑡) = 4 + 12𝑡 − 16𝑡2, where 𝑡 is measured in sec. since the ball was thrown. a. Find the solutions to the equation 0 = 4 + 12𝑡 − 16𝑡2.

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b. What do the solutions tell us about the tennis ball?

Review Problem

3. Consider the equation 4𝑥2 − 4𝑥 − 15 = 0.

ip t

a. Identify the values of 𝑎, 𝑏, and 𝑐 that you would substitute into the quadratic formula to solve the equation.

b. Evaluate each expression using the values of 𝑎, 𝑏, and 𝑐. 4𝑎𝑐

𝑏2

𝑏2 − 4𝑎𝑐

−𝑏

− 𝑏 ± �𝑏2−4𝑎𝑐

�𝑏2−4𝑎𝑐

2𝑎

𝑡 =

�𝑏2−4𝑎𝑐 2𝑎

c. The solutions to the equation are 𝑥 = −32 and 𝑥 = 52. Do these match the values of the last expression you evaluated in the previous question?

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Unit 9, Lesson 16: Using Quadratic Expressions in Vertex Form to Solve Problems

Warm-Up: Values of a Function Here are graphs that represent two functions, 𝑓 and 𝑔, defined by:

op y

𝑓(𝑥) = (𝑥 − 4)2 + 1

ip t

𝑔(𝑥) = − (𝑥 − 12)2 + 7

1. 𝑓(1) can be expressed in words as “the value of 𝑓 when 𝑥 is 1.” Find or compute:

b. 𝑓(3)

a. the value of 𝑓 when 𝑥 is 1

c. 𝑓(10)

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2. Can you find an 𝑥 value that would make 𝑓(𝑥): a. Less than 1?

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b. Greater than 10,000? 3. 𝑔(9) can be expressed in words as “the value of 𝑔 when 𝑥 is 9.” Find or compute:

a. value of 𝑔 when 𝑥 is 9

ip t

b. 𝑔(13)

c. 𝑔(2)

4. Can you find an 𝑥 value that would make 𝑔(𝑥):

a. Greater than 7?

b. Less than −10,000?

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Exploration Activity: Maximums and Minimums 1. The graph that represents 𝑝(𝑥) = (𝑥 − 8)2 + 1 has its vertex at (8, 1). Here is one way to show, without graphing, that (8, 1) corresponds to the minimum value of . • When 𝑥 = 8, the value of (𝑥 − 8)2 is 0 because (8 − 8)2 = 02 = 0.

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• Squaring any number always results in a positive number, so when 𝑥 is any value other than 8, (𝑥 − 8) will be a number other than 0, and when squared, (𝑥 − 8)2 will be positive.

• Any positive number is greater than 0, so when 𝑥 ≠ 8, the value of (𝑥 − 8)2 will be greater than when 𝑥 = 8. In other words, 𝑝 has the least value when 𝑥 = 8.

ip t

Use similar reasoning to explain why the point (4, 1) corresponds to the maximum value of 𝑞, defined by 𝑞(𝑥) = −2(𝑥 − 4)2 + 1.

2. Here are some quadratic functions and the coordinates of the vertex of the graph of each. Determine if the vertex corresponds to the maximum or the minimum value of the function. Be prepared to explain how you know. Coordinates of the Vertex

𝑓(𝑥) = −(𝑥 − 4)2 + 6

(4, 6)

Equation

𝑔(𝑥) = (𝑥 + 7)2 − 3

(−7, −3)

𝑘(𝑥) = 𝑥2 − 6𝑥 − 3

(3, −12)

ℎ(𝑥) = 4(𝑥 + 5)2 + 7 𝑚(𝑥) = −𝑥2 + 8𝑥

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Maximum or Minimum?

(− 5, 7) (4, 16)

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Guided Activity: All the World’s a Stage A function, 𝐴, defined by 𝑝(600 − 75𝑝), describes the revenue collected from the sales of tickets for performance A, a musical.

op y

The graph represents a function 𝐵 that models the revenue collected from the sales of tickets for performance B, a Shakespearean comedy.

In both functions, 𝑝 represents the price of one ticket, and both revenues and prices are measured in dollars.

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Without creating a graph of 𝐴, determine which performance gives the greater maximum revenue when tickets are 𝑝 dollars each. Explain or show your reasoning.

Lesson Summary

Any quadratic function has either a maximum or a minimum value, which can be determined by observing the vertex of its graph. Shown are graphs representing functions 𝑓 and 𝑔, defined by 𝑓(𝑥) = −(𝑥 + 5)2 + 4 and 𝑔(𝑥) = 𝑥2 + 6𝑥 − 1, respectively. 488 | Unit 9

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• The vertex of the graph of 𝑔 is ( −3, −10), and the graph is a parabola that opens upward.

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• The vertex of the graph of 𝑓 is (−5, 4), and the graph is a parabola that opens downward.

• No other points on the graph are higher • No other points on the graph are lower than (−5, 4), so 𝑓 has a maximum of 4, than (−3, −10), so 𝑔 has a minimum of and that occurs when 𝑥 = −5. − 10, and that occurs when 𝑥 = −3.

ip t

Since a quadratic expression in vertex form, written as 𝑎(𝑥 − ℎ)2 + 𝑘, can reveal the vertex (ℎ, 𝑘) of the graph, the function does not have to be graphed to determine the vertex. The vertex form also reveals whether the vertex corresponds to a maximum or a minimum value of the function.

To determine if (−3, −10) is a minimum or maximum of 𝑔, rewrite 𝑥2 + 6𝑥 − 1 in vertex form, which is (𝑥 + 3)2 − 10.

• When 𝑥 = − 3, (𝑥 + 3) is 0, so (𝑥 + 3)2 is also 0.

• When 𝑥 ≠ −3, the expression (𝑥 + 3) will be a nonzero number, and (𝑥 + 3)2 will be positive. Because a squared number cannot have a value less than 0, (𝑥 + 3)2 has the least value when 𝑥 = −3. To determine if (−5, 4) is a minimum or maximum of 𝑓, use the squared term in −(𝑥 + 5)2 + 4.

• When 𝑥 = −5, (𝑥 + 5) is 0, so (𝑥 + 5)2 is also 0.

• When 𝑥 is not −5, the expression (𝑥 + 5) will be nonzero, so (𝑥 + 5)2 will be positive. However, the expression −(𝑥 + 5)2 has a negative coefficient of −1. Multiplying (𝑥 + 5)2 by a negative number results in a negative number.

• Because a negative number is always less than 0, the value of −(𝑥 + 5)2 + 4 will always be less when 𝑥 ≠ −5 than when 𝑥 = −5. This means 𝑥 = −5 gives the greatest value of 𝑓.

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In addition to this reasoning, the value of 𝑎 can also indicate whether the vertex of a quadratic function is a maximum or a minimum value. Recall that when 𝑎 > 0, the graph of a quadratic function opens upward. Therefore, the vertex would reveal the minimum value of the function. When 𝑎 < 0, the graph opens downward, indicating the vertex reveals the maximum value of the function.

Practice Problems

ip t

op y

1. Here is a graph of a quadratic function 𝑓(𝑥). What is the minimum value of 𝑓(𝑥)?

2. The graph that represents 𝑓(𝑥) = (𝑥 + 1)2 − 4 has its vertex at (−1, −4).

Explain how we can tell from the expression (𝑥 + 1)2 − 4 that −4 is the minimum value of 𝑓 rather than the maximum value.

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3. Each expression here defines a quadratic function. Find the vertex of the graph of the function. Then, state whether the vertex corresponds to the maximum or the minimum value of the function. a. (𝑥 − 5)2 + 6

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b. (𝑥 + 5)2 − 1

c. −2(𝑥 + 3)2 − 10

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d. 3(𝑥 − 7)2 + 11

f. (𝑥 + 1)2

e. −(𝑥 − 2)2 − 2

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Review Problems 4. Match each equation to the number of solutions it has. Number of Solutions

Equation

No solutions

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(𝑥 − 1)(𝑥 − 5) = 5

1 solution

𝑥2 − 2𝑥 = −1

2 solutions

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5. Which equation has irrational solutions?

C. 4𝑥2 − 1 = 0

D. 9(𝑥 + 3)2 = 27

A. 100𝑥2 = 9

B. 9(𝑥 − 1)2 = 4

(𝑥 − 5)2 = −25

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Unit 9, Lesson 17: Using Quadratic Equations to Model Situations and Solve Problems

Warm-Up: Equations of Two Lines and a Curve 1. Write an equation representing the line that passes through each pair of points.

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a. (3, 3) and (5, 5)

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b. (0, 4) and (−4, 0)

2. Solve this equation: 𝑥 + 1 = (𝑥 − 2)2 − 3. Show your reasoning.

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Collaborative Activity: The Dive The function, ℎ, defined by ℎ(𝑡) = −5𝑡2 + 10𝑡 + 7.5, models the height of a diver above the water, in meters (m), 𝑡 seconds (sec.) after the diver leaves the board. For each question, explain how you know.

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1. How high above the water is the diving board?

2. When does the diver hit the water?

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3. At what point during her descent toward the water is the diver at the same height as the diving board?

4. When does the diver reach the maximum height of the dive?

5. What is the maximum height the diver reaches during the dive?

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Guided Activity: A Linear Function and a Quadratic Function Here are graphs of a linear function and a quadratic function. The quadratic function is defined by the expression (𝑥 − 4)2 − 5.

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Find the coordinates of 𝑃, 𝑄, and 𝑅 without using graphing technology. Show your reasoning.

Lesson Summary

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Certain real-world situations can be modeled by quadratic functions, and these functions can be represented by equations. Sometimes, several skills are needed to make sense of these situations. A mathematical model, combined with having the skills to use the model to answer questions, offers useful or interesting insights about the situation.

A model for the height of a launched object, ℎ, as a function of time since it was launched, 𝑡, is defined by ℎ(𝑡) = −4.9𝑡2 + 28𝑡 + 2.1. The questions shown can be answered about the object’s flight.

• From what height was the object launched? An expression in standard form can help with this question. Or, evaluate ℎ(0) to find the answer.

• At what time did it hit the ground? When an object hits the ground, its height is 0, so find the zeros using 1 of the methods learned recently: graphing, rewriting in factored form, completing the square, or using the quadratic formula.

• What was its maximum height, and at what time did it reach the maximum height? Rewrite the expression in vertex form, or use the zeros of the function or a graph. © Accelerate Learning Inc. - All Rights Reserved

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Sometimes, relationships between quantities can be effectively communicated with graphs and expressions rather than with words. For example, these graphs represent a linear function, 𝑓, and a quadratic function, 𝑔, with the same variables for their input and output.

• Will the 2 functions ever have the same value? Yes. The graphs reveal they intersect in 2 places.

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If the expressions that define these functions are known, knowledge of quadratic equations can be used to answer similar questions as shown.

• If so, at what input values does that happen? What are the output values they have in common?

Practice Problems

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To find out, write and solve the equation 𝑓(𝑥) = 𝑔(𝑥).

1. The function ℎ represents the height of an object 𝑡 sec. after it is launched into the air. The function is defined by ℎ(𝑡) = −5𝑡 + 20𝑡 + 18. Height is measured in m.

Answer each question without graphing. Explain or show your reasoning.

a. After how many sec. does the object reach a height of 33 m?

b. When does the object reach its maximum height?

c. What is the maximum height the object reaches?

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2. The graphs that represent a linear function and a quadratic function are shown here. The quadratic function is defined by 2𝑥2 − 5𝑥.

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Find the coordinates of 𝑅 without using graphing technology. Show your reasoning.

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3. Diego finds his neighbor’s baseball in his yard, about 10 feet (ft.) away from a fivefoot fence. He wants to return the ball to his neighbors, so he tosses the baseball in the direction of the fence.

Function ℎ, defined by ℎ(𝑥) = −0.078𝑥2 + 0.7𝑥 + 5.5, gives the height of the ball as a function of the horizontal distance away from Diego.

Does the ball clear the fence? Explain or show your reasoning.

Review Problems

4. Here are two quadratic functions: 𝑓(𝑥) = (𝑥 + 5)2 + 12 and 𝑔(𝑥) = (𝑥 + 5)2 + 1.

Andre says that both 𝑓 and 𝑔 have a minimum value, and that the minimum value of 𝑓 is less than that of 𝑔. Do you agree? Explain your reasoning.

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5. The following expressions all define the same quadratic function. 𝑥2 + 2𝑥 − 8 (𝑥 + 4)(𝑥 − 2) (𝑥 + 1)2 − 9

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a. What is the 𝑦-intercept of the graph of the function? b. What are the 𝑥-intercepts of the graph?

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d. Sketch a graph of the quadratic function without using technology. Make sure the 𝑥-intercepts, 𝑦-intercept, and vertex are plotted accurately.

c. What is the vertex of the graph?

6. Function 𝑝 is defined by the equation 𝑝(𝑥) = (𝑥 + 10)2 − 3. Function 𝑞 is represented by this graph.

Which function has the smaller minimum? Explain your reasoning.

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Unit 10: Sequences

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Unit 10, Lesson 1: A Towering Sequence

Warm-Up: What’s Next? Here is a rule for making a list of numbers: Each number is 1 less than twice the previous number.

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Pick a number to start with, and then follow the rule to build a list of 5 numbers.

Exploration Activity: Towers of Hanoi

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This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

In the Tower of Hanoi puzzle, a set of discs sits on a peg, while there are 2 other empty pegs.

A move in the Tower of Hanoi puzzle involves taking a disc and moving it to another peg. To move a disc, click on it, and then click on the peg where you want it to go. You cannot drag a disc. There are two rules: • Only move 1 disc at a time.

• Never put a larger disc on top of a smaller one.

You complete the puzzle by building the complete tower on a different peg than the starting peg.

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1. Using 3 discs, complete the puzzle. What is the smallest number of moves you can find?

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2. Using 4 discs, complete the puzzle. What is the smallest number of moves you can find?

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3. Jada says she used the solution for 3 discs to help her solve the puzzle for 4 discs. Describe how this might happen.

4. How many moves do you think it will take to complete a puzzle with 5 discs?

5. How many moves do you think it will take to complete a puzzle with 7 discs?

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Exploration Activity: Checker Jumping Puzzle

This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

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Some checkers are lined up, with blue on one side, red on the other, and with one empty space between them. A move in this checker game pushes any checker forward 1 space, or jumps over any 1 checker of the other color. Jumping the same color is not allowed, moving backward is not allowed, and 2 checkers cannot occupy the same space.

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Drag the checkers to move them.

You complete the puzzle by switching the colors completely: ending up with blue on the right, red on the left, and with 1 empty space between them.

1. Using 1 checker on each side, complete the puzzle. What is the smallest number of moves needed?

2. Using 3 checkers on each side, complete the puzzle. What is the smallest number of moves needed?

3. Make guesses about the number of moves for 2 and 4 checkers on each side, and then test your guesses.

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4. Noah says he used the solution for 3 checkers on each side to help him solve the puzzle for 4 checkers. Describe how this might happen.

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5. How many moves do you think it will take to complete a puzzle with 7 checkers on each side?

Lesson Summary

The study of mathematics often uses lists of numbers presented in a specific order with a predictable pattern called a sequence.

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A sequence is a list of numbers, possibly going on forever, such as all the odd positive integers arranged in order: 1, 3, 5, 7, . . .

There are many ways to define a sequence, but one way is to describe how each term relates to the term before it. For example, the sequence 3, 5, 7, 9, 11, . . . can be described this way: the starting term is 3, and then each following term is 2 more than the term before it. The sequence 1, 5, 13, 29, 61, . . . can be described this way: the starting term is 1, and then each following term is the sum of 3 and twice the previous term.

Throughout this unit, several types of sequences will be explored, as will various ways to represent them.

Practice Problems

1. Here is a rule to make a list of numbers: Each number is the sum of the previous two numbers. Start with the numbers 0 and 1, and then follow the rule to build a sequence of 10 numbers.

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2. A sequence starts 1 , 1 , 1 , . . . 2 4 8

a. Give a rule that the sequence could follow.

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b. Follow your rule to write the next 3 terms in the sequence.

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a. Give the next 3 terms in the sequence.

3. A sequence of numbers follows the rule: multiply the previous number by − 2 and add 3. The fourth term in the sequence is − 7.

b. Give the 3 terms that came before − 7 in the sequence.

4. A sequence starts 0, 5, . . .

a. Give a rule the sequence could follow and the next 3 terms for that rule.

b. Give a different rule the sequence could follow and the next 3 terms for that rule.

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Review Problems 5. Which equation is equivalent to 𝑥2 − 3𝑥 = 7 but has a perfect square on one side? 4

A. 𝑥2 − 3𝑥 + 3 = 19 4

B. 𝑥2 − 3𝑥 + 3 = 10 4

C. 𝑥2 − 3𝑥 + 9 = 16 4

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D. 𝑥2 − 3𝑥 + 9 = 7 4

6. Without using graphing technology, sketch a graph that represents each quadratic function. Make sure the 𝑥-intercepts, 𝑦-intercept, and vertex are plotted accurately. 𝑔(𝑥) = 𝑥2 − 4𝑥 + 3

ℎ(𝑥) = 𝑥2 − 11𝑥 + 28

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𝑓(𝑥) = 𝑥2 + 4𝑥 + 3

7. Function 𝑓 is defined by the expression

a. Evaluate 𝑓(12).

5 . 𝑥 − 2

b. Explain why 𝑓(2) is undefined. c. Give a possible domain for 𝑓.

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Unit 10, Lesson 2: Introducing Geometric Sequences

Warm-Up: Notice and Wonder: A Pattern in Lists

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What do you notice? What do you wonder?

• 40, 120, 360, 1080, 3240

• 2, 8, 32, 128, 512

• 256, 192, 144, 108, 81

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• 1000, 500, 250, 125, 62.5

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Exploration Activity: Paper Slicing Clare takes a piece of paper, cuts it in half, and then stacks the pieces. She takes the stack of two pieces, and then cuts in half again to form four pieces, stacking them. She keeps repeating the process.

Number of Cuts

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1. The original piece of paper has a length of 8 inches (in.) and a width of 10 in. Complete the table. Number of Pieces

Area (sq. in.) of Each Piece

1 2

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3 5

2. Describe in words how you can use the results after 5 cuts to find the results after 6 cuts.

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3. On the given axes, sketch a graph of the number of pieces as a function of the number of cuts. How can you see on the graph how the number of pieces is changing with each cut?

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4. On the given axes, sketch a graph of the area of each piece as a function of the number of cuts. How can you see how the area of each piece is changing with each cut?

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Collaborative Activity: Complete the Sequence 1. Complete each geometric sequence. a. 1.5, 3, 6, ___, 24, ___

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b. 40, 120, 360, ___, ___

c. 200, 20, 2, ___, 0.02, ___ d. 1 , ___, 9 , 27 , ___ 7

e. 24, 12, 6, ___, ___

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2. For each sequence, find its growth factor.

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Lesson Summary Consider the sequence 2, 6, 18, . . . How would you describe how to calculate the next term from the previous?

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In this case, each term in this sequence is 3 times the term before it.

A way to describe this sequence would be, “The starting term is 2, and the current term = 3 ⋅ previous term.” This is an example of a geometric sequence.

A geometric sequence is a sequence of numbers in which each consecutive pair of numbers has a common ratio.

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In a geometric sequence, the value of each term is the value of the previous term multiplied by a constant. If the constant and starting value are known, then the value of all other terms in the geometric sequence can be determined.

This constant multiplier (the 3 in the example) is often called the sequence’s growth factor or common ratio. To find it, divide 2 consecutive terms. This can also help determine whether a sequence is geometric. The term growth factor may be familiar because it represents a similar relationship to the exponential functions studied in a previous unit in this course.

Not all sequences are geometric. Upcoming lessons will explore sequences that follow a pattern but not necessarily one with a growth factor. For example, the sequence 1, 3, 5, 7, 9, . . . is not a geometric sequence because the ratios between consecutive terms are not equivalent: 3 ≠ 5 ≠ 7 . The sequence 100, 20, 4, 1

0.8, . . ., however, is geometric because when each term is divided by the previous term, the common ratio is 0.2 each time: 20 = 4 = 0.8 = 0.2. 100

Practice Problems

1. Here are the first two terms of a geometric sequence: 2, 4. What are the next 3 terms?

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2. What is the growth factor of each geometric sequence? a. 1, 1, 1, 1, 1

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b. 256, 128, 64 c. 18, 54, 162

d. 0.8, 0.08, 0.008

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e. 0.008, 0.08, 0.8

3. A person owes $1,000 on a credit card that charges an interest rate of 2% per month.

Complete this table showing the credit card balance each month if they do not make any payments.

Month

Total bill in dollars

1,000

1,040.40

2 4

1,020

5 6 7 8

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Review Problems 4. Here is a rule to make a list of numbers: Each number is 4 less than 3 times the previous number.

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a. Starting with the number 10, build a sequence of 5 numbers.

b. Starting with the number 1, build a sequence of 5 numbers.

5. A sequence starts 1, − 1, . . .

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c. Select a different starting number and build a sequence of 5 numbers.

a. Give a rule the sequence could follow and the next 3 terms.

b. Give a different rule the sequence could follow and the next 3 terms.

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Unit 10, Lesson 3: Different Types of Sequences

Warm-Up: Remembering Function Notation Consider the function 𝑓 given by 𝑓(𝑛) = 3𝑛 − 7. This function takes an input, multiplies it by 3, and then subtracts 7.

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Evaluate mentally. • 𝑓(10)

• 𝑓(10) − 1

• 𝑓(5) − 𝑓(4)

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• 𝑓(10 − 1)

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Exploration Activity: Three Sequences Here are the values of the first 5 terms of 3 sequences: •

•

𝐴: 30, 40, 50, 60, 70, . . .

𝐵: 0, 5, 15, 30, 50, . . .

𝐶: 1, 2, 4, 8, 16, . . .

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•

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1. For each sequence, describe a way to produce a new term from the previous term.

2. If the patterns you described continue, which sequence has the second greatest value for the 10th term?

3. Which of these could be geometric sequences? Explain how you know.

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Guided Activity: Different Types of Sequences Complete the table by determining whether each sequence is arithmetic, geometric, or neither. • For each arithmetic sequence, identify the common difference, 𝑑.

Sequence

Type

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Arithmetic sequence Geometric sequence Neither

56, 49, 42, 35, . . .

Arithmetic sequence Geometric sequence Neither

− 99, 11, − 11 , 11 , . . . 81

Arithmetic sequence Geometric sequence Neither Arithmetic sequence Geometric sequence Neither

576, 288, 72, 9, . . . 9

Common Difference or Growth Factor

7, 21, 63, 189, 567, . . .

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• For each geometric sequence, identify the growth factor, 𝑟.

− 35, − 21, − 7, 7, . . .

1 , 3 , 9 , 27 , . . . 5 5 5 5

23 , 20 , 17 , 14 , . . . 11 11 11 11

Arithmetic sequence Geometric sequence Neither Arithmetic sequence Geometric sequence Neither Arithmetic sequence Geometric sequence Neither

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Collaborative Activity: Representing a Sequence Jada and Mai are trying to decide what type of sequence this could be. Value

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Term Number

• Jada says: “I think this sequence is geometric because in the value column each row is 3 times the previous row.”

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• Mai says: “I don’t think it is geometric. I graphed it and it doesn’t look geometric.”

Do you agree with Jada or Mai? Explain or show your reasoning.

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Lesson Summary Consider the sequence 2, 5, 8, . . . How would you describe how to calculate the next term from the previous?

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In this case, each term in this sequence is 3 more than the term before it.

A way to describe this sequence would be, “The starting term is 2, and the current term = previous term + 3.” This is an example of an arithmetic sequence.

An arithmetic sequence is a sequence of numbers in which each consecutive pair of numbers has a common difference, 𝑑.

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An arithmetic sequence is one where the value of each term is the value of the previous term with a constant added. If the constant and the starting value are known, any term in the arithmetic sequence can be determined.

For example, each term in this sequence is 3 more than the term before it. To find this constant, sometimes called the rate of change or common difference, subtract the consecutive terms. This can also help determine whether a sequence is arithmetic.

For example, the sequence 3, 6, 12, 24, . . . is not an arithmetic sequence because 6 − 3≠12 − 6≠24 − 12. But the sequence 100, 80, 60, 40, . . . is arithmetic because the differences of the consecutive terms are the same: 80 − 100 = 60 − 80 = 40 − 60 = − 20. This means that the common difference in this arithmetic sequence is − 20.

It is important to remember that although the last 2 lessons have introduced geometric and arithmetic sequences, there are many other sequences that are neither geometric nor arithmetic.

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Practice Problems 1. Here are the first two terms of some different arithmetic sequences. What are the next three terms of each sequence? a. −2, 4

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b. 11, 111

c. 5, 7.5

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d. 5, −4

b. 2, 4, 16, . . .

a. 200, 40, 8, . . .

2. For each sequence, decide whether it could be arithmetic, geometric, or neither.

c. 10, 20, 30, . . .

d. 100, 20, 4, . . . e. 6, 12, 18, . . .

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3. Complete each arithmetic sequence with its missing terms, and then state the rate of change for each sequence. a. −3, − 2, ___, ___, 1

b. ___, 13, 25, ___, ___

d. 92, ___, ___, ___, 80 4. A sequence starts with the terms 1 and 10.

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c. 1, .25, ___, −1.25, ___

a. Find the next two terms if it is arithmetic: 1, 10, ___, ___.

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b. Find the next two terms if it is geometric: 1, 10, ___, ___.

1, 10, ___, ___.

c. Find two possible next terms if it is neither arithmetic nor geometric:

Review Problems

5. Complete each geometric sequence with the missing terms. Then find the growth factor for each.

a. ___, 5, 25, ___, 625

b. −1, ___, −36, 216, ___

c. 10, 5, ___, ___, 0.625

d. ___, ___, 36, − 108, ___ e. ___, 12, 18, 27, ___

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6. The first term of a sequence is 4.

a. Choose a growth factor and list the next 3 terms of a geometric sequence.

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b. Choose a different growth factor and list the next 3 terms of a geometric sequence.

7. Here is a rule that can be used to build a sequence of numbers once a starting number is chosen: Each number is two times three less than the previous number.

a. Starting with the number 0, build a sequence of 5 numbers.

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b. Starting with the number 3, build a sequence of 5 numbers.

c. Can you choose a starting point so that the first 5 numbers in your sequence are all positive? Explain your reasoning.

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Unit 10, Lesson 4: Sequences Are Functions

Warm-Up: Bowling for Triangles - Part 1

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Describe how to produce one step of the pattern from the previous step.

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Exploration Activity: Bowling for Triangles - Part 2

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Here is a visual pattern of dots. The number of dots, 𝐷(𝑛), is a function of the step number 𝑛.

𝒏

𝑫(𝒏) 1

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2. Complete the table for Steps 1 to 5.

1. What values make sense for 𝑛 in this situation? What values don’t make sense for 𝑛?

𝐷(1) + 2 = 3 𝐷(2) + 3 = 6

3. Following the pattern in the table, write an equation for 𝐷(𝑛) in terms of the previous step. Be prepared to explain your reasoning.

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Guided Activity: Exploring Representations of Sequences 1. The graphs of 2 sequences are shown on the coordinate plane. a. Complete the table for each sequence. Term Sequence Sequence Number A B

1 2

−1

2 3

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4 5

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b. For sequence A, define the sequence recursively using function notation.

c. For sequence B, define the sequence recursively using function notation.

d. Which of these is a geometric sequence? Explain how you know.

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Collaborative Activity: Let’s Define Some Sequences Use the first 5 terms of each sequence to state if the sequence is arithmetic, geometric, or neither. Next, define the sequence recursively using function notation.

op y

1. 𝐴: 30, 40, 50, 60, 70, . . .

2. 𝐵: 80, 40, 20, 10, 5, 2.5, . . .

2 4 8

4. 𝐷: 1, 1 , 1 , 1 , 1 , . . .

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3. 𝐶: 1, 2, 4, 8, 16, 32, . . .

5. 𝐸: 20, 13, 6, −1, −8, . . .

6. 𝐹: 1, 3, 7, 15, 31, . . .

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Lesson Summary When working with sequences, it is often useful to determine an equation or a function that can be used to summarize the patterns of the sequence. The patterns of arithmetic and geometric sequences lend themselves to being modeled by functions. However, it is important to remember that sequences are discrete values with a domain of integers, whereas functions that can be used to model the patterns of sequences have a domain of all real numbers.

op y

Sequences can be defined recursively. That is, a sequence can be described using an equation, or a series of equations, that can be used to calculate the next term in a sequence if the previous term is known. Consider the sequence 6, 10, 14, 18, 22, . . . . This is an arithmetic sequence where each term is 4 more than the previous term.

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• Because sequences are functions, this sequence can be named 𝑓, and then it can be written using function notation as 𝑓(𝑛) = 𝑓(𝑛 − 1) + 4. Here, 𝑓(𝑛) is the term, 𝑓(𝑛 − 1) is the previous term, and + 4 is the rate of change, or common difference because 𝑓 is an arithmetic sequence.

• When a function is defined recursively, the first term, 𝑓(1), must also be identified. Without that, there would be no way of knowing if the sequence defined by 𝑓(𝑛) = 𝑓(𝑛 − 1) + 4 started with 6, 81, or any other number. Here, one possible initial condition is 𝑓(1) = 6. In some cases, it may make more sense to number the terms starting with 0, using 𝑓(0) = 6. This concept will be addressed more closely in upcoming lessons.

Combining this information for the sequence 6, 10, 14, 18, 22, . . ., the recursive definition is 𝑓(1) = 6, 𝑓(𝑛) = 𝑓(𝑛 − 1) + 4 for 𝑛 ≥ 2, where 𝑛 is an integer. Recursive definitions must always include the 𝑛 ≥ 2 since the value of 𝑓(1) is defined separately and the other terms in the sequence are generated by inputting integers larger than 1 into the definition.

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Practice Problems 1. A graph of a sequence is shown.

ip t

b. Define the sequence recursively using function notation.

op y

a. Explain how you know whether this sequence is arithmetic or geometric.

2. Match each sequence with one of the definitions. Note that only the part of the definition showing the relationship between the current term and the previous term is given so as not to give away the solutions. Definition

6, 12, 18, 24

𝑎(𝑛) = 7∙𝑎(𝑛 − 1)

Sequence

2, 14, 98, 686

160, 80, 40, 20

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𝑏(𝑛) = 1 ∙𝑏(𝑛 − 1) 2

𝑐(𝑛) = 𝑐(𝑛 − 1) + 6

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3. Write the first five terms of each sequence. Determine whether each sequence is arithmetic, geometric, or neither. a. 𝑎(1) = 7,𝑎(𝑛) = 𝑎(𝑛 − 1) − 3 for 𝑛 ≥ 2.

c. 𝑐(1) = 3,𝑐(𝑛) = 10 ∙ 𝑐(𝑛 − 1) for 𝑛 ≥ 2.

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b. 𝑏(1) = 2,𝑏(𝑛) = 2 ∙ 𝑏(𝑛 − 1) − 1 for 𝑛 ≥ 2.

Review Problems

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d. 𝑑(1) = 1,𝑑(𝑛) = 𝑛 ∙ 𝑑(𝑛 − 1) for 𝑛 ≥ 2.

4. The first 5 terms of some sequences are given. State a rule that each sequence could follow.

a. 2, 4, 6, 8, 10

b. 5, 7, 9, 11, 13

c. 50, 25, 0, −25, −50 d. 1 , 1, 3, 9, 27 3

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5. Function 𝑓 is defined by 𝑓(𝑥) = 2𝑥 − 7 and 𝑔 is defined by 𝑔(𝑥) = 5𝑥. a. Find 𝑓(3), 𝑓(2), 𝑓(1), 𝑓(0), and 𝑓( −1).

ip t

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b. Find 𝑔(3), 𝑔(2), 𝑔(1), 𝑔(0), and 𝑔( −1).

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Unit 10, Lesson 5: Representing Sequences

Warm-Up: Reading Representations For each sequence shown, find either the growth factor or rate of change. Be prepared to explain your reasoning.

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1. 5, 15, 25, 35, 45, . . .

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2. Starting at 10, each new term is 5 less than the previous term.

4. 𝑔(1) = −5, 𝑔(𝑛) = 𝑔(𝑛 − 1)⋅( −2) for 𝑛 ≥ 2 𝒏

𝒇(𝒏) 0

0.1

0.3

3 5

0.2 0.4

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Collaborative Activity: Matching Recursive Definitions Take turns with your partner to match a sequence with a recursive definition. It may help to first figure out if the sequence is arithmetic or geometric. • For each match that you find, explain to your partner how you know it’s a match.

op y

• For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement. There is one sequence and one definition that do not have matches. Create their corresponding match. Definitions

Sequences

𝐺(1) = 18, 𝐺(𝑛) = 1 ∙𝐺(𝑛 − 1), 𝑛 ≥ 2

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1. 3, 6, 12, 24 2. 18, 36, 72, 144

4. 18, 13, 8, 3

5. 18, 9, 4.5, 2.25

𝐻(1) = 3, 𝐻(𝑛) = 5∙𝐻(𝑛 − 1), 𝑛 ≥ 2

3. 3, 8, 13, 18

6. 18, 20, 22, 24

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𝐽(1) = 3, 𝐽(𝑛) = 𝐽(𝑛 − 1) + 5, 𝑛 ≥ 2

𝐾(1) = 18, 𝐾(𝑛) = 𝐾(𝑛 − 1) − 5, 𝑛 ≥ 2 𝐿(1) = 18, 𝐿(𝑛) = 2∙𝐿(𝑛 − 1), 𝑛 ≥ 2 𝑀(1) = 3, 𝑀(𝑛) = 2∙𝑀(𝑛 − 1), 𝑛 ≥ 2

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Collaborative Activity: Info Gap: Ways to Represent a Sequence Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

1. Silently read your card and think about what information you need to answer the question. 2. Ask your partner for the specific information that you need.

3. Explain to your partner how you are using the information to solve the problem.

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3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”

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1. Silently read the information on your card.

If your teacher gives you the problem card:

If your teacher gives you the data card:

5. Share the data card, and discuss your reasoning.

5. Read the data card, and discuss your reasoning.

4. Read the problem card, and solve the problem independently.

4. When you have enough information, share the problem card with your partner, and solve the problem independently.

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Lesson Summary There are several ways to represent a sequence. Each representation provides a different view of the same sequence. • A list of terms: Consider the terms for a sequence 𝐷: 4, 7, 10, 13, 16, . . .

• A table: A table lists the term number, 𝑛, and value for each term, 𝐷(𝑛). It can sometimes be easier to detect or analyze patterns when using a table.

𝒏

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Based on the list, it can be determined that this sequence is arithmetic by noting that the difference between consecutive terms is always 3, so this sequence has a rate of change of 3.

𝑫(𝒏)

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• A graph: The graph of a sequence is a set of points because a sequence is a function whose domain is a part of the integers number system. For an arithmetic sequence, these points lie on a line because arithmetic sequences are a type of linear function with a constant rate of change.

• An equation: Sequences can be defined recursively using function notation to make an equation. For the sequence 4, 7, 10, 13, 16, . . ., the starting term is 4, and the rate of change, or common difference, is 3, so the sequence is represented by 𝐷(1) = 4, 𝐷(𝑛) = 𝐷(𝑛 − 1) + 3 for 𝑛 ≥ 2. This type of definition explains how to find any term if the previous term is known. It is not as helpful in calculating terms where 𝑛 is large, such as 𝐷(100). Some sequences do not have recursive definitions, but geometric and arithmetic sequences always do. Sometimes a small amount of information can provide a lot of information about a function or a sequence. For example, consider a geometric sequence, 𝐻, with a growth factor of 2 and a starting term of 20.25. From here, the terms in the sequence after 3

20.25 can be determined to be 13.5, 9, 6, 4, and so on because in a geometric sequence, the current term is multiplied by the growth factor to get to the next term.

Using the same strategy, 𝐻(𝑛) can be represented using an equation, a table of values, or a graph, all that show that 𝐻 is not linear. Instead, it is exponential, as are all geometric sequences. The equation that can be used to represent 𝐻 is 𝐻(1) = 20.25, 𝐻(𝑛) = 𝐻(𝑛 − 1) ⋅ 2 for 𝑛 ≥ 2. 534 | Unit 10

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Practice Problems 1. Here is the recursive definition of a sequence: 𝑓(1) = 10, 𝑓(𝑛) = 𝑓(𝑛 − 1) − 1.5 for 𝑛 ≥ 2. a. Is this sequence arithmetic, geometric, or neither?

op y

b. List at least the first five terms of the sequence.

f(n)

12 10 8 6

c. Graph the value of the term 𝑓(𝑛) as a function of the term number 𝑛 for at least the first five terms of the sequence.

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n 1

2. An arithmetic sequence 𝑘 starts 12, 6, . . .

a. Write a recursive definition for this sequence.

b. Graph at least the first five terms of the sequence.

f(n)

14 12 10 8 6 4 2 -2

-4 -6 -8 -10 -12 -14

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3. An arithmetic sequence 𝑎 begins 11, 7, . . .

a. Write a recursive definition for this sequence using function notation.

f(n)

c. Explain how to use the recursive definition to find 𝑎(100). (Don’t actually determine the value.)

10 5

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b. Sketch a graph of the first 5 terms of 𝑎.

-5

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4. A geometric sequence 𝑔 starts 80, 40, . . .

-10

a. Write a recursive definition for this sequence using function notation.

b. Use your definition to make a table of values for 𝑔(𝑛) for the first 6 terms.

c. Explain how to use the recursive definition to find 𝑔(100). (Don’t actually determine the value.)

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Review Problems 5. Match each recursive definition with one of the sequences. Definition

Sequence

a. ℎ(1) = 1, ℎ(𝑛) = 2 ∙ ℎ(𝑛 − 1) + 1 for 𝑛 ≥ 2

80, 40, 20, 10, 5

b. (1) = 1, 𝑝(𝑛) = 2 ∙ 𝑝(𝑛 − 1) for 𝑛 ≥ 2

op y

1, 2, 4, 8, 16

c. 𝑎(1) = 80, 𝑎(𝑛) = 1 ∙ 𝑎(𝑛 − 1) for 𝑛 ≥ 2

1, 3, 7, 15, 31

6. For each sequence, decide whether it could be arithmetic, geometric, or neither.

ip t

a. 25, 5, 1, . . .

c. 4, 9, 16, . . .

b. 25, 19, 13, . . .

d. 50, 60, 70, . . . e. 1 , 3, 18, . . .

For each sequence that is neither arithmetic nor geometric, how can you change a single number to make it an arithmetic sequence? A geometric sequence?

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Unit 10, Lesson 6: The 𝒏th Term Warm-Up: Which One Doesn’t Belong: Repeated Operations 1. Which one doesn’t belong?

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A. 5 + 2 + 2 + 2 + 2 + 2 + 2

B. 5 + 6 ∙ 2 C. 5 ∙ 26

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D. 5 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2

Guided Activity: More Paper Slicing

1. Clare takes a piece of paper with length 8 inches (in.) and width 10 in. and cuts it in half. Then she cuts it in half again, and again . . . 1

𝑛

a. Instead of writing a recursive definition, Clare writes 𝐶(𝑛) = 80 ∙ � 2 � , where 𝐶 is

the area, in square inches (sq. in.), of the paper after 𝑛 cuts. Explain where the different terms in her expression came from.

b. Approximately what is the area of the paper after 10 cuts?

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2. Kiran takes a piece of paper with length 8 in. and width 10 in. and cuts away 1 in. of the width. Then he does it again, and again . . . a. Complete the table for the area of Kiran’s paper 𝐾(𝑛), in sq. in., after 𝑛 cuts. 𝒏

𝑲(𝒏)

1 2

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80 − 8 − 8 = 80 − 8(2) = 64

4 5

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b. Kiran says the area after 6 cuts, in sq. in., is 80 − 8 ∙ 6. Explain where the different terms in his expression came from.

c. Write a definition for 𝐾(𝑛) that is not recursive.

3. Which is larger, 𝐾(6) or 𝐶(6)?

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Collaborative Activity: A Sierpinski Triangle

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A Sierpinski triangle can be created by starting with an equilateral triangle, breaking the triangle into 4 congruent equilateral triangles, and then removing the middle triangle. Starting from a single black equilateral triangle:

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1. Let 𝑆 be the number of black triangles in step 𝑛. Define 𝑆(𝑛) recursively.

2. Andre and Lin are asked to write an equation for 𝑆 that isn’t recursive. Andre writes 𝑆(𝑛) = 3𝑛 for 𝑛 ≥ 0 while Lin writes 𝑆(𝑛) = 3𝑛−1 for 𝑛 ≥ 1. Whose equation do you think is correct? Explain or show your reasoning.

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Lesson Summary Consider an arithmetic sequence 𝑓: 6, 10, 14, 18, 22, . . .

op y

In this sequence, each term is 4 more than the previous term. One recursive definition of this sequence is 𝑓(1) = 6, 𝑓(𝑛) = 𝑓(𝑛 − 1) + 4 for 𝑛 ≥ 2. This function could also be written 𝑓(0) = 6, 𝑓(𝑛) = 𝑓(𝑛 − 1) + 4, for 𝑛 ≥ 1 because it generates the same sequence. Neither of these definitions is better than the other, but they depend on how the “first term” of the sequence is defined: 𝑓(1) or 𝑓(0). Let’s use 𝑓(1) for now.

𝒏 1

𝒇(𝒏)

6 + 0 = 6 + 4(0) = 6

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The table of values for sequence 𝑓 is shown. Notice how for each term, the number of 4s added to the expression increases by 1. In fact, the number of 4s is 𝑛 − 1 for any term 𝑛.

Although defining a sequence recursively works to calculate the current term using the previous term, it is problematic when trying to determine 𝑓(100) because it would mean calculating all the terms up to 𝑓(99) to get there. For sequences, there is a better way.

6 + 4 = 6 + 4(1) = 10

Looking carefully at the pattern in the 3 6 + 4 + 4 = 6 + 4(2) = 14 table, the 𝑛th term of the sequence could be written as 𝑓(𝑛) = 6 + 4(𝑛 − 1) for 𝑛 ≥ 1. 4 6 + 4 + 4 + 4 = 6 + 4(3) = 18 This is sometimes called an explicit formula of a sequence, but it’s just a way to 5 6 + 4 + 4 + 4 + 4 = 6 + 4(4) = 22 calculate the value of the 𝑛th term without having to calculate all the terms that came before it. Need to know 𝑓(100)? Just compute 6 + 4(100 − 1).

Defining an arithmetic sequence this way takes advantage of the fact that this type of sequence is a linear function with a starting value (in this case, 6) and a rate of change (in this case, 4). If the sequence was defined as starting at 𝑛 = 0 where 𝑓(0) = 6, the equation for the 𝑛th term would be 𝑓(𝑛) = 6 + 4(𝑛) for 𝑛 ≥ 0. Geometric sequences behave the same way but with repeated multiplication. The geometric sequence 𝑔: 3, 15, 75, 375, . . . can be written as 3, 3 ⋅ 5, 3 ⋅ 5 ⋅ 5, 3 ⋅ 5 ⋅ 5 ⋅ 5, . . . This means if 𝑔(0) = 3, and the 𝑛th term can be written as 𝑔(𝑛) = 3 ⋅ 5𝑛.

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Practice Problems 1. A sequence is defined by 𝑓(0) = − 20, 𝑓(𝑛) = 𝑓(𝑛 − 1) − 5 for 𝑛 ≥ 1. a. Explain why 𝑓(1) = − 20 − 5.

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b. Explain why 𝑓(3) = − 20 − 5 − 5 − 5.

c. Complete the expression: 𝑓(10) = − 20 − _______. Explain your reasoning.

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2. A sequence is defined by 𝑓(0) = − 4, 𝑓(𝑛) = 𝑓(𝑛 − 1) − 2 for 𝑛 ≥ 1. Write a definition for the 𝑛th term of the sequence.

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Review Problems 3. Here is the recursive definition of a sequence: 𝑓(1) = 3, 𝑓(𝑛) = 2 ∙ 𝑓(𝑛 − 1) for 𝑛 ≥ 2.

b. Graph the value of the term as a function of the term number.

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c. Is the sequence arithmetic, geometric, or neither? Explain how you know.

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a. Find the first 5 terms of the sequence.

4. Here is a graph of sequence 𝑀. Define 𝑀 recursively using function notation.

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5. Write the first five terms of each sequence. Determine whether each sequence is arithmetic, geometric, or neither. a. 𝑎(1) = 5, 𝑎(𝑛) = 𝑎(𝑛 − 1) + 3 for 𝑛 ≥ 2.

c. 𝑐(1) = 3, 𝑐(𝑛) = − 𝑐(𝑛 − 1) + 1 for 𝑛 ≥ 2.

op y

b. 𝑏(1) = 1, 𝑏(𝑛) = 3 ⋅ 𝑏(𝑛 − 1) for 𝑛 ≥ 2.

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d. 𝑑(1) = 5, 𝑑(𝑛) = 𝑑(𝑛 − 1) + 𝑛 for 𝑛 ≥ 2. 6. Here is the graph of a sequence.

a. Is this sequence arithmetic or geometric? Explain how you know.

b. List at least the first five terms of the sequence.

c. Write a recursive definition of the sequence.

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Unit 10, Lesson 7: Arithmetic Sequences and Linear Functions

Warm-Up: Math Talk: Multiplying Fractions 3

𝑥

1. For the function 𝑓(𝑥) = 32 ∙ � 4 � , evaluate mentally:

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a. 𝑓(0)

b. 𝑓(1)

d. 𝑓(3)

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c. 𝑓(2)

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Guided Activity: Comparing Arithmetic Sequences and Linear Functions 1. A graph of a sequence and a linear function are shown. Sequence

Linear Function

Term

40 40

Term Number

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ip t

-20

a. Write the recursive definition for the sequence.

b. Write the explicit formula for the sequence.

c. Write the equation of the line that represents the linear function.

d. What is the domain of the sequence?

e. What is the domain of the linear function?

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2. Complete the statements. discrete. continuous.

The domain of an arithmetic sequence is

integers. rational numbers. irrational numbers. discrete continuous

The domain of a linear function that represents an arithmetic

discrete continuous

sequence is

The domain of a linear function is typically

op y

and is a subset of the set

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3. The share of US households that used social media in 2005 was 5%. Each year between 2005 and 2019, the share of US households that used social media increased 6.1%.

a. Discuss with your partner if the domain is discrete or continuous. Summarize your discussion.

b. Determine if the context is best represented by an arithmetic sequence or a linear function. If the context is a linear function, define the variables, and if necessary, state the domain.

c. Write the explicit formula for the sequence or the equation of the line that represents the linear function based on your choice in part B.

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4. The first time Geoff completed a task, it took him 9.7 hours (hr.). Each subsequent time Geoff completed the same task, the time it took was reduced by 0.4 hr. a. Determine if the context is best represented by an arithmetic sequence or a linear function. Arithmetic sequence

op y

Linear function

a. Write the explicit formula for the sequence or the equation of the line that represents the linear function. If the context is a linear function, define the variables, and if necessary, state the domain.

ip t

5. In 1980, the average cost of a pound of chocolate chip cookies was $1.28. For the years since 1980, the cost increased 5 cents a year. a. Determine whether the context is best represented by a linear function or an arithmetic sequence. Then, justify your choice.

Best Representation of Context

Justification

Arithmetic sequence Linear function

b. Write the explicit formula for the sequence or the equation of the line that represents the linear function. If the context is a linear function, define the variables, and if necessary, state the domain.

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6. Julian has a collection of party balloons filled with helium. He determines that the collection can lift an object that weighs 12 grams (g). Each additional party balloon filled with helium can lift 0.5 g. a. Write the explicit formula for the sequence that represents the context.

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op y

b. Explain why the context cannot be modeled by a linear function.

Collaborative Activity: Modeling Real-World Contexts Work with your partner to complete the following.

1. When Isaiah got contact lenses, his doctor told him that he can wear them 2 hr. the first day, and then increase the length of time by 30 minutes (min.) each day.

a. Determine whether the context is best represented by a linear function or an arithmetic sequence. Then, justify your choice.ye Justification

Best Representation of Context Arithmetic sequence Linear function

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2. An ounce (oz.) of chia seeds contains 4.7 g of protein.

a. Determine whether the context is best represented by a linear function or an arithmetic sequence. Then, justify your choice. Best Representation of Context

Justification

op y

Arithmetic sequence Linear function

ip t

3. A lecture hall has semicircular seating. The first row has 23 seats. The second row has 27 seats.

a. Determine whether the context is best represented by a linear function or an arithmetic sequence. Then, justify your choice.

Best Representation of Context

Justification

Arithmetic sequence Linear function

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Lesson Summary A domain is considered discrete when the input values consist of a set of values that are distinct and separate on a finite or infinite interval. Contexts that grow by a common difference with a discrete domain are best represented by an arithmetic sequence, usually described with an explicit formula. Recursive definitions can also be used, but they require the previous term(s) to be known, which can get cumbersome, depending on the context.

op y

Consider the following example.

An athletic booster club is selling plastic cups. One cup is 5.25 inches (in.) tall. Each cup that is added to the stack adds 1.3 in. to the height of the stack.

This situation is best represented by an arithmetic sequence because the domain of the function is discrete. That is, the input values are the numbers of cups, which are the distinct integer values 1, 2, 3, 4, 5, . . . Therefore, this context is best represented by the explicit formula shown.

ip t

𝑓(𝑛) = 5.25 + 1.3(𝑛 − 1), 𝑛 ≥ 1

A domain is considered continuous when the input values consist of a set of values and can take on any value within a finite or infinite interval. Contexts that grow by a common difference with a continuous domain are best represented by a linear function. Consider the following example.

Trixie is traveling by car to her vacation destination. Once she reaches the interstate highway 21.6 miles (mi.) into her journey, she drives at a speed of 70 miles per hour (mph).

This situation is best represented by a linear function because the domain of the function is continuous. The input values are any increment of time that Trixie has been traveling, which can be divided continuously into smaller values in between integers. Therefore, this context is best represented by an equation of a linear function, as shown. 𝑦 = 21.6 + 70𝑥

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Practice Problems 1. The share of US households that used a dishwasher in 1964 was 11.8%. From 1964 to 1978, the share of US households that used dishwashers increased 2.5% each year.

op y

a. Determine whether the context is best represented by a linear function or an arithmetic sequence. Then, justify your choice. Best Representation of Context

Justification

Arithmetic sequence Linear function

ip t

b. Write the explicit formula for the sequence or the equation of the line that represents the linear function. If the context is a linear function, define the variables and, if necessary, state the domain.

2. Chicago-style pizza is well-known around the world as excellent deep-dish pizza. The cost of a medium pizza at a well-known pizzeria in Chicago is $20.00, and each topping costs $2.10.

a. Determine whether the context is best represented by a linear function or an arithmetic sequence. Then, justify your choice. Justification

Best Representation of Context Arithmetic sequence Linear function

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3. Diego is making a stack of pennies. He starts with 5 and then adds them 1 at a time. A penny is 1.52 millimeter (mm) thick. a. Complete the table with the height of the stack, ℎ(𝑛), in mm, after 𝑛 pennies have been added. b. Does ℎ(1.52) make sense? Explain how you know.

𝒏 0 1

𝒉(𝒏) 7.6

op y

Review Problems

ip t

a. List the first 5 terms of the sequence.

4. Here is the recursive definition of a sequence: 𝑓(1) = 35, 𝑓(𝑛) = 𝑓(𝑛 − 1) − 8 for 𝑛 ≥ 2.

b. Graph the value of each term as a function of the term number.

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op y

5. Here is a graph of sequence 𝑞. Define 𝑞 recursively using function notation.

6. Here is a recursive definition for a sequence 𝑓: 𝑓(0) = 19, 𝑓(𝑛) = 𝑓(𝑛 − 1) − 6 for 𝑛≥1. The definition for the 𝑛th term is 𝑓(𝑛) = 19 − 6∙𝑛 for 𝑛≥0.

ip t

a. Explain how you know that these definitions represent the same sequence.

b. Select a definition to calculate 𝑓(20), and explain why you chose it.

7. An arithmetic sequence 𝑗 starts 20, 16, . . . Explain how you would calculate the value of the 500th term.

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Unit 10, Lesson 8: Geometric Sequence and Exponential Functions

Warm-Up: Describing Growth

16, 24, 36, 54, 81

op y

1. Here is a geometric sequence. What is the growth factor?

ip t

2. One way to describe its growth is to say it’s growing by ____% each time. What number goes in the blank for the sequence 16, 24, 36, 54, 81? Be prepared to explain your reasoning.

Collaborative Activity: Geometric Sequence or Exponential Function?

Discuss each context with your partner, and determine whether the context would best be modeled by a geometric sequence or an exponential function. 1. A photo with a width of 4 inches (in.) was enlarged 5 times. Each time it is enlarged, the photo grows by a factor of 1.25.

Geometric sequence

Exponential function

2. In 2006, the total revenue, in billions of dollars, from streaming music was 0.1. Each subsequent year, the rate of growth of the total revenue was 41%. Geometric sequence

Exponential function

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3. A tennis tournament has 128 players. When a player loses a match, they are eliminated from the tournament. The table shows the number of players for each round.

Number of Players

128

op y

Round

Geometric sequence Exponential function

Weight (lb.)

1.2

1.44

1.728

2.0736

Length (in.)

Some of his data is shown.

ip t

4. A fishery biologist was catching and releasing bass from Toledo Bend Reservoir on the Texas-Louisiana state line. He recorded the length of the bass, in inches, and the weight of the bass, in pounds (lb.). The length of the bass the biologist caught ranged from 12 in. to 20 in.

Geometric sequence

Exponential function

5. In 1970, the amount of agricultural land in Brazil was 1.32 hectares per person. Since 1970, the rate of decay of the amount of agricultural land in Brazil, in number of hectares per person, is 1% for every year. Geometric sequence

Exponential function

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6. The first time Angelica completed a soccer drill, it took her 16 minutes (min.). Each subsequent time Angelica completed the same drill, it took 10% less time than the previous time. Geometric sequence Exponential function

op y

7. A homeowners’ association charges fines for breaking rules. The first violation is $20. The fine for each additional violation is doubled. Geometric sequence

Exponential function

ip t

8. Summarize how you determined whether a context would be best modeled by a geometric sequence or an exponential function.

Guided Activity: Modeling Contexts with Sequences and Functions

Write a function or sequence that will model each context. • If writing a sequence, use an explicit formula. • If writing a function, state the definition of the variables.

• For either a function or a sequence, state any constraints.

1. A fishery biologist was catching and releasing bass from Toledo Bend Reservoir on the Texas-Louisiana state line. He recorded the length of the bass, in inches, and the weight of the bass, in pounds (lb.). The length of the bass the biologist caught ranged from 12 in. to 20 in. © Accelerate Learning Inc. - All Rights Reserved

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Some of his data is shown.

Length (in.)

Weight (lb.)

1.2

1.44

1.728

2.0736

Stage 2

Stage 3

ip t

Stage 1

op y

2. Three stages of an eight-stage pattern are shown.

3. In 2015, 4% of US households were using light-emitting diode (LED) bulbs. Each subsequent year since 2015, the percentage of households using LED bulbs increased by a ratio of 1.64.

4. A photo with a width of 4 in. was enlarged 5 times. Each time it is enlarged, the photo grows by a factor of 1.25.

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Lesson Summary The relationship between exponential functions and geometric sequences is similar to the relationship between linear functions and arithmetic sequences.

op y

Contexts that grow by a common ratio with a discrete domain are best represented by a geometric sequence, usually described with an explicit formula. Consider the following example.

A tennis ball is dropped from a height of 4.2 feet (ft.). It rebounds to a height that is 55% of the previous height after each subsequent bounce.

ip t

This situation is best represented by a geometric sequence because the domain of the function is discrete. That is, the input values are the number of bounces, which are the distinct integer values 1, 2, 3, 4, 5, . . . Therefore, this context is best represented by the explicit formula shown.

𝑓(𝑛) = 4.2(0.55)𝑛 − 1, 𝑛≥1

Contexts that grow by a common ratio with a continuous domain are best represented by an exponential function. Consider the following example.

In 2011, the share of the US population that was exposed to air pollution at concentrations of 10 micrograms per cubic meter (mcg per cu. m) was 46.5%. Each subsequent year, the rate of decay of the share of the US population exposed to air pollution at concentrations of 10 mcg per cu. m was 32%. This situation is best represented by an exponential function because the domain of the function is continuous. The input values are any increment of time, in years, since 2011, which can be divided continuously into smaller values. 𝑦 = 0.4650.68𝑛

The domain in this context is limited to years since 2011, which represents the initial time. This could be represented by 0 ≤ 𝑛 ≤ (current year − 2011). © Accelerate Learning Inc. - All Rights Reserved

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Carefully consider the domain in context to determine whether it is best represented by an exponential function or a geometric sequence. Also, be sure to note any restrictions on the domain.

Practice Problems 1. Write a function or sequence that will model the context. State any constraints.

op y

In 2016, 9 million smart speakers were sold worldwide. Each subsequent year, the number of smart speakers sold worldwide, in millions, increased by a ratio of 1.77.

2. Write a function or sequence that will model the context. State any constraints.

ip t

A homeowners’ association charges fines for breaking rules. The first violation is $20. The fine for each additional violation is doubled.

Review Problems

a. An arithmetic sequence has 𝑎(1) = 4 and 𝑎(2) = 16. Explain or show how to find the value of (15).

400 300 200 100 0

x 1

Term number

500

Value

3. A geometric sequence 𝑔 starts at 500 and has a growth factor of 0.6. Sketch a graph of 𝑔 showing the first 5 terms.

b. A geometric sequence has 𝑔(0) = 4 and 𝑔(1) = 16. Explain or show how to find the value of 𝑔(15).

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op y

ip t

Unit 11: Reasoning with Parent Functions

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Unit 11, Lesson 1: Classifying Parent Functions

Warm-Up: Revisiting Transformations of a Triangle y

Triangle 𝐾𝑁𝐵 is shown on the coordinate plane.

op y

1. Reflect ∆𝐾𝑁𝐵 across the 𝑦-axis, and then translate it down 2 units. Label the image ∆𝐾”𝑁”𝐵”.

ip t

2. Explain the relationship between the preimage and the image.

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Exploration Activity: Parent Functions Gallery Walk In a function family, the parent function is the simplest function that preserves the characteristics of the family.

Linear Function Table

Parent Function

op y

This activity revisits 5 parent functions explored during this course, each represented as an equation, a table, and a graph. Examine each parent function, and answer the questions that follow. Graph 10 y

-10

-5

x 0

-5

-10

𝑦 = 𝑥

𝒚 −2 −1 0 1 2 3

ip t

𝒙 −2 −1 0 1 2 3

1. Use the representations of the parent function to answer the following for the linear function family.

a. How can a graph be used to classify a function as linear?

b. How can an equation be used to classify a function as linear?

c. How can a table of values be used to classify a function as linear?

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Quadratic Function Table

Parent Function

Graph 10 y

𝑦 = 𝑥2

1 2 3

𝒚 4 1 0

x -10

1 4 9

-5

op y

𝒙 −2 −1 0

-5

-10

2. Use the representations of the parent function to answer the following for the quadratic function family.

ip t

a. How can a graph be used to classify a function as quadratic?

b. How can an equation be used to classify a function as quadratic?

c. How can a table of values be used to classify a function as quadratic?

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Absolute Value Function Table

Parent Function

Graph 10 y

𝒙

𝒚

𝑦 = |𝑥|

2 1 0 1 2 3

x -10

-5

op y

−2 −1 0 1 2 3

-5

-10

3. Use the representations of the parent function to answer the following for the absolute value function family.

ip t

a. How can a graph be used to classify a function as an absolute value?

b. How can an equation be used to classify a function as an absolute value?

c. How can a table of values be used to classify a function as an absolute value?

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𝒙

𝒚

−2

1 4

𝑥

𝑦 = � 1 � 2

−1

𝒚 4

1 2

1 8

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1 4

Graph

2 1

-10

−2

-5

ip t

𝒙

-5

Exponential Decay Function Table

Parent Function

base � 1 �

x -10

Exponential decay

op y

𝑦 = 2𝑥

1 2

−1

Exponential growth base (2)

Graph

Parent Function

Exponential Growth Function Table

x -10

-5

-10

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4. Use the representations of the parent function to answer the following for an exponential function family. a. How can a graph be used to classify a function as exponential?

op y

b. How can an equation be used to classify a function as exponential?

c. How can a table of values be used to classify a function as exponential?

d. Describe the differences to look for when distinguishing between an exponential growth function and an exponential decay function from each representation.

ip t

Graph:

Equation:

Table of values:

Guided Activity: Classifying Functions

Like geometric figures, functions can be transformed by translations, dilations, and reflections. In the examples shown in the table, the quadratic parent function is translated vertically, translated horizontally, and reflected over the 𝑥-axis. 570 | Unit 11

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Quadratic parent function 𝑦 = 𝑥2

Translated vertically 𝑦 = 𝑥2 − 2

x -5

-10

Translated horizontally 𝑦 = (𝑥 − 5)2

Reflected over the 𝑥-axis 𝑦 = − (𝑥)2

ip t

-5

-10

-5

op y

-10

-5

x 0

-5

-10

-5

1. What stayed the same in each of the equations?

2. What stayed the same in each of the graphs?

3. Classify each function into its function family based on the characteristics you observed about the parent functions.

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Function

Function Family Linear Quadratic Absolute value

𝑦 = |𝑥 + 9|

Exponential growth

op y

Exponential decay Linear

Quadratic 2

Absolute value

Exponential growth

𝑦 = � 1 �

𝑥

Exponential decay

ip t

10 y

Linear

Quadratic

-5

-10

-5

Absolute value Exponential growth Exponential decay

-10

𝒙

𝒚

Linear

−3

Absolute value

−6

−1

−5

−3 3 6

−7

Quadratic

Exponential growth Exponential decay

−9

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Lesson Summary The functions studied throughout this course can be classified into families, where the simplest function that has all the characteristics of the family is known as the parent function.

𝒚 = −𝒙𝟐

ip t

Like geometric figures, parent functions can be translated, reflected, or dilated to create a new function in the same family. The function 𝑦 = −𝑥2 is a reflection of the parent function over the 𝑥-axis, and the function 𝑦 = 𝑥2 − 3 is a translation of the parent function down 3 units. The graphs of both functions are shown.

op y

For example, the quadratic family of functions are U-shaped, opening upward or downward, with a highest or lowest point known as the vertex. The parent function for the quadratic family of functions is 𝑦 = 𝑥2, whose graph opens upward with the vertex at (0, 0), as shown.

𝒚 = 𝒙𝟐 − 𝟑 y

Note that like transformed geometric figures, the shape of the parent function is retained after transformations are applied to it.

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Practice Problems 1. Classify each function into its function family based on the characteristics you have observed about the parent functions. Function

Function Family

op y

𝑦 = − 𝑥 − 23 𝑦 = 2(𝑥 − 7)

𝑦 = 𝑥2 + 31 y

ip t

-5

-10

2. Bruce and Keita are determining the function family for the function represented in the table shown.

Bruce thinks that the function could be a member of the linear family, and Keita thinks the function could be an absolute value. Explain which student is correct.

𝒙

𝒚

−2

−3

−1

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−2

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3. The graph of a function is shown.

a. Determine the function family for the graph. Explain your reasoning.

op y

b. The graph is of a transformation that has been applied to the parent function. Explain whether the transformation is a translation, a dilation, or a reflection.

ip t

Review Problems

4. A system of linear equations is graphed on the coordinate plane as shown.

Use the graph to determine the solution to the system of linear equations.

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Unit 11, Lesson 2: Determining Which Function Type Best Models a Real-World Context

Warm-Up: Reading Books

Month Total Books Read

op y

Mr. Underwood recorded the total number of books read optionally by his students each month. The table of values for the data is shown. 5

ip t

Stevie, one of Mr. Underwood’s students, realized that the table of values can be modeled by one of the types of functions they have studied this year. Which type of function can be used to model the data?

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Exploration Activity: Card Sort 1. Sort the cards to match each function type with the table of values, graph, and written description. Function Type

Table of Values

Graph

op y

Linear

Written Description

Quadratic

Exponential

ip t

2. Discuss with your partner the strategies that can be used to determine the type of function from the graph.

3. Discuss with your partner the strategies that can be used to determine the type of function from the table of values. Summarize your discussion.

4. Discuss with your partner the strategies that can be used to determine the type of function from the written description. Summarize your discussion.

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Collaborative Activity: Gallery Walk 1. The international poverty line is the base that determines whether someone in the world is living in poverty. In 2021, the international poverty line was $1.90 per day.

op y

A typical year contains 261 working days. What is the yearly salary for someone living at the international poverty line?

2. Each table or written description gives information about the percentage of people living at or below the international poverty line for the given time period, where 0 represents the first year in the given time period.

ip t

• Use the information provided on each card to determine whether a linear, quadratic, or exponential function best models the table or written description. • On your paper, include the function type and a justification for your choice.

• If your group has a table of values, then create a written description for the same information.

• If your group has a written description, then create a table of values for the same information.

• During the gallery walk, complete the table by recording the information provided by each group.

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Country

Function Type

Useful Information to Determine Function Type

op y

Bangladesh

China

ip t

Republic of the Congo

Peru

Ghana

Ethiopia

Romania

Rwanda

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Lesson Summary Linear, quadratic, and exponential functions can be used to model data in realworld situations. The characteristics or key features of the graph, table of values, or description of the situation can be used to determine the function that best models the data.

op y

• The graph of a linear function is a straight line, and the table of values shows the 𝑦-values increase or decrease at a constant rate.

• The graph of a quadratic function is U-shaped, and the table of values shows the 𝑦-values decrease to a minimum value and then increase or increase to a maximum value and then decrease.

• The graph of an exponential growth function increases slowly at first and then quickly, whereas the graph of an exponential decay function decreases quickly at first and then slowly. The table of values for an exponential function shows the 𝑦-values change by a common factor over equal intervals of 𝑥.

Table of Values

0.8

0.47

0.92

11 14

0.96 0.8

Milk consumed per person per day (cups)

𝒚

Graph

𝒙

ip t

The graph, table of values, and description of dairy milk consumption in the United States, 𝑦, in cups of milk consumed per person per day, since 1900 are shown, where 𝑥 is the number of decades since 1900. Description

The consumption of milk by US consumers in the decades since 1900 reached a maximum of 0.96 cups per person per day and then began to decrease.

Number of decades since 1900

0.47

Because the table, graph, and description show that the data increases to a maximum value and then decreases, forming a U shape, a quadratic function would best model the data.

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Practice Problems 1. The populations, in millions, of the United States and India for the years since 2000 are shown. United States

India

202

209.2

223.6

12 14 16

238

245.2 252.4

259.6

Years Since 2000

230.8

ip t

216.4

op y

Population (millions)

Years since 2000

a. Explain whether a linear, quadratic, or exponential function best models the population data for the United States.

b. Explain whether a linear, quadratic, or exponential function type best models the population data for India.

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2. The US Department of Agriculture (USDA) compiles data on average fruit prices. From 2013 to 2022, the average price of bananas decreased from 60.9 cents per pound (lb.) to a minimum of 56.8 cents per lb. in 2018 before it began increasing again.

Weeks after Sahara Started Removing Money

Amount of Money in Piggy Bank (dollars)

100

ip t

3. Sahara has $100 saved in her piggy bank. She begins to take out money each week. The amount of money, in dollars, in her piggy bank at the end of each week is shown in the table.

op y

Amrit thinks that the USDA data can best be modeled by a quadratic function, but he’s not sure how to explain his reasoning. Help Amrit support his answer by explaining why a quadratic function best models the data.

Explain whether a linear, quadratic, or exponential function type best models the money in Sahara’s piggy bank.

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1 3 4 5

90 70 60 50

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Review Problems 4. Sebastian graphed the linear inequality 𝑦 ≤ 1 𝑥 + 4, but he made an error.

Sebastian’s graph is shown.

op y

Explain Sebastian’s error.

ip t

5. A tennis ball is hit straight up in the air, and its height, in feet (ft.) above the ground, is modeled by the equation 𝑓(𝑡) = 4 + 12𝑡 − 16𝑡2, where 𝑡 is measured in seconds (sec.) since the ball was thrown.

a. Find the solutions to 6 = 4 + 12𝑡 − 16𝑡2 without graphing. Show your reasoning.

b. What do the solutions say about the tennis ball?

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Unit 11, Lesson 3: Modeling Real-World Contexts

Warm-Up: Key Features Key features for 3 different functions are described in the table. Select the function that is described. Function

Increasing interval(s): (−∞, ∞)

𝑥-intercept(s): � −1 , 0� 4

𝑦-intercept(s): (0, 1)

𝑦 = � 1 � 3

𝑦 = 4𝑥 + 1

𝑦 = (𝑥 − 2)2

ip t

Increasing interval(s): None

Decreasing interval(s): (−∞, ∞)

𝑥-intercept(s): None

𝑦-intercept(s): (0, 1)

Increasing interval(s): (2, ∞)

Decreasing interval(s): (−∞, 2)

𝑥-intercept(s): (2, 0)

𝑦-intercept(s): (0, 4)

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𝑥

Decreasing interval(s): None

op y

Key Features

𝑦 = � 1 � 3

𝑥

𝑦 = 4𝑥 + 1

𝑦 = (𝑥 − 2)2 𝑦 = � 1 � 3

𝑥

𝑦 = 4𝑥 + 1

𝑦 = (𝑥 − 2)2

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Collaborative Activity: Comparing Functions from Different Representations Four functions are shown using different representations. Function D

Function W

op y

-2

−6

0 1

1 8

𝒚

−1

x 2

Function S

Function L

𝒙

-1

ip t

-2

𝒙

𝒚

−1 5 7

7 5

49 25

343 125

1. Complete each statement.

a. Function D is

an exponential decay function. an exponential growth function. a quadratic function that opens up. a quadratic function that opens down. a linear function with a positive slope. a linear function with a negative slope.

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an exponential decay function. an exponential growth function. a quadratic function that opens up.

b. Function L is

a quadratic function that opens down. a linear function with a positive slope.

op y

a linear function with a negative slope.

an exponential decay function.

an exponential growth function.

a quadratic function that opens up.

c. Function S is

a quadratic function that opens down. a linear function with a positive slope.

ip t

a linear function with a negative slope.

an exponential decay function.

a quadratic function that opens up. a quadratic function that opens down.

d. Function W is

an exponential growth function.

a linear function with a positive slope.

a linear function with a negative slope.

2. Discuss with your partner which strategies you used to determine whether a table of values represents a linear, quadratic, or exponential function. Summarize your discussion.

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Guided Activity: Determining Functions to Model Real-World Contexts 1. In 1908, 10% of US households had electricity. The table shows the percentage of US households that had electricity for the years since 1908. The Great Depression began in August 1929, so the percentage of US households who had electricity stalled at 68% for a period of time. 0

Percentage of US Households

op y

Years Since 1908

a. Based on the provided information, define the variables for the context.

ip t

b. Determine the constraints for each variable.

c. Based on the data, explain which function type, linear, quadratic, or exponential, would be the least likely to model the data well?

Each scatter plot shows the data with a different function type fit to the data.

Exponential Percentage of US households

Percentage of US households

Quadratic

Percentage of US households

Linear

Years since 1908

d. Explain why it appears that both a quadratic and an exponential model better fit the data.

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2. The highest recorded temperatures for a town in Japan are shown in the table of values, where 𝑥 represents the month of the year and 𝑦 represents the temperature, in degrees Fahrenheit. 𝒙

𝒚

An exponential decay function An exponential growth function

A quadratic function that opens up A quadratic function that opens down

op y

a. Which type of function best models the data?

ip t

A linear function with a positive slope

A linear function with a negative slope

□ Slope

b. Select the key features that helped you identify the type of function.

□ Minimum

□ Maximum

□ Symmetry

□ 𝑥-intercept(s) □ 𝑦-intercept

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□ Interval(s) where the function is positive or negative □ Interval(s) where the function is increasing or decreasing

op y

c. Determine the constraints, if any, on the variables.

3. A research scientist found that under a controlled environment the area, in square inches (sq. in.), of algae in a pond changed each week. Her data is shown. Week of the Study Area (sq. in.)

384

ip t

a. Which type of function best models the data?

An exponential decay function

An exponential growth function

A quadratic function that opens up

A quadratic function that opens down

A linear function with a positive slope

A linear function with a negative slope

b. Select the key features that helped you identify the type of function.

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□ Symmetry □ 𝑥-intercept(s) □ 𝑦-intercept

□ End behavior □ Constant percent rate of change

op y

□ Interval(s) where the function is positive or negative

□ Interval(s) where the function is increasing or decreasing

ip t

c. Determine the constraints, if any, on the variables.

Years since 1990

4. The sales, in millions of dollars, of walking shoes for the years since 1990 are shown in the table.

Sales (in millions of dollars)

3141.1

3278.6

3415.8

3553

3690.2

3827.4

3964.6

a. Which type of function best models the data?

An exponential decay function An exponential growth function A quadratic function that opens up A quadratic function that opens down A linear function with a positive slope A linear function with a negative slope

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b. Select the key features that helped you identify the type of function.

□ Slope □ Minimum □ Maximum □ Symmetry

op y

□ 𝑥-intercept(s) □ 𝑦-intercept

□ Constant percent rate of change

□ End behavior

ip t

□ Interval(s) where the function is positive or negative

□ Interval(s) where the function is increasing or decreasing

c. Determine the constraints, if any, on the variables.

Lesson Summary

Linear, quadratic, and exponential functions can be used to model real-world data. Key features and characteristics can be compared to determine which function type best models the data. Note any constraints in a real-world context that might affect the viability of any part of the model. The data for the revenue 𝑦, in millions of dollars, of 3 different companies, where 𝑥 is the number of years after 2010, is shown.

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𝒚

0 2 4 6 8 10 12 14 16 18 20

𝒙

0.9 1.2 1.44 1.7 2.1 2.1 2.5 2.8 3.1 3.33 3.7

Company B

𝒚

0 2 4 6 8 10 12 14 16 18 20

0.2 0.29 0.41 0.6 0.86 1.24 1.78 2.57 3.7 5.3 7.7

𝒙

𝒚

0 2 4 6 8 10 12 14 16 18 20

𝒙

Company B

1.2 1.9 2.48 2.9 3.24 3.44 3.5 3.44 3.24 2.9 2.5

op y

Company A

16 18

2.5

Years since 2000

Revenue (Millions of dollars)

1.25

7.5

2.5

ip t

Revenue (Millions of dollars)

3.75

Revenue (Millions of dollars)

4.5 4 3.5 3 2.5 2 1.5 1 0.5

16 18

Years since 2000

16 18

Years since 2000

• Company A’s revenue increases at close to a constant rate throughout the 20 years, so a linear function would best model company A’s revenue.

• For company B, the revenue increases slowly at first and then quickly, so the revenue more closely models an increase by a common factor. Thus, an exponential function would best model company B’s revenue. • Company C’s revenue increases to a maximum of $3.5 million and then decreases, so a quadratic function would best model the revenue for company C.

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Practice Problems 1. Match the context from the Guided Activity with the equation that models it.

𝑦 = 6(4)𝑥

𝑦 = 6(4)𝑥−1

1372𝑥 + 10𝑦 = 31414

□ □ □ □ □ □

𝑦 = −1(𝑥 − 7)2 + 73

□ □ □ □ □ □

ip t

−1372𝑥 + 10𝑦 = 31414

□ □ □ □ □ □

Sales of Walking Shoes

op y

𝑦 = (𝑥 − 7)2 + 73

Algae on a Pond

Equation

Highest Recorded Temperatures for a Town in Japan

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2. The swift fox is native to Montana. However, due to habitat loss, the swift fox was eliminated from the state until restoration efforts began. The number of swift foxes in Montana, in hundreds, for the years since 1960 are shown in the table. Years since 1960

Number of Swift Foxes (hundreds)

6.25

10 4

2.25

20 1

0.25

30 0

0.25

40 1

2.25

op y

a. Which function best models the number of swift foxes in Montana, 𝑦, in hundreds, 𝑥 years since 1960? A. 𝑦 = 3.6(0.96)𝑥

D. 𝑦 = −0.1𝑥2 + 0.6𝑥 − 9

ip t

C. 𝑦 = 0.1𝑥2 −0.6𝑥 + 9

B. 𝑦 = −0.1𝑥 + 4.4

b. Explain your reasoning for your answer to part A.

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Review Problem 3. Nara solved a linear equation, but she made an error. Her work is shown. a. Explain Nara’s error.

Step

Work

2� 1 𝑥 + 5� = 6𝑥 − 25

2 b. Determine the correct solution to the equation 2� 1 𝑥 + 5� = 6𝑥 − 25.

𝑥 + 5 = 6𝑥 − 25 −5𝑥 + 5 = −25

op y

𝑥 = 6

ip t

−5𝑥 = −30

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Unit 11, Lesson 4: Transformations Using Horizontal and Vertical Translations

Warm-Up: Translations of a Geometric Figure

op y

Triangles 𝐶𝑀𝐾 and 𝐷𝑁𝐹 are shown on the coordinate plane, where ∆𝐷𝑁𝐹 is the image of ∆𝐶𝑀𝐾 after a sequence of transformations.

ip t

1. Describe the sequence of transformations that maps ∆𝐶𝑀𝐾 onto ∆𝐷𝑁𝐹.

2. Juan states that ∆𝐶𝑀𝐾 and ∆𝐷𝑁𝐹 are congruent. Katy states they are similar but not congruent. Explain which student is correct.

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Exploration Activity: Horizontal Transformations Work with your partner to complete the following. 1. The graphs of 𝑦 = 𝑥2 and 𝑦 = (𝑥 − 3)2 are shown. 4

(2, 4)

y (1, 4)

(5, 4)

3 2 1 (3, 0)

-2

-1

(0, 0)

op y

(−2, 4)

ip t

a. What do you notice about the graphs of 𝑦 = 𝑥2 and 𝑦 = (𝑥 − 3)2?

b. Work with your partner to describe how to translate the graph of 𝑦 = 𝑥2 so that it coincides with the graph of 𝑦 = (𝑥 − 3)2.

c. The table of values for 𝑦 = 𝑥2 and 𝑦 = (𝑥 − 3)2 is shown.

𝒚 = 𝒙𝟐

𝒚

𝒙

−2 2

4 4

1 5

What do you notice in comparing the table of values?

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𝒚 = (𝒙 − 𝟑)𝟐

𝒚 4 0 4

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d. Discuss with your partner the connections between the 𝑥-values in the table of values in part C and your description of the transformation in part B. Summarize your discussion.

(−5, 4)

2. The graphs of 𝑦 = 𝑥2 and 𝑦 = (𝑥 + 3)2 are shown.

4 (−1, 4)

(2, 4)

3 2

op y

a. Work with your partner to describe how to translate the graph of 𝑦 = 𝑥2 so that it coincides with the graph of 𝑦 = (𝑥 + 3)2.

(−2, 4)

(−3, 0)

-5

-4

-3

(0, 0)

-2

-1

x 2

𝒙

−5

−1

ip t

−2

𝒚

𝒙

𝒚 = 𝒙𝟐

b. The table of values for 𝑦 = 𝑥2 and 𝑦 = (𝑥 + 3)2 is shown.

−3

𝒚 = (𝒙 + 𝟑)𝟐

𝒚 4 0 4

Discuss with your partner the connections between the 𝑥-values in the table of values and your description of the transformation in part A. Summarize your discussion.

c. Complete the statements.

The graph of 𝑦 = (𝑥 + 3)2 is a

horizontal translation vertical translation

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of the graph of 𝑦 = 𝑥2.

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For the graph of 𝑦 = 𝑥2 to coincide with the graph of 𝑦 = (𝑥 + 3)2, it is translated up down left right

3 units. adding 3 to subtracting 3 from

x-value. y-value.

each

for each corresponding

x-value y-value

op y

The translation can be seen in the table of values by

ip t

Collaborative Activity: Vertical Transformations The graphs of 𝑦 = 𝑑(𝑥) and 𝑦 = ℎ(𝑥) are shown.

1. When a graph is translated left or right, the 𝑥-values are affected. Discuss with your partner which values are affected by translating the graph of 𝑦 = 𝑑(𝑥) up 5 units to create 𝑦 = ℎ(𝑥). Summarize your discussion.

h(x)

10 8 6 4 2

d(x) x

-10

-8

-6

-4

-2

-2 -4 -6

2. On the coordinate plane, draw the graph of the function 𝑘(𝑥) by translating 𝑑(𝑥) down 3 units.

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3. The table of values for 𝑦 = 𝑑(𝑥), 𝑦 = ℎ(𝑥), and 𝑦 = 𝑘(𝑥) is shown.

−2 0

𝒚

𝒙

−2

𝒚 = 𝒉(𝒙)

𝒚

𝒙

−2

𝒚 = 𝒌(𝒙)

𝒚 1

−3

op y

𝒙

𝒚 = 𝒅(𝒙)

4. Complete the statements.

horizontal translation vertical translation

a. The graph of 𝑦 = ℎ(𝑥) is a

ip t

Discuss with your partner how the translation in the graph is shown in the 𝑦-values of the table.

of the graph of 𝑦 = 𝑑(𝑥).

b. For the graph of 𝑦 = 𝑑(𝑥) to coincide with the graph of 𝑦 = ℎ(𝑥), it is translated 5 units.

up down left right

c. The translation can be seen in the table of values by

each

x-value y-value

adding 5 to subtracting 5 from

of the function 𝑑(𝑥) for each corresponding

x-value y-value

shown in the table.

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d. The graph of 𝑦 = 𝑘(𝑥) is a

horizontal translation vertical translation

of the graph of 𝑦 = 𝑑(𝑥).

e. For the graph of 𝑦 = 𝑑(𝑥) to coincide with the graph of 𝑦 = 𝑘(𝑥), it is translated 3 units.

op y

up down to the left to the right

f. The translation can be seen in the table of values by x-value y-value

of the function 𝑑(𝑥) for each corresponding

each

adding 3 to subtracting 3 from

ip t

shown in the table.

x-value y-value

Guided Activity: Expressing Translations Using Function Notation

The translations of graphs can also be expressed using function notation. The parent function 𝑦 = 𝑥2 can be written as 𝑓(𝑥) = 𝑥2.

1. Notice that the horizontal translation of 3 units to the right is written as 𝑦 = (𝑥 − 3)2, where (𝑥 − 3) takes the place of the 𝑥. To write this translation using function notation, substitute (𝑥 − 3) for 𝑥.

f(x) = x2 x−3

𝑓(_________) = (______________)2

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2. A vertical translation of 5 units up is written 𝑦 = 𝑥2 + 5, where 5 is added to the parent function. To write this translation using function notation, add 5 to 𝑓(𝑥). 𝑓(𝑥)______ = 𝑥2________

3. Work with your partner to complete the table. The first 2 rows have been completed as examples. Parent Function

Shift

𝑦 = |𝑥|

𝑓(𝑥) = |𝑥|

Right 5

𝑓(𝑥 − 5) = |𝑥 − 5|

𝑦 = |𝑥|

𝑓(𝑥) = |𝑥|

Up 2

𝑓(𝑥) + 2 = |𝑥| + 2

ip t Right 1

𝑦 = 3𝑥

𝑥

Down 4

𝑦 = 𝑥 1

C Left 7

𝑦 = 𝑥2

𝑦 = � 2 �

Transformed Function

op y

Original Equation

Up 6

4. The graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥) are shown.

a. How does the graph of 𝑓(𝑥) compare to the graph of 𝑔(𝑥)?

12 10 8 6 4

g(x)

f(x)

x -16 -14

-12 -10

-8

-6

-4

-2

-2 -4 -6

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b. Two function descriptions are shown. Discuss with your partner which function description matches the transformation on the graph. Then, justify your choice.

Function Description

Justification

𝑔(𝑥) = 𝑓(𝑥 + 6)

op y

𝑔(𝑥) = 𝑓(𝑥 − 6)

5. The table of values for the function 𝑦 = 𝐵(𝑥) is shown.

a. Describe the effect that the transformation of 𝐵(𝑥) − 2 will have on the 𝐵-value for each corresponding 𝑥-value in the table.

𝑩

−4

−6

−1 14

− 27

ip t

b. Describe the effect that the transformation of 𝐵(𝑥) + 7 will have on the 𝐵-value for each corresponding 𝑥-value in the table.

𝒙

Lesson Summary

Similar to geometric figures, functions can be transformed by applying translations, reflections, or dilations to a parent function. This lesson explored translations of parent functions that create new functions.

A translation is when every point in the function is moved in the same direction and by the same distance.

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Translations can be represented using function notation. The effects of the translation on the graph, table of values, and 𝑓(𝑥) for each translation are shown in the table. Graph

Table of Values

𝑘 > 0

Translates |𝑘| units to the right

Add 𝑘 to the 𝑥-values

𝑓(𝑥 − 𝑘)

𝑘 < 0

op y

𝒌-value

Translates |𝑘| units to the left

Subtract |𝑘| from the 𝑥-values or

Add 𝑘 to the 𝑥-values

Subtract |𝑘| from the 𝑦-values

Function Notation

𝑘 < 0

Translates down |𝑘| units

𝑓(𝑥) + 𝑘

𝑘 > 0

Translates up 𝑘 units

Add 𝑘 to the 𝑦-values

ip t

𝑓(𝑥) + 𝑘

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For example, 4 translations of the parent function 𝑓(𝑥) = 𝑥2 are shown in the table. For each, the graph, a verbal description, a description using function notation, and the equation of the function are shown. y

Translation right 3 units 𝑓(𝑥 − 3) 𝑦 = (𝑥 − 3)2

ip t

Translation left 3 units 𝑓(𝑥 + 3) 𝑦 = (𝑥 + 3)2

Translation up 3 units

op y

𝑓(𝑥) + 3 𝑦 = 𝑥2 + 3

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Translation down 3 units 𝑓(𝑥) − 3 𝑦 = 𝑥2 − 3

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Practice Problems 1. Complete the statement to describe how the graph of 𝑔(𝑥) = 𝑓(𝑥) + 6 compares to the graph of 𝑓(𝑥) = |𝑥 + 8|. horizontal vertical

shift of the graph of 𝑓(𝑥). up from down from to the right of to the left of

op y

The graph of 𝑔(𝑥) represents a

The graph of 𝑔(𝑥) is shifted _______ units

the graph of 𝑓(𝑥).

2. Describe the effect on the 𝑥-values of 𝐻(𝑥) = |𝑥 + 2| for the transformation 𝐾(𝑥) = 𝐻(𝑥 − 7). 2

ip t

3. Describe the effect on the 𝑦-values of 𝑤(𝑥) = 5 𝑥 − 9 for the transformation 𝑚(𝑥) = 𝑤(𝑥) + 5.

4. Describe how the graph of 𝑃(𝑥) = (𝑅𝑥 + 11) compares to the graph of 𝑅(𝑥) = 𝑥2 + 5.

Review Problem

a. Where is the vertex of the graph that represents 𝑦 = (𝑥 − 3)2 + 5? b. Does the graph open up or down? Explain how you know.

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Unit 11, Lesson 5: Transformations by a Factor

Warm-Up: Notice and Wonder The graphs of 𝑓(𝑥) = |𝑥| and 𝑔(𝑥) = 2|𝑥| are shown on the coordinate plane.

op y

ip t

What do you notice? What do you wonder?

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Exploration Activity: Exploring Transformations by a Factor 1. The functions 𝐾(𝑥), 𝑀(𝑥), and 𝑁(𝑥) are shown on the coordinate plane, where 𝐾(𝑥) = 𝑥2, 𝑀(𝑥) = 1 𝑥2, and 𝑁(𝑥) = 1 𝑥2. The table of values for the functions are also 2 5 shown.

9 8 7

−3

M(x)

K(x)

−2

N(x)

-8 -7 -6 -5 -4 -3 -2 -1 0 -1

-3

-4

1 2

−1

1 5

1 2

1 5

9 2

9 5

-2

𝒚

−1 0

𝒙

9 5

𝒚

−9

0 1

𝒚 = 𝑵(𝒙)

9 2

−3

−2

ip t

3 1

𝒙

−1

4 2

𝒚

𝒙

𝒚 = 𝑴(𝒙)

op y

𝒚 = 𝑲(𝒙)

0 2

−2 0 2

4 5

a. Discuss with your partner the effect that the function’s coefficient has on the 𝑦-values.

4 5

b. Complete the statements.

The coefficient of 𝑀(𝑥) = 1 𝑥2 is ______. The 𝑦-values of 𝑀(𝑥) can be found by 2 multiplying the 𝑦-values of 𝐾(𝑥) by _______ for each corresponding 𝑥-value. The coefficient of 𝑁(𝑥) = 1 𝑥2 is ______. The 𝑦-values of 𝑁(𝑥) can be found by 5 multiplying the 𝑦-values of 𝐾(𝑥) by _______ for each corresponding 𝑥-value.

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c. Complete the statements. vertical stretch vertical compression

The coefficient of 𝑀(𝑥) = 1 𝑥2 resulted in a 2

of 𝐾(𝑥) by a factor of 1 . The coefficient of 𝑁(𝑥) = 1 𝑥2 resulted in 2

vertical stretch vertical compression

by a factor of 1 . 5

op y

2. The functions 𝐻(𝑥) and 𝑇(𝑥) are shown on the graph, where 𝐻(𝑥) = 𝑥2 and 𝑇(𝑥) = 2𝑥2. The table of values for the functions is also shown. y

𝒚 = 𝑯(𝒙)

𝒙

𝒚

−1 1

ip t

−2

𝒙

𝒚

−1

0 4

−2 0 2

8 0 2 8

Complete the statements.

𝒚 = 𝑻(𝒙)

The coefficient of 𝑇(𝑥) = 2𝑥2 is _____. The 𝑦-values of 𝑇(𝑥) can be found by multiplying the 𝑦-values of 𝐻(𝑥) by _____ for each corresponding 𝑥-value. The graph of 𝑇(𝑥) is a vertical ______________ of the graph of 𝐻(𝑥).

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3. The functions 𝐴(𝑥), 𝐷(𝑥), and 𝐿(𝑥) are shown on the graph, where 𝐴(𝑥) = 𝑥2, 𝐷(𝑥) = − 𝑥2, and 𝐿(𝑥) = −4𝑥2. The table of values for the functions is also shown. y

6 5

A(x)

4 2 1

x 1

D(x)

-4

−2

−1

-2 -3

𝒚

−3

-5 -4 -3 -2 -1 0 -1

𝒙

-5

L(x)

-6 -7

1 1

𝒙

−3

𝒚

−9

4 9

𝒚 = 𝑳(𝒙)

𝒙

−3

−2

−4

−2

−1

−9

ip t

𝒚 = 𝑫(𝒙)

𝒚

−36 −16

op y

𝒚 = 𝑨(𝒙)

−4

−1 1

2 3

−4 0

−4

−16 −36

a. Discuss with your partner the effect that the function’s coefficient has on the 𝑦-values. b. Complete the statements.

The coefficient of 𝐷(𝑥) = −𝑥2 is _____. The 𝑦-values of 𝐷(𝑥) can be found by multiplying the 𝑦-values of 𝐴(𝑥) by _____ for each corresponding 𝑥-value.

The coefficient of 𝐿(𝑥) = −4𝑥2 is _____. The 𝑦-values of 𝐿(𝑥) can be found by

multiplying the 𝑦-values of 𝐴(𝑥) by _____ for each corresponding 𝑥-value. Both

𝐷(𝑥) and 𝐿(𝑥) are ___________________________ of 𝐴(𝑥) across the 𝑥-axis.

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4. The functions 𝑓(𝑥), 𝑔(𝑥), and ℎ(𝑥) are shown on the graph, where 𝑓(𝑥) = |𝑥|, 𝑔(𝑥) = 3|𝑥|, and ℎ(𝑥) = 0.75|𝑥|. The table of values for the functions is also shown.

−1 0

2 3

2 1

0 1

2 3

𝒚

−3

−2

−3

−1

−2

−1

2 1

𝒙

6 3

𝒚

9 4

3 2 3 4

8 7 6

f(x)

4 3 2

h(x)

-6 -5 -4 -3 -2 -1 0 -1

3 2

g(x)

3 4

op y

−2

𝒙

−3

𝒚

𝒚 = 𝒉(𝒙)

x 5

-2 -3 -4

ip t

𝒙

𝒚 = 𝒈(𝒙)

9 4

𝒚 = 𝒇(𝒙)

Values that Changed from 𝒇(𝒙) =|𝒙|

Function

Discuss with your partner the effect that the function’s coefficient has on the table of values and the graph. Then, complete the table. Description of the Change

Effect on Graph of 𝒇(𝒙) = |𝒙|

𝑥-values 𝑦-values

Divided by _____ Multiplied by _____

Vertical stretch Vertical compression

ℎ(𝑥) = 0.75|𝑥|

𝑥-values 𝑦-values

Divided by _____ Multiplied by _____

Vertical stretch Vertical compression

𝑔(𝑥) = 3|𝑥|

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Guided Activity: The Effect of Transformations of Factors 1. Complete the table by determining which values change in the table of values of 𝑓(𝑥), and then describe the effect of the factor on the table of values and the graph. Function Notation

𝒌-value

Which values change in the table of values?

𝑘 < − 1

𝑥-values 𝑦-values

𝑘𝑓(𝑥)

𝑘 = −1

𝑥-values 𝑦-values

𝑘𝑓(𝑥)

−1 < 𝑘 < 0

𝑘𝑓(𝑥)

0 < 𝑘 < 1

ip t

op y

𝑘𝑓(𝑥)

𝑥-values 𝑦-values

𝑘 > 1

𝑥-values 𝑦-values 𝑥-values 𝑦-values

𝑘𝑓(𝑥)

Effect of the Factor

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Lesson Summary Recall that functions can undergo transformations like geometric figures do. For a given function, 𝑓(𝑥), the function 𝑦 = 𝑘𝑓(𝑥) will be a dilation on the 𝑦-values, resulting in a vertical stretch or compression by a factor of |𝑘|. • To visualize a vertical stretch, imagine pulling the ends of the function vertically to stretch it out. As a result, the graph looks narrower than the original function.

op y

• Similarly, to visualize a vertical compression, imagine compressing or squeezing the function from top to bottom. As a result, the graph appears wider than the original function because of vertical compression.

• If 𝑘 < 0, the function will be reflected across the 𝑥-axis.

𝒙

𝒚

−2

−1

−3

−27

−1

−2 0

2 3

1 4 9

1 2 3

𝒚

𝒙

−12

−3

𝒚 = 𝒈(𝒙)

𝒚 = 𝒇(𝒙)

ip t

The graphs and tables of values for 𝑓(𝑥) = 𝑥2 and 𝑔(𝑥) = −3𝑥2 are shown.

−1

−12 −27

The table of values for 𝑔(𝑥) is found by multiplying the 𝑦-values of 𝑓(𝑥) by −3 for the corresponding 𝑥-values. This is a dilation on the 𝑦-values. The graph of 𝑔(𝑥) is a vertical stretch of the graph of 𝑓(𝑥) by a factor of 3. Since 𝑘 < 0, the graph of 𝑓(𝑥) is also reflected across the 𝑥-axis. © Accelerate Learning Inc. - All Rights Reserved

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Practice Problems 1. The table of values of the function 𝑦 = 𝑊(𝑥) is shown. 𝒙 𝑾

−9 −12

−6 −10

−3 −8

0 −6

3 −4

6 −2

op y

Complete the sentence to describe the effect on the 𝑥-values for the function 𝑦 = 4𝑊(𝑥). divided multiplied

The 𝑦-values for 𝑊(𝑥) will be

9 0

by 4 for the corresponding 𝑥-values.

2. The functions 𝑓(𝑥) and 𝑔(𝑥) are shown on the graph, where 𝑔(𝑥) = − 𝑓(𝑥). y

ip t

10 8

f(x)

-6

-4

-2

-12 -10 -8

-2

-4 -6 -8

g(x)

-10

-12

Complete the sentence to describe the effect on the 𝑦-values for the function 𝑦 = 𝑓(𝑥). The 𝑦-values for 𝑓(𝑥) will be

divided multiplied

616 | Unit 11

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by −1.

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3. The table of values for 𝑦 = 𝐵(𝑥) is shown. 𝒙 𝑩

−6 22

−4 6

−2 −2

0 −2

2 6

Complete the table of values for 𝑅(𝑥) = 1 𝐵(𝑥).

4 22

6 46

op y

𝒙 𝑹

Review Problems

4. Mai is solving the equation 𝑥2 + 12𝑥 = 13. She writes: 𝑥2 + 12x = 13

ip t

(𝑥 + 6)2 = 49

𝑥 = 1

𝑥 = −13

Jada looks at Mai’s work and is confused. She doesn’t see how Mai got her answer.

Complete Mai’s missing steps to help Jada see how Mai solved the equation.

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5. Select all expressions that are perfect squares.

□ 9𝑥 + 24𝑥 + 16 2

□ 2𝑥 + 20𝑥 + 100 2

□ (7 − 3𝑥)

□ (5𝑥 + 4)(5𝑥 − 4)

op y

□ (1 − 2𝑥)(−2𝑥 + 1) □ 4𝑥 + 6𝑥 + 94

ip t

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Unit 11, Lesson 6: Types of Transformations

Warm-Up: Transformations

Direction(s) of Transformation

Function

op y

Four transformations of the function 𝑦 = 𝑁(𝑥) are shown in the table. Complete the table by selecting the direction(s) and type(s) of transformation. Type(s) of Transformation Stretch

Horizontal

Reflection

Translation

Vertical

𝑦 = − 2𝑁(𝑥)

Compression

Reflection

Horizontal

Translation

Vertical

𝑦 = 𝑁(𝑥 − 3)

ip t

Stretch

𝑦 = 𝑁(𝑥 + 1) + 4

𝑦 = − 2 𝑁(𝑥) 5

Compression Stretch

Vertical

Reflection

Horizontal

Translation Compression Stretch

Vertical

Reflection

Horizontal

Translation Compression

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Guided Activity: Describing Transformations 1. The functions 𝑓(𝑥) and 𝑔(𝑥) are shown on the graph.

20 15

Work with your partner to complete the table to describe 𝑔(𝑥).

𝑓(𝑥 − 𝑘)

-25

-20

-15

-10

-5

x 15

-5

−6

-10

-15 -20

ip t

𝑓(𝑥) + 𝑘

g(x)

𝑘𝑓(𝑥)

𝒌-value

f(x)

op y

Type of Transformation

2. The table of values for 𝐵(𝑥) and 𝑅(𝑥) is shown, where 𝑅(𝑥) is a transformation of 𝐵(𝑥).

Describe how the graph of 𝑦 = 𝐵(𝑥) was transformed to create 𝑦 = 𝑅(𝑥). Include the 𝑘-value in your description.

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𝒚 = 𝑩(𝒙)

𝒙

𝒚

𝒚 = 𝑹(𝒙)

𝒙

𝒚

−2

−15

−4

−15

−7

−2

−7

−1 1 2

−11 −3

−3

−1

−11 −3 1

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3. The graph of the function 𝑊(𝑥) is shown.

a. Write the function, 𝑊(𝑥), that represents the line.

4 2 -4

-2

b. Draw a line that is parallel to 𝑊(𝑥) with a 𝑦-intercept of 2. Label this line 𝑀(𝑥).

x 2

-2 -4 -6

c. Describe how 𝑊(𝑥) could be shifted to coincide with 𝑀(𝑥).

op y

-8

-10

d. Write the function 𝑀(𝑥) in terms of 𝑊(𝑥).

ip t

4. The functions 𝑐(𝑥) and 𝑟(𝑥) are shown on the graph. a. Write each function in vertex form.

𝒓(𝒙)

𝒄(𝒙)

b. Describe how 𝑐(𝑥) was transformed to create 𝑟(𝑥).

16 14 12

c(x)

10 8

r(x)

6 4 2 -12 -10 -8 -6 -4 -2 0 -2 2

x 2

8 10 12

-4 -6

c. Discuss with your partner how a transformation of a quadratic function can be determined using the vertex form of the function. Summarize your discussion.

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Collaborative Activity: Describing Transformations Work with your partner to complete the following.

1. The functions 𝑇(𝑥) and 𝑀(𝑥) are shown on the graph, where 𝑀(𝑥) is a transformation of 𝑇(𝑥).

12 10

Write the transformation by completing the equation.

op y

6 4

-2

-6

ip t

-8

2. The table of values for 𝐵(𝑥) and 𝑅(𝑥) is shown, where 𝑅(𝑥) is a transformation of 𝐵(𝑥). Write a description of the transformation.

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T(x)

M(x)

𝑀(𝑥) = 𝑇(𝑥) + __________

622 | Unit 11

-4

-8 -10

𝒙

−1

−5

𝒚 = 𝑹(𝒙)

𝒚

-6

9 7

-4

𝒚 = 𝑩(𝒙)

𝒙

𝒚

−3

−7

8 8

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Lesson Summary When a given function, 𝑓(𝑥), is transformed to create a new function, 𝑔(𝑥), the type of transformation applied can be determined from the tables of values or the graphs. The table summarizes the types of transformations and their effects on the table of values and the graph of a function compared to the given function.

𝑓(𝑥) + 𝑘

op y

Effect on Graph

For 𝑘 > 0, |𝑘| is added For 𝑘 > 0, the graph is to the 𝑥-values. translated |𝑘| units to the right. For 𝑘 < 0, |𝑘| is subtracted from the For 𝑘 < 0, the graph is 𝑥-values. translated |𝑘| units to the left. For 𝑘 < 0, |𝑘| is subtracted from to the 𝑦-values.

For 𝑘 < 0, the graph is translated |𝑘| units down.

Vertical translation

𝑓(𝑥 − 𝑘)

Effect on Table of Values

Horizontal translation

Function Notation

ip t

Type of Transformation

us − 𝑓(𝑥)

Reflection

Dilation of 𝑦-coordinates

For 𝑘 > 0, the graph is For 𝑘 > 0, |𝑘| is added translated |𝑘| units up. to the 𝑦-values.

𝑘𝑓(𝑥)

The 𝑦-values are multiplied by −1.

The 𝑦-values are multiplied by 𝑘.

9798888261170_Math Nation_Illustrative Math_Algebra 1 Volume 2_Version2_652 pages.indb 623

The graph is reflected across the 𝑥-axis.

For 𝑘 < − 1 or 𝑘 > 1, the graph is stretched vertically by a factor of |𝑘|. For −1 < 𝑘 < 1, the graph is compressed vertically by a factor of |𝑘|.

Unit 11 | 623

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Practice Problems 1. The table of values for 𝑓(𝑥) and 𝑔(𝑥) is shown, where 𝑔(𝑥) is a transformation of 𝑓(𝑥). Write a description of the transformation.

𝒚 = 𝒇(𝒙)

𝒙

−2

𝒚

𝒙

−12

−2

−6

𝒚

−1 5

op y

𝒚 = 𝒈(𝒙)

2.5 4

−1

x 0

Neil said that 𝑚(𝑥) is a vertical translation of 𝑏(𝑥). Explain whether Neil is correct.

−12

−6

ip t

2. The graphs of the functions 𝑏(𝑥) and 𝑚(𝑥) are shown, where 𝑚(𝑥) is a transformation of 𝑏(𝑥).

2.5

−3

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3. The table of values for 𝑟(𝑥) and 𝑝(𝑥) is shown, where 𝑝(𝑥) is a transformation of 𝑟(𝑥). Write a description of the transformation.

𝒚 = 𝒓(𝒙)

𝒙

𝒚 = 𝒑(𝒙)

𝒚

𝒙

−2.5

13.5

1.5

13.5

−4

op y

−1

𝒚

Review Problems

ip t

4. Technology required. Two rocks are launched straight up in the air. This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

• The height of Rock A is given by the function 𝑓, where 𝑓(𝑡) = 4 + 30𝑡 − 16𝑡2.

• The height of Rock B is given by function 𝑔, where 𝑔(𝑡) = 5 + 20𝑡 − 16𝑡2.

In both functions, 𝑡 is time measured in seconds and height is measured in feet. Use graphing technology to graph both equations.

a. What is the maximum height of each rock?

b. Which rock reaches its maximum height first? Explain how you know.

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5. The graph shows the number of grams of a radioactive substance in a sample at different times after the sample was first analyzed.

op y

a. What is the average rate of change for the substance during the 10 year period?

ip t

b. Is the average rate of change a good measure for the change in the radioactive substance during these 10 years? Explain how you know.

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Unit 11, Lesson 7: Writing Equations of Transformed Functions

Warm-Up: Identify the Range 1. Identify the range of the function 𝑦 =−3𝑥 + 6 when the domain is {−1, 2, 4}.

op y

A. {−6, 0, 9} B. {−9}

C. {3, −12, −18}

D. {−3, 12}

ip t

E. {−6, 9}

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Exploration Activity: Describing Transformations with an Equation 1. The functions 𝐵(𝑥) and 𝐻(𝑥) are shown on the graph. The table of values for 𝐵(𝑥) and 𝐻(𝑥) is shown, where 𝐻(𝑥) is a transformation of 𝐵(𝑥). 10

8 7 6 5 3 1

-4

-5

-3

𝒙

𝒚

𝒙

𝒚

−1

−9

−4

−1

ip t

-2

𝒚 = 𝑯(𝒙)

-5 -4 -3 -2 -1 0 -1

𝒚 = 𝑩(𝒙)

op y

-6

-7

-8 -9

-10

628 | Unit 11

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a. Work with your partner to complete the table to describe 𝐻(𝑥). Type of Transformation 𝑘𝐵(𝑥)

𝒌-value

𝐵(𝑥 − 𝑘)

op y

𝐵(𝑥) + 𝑘

b. Describe the effect of the transformation of 𝐵(𝑥) on the graph of 𝐻(𝑥).

ip t

𝐻(𝑥) = ______________

c. Complete the equation that describes the function 𝐻(𝑥) in terms of 𝐵(𝑥).

Collaborative Activity: Describing Transformations

1. The functions 𝑀(𝑥) and 𝑇(𝑥) are shown on the graph, where 𝑀(𝑥) is a transformation of 𝑇(𝑥).

T(x)

Write an equation to describe the function 𝑀(𝑥) in terms of 𝑇(𝑥).

3 2 1 -6 -5

-4 -3 -2 -1 0 -1

x 1

-2 -3 -4 -5

M(x)

-6 -7 -8

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2. The table of values for 𝐵(𝑥) and 𝑅(𝑥) is shown, where 𝑅(𝑥) is a transformation of 𝐵(𝑥).

𝒚 = 𝑩(𝒙)

𝒙

a. Write a description of the transformation.

b. Complete the table to describe 𝑅(𝑥).

𝑘𝐵(𝑥)

−1

−5

𝒌-value

𝒚

−3

−7

𝐵(𝑥 − 𝑘)

𝒙

op y

Type of Transformation

𝒚

9 7

𝒚 = 𝑹(𝒙)

𝐵(𝑥) + 𝑘

ip t

c. Write an equation to describe the function 𝑅(𝑥) in terms of 𝐵(𝑥).

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Guided Activity: Applying Multiple Transformations 1. Given a graph and table of values of an original function and its transformation, identify the 2 transformations from the original graph to the new graph. Then, write an equation that describes the transformation of the original function. The graph of the original function, 𝑓(𝑥), is black, and the graph of the new function, 𝑔(𝑥), is pink. a.

Tables

Original Function

New Function

𝒙

𝒚

−6

−7

ip t

−2

−5

4 3 2 (−5, 1)

𝒚

op y

Graph

-7 -6 -5 -4 -3 -2 -1 0 (0, 10) 2 -1

−1 1 2

−4

−3

1 2 5

Equation

Transformations

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b. Graph

10 8

𝒙

𝒚

−4

x 0 (0, 0)

−5

−3

−2 −1

8 0 4 8

Equation

Transformations

ip t

(−3, 0)

𝒚

-2

𝒙

−1

-4

New Function

−2

-6

Original Function

op y

Tables

2. Discuss the following questions with your partner. Summarize your discussions.

a. How can the graph be used to determine the types of transformations?

b. How do you determine from the graph of a new function that the 𝑦-values of the original function were multiplied by a factor?

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Lesson Summary Parent functions can undergo multiple transformations, which can be described by an equation. To write an equation that represents a transformed function, determine the transformation(s) applied to the parent function by analyzing the graph or table of values that represents the transformation. For example, the parent function 𝑓(𝑥) has had 2 transformations applied, resulting in 𝑔(𝑥), as shown by the graph and table of values. y

op y

𝒚 = 𝒇(𝒙)

9 8

𝒙

g(x)

−2

6 5

4 3 2 1

-8 -7 -6 -5 -4 -3 -2 -1 0 -1

ip t

f(x)

𝒚

𝒚 = 𝒈(𝒙)

𝒙

𝒚

−4

−5

−3

−2

−1

2 4

Both the graph and the 𝑥-values in the table show that the function 𝑔(𝑥) has shifted left 3 units from 𝑓(𝑥), which can be represented by 𝑓|𝑥 + 3| or 𝑦 = |𝑥 + 3|. Also, the representations show that the 𝑦-values have been dilated by a factor of 2, which can be represented by 2𝑓|𝑥| or 𝑦 = 2|𝑥|. Therefore, 𝑔(𝑥) can be described by either the equation 𝑔(𝑥) = 2𝑓|𝑥 + 3| or 𝑦 = 2|𝑥 + 3|.

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Practice Problems 1. The table of values for 𝑓(𝑥) and 𝑔(𝑥) is shown, where 𝑔(𝑥) is a transformation of 𝑓(𝑥).

𝒚 = 𝒇(𝒙)

𝒙

Write a description of the transformation, including an equation.

𝒚

−2

−1

2.5

−3

2.5

−12

op y

−6

−1

8 7

ip t

-6 -5

p(x)

n(x)

4 3 2 1

-4 -3 -2 -1 0 -1

x 1

-2 -3

𝒙

−12

Write a description of the transformation, including an equation.

𝒚

−2 1

2. The graphs of functions 𝑛(𝑥) and 𝑝(𝑥) are shown, where 𝑝(𝑥) is a transformation of 𝑛(𝑥).

𝒚 = 𝒈(𝒙)

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3. Given the graph and table of values of an original function and its transformation, identify the 2 transformations from the original graph, 𝑓(𝑥), to the new graph, 𝑔(𝑥). Then, write an equation to describe the transformation. Graph 7

Tables

6 5

𝒙

−2

−1

(0, 0) -3

-2

-1

−1 0 1

𝒚 3

3 2

Equation

Transformations

−2

ip t

-4

𝒙

𝒚

2 (0, 1) 1

New Function

op y

Original Function

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Review Problem 4. The function 𝑓 is defined by 𝑓(𝑥) = (𝑥 + 1)(𝑥 + 6). a. What are the 𝑥-intercepts of the graph of 𝑓?

op y

b. Find the coordinates of the vertex of the graph of 𝑓. Show your reasoning.

ip t

c. Sketch a graph of 𝑓.

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Unit 11, Lesson 8: Comparing Key Features of Functions

Warm-Up: From Least to Greatest

•

210

•

102 92

•

op y

List these quantities in order, from least to greatest, without evaluating each expression. Be prepared to explain your reasoning.

ip t

Exploration Activity: Which One Grows Faster?

• In Pattern A, the length and width of the rectangle grow by one small square from each step to the next. • In Pattern B, the number of small squares doubles from each step to the next.

Pattern A

• In each pattern, the number of small squares is a function of the step number, 𝑛.

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1. Write an equation to represent the number of small squares at Step 𝑛 in Pattern A. 2. Is the function linear, quadratic, or exponential? Step Number

Number of Small Squares

3. Complete the table.

op y

1 2

Pattern B

3 4

ip t

4. Write an equation to represent the number of small squares at Step 𝑛 in Pattern B.

5. Is the function linear, quadratic, or exponential?

6. Complete the table.

7. How would the two patterns compare if they continue to grow? Make 1–2 observations.

6 7 8

Step Number

Number of Small Squares

0 1 2 3 4 5 6 7 8

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Collaborative Activity: Comparing Key Features of Functions Work with your partner to complete the following. y

1. Two functions are shown on the coordinate plane.

op y

a. Which function has the larger 𝑥-intercept? b. Which function has the smaller 𝑦-intercept?

ip t

c. Complete the statements.

Function D is Function H is

Both functions are

positive over the interval (−∞, −2).

Neither function is

Function D is

Function H is

Both functions are

negative over the interval (−2, ∞).

Neither function is

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d. Which function’s right end behavior could be described by the phrase “as 𝑥→∞, 𝑓(𝑥)→ − ∞”?

2. The table shows 3 different functions. Function B

op y

e. Michelle said that the function D will eventually exceed function H as the values of 𝑥 increase because it is an exponential function. Explain the error in Michelle’s reasoning.

Function R

𝑦 + 2 = 5(𝑥 + 1)

Function L

9 8

ip t

7 6

𝒙

𝒚

−1

−3

2 1

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1

x 1

-2 -3 -4 -5

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a. Determine the type of function that each could represent. Function B

Function L

Function R

Linear

Quadratic

Absolute value

op y

Linear

b. Which key features are the same for all 3 functions?

□ Domain □ 𝑥-intercepts

□ 𝑦-intercepts

□ Right end behavior

ip t

□ Range

□ Left end behavior

□ Intervals where the function is positive

□ Intervals where the function is negative □ Intervals where the function is increasing

□ Intervals where the function is decreasing

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c. Which function(s) include(s) all real numbers in the range?

d. Complete the statements. B. The 𝑥-intercept of Function

is greater than

B. of Function

the 𝑥-intercepts

op y

is less than is in between

R. is less than is equal to

is greater than

ip t

The 𝑦-intercept of Function L

of Function R.

the 𝑦-intercept

Lesson Summary

A key characteristic of a function with exponential growth is that it will always overtake the growth of linear, absolute value, and quadratic functions. All these function types can have right end behavior that as 𝑥→∞, 𝑓(𝑥)→∞. Yet, the values of 𝑓(𝑥) for the exponential function will eventually surpass those of the other function types.

For example, as 𝑛 increases by 1, the exponential expression 2𝑛 always increases by a factor of 2. Consider these comparisons of exponential growth to quadratic and linear growth. • The quadratic expression 3𝑛2 increases by different factors, depending on 𝑛, but these factors get smaller. For example, when 𝑛 increases from 2 to 3, the factor is 27 12 or 2.25. When 𝑛 increases from 6 to 7, the factor is 147 or about 1.36. As 𝑛 increases 108

to larger and larger values, 3𝑛2 grows by a factor that gets closer and closer to 1.

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• Functions that grow linearly increase by a constant rate, by adding the same amount for every equal-sized increase in 𝑛. Increase by a factor (multiplication) will always overtake increase by addition.

op y

Along with right end behavior, the additional key features of functions learned throughout this course include domain/range, intercepts, left end behavior, positive/ negative intervals, and increasing/decreasing intervals. Any or all key features can be used to compare functions represented in different ways. Use the given representation(s) to translate to other representations of functions as needed to make the comparisons required to solve problems.

Practice Problems

1. Function 𝑓 is defined by 𝑓(𝑥) = 1.5𝑥. Function 𝑔 is defined by 𝑔(𝑥) = 500𝑛2 + 345𝑥.

ip t

a. Which function is quadratic? Which one is exponential?

b. The values of which function will eventually be greater for larger and larger values of 𝑥?

2. The table shows 2 functions.

Function A

Function B

a. Which function has the smallest 𝑦-intercept?

Function A

Function B

𝑦 = 6(0.75)𝑥

b. As 𝑥→∞, which function has larger values of 𝑦? Function A

Function B

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c. Compare the ranges of the functions.

d. On the interval where Function A is negative, what type of values does Function B have?

Review Problems

ip t

4. Bobby incorrectly graphed the function 𝑓(𝑥) = −1 |𝑥 + 5| −3. His graph is shown.

op y

3. Create a table of values to show that the exponential expression 3(2)𝑥 eventually overtakes the quadratic expression 3𝑥2 + 2𝑥.

Explain the error(s) Bobby made.

5. A bicycle costs $240 and it loses 3 of its value 5 each year.

a. Write expressions for the value of the bicycle, in dollars, after 1, 2, and 3 years.

b. When will the bike be worth less than $1? c. Will the value of the bike ever be 0? Explain your reasoning.

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