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VOLUME 1

ALGEBRA 1 SOUTH CAROLINA

ISBN: 979-8-89353-909-7 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). 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 1: Linear Equations and Inequalities...................................... 1 Lesson 1: Planning a Pizza Party............................................. 3 Lesson 2: Writing Equations to Model Relationships..................11 Lesson 3: Equations and Their Graphs....................................21 Lesson 4: Equivalent Expressions and Equations......................31 Lesson 5: Explaining Steps for Rewriting Equations..................43 Lesson 6: Solving Multistep One-Variable Equations..................51 Lesson 7: Solving for a Specified Variable...............................59 Lesson 8: Solving Equations and Rearranging Formulas.............69 Lesson 9: Rearranging Equations of Lines................................79 Lesson 10: Connecting Equations to Graphs – Part 1................89 Lesson 11: Connecting Equations to Graphs – Part 2.............. 101 Lesson 12: Graphing Two-Variable Linear Inequalities – Part 1.... 109 Lesson 13: Graphing Two-Variable Linear Inequalities – Part 2.... 121 Lesson 14: Solving Problems with Two-Variable Inequalities....... 133 Unit 2: Two-Variable Statistics.................................................. 149 Lesson 1: Summarizing Data in Two-Way Tables.................... 151 Lesson 2: Interpreting Joint and Marginal Frequencies............ 161 Lesson 3: Exploring Possible Associations in Categorical ......... 169 Data Using Various Displays Lesson 4: Relative Frequency Tables..................................... 177

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Lesson 5: Digging Deeper into Associations in Categorical Data.............................................. 189 Lesson 6: Interpreting Relative Frequency Tables................... 199 Lesson 7: Linear Models for Bivariate Quantitative Data.......... 209 Lesson 8: Fitting Lines to Data............................................. 221 Lesson 9: Introducing Correlation Coefficient......................... 231 Lesson 10: Using the Correlation Coefficient.......................... 237 Lesson 11: Interpreting Linear Models.................................. 245 Unit 3: Functions....................................................................... 253 Lesson 1: Describing and Graphing Situations........................ 255 Lesson 2: Function Notation................................................ 263 Lesson 3: Interpreting and Using Function Notation................ 271 Lesson 4: Using Function Notation to Describe Rules............... 279 Lesson 5: Features of Graphs.............................................. 289 Lesson 6: Using Graphs to Find Average Rate of Change......... 299 Lesson 7: Interpreting and Creating Graphs.......................... 307 Lesson 8: Comparing Graphs............................................... 319 Lesson 9: Exploring Domain and Range................................ 327 Lesson 10: Identifying Domain and Range from Different Representations................................................. 335 Lesson 11: Introducing Set and Interval Notation................... 343 Lesson 12: Digging Deeper into Domain and Range................ 353 Unit 4: Systems of Equations..................................................... 361 Lesson 1: Introduction to Systems of Linear Equations............ 363 Lesson 2: Solving Systems by Graphing................................ 373 Lesson 3: Solving Systems by Substitution............................ 381 Lesson 4: Solving Systems by Elimination – Part 1................. 391

iv | Table of Contents

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Lesson 5: Solving Systems by Elimination – Part 2................. 401 Lesson 6: Solving Systems Algebraically............................... 411 Lesson 7: Systems of Linear Equations and Their Solutions...... 421 Lesson 8: Solving Equations in the Form f(x) = g(x)............... 431 Unit 5: Linear and Absolute Value Functions............................. 441 Lesson 1: Identifying Key Features of Lines from Different Representations.................................................. 443 Lesson 2: Relating Key Features of Lines to Real-World Contexts............................................................ 451 Lesson 3: Graphing and Interpreting Key Features of Lines...... 461 Lesson 4: Solving Problems Modeled with Linear Functions...... 469 Lesson 5: Introducing Absolute Value Functions..................... 477 Lesson 6: Exploring Absolute Value Functions........................ 483 Lesson 7: Solving One-Variable Absolute Value Equations........ 493 Lesson 8: Graphing Absolute Value Functions........................ 503 Lesson 9: Interpreting Key Features of Absolute Value Functions................................................... 513 Lesson 10: Digging Deeper into Absolute Value Functions........ 523 Lesson 11: Comparing Key Features of Functions................... 535

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vi | Table of Contents

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Unit 1: One-Variable Statistics

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Unit 1, Lesson 1: Planning a Pizza Party

Warm-Up: A Main Dish and Some Side Dishes Here are some letters and what they represent. All costs are in dollars. •

𝑚 represents the cost of a main dish.

•

𝑠 represents the cost of a side dish.

• •

𝑛 represents the number of side dishes. 𝑡 represents the total cost of a meal.

1. Discuss with a partner: What does each equation mean in this situation? a. 𝑚 = 7.50 b. 𝑚 = 𝑠 + 4.50 c. 𝑛𝑠 = 6 d. 𝑚 + 𝑛𝑠 = 𝑡 2. Write a new equation that could be true in this situation.

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Exploration Activity: How Much Will It Cost? Imagine your class is having a pizza party. Work with your group to plan what to order and to estimate what the party would cost.

1. Record your group’s plan and cost estimate. What would it take to convince the class to go with your group’s plan? Be prepared to explain your reasoning.

2. Write down one or more expressions that show how your group’s cost estimate was calculated.

3. a. In your expression(s), are there quantities that might change on the day of the party? Which ones?

b. Rewrite your expression(s), replacing the quantities that might change with letters. Be sure to specify what the letters represent.

4 | Unit 1

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Collaborative Activity: What Are Constraints? A constraint is something that limits what is possible or reasonable in a situation. For example, one constraint in a pizza party might be the number of slices of pizza each person could have, 𝑠. We can write 𝑠 < 4 to say that each person gets fewer than 4 slices. 1. Look at the expressions you wrote when planning the pizza party earlier. a. Choose an expression that uses one or more letters.

b. For each letter, determine what values would be reasonable. (For instance, could the value be a non-whole number? A number greater than 50? A negative number? Exactly 2?)

2. Write equations or inequalities that represent some constraints in your pizza party plan. If a quantity must be an exact value, use the = symbol. If it must be greater or less than a certain value to be reasonable, use the < or > symbol.

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Lesson Summary Expressions, equations, and inequalities are mathematical models. A model is a mathematical or statistical representation of a problem from science, technology, engineering, work, or everyday life, used to solve problems and make decisions. Often, models have constraints. A constraint is a limitation on the possible values of variables in a model, often expressed by an equation or inequality or by specifying that the value must be an integer. For example, some possible quantities and corresponding constraints for planning a birthday party are listed. Quantities • The number of guests

Constraints • 20 people maximum

• The cost of food and drinks • $5.50 per person • The cost of birthday cake

• The cost of entertainment • The total cost

• $40 for a large cake

• $15 for music and $27 for games • No more than $180 total cost

In algebra, variables can be used to represent these quantities. For example, while 42 can represent the total cost of entertainment, in dollars, the variable n can represent the number of people, and the variable C can represent the total cost of the party. Expressions can also be written using these numbers and variables. For instance, the expression 5.50𝑛 is a concise way to express the overall cost of food if it costs $5.50 per guest and there are 𝑛 guests.

Sometimes, a constraint is an exact value. For instance, the cost of music is $15. Other times, a constraint is a boundary or a limit. For instance, the total cost must be

6 | Unit 1

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no more than $180. Symbols such as <, >, ≤, ≥, and = can be used to express the same constraints from earlier. Equations can be used to show the relationships between different quantities and constraints. For example, the total cost of the party is the sum of the costs of food, cake, and entertainment. Two equations that can be used to represent this relationship are shown.

Quantities • The number of guests • The cost of food and drinks • The cost of birthday cake • The cost of entertainment • The total cost

Constraints • 𝑛 ≤ 20 • 5.50𝑛 • 40

• 15 + 27

• 𝐶 ≤ 180

𝐶 = 5.50𝑛 + 40 + 15 + 27 or 𝐶 = 5.50𝑛 + 82

Deciding how to use numbers and variables to represent quantities, relationships, and constraints is an important part of mathematical modeling. A model such as 𝐶 = 5.50𝑛 + 82 can be an efficient way to make estimates or predictions. When a quantity or a constraint changes, or when an unknown value needs to be determined, the model can be adjusted to make a new calculation instead of repeating a series of calculations.

Practice Problems 1. The videography team entered a contest and won a monetary prize of $1,350.

Which expression represents how much each person would get if there were 𝑥 people on the team? A.

1350

𝑥

B. 1350 + 𝑥 C. 1350 5

D. 1350 − 𝑥

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2. To support a local senior citizens center, a student club sent a flyer home to the students in the school. The flyer said, “Please bring in money to support the senior citizens center. Paper money and coins accepted!” Their goal is to raise 𝑇 dollars. Match each quantity to an expression that describes it.

Description

Expression

a. the dollar amount the club would have if they reached half of their goal b. the dollar amount the club would have if every student at the school donated 50 cents to the cause c. the dollar amount the club could donate if they made $50 more than their goal

d. the dollar amount the club would still need to raise to reach its goal after every student at the school donated 50 cents

e. the dollar amount the club would have if half of the students at the school each gave 50 cents

𝑇 + 50 0.5𝑇 0.25𝑛 0.5𝑛 𝑇 − 0.5𝑛

3. Each of the 10 students in the baking club made 2 chocolate cakes for a fundraiser. They all used the same recipe, using 𝐶 cups of flour in total. Write an expression that represents the amount of flour required for one cake.

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4. A student club started a fundraising effort to support animal rescue organizations. The club sent an information flyer home to the students in the school. It says, “We welcome donations of any amount, including any change you could spare!” Their goal is to raise 𝑇 dollars, and to donate to a cat shelter and a dog shelter. Match each quantity to an expression that describes it. Description

Expression

a. the dollar amount the club would have if they reached one-fourth of their goal

3 𝑛 · 1 4 2

b. the dollar amount the club would have if every student at the school donated a quarter to the cause c. the dollar amount the club could donate to the cat shelter if they reached their goal and gave a quarter of the total donation to a dog shelter d. the dollar amount the club would still need to raise to reach its goal after every student at the school donated a quarter e. the dollar amount the club would have if three-fourths of the students at the school each gave 50 cents

1 𝑇 4

𝑇 − 1𝑛 4

3 𝑇 4 1 𝑛 4

5. A softball team is ordering pizza to eat after their tournament. They plan to order cheese pizzas that cost $6 each and four-topping pizzas that cost $10 each. They order 𝑐 cheese pizzas and 𝑓 four-topping pizzas. Which expression represents the total cost of all of the pizzas they order? A. 6 + 10 B. 𝑐 + 𝑓

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Unit 1, Lesson 2: Writing Equations to Model Relationships

Warm-Up: Finding a Relationship Here is a table of values. The two quantities, 𝑥 and 𝑦, are related. 𝒙

𝒚

1 5 7

24 48

What are some strategies you could use to find a relationship between 𝑥 and 𝑦? Brainstorm as many ways as possible.

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Exploration Activity: A Platonic Relationship These three figures are called Platonic solids. Tetrahedron

Cube

Dodecahedron

The table shows the number of vertices, edges, and faces for the tetrahedron and dodecahedron.

Tetrahedron Cube Dodecahedron

Faces

Vertices

Edges

1. Complete the missing values for the cube. Then, make at least two observations about the number of faces, edges, and vertices in a Platonic solid.

2. There are some interesting relationships between the number of faces (𝐹), edges (𝐸), and vertices (𝑉) in all Platonic solids. For example, the number of edges is always greater than the number of faces, or 𝐸 > 𝐹. Another example: The number of edges is always less than the sum of the number of faces and the number of vertices, or 𝐸 < 𝐹 + 𝑉.

There is a relationship that can be expressed with an equation. Can you find it? If so, write an equation to represent it.

12 | Unit 1

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Guided Activity: Car Prices The tax on the sale of a car in Michigan is 6% At a dealership in Ann Arbor, a car purchase also involves $120 in miscellaneous charges. 1. There are several quantities in this situation: the original car price, sales tax, miscellaneous charges, and total price. Write an equation to describe the relationship between all the quantities when: a. The original car price is $9,500. b. The original car price is $14,699. c. The total price is $22,480. d. The original price is 𝑝.

2. How would each equation you wrote change if the tax on car sales is 𝑟% and the miscellaneous charges are 𝑚 dollars?

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Collaborative Activity: What Are the Relationships? 1. The table represents the relationship between the base length and the height of some parallelograms. Both measurements are in inches (in.). What is the relationship between the base length and the height of these parallelograms?

Base length (in.)

Height (in.)

2 4 6

24 12 8

2. Visitors to a carnival are invited to guess the number of beans in a jar. The person who guesses the correct number wins $300. If multiple people guess correctly, the prize will be divided evenly among them. What is the relationship between the number of people who guess correctly and the amount of money each person will receive?

3. A 2 -gallon (gal.) jug of milk can fill 8 cups (c.), while 32 fluid ounces (fl. oz.) of milk can fill 4 c.

What is the relationship between number of gallons and ounces? If you get stuck, try creating a table.

14 | Unit 1

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Lesson Summary Sometimes, the relationship between 2 quantities is easy to see. For instance, the perimeter of a square is always 4 times the side length of the square. If 𝑃 represents the perimeter and 𝑠 represents the side length, then the relationship between the 2 measurements (in the same unit) can be expressed as 𝑃 = 4𝑠, or 𝑠 = 𝑃 . 4

Other times, the relationship between quantities might take a bit of work to figure out, by doing calculations several times or by looking for a pattern. Consider 2 examples. 1. A plane departed from New Orleans and is heading to San Diego. The table shows its distance from New Orleans, 𝑥, in miles (mi.), and its distance from San Diego, 𝑦, in mi., at some points along the way.

Distance from New Orleans (mi.)

Distance from San Diego (mi.)

100

1,500

500

1,100

What is the relationship between the 2 distances? Is there a pattern in how each quantity is changing? Can the missing values be determined? • Notice that every time the distance from New Orleans increases by some number of mi., the distance from San Diego decreases by the same number of mi., and that the sum of the 2 values is always 1,600 mi.

300 _____ 900

1,450 𝑥

1,300 1,020 700

_____ 𝑦

This relationship can be expressed with any of the 3 equations shown. 𝑥 + 𝑦 = 1,600

𝑦 = 1,600 − 𝑥

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𝑥 = 1,600 − 𝑦

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2. A company decides to donate $50,000 to charity. The company will select up to 20 charitable organizations nominated by its employees. Each organization selected will receive an equal donation. What is the relationship between the number of selected organizations, 𝑛, and the dollar amount each of them will receive, 𝑑?

• If 5 organizations are selected, each one receives $10,000. If 10 organizations are selected, each one receives $5,000. If 20 organizations are selected, each one receives $2,500.

• Notice the pattern here. 10,000 is 50,000 , 5,000 is 50,000 , and 2,500 is 50,000 . 5

From the pattern, it can be generalized that the amount each organization receives, 𝑑, is 50,000 divided by the number of organizations selected, 𝑛, so 𝑑 =

50,000 . 𝑛

Practice Problems 3. Jada plans to serve milk and healthy cookies for a book club meeting. She is preparing 12 ounces (oz.) of milk and 4 cookies per person. Including herself, there are 15 people in the club. A package of cookies contains 24 cookies and costs $4.50.

A 1-gal. jug of milk contains 128 oz. and costs $3. Let 𝑛 represent number of people in the club, 𝑚 represent the ounces of milk, 𝑐 represent the number of cookies, and 𝑏 represent Jada’s budget in dollars.

Select all of the equations that could represent the quantities and constraints in this situation.

□ 𝑚 = 12(15)

□ 3𝑚 + 4.5𝑐 = 𝑏 □ 4𝑛 = 𝑐

□ 4(4.50) = 𝑐

□ 𝑏 = 2(3) + 3(4.50) 16 | Unit 1

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4. A student on the track team runs 45 minutes (min.) each day as a part of her training. She begins her workout by running at a constant rate of 8 miles per hour (mph) for � min., then slows to a constant rate of 7.5 mph for 𝑏 min.

Which equation describes the relationship between the distance she runs in mi., 𝐷, and her running speed, in mph? A. 𝑎 + 𝑏 = 45

B. 8𝑎 + 7.5𝑏 = 𝐷 𝑎

𝑏

C. 8 �60 � + 7.5 �60 � = 𝐷

D. 8(45 − 𝑏) + 7.5𝑏 = 𝐷

5. A landscaping company is delivering crushed stone to a construction site. The table shows the total weight in pounds (lb.), 𝑊, of 𝑛 loads of crushed stone.

Number of Loads of Crushed Stone

Total Weight (lb.)

Which equation could represent the total weight, in lb., for 𝑛 loads of crushed stone?

2,000

B. 𝑊 = 6,000 − 2,000𝑛

6,000

A. 𝑊 =

6,000 𝑛

C. 𝑊 = 2,000𝑛

4,000

D. 𝑊 = 𝑛 + 2,000

6. Tyler needs to complete this table for his consumer science class. He knows that 1 tablespoon contains 3 teaspoons and that 1 cup contains 16 tablespoons. a. Complete the missing values in the table. b. Write an equation that represents the number of teaspoons, 𝑡, contained in a cup, 𝐶. © Accelerate Learning Inc. - All Rights Reserved

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Number of Teaspoons

Number of Tablespoons

12 48

Number of Cups 2 3

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7. The volume of dry goods, like apples or peaches, can be measured using bushels, pecks, and quarts (qt.). A bushel contains 4 pecks, and a peck contains 8 qt. What is the relationship between number of bushels, 𝑏, and the number of qt., 𝑞? If you get stuck, try creating a table.

8. Priya is hosting a poetry club meeting this week and plans to have fruit punch and cheese for the meeting. She is preparing 8 oz. of fruit punch per person and 2 oz. of cheese per person. Including herself, there are 12 people in the club. A package of cheese contains 16 oz. and costs $3.99. A 1-gal. jug of fruit punch contains 128 oz. and costs $2.50. Let 𝑝 represent number of people in the club, 𝑓 represent the ounces of fruit punch, 𝑐 represent the ounces of cheese, and 𝑏 represent Priya’s budget in dollars.

Select all of the equations that could represent the quantities and constraints in this situation.

□ 𝑓 = 8 · 12

□ 𝑐 = 2 · 3.99

□ 2 · 3.99 + 2.50 = 𝑏 □ 2𝑝 = 𝑐

□ 8𝑓 + 2𝑐 = 𝑏

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Review Problem 9. In a trivia contest, players form teams and work together to earn as many points as possible for their team. Each team can have between 3 and 5 players. Each player can score up to 10 points in each round of the game. Elena and four of her friends decided to form a team and play a round. Write an expression, an equation, or an inequality for each quantity described here. If you use a variable, specify what it represents. a. the number of points that Elena’s team earns in one round

b. the number of points Elena’s team earns in one round if every player scores between 6 and 8 points c. the number of points Elena’s team earns if each player misses one point

d. the number of players in a game if there are 5 teams of 4 players each e. the number of players in a game if there are at least 3 teams

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Unit 1, Lesson 3: Equations and Their Graphs

Warm-Up: Which One Doesn’t Belong: Hours and Dollars Which one doesn’t belong? A

Collaborative Activity: Calories from Protein and Fat One gram (g) of protein contains 4 calories (cal.). One g of fat contains 9 cal. A snack has 60 cal. from 𝑝 g of protein and 𝑓 g of fat. The equation 4𝑝 + 9𝑓 = 60 represents the relationship between these quantities.

1. Determine if each pair of values could be the number of g of protein and fat in the snack. Be prepared to explain your reasoning. a. 5 g of protein and 2 g of fat

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b. 10.5 g of protein and 2 g of fat

c. 8 g of protein and 4 g of fat

2. If there are 6 g of fat in the snack, how many g of protein are there? Show your reasoning.

3. In this situation, what does a solution to the equation 4𝑝 + 9𝑓 = 60 tell us? Give an example of a solution.

Collaborative Activity: Snacks in Bulk To get snacks for a class trip, Clare went to the “bulk” section of the grocery store, where she could buy any quantity of a product and the prices are usually good. Clare purchased some salted almonds at $6 a pound (lb.) and some dried figs at $9 per lb. She spent $75 before tax.

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1. If she bought 2 lb. of almonds, how many lb. of figs did she buy?

2. If she bought 1 lb. of figs, how many lb. of almonds did she buy?

3. Write an equation that describes the relationship between lb. of figs and lb. of almonds that Clare bought, and the dollar amount that she paid. Be sure to specify what the variables represent.

4. Here is a graph that represents the quantities in this situation. a. Choose any point on the line, state its coordinates, and explain what it tells us.

b. Choose any point that is not on the line, state its coordinates, and explain what it tells us.

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Guided Activity: Graph It! 1. A student has a savings account with $475 in it. She deposits $125 of her paycheck into the account every week. Her goal is to save $7,000 for college. a. How much will be in the account after 3 weeks?

b. How long will it take before she has $1,350?

c. Write an equation that represents the relationship between the dollar amount in her account and the number of weeks since she started depositing $125 each week.

d. Graph your equation using graphing technology. Mark the points on the graph that represent the amount after 3 weeks, and also the week she has $1,350. Write down the coordinates.

e. How long will it take her to reach her goal?

2. A 450-gallon (gal.) tank full of water is draining at a rate of 20 gal. per minute (min.). a. How many gal. will be in the tank after 7 min.?

b. How long will it take for the tank to have 200 gal.?

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c. Write an equation that represents the relationship between the gal. of water in the tank and min. the tank has been draining.

d. Graph your equation using graphing technology. Mark the points on the graph that represent the gallons after 7 min. and the time when the tank has 200 gal. Write down the coordinates.

e. How long will it take until the tank is empty?

Lesson Summary An equation that contains 2 unknown quantities, or 2 quantities that vary, is called an equation in 2 variables. A solution to such an equation is a pair of numbers that makes the equation true. Suppose Tyler spends $45 on T-shirts and socks. Each T-shirt costs $10, and each pair of socks costs $2.50. If 𝑡 represents the number of T-shirts and 𝑝 represents the number of pairs of socks that Tyler buys, then the equation 10𝑡 + 2.50𝑝 = 45 represents the situation. This is an equation in 2 variables. More than one pair of values for 𝑡 and 𝑝 makes the equation true. Three examples are shown. 𝑡 = 3 and 𝑝 = 6 10(3) + 2.50(6) = 45 30 + 15 = 45 45 = 45

𝑡 = 4 and 𝑝 = 2 10(4) + 2.50(2) = 45 40 + 5 = 45 45 = 45

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𝑡 = 2 and 𝑝 = 10 10(2) + 2.50(10) = 45 20 + 25 = 45 45 = 45

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• In this situation, one constraint is that the combined cost of shirts and socks must equal $45. Solutions to the equation are pairs of 𝑡- and 𝑝-values that satisfy this constraint. • Combinations such as 𝑡 = 1 and 𝑝 = 10, or 𝑡 = 2 and 𝑝 = 7, are not solutions because they don’t meet the constraint. When these pairs of values are substituted into the equation, they result in statements that are false. Like an equation, a graph can also give information about the relationship between quantities and the constraints on them. Suppose beans and rice are being purchased to feed a large group of people, and $120 total will be spent on the 2 ingredients. Beans cost $2 per lb., and rice costs $0.50 per lb. If 𝑥 represents lb. of beans, and 𝑦 represents lb. of rice, the equation 2𝑥 + 0.50𝑦 = 120 can represent the constraints in this situation. The graph of 2𝑥 + 0.50𝑦 = 120 shows a straight line. • Each point on the line is a pair of 𝑥- and 𝑦-values that makes the equation true and is therefore a solution. Each is also a pair of values that satisfies the constraints in the situation. • The point (10,200) is on the line. If 10 lb. of beans and 200 lb. of rice are purchased, the cost will be 2(10) + 0.50(200), which equals $120.

• The points (60, 0) and (45, 60) are also on the line. If only beans are purchased, a total of 60 lb. of them, and no rice, the total cost is $120. If 45 lb. of beans and 60 lb. of rice are purchased, the total cost is also $120. Points that are not on the line are not solutions because they don’t satisfy the constraints, but they still have meaning in the situation. • The point (20, 80) is not on the line. Buying 20 lb. of beans and 80 lb. of rice costs 2(20) + 0.50(80) or $80, which does not equal $120. This combination costs less than the amount that was planned for spending. • The point (70, 180) represents buying 70 lb. of beans and 180 lb. of rice. It will cost 2(70) + 0.50(180) or $230, which is over the budget of $120.

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Practice Problems 1. Diego is collecting dimes and nickels in a jar. He has collected $22.25 so far. The relationship between the numbers of dimes and nickels, and the amount of money in dollars is represented by the equation 0.10𝑑 + 0.05𝑛 = 22.25. Select all the values (𝑑, 𝑛) that could be solutions to the equation.

□ (0, 445)

□ (0.50, 435) □ (233, 21) □ (118, 209) □ (172, 101)

2. Volunteer drivers are needed to bring 80 students to the championship baseball game. Drivers either have cars, which can seat 4 students, or vans, which can seat 6 students. The equation 4𝑐 + 6𝑣 = 80 describes the relationship between the number of cars, 𝑐, and number of vans, 𝑣, that can transport exactly 80 students. Select all statements that are true about the situation.

□ If 12 cars go, then 2 vans are needed. □ The pair 𝑐 = 14 and 𝑣 = 4 is a solution to the equation.

□ If 6 cars go and 11 vans go, there will be extra space. □ 10 cars and 8 vans isn’t enough to transport all the students. □ If 20 cars go, no vans are needed. □ 8 vans and 8 cars are numbers that meet the constraints in this situation.

Book 1.indb 27

Unit 1 | 27

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3. Here is a graph of the equation 𝑥 + 3𝑦 = 6.

Select all coordinate pairs that represent a solution to the equation.

□ (0, 2) □ (0, 6) □ (2, 6) □ (3, 1) □ (4, 1) □ (6, 2)

4. A theater is selling tickets to a play. Adult tickets cost $8 each and children’s tickets cost $5 each. They collect $275 after selling 𝑥 adult tickets and 𝑦 children’s tickets. What does the point (30, 7) mean in this situation?

Review Problems 5. The speed of an object can be found by taking the distance it travels and dividing it by the time it takes to travel that distance. An object travels 100 feet (ft.) in 2.5 seconds (sec.). Let the speed, 𝑆, be measured in ft. per sec. Write an equation to represent the relationship between the three quantities (speed, distance, and time).

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6. In the 21st century, people measure length in ft. and meters (m). At various points in history, people measured length in hands, cubits, and paces. There are 9 hands in 2 cubits. There are 5 cubits in 3 paces. a. Write an equation to express the relationship between hands, ℎ, and cubits, 𝑐.

b. Write an equation to express the relationship between hands, ℎ, and paces, 𝑝.

7. The table shows the amount of money, 𝐴, in a savings account after 𝑚 months.

Select all the equations that represent the relationship between the amount of money, 𝐴, and the number of months, 𝑚.

□ 𝐴 = 100𝑚

□ 𝐴 = 100(𝑚 − 5)

□ 𝐴 − 700 = 100𝑚

Number of Months

Dollar Amount

1,200

1,300

1,400

1,500

□ 𝐴 − 1,200 = 100𝑚 □ 𝐴 = 700 + 100𝑚

□ 𝐴 = 1,200 + 100𝑚

□ 𝐴 = 1,200 + 100(𝑚 − 5)

Book 1.indb 29

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Book 1.indb 30

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Unit 1, Lesson 4: Equivalent Expressions and Equations

Warm-Up: Two Expressions Your teacher will assign you one of these expressions: 𝑛2 − 9 2(4 − 3)

(𝑛 + 3) ·

𝑛 −3 8−3 · 2

1. Evaluate your expression when 𝑛 is: a. 5

b. 7

c. 13

d. −1

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Unit 1 | 31

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Exploration Activity: Much Ado about Ages 1. Write as many equations as possible that could represent the relationship between the ages of the two children in each family described. Be prepared to explain what each part of your equation represents. a. In family A, the youngest child is 7 years younger than the oldest, who is 18.

b. In family B, the middle child is 5 years older than the youngest child.

2. Tyler thinks that the relationship between the ages of the children in family B can be described with 2𝑚 − 2𝑦 =10, where 𝑚 is the age of the middle child and 𝑦 is the age of the youngest. Explain why Tyler is right.

3. Are any of these equations equivalent to one another? If so, which ones? Explain your reasoning. 3𝑎 + 6 = 15

32 | Unit 1

Book 1.indb 32

3𝑎 = 9

𝑎+2=5

1𝑎 = 1 3

19/07/24 5:07 AM

Collaborative Activity: What’s Acceptable? Noah is buying a pair of jeans and using a coupon for 10% off. The total price is $56.70, which includes $2.70 in sales tax. Noah’s purchase can be modeled by the equation shown.

1. Discuss with a partner:

𝑥 − 0.1𝑥 + 2.70 = 56.70

a. What does the solution to the equation mean in this situation?

b. How can you verify that 70 is not a solution but is the solution?

2. Here are some equations that are related to 𝑥 − 0.1𝑥 + 2.70 = 56.70. Each equation is a result of performing one or more moves on that original equation. Each can also be interpreted in terms of Noah’s purchase. For each equation, determine either what move was made or how the equation could be interpreted. Then, check if 60 is the solution of the equation. Some examples are given.

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Equation A

100𝑥 − 10𝑥 + 270 = 56.70

What was done?

Interpretation?

The price is expressed in cents instead of dollars.

Same Solution?

Equation B

𝑥 − 0.1𝑥 = 54

What was done?

Subtract 2.70 from both sides of the equation.

Interpretation?

Same Solution?

Equation C

0.9𝑥 + 2.70 = 56.70

What was done?

Interpretation?

10% off means paying 90% of the original price. 90% of the original price plus sales tax is $56.70.

Same Solution?

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3. Here are some other equations. For each equation, determine what move was made or how the equation could be interpreted. Then, check if 60 is the solution to the equation. Equation D

𝑥 − 0.1𝑥 = 56.70

What was done?

Interpretation?

The price after using the coupon for 10% off and before sales tax is $56.70.

Same Solution?

Equation E

𝑥 − 0.1𝑥 = 59.40

What was done?

Subtract 2.70 from the left and add 2.70 to the right.

Interpretation?

Same Solution?

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Equation F

2(𝑥 − 0.1𝑥 +2.70) = 56.70

What was done?

Interpretation?

The price of 2 pairs of jeans, after using the coupon for 10% off and paying sales tax, is $56.70.

Same Solution?

4. Which of the six equations are equivalent to the original equation? Be prepared to explain how you know.

Lesson Summary Suppose 2 packs of markers and a $0.50 glue stick totaled $6.10. If 𝑝 is the dollar cost of 1 pack of markers, the equation 2𝑝 +0.50 = 6.10. represents this purchase. The solution to this equation is 2.80. A friend bought 6 of the same packs of markers and 3 glue sticks and paid $18.30. The equation 6𝑝 + 1.50 = 18.30 represents this purchase. The solution to this equation is also 2.80. The equations 2𝑝 + 0.50 = 6.10 and 6𝑝 + 1.50 = 18.30 are equivalent equations.

Equations that have the exact same solutions are equivalent equations.

Besides 2.80, no other values of 𝑝 make either equation true. Only the price of $2.80 per pack of markers satisfies the constraint in each purchase.

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As explored in previous courses, there are certain moves that can be made to write equivalent equations. In this example, the second equation, 6𝑝+1.50 = 18.30, is a result of multiplying both sides of the first equation by 3. Buying 3 times as many markers and glue sticks means paying 3 times as much money. The unit price of the markers hasn’t changed. Some other equations that are equivalent to 2𝑝 + 0.50 = 6.10 are shown, along with the moves that led to these equations. 2𝑝 + 4 = 9.60 Add 3.50 to each side of the original equation.

2𝑝 = 5.60 Subtract 0.50 from each side of the original equation. 1 (2𝑝 + 0.50) = 3.05 2

2(𝑝 + 0.25) = 6.10

Multiply each side of the original equation by 1 . 2

Apply the distributive property to rewrite the left side.

In each case, the move is acceptable because it doesn’t change the equality of the 2 sides of the equation. If 2𝑝 + 0.50 has the same value as 6.10, then multiplying 2𝑝 + 0.50 by 1 and multiplying 6.10 by 1 keeps the 2 sides equal. 2

These moves (applying the distributive property, adding the same amount to both sides, dividing each side by the same number, and so on) might be familiar because they use properties of equality that are applied when solving equations, which were explored in prior courses. Solving an equation essentially involves writing a series of equivalent equations that eventually isolate the variable to 1 side. The same properties can be applied to generate equivalent equations.

Practice Problems 1. Which equation is equivalent to the equation 6𝑥 + 9 = 12? A. 𝑥 + 9 = 6

B. 2𝑥 + 3 = 4 C. 3𝑥 + 9 = 6

D. 6𝑥 + 12 = 9

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2. Select all the equations that have the same solution as the equation 3𝑥 − 12 = 24.

□ 15𝑥 − 60 = 120 □ 3𝑥 = 12 □ 3𝑥 = 36

□ 𝑥−4=8

□ 12𝑥 − 12 = 24 3. Jada has a coin jar containing 𝑛 nickels and 𝑑 dimes worth a total of $3.65. The equation 0.05𝑛 + 0.1𝑑 = 3.65 is one way to represent this situation. Which equation is equivalent to the equation 0.05𝑛 + 0.1𝑑 = 3.65? A. 5𝑛 + 𝑑 = 365

B. 0.5𝑛 + 𝑑 = 365

C. 5𝑛 + 10𝑑 = 365

D. 0.05𝑑 + 0.1𝑛 = 365

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4. Select all the equations that have the same solution as 2𝑥 − 5 = 15.

□ 2𝑥 = 10 □ 2𝑥 = 20

□ 2(𝑥 − 5) = 15 □ 2𝑥 − 20 = 0

□ 4𝑥 − 10 = 30 □ 15 = 5 − 2𝑥

Review Problems 5. A basketball coach purchases bananas for the players on his team. The table shows total price in dollars ($), P, of 𝑛 bananas. Number Total Price ($) of Bananas Which equation could represent the total price in dollars for 𝑛 bananas? 7 4.13 A. P = 0.59𝑛 8 4.72 B. P = 5.90 − 0.59𝑛 C. P = 5.90 𝑛

5.31

5.90

D. P = 𝑛 +0.59

Book 1.indb 39

Unit 1 | 39

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6. Kiran is collecting dimes and quarters in a jar. He has collected $10.00 so far and has 𝑑 dimes and 𝑞 quarters. The relationship between the numbers of dimes and quarters, and the amount of money in dollars is represented by the equation 0.1𝑑 + 0.25𝑞 = 10. Select all the values (𝑑, 𝑞) that could be solutions to the equation.

□ (100,0) □ (20,50) □ (50,20) □ (0,100) □ (10,36) 7. Here is a graph of the equation 3𝑥 − 2𝑦 = 12.

Select all coordinate pairs that represent a solution to the equation.

□ (2, −3) □ (4, 0)

□ (5, −1) □ (0, −6) □ (2, 3)

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8. Jada bought some sugar and strawberries to make strawberry jam. Sugar costs $1.80 per pound (lb.), and strawberries cost $2.50 per lb. Jada spent a total of $19.40. Which point on the coordinate plane could represent the pounds of sugar and strawberries that Jada used to make jam? A. Point A B. Point B C. Point C D. Point D

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Book 1.indb 42

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Unit 1, Lesson 5: Explaining Steps for Rewriting Equations

Warm-Up: Math Talk: Could It Be Zero? 1. Is 0 a solution to each equation? a. 4(𝑥 + 2) = 10

b. 12 − 8𝑥 = 3(𝑥 + 4)

c. 5𝑥 = 2 𝑥

d. 6𝑥 + 1 = 8

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Collaborative Activity: Explaining Acceptable Moves Here are some pairs of equations. While one partner listens, the other partner should: • Choose a pair of equations from column A. Explain why, if 𝑥 is a number that makes the first equation true, then it also makes the second equation true. • Choose a pair of equations from column B. Explain why the second equation is no longer true for a value of 𝑥 that makes the first equation true. Then, switch roles until you run out of time or you run out of pairs of equations.

16 = 4(9 − 𝑥)

9𝑥 = 5𝑥 + 4

5𝑥 = 24 + 2𝑥

1 2𝑥 − 8 = 9

−3(2𝑥 + 9) = 12

6𝑥 − 6 = 3𝑥

5𝑥 = 3 − 𝑥

−11(𝑥 − 2) = 8

18 = 3𝑥 − 6 + 𝑥

4 − 5𝑥 = 24

16 = 36 − 4𝑥

3𝑥 = 24

2𝑥 + 9 = −4

5𝑥 = −𝑥 + 3

18 = 4𝑥 − 6

44 | Unit 1

Book 1.indb 44

14𝑥 = 4

𝑥 − 8 = 18 𝑥 − 1 = 3𝑥

𝑥 − 2 = 8 + 11 5𝑥 = 20

19/07/24 5:07 AM

Collaborative Activity: It Doesn’t Work! Noah is having trouble solving two equations. In each case, he took steps that he thought were acceptable but ended up with statements that are clearly not true. Analyze Noah’s work on each equation and the moves he made. Were they acceptable moves? Why do you think he ended up with a false equation? Discuss your observations with your group and be prepared to share your conclusions. If you get stuck, consider solving each equation. 1.

𝑥 + 6 = 4𝑥 + 1 − 3𝑥 𝑥 + 6 = 4𝑥 − 3𝑥 + 1 𝑥+6=𝑥+1 6=1

2(5 + 𝑥) −1 = 3𝑥 + 9

10 + 2𝑥 − 1 = 3𝑥 + 9 2𝑥 − 1 = 3𝑥 − 1 2𝑥 = 3𝑥 2=3

Book 1.indb 45

Original equation Apply the commutative property. Combine like terms. Subtract 𝑥 from each side.

Original equation Apply the distributive property. Subtract 10 from each side. Add 1 to each side. Divide each side by 𝑥.

Unit 1 | 45

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Lesson Summary When solving an equation, the result is sometimes a false equation instead of a solution. Two attempts to solve the equation 4(𝑥 + 1) = 4𝑥 are shown.

4(𝑥 + 1) = 4𝑥 Original equation 4(𝑥 + 1) = 4𝑥 Original equation 𝑥 + 1 = 𝑥 Divide each side by 4. 4𝑥 + 4 = 4𝑥 Apply the distributive property. 1=0 Subtract 𝑥 from each side. 4=0 Subtract 4𝑥 from each side.

Each attempt shows acceptable moves, but the final equation is a false statement. • When solving an equation, start by assuming that there is at least 1 value that makes the equation true. The equation 4(𝑥 + 1) = 4𝑥 can be interpreted as “4 groups of (𝑥 + 1) are equal to 4 groups of 𝑥.” There are no values of 𝑥 that can make this true.

• For instance, if 𝑥 = 10, then 𝑥 + 1 = 11. It’s not possible that 4 times 11 is equal to 4 times 10. Likewise, 1.5 is 1 more than 0.5, but 4 groups of 1.5 cannot be equal to 4 groups of 0.5. Because of this, the moves made to solve the equation would not lead to a solution. The equation 4(𝑥 + 1) = 4 has no solutions.

An attempt to solve the equation 2𝑥 − 5 =

𝑥 − 20 is shown. 4

Each step in the process seems acceptable, but the last equation is a false statement. It is not easy to tell from the original equation whether it has a solution, but when looking at the equivalent equation 8𝑥 = 𝑥, it’s more apparent that 0 could be a solution. When 𝑥 is 0, the final equation becomes 0 = 0, which is a true statement.

46 | Unit 1

Book 1.indb 46

2𝑥 − 5 =

𝑥 − 20 4

Original equation

8𝑥 − 20 = 𝑥 − 20 Multiply each side by 4. 8𝑥 = 𝑥 8=1

Add 20 to each side.

Divide each side by 𝑥.

19/07/24 5:07 AM

The last move in the solving process was dividing by 𝑥. Because 0 could be the value of 𝑥 and dividing by 0 gives an undefined number, dividing by the variable is not recommended when solving an equation. Instead, either solution method shown could be used to solve the equation 8𝑥 = 𝑥. 8𝑥 = 𝑥 7𝑥 = 0 𝑥=0

Subtract 𝑥 from each side. Divide each side by 7.

8𝑥 = 𝑥 0 = −7𝑥 0=𝑥

Subtract 8𝑥 from each side. Divide each side by −7.

Practice Problems 1. Match each equation with an equivalent equation. Some of the answer choices are not used. Equation

Equivalent Equation

a. 3𝑥 + 6 = 4𝑥 + 7

9𝑥 = 4𝑥 + 7

b. 3(𝑥 + 6) = 4𝑥 + 7 c. 4𝑥 + 3𝑥 = 7 − 6

3𝑥 + 18 = 4𝑥 + 7

2. Mai says that equations A and B have the same solution.

03𝑥 = 4𝑥 + 7 3𝑥 − 1 = 4𝑥 7𝑥 = 1

• Equation A: −3(𝑥 + 7) = 24 • Equation B: 𝑥 + 7 = −8

Book 1.indb 47

Unit 1 | 47

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Which statement explains why this is true? A. Adding 3 to both sides of Equation A gives 𝑥 + 7 = −8.

B. Applying the distributive property to Equation A gives 𝑥 + 7 = −8.

C. Subtracting 3 from both sides of Equation A gives 𝑥 + 7 = −8. D. Dividing both sides of Equation A by −3 gives 𝑥 + 7 = −8.

3. Is 0 a solution to 2𝑥 + 10 = 4𝑥 + 10? Explain or show your reasoning.

4. Kiran says that a solution to the equation 𝑥 + 4 = 20 must also be a solution to the equation 5(𝑥 + 4) = 100. Write a convincing explanation as to why this is true.

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Review Problems 5. The entrepreneurship club is ordering potted plants for all 36 of its sponsors. One store charges $8.50 for each plant plus a delivery fee of $20. The equation 320 = 𝑥 + 7.50(36) represents the cost of ordering potted plants at a second store. What does the 𝑥 represent in this situation?

A. The cost for each potted plant at the second store B. The delivery fee at the second store C. The total cost of ordering potted plants at the second store D. The number of sponsors of the entrepreneurship club

6. The environmental science club is printing T-shirts for its 15 members. The printing company charges a certain amount for each shirt plus a setup fee of $20. If the T-shirt order costs a total of $162.50, how much does the company charge for each shirt?

7. The graph shows the relationship between temperature in degrees Celsius (℃) and temperature in degrees Fahrenheit (℉).

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Unit 1 | 49

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a. Mark the point on the graph that shows the temperature in Celsius when it is 60℉. b. Mark the point on the graph that shows the temperature in Fahrenheit when it is 60℃. c. Water boils at 100℃. Use the graph to approximate the boiling temperature in Fahrenheit, or to confirm it, if you know what it is.

d. The equation that converts Fahrenheit to Celsius is 𝐶 = 9 (𝐹−32). Use it to calculate the temperature in Celsius when it is 60℉. (This answer will be more exact than the point you found in the first part.)

8. Select all the equations that have the same solution as 2𝑥−5 = 15.

□ 2𝑥 = 10 □ 2𝑥 = 20

□ 2(𝑥−5) = 15 □ 2𝑥−20 = 0

□ 4𝑥−10 = 30 □ 15 = 5−2𝑥

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Unit 1, Lesson 6: Solving Multistep One-Variable Equations

Warm-Up: Solving Two-Step Equations 1. Complete the statements using the words from the bank. Words may be used more than once.

coefficient

constant

variable

a. In the expression 7𝑥 − 2, 𝑥 is the _____________________, 7 is a

_____________________, and −2 is a _____________________.

b. When solving an equation, the goal is to isolate the _____________________. 2. Solve the equations, and verify your solutions using substitution. a. 8 − 1.5𝑦 = 2

b. 21 = 7(2� + 5)

Book 1.indb 51

Unit 1 | 51

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Collaborative Activity: Solving Multistep Equations Javier and Arnav solved the same two equations, but they used different first steps. 1. In the table, each student’s first step for solving each equation is started in blue. Work with your partner to finish solving the equations, continuing from each student’s first step. Equation

Javier

Arnav

47 − 5.7𝑚 = 2.3𝑚 + 52

+ 5.7𝑚 + 5.7𝑚

−47

−1 (28𝑝 + 60) = −6𝑝 + 1

�− 1 � (28𝑝) + �− 1 � (60) = −6𝑝 + 1 4

− 1 (28𝑝 + 60) 4

−1 4

−6𝑝 + 1 −1 4

2. Discuss with your partner why both students had the same solution for each equation even though they used different strategies.

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3. Discuss which strategies you preferred to start with when solving each of these equations. Summarize your discussion.

Guided Activity: Solving Multistep Equations Solve each equation. Justify the steps in each solution method, and verify the solution using substitution. 1. 4.7𝑦 + 1.6 − 0.2𝑦 = 6.6

2. 49 − 6𝑑 = 2 (28𝑑 − 10)

62𝐾 − 27 = 8𝑘 + 7 11

Book 1.indb 53

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Lesson Summary When solving equations, the goal is to isolate the variable to find the value of the variable that makes the equation true. Apply properties of operations and properties of equality to justify the moves, or steps, used to solve a multistep equation by writing a series of equivalent equations until the variable is isolated. • If the resulting equation is a variable equal to a known value, the value on the opposite side from the variable is the solution. • If the resulting equation is false, there is no solution to the equation. • If the resulting equation is 2 known values set equal to each other, there are infinitely many solutions that make the original equation true. If a solution is found, verify it by substituting the value for the variable in the original equation. Apply the order of operations to evaluate the equation and ensure the result is a true equation. With multistep linear equations, it may be helpful to apply the distributive property or combine like terms on one side of the equation first, before applying properties of equality.

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Practice Problems

1. Solve 8 + 5 (16 + 9𝑔) = 35𝑔.

2. Solve 3 + 29𝑧 + 1 = 7𝑧 −3 .

3. Solve −11.3 − 27.7ℎ + 7.8 = −5.2ℎ + 32.5.

Book 1.indb 55

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4. Gracyn was solving an equation, but when she checked her solution, she noticed it was incorrect. She knows she made a mistake but cannot find it. Gracyn’s work is shown. Gracyn’s Work 1

Corrected Solution

−2 (4𝑦 + 5) = 13−8𝑦 + 2 1

−2 (4𝑦 + 5) = 15−8𝑦 −2𝑦 + 5 = 15−8𝑦 −2𝑦 = 10−8𝑦 6𝑦 = 10 10

𝑦 = 6

a. Circle the step where Gracyn made the mistake.

b. Describe Gracyn’s error.

c. Find the correct solution to the equation using the space provided in the table.

d. Verify that your solution is correct using substitution.

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Review Problems 5. An artist is selling children’s crafts. Necklaces cost $2.25 each, and bracelets cost $1.50 per each.

Select all the combinations of necklaces and bracelets that the artist could sell for exactly $12.00.

□ 5 necklaces and 1 bracelet

□ 2 necklaces and 5 bracelets □ 3 necklaces and 3 bracelets □ 4 necklaces and 2 bracelets □ 3 necklaces and 5 bracelets

□ 6 necklaces and no bracelets

□ No necklaces and 8 bracelets 6. The drama club is printing t-shirts for its members. The printing company charges a certain amount for each shirt plus a setup fee of $40. There are 21 students in the drama club. a. If there are 21 students in the club and the t-shirt order costs a total of $187, how much does each t-shirt cost? Show your reasoning.

b. The equation 201.50 = 𝑓 + 6.50(21) represents the cost of printing the shirts at a second printing company. Find the solution to the equation and state what it represents in this situation.

Book 1.indb 57

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7. Kiran’s family is having people over to watch a football game. They plan to serve sparkling water and pretzels. They are preparing 12 ounces (oz.) of sparkling water and 3 oz. of pretzels per person. Including Kiran’s family, there will be 10 people at the gathering. A bottle of sparkling water contains 22 oz. and costs $1.50. A package of pretzels contains 16 oz. and costs $2.99. Let 𝑛 represent number of people watching the football game, 𝑠 represent the oz. of sparkling water, 𝑝 represent the oz. of pretzels, and 𝑏 represent Kiran’s budget in dollars. Which equation best represents Kiran’s budget? A. 12𝑆 + 3𝑝 = 𝑏

B. 12 · 10 + 3 · 10 = 𝑏 C. 1.50𝑆 + 2.99𝑝 = 𝑏

D. 1.50 · 6 + 2.99 · 2 = 𝑏 8. A donut shop made 12 dozen donuts to give to a school’s math club.

Which expression represents how many donuts each student would get if the donuts were equally distributed and there were 𝑥 students in math club? 𝑥

A. 12

B. 12 · 𝑥

C. (12 · 12) · 𝑥 D. 12 · 12

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Unit 1, Lesson 7: Solving for a Specified Variable

Warm-Up: Which Equations? 1. The table shows the relationship between the base length, 𝑏, and the area, 𝐴, of some parallelograms. All the parallelograms have the same height. Base length is measured in inches (in.), and area is measured in square inches (sq. in.). Complete the table. Base Length(in.)

Area(sq. in.)

4.5

11 2

46.5 2. Decide whether each equation could represent the relationship between 𝑏 and 𝐴. Be prepared to explain your reasoning.

□ 𝑏 = 3𝐴 □ 𝑏 = 𝐴 3

□ 𝐴 = 𝑏

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Collaborative Activity: Filling and Emptying Tanks 1. Tank A initially contained 124 liters (L) of water. It is then filled with more water, at a constant rate of 9 L per minute (min.). How many liters of water are in Tank A after the following amounts of time have passed? a. 4 min. b. 80 seconds (sec.) c. 𝑚 min. 2. How many min. have passed, 𝑚, when Tank A contains the following amounts of water? a. 151 L b. 191.5 L c. 270.25 L d. 𝑝 L

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3. Tank B, which initially contained 80 L of water, is being drained at a rate of 2.5 L per min. How many L of water remain in the tank after the following amounts of time? a. 30 sec. b. 7 min. c. 𝑡 min. 4. For how many min., 𝑡, has the water been draining when Tank B contains the following amounts of water? a. 75 L b. 32.5 L c. 18 L d. 𝑣 L

Guided Activity: Cargo Shipping An automobile manufacturer is preparing a shipment of cars and trucks on a cargo ship that can carry 21,600 tons. The cars weigh 3.6 tons each and the trucks weigh 7.5 tons each. © Accelerate Learning Inc. - All Rights Reserved

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1. Write an equation that represents the weight constraint of a shipment. Let 𝑐 be the number of cars and 𝑡 be the number of trucks.

2. For one shipment, trucks are loaded first and cars are loaded afterwards. (Even though trucks are bulkier than cars, a shipment can consist of all trucks as long as it is within the weight limit.) Find the number of cars that can be shipped if the cargo already has: a. 480 trucks b. 1,500 trucks c. 2,736 trucks d. 𝑡 trucks 3. For a different shipment, cars are loaded first, and then trucks are loaded afterwards. a. Write an equation you could enter into a calculator or a spreadsheet tool to find the number of trucks that can be shipped if the number of cars is known.

b. Use your equation and a calculator or a computer to find the number of trucks that can be shipped if the cargo already has 1,000 cars. What if the cargo already has 4,250 cars? 62 | Unit 1

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Lesson Summary A relationship between quantities can be described in more than one way. Some ways are more helpful than others, depending on what needs to be determined. Consider the angles of an isosceles triangle, for example. • The 2 angles near the horizontal side have equal measures in degrees, 𝑎. • T he sum of the angles in a triangle is 180°, so the relationship between the angles can be expressed as 𝑎 + 𝑎 + 𝑏 = 180.

Suppose the value of 𝑏 is known and 𝑎 needs to be determined. The equation can be rearranged using properties of operations and properties of equality, as needed, to isolate 𝑎, as shown.

𝑎 + 𝑎 + 𝑏 2𝑎 + 𝑏 2𝑎

= = =

𝑎 + 𝑎 + 𝑏 2𝑎 + 𝑏 𝑏

= = =

𝑎

180 180 180−𝑏

180−𝑏 2

180 180 180−2𝑎

This equation is equivalent to the first one. Now, to find 𝑎, substitute any value of 𝑏 into this equation, and evaluate the expression on the right side. Likewise, an equivalent equation can be written to make it easier to find 𝑏 when 𝑎 is known, as shown.

Rearranging an equation to isolate 1 variable is called solving for a variable. Solving for a variable before substituting any known values can make it easier to test different values of a variable to see how they affect the other variable. In this example, the equation was solved for 𝑎 and for 𝑏. All 3 equations are equivalent. What information is known and what information needs to be determined should guide which equation is used when solving problems.

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Practice Problems 1. Here is a linear equation in two variables:2𝑥 + 4𝑦−31 = 123. Solve the equation, first for 𝑥 and then for 𝑦.

2. A chef bought $17.01 worth of ribs and chicken. Ribs cost $1.89 per pound (lb.) and chicken costs $0.90 per lb. The equation 0.9𝑐 + 1.89𝑟 = 17.01 represents the relationship between the quantities in this situation. Show that each of the following equations is equivalent to 0.9𝑐 + 1.89𝑟 = 17.01. Then, explain when it might be helpful to write the equation in these forms. a. 𝑐 = 18.9 − 2.1𝑟

b. 𝑟 = − 10 𝑐 + 9 21

3. A car has a 16-gallon (gal.) fuel tank. When driven on a highway, it has a gas mileage of 30 miles per gallon (mpg). The gas mileage (also called “fuel efficiency”) tells us the number of miles (mi.) the car can travel for a particular amount of fuel (one gal. of gasoline, in this case). After filling the gas tank, the driver got on a highway and drove for a while. a. How many mi. has the car traveled if it has the following amounts of gas left in the tank? 15 gal.

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10 gal. 2.5 gal.

b. Write an equation that represents the relationship between the distance the car has traveled in mi., 𝑑, and the amount of gas left in the tank in gal., 𝑥. c. How many gal. are left in the tank when the car has traveled the following distances on the highway? 90 mi. 246 mi. d. Write an equation that makes it easier to find the amount of gas left in the tank, 𝑥, if we know the car has traveled 𝑑 mi. 4. Noah is helping to collect the entry fees at his school’s sports game. Student entry costs $2.75 each and adult entry costs $5.25 each. At the end of the game, Diego collected $281.25. Select all equations that could represent the relationship between the number of students, 𝑠, the number of adults, 𝑎, and the dollar amount received at the game.

□ 281.25 − 5.25𝑎 = 2.75𝑠 □ 𝑎 = 53.57 − 2.75 𝑠 5.25

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□ 281.25 + 2.75𝑎 = 𝑠 □ 281.25 + 5.25𝑠 = 𝑎 Review Problems 5. Elena says that equations A and B are not equivalent. • Equation A: 13−5𝑥 = 48 • Equation B: 5𝑥 = 35

Write a convincing explanation as to why this is true.

6. Lin says that a solution to the equation 2𝑥−6 = 7𝑥 must also be a solution to the equation 5𝑥−6 = 10𝑥. Write a convincing explanation about why this is true.

7. Han is solving an equation. He took steps that are acceptable but ended up with equations that are clearly not true.

5𝑥 + 6

= 7𝑥 + 5−2𝑥

Original equation

5𝑥 + 6

= 5𝑥 + 5

Combine like terms.

5𝑥 + 6

= 7𝑥−2𝑥 + 5

6 = 5

Apply the commutative property.

Subtract 5𝑥 from each side.

What can Han conclude as a result of these acceptable steps? A. There’s no value of 𝑥 that can make the equation 5𝑥 + 6 = 7𝑥 + 5−2𝑥 true. 66 | Unit 1

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B. Any value of 𝑥 can make the equation 5𝑥 + 6 = 7𝑥 + 5−2𝑥 true. C. 𝑥 = 6 is a solution to the equation 5𝑥 + 6 = 7𝑥 + 5−2𝑥. D. 𝑥 = 5 is a solution to the equation 5𝑥 + 6 = 7𝑥 + 5−2𝑥.

8. A catering company is setting up for a wedding. They expect 150 people to attend. They can provide small tables that seat 6 people and large tables that seat 10 people. a. Find a combination of small and large tables that seats exactly 150 people. b. Let 𝑥 represent the number of small tables and 𝑦 represent the number of large tables. Write an equation to represent the relationship between 𝑥 and 𝑦. c. Explain what the point (20, 5) means in this situation. d. Is the point (20, 5) a solution to the equation you wrote? Explain your reasoning.

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Unit 1, Lesson 8: Solving Equations and Rearranging Formulas

Warm-Up: Faces, Vertices, and Edges In an earlier lesson, you saw the equation 𝑉 + 𝐹 − 2 = 𝐸, which relates the number of vertices, faces, and edges in a Platonic solid. 1. Write an equation that makes it easier to find the number of vertices in each of the Platonic solids described: a. An octahedron (shown here), which has 8 faces. b. An icosahedron, which has 30 edges. 2. A Buckminsterfullerene (also called a “Buckyball”) is a polyhedron with 60 vertices. It is not a Platonic solid, but the numbers of faces, edges, and vertices are related the same way as those in a Platonic solid. Write an equation that makes it easier to find the number of faces a Buckyball has if we know how many edges it has.

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Collaborative Activity: Find the Error Sandra, Jerry, Marly, and Xavier solved the same equation, and each found different solutions. Each student’s work and solution are shown in the table. 1. Work with your partner to examine the work shown and determine who solved the equation correctly. Then, for the students who made errors, circle the step where the error was made and explain the error. Name

Student’s Work and Solution 9 − 3 (16 − 2𝑤) = 1 𝑤 + 11 4 3 3 (16 − 2𝑤) = 1 𝑤 + 2 4 3

Sandra

Is the solution correct? o Yes o No Error Explanation

12 − 3 𝑤 = 1 𝑤 + 2 2

10 − 3 𝑤 = 1 𝑤 2

10 = 11𝑤

Name

60 = 𝑤 11

Student’s Work and Solution 9 − 3 (16 − 2𝑤) = 1 𝑤 + 11 4

9 − 12 + 3 𝑤 = 1 𝑤 + 11

Jerry

Is the solution correct? o Yes o No Error Explanation

−3 + 3 𝑤 = 1 𝑤 + 11 2

−3 = 7 𝑤 + 11 6

−14 = 7 𝑤 6

−12 = 𝑤

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Name

Student’s Work and Solution 9 − 3 (16 − 2𝑤) = 1 𝑤 + 11 4

9 − 12 + 3 𝑤 = 1 𝑤 + 11

Marly

Is the solution correct? o Yes o No Error Explanation

−3 + 3 𝑤 = 1 𝑤 + 11 2

3 𝑤 = 1 𝑤 + 14 2 3 7 𝑤 = 14 6

Name

𝑤 = 12

Student’s Work and Solution 9 − 3 (16 − 2𝑤) = 1 𝑤 + 11 4

9 − 12 − 3 𝑤 = 1 𝑤 + 11

Xavier

Is the solution correct? o Yes o No Error Explanation

−3 − 3 𝑤 = 1 𝑤 + 11 2

−3 𝑤 = 1 𝑤 + 14 2

−5 𝑤 = 14 6

𝑤 = −16.8

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Guided Activity: Rearranging Formulas The formula for the area of a trapezoid is 𝐴 = 1 (𝑏1 + 𝑏2)ℎ, where 𝑏1 and 𝑏2 are the 2 lengths of the bases, and ℎ is the height of the trapezoid. Stacey knows the area of the trapezoid shown is 12.5 square units and uses the values given to determine the height, ℎ, of the trapezoid. Her work is shown.

4 3

h 6

1. Complete the table to explain each step of Stacey’s work. Stacey’s Work

Explanation of Steps

𝐴 = 1 (𝑏1 + 𝑏2)ℎ

Formula for area of a trapezoid

12.5 = 1 (4 + 6)ℎ 2

Substitute known values

25 = (4 + 6)ℎ 25 = ℎ (4 + 6) 25 = ℎ (10)

2.5 = ℎ

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2. Stacey wants to rearrange the area formula for a trapezoid to isolate the variable ℎ before replacing the values of the other variables. a. Use Stacey’s work on the previous problem to help solve the equation for ℎ.

b1 c

h b2

𝐴 = 1 (𝑏1 + 𝑏2)ℎ 2

b. Check your work by substituting the values from the previous question. If correct, ℎ = 2.5.

c. Explain the similarities and differences between rearranging the formula to isolate ℎ and Stacey’s work to find the value of ℎ. Similarities

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Differences

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3. The formula for the total surface area of a cylinder, 𝑆, is 𝑆 = 2𝐵 + 𝑃ℎ, where 𝐵 is the area of the base, 𝑃 is the perimeter of the base, and ℎ is the height of the cylinder.

Rearrange the formula for the surface area of a cylinder to isolate the variable ℎ.

𝑆 = 2𝐵 + 𝑃ℎ

Lesson Summary Just as rearranging an equation sometimes makes it easier to solve for a variable of interest, the same skill can be applied to rearrange formulas used in math, science, and other disciplines in order to solve for a variable of interest. Solving for a variable of interest in a multivariable equation (or formula) requires the same application of properties of operations and properties of equality as when solving a one-variable equation. For example, the formula for the perimeter of a rectangle is 𝑃 = 2𝑙 + 2𝑤, where 𝑙 is the length of the rectangle and 𝑤 is the width. Steps to rearrange the formula to isolate 𝑙 are shown and described in the table. Equivalent Equation

Explanation of Steps

𝑃 = 2𝑙 + 2𝑤

Given formula

𝑃 − 2𝑤 = 2𝑙 𝑃−2𝑤 = 𝑙 2

𝑃 − 𝑤 = 𝑙 2

Subtracted 2𝑤 from both sides of the equation Divided both sides of the equation by 2 Distributed division by 2 to both terms on the left side

𝑃 = 2𝑙 + 2𝑤 and 𝑙 = 𝑃 −𝑤 are equivalent equations that both express the relationship 2

between a rectangle’s perimeter, length, and width.

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Practice Problems 1. The formula for force, 𝐹, is mass, 𝑚, times acceleration, 𝑎, or 𝐹 = 𝑚𝑎. Solve for mass, 𝑚.

2. The formula 𝐶 = 2𝜋𝑟 is used to find the circumference, 𝐶, of a circle with radius 𝑟. Solve the equation for the radius,.

3. The formula to convert degrees Fahrenheit, 𝐹, into degrees Celsius, 𝐶, is 𝐶 = 5 (𝐹−32). Solve for 𝐹. 9

4. 𝑉 = 𝜋𝑟2ℎ is an equation to calculate the volume of a cylinder, 𝑉, where 𝑟 represents the radius of the cylinder and ℎ represents its height.

Which equation allows us to easily find the height of the cylinder because it is solved for ℎ? A. 𝑟2ℎ = 𝑉 𝜋

B. ℎ = 𝑉− 𝜋𝑟2 C. ℎ = 𝑉 2 𝜋𝑟

D. 𝜋ℎ = 𝑉2 𝑟

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Review Problems 5. The density of an object can be found by taking its mass and dividing by its volume. Write an equation to represent the relationship between the three quantities (density, mass, and volume) in each situation. Let the density, 𝐷, be measured in grams/cubic centimeters (or g/cm3). a. The mass is 500 grams (g) and the volume is 40 cubic centimeters (cu. cm). b. The mass is 125 g and the volume is 𝑣 cu. cm. c. The volume is 1.4 cu. cm and the density is 80 g/cu. cm. d. The mass is 𝑚 grams and the volume is 𝑣 cu. cm. 6. In a video game, players form teams and work together to earn as many points as possible for their team. Each team can have between 2 and 4 players. Each player can score up to 20 points in each round of the game. Han and three of his friends decided to form a team and play a round. Write an expression, an equation, or an inequality for each quantity described here. If you use a variable, specify what it represents. a. the allowable number of players on a team

b. the number of points Han’s team earns in one round if every player earns a perfect score

c. the number of points Han’s team earns in one round if no players earn a perfect score 76 | Unit 1

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d. the number of players in a game with six teams of different sizes: two teams have 4 players each and the rest have 3 players each e. the possible number of players in a game with eight teams

7. A student on the cross-country team runs 30 minutes (min.) a day as a part of her training. Write an equation to describe the relationship between the distance she runs in miles (mi.), 𝐷, and her running speed, in miles per hour (mph), when she runs: a. at a constant speed of 4 mph for the entire 30 min.

b. at a constant speed of 5 mph the first 20 min., and then at 4 mph the last 10 min. c. at a constant speed of 6 mph the first 15 min., and then at 5.5 mph for the remaining 15 min. d. at a constant speed of 𝑎 mph the first 6 min., and then at 6.5 mph for the remaining 24 min. e. at a constant speed of 5.4 mph for 𝑚 min., and then at 𝑏 mph for 𝑛 min.

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Unit 1, Lesson 9: Rearranging Equations of Lines

Warm-Up: Solving Equations 1. Solve each equation by isolating the variable. Show your work. a. 14 = 3𝑥−28 b. 𝑥 + 19 = 6(𝑥−11) c. 27 + 9𝑦 = −12

Guided Activity: Different Forms of Linear Equations Three linear equations and their graphs are shown. 𝒚 = 3 𝒙 − 𝟐 4

(𝒚−𝟏) = 3 (𝒙 − 𝟒)

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𝟑𝒙−𝟒𝒚 = 𝟖

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1. Discuss with your partner what you notice about the graphs and equations.

The equations of lines can be written in different but equivalent forms. Each form of the equation relates to different features of the graph, including the slope, 𝑦-intercept, and other points on the line. One form for the equation of a line is slope-intercept form. 𝒚 = 3 𝒙 − 𝟐

Slope-Intercept Form

5 y 4 3 2

𝒚 = 𝒎𝒙 + 𝒃

where 𝑚 is the slope and 𝑏 is the 𝑦-intercept

1 -5 -4

-3

-2

+4 +3

-1

0 -1 -2 -3

x 1

(0, –2)

-4 -5

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2. Discuss with your partner why this form is called slopeintercept form. Summarize your discussion.

A second form for the equation of a line is called point-slope form. (𝒚 − 𝟏) = 3 (𝒙 − 𝟒)

Point-Slope Form

4 3 2

(𝒚−𝒚𝟏) = 𝒎(𝒙−𝒙𝟏)

(4, 1)

1 -5 -4

where 𝑚 is the slope and (𝑥1,𝑦1) is any point on the line

-3

-2

+4 +3

-1

0 -1

x 1

-2 -3 -4 -5

Point-slope form can be written using the slope and any point on the line.

3. Compare the equation and the graph of the line. What do you notice about the signs of the coordinates of the green point used to write the equation compared to the values in the equation?

4. Katrina wrote a different equation for the same line in point-slope form using the 𝑦-intercept,(0,−2). Her equation is shown. (𝑦 + 2) = 3 (𝑥−0) 4

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b. Discuss with your partner how the equation (𝑦 + 2) = 3 (𝑥−0) could be rewritten in 4 slope-intercept form.

c. Rewrite the equation (𝑦 + 2) = 3 (𝑥−0) in slope-intercept form by isolating the 4 variable𝑦.

d. Rewrite the original point-slope equation, (𝑦−1) = 3 (𝑥−4), in slope-intercept form. 4

e. What do you notice about the 2 equations you just generated?

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The third form for the equation of a line is called standard form. Standard Form

𝟯𝒙 − 𝟒𝒚 = 𝟖 5

4 3 2

𝑨𝒙 + 𝑩𝒚 = 𝑪

where 𝐴 and 𝐵 cannot both be zero

1 -5 -4

-3

-2

-1

0 -1

x 1

-2 -3 -4 -5

5. Rewrite the equation 3𝑥−4𝑦 = 8 in slope-intercept form by isolating the variable𝑦.

6. Use the equation 𝑓(𝑥) = 2 𝑥−4 to complete the following. 5

a. What is the form of the linear equation given?

b. Rewrite the equation in standard form.

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Collaborative Activity: Equivalent Forms Twelve linear equations are given in various forms.

4𝑥−6𝑦 = 24

−4𝑥−6𝑦 = 24

−6𝑥 + 4𝑦 = 24

6𝑥 + 4𝑦 = 24

𝑦 = −2 𝑥−4

𝑓(𝑥) = −2 𝑥−6

𝑦 = 3 𝑥 + 6

𝑓(𝑥) = 2 𝑥−4

(𝑦 + 6) = −3 (𝑥−8)

(𝑦−6) = 3 𝑥

(𝑦 + 6) = 2 (𝑥 + 3)

(𝑦−6) = −2 (𝑥 + 15)

1. Using the equations, work with your partner to complete the table so that each row contains a set of equivalent equations, one written in each form. Not all equations will be used. Equivalent Equations

Standard Form

Slope-Intercept Form

Point-Slope Form

Set 1

Set 2

Set 3

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Lesson Summary Previous courses introduced the slope-intercept form for the equation of a line. The slope-intercept form of a line is 𝑦 = 𝑚𝑥 + 𝑏, where 𝑚 represents the slope of the line and 𝑏 represents the 𝑦-intercept. The slope of a line is the ratio of the change in the vertical direction to change in the horizontal direction, often expressed as △𝑦 change in 𝑦 , or . △� change in � The 𝒚-intercept is the value of 𝑦 at the point where a line or graph intersects the 𝑦-axis. The value of � is 0 at this point. In this lesson, 2 additional forms of linear equations were introduced. • The point-slope form of the equation of a line is (𝑦 − 𝑦1) = 𝑚(𝑥 − 𝑥1), where 𝑚 is the slope and (𝑥1,𝑦1) is any point on the line.

• The standard form of the equation of a line is 𝐴𝑥 + 𝐵𝑦 = 𝐶, where 𝐴 and 𝐵 cannot both be zero.

By using key features revealed in the form of the linear equation given, along with properties of operations and properties of equality, linear equations can be rewritten in other equivalent forms.

Practice Problems 1. Use the equation (𝑦 − 7) = −1 (𝑥 + 2) to complete the following. 3

a. What is the form of the linear equation given?

b. Rewrite the equation in standard form.

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2. Use the equation 6𝑥 − 9𝑦 = −12 to complete the following. a. What is the form of the linear equation given?

b. Rewrite the equation in slope-intercept form.

3. Select all of the equations that are equivalent to (𝑦 + 3) = −1 𝑥. 4

□ 𝑦 = −14 𝑥

o 𝑦 = −1 𝑥 − 3 4

o (𝑦−3) = −1 (𝑥 − 1) o 𝑥 + 4𝑦 = 3

o 𝑥 + 4𝑦 = −12 4. Is the equation (𝑦−5) = 1 (𝑥 + 4) equivalent to 𝑦 = 1 𝑥 + 7? Explain your thinking. 2

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Review Problems 5. A car traveled 180 miles (mi.) at a constant rate. a. Complete the table to show the rate at which the car was traveling if it completed the same distance in each number of hours (hr.).

b. Write an equation that would make it easy to find the rate at which the car was traveling in miles per hour (mph), 𝑟, if it traveled for 𝑡 hr.

Travel Time (hr.)

Rate of Travel (mph)

4.5 3

2.25

6. A group of 280 elementary school students and 40 adults are going on a field trip. They are planning to use two different types of buses to get to the destination. The first type of bus holds 50 people and the second type of bus holds 56 people. Andre says that 3 of the first type of bus and 3 of the second type of bus will hold all of the students and adults going on the field trip. Is Andre correct? Explain your reasoning.

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7. To grow properly, each tomato plant needs 1.5 square feet (sq. ft.) of soil and each broccoli plant needs 2.25 sq. ft. of soil. The graph shows the different combinations of broccoli and tomato plants in an 18 sq. ft. plot of soil.

Match each point to the statement that describes it. Point a. Point A b. Point B c. Point C d. Point D

88 | Unit 1

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Statement The soil is fully used when 6 tomato plants and 4 broccoli plants are planted. Only broccoli was planted, but the plot is fully used and all plants can grow properly. After 3 tomato plants and 2 broccoli plants were planted, there is still extra space in the plot. With 4 tomato plants and 6 broccoli plants planted, the plot is overcrowded.

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Unit 1, Lesson 10: Connecting Equations to Graphs – Part 1

Warm-Up: Games and Rides Jada has $20 to spend on games and rides at a carnival. Games cost $1 each and rides are $2 each. 1. Which equation represents the relationship between the number of games, 𝑥, and the number of rides, 𝑦, that Jada could do if she spends all her money? A. 𝑥 + 𝑦 = 20

B. 2𝑥 + 𝑦 = 20 C. 𝑥 + 2𝑦 = 20

2. Explain what each of the other two equations could mean in this situation.

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Collaborative Activity: Graphing Games and Rides Two equations are shown. Each represents the relationship between the number of games, 𝑥, the number of rides, 𝑦, and the dollar amount that a student is spending on games and rides at different amusement parks. Equation 1: 𝑦 − 5 = −2.50 (𝑥 − 4) Equation 2: 𝑥 + 4𝑦 = 28

1. Use equation 1, 𝑦 − 5 = −2.50 (𝑥−4), and the coordinate plane shown to complete the following. a. How many rides can the student get on if they don’t play any games?

On the coordinate plane, mark the point that represents this situation, and label the point with its coordinates. b. How many games can the student play if they don’t get on any rides?

On the coordinate plane, mark the point that represents this situation, and label the point with its coordinates.

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c. Draw a line to connect the 2 points using a straightedge. d. Complete the statements. If the student plays no games, they can get on ____ rides. For every additional game that the student plays, 𝑥, the possible number of rides they can go on, 𝑦, increases decreases

by _______.

e. What is the slope of the graph?

f. Where does the graph intersect the vertical axis?

g. Rearrange the equation to solve for 𝑦. h. Describe the connections, if any, that you notice between the equation from part G and the graph.

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2. Use equation 2, 𝑥 + 4𝑦 = 28, and the coordinate plane shown to complete the following. a. How many rides can the student get on if they don’t play any games?

On the coordinate plane, mark the point that represents this situation, and label the point with its coordinates. b. How many games can the student play if they don’t get on any rides?

On the coordinate plane, mark the point that represents this situation, and label the point with its coordinates. c. Draw a line to connect the 2 points using a straightedge. d. Complete the statements. If the student plays no games, they can get on ____ rides. For every additional game that the student plays, 𝑥, the possible number of rides they can go on, 𝑦, increases decreases

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by _______.

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e. What is the slope of the graph?

f. Where does the graph intersect the vertical axis?

g. Rearrange the equation to solve for 𝑦.

h. Describe the connections, if any, that you notice between the equation from part G and the graph.

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Guided Activity: Nickels and Dimes Andre’s coin jar contains 85 cents. There are no quarters or pennies in the jar, so the jar has all nickels, all dimes, or some of each. 1. Write an equation that relates the number of nickels, 𝑛, the number of dimes, 𝑑, and the amount of money, in cents, in the coin jar.

2. Graph your equation on the coordinate plane. Be sure to label the axes.

3. How many nickels are in the jar if there are no dimes?

4. How many dimes are in the jar if there are no nickels?

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Lesson Summary Linear equations can be written in different forms. Some forms more readily provide key information about the relationship between quantities or about the graph of the equation. Suppose an athlete wants to burn 700 calories (cal.) a day by running and swimming. He burns 17.5 cal. per minute (min.) of running and 12.5 cal. per min. of freestyle swimming. Let 𝑥 represent the number of min. of running, and let 𝑦 represent the number of min. of swimming. To represent the combination of running and swimming that would allow him to burn 700 calories, an equation in standard form, 𝐴𝑥 + 𝐵𝑦 = 𝐶, can be written. 17.5𝑥 + 12.5𝑦 = 700

The more min. he runs, the fewer min. he has to swim to meet his goal. In other words, as 𝑥 increases, 𝑦 decreases. The graph of the equation will slope down from left to right. • If the athlete only runs and doesn’t swim, how many min. will he need to run for? Substitute 0 for 𝑦 to find 𝑥, as shown. On a graph, this combination of times is the point (40, 0), which is the 𝑥-intercept. • If he only swims and doesn’t run, how many min. will he need to swim for? Substitute 0 for 𝑥 to find 𝑦, as shown.

On a graph, this combination of times is the point (0, 56), which is the 𝑦-intercept.

Book 1.indb 95

17.5𝑥 + 12.5(0)

700

17.5𝑥

700

𝑥

700 17.5

𝑥

17.5(0) + 12.5𝑦

700

12.5𝑦

700

𝑦

700 12.5

𝑦

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• If the athlete wants to know how many min. he will need to swim for if he runs for 15 min., he can substitute each of these values for 𝑥 in the equation and find 𝑦, or he can first solve the equation for 𝑦, as shown.

700

12.5𝑦

𝑦

700 − 17.5𝑥

17.5𝑥 + 12.5𝑦

Notice that 𝑦 = 56−1.4𝑥, or 𝑦 = −1.4𝑥 + 56, is written in slope-intercept form, 𝑦 = 𝑚𝑥 + 𝑏.

𝑦

700 − 17.5𝑥 12.5

56 − 1.4𝑥

The coefficient of 𝑥, −1.4, is the slope of the graph. It means that as 𝑥 increases by 1, 𝑦 decreases by 1.4. For every additional min. of running, the athlete can swim for 1.4 fewer min. The constant term, 56, shows where the graph intersects the 𝑦-axis. This is the number of min. that the athlete will need to swim if he doesn’t do any running. When solved for 𝑦, both equations reveal that if he runs for 15 min., he needs to swim for 35 min. to burn 700 cal. This means that the point (15, 35) is another combination (or solution) that satisfies the goal of burning a total of 700 cal. This point can be used to write an equation in point-slope form, 𝑦−𝑦1 = 𝑚(𝑥−𝑥1), as shown. 𝑦 − 35 = −1.4 (𝑥−15)

The 3 equations, 17.5𝑥 + 12.5𝑦 = 700, 𝑦 = −1.4𝑥 + 56, and 𝑦 − 35 = −1.4(𝑥−15), are equivalent, so they have the same solutions and are represented by the same graph, shown.

y 60 55

(0,56)

50 45 40

(15,35)

35 30 25 20 15 10 5 0

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(40,0) 5

10 15 20 25 30 35 40 45 50

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Practice Problems 1. Andre bought a new bag of cat food. The next day, he opened it to feed his cat. The graph shows how many ounces (oz.) were left in the bag on the days after it was bought. a. How many oz. of food were in the bag 12 days after Andre bought it?

b. How many days did it take for the bag to contain 16 oz. of food?

c. How much did the bag weigh before it was opened?

d. About how many days did it take for the bag to be empty?

2. A little league baseball team is ordering hats. The graph shows the relationship between the total cost, in dollars, and the number of hats ordered. What does the slope of the graph tell us in this situation? A. It tells us that there is a fixed cost of approximately $35 for ordering hats. B. It tells us the amount that the total cost increases for each additional hat ordered.

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C. It tell us that when 9 hats are ordered, the total cost is approximately $160. D. It tells us that when the number of hats ordered increases by 10, the total cost increases by approximately $175. 3. A group of hikers is progressing steadily along an uphill trail. The graph shows their elevation (or height above sea level), in feet, at each distance from the start of the trail, in miles. a. What is the slope of the graph? Show your reasoning.

b. What does the slope tell us about this situation?

c. Write an equation that represents the relationship between the hikers’ distance from the start of the trail, 𝑥, and their elevation,𝑦.

d. Does the equation 𝑦 − 250𝑥 = 500 represent the same relationship between the distance from the start of trail and the elevation? Explain your reasoning.

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4. A kindergarten teacher bought $21 worth of stickers and cardstock for his class. The stickers cost $1.50 a sheet and the cardstock cost $3.50 per pack. The equation 1.5𝑠 + 3.5𝑐 = 21 represents the relationship between sheets of stickers, 𝑠, packs of cardstocks, 𝑐, and the dollar amount a kindergarten teacher spent on these supplies. a. Explain how we can tell that this graph represents the given equation.

b. What do the vertical and horizontal intercepts, (0, 6) and (14, 0), mean in this situation?

Review Problems 5. In physics, the equation 𝑃𝑉 = 𝑛𝑅𝑇 is called the ideal gas law. It is used to approximate the behavior of many gases under different conditions. 𝑃, 𝑉, and 𝑇 represent pressure, volume, and temperature, 𝑛 represents the number of moles of gas, and 𝑅 is a constant for the ideal gas. Which equation is solved for 𝑇? A. 𝑃𝑉 = 𝑛𝑇 𝑅

B. 𝑃𝑉 = 𝑇 𝑛𝑅

C. 𝑇 = 𝑃𝑉−𝑛𝑅 D. 𝑃𝑉𝑛𝑅 = 𝑇

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6. Explain why the equation 2(3𝑥 − 5) = 6𝑥 + 8 has no solutions. 7. Consider the equation 3𝑎 + 0.1𝑛 = 123. If we solve this equation for 𝑛, which equation would result? A. 0.1𝑛 = 123 − 3𝑎

B. 𝑛 = 123 − 3𝑎 − 0.1 C. 𝑛 = 1,230 − 30𝑎 D. 3𝑎 − 123 = 𝑛 0.1

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Unit 1, Lesson 11: Connecting Equations to Graphs – Part 2

Warm-Up: Rewrite These! Rewrite each quotient as a sum or a difference. 1. 4𝑥 − 10 2

2. 1 − 50𝑥 −2

5(𝑥 + 10) 25 1 𝑥 + 5 5

Exploration Activity: Graphs of 2 Equations Two graphs that represent different situations are shown.

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1. The first graph represents 𝑎 = 450−20𝑡, which describes the relationship between the amount of water in a tank, in gallons (gal.), and time, in minutes (min.). a. Where on the graph can the values 450 and −20 be found? 450:

−20: b. What do these numbers mean in this situation? 450: −20: 2. The second graph represents 6𝑥 + 9𝑦 = 75. It describes the relationship between pounds (lb.) of almonds and figs and the dollar amount that Conroy spent on them. Samir says, “I am not sure the graph represents 6𝑥 + 9𝑦 = 75 because I don’t see the 6, 9, or 75 on the graph.” Explain how to show Samir that the graph does represent this equation.

3. Write an equation in point-slope form for either situation. Explain how your equation is represented by the graph.

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Collaborative Activity: Slope Match Match each of the equations with the slope, 𝑚, and 𝑦-intercept, 𝑏, of its graph. 1. −4𝑥 + 3𝑦 = 3

a. 𝑚 = 3, 𝑏 = (0, 1)

5. 1.5𝑥−0.5𝑦 = 1

b. 𝑚 = 4 , 𝑏 = (0, 1)

6. 8𝑥 + 2𝑦 = 16

c. 𝑚 = 4 , 𝑏 = (0, −2)

7. 1 𝑦 = 1 + 𝑥

d. 𝑚 = −4, 𝑏 = (0, 8)

8. −4𝑥 = −3𝑦 + 6

e. 𝑚 = 3, 𝑏 = (0, −2)

3 3

Lesson Summary Two situations and equations that represent them are shown. Situation 1: Mai receives a $40 bus pass. Each school day, she spends $2.50 to travel to and from school. Let 𝑛 be the number of school days since Mai received the pass, and let 𝑑 be the dollar amount remaining on the pass. 𝑑 = 40−2.50𝑛

Book 1.indb 103

Situation 2: A school club is raising money by selling popcorn and iced tea. The club is charging $3 per bag of popcorn and $1.50 per cup of iced tea, and they plan to make $60.

Let 𝑝 be the bags of popcorn sold, and let 𝑡 be the cups of iced tea sold. 3𝑝 + 1.50𝑡 = 60

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The graphs of the equations and some coordinates are shown.

The first equation is essentially in slope-intercept, with the order of the terms on the right side swapped. Instead of 𝑦 = 𝑚𝑥 + 𝑏, the equation is given in the form 𝑦 = 𝑏 + 𝑚𝑥. The slope and the 𝑦-intercept can be observed from the equation and the graph. • The graph intersects the vertical axis at 40, and −2.50 is the slope of the line.

• Every time 𝑛 increases by 1, 𝑑 decreases by 2.50. In other words, with each passing school day, the dollar amount on Mai’s bus pass drops by $2.50. The numbers in the second equation are not as apparent on the graph. The values where the line intersects the vertical and horizontal axes, 40 and 20, do not appear in the equation. Reasoning can be used to determine where they come from.

• If 𝑝 is 0 (no popcorn is sold), the club would need to sell 40 cups of iced tea to make $60 because 40(1.50) = 60.

• If 𝑡 is 0 (no iced tea is sold), the club would need to sell 20 bags of popcorn to make $60 because 20(3) = 60. What about the slope of the second graph? It can be computed from the graph, but it is not shown in the equation 3𝑝 + 1.50𝑡 = 60. Notice that in the first equation, the variable 𝑑 was isolated. The second equation can be rewritten to isolate 𝑡, as shown. Now, the numbers in the equation can be more easily related to the graph. The 40 is where the graph intersects the vertical axis, and −2 is the slope. The slope reveals that as 𝑝 increases by

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3𝑝 + 1.50𝑡 1.50𝑡

= =

60 60−3𝑝

𝑡

40−2𝑝

𝑡

60 − 3𝑝 −2

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1, 𝑡 decreases by 2. In other words, for every additional bag of popcorn sold, the club can sell 2 fewer cups of iced tea.

By using the intercepts, or any other points on the graph, the equation of a line can also be written in point-slope form. Consider situation 1, for example. Using the slope that has already been determined, −2.50, and the point (2, 35) from the graph, the equation 𝑦−35 = −2.50(𝑥−2)can also be used to represent the situation.

Practice Problems 1. What is the slope of the graph of 5𝑥−2𝑦 = 20? A. −10 B. −2 5

C. 5 2

D. 5 2. What is the 𝑦-intercept of each equation? a. 𝑦 = 6𝑥 + 2

b. 10𝑥 + 5𝑦 = 30 c. 𝑦−6 = 2(3𝑥−4)

3. Han wanted to find the intercepts of the graph of the equation 10𝑥 + 4𝑦 = 20. He decided to put the equation in slope-intercept form first. Here is his work:

(1 )

Book 1.indb 105

10𝑥 + 4𝑦 = 20 4𝑦 = 20−10𝑥 𝑦 = 5−10𝑥

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a. What error did Han make?

b. What are the 𝑥- and 𝑦-intercepts of the line? Explain or show your reasoning.

4. Which graph represents the equation 𝑦 − 6 = 3 (𝑥−4)? Explain how you know. 4

Review Problems 5. The graph shows how much money Priya has in her savings account weeks after she started saving on a regular basis. a. How much money does Priya have in the account after 10 weeks? b. How long did it take her to save $200?

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c. How much money did Priya have in her savings account when she started to save regularly?

d. Write an equation to represent the dollar amount in her savings account and the number of weeks of saving. Be sure to specify what each variable represents.

6. Noah orders an extra-large pizza. It costs $12.49 for the pizza plus $1.50 for each topping. He orders an extra-large pizza with 𝑡 toppings that costs a total of 𝑑 dollars. Select all of the equations that represent the relationship between the number of toppings 𝑡 and total cost 𝑑 of the pizza with 𝑡 toppings.

□ 12.49 + 𝑡 = 𝑑

□ 12.49 + 1.50𝑡 = 𝑑

□ 12.49 + 1.50𝑑 = 𝑡

□ 12.49 = 𝑑 + 1.50𝑡 □ 𝑡 = 𝑑 −12.49 1.5 □ 𝑡 = 𝑑 − 12.49 1.5

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7. A school sells adult tickets and student tickets for a play. It collects $1,400 in total. The graph shows the possible combinations of the number of adult tickets sold and the number of student tickets sold.

What does the vertical intercept (0, 200) tell us in this situation?

A. It tells us the decrease in the sale of adult tickets for each student ticket sold. B. It tells us the decrease in the sale of student tickets for each adult ticket sold. C. It tells us that if no adult tickets were sold, then 200 student tickets were sold. D. It tells us that if no student tickets were sold, then 200 adult tickets were sold.

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Unit 1, Lesson 12: Graphing Two-Variable Linear Inequalities – Part 1

Warm-Up: Less Than, Equal To, or More Than 12? Consider the expression 2𝑥 + 3𝑦.

1. Decide if the values in each ordered pair, (𝑥, y), make the value of the expression less than, greater than, or equal to 12. a. (0, 5)

b. (6, 0)

c. (−1, −1)

d. (−5, 10)

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Guided Activity: Two-Variable Linear Inequalities y

1. The graph of the two-variable linear inequality 𝑦 ≥ − 1 𝑥 + 2 is shown.

7 6

5 4

a. The orange line is called the boundary line of the inequality. Write the equation of the boundary line in slope-intercept form.

3 2 1 -8 -7 -6 -5 -4 -3 -2 -1 0 -1

x 1

3 4

-2 -3 -4

b. How does the equation of the boundary line relate to the inequality?

-5 -6 -7

c. Select one ordered pair for each of the criteria given in the table. Then, substitute each ordered pair into the inequality. An Ordered Pair in the Shaded Region

An Ordered Pair in the Unshaded Region

An Ordered Pair on the Boundary Line

Coordinates of the Point Substitute the Point into the Inequality 𝒚 � − 1𝒙 � 2 4

Is it a solution to 𝒚 � − 1 𝒙� 2 4

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Yes No

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d. Describe which part(s) of the graph represent the solution set of the inequality 𝑦 ≥ − 1 𝑥 + 2 . 4

e. Which part of the graph of the solution set of 𝑦 ≥ − 1 𝑥 + 2 corresponds to each 4 mathematical statement? Mathematical Statement

Graphical Representation Boundary line Shaded region

𝑦 = − 1 𝑥 + 2 4

Boundary line Shaded region

𝑦 > − 1 𝑥 + 2 4

f. Discuss with your partner how you think the graph of 𝑦 ≤− 1 𝑥 + 2 would look 4 different from the graph of 𝑦 ≥ − 1 𝑥 + 2. 4

Two linear inequalities and their graphs are shown. 𝒚 � −𝟐𝒙 � 𝟑

𝒚 � −𝟐𝒙 � 𝟑

-6 -5 -4 -3 -2 -1 0 -1

-6

-5 -4 -3

-2

-1 0 -1

-2

-3

-4

-5

-6

g. Discuss with your partner why you think the boundary line on the graph of 𝑦 ≥ −2𝑥 + 3 is solid while the boundary line on the graph of 𝑦 > −2𝑥 + 3 is dotted.

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h. Explain how the solid and dotted lines on the graphs of two-variable inequalities relate to closed and open circles on the graphs of one-variable inequalities.

2. Complete the statements. If the points on the boundary line are included in the solution set, the boundary line is

dotted. solid.

If the points on the boundary line are not included in the solution

set, the boundary line is

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dotted. solid.

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Collaborative Activity: Sketching Solutions to Inequalities 1. Here is a graph that represents solutions to the equation 𝑥−𝑦 = 5.

Sketch 4 quick graphs representing the solutions to each of these inequalities.

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2. For each graph, write an inequality whose solutions are represented by the shaded part of the graph.

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Lesson Summary The equation 𝑥 + 𝑦 = 7 is a two-variable equation. Its solutions are all pairs of 𝑥- and 𝑦-values whose sum is 7. The pairs 𝑥 = 0, 𝑦 = 7 and 𝑥 = 5, 𝑦 = 2 are examples.

• All the solutions to 𝑥 + 𝑦 = 7 can be represented by the graph of the equation on a coordinate plane.

• The graph is a line, where all the points on the line are solutions to 𝑥 + 𝑦 = 7. The inequality 𝑥 + 𝑦 ≤ 7 is a two-variable inequality. Its solutions are all pairs of 𝑥- and 𝑦-values whose sum is 7 or less than 7. This means the solutions include all the (𝑥, 𝑦) pairs that are solutions to the equation 𝑥 + 𝑦 = 7, but also many other pairs of 𝑥- and 𝑦-values that add up to a value less than 7. The pairs 𝑥 = 4, 𝑦 = −7 and 𝑥 = −6, 𝑦 = 0 are examples.

On a coordinate plane, the solution to 𝑥 + 𝑦 ≤ 7 includes the line that represents 𝑥 + 𝑦 = 7. Plotting some other (𝑥, 𝑦) pairs that make the inequality true, such as (4, −7) and (−6, 0), shows that these points fall on one side of the line. In contrast, (𝑥, 𝑦) pairs that make the inequality false fall on the other side of the line. Shade the region on one side of the line to indicate that all points within that region are solutions.

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What about the inequality 𝑥 + 𝑦 < 7?

The solution is all pairs of 𝑥- and 𝑦-values whose sum is less than 7. This means pairs like 𝑥 = 0, 𝑦 = 7 and 𝑥 = 5, 𝑦 = 2 are not solutions.

On a coordinate plane, the solution does not include points on the line that represents 𝑥 + 𝑦 = 7 because those points are (𝑥, 𝑦) pairs whose sum is 7.

To show that the points on the boundary line are excluded, use a dashed or dotted line. All points below that line are (𝑥, 𝑦) pairs that make 𝑥 + 𝑦 < 7 true. The region on that side of the line can be shaded to show that that region contains the solutions.

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Practice Problems 1. Here is a graph of the equation 2𝑦 − 𝑥 = 1.

a. Are the points (0, 1 ) and (−7, −3) solutions to 2

the equation? Explain or show how you know.

b. Check if each of these points is a solution to the inequality 2𝑦 − 𝑥 > 1. (0, 2)

(8, 1 ) 2

(−6, 3)

(−7, −3)

c. Shade the region that represents the solution set to the inequality 2𝑦 − 𝑥 > 1.

d. Are the points on the line included in the solution set? Explain how you know.

2. Select all coordinate pairs that are solutions to the inequality 5𝑥 + 9𝑦 < 45.

□ (0, 0) □ (5, 0) □ (9, 0) □ (0, 5) □ (0, 9) □ (5, 9)

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3. Consider the linear equation 2𝑦 − 3𝑥 = 5.

a. The pair (−1, 1) is a solution to the equation. Find another (𝑥, y) pair that is a solution to the equation.

b. Are (−1, 1) and (4,1) solutions to the inequality 2𝑦 − 3𝑥 < 5? Explain how you know.

c. Explain how to use the answers to the previous questions to graph the solution set to the inequality 2𝑦 − 3𝑥 < 5.

4. The boundary line on the graph represents the equation 5𝑥 + 2𝑦 = 6. Write an inequality that is represented by the graph.

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5. Choose the inequality whose solution set is represented by this graph. A. 𝑥 − 3𝑦 < 5 B. 𝑥 − 3𝑦 ≤ 5 C. 𝑥 − 3𝑦 > 5 D. 𝑥 − 3𝑦 ≥ 5 Review Problems 6. Elena is planning to go camping for the weekend and has already spent $40 on supplies. She goes to the store and buys more supplies. Which inequality represents 𝑑, the total amount in dollars that Elena spends on supplies? A. 𝑑 > 40 B. 𝑑 ≥ 40 C. 𝑑 < 40 D. 𝑑 ≤ 40 7. Solve −𝑥 < 3. Explain how to find the solution set.

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Unit 1, Lesson 13: Graphing Two-Variable Linear Inequalities – Part 2

Warm-Up: Landscaping Options A homeowner is making plans to landscape her yard. She plans to hire professionals to install grass sod in some parts of the yard and flower beds in other parts. Grass sod installation costs $2 per square foot (sq. ft.) and flower bed installation costs $12 per sq. ft. Her budget for the project is $3,000.

1. Write an equation that represents the sq. ft. of grass sod, 𝑥, and the sq. ft. of flower beds, 𝑦, that she could afford if she used her entire budget. 2. On the coordinate plane, sketch a graph that represents your equation. Be prepared to explain your reasoning.

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Exploration Activity: Rethinking Landscaping The homeowner is worried about the work needed to maintain a grass lawn and flower beds, so she is now looking at some lowmaintenance materials. She is considering artificial turf, which costs $15 per sq. ft. to install, and gravel, which costs $3 per sq. ft. She may use a combination of the two materials in different parts of the yard. Her budget is still $3,000. Here is a graph representing some constraints in this situation.

1. The graph shows a line going through (500, 100). a. In this situation, what does the point (500, 100) mean?

b. Write an equation that the line represents.

c. What do the solutions to the equation mean?

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2. The point (600, 200) is located to the right and above the line. a. Does that combination of turf and gravel meet the homeowner’s constraints? Explain or show your reasoning.

b. Choose another point in the same region (to the right and above the line). Check if the combination meets the homeowner’s constraints.

3. The point (200, 100) is located to the left and below the line. a. Does that combination of turf and gravel meet the homeowner’s constraints? Explain or show your reasoning.

b. Choose another point in the same region (to the left and below the line). Check if the combination meets the homeowner’s constraints.

4. Write an inequality that represents the constraints in this situation. Explain what the solutions mean and show the solution region on the graph.

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Guided Activity: The Saturday Market A vendor at the Saturday Market makes $9 profit on each necklace she sells and $5 profit on each bracelet. 1. Find a combination of necklaces and bracelets that she could sell and make: a. exactly $100 profit

b. more than $100 profit

2. Write an equation whose solution is the combination of necklaces and bracelets she could sell and make exactly $100 profit.

3. Write an inequality whose solutions are the combinations of necklaces and bracelets she could sell and make more than $100 profit.

4. Graph the solutions to your inequality.

5. Is (3, 18.6) a solution to the inequality? Explain your reasoning.

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Collaborative Activity: Charity Concerts A popular band is trying to raise at least $20,000 for charity by holding multiple concerts at a park. It plans to sell tickets at $25 each. For each 2-hour concert, the band would need to pay the park $1,250 in fees for security, cleaning, and traffic services. The band needs to find the combinations of number of tickets sold, 𝑡, and number of concerts held, 𝑐, that would allow it to reach its fundraising goal. 1. Write an inequality to represent the constraints in this situation.

2. Graph the solutions to the inequality on the coordinate plane.

3. Name two possible combinations of number of tickets sold and number of concerts held that would allow the band to meet its goal.

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4. Which combination of tickets and concerts would mean more money for charity: a. 1,300 tickets and 10 concerts, or 1,300 tickets and 5 concerts?

b. 1,600 tickets and 16 concerts, or 1,200 tickets and 9 concerts?

c. 2,000 tickets and 4 concerts, or 2,500 tickets and 10 concerts?

Lesson Summary Two-variable inequalities can represent constraints in real-life situations. Graphing their solutions helps to solve problems. Suppose a cafe is purchasing coffee and tea from a supplier and can spend up to $1,000 total. Coffee beans cost $12 per kilogram (kg), and tea leaves costs $8 per kg. Buying 𝑐 kg of coffee beans and 𝑡 kg of tea leaves will therefore cost 12𝑐 + 8𝑡. The inequality 12𝑐 + 8𝑡 ≤ 1,000 can be used to represent the budget constraint.

The solution to this inequality is any pair of 𝑐 and 𝑡 that makes the inequality true. In context, it is any combination of the kg of coffee and tea that the cafe can order without going over the $1,000 budget.

Different pairs of 𝑐 and 𝑡 can be tested to see which combinations satisfy the constraint, but it would be difficult to capture all the possible combinations this way. Instead, graph a related equation, 12𝑐 + 8𝑡 = 1,000, and then determine which region represents all possible solutions. The graph of the equation is shown.

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To determine the solution region, select a point on the line and a point on each side of the line, and see if the pairs of values produce true statements. Three examples are shown. A point on the line:

A point below the line:

A point above the line:

(30, 80)

(20, 40)

(70, 100)

12(30) + 8(80) ≤ 1,000 360 + 640 ≤ 1,000 1,000 ≤ 1,000 This is true.

12(20) + 8(40) ≤ 1,000 12(70) + 8(100) ≤ 1,000 240 + 320 ≤ 1,000 840 + 800 ≤ 1,000 560 ≤ 1,000 1,640 ≤ 1,000 This is true.

This is false.

The points on the line and in the region below the line are solutions to the inequality. Shade the solution region. It’s easy to read solutions from the graph. For example, without any computation, it’s apparent that (50, 20) is a solution because it falls within the shaded region. If the cafe orders 50 kg of coffee and 20 kg of tea, the cost will be less than $1,000.

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Practice Problems 1. To qualify for a loan from a bank, the total in someone’s checking and savings accounts together must be $500 or more. a. Which of these inequalities best represents this situation? A. 𝑥 + 𝑦 < 500

B. 𝑥 + 𝑦 ≤ 500 C. 𝑥 + 𝑦 > 500 D. 𝑥 + 𝑦 ≥ 500

b. Complete the graph so that it represents solutions to an inequality representing this situation. (Be clear about whether you want to use a solid or dashed line.)

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2. The soccer team is selling bags of popcorn for $3 each and cups of lemonade for $2 each. To make a profit, they must collect a total of more than $120. a. Write an inequality to represent the number of bags of popcorn sold, 𝑝, and the number of cups of lemonade sold, 𝑐, in order to make a profit. b. Graph the solution set to the inequality on the coordinate plane.

c. Explain how we could check if the boundary is included or excluded from the solution region.

3. Tyler filled a small jar with quarters and dimes and donated it to his school’s charity club. The club member receiving the jar asked, “Do you happen to know how much is in the jar?” Tyler said, “I know it’s at least $8.50, but I don’t know the exact amount.” a. Write an inequality to represent the relationship between the number of dimes, 𝑑, the number of quarters, 𝑞, and the dollar amount of the money in the jar. b. Graph the solution set to the inequality and explain what a solution means in this situation.

c. Suppose Tyler knew there are 25 dimes in the jar. Write an inequality that represents how many quarters could be in the jar.

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4. Tickets to the aquarium are $11 for adults and $6 for children. An after-school program has a budget of $200 for a trip to the aquarium. If the boundary line in each graph represents the equation 11𝑥 + 6𝑦 = 200, which graph represents the cost constraint in this situation?

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Review Problems 5. Andre is solving the inequality 14𝑥 + 3 ≤ 8𝑥 + 3. He first solves a related equation. 14𝑥 + 3 = 8𝑥 + 3 14𝑥 = 8𝑥 8 = 14

This seems strange to Andre. He thinks he probably made a mistake. What was his mistake?

6. Select all the values of 𝑥 that are solutions to the inequality −7−(3𝑥 + 2) < −8(𝑥 + 1).

□ 𝑥 = −0.2 □ 𝑥 = −0.1 □ 𝑥 = 0

□ 𝑥 = 0.1 □ 𝑥 = 0.2 □ 𝑥 = 0.3

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7. Here is a graph of the equation 6𝑥 + 2𝑦 = −8.

a. Are the points (1.5, −4) and (0, −4) solutions to the equation? Explain or show how you know.

b. Check if each of these points is a solution to the inequality 6𝑥 + 2𝑦 ≤ −8: (−2, 2)

(4, −2)

(0, 0)

(−4, −4)

c. Shade the solutions to the inequality.

d. Are the points on the line included in the solution region? Explain how you know.

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Unit 1, Lesson 14: Solving Problems with Two-Variable Inequalities

Warm-Up: Graphing Inequalities with Technology Use graphing technology to graph the solution region of each inequality and sketch each graph. Adjust the graphing window as needed to show meaningful information.

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Collaborative Activity: Solving Problems with Two-Variable Inequalities Three situations are described below. There are two questions about each situation. For each question that you work on: • Write an inequality to describe the constraints. Specify what each variable represents. • Use graphing technology to graph the inequality. Sketch the solution region on the coordinate plane and label the axes. • Name one solution to the inequality and explain what it represents in that situation. • Answer the question about the situation.

Bank Accounts 1. A customer opens a checking account and a savings account at a bank. They will deposit a maximum of $600, some in the checking account and some in the savings account. (They might not deposit all of it and keep some of the money as cash.) a. Inequality:

b. Graph:

c. Solution and interpretation:

d. If the customer deposits $200 in their checking account, what can you say about the amount they deposit in their savings account?

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2. The bank requires a minimum balance of $50 in the savings account. It does not matter how much money is kept in the checking account. a. Inequality:

b. Graph:

c. Solution and interpretation:

d. If the customer deposits no money in the checking account but is able to maintain both accounts without penalty, what can you say about the amount deposited in the savings account?

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Concert Tickets 1. Two kinds of tickets to an outdoor concert were sold, lawn tickets and seat tickets. Fewer than 400 tickets in total were sold. a. Inequality:

b. Graph:

c. Solution and interpretation:

d. If you know that exactly 100 lawn tickets were sold, what can you say about the number of seat tickets?

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2. Lawn tickets cost $30 each and seat tickets cost $50 each. The organizers want to make at least $14,000 from ticket sales. a. Inequality:

b. Graph:

c. Solution and interpretation:

d. If you know that exactly 200 seat tickets were sold, what can you say about the number of lawn tickets?

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Advertising Packages 1. An advertising agency offers two packages for small businesses who need advertising services. A basic package includes only design services. A premium package includes design and promotion. The agency’s goal is to sell at least 60 packages in total. a. Inequality:

b. Graph:

c. Solution and interpretation:

d. If the agency sells exactly 45 basic packages, what can you say about the number of premium packages it needs to sell to meet its goal?

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2. The basic advertising package has a value of $1,000 and the premium package has a value of $2,500. The goal of the agency is to sell more than $60,000 worth of small-business advertising packages. a. Inequality:

b. Graph:

c. Solution and interpretation:

d. If you know that exactly 10 premium packages were sold, what can you say about the number of basic packages the agency needs to sell to meet its goal?

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Collaborative Activity: Representation of Inequalities Your teacher will give you a set of cards. Take turns with your partner to match a group of 4 cards that contain: a situation, an inequality that represents it, a graph that represents the solution region, and a solution written as a coordinate pair. For each match that you find, explain to your partner how you know it’s a match. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

Record your matches. Group 1

Group 2

• situation: perimeter of a rectangle

• situation: jar of coins

• inequality:

• a solution:

• sketch of graph:

Group 3

Group 4

• situation: honey and jam

• situation: a school trip

• inequality:

• a solution:

• sketch of graph:

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Lesson Summary To find the solution to 𝑥 − 𝑦 > 5, start by graphing the related equation 𝑥 − 𝑦 = 5. When identifying the solution region, it’s important not to assume that the solution will be above the line because of a > symbol or below the line because of a < symbol.

Instead, test points on the line and on either side of the line to see if they are solutions. For 𝑥 − 𝑦 > 5, points on the line and above the line are not solutions to the inequality because the (𝑥, 𝑦) pairs make the inequality false. Points that are below the line are solutions. Therefore, the lower region gets shaded.

Graphing technology can be used to graph the solution to a two-variable inequality. Many graphing tools show the solution region, as shown here. Some tools, however, may require the inequalities to be in slopeintercept form (or another form) in order to display the solution region. Be sure to learn how to use the graphing technology available in your classroom.

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Although graphing by using technology is efficient, the graph needs to be analyzed with care. Some things to consider are listed. • The graphing window: If the graphing window is too small, the solution region or the boundary line may be only minimally visible, as shown here. • The meaning of solution points in the situation: For example, if 𝑥 and 𝑦 represent the lengths of 2 sides of a rectangle, then only positive values of 𝑥 and 𝑦 (or points in the first quadrant) make sense in the situation, even though all solutions to the mathematical inequality will be represented on the graph using technology.

Practice Problems 1. This year, students in the 9th grade are collecting dimes and quarters for a school fundraiser. They are trying to collect more money than the students who were in the 9th grade last year. The students in 9th grade last year collected $143.88. Using 𝑑 to represent the number of dimes collected and 𝑞 to represent the number of quarters, which statement best represents this situation? A. 0.25𝑑 + 0.1𝑞 ≥ 143.88

B. 0.25𝑞 + 0.1𝑑 ≥ 143.88 C. 0.25𝑑 + 0.1𝑞 > 143.88 D. 0.25𝑞 + 0.1𝑑 > 143.88

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2. A farmer is creating a budget for planting soybeans and wheat. Planting soybeans costs $200 per acre and planting wheat costs $500 per acre. He wants to spend no more than $100,000 planting soybeans and wheat. a. Write an inequality to describe the constraints. Specify what each variable represents.

b. Name one solution to the inequality and explain what it represents in that situation.

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3. Elena is ordering dried chili peppers and corn husks for her cooking class. Chili peppers cost $16.95 per pound (lb.) and corn husks cost $6.49 per lb. Elena spends less than $50 on 𝑑 lb. of dried chili peppers and ℎ lb. of corn husks. Here is a graph that represents this situation. a. Write an inequality that represents this situation.

b. Can Elena purchase 2 lb. of dried chili peppers and 4 lb. of corn husks and spend less than $50? Explain your reasoning.

c. Can Elena purchase 1.5 lb. of dried chili peppers and 3 lb. of corn husks and spend less than $50? Explain your reasoning.

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4. Which inequality is represented by the graph? A. 4𝑥 − 2𝑦 > 12 B. 4𝑥 − 2𝑦 < 12 C. 4𝑥 + 2𝑦 > 12 D. 4𝑥 + 2𝑦 < 12

Review Problems 5. Jada has a sleeping bag that is rated for 30 degrees Fahrenheit (℉). This means that if the temperature outside is at least 30℉, Jada will be able to stay warm in her sleeping bag. a. Write an inequality that represents the outdoor temperature at which Jada will be able to stay warm in her sleeping bag.

b. Write an inequality that represents the outdoor temperature at which a thicker or warmer sleeping bag would be needed to keep Jada warm.

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6. Here is a graph of the equation 2𝑥 − 3𝑦 = 15.

a. Are the points (1.5, −4) and (4, −4) solutions to the equation? Explain or show how you know.

b. Check if each of these points is a solution to the inequality 2𝑥 − 3𝑦 < 15: (0, −5) (4, −2) (2, −4) (5, −1)

c. Shade the solutions to the inequality.

d. Are the points on the line included in the solution region? Explain how you know.

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7. A store sells notepads in packages of 24 and packages of 6. The organizers of a conference need to prepare at least 200 notepads for the event. a. Would they have enough notepads if they bought these quantities? i. Seven packages of 24 and one package of 6

ii. Five packages of 24 and fifteen packages of 6

b. Write an inequality to represent the relationship between the number of large and small packages of notepads and the number of notepads needed for the event.

c. Use graphing technology to graph the solution set to the inequality. Then, use the graph to name two other possible combinations of large and small packages of notepads that will meet the number of notepads needed for the event.

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Unit 2: Two-Variable Statistics

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Unit 2, Lesson 1: Summarizing Data in Two-Way Tables

Warm-Up: Utensil and Paper Preferences Several students are surveyed about whether they prefer writing with a pen or a pencil and they are also asked whether they prefer lined paper or unlined paper. Some of the results are shown. • The survey included 100 different students.

• 40 students said they prefer using pen more than pencil.

• 45 students said they prefer using unlined paper more than lined paper. • 10 students said they prefer lined paper and pen.

• 45 students said they prefer pencil and lined paper. For each part, explain or show your reasoning.

1. How many students prefer using pencil more than pen?

2. How many students prefer using pen and unlined paper?

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3. How many students prefer using pencil and unlined paper?

Exploration Activity: Fruit Fly Mutations A scientist is trying to determine the role of specific genes by looking at traits of fruit flies. The offspring of two fruit flies are examined to determine the color of their eyes and whether they have curled wings or standard wings. Eighty offspring are randomly selected, and the results are recorded in the two-way table. Curled Wings

Standard Wings

Red Eyes

White Eyes

1. Describe what the 17 in the table represents.

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2. How many selected fly offspring had white eyes? Explain or show your reasoning.

3. How many selected fly offspring had standard wings? Explain or show your reasoning.

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Collaborative Activity: Info Gap: Running to the Dentist 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 data card: 1. Silently read the information on your card. 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!) 3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”

If your teacher gives you the problem card: 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.

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.

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 In statistics, the term variable has a slightly different meaning than it has in algebra. A variable in statistics is a characteristic of individuals in a population that can take on different values.

Data from 2 categorical variables about a single subject can be organized using a two-way table. A categorical variable takes on values which can be divided into groups or categories.

A two-way table is a way of organizing data from two categorical variables in order to investigate the association between them.

For example, this two-way table shows the results from 170 responses to a survey asking people their age group and whether or not they have a cell phone. Has a Cell Phone

Does Not Have a Cell Phone

10–12 Years Old

13–15 Years Old

16–18 Years Old

• The 38 in the table means that 38 of the 170 people surveyed are in the 13–15-yearold age range and have cell phones. • The two-way table also shows that 55 of the people surveyed do not have cell phones, since 35 + 12 + 8 = 55. © Accelerate Learning Inc. - All Rights Reserved

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The categories for a single variable should not overlap. For example, a person cannot be 10–12 years old and 13–15 years old at the same time. Each individual is included in only one of the boxes in the table rather than in multiple boxes.

Practice Problems 1. This two-way table shows the results of asking students if they prefer to have gym class in the morning or the afternoon.

Morning

Afternoon

Total

Grade 6

Grade 8

Grade 10

Total

100

a. How many students participated in the survey?

b. How many students in grade 8 prefer to have gym in the morning?

c. How many grade 10 students participated in the survey?

d. How many students prefer to have gym in the afternoon?

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2. A random sample of adults are asked about their preferences for a first dinner date with someone. Complete the two-way table so that it has the characteristics listed. • 122 people responded to the survey.

• 50 of the people who said they order dessert said they also prefer to split the check. • 68 people prefer splitting the check.

• 56 people prefer to skip dessert rather than ordering one. Order Dessert

No Dessert

Split the Check

One Person Pays

3. Students in the 7th, 8th, and 9th grade were asked whether they prefer to write in pen or pencil. • 40 students prefer to write in pen.

• 60 students prefer to write in pencil.

Create values that could represent the number of students in the 7th, 8th, and 9th grade that responded to the survey.

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4. A recent study observed the number of bike riders ages 0 to 20 that wear helmets. The results are represented in the table. Wear Helmet

Did Not Wear Helmet

0-5 Years

6-10 Years

11-15 Years

16-20 Years

Make an observation based on a data value that is not in the table. Explain your reasoning.

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Review Problems 5. Select all the points that are on the graph of the equation 4𝑦 − 6𝑥 = 12.

□ (−4, −3)

□ (−1, −1.5) □ (0, −2) □ (0, 3)

□ (3, −4) □ (6, 4)

6. Which equation is equivalent to the equation 5𝑥 + 30 = 45? A. 35𝑥 = 45 B. 5𝑥 = 75

C. 5(𝑥 + 30) = 45 D. 5(𝑥 + 6) = 45

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7. Diego’s age, 𝑑, is 5 more than 2 times his sister’s age, 𝑠. This situation is represented by the equation 𝑑 = 2𝑠 + 5. Which equation is equivalent to the equation 𝑑 = 2𝑠 + 5? A. 𝑑 = 2(𝑠 + 5) B. 𝑑 − 5 = 2𝑠

C. 𝑑 − 2 = 𝑠 + 5 D. 𝑑 = 𝑠 + 5 2

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Unit 2, Lesson 2: Interpreting Joint and Marginal Frequencies

Warm-Up: Creating a Two-Way Table 1. A survey conducted at Ambidextrous High School asked all 1,700 students to indicate their grade level and whether they are left-handed or right-handed. Only 59 of the 491 freshmen are left-handed. Out of the 382 students in the sophomore class, 289 of them are right-handed. There are 433 students in the junior class, and 120 of them are lefthanded. There are 307 right-handed seniors. Complete the two-way frequency table to display the results of the survey.

Dominant Hand Survey

Total Righthanded Lefthanded Total

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Guided Activity: Interpreting Joint and Marginal Frequencies 1. In 2016, the summer Olympics were held in Rio de Janeiro, Brazil. The US won the most medals with a total of 121. China came in second, winning 70 total medals. This was a much higher total than the 9 medals China won 2 years later in the winter Olympics held in Pyeongchang, South Korea, in 2018. Norway led the medal count in 2018 with 39, a much higher number than the 4 total medals Norway won in 2016. Canada came in third in the 2018 winter Olympics, winning 29 total medals. a. Work with your partner to use the data to construct a two-way frequency table showing how the different countries’ medal counts compare to each other. US

China

Norway

Canada

Summer 2016

217

Winter 2018 Total

Total

144

317

b. How many total medals did the US win in 2016 and 2018 combined?

c. Which team won the most medals in the two events combined?

d. How many medals did Norway win in 2018?

e. Complete the statement. ________________ won slightly more medals in 2018 than in 2016.

f. What does the joint relative frequency, 121 , represent in this context? 317

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2. Work with your partner to write questions that can be answered using the data organized in the table. a. Write at least 1 question that must be answered with a joint frequency.

b. Write at least 1 question that must be answered with a marginal frequency.

Collaborative Activity: Interpreting Joint and Marginal Frequencies Mono Lake, in California, is a saltwater lake. Its salinity, or the concentration of salt, has varied over time. The salinity of Mono Lake was at an all-time high in 1982, at 99 grams per liter (g/L), as a result of diverting water that fed the lake. Since then, efforts to control water diversion at Mono Lake have slowly returned the salinity of the lake to normal. In 2008, the lake’s salinity was 79.8 g/L.

Two of the main organisms that live in the lake are brine shrimp and alkali flies. In 1982, there were an estimated 2,000 brine shrimp per liter of lake water. In 2008, there were 3,000 brine shrimp per liter of lake water. In 1982, there were 1,200 alkali flies per liter of lake water. That number nearly tripled to 3,500 alkali flies per liter by 2008. 1. Construct a two-way frequency table that summarizes the given information. Salinity of 99 g/L in 1982

Salinity of 79.8 g/L in 2008

Total

Brine shrimp Alkali flies Total

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2. Work with your partner to answer the following questions. a. How many alkali flies per liter were found in Mono Lake when the lake’s salinity was 99 g/L? b. When the salinity of Mono Lake was 79.8 g/L, how many brine shrimp and alkali flies per liter were found?

c. Which is greater, the combined total of alkali flies or the combined total of brine shrimp in these 2 samples?

d. What was the salinity of Mono Lake when there were 3,500 alkali flies per liter? e. Was the population of brine shrimp smaller when the salinity of Mono Lake was 99 g/L or 79.8 g/L?

Lesson Summary Two-way frequency tables contain marginal frequencies and joint frequencies. A joint frequency is found at the intersection, or joint, of a row and column. In a two-way table, a joint frequency is the number of times that a combination of 2 conditions occurs.

The total frequency for a category, which is the total frequency for any row or column in a two-way frequency table, is called a marginal frequency. A marginal frequency is found at the margin, or edge, of a column or row.

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For example, in the two-frequency table shown, which displays the results of a survey given to a group of teenagers about the effect of social media and the amount of time they spend with friends in person, the marginal frequencies are in the green cells, and the joint frequencies are in the orange cells.

Time Spent with Friends In Person

Effect of Social Media on Teenagers Mostly Positive

Mostly Negative

Neutral

Total

Too Much Too Little

Just Right

110

Total

113

219

211

105

186

311

269

409

741

Practice Problems 1. A group of teenagers were surveyed about the effect of social media and the amount of time they spend with friends in person. The two-way frequency table shows the results of the survey.

Time Spent with Friends In Person

Effect of Social Media on Teenagers

Too Much Too Little Just Right Total

Mostly Positive

Mostly Negative

Neutral

Total

110 219

105

269

211

311

741

113

186

409

a. What is the frequency for those who said the effect of social media was neutral?

b. How many teens reported spending too much time with friends in person?

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c. Do the values used to answer parts A and B represent joint or marginal frequencies?

d. How many people reported that the effect of social media is mostly negative and that they spend too much time with friends in person?

e. How many people reported that the effect of social media is mostly negative and that they spend too little time with friends in person?

f. Do the values used to answer parts D and E represent joint or marginal frequencies?

2. Corbin studies birth weight, in pounds (lb.), and height, in inches (in.), as part of his research at Florida School of Traditional Midwifery in Gainesville, Florida. The two-way table shows data from births in the month of June 2021 at the University of Florida (UF) Health Shands Hospital. Births from UF Health Shands Hospital (June 2021) < 5 lb. < 16 in. 16–20 in. >20 in. Total

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14 0

5–8 lb.

97 8

>8 lb.

Total

101

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a. Circle the smallest marginal frequency in the table. b. Underline the largest joint frequency in the table. c. Describe what the value 14 means in the context of the data.

d. Describe what the value 101 means in the context of the data. Review Problems 3. Large cheese pizzas cost $5 each and large one-topping pizzas cost $6 each.

Write an equation that represents the total cost, 𝑇, of 𝑐 large cheese pizzas and 𝑑 large one-topping pizzas.

4. Elena bikes 20 minutes (min.) each day for exercise.

Write an equation to describe the relationship between her distance in miles (mi.), 𝐷, and her biking speed, in miles per hour (mph), when she bikes: a. at a constant speed of 13 mph for the entire 20 min.

b. at a constant speed of 15 mph for the first 5 min., then at 12 mph for the last 15 min. c. at a constant speed of 𝑀 mph for the first 5 min., then at 𝑁 mph for the last 15 min.

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5. The robotics team needs to purchase $350 of new equipment. Each of the 𝑥 students on the team plans to fundraise and contribute equally to the purchase. Which expression represents the amount that each student needs to fundraise? A. 350 − 𝑥 B. 350 + 𝑥 C. 350 𝑥

D. 350 ⋅ 𝑥

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Unit 2, Lesson 3: Exploring Possible Associations in Categorical Data Using Various Displays

Warm-Up: Sports and Musical Instruments For a survey, students in a class answered these questions: • Do you play a sport? • Do you play a musical instrument?

1. The table shows some results from the survey. Complete the table, assuming that all students answered both questions. Plays Instrument Plays Sport

Does Not Play Instrument

Total

Does Not Play Sport

Total

2. To the nearest percentage point, what percentage of students who play a sport don’t play a musical instrument?

3. To the nearest percentage point, what percentage of students who don’t play a sport also don’t play a musical instrument?

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Exploration Activity: Card Sorts: Matching Representations Your teacher will hand out cards. Some cards show two-way tables. Some cards show bar graphs like the one shown.

Some cards show segmented bar graphs, like the one shown.

The bar graphs and segmented bar graphs on the cards have their labels removed. 1. Put all the cards that describe the same situation in the same group. 2. One of the groups does not have a two-way table. Make a two-way table for the situation described by the other graphs in the group.

3. Label the bar graphs and segmented bar graphs so that the category represented by each bar is indicated. 4. Describe in your own words the kind of information shown by a segmented bar graph.

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Guided Activity: Building Another Type of Two-Way Table A two-way table is shown for data about cell phone usage among children 10 to 18 years old.

10 to 12 Years Old 13 to 15 Years Old 16 to 18 Years Old Total

Has Cell Phone

Does Not Have Cell Phone

Total

115

170

1. Complete the table. In each row, the entries for “Has Cell Phone” and “Does Not Have Cell Phone” should total 100%. Round entries to the nearest whole percentage. Has Cell Phone

10 to 12 Years Old 13 to 15 Years Old

Does Not Have Cell Phone

Total

42%

16 to 18 Years Old

17%

100%

This is still a two-way table. Instead of showing frequency, this table shows relative frequency.

2. Discuss with your partner why the two-way relative frequency table doesn’t include a “Total” row at the bottom. 3. Explain whether there appears to be an association between age and cell phone use. Use the two-way relative frequency table to help illustrate your reasoning.

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Collaborative Activity: Sports and Music Association 1. Complete the table to create a two-way relative frequency table for the data from the Warm-Up. The table will show relative frequencies by row.

Plays Instrument

Does Not Play Instrument

Row Total

Plays Sport

100%

Does Not Play Sport

100%

2. Complete the table to create a two-way relative frequency table for the data from the Warm-Up. The table will show relative frequencies by column.

Plays Instrument

Does Not Play Instrument

100%

Plays Sport Does Not Play Sport Column Total

3. Based on the 2 relative frequency tables, explain whether you think there is an association between playing a sport and playing a musical instrument.

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Lesson Summary Two-way tables can be used to investigate possible associations between categorical variables. For example, this two-way frequency table shows the results of a study on meditation and athletes’ state of mind before a track meet. Two-way tables and other displays explored in this lesson, such as bar graphs and segmented bar graphs, can be used to determine whether there is a possible association between meditating and being calm.

Meditated

Did Not Meditate

Total

Calm Agitated Total

An association is a way to describe the form, direction, or strength of the relationship between the two variables in a bivariate data set.

• Notice that the number of athletes who did not meditate is small compared to the number who did meditate (29 compared to 68, as shown in the table). The proportion of calm meditators and calm non-meditators can also be analyzed to determine possible associations. These proportions are relative frequencies and can be organized into a relative frequency table, like the one shown.

Calm Agitated Total

Meditated

Did Not Meditate

66%

28%

34%

100%

72%

100%

A proportion of a data set is the fraction of the data in a given category.

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The relative frequency of a category tells us the proportion at which the category occurs in the data set. It is expressed as a fraction, a decimal, or a percentage of the total number. A relative frequency table is a version of a two-way table in which the value in each cell is divided by the total number of responses in the entire table or by the total number of responses in a row or a column..

The relative frequencies can also be presented in a segmented bar graph, like the one shown. Is there an association between meditation and state of mind? Both the two-way relative frequency table and the segmented bar graph show that about 66% of athletes who meditated were calm, whereas only about 27% of those who did not meditate were calm. This does not necessarily mean that meditation causes calmness. It could be the other way around, that calm athletes are more inclined to meditate. However, this does suggest that there is an association between meditating and calmness. Relative frequencies and associations in categorical data will be explored more fully in upcoming lessons.

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Practice Problems 1. A scientist wants to know if the color of the water affects how much animals drink. The average amount of water each animal drinks was recorded in milliliters (mL) for a week and then graphed.

Blue Water Green Water Total

Cat Intake (mL)

Dog Intake (mL)

Total (mL)

210

1,200

1,410

410

2,300

2,710

200

1,100

1,300

Is there evidence to suggest an association between water color and animal?

2. Workers at an advertising agency are interested in people’s TV viewing habits. They take a survey of people in two cities to try to find patterns in the types of shows they watch. The results are recorded in a table and shown in a segmented bar graph.

Chicago Topeka

Reality

News

Comedy

Drama

Is there evidence to suggest an association of different viewing habits? If so, explain.

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Review Problems 3. Members of the band sold juice and popcorn at a college football game to raise money for an upcoming trip. The band raised $2,000. The amount raised is divided equally among the 𝑚 members of the band. Which equation represents the amount, 𝐴, each member receives? A. 𝐴 = 𝑚

2000

B. 𝐴 = 2000 𝑚

C. 𝐴 = 2,000𝑚

D. 𝐴 = 2,000 − 𝑚 4. Elena has $225 in her bank account. She takes out $20 each week for 𝑤 weeks. After 𝑤 weeks she has 𝑑 dollars left in her bank account.

Write an equation that represents the amount of money left in her bank account after 𝑤 weeks.

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Unit 2, Lesson 4: Relative Frequency Tables

Warm-Up: Notice and Wonder: Teacher Degrees Several adults in a school building were asked about their highest degree completed and whether they were a teacher.

Associate Degree Bachelor’s Degree Master’s Degree or Higher What do you notice? What do you wonder?

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Teacher

Not a Teacher

16%

44%

20%

52%

64%

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Exploration Activity: City Cat, Country Cat 200 people were asked if they prefer dogs or cats, and whether they live in a rural or urban setting. The actual values collected from the survey are in the first table.

Cat Dog Total

Urban

Rural

Total

134

104 200

The next table shows what percentage of the 200 total people included are represented by each combination of categories. The segmented bar graph represents the same information graphically.

Cat Dog

Urban

Rural

27%

21%

40%

12%

The next table shows the percentage of each column that had a certain pet preference in a column relative frequency table. The segmented bar graph represents the same information graphically.

Cat Dog

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Urban

Rural

40%

64%

60%

36%

19/07/24 5:07 AM

The last table shows the percentage of each row that live in a certain area in a row relative frequency table. The segmented bar graph represents the same information graphically.

Cat Dog

Urban

Rural

56%

44%

77%

23%

1. For each relative frequency table, select a percentage and explain how numbers from the original table were used to get the percentage.

2. What percentage of those surveyed live in an urban area and prefer dogs?

3. Among the people surveyed who prefer dogs, what percentage of them live in an urban setting?

4. What percentage of people surveyed who live in an urban setting prefer dogs?

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5. How many of the people responded that they prefer dogs and live in an urban setting?

6. Among the people surveyed, are there more people who prefer dogs or cats?

7. Your pet food company has access to a billboard in a rural setting. Would you recommend advertising dog food or cat food on this billboard? Which table did you use to make this decision? Explain your reasoning.

Collaborative Activity: Analyzing a Study with Two Treatments In an experiment to test the effectiveness of vitamin C on the length of colds, two groups of people with colds are given a pill to take once a day. The pill for one of the groups contains 1,000 milligrams (mg) of vitamin C, while the other group takes a placebo pill. The researchers record the results in a table.

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Group A Group B Cold lasts less than a week Cold lasts a week or more

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1. First, the researchers want to know what percentage (to the nearest whole percent) of people are in each combination of categories. Fourteen percent of all the participants had a cold that lasted less than a week and were in group A. What percentage of all the participants had a cold that lasted less than a week and were in group B? Complete the rest of the relative frequency table with the corresponding percentages. Group A Cold lasts less than a week Cold lasts a week or more

Group B

14% �16 + 27 + 17 + 53 ≈0.14�

2. Next, the researchers notice that, among participants who had colds that lasted less than a week, 37% were in group A. Among participants who had colds that lasted a week or more, what percentage were in group B? Complete the table with the corresponding percentages. Group A Cold lasts less than a week Cold lasts a week or more

37% � 16 + 27 ≈0.37�

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Group B

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3. Finally, the researchers notice that, among the participants in group A, 48% had colds that lasted less than one week. Among the participants in group B, how many had colds that lasted a week or more? Complete the table with the corresponding percentages. Group A Cold lasts less than a week Cold lasts a week or more

Group B

48% � 16 + 17 ≈0.48�

4. To understand the results, the researchers want to know: Among people whose colds lasted less than a week, what percentage are in group B? Explain your reasoning.

5. If the researchers believe that vitamin C has a small effect on the length of a cold, which group most likely got the pills containing vitamin C? Explain your reasoning.

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Lesson Summary Converting two-way tables to relative frequency tables can help reveal patterns in paired categorical variables. Relative frequency tables are created by dividing the value in each cell of a two-way table by the total number of responses in the entire table, or by the total responses in a row or a column. Depending on what patterns are important, different types of relative frequency tables are used. To examine how individual combinations of the categorical variables relate to the whole group, divide each value in a two-way table by the total number of responses in the entire table to find the relative frequency.

For example, this two-way table displays the condition of each textbook and its price for 120 of the books at a college bookstore.

New Used

$𝟭𝟬 or Less

More Than $𝟭𝟬 but Less Than $𝟯𝟬

$𝟯𝟬 or More

$𝟭𝟬 or Less

More Than $𝟭𝟬 but Less Than $𝟯𝟬

$𝟯𝟬 or More

0.025

0.075

0.225

A two-way relative frequency table is created by dividing each number in the two-way table by 120, because there are 120 values (3 + 9 + 27 + 33 + 36 + 12) in this data set. The resulting two-way relative frequency table can be represented using fractions, decimals, or percentages.

New Used

0.275

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0.300

0.100

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• This two-way relative frequency table allows shows what proportion of the total is represented by each number in the two-way table. • The number 33 in the original two-way table represents the number of used books that sell for $10 or less, which is 27.5% of all the books in the data set. • This two-way relative frequency table shows that there are very few (2.5%) new books that are inexpensive, and that 10% of the books in the bookstore are both expensive and in used condition.

In other situations, it makes sense to examine row or column proportions in a relative frequency table. For example, to convert the original two-way table to a column relative frequency table using column proportions, divide each value by the sum of the column.

New Used

$𝟭𝟬 or Less

More Than $𝟭𝟬 but Less Than $𝟯𝟬

$𝟯𝟬 or More

0.083

0.2

0.692

0.917

0.8

0.308

• This shows that about 91.7% of the books that are sold for $10 or less are in used 33

condition ( 3 + 33

≈ 0.917).

• Notice that each column of this column relative frequency table reveals the proportions of the books in each price category that are in each condition, and the relative frequencies in each column sum to 1.

• In particular, this shows that most of the inexpensive and moderately priced books are used, and most of the expensive books are new.

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Practice Problems 1. A teacher asks their students whether they studied for a quiz, then scores the quiz. A relative frequency table displays some of the information they collected.

Passed Quiz Failed Quiz What does the 86% represent?

Studied

Did Not Study

86%

14%

46%

54%

2. The accessory choices of 143 people are recorded in the table.

Wearing a Belt No Belt

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Wearing a Watch

No Watch

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Create a relative frequency table that could be used to show the percentages of belt wearers who wear a watch or not, as well as the percentages of people without belts who wear a watch or not. Wearing a Watch

No Watch

Wearing a Belt No Belt 3. Scientists give two different treatments to people who have the flu and determine if their health improves. The results for the test are in the two-way table.

Improved Health No Improvement

Treatment 1

Treatment 2

a. What percentage of people receiving treatment 1 had improved health? b. What percentage of people receiving treatment 2 had improved health? 4. A group of people are surveyed about whether they have any brothers or sisters or are an only child, and whether they have any pets.

Have Pets No Pets

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Have Sibling

Only Child

105

141

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Which value could go in the blank cell so that the percentage of only children that have no pets is 37.5%? A. 63 B. 82

C. 175 D. 205 Review Problems 5. Many adults are selected at random to respond to a survey about their favorite season and whether they have allergies or not. The two-way table summarizes the results from the survey. a. Which season is the least popular in this group?

Winter Spring Summer Fall

Allergies

No Allergies

12 33

13 35

b. How many more people have allergies than people without allergies in this group?

c. How many people were surveyed in this group?

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6. A random sample of people are asked about their preferences regarding home decoration and their interest in fashion. Complete the two-way table so it has the characteristics listed. •

150 people responded to the survey.

•

33 of the responders who prefer neutral decorations also do not pay attention to fashion.

•

70% of the responders do not pay attention to fashion.

20 of the responders who pay attention to fashion also prefer colorful decorations in their homes. Prefer Colorful Decorations

Prefer Neutral Decorations

Pay Attention to Fashion Do Not Pay Attention to Fashion

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Unit 2, Lesson 5: Digging Deeper into Associations in Categorical Data

Warm-Up: Cake or Pie The table displays the dessert preference and dominant hand (left- or right-handed) for a sample of 300 people.

Left-handed Right-handed Total

Prefers Cake

Prefers Pie

Total

100

180

200

270

300

For each of the calculations, describe the interpretation of the percentage in terms of the situation.

1. 10% from 100 = 0.1 180

2. 67% from 270 ≈ 0.67 90

3. 30% from 300 = 0.3

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Guided Activity: Associations in Categorical Data 1. The two-way table displays data about 55 different locations. Scientists have a list of possible chemicals that may influence the health of the coral. They first look at how nitrate concentration might be related to coral health. The table displays the health of the coral (healthy or unhealthy) and nitrate concentration (low or high).

Healthy Unhealthy Total

Low Nitrate Concentration

High Nitrate Concentration

Total

a. Complete the two-way relative frequency table for the data in the two-way table in which the relative frequencies are based on the total for each column. Low Nitrate Concentration

High Nitrate Concentration

100%

Healthy Unhealthy Total

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b. When there is a low nitrate concentration, which had a higher relative frequency, healthy or unhealthy coral?

c. When there is a high nitrate concentration, is there a higher relative frequency of healthy or unhealthy coral?

d. Based on this data, is there a possible association between coral health and the level of nitrate concentration? Explain your reasoning.

e. The scientists next look at how silicon dioxide concentration might be related to coral health. The relative frequencies based on the total for each column are shown in the table.

Healthy Unhealthy Total

Low Silicon Dioxide Concentration

High Silicon Dioxide Concentration

44%

46%

100%

56%

54%

Based on this data, is there a possible association between coral health and the level of silicon dioxide concentration? Explain your reasoning.

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2. Jada surveyed 300 people from various age groups about their shoe preferences. The two-way table summarizes the results of the survey. Prefers Sneakers without Laces

Prefers Sneakers with Laces

Prefers Shoes That Are Not Sneakers

Total

108

156

114

129

300

4–10 Years Old 11–17 Years Old 18–24 Years Old Total

Jada concludes that there is a possible association between age and shoe preference. Is Jada’s conclusion reasonable? Explain your reasoning.

3. The two-way table summarizes data on writing utensil preference and the dominant hand for a sample of 100 people.

Prefers Pen Prefers Pencil Total

Left-handed

Right-handed

Total

100

Is there a possible association between the dominant hand and writing utensil preference? Explain your reasoning.

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Collaborative Activity: Colored Erasers An eraser factory has 5 machines. One machine makes the eraser shapes. Then, each shaped eraser goes through the red machine, blue machine, yellow machine, or green machine to have one side colored. The manager notices that the uncolored side of some of the erasers is flawed at the end of the process and wants to know which machine(s) need(s) to be fixed, the shape machine or some of the coloring machines. The manager collected data on the number of flawed and unflawed erasers of each color. The data is shown in the two-way table.

Unflawed Flawed Red Blue Yellow Green Total

285

223

195

120

823

Total 300

240

200

260

177

1,000

1. Work with a partner. Each of you should make 1 relative frequency table for the data in the table. One relative frequency table should show the relative frequency by row. The other relative frequency table should show the relative frequency by column. Round to the nearest whole percentage.

2. Explain whether flawed erasers are associated with certain colors, and, if so, which colors.

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Lesson Summary An association between 2 variables means that the 2 variables are statistically related to each other. For example, most would expect that ice cream sales would be higher on sunny days than on snowy days. If sales were higher on sunny days than on snowy days, then it could be stated that there is a possible association between ice cream sales and whether it is sunny or snowing out. When dealing with categorical variables, row or column relative frequency tables are often used to look for associations in the data. A two-way table is shown displaying the ice cream cone sales and the weather conditions at a particular creamery for 41 days.

Sold Fewer Than 50 Cones Sold 50 Cones or More Total

Sunny Day

Snowy Day

Total

Noticing a pattern in the raw data can be difficult, especially when the row or column totals are different for different categories, so the data should be converted into a row or column relative frequency table to more easily compare the categories. For the creamery, notice that the number of days with low sales is about the same for the 2 weather conditions, which contradicts the original assumption. In this case, it makes sense to look at the percentage of days that saw high sales under each weather condition separately. That is, consider the column relative frequencies. • From the column relative frequency table, it’s clear that most of the sunny days resulted in at least 50 cones sold (73%), while most of the snowy days resulted in fewer than 50 cones sold (64%). • Because these percentages are quite different, this suggests there is an association between the weather conditions and the number of cone sales. 194 | Unit 2

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Sunny Day Snowy Day Sold Fewer Than 𝟱𝟬 Cones Sold 𝟱𝟬 Cones or More Total

27%

64%

73%

36%

100%

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A bakery might wonder if the weather conditions impact their muffin sales as well. • For the bakery, it seems there is not an association between weather conditions and muffin sales, since the percentage of days with low sales are very similar under the different weather conditions, and the percentages are also close on days when they sold many muffins.

Sunny Day Snowy Day Sold Fewer Than 𝟱𝟬 Muffins Sold 𝟱𝟬 Muffins or More Total

32%

35%

68%

65%

100%

Using row or column relative frequency tables helps organize data so that columns or rows that represent different categories for a variable can easily be compared. This comparison can be accomplished using a two-way table or a two-way relative frequency table, but it requires accounting for the differences in the number of data values in a given category.

Practice Problems 1. Which value would best fit in the missing cell to suggest there is no evidence of an association between the variables? Digital Watch

Analog Watch

Displays the Date No Date Display A. 9

B. 18 C. 27 D. 54

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2. The relative frequency table shows the percentage of each type of art (painting or sculpture) in a museum that would classify in the different styles (modern or classical). Based on these percentages, is there evidence to suggest an association between the variables? Explain your reasoning.

Modern

Classical

41%

59%

Car

Truck

812

233

Paintings Sculptures

3. An automobile dealership keeps track of the number of cars and trucks they have for sale, as well as whether they are new or used. Based on the data, does there appear to be an association between the type of automobile and whether it is new or used? Explain your reasoning.

38%

New Used

422

62%

Review Problems 4. A survey is given to 1,432 people about whether they take daily supplemental vitamins and whether they eat breakfast on a regular basis. The results are shown in the table. Create a relative frequency table that shows the percentage of the entire group that is in each cell.

Take Daily No Daily Vitamins Vitamins Eat Breakfast No Breakfast

Take Daily Vitamins

384 268

476 304

No Daily Vitamins

Eat Breakfast No Breakfast

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5. Several college students are surveyed about their college location and preferred locations for a spring break trip. College Near the Coast

College Away From the Coast

Beach Break Ski Break

a. What percentage of people who prefer to spend spring break at the beach go to a college away from the coast?

b. What percentage of people who prefer to spend spring break skiing go to a college away from the coast?

6. A group of people are surveyed about whether they prefer to bike or run to exercise, and whether they prefer summer or winter weather. The results are in the table.

Summer Winter

Bike

Run

108

212 98

What value could go in the blank cell so that the percentage of people who like to bike and also prefer winter weather is 10%?

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Unit 2, Lesson 6: Interpreting Relative Frequency Tables

Warm-Up: Social Media and Friends The two-way frequency table shown summarizes the data explored in an earlier lesson comparing teenagers’ perceptions about the effect of social media and time spent with their friends in person.

Time Spent with Friends In Person

Effect of Social Media on Teenagers

Too Much Too Little Just Right Total

Mostly Positive

Mostly Negative

Neutral

Total

110

113

219

105

269

211

311

741

The same data is displayed in a relative frequency table.

186

409

Time Spent with Friends In Person

Effect of Social Media on Teenagers

Too Much Too Little Just Right Total

Mostly Positive

Mostly Negative

Neutral

Total

2.97%

2.83%

2.7%

8.5%

11.74%

10.39%

14.17%

29.55%

28.48%

41.97%

14.84%

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15.25%

25.10%

36.3%

55.20%

100.00%

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1. Identify a joint frequency and a marginal frequency from the first table.

2. What are the corresponding percentages in the relative frequency table?

3. Discuss with your partner how the percentages in the relative frequency table were determined.

Exploration Activity: Joint, Marginal, and Conditional Relative Frequencies The values you chose for question 2 of the Warm-Up are a joint relative frequency and a marginal relative frequency.

1. What is the sum of the joint relative frequencies?

2. What is the sum of the marginal relative frequencies for the variable “Effect of Social Media on Teenagers”?

3. What is the sum of the joint relative frequencies for those who believe that the effect of social media is mostly positive?

4. What is the sum of the joint relative frequencies for those who believe that they spend too much time with friends in person?

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5. Use only the values from the joint frequency table to answer the following questions. a. What percentage of those who spend too much time with friends in person believe that the effect of social media has been mostly positive?

b. What percentage of those who spend too much time with friends in person believe that the effect of social media has been mostly negative?

The percentages calculated in parts A and B are conditional relative frequencies. 6. Complete each statement.

a. 55.20% represents a b. 11.74% represents a

joint marginal conditional

relative frequency.

joint marginal conditional

relative frequency.

c. The ratio of those who believe the effect of social media on teenagers is neutral to those who spend just the right amount of time with friends in person represents a

joint marginal conditional

d. 59.80% represents a

relative frequency.

joint marginal conditional

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relative frequency.

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Guided Activity: Interpreting Two-Way Relative Frequency Tables Data was collected from a 2019 random sample of traffic accidents in Chicago. Data on the road conditions at the time of the accidents (dry, wet/slushy, icy) and the amount of damage caused ($500 or less, $501–$1,500, more than $1,500) is summarized. • 27.76% of the accidents involved damage of $500 or less.

• 28.77% of the accidents involved damage of $501 to $1,500. • 78.43% of the accidents involved dry road conditions.

• 20.40% of the accidents involved wet/slushy road conditions.

• Of the accidents with damage of more than $1,500, 69.77% involved dry road conditions. • Of the accidents with damage of $501–$1,500, 80.22% involved dry road conditions. • 24.79% of the accidents with icy road conditions had damage of $501–$1,500.

• 10.26% of the accidents with icy road conditions had damage of $500 or less. 1. Use the data provided to construct a two-way relative frequency table. Round values to the nearest hundredth. Damage

Road conditions

$𝟱𝟬𝟬 or less

More than $𝟭,𝟱𝟬𝟬

Total

Dry Wet/slushy Icy Total

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$𝟱𝟬𝟭–$𝟭,𝟱𝟬𝟬

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2. Interpret the joint relative frequency for dry road conditions with damage of $500 or less.

3. Interpret the marginal relative frequency for traffic accidents that had icy road conditions.

4. When the road conditions are icy, is the likelihood of the different amounts of damage the same? Explain your reasoning using conditional relative frequencies.

5. Does there appear to be an association between road conditions and the amount of damage caused?

6. Does this mean that more hazardous road conditions cause more damage in traffic accidents?

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Lesson Summary A two-way relative frequency table shows the joint relative frequencies and marginal relative frequencies of 2 categorical variables. A joint relative frequency is the ratio of the joint frequency and the total number of data points.

• All of the joint relative frequencies in a two-way relative frequency table sum to 100%.

• Marginal relative frequency is the ratio of the marginal frequency to the total number of data points. Each column or row of marginal relative frequencies sums to 100%. Conditional relative frequency is not overtly displayed in a two-way relative frequency table, but it can be found with a simple calculation. A conditional relative frequency is the ratio of a joint relative frequency to a marginal relative frequency. Equivalently, it is the ratio of a joint frequency to a marginal frequency. Conditional relative frequencies can also be used to analyze associations between categorical variables. When interpretating data with two-way frequency tables, it’s important to remember that if there is an association between the variables, it does not mean that one causes the other. Two variables can be associated without having a causal relationship.

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Practice Problems 1. The General Social Survey (GSS) interviews a random sample of Americans every two years. In one question, respondents are asked to rate their general happiness level. Sample results for 1972, 1998, and 2021 are summarized in the table. Year

General happiness

Very happy

𝟭𝟵𝟳𝟮

𝟭𝟵𝟵𝟴

𝟮𝟬𝟮𝟭

Total

855

1,575

2,308

4,738

1,606

2,806

4,014

486

Pretty happy Not too happy Total

265

891

340

783

2,160

923

1,528 8,426

After reviewing the two-way frequency table, Diana said, “It seems like people in 2021 were a lot happier than in 1972.” a. Why do you think Diana said this?

b. Explain why this may not be an accurate conclusion to draw from the data.

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2. Use the data from question 1 to create a two-way relative frequency table. Round values to the nearest tenth of a percent. Year

General happiness

Very happy

𝟭𝟵𝟳𝟮

𝟭𝟵𝟵𝟴

𝟮𝟬𝟮𝟭

Total

Pretty happy Not too happy Total

3. What can be concluded about the general happiness of Americans from the two-way relative frequency table created in question 2?

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Review Problems 4. Bananas cost $0.50 each, and apples cost $1.00 each.

Select all the combinations of bananas and apples that Elena could buy for exactly $3.50.

□ 2 bananas and 2 apples □ 3 bananas and 2 apples □ 1 banana and 2 apples □ 1 banana and 3 apples

□ 5 bananas and 2 apples □ 5 bananas and 1 apple

5. A group of 280 elementary school students and 40 adults are going on a field trip. They are planning to use two different types of buses to get to the destination. The first type of bus holds 50 people and the second type of bus holds 56 people. Andre says that 3 of the first type of bus and 3 of the second type of bus will hold all of the students and adults going on the field trip. Is Andre correct? Explain your reasoning.

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Unit 2, Lesson 7: Linear Models for Bivariate Quantitative Data

Warm-Up: Notice and Wonder: Crowd Noise What do you notice? What do you wonder? 𝑦 = 1.5𝑥 + 22.7

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Exploration Activity: Orange You Glad We’re Boxing Fruit? This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

Use the Desmos graphing calculator to help answer the questions.

1. Watch the video and record the weight for the number of oranges in the table.

2. Adjust the window settings as needed to fit the scatter plot of the data.

3. Drag the moveable points to reposition the line through the points so it fits the data well.

4. Estimate a value for the slope of the line. What does the value of the slope represent?

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5. Estimate the weight of a box containing 11 oranges. Will this estimate be close to the actual value? Explain your reasoning.

6. Estimate the weight of a box containing 50 oranges. Will this estimate be close to the actual value? Explain your reasoning.

7. Estimate the coordinates for the vertical intercept of the line you drew. What does the 𝑦-coordinate for this point represent?

8. Which point(s) are best fit by your linear model? How did you decide?

9. Which point(s) are fit the least well by your linear model? How did you decide?

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Guided Activity: Food Markup The scatter plot shows the sale price of several food items, 𝑦, and the cost of the ingredients used to produce those items, 𝑥, as well as a line that models the data. The line is also represented by the equation 𝑦 = 3.48𝑥 + 0.76.

1. What is the predicted sale price of an item that has ingredients that cost $1.50? Explain or show your reasoning.

2. What is the predicted ingredient cost of an item that has a sale price of $7? Explain or show your reasoning.

3. What is the slope of the linear model? What does that mean in this situation?

4. What is the 𝑦-intercept of the linear model? What does this mean in this situation? Does this make sense?

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Collaborative Activity: The Slope Is the Thing 1. Four linear models and their corresponding scatter plots are shown in the table. Complete the table by interpreting the slope and 𝑦-intercept of each linear model in the situations represented by the scatter plots. When interpreting slope, use the horizontal axis for 𝑥 and the vertical axis for 𝑦. When interpreting the 𝑦-intercept, assume the linear model continues to hold beyond the points shown. Linear Model and Scatter Plot 𝑦 = − 9.25𝑥 + 400

Interpretation of the Slope

Interpretation of the 𝒚-intercept

𝑦 = 0.44𝑥 + 0.04

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Linear Model and Scatter Plot 𝑦 = 4𝑥 + 87

Interpretation of the Slope

Interpretation of the 𝒚-intercept

𝑦 = − 2.4𝑥 + 25.0

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Lesson Summary While working on problems in math class, it can be easy to forget that reality is somewhat messy. Not all oranges weigh exactly the same amount, beans have different lengths, and even the same person running the same race multiple times will probably have different finishing times. To better understand these messy situations, more precise mathematical models are used to approximate them. Additional results can be interpolated or extrapolated from the models, with caution, to predict or estimate results beyond or between the data values that were used to build the model. Keep in mind that reality will vary a little bit from what mathematical models predict, and that the model will usually only be useful for a finite interval of the domain. For quantitative bivariate data, linear equations can also be used to model the relationship. Bivariate data is data that measures two characteristics of a population.

As explored in previous courses, bivariate quantitative data can be represented using a scatter plot. A scatter plot is a graph in the coordinate plane representing a set of bivariate numerical data that is used to observe the relationship between two variables. Each point on a scatter plot represents a specific pair of values from the raw data. The 𝑥- and 𝑦-values are used to plot each data point on the coordinate plane. When all the points in a data set are represented, a line of fit is often used to model the relationship between the 2 characteristics in the data set. A line of fit is a line drawn on a scatter plot to estimate the relationship between two sets of data. A line of fit is also known as a trend line. While different lines of fit can be drawn to model the relationship, the goal is to draw the line of best fit that most accurately models the relationship. This line can then be used as a model to make predictions.

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For example, the data in the scatter plot shown represents the price of a package of broccoli in relation to its weight. The data can be modeled by a line given by the equation 𝑦 = 0.46𝑥 + 0.92. The data does not all fall on the line because there may be factors other than weight that affect the price, such as the quality of the broccoli, the region where the package is sold, and any store-specific discounts. The slope and 𝑦-intercept of the model can be interpreted in context of the data.

• The 𝑦-intercept of the line represents the price for the package without any broccoli, which might include the cost of things like preparing the package and shipping costs for getting the vegetable to the store. In many situations, the data may not follow the same linear model farther away from the original given data, especially as 1 variable gets close to 0. For this reason, the 𝑦-intercept should always be interpreted in context to determine if it is reasonable.

• The slope can be interpreted as the approximate increase in price of the package for every 1 pound (lb.) of broccoli added to the package.

The equation can be used to predict additional values for the price of packages of broccoli that have weights near those observed in the data set. For example, even though the data does not include the price of a package that contains 1.7 lb. of broccoli, through interpolation, it can be predicted that the price is about $1.70 based on the equation of the line, since 0.46⋅1.7 + 0.92 ≈ 1.70.

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Practice Problems 1. The scatter plot shows the number of times a player came to bat and the number of hits they had. The scatter plot includes a point at (318, 80). Describe the meaning of this point in this situation.

2. The scatter plot shows the number of minutes (min.) people had to wait for service at a restaurant and the number of staff available at the time. A line that models the data is given by the equation 𝑦 = − 1.62𝑥 + 18, where 𝑦 represents the wait time, and 𝑥 represents the number of staff available. a. The slope of the line is − 1.62. What does this mean in this situation? Is it realistic?

b. The 𝑦-intercept is (0, 18). What does this mean in this situation? Is it realistic?

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3. A taxi driver records the time required to complete various trips and the distance for each trip. The best fit line is given by the equation 𝑦 = 0.467𝑥 + 0.417, where 𝑦 represents the distance in miles (mi.), and 𝑥 represents the time for the trip in min. a. Use the best fit line to estimate the distance for a trip that takes 20 min. Show your reasoning.

b. Use the best fit line to estimate the time for a trip that is 6 mi. long. Show your reasoning.

Review Problems 4. Data is collected about the number of wins and losses by a random sample of teams with an animal mascot and those with another kind of mascot. The column relative frequencies are shown in the table. Based on the information in the table, is there an association between the variables? Explain your reasoning.

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Wins Losses

Animal Mascot

Other Type of Mascot

74%

49%

26%

51%

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5. A random selection of indoor and outdoor pool managers are surveyed about the number of people in each age group that swim there. Results from the survey are displayed in the two-way table. Based on the data, does there appear to be an association between pool type and age group? Explain your reasoning.

6. Data from a random sample of people are collected about how they watch movies in the genres of action or mystery. Which value would best fit in the missing cell to suggest there is no association between the genre and how the movies are watched? A. 19 B. 89

Younger Than 18 18 or Older

Action Mystery

Outdoor Pool

Indoor Pool

317

352

163

Streaming Movies

Disc Rental

526

147

317

C. 147 D. 320

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Unit 2, Lesson 8: Fitting Lines to Data

Warm-Up: Selecting the Best Line Which of the lines is the best fit for the data in each scatter plot? Explain your reasoning.

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Collaborative Activity: Card Sort: Data Patterns Your teacher will give you a set of cards that show scatter plots.

1. Arrange all the cards in three different ways. Ensure that you and your partner agree on the arrangement before moving on to the next one. Sort all the cards in order from: a. best to worst for representing with a linear model

b. least to greatest slope of a linear model that fits the data well

c. least to greatest 𝑦-intercept of a linear model that fits the data well 2. For each card, write a sentence that describes how 𝑦 changes as 𝑥 increases and whether the linear model is a good fit for the data or not.

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Guided Activity: Fitting Lines with Technology The weight of ice cream sold in a day at a small store, in pounds (lb.), 𝑥, and the average temperature outside during the day, in degrees Celsius (℃), 𝑦, are recorded in the table. 𝒙

𝒚

20 6

4.5

6.5

3.5

21.5 7.5

19.5 6.5

21 7

18 5

This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

1. For this data, create a scatter plot and a line that fits the data well. 2. Use technology to compute the best fit line. Round any numbers to 2 decimal places.

3. What are the values for the slope and 𝑦-intercept for the best fit line? What do these values mean in this situation?

4. Use the best fit line to predict the 𝑦 value when 𝑥 is 10. Is this a good estimate for the data? Explain your reasoning.

5. Your teacher will assign a table of data from the previous activity. Following your teacher’s directions, use technology and the table of data to create a scatter plot that also shows the line of best fit, and then interpret the slope and 𝑦-intercept. © Accelerate Learning Inc. - All Rights Reserved

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Table for Card A

Table for Card B

Table for Card C

Table for Card D

Table for Card E

𝒙

𝒚

𝒙

𝒚

𝒙

𝒚

𝒙

𝒚

𝒙

2.2

11.332

2.2

7.142

2.2

11.5

2.2

3.3

4.5

3.4

10.741

3.5

8.19

3.5

9.2

3.3

3.8

6.5

3.3

3.6

3.9

5.7

3.3

3.6

3.8

3.9 4

11.86

10.848

10.716

10.628

10.584 10.54

3.3

6.11

8.088

3.6

8.346

3.9

8.604

3.8 4

2.92

8.69

4.9

10.144

4.9

9.464

4.4 4.7

4.9

8.7

5.1

7.7

5.1

10.056

5.1

9.636

5.1

5.5

9.88

5.5

9.98

5.5

5.2

6.7

5.5

8.5

6.6 7.7

10.5

9.753

8.6

6.6

10.313

7.7

9.5

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5.2

10.1

5.7

5.2

9.55

9.722

4.9 5

5.2

5.8

10.32

9.396

6.6

10.926

6.6

8.912

7.7

11.872

7.7

9.66

9.22

10.5

9.292

4.4

4.8

4.7

10.2

10.41

11.27

5.8 6

7.1 6.5

6.1

5.9

6.3

10.232

4.8

3.9

3.8

4.8

4.7

10.188

10.5

13.6

4.8

8.1

3.6

11.5

4.7

3.9

8.3

4.5

9.034

4.8

3.8

8.7

9.5

6.9

4.4

4.5

3.6

3.3

8.5

4.5

10.364 10.32

9.3

6.5

9.12

4.4

4.5

3.3

7.9

6.5 7

13.9

4.4

4.5

𝒚

4.7

4.9 5

5.7

5.1

4.9

5.5

5.5 4.3

3.9

5.2 5.5 6

11.5

13.2

11.5

12.2

11.2 12.5 13

1.3

6.6

13.1

0.5

7.7

14.813

1.9

13.5

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Lesson Summary Some data sets appear to have a linear relationship, in which case finding an equation of a line that fits the data can help you understand the association between the variables.

An association is a way to describe the form, direction, or strength of the relationship between the two variables in a bivariate data set. For numerical data, descriptions include linear or nonlinear, positive or negative, strong or weak.

The scatter plot shown appears to have a linear relationship. When a linear function seems useful for modeling a data set, it’s important to find a linear function that’s close to the data. The line should have a 𝑦-intercept and slope that follow the shape of the data in the scatter plot as closely as possible. Technology can help to quickly find a line of best fit for the data, as well as the equation of the line, which can be used to analyze the situation.

Other data sets may follow nonlinear trends or lack any apparent trend at all, such as on the scatter plot shown. For these scatter plots, a linear function may not be as accurate or useful for modeling the relationship or making predictions.

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Practice Problems This activity requires the use of an applet, so please make your way over to the digital platform to find the links.

1. Technology required. 𝒙

𝒚

102

115

107

122

125

127

101

104

120

a. Use graphing technology to create a scatter plot and find the best fit line.

127

b. What does the best fit line estimate for the 𝑦-value when 𝑥 is 100? 2. Technology required. a. What is the equation of the line of best fit? Round numbers to 2 decimal places.

b. What does the equation estimate for 𝑦 when 𝑥 is 2.3? Round to 3 decimal places.

c. How does the estimated value compare to the actual value from the table when 𝑥 is 2.3?

𝒙

𝒚

2.3

6.2

3.1

4.7

2.8 3

3.5 3.8

5.7 3.2 3

2.8

d. How does the estimated value compare to the actual value from the table when 𝑥 is 3? 226 | Unit 2

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3. Which of these scatter plots are best fit by the shown linear model?

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Review Problems 4. A seed is planted in a glass pot and its height is measured in centimeters (cm) every day. The best fit line is given by the equation 𝑦 = 0.404𝑥 − 5.18, where 𝑦 represents the height of the plant above ground level, and 𝑥 represents the number of days since it was planted. a. What is the slope of the best fit line? What does the slope of the line mean in this situation? Is it reasonable?

b. What is the 𝑦-intercept of the best fit line? What does the 𝑦-intercept of the line mean in this situation? Is it reasonable?

5. At a restaurant, the total bill and the percentage of the bill left as a tip are represented in the scatter plot. The best fit line is represented by the equation 𝑦 = − 0.632𝑥 + 27.1, where 𝑥 represents the total bill in dollars, and 𝑦 represents the percentage of the bill left as a tip. a. What does the best fit line estimate for the percentage of the bill left as a tip when the bill is $15? Is this reasonable? b. What does the best fit line predict for the percentage of the bill left as a tip when the bill is $50? Is this reasonable?

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6. A recent study investigated the amount of battery life remaining in alkaline batteries of different ages. The scatter-plot shows this relationship between the different alkaline batteries tested. The scatter plot includes a point at (7, 15). Describe the meaning of this point in this situation.

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Unit 2, Lesson 9: Introducing Correlation Coefficient

Warm-Up: Which One Doesn’t Belong: Linear Models Which one doesn’t belong?

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Exploration Activity: Card Sort: Scatter Plot Fit Your teacher will give you a set of cards that show scatter plots of data. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories. Then, sort the cards into 2 categories in a different way. Be prepared to explain the meaning of your new categories.

Collaborative Activity: Matching Correlation Coefficients A set of correlation coefficients is shown. • 𝑟 = − 1

• 𝑟 = − 0.95

• 𝑟 = − 0.74

• 𝑟 = − 0.06

• 𝑟 = 0.48

• 𝑟 = 0.65

• 𝑟 = 0.9 • 𝑟 = 1

1. Take turns with your partner to match a scatter plot with a correlation coefficient. 2. For each match you find, explain to your partner how you know it’s a match. 3. For each match your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

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Lesson Summary While mathematical tools can help find the best fit line for a data set, the strength of a linear relationship still needs to be determined. Scatter plots of data points that are close to the best fit line are better modeled by the line than scatter plots of data points that are farther from the line. The correlation coefficient is a convenient number that can be used to evaluate how well a line fits the data. The correlation coefficient is a number between 1 and −1 that describes the strength and direction of a linear association between two numerical variables. The sign of the correlation coefficient is the same as the sign of the slope of the best fit line. The closer the correlation coefficient is to 0, the weaker the linear relationship. When the correlation coefficient is closer to 1 or −1, the linear model fits the data better.

The correlation coefficient is usually represented by the letter 𝑟. Several scatter plots and their corresponding correlation coefficients are shown.

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While it is possible to try to fit a linear model to any data, the scatter plot needs to be analyzed to see if there is a possible linear trend. The correlation coefficient can be used to help determine whether the linear model makes sense to use to estimate the situation. In some cases, another type of function might be a better fit for the data, or the variables being examined may be uncorrelated, in which case other connections using other variables should be explored.

Practice Problems 1. Select all the values for 𝑟 that indicate a positive slope for the line of best fit. o 1

o −1 o 0.5

o − 0.5 o 0

o 0.8

o − 0.8 2. The correlation coefficient, 𝑟, is given for several different data sets. Which value for 𝑟 indicates the strongest correlation? A. 0.01

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3. Which of the values is the best estimate of the correlation coefficient for the line of best fit shown in the scatter plot? A. − 0.9 B. − 0.4 C. 0.4 D. 0.9 Review Problem 4. Technology required. This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

A study investigated the relationship between the amount of daily food waste measured, in pounds (lb.), 𝑦, and the number of people in a household, 𝑥. The data in the table displays the results of the study.

Number of People in Household

Food Waste (lb.)

Use graphing technology to create the line of best fit for the data in the table.

2 3

3.4

8.9

a. What is the equation of the line of best fit for this data? Round numbers to 2 decimal places.

b. What is the slope of the line of best fit? What does it mean in this situation? Is this realistic?

c. What is the 𝑦-intercept of the line of best fit? What does it mean in this situation? Is this realistic? 236 | Unit 2

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4 4 4

2.5 4.7 3.5 4

5.3

7.8

5 6 8

4.6 3.2 12

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Unit 2, Lesson 10: Using the Correlation Coefficient Warm-Up: Putting the Numbers in Context Match the variables to the scatter plot you think they best fit. Be prepared to explain your reasoning. 𝒙-variable

𝒚-variable

daily low temperature in Celsius for Denver, CO

boxes of cereal in stock at a grocery in Miami, FL

average number of free throws shot in a season

basketball team score per game

measured student height in feet

measured student height in inches

average number of minutes spent in a waiting room

hospital satisfaction rating

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Exploration Activity: Never Know How Far You’ll Go Priya takes note of the distance the car drives, in miles (mi.), 𝑥, and the time it takes to get to the destination for many trips, in minutes (min.), 𝑦.

1. Distance is one factor that influences the travel time of Priya’s car trips. What are some other factors?

Distance (mi.)

Travel Time (min.)

10 2. Which of these factors (including distance) most likely has the most consistent influence for all the car trips? Explain your reasoning.

15 20 25 26 30 32

3. Use technology to create a scatter plot of the data and add the best fit line to the graph.

4. What do the slope and 𝑦-intercept for the line of best fit mean in this situation?

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40 50 65 78

15 22 23 25 28 36 35 37 51 70 72

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5. Use technology to find the correlation coefficient for this data. Based on the value, how would you describe the strength of the linear relationship?

6. How long do you think it would take Priya to make a trip of 90 mi. if the linear relationship continues? If she drives 90 mi., do you think the prediction you made will be close to the actual value? Explain your reasoning.

Collaborative Activity: Correlation Zoo For each situation, describe the association between the variables based on the correlation coefficient. Make sure to mention whether the association is strong, moderate, weak, or none, as well as whether it is a positive association or negative association. 1. Number of steps taken per day and number of kilometers walked per day 𝑟 = 0.92

2. Temperature of a rubber band and distance that the rubber band can stretch 𝑟 = 0.84

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3. Car weight and distance traveled using a full tank of gas 𝑟 = − 0.86

4. Average fat intake per citizen of a country and average cancer rate of the country 𝑟 = 0.73

5. Score on a science exam and number of words written on the essay question 𝑟 = 0.28

6. Average time spent listening to music per day and average time spent watching TV per day 𝑟 = − 0.17

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Lesson Summary The value for the correlation coefficient can be used to determine the strength and direction of the association between the 2 variables represented on the scatter plot. • When both variables increase together, they have a positive association. • When one variable tends to increase as the other variable tends to decrease, the variables have a negative association. • When the data is tightly clustered around the best fit line, there is a strong association. • When the data is loosely spread around the best fit line, there is a weak association. • A moderate association would be somewhere in between. • When the data seems to have no clustering or best fit line, there is no association.

A correlation coefficient with a value near 1 suggests a strong, positive association between the variables. This means that most of the data tends to be tightly clustered around a line, and that as one of the variables increases in value, the other does as well. • The number of schools in a community and the population of the community are an example of variables that have a strong, positive association. Where there is a large population, there are usually a large number of schools, and small communities tend to have fewer schools, so the association is positive. These variables are closely tied together, so the association is strong. Similarly, a correlation coefficient near − 1 suggests a strong, negative association between the variables. Again, most of the data tends to be tightly clustered around a line, but now, as one value increases, the other decreases. • The time elapsed since leaving home and the distance left to reach school has a strong, negative association. As the time elapsed increases, the distance to school tends to decrease, so this is a negative association. Again, the variables are closely, linearly related, so this is a strong correlation.

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The context of the situation should be considered when determining whether the association is strong or weak. In physics, measuring with precise instruments, a correlation coefficient of 0.8 may not be considered strong. In social sciences, collecting data through surveys, a correlation coefficient of 0.8 may be considered very strong. Weaker correlations mean there may be other reasons the data varies besides the connection between the 2 variables represented. For example, a sample population’s number of pets and number of siblings have a weak correlation. There may be some relationship, but there are many other factors that account for the variability in the number of pets someone has besides their number of siblings.

Practice Problems 1. The number of hours worked, 𝑥, and the total dollars earned, 𝑦, have a strong positive relationship. Explain what it means to have a strong positive relationship in this situation.

2. The number of minutes on the phone and the customer satisfaction rating have a weak negative relationship. Explain what it means to have a weak negative relationship in this context.

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3. Technology required. Use a graphing calculator to answer the questions. This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

𝒙

𝒚

4.6

7.5

7.2

8.4

8.3

8.2

a. What is an equation of the line of best fit?

9.1

10.2

10.3

11.4 9.9

11.4 11

12.5

b. What is the value of the correlation coefficient?

4. Elena collects data to investigate the relationship between the number of bananas she buys at the store, 𝑥, and the total cost of the bananas, 𝑦. Which value for the correlation coefficient is most likely to match a line of best fit of the form 𝑦 = 𝑚𝑥 + 𝑏 for this situation? A. − 0.9 B. − 0.4 C. 0.4 D. 0.9

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Review Problems 5. The correlation coefficient, 𝑟, is given for several different data sets. Which value for 𝑟 indicates the weakest correlation? A. 0.01 B. 0.5

C. − 0.99 D. 1

6. Which of the following is the best estimate of the correlation coefficient for the data shown in the scatter plot? A. − 0.9 B. − 0.4 C. 0.4 D. 0.9

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Unit 2, Lesson 11: Interpreting Linear Models

Warm-Up: Fitting a Line

a. Which line fits the data better, the solid orange line or the dashed green line?

High temperature (°F)

1. The scatter plot shows the relationship between the day of the month and the high temperature, in degrees Fahrenheit (℉), for February 2023 in Lexington, Kentucky.

60 50 40 30 20 10

b. Explain your reasoning.

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Day of the month

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Collaborative Activity: High Temperature and Rainfall The table shows the mean high temperature in January, in ℉, and the mean amount of rainfall per year, in inches (in.), for 20 cities in Louisiana.

Alexandria Baton Rouge Bossier City Converse Grand Coteau Houma Kenner Lafayette Lake Charles Logansport

Mean High Temp in Jan. (°𝗙)

60.9

60.7

51.8

51.2

61.2

62.3

57.8

60.5

57.5

52.0

City

Mean Rain per Year (in.)

Mean High Temp in Jan. (°𝗙)

65.2

Marrero Monroe

54.0

New Iberia

62.5

Oakdale

62.6

Paradis

65.8

Ruston

54.8

Shreveport

51.4

Slidell

62.9

Winnsboro

The scatter plot shown represents the data from the table with a linear equation fit to model the relationship.

59.7

New Orleans

57.1

Average January high temperature

City

Mean Rain per Year (in.)

66 64 62 60 58 56 54 52 50 48 x

52 54

Average rainfall per year (in.)

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1. The screenshot shows the slope, 𝑦-intercept, and correlation coefficient associated with the linear relationship modeled on the scatter plot.

y 1 ~ mx1 + b

Statistics

r ² = 0.6772 r = 0.8229

a. Write the equation of the line of best fit shown.

Residuals

plot

Parameters

m = 0.487001

b = 31.2142

b. Interpret the meaning of each value listed in the table. Value

Meaning

Slope, 𝑚

𝑦-intercept, 𝑏

Correlation coefficient, 𝑟 2. Use the model to predict the mean high temperature in January for a Louisiana city that has a mean of 59.0 in. of rainfall per year.

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3. Suppose there is a city in Louisiana that averages 48.5 in. of rainfall per year.

a. Use the equation to predict the mean high temperature in January for that city.

b. Explain whether this January high temperature seems reasonable.

c. Discuss with your partner whether you think the model can be extrapolated to a city in Louisiana averaging 48.5 in. of rainfall per year. Summarize your discussion.

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Collaborative Activity: Hurricanes in Florida since 2011 y

Number of hurricanes or tropical storms to hit Florida

Michaela wondered whether there was a trend between the number of hurricanes and tropical storms that hit Florida and time. She used data for the number of hurricanes and tropical storms to hit Florida from 2011–2020 to create the scatter plot shown.

1. The equation 𝑦 = 0.612𝑥 + 2.75 models this relationship.

Years since 2011

a. Interpret the meaning of the slope and 𝑦-intercept in context. Value

Meaning

Slope, 𝑚

𝑦-intercept, 𝑏 b. Explain why it does not make sense for the slope or 𝑦-intercept to be decimal values in this context.

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2. Use the model for the relationship, 𝑦 = 0.612𝑥 + 2.75, to complete the following.

a. Predict the number of hurricanes and tropical storms expected to hit Florida in 2023.

b. Explain whether it seems reasonable for the number of hurricanes and tropical storms to hit Florida to continue increasing at this rate each year.

c. Discuss with your partner how the model could be used to estimate the number of hurricanes and tropical storms that hit Florida in 2008. Summarize your discussion.

Lesson Summary Mathematical tools such as scatter plots can be helpful in determining whether there may be a linear relationship between bivariate data. If there appears to be a linear relationship, the line of best fit and its correlation coefficient can confirm the direction (positive or negative) and strength (strong, moderate, weak, or none) of the association. When taking these features into account, along with interpretations of the slope and 𝑦-intercept in context, the model can be used to make predictions and solve problems. • When the model is used to estimate values or make predictions within the given range of data used to create the model, this is called interpolation. • When the model is used to predict future or historical values outside the range of data used to create the model, this is called extrapolation. Use caution when extrapolating from the data, as it’s not always reasonable to assume that the trends observed within the data used to create the model will extend outside of that model. For example, when examining data related to cell phone usage collected from 2000 to 2020, it would be unreasonable to extrapolate to 1950, when cell phones did not exist.

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Practice Problems 1. The scatter plot shows the finishing times for the Olympic gold medalist in the men’s 100-meter (m) dash for the Olympic Games since 1960.

2. The equation of the line of best fit shown on the scatter plot in question 1 is 𝑦 = 10.2217 − 0.0098𝑥, where 𝑦 is the finishing time of the Olympic gold medalist in the men’s 100 m dash, and 𝑥 is the number of years since 1960.

10.5

Finishing time of Olympic gold medalists (seconds)

Dana says that the strength and direction of the linear association between the two variables is moderately negative. Explain whether or not you agree with Dana.

9.5

8.5

8 x

Years since 1960

a. Interpret the 𝑦-intercept of the linear equation modeling the data. b. Explain the difference between the finishing time for the Olympic gold medal winner in the men’s 100 m dash and the 𝑦-intercept of the linear equation. c. Interpret the slope of the linear equation that models the data.

d. The Summer Olympics, which includes the 100 m dash, occurs every 4 years. By how many seconds does the model predict the first place finishing time will decrease every 4 years? e. Explain whether or not it’s reasonable to use this model to predict the first place finishing time in the men’s 100 m dash at the 2050 Summer Olympics. © Accelerate Learning Inc. - All Rights Reserved

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Review Problems 3. Select all the values for 𝑟 that indicate a strong, negative relationship for the line of best fit. o 1

o − 0.97 o − 0.45 o 0.53 o 0.9

o − 0.8 o −1

4. Noah collects data to investigate the relationship between the number of runs scored by his favorite baseball team, 𝑥, and the number of runs scored by his high school baseball team, 𝑦. Which value for the correlation coefficient is most likely to match a line of best fit of the form 𝑦 = 𝑚𝑥 + 𝑏 for this situation? A. −1 B. 0

C. 0.7 D. 1

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Unit 3: Functions

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Unit 3, Lesson 1: Describing and Graphing Situations

Warm-Up: Bagel Shop A customer at a bagel shop is buying 13 bagels. The shopkeeper says, “That would be $16.25.”

Jada, Priya, and Han, who are in the shop, all think it is a mistake. • Jada says to her friends, “Shouldn’t the total be $13.25?” • Priya says, “I think it should be $13.00.”

• Han says, “No, I think it should be $11.25.”

Explain how the shopkeeper, Jada, Priya, and Han could all be right.

Number of Bagels 1 2 3 4 5 6 7 8 9 10 11

Your teacher will give you instructions for completing the table.

12 13

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Exploration Activity: Be Right Back! Three days in a row, a dog owner tied his dog’s 5-foot-long leash to a post outside a store while he ran into the store to get a drink. Each time, the owner returned within minutes (min.). The dog’s movement each day is described here.

• Day 1: The dog walked around the entire time while waiting for its owner. • Day 2: The dog walked around for the first min., and then laid down until its owner returned. • Day 3: The dog tried to follow its owner into the store but was stopped by the leash. Then, it started walking around the post in one direction. It kept walking until its leash was completely wound up around the post. The dog stayed there until its owner returned. • Each day, the dog was 1.5 feet (ft.) away from the post when the owner left.

• Each day, 60 seconds (sec.) after the owner left, the dog was 4 ft. from the post. Your teacher will assign one of the days for you to analyze. Sketch a graph that could represent the dog’s distance from the post, in ft., as a function of time, in sec., since the owner left. Day _______

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Collaborative Activity: Talk about a Function Two pairs of quantities from a situation similar to one you’ve seen in this lesson are described. Each pair has a relationship that can be defined as a function. • Time, in sec., since a dog owner left his dog to go into the store and the total number of times the dog has barked • Time, in sec., since the owner left his dog to go into the store and the total distance, in ft., that the dog has walked while waiting

Choose 1 pair of quantities, and express their relationship as a function. 1. In that function, which variable is independent? Which one is dependent?

2. Write a sentence of the form “_____ is a function of _____.”

3. Sketch a possible graph of the relationship on the coordinate plane. Be sure to label and indicate a scale on each axis, and be prepared to explain your reasoning.

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Lesson Summary This lesson revisited the concepts of relations and functions explored in prior courses. A relation is a set of input-output pairs.

A function is a mathematical relation for which each element of the domain corresponds to exactly one element of the range. In a relation or function, the input is the independent variable, and the output is the dependent variable. The independent variable is used to calculate the value of another variable.

The dependent variable is a variable representing the output of a function. Consider the relationship between the amount of time since a plane takes off, in sec., and the plane’s height above the ground, in ft. • These 2 quantities form a function if time is the independent variable (the input) and height is the dependent variable (the output) because after any number of sec. since takeoff, it is only possible for the plane to be at 1 specific height above the ground. For example, 50 sec. after takeoff, the plane might have a height of 180 ft. At that moment, it cannot be simultaneously 180 ft. and 95 ft. above the ground. For any input, there is only 1 possible output, so the height of the plane is a function of the time since takeoff. • The 2 quantities do not form a function, however, if height is the input and time is the output because the plane can be at the same height for multiple lengths of time since takeoff. For instance, when the plane is 1,500 ft. above ground, it is possible that 300 sec. have passed. It is also possible that 425 sec., 275 sec., or some other amounts of time have passed. For any input, there are multiple possible outputs, so the time since takeoff is not a function of the height of the plane.

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Functions can be represented in many ways, including with a verbal description, a table of values, a graph, a mapping diagram, an expression or an equation, or a set of ordered pairs. When a function is represented with a graph, each point on the graph is a specific input-output pair. The graph shown represents the height of a plane as a function of time since takeoff.

The point (125, 400) means that 125 sec. after takeoff, the height of the plane is 400 ft.

Practice Problems 1. The relationship between the amount of time a car is parked, in hours (hr.), and the cost of parking, in dollars, can be described with a function. a. Identify the independent variable and the dependent variable in this function.

b. Describe the function with a sentence of the form “_____ is a function of _____.” c. Suppose it costs $3 per hr. to park, with a maximum cost of $12. Sketch a possible graph of the function. Be sure to label the axes.

d. Identify one point on the graph and explain its meaning in this situation.

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2. The prices of different burgers are shown on this sign. Based on the information from the menu, is the price of a burger a function of the number of patties? Explain your reasoning.

3. The distance a person walks, 𝑑, in kilometers (km), is a function of time, 𝑡, in min., since the walk begins. Select all true statements about the input variable of this function.

□ Distance is the input. □ Time of day is the input. □ Time since the person starts walking is the input. □ 𝑡 represents the input.

□ 𝑑 represents the input.

□ The input is not measured in any particular unit. □ The input is measured in hr. □ For each input, there are sometimes two outputs.

4. It costs $3 per hr. to park in a parking lot, with a maximum cost of $12.

Explain why the amount of time a car is parked is not a function of the parking cost.

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Review Problem 5. A recent survey investigated the relationship between the number of traffic tickets a person received and the cost of the person’s car insurance. The scatter plot displays the relationship.

The line that models the data is given by the equation 𝑦 = 73𝑥 + 146.53, where 𝑥 represents the number of traffic tickets, and 𝑦 represents the cost of car insurance. a. The slope of the line is 73. What does this mean in this situation? Is it realistic?

b. The 𝑦-intercept is (0, 146.53). What does this mean in this situation?

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Unit 3, Lesson 2: Function Notation

Warm-Up: Back to the Post! Here are the graphs of some situations you saw before. Each graph represents the distance of a dog from a post as a function of time since the dog owner left to purchase something from a store. Distance is measured in feet (ft.) and time is measured in seconds (sec.).

Day 1

1. Use the given graphs to answer these questions about each of the three days. a. How far away was the dog from the post 60 sec. after the owner left? Day 1:

Day 2:

Day 3:

Day 2

b. How far away was the dog from the post when the owner left? Day 1:

Day 2:

Day 3:

c. The owner returned 160 sec. after he left. How far away was the dog from the post at that time? Day 1:

Day 2:

Day 3

Day 3:

d. How many sec. passed before the dog reached the farthest point it could reach from the post? Day 1:

Day 2:

Day 3:

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2. Consider the statement, “The dog was 2 ft. away from the post after 80 sec.” Do you agree with the statement?

3. What was the distance of the dog from the post 100 sec. after the owner left?

Guided Activity: A Handy Notation Let’s name the functions that relate the dog’s distance from the post and the time since its owner left: function 𝑓 for Day 1, function 𝑔 for Day 2, function ℎ for Day 3. The input of each function is time in sec., 𝑡. 1. Use function notation to complete the table.

Day 1

Day 2

Day 3

a. Distance from post 60 sec. after the owner left b. Distance from post when the owner left c. Distance from post 150 sec. after the owner left 2. Describe what each expression represents in this context. a. 𝑓(15) b. 𝑔(48) c. ℎ(𝑡) 264 | Unit 3

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3. The equation 𝑔(120) = 4 can be interpreted to mean: “On Day 2, 120 sec. after the dog owner left, the dog was 4 ft. from the post.” What does each equation mean in this situation? a. ℎ(40) = 4.6

b. 𝑓(𝑡) = 5

c. 𝑔(𝑡) = 𝑑

Collaborative Activity: Birthdays

Rule 𝐵 takes a person’s name as its input, and gives their birthday as the output. Input

Output

Input

Output

Abraham Lincoln

February 12

August 26

Katherine Johnson

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Rule 𝑃 takes a date as its input and gives a person with that birthday as the output.

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1. Complete each table with three more examples of input-output pairs. 2. If you use your name as the input to 𝐵, how many outputs are possible? Explain how you know.

3. If you use your birthday as the input to 𝑃, how many outputs are possible? Explain how you know.

4. Only one of the two relationships is a function. The other is not a function. Which one is which? Explain how you know.

5. For the relationship that is a function, write two input-output pairs from the table using function notation.

Lesson Summary Since mathematicians often work with multiple functions at a time, function notation is used to name specific functions. Function notation is a notation that describes a function. For a function f when x is a member of the domain, the symbol f(x) denotes the corresponding member of the range.

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The notation 𝑦 = 𝑓(𝑥) defines a function named 𝑓. This indicates “𝑓 is a function of 𝑥.” The letter 𝑥 represents an input value, or independent variable. The letter 𝑦, or 𝑓(𝑥), represents the output value, or dependent variable. The notation 𝑓(𝑥) is read as “𝑓 of 𝑥.”

Name of function

Input of function f (x)

Output of function

A function can be named using any letter. Consider the function 𝑓(𝑤) = − 7𝑤2 − 10𝑤 + 3.

• To find the value 𝑓(10), substitute 10 for 𝑤, and evaluate. When 10 is the input, the output of the function is − 797. Using function notation, this is written 𝑓(10) = − 797.

• This same function can be represented in other ways. Shown are a table of values and a graph of 𝑓. The function can also be evaluated from each of these representations. Table of Values

Graph f (–1, 6)

𝒘

6 5

− 597

− 1,422

−1

𝒇(𝒘)

− 15 − 10

− 45

(–0.5, 6.25)

4 3

(0, 3)

1 -6 -5

-4 - 3

-2

-1

0 -1

-2 -3 -4

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To illustrate the efficiency of function notation, the graphs of 2 functions are shown. Each represents the cost of riding in a taxi from different companies, Friendly Rides and Great Cabs. For each taxi, the cost of a ride is a function of the distance traveled. The input is distance, in miles (mi.), and the output is cost, in dollars. • The point (2, 5.70) on 1 graph represents the cost of riding a Friendly Rides taxi for 2 mi.

• The point (2, 4.25) on the other graph represents the cost of riding a Great Cabs taxi for 2 mi. This information can be conveyed more efficiently using function notation.

• Name the function for Friendly Rides function 𝑓 and the function for Great Cabs function 𝑔. • To refer to the cost of riding each taxi for 2 mi, write 𝑓(2) and 𝑔(2).

• To say that a 2 mi. trip with Friendly Rides will cost $5.70, write 𝑓(2) = 5.70.

• To say that a 2 mi. trip with Great Cabs will cost $4.25, write 𝑔(2) = 4.25.

The function notation is a concise way to refer to a function and describe its input and output, which can be very useful. Throughout this and future courses, function notation will be used to talk about functions.

Practice Problems 1. Function 𝐶 takes time for its input and gives a student’s Monday class for its output. a. Use function notation to represent: A student has English at 10:00.

b. Write a statement to describe the meaning of 𝐶�11:15� = chemistry.

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2. The height of water in a bathtub, 𝑤, is a function of time, 𝑡. Let 𝑃 represent this function. Height is measured in inches (in.) and time in minutes (min.). Match each statement in function notation with a description. Description

Statement

a. After 20 min., the bathtub is empty.

𝑃(0) = 0

b. The bathtub starts out with no water.

𝑃(4) = 10

c. After 10 min., the height of the water is 4 in.

𝑃(10) = 4

d. The height of the water is 10 in. after 4 min.

𝑃(20) = 0

3. Function 𝑓 gives the distance of a dog from a post, in ft., as a function of time, in sec., since its owner left. Find the value of 𝑓(20) and of 𝑓(140).

4. Function 𝐶 gives the cost, in dollars, of buying 𝑛 apples. What does each expression or equation represent in this situation? a. 𝐶(5) = 4.50 b. 𝐶(2)

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Review Problems 5. A number of identical cups are stacked up. The number of cups in a stack and the height of the stack in centimeters (cm) are related. a. Can we say that the height of the stack is a function of the number of cups in the stack? Explain your reasoning.

b. Can we say that the number of cups in a stack is a function of the height of the stack? Explain your reasoning.

6. In a function, the number of cups in a stack is a function of the height of the stack in cm. a. Sketch a possible graph of the function on the coordinate plane. Be sure to label the axes. b. Identify one point on the graph and explain the meaning of the point in the situation.

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Unit 3, Lesson 3: Interpreting and Using Function Notation

Warm-Up: Observing a Drone Here is a graph that represents function 𝑓, which gives the height of a drone, in meters (m), 𝑡 seconds (sec.) after it leaves the ground.

Decide which function value is greater. 1. 𝑓(0) or 𝑓(4) 2. 𝑓(2) or 𝑓(5) 3. 𝑓(3) or 𝑓(7) 4. 𝑓(𝑡) or 𝑓(𝑡 + 1)

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1. What does each equation tell us about smartphone ownership? a. 𝑃(17) = 2,320 b. 𝑃( − 10) = 0 2. Use function notation to represent each statement. a. In 2010, the number of people who owned a smartphone was 296,600,000. b. In 2015, about 1.86 billion people owned a smartphone. 3. Mai is curious about the value of 𝑡 in 𝑃(𝑡) = 1,000.

a. What would the value of 𝑡 tell Mai about the situation? b. Is 4 a possible value of 𝑡 here?

4. Use the information you have so far to sketch a graph of the function.

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Collaborative Activity: Boiling Water The function 𝑊 gives the temperature, in degrees Fahrenheit (℉), of a pot of water on a stove 𝑡 minutes (min.) after the stove is turned on.

1. Take turns with your partner to explain the meaning of each statement in this situation. When it’s your partner’s turn, listen carefully to their interpretation. If you disagree, discuss your thinking and work to reach an agreement. a. 𝑊(0) = 72

b. 𝑊(5) > 𝑊(2) c. 𝑊(10) = 212

d. 𝑊(12) = 𝑊(10)

e. 𝑊(15) > 𝑊(30) f. 𝑊(0) < 𝑊(30)

2. If all statements in the previous question represent the situation, sketch a possible graph of function 𝑊.

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Lesson Summary What does a statement like 𝑝(3) = 12 mean? On its own, 𝑝(3) = 12 only reveals that when the function 𝑝 takes 3 as its input, its output is 12. If the quantities that the input and output represent are known, however, there is much more about the situation that the function represents. • If function 𝑝 gives the perimeter of a square whose side length is 𝑥 and both measurements are in inches (in.), then 𝑝(3) = 12 can be interpreted to mean “a square whose side length is 3 in. has a perimeter of 12 in.”

Statements like 𝑝(𝑥) = 32 can be interpreted to mean “a square with side length 𝑥 has a perimeter of 32 in.,” which implies that 𝑥 must be 8 in., so this can be rewritten as 𝑝(8) = 32.

• If function 𝑝 gives the number of followers, in thousands, 𝑥 months after a person started publishing content on social media, then 𝑝(3) = 12 means “3 months after the person started publishing content, they have 12,000 followers.”

It is important to pay attention to the units of measurement when analyzing a function to correctly interpret the situation. If it is overlooked that 𝑝(𝑥) is measured in thousands, 𝑝(𝑥) = 36 could be misinterpreted to mean “there are 36 followers after 𝑥 months,” when it actually means “there are 36,000 followers after 𝑥 months.”

A graph of a function can similarly help with interpreting statements in function notation.

Function 𝑓 gives the depth, in inches (in.), of water in a tub as a function of time, 𝑡, in min., since the tub started being drained. A graph of 𝑓 is shown.

Each point on the graph has the coordinates (𝑡,𝑓(𝑡)), where the first value is the input of the function and the second value is the output. • • •

𝑓(2) represents the depth of water 2 min. after the tub started being drained. The graph passes through (2, 5), so the depth of water is 5 in. when 𝑡 = 2. The equation 𝑓(2) = 5 captures this information. 𝑓(0) gives the depth of the water when the draining began, when 𝑡 = 0. The graph shows the depth of water to be 6 in. at that time, which can be expressed as 𝑓(0) = 6. 𝑓(𝑡) = 3 means that 𝑡 min. after the tub started draining, the depth of the water is 3 in. The graph shows that this happens when 𝑡 = 6.

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Practice Problems 1. Function 𝑓 gives the temperature, in degrees Celsius (℃), 𝑡 hours (hr.) after midnight. Choose the equation that represents the statement: “At 1:30 p.m., the temperature was 20℃.” A. 𝑓(1:30) = 20 B. 𝑓(1.5) = 20

C. 𝑓(13:30) = 20 D. 𝑓(13.5) = 20

2. Tyler filled up his bathtub, took a bath, and then drained the tub. The function 𝐵 gives the depth of the water, in in., 𝑡 min. after Tyler began to fill the bathtub. Explain the meaning of each statement in this situation. a. 𝐵(0) = 0 b. 𝐵(1) < 𝐵(7) c. 𝐵(9) = 11 d. 𝐵(10) = 𝐵(22) e. 𝐵(20) > 𝐵(40)

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3. Function 𝑓 gives the temperature, in ℃, 𝑡 hr. after midnight.

Use function notation to write an equation or expression for each statement. a. The temperature at 12 p.m.

b. The temperature was the same at 9 a.m. and at 4 p.m.

c. It was warmer at 9 a.m. than at 6 a.m.

d. Some time after midnight, the temperature was 24℃. 4. Select all points that are on the graph of 𝑓 if we know that 𝑓(2) = − 4 and 𝑓(5) = 3.4.

□ ( − 4, 2) □ (2, − 4) □ (3.4, 5) □ (5, 3.4) □ (2, 5)

5. Write three statements that are true about this situation. Use function notation. Function 𝑓 gives the distance of a dog from a post, in feet (ft.), as a function of time, 𝑡, in sec., since its owner left. Use the = sign in at least one statement and the < sign in another statement. 276 | Unit 3

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Review Problem 6. Technology required. This activity requires the use of an applet, so please make your way over to the digital platform to find the link. Data about the outside temperature in degrees Fahrenheit (℉), 𝑥, and gas used, in therms, 𝑦, for heating a building are given in the table. Use a graphing calculator to create the line of best fit for the data. a. What is the equation of the line of best fit for this data? Round numbers to the nearest whole number.

Temperature (°F)

Gas Usage (therms)

5,686

5,805

62 67 70 73

b. What is the slope of the line of best fit? What does it mean in this situation?

74 74 75 73

c. What does the line of best fit estimate for gas usage when the outside temperature is 59℉?

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65 59 58 62 64 67 70 73

7,373 5,636 3,782 3,976 3,351 3,396 2,936 3,078 4,549 7,022 6,106 4,566 4,608 5,790 6,501 3,843 Unit 3 | 277

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Unit 3, Lesson 4: Using Function Notation to Describe Rules

Warm-Up: Make It True Consider the equation 𝑞 = 4 + 0.8𝑝.

1. What value of 𝑞 would make the equation true when: a. 𝑝 is 7?

b. 𝑝 is 100?

2. What value of 𝑝 would make the equation true when: a. 𝑞 is 12?

b. 𝑞 is 60? Be prepared to explain or show your reasoning.

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Collaborative Activity: Four Functions Here are descriptions and equations that represent four functions. 1. Match each equation with a verbal description that represents the same function. Record your results. Equation 𝑓(𝑥) = 3𝑥 − 7 𝑔(𝑥) = 3(𝑥 − 7) ℎ(𝑥) = 3𝑥 − 7 𝑘(𝑥) = 𝑥−7 3

Description A. To get the output, subtract 7 from the input, and then divide the result by 3.

B. To get the output, subtract 7 from the input, and then multiply the result by 3.

C. To get the output, multiply the input by 3, and then subtract 7 from the result.

D. To get the output, divide the input by 3, and then subtract 7 from the result.

2. For one of the functions, when the input is 6, the output is − 3. Which is that function: 𝑓, 𝑔, ℎ, or 𝑘? Explain how you know.

3. Which function value, 𝑓(𝑥), 𝑔(𝑥), ℎ(𝑥), or 𝑘(𝑥), is the greatest when the input is 0? What about when the input is 10?

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Guided Activity: Rules for Area and Perimeter 1. A square that has a side length of 9 centimeters (cm) has an area of 81 square centimeters (sq. cm). The relationship between the side length and the area of the square is a function. a. Complete the table with the area for each given side length. Then, write a rule for a function, A, that gives the area of the square, in sq. cm, when the side length is 𝑠 cm. Use function notation.

Side Length (cm)

Area (sq. cm)

1 2 4 6 𝑠

b. What does A(2) represent in this situation? What is its value?

c. On the coordinate plane, sketch a graph of this function.

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2. A roll of paper that is 3 feet (ft.) wide can be cut to any length. a. If we cut a length of 2.5 ft., what is the perimeter of the paper?

b. Complete the table with the perimeter for each given side length. Then, write a rule for a function, 𝑝, that gives the perimeter of the paper, in ft., when the side length, in ft., is 𝑙. Use function notation.

Side Length (ft.)

Perimeter (ft.)

1 2

6.3 11 𝑙

c. What does 𝑃(11) represent in this situation? What is its value?

d. On the coordinate plane, sketch a graph of this function.

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Collaborative Activity: Data Plans A college student is choosing between two data plans for her new cell phone. Both plans include an allowance of 2 gigabytes (GB) of data per month. The monthly cost of each option can be seen as a function and represented with an equation. • Option A: A(𝑥) = 60

• Option B: 𝐵(𝑥) = 10𝑥 + 25

In each function, the input, 𝑥, represents the GB of data used over the monthly allowance. 1. The student decides to find the values of A(1) and 𝐵(1) and compare them. What are those values?

2. After looking at some of her past phone bills, she decided to compare A(7.5) and 𝐵(7.5). What are those values? 3. Describe each data plan in words.

4. Graph each function on the same coordinate plane. Then, explain which plan you think she should choose.

5. The student only budgeted $50 a month for her cell phone. She thought, “I wonder how many additional GB of data I would have for $50 if I go with Option B?” and wrote 𝐵(𝑥) = 50. What is the answer to her question? Explain or show how you know.

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Lesson Summary Some functions are defined by rules that specify how to compute the output from the input. Knowing the rule that defines a function can be useful because it helps us to do each of the following. • Find the output when the input is known. ◦ ◦

If the rule 𝑓(𝑥) = 5(𝑥 + 2) defines 𝑓, then find 𝑓(100) by evaluating 5(100 + 2).

If 𝑚(𝑥) = 3 − 1 𝑥 defines function 𝑚, then find 𝑚(10) by evaluating 3 − 1 𝑥(10). 2

• Create a table of values.

Tables of values representing functions 𝑓 and 𝑚 are shown.

𝒙

𝒇(𝒙)

𝒙

𝒎(𝒙)

2 3 4

20 25 30

2 4

• Graph the function. The values along the horizontal axis represent the inputs, and the values along the vertical axis represent the outputs. For function 𝑓, the values of 𝑓(𝑥) are the values along the vertical axis, which are often labeled 𝑦, so 𝑦 = 𝑓(𝑥). Because 𝑓(𝑥) is defined by the expression 5(𝑥 + 2), graph 𝑦 = 5(𝑥 + 2).

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For function 𝑚, write 𝑦 = 𝑚(𝑥), and graph 𝑦 = 3 − 1 𝑥. 2

• Find the input when the output is known. Suppose the output of function 𝑓 is 65 at some value of 𝑥, or 𝑓(𝑥) = 65, and that value needs to be determined. Because 𝑓(𝑥) is equal to 5(𝑥 + 2), write 5(𝑥 + 2) = 65, and solve for 𝑥, as shown. 5(𝑥 + 2) = 65 𝑥 + 2 = 13 𝑥 = 11

Each function described in this summary is a linear function because the value of the function changes by a constant rate and its graph is a line. A linear function is a function that has a constant rate of change.

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Practice Problems 1. Function 𝑃 represents the perimeter, in inches (in.), of a square with side length 𝑥 in. a. Complete the table. 𝒙

𝑷(𝒙)

b. Write an equation to represent function 𝑃. c. Sketch a graph of function 𝑃.

2. Functions 𝑓 and 𝐴 are defined by these equations. 𝑓(𝑥) = 80 − 15𝑥

𝐴(𝑥) = 25 + 10𝑥

Which function has a greater value when 𝑥 is 2.5?

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3. An equilateral triangle has three sides of equal length. Function 𝑃 gives the perimeter of an equilateral triangle of side length 𝑠. a. Find 𝑃(2)

b. Find 𝑃(10) c. Find 𝑃(𝑠) 4. Function 𝑔 is represented by the graph.

For what input value or values is 𝑔(𝑥) = 4? A. 2

B. − 2 and 2 C. 16

D. none 5. Function 𝑃 gives the perimeter of an equilateral triangle of side length 𝑠. It is represented by the equation 𝑃(𝑠) = 3𝑠. a. What does 𝑃(𝑠) = 60 mean in this situation?

b. Find a value of 𝑠 to make the equation 𝑃(𝑠) = 60 true.

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Review Problems 6. Function 𝐶 gives the cost, in dollars, of buying 𝑛 apples.

Which statement best represents the meaning of 𝐶(10) = 9? A. The cost of buying 9 apples B. The cost of 9 apples is $10.

C. The cost of 10 apples. D. Ten apples cost $9.

7. Function 𝐺 takes a student’s first name for its input and gives the number of letters in the first name for its output. a. Describe the meaning of 𝐺(Jada) = 4. b. Find the value of (Diego).

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Unit 3, Lesson 5: Features of Graphs

Warm-Up: Walking Home Diego is walking home from school at a constant rate. This graph represents function 𝑑, which gives his distance from home, in kilometers (km), 𝑚 minutes (min.) since leaving the school.

Use the graph to find or estimate: 1. 𝑑(0) 2. 𝑑(12) 3. the solution to 𝑑(𝑚) = 1 4. the solution to 𝑑(𝑚) = 0

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Exploration Activity: A Toy Rocket and a Drone A toy rocket and a drone were launched at the same time. Here are the graphs that represent the heights of two objects as a function of time since they were launched. Height is measured in meters (m) above the ground and time is measured in seconds (sec) since launch. 1. Analyze the graphs and describe, as precisely as you can, what was happening with each object. Your descriptions should be complete and precise enough that someone who is not looking at the graph could visualize how the objects were behaving.

2. Which parts or features of the graphs show important information about each object’s movement? List the features or mark them on the graphs.

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Collaborative Activity: The Jump In a bungee jump, the height of the jumper is a function of time since the jump begins. Function ℎ defines the height, in m, of a jumper above a river, 𝑡 sec. since leaving the platform. Here is a graph of function ℎ, followed by five expressions or equations and five graphical features.

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•

ℎ(0)

• first dip in the graph

•

ℎ(𝑡) = 0

• vertical intercept

•

ℎ(4)

• first peak in the graph

•

ℎ(𝑡) = 80

• horizontal intercept

•

ℎ(𝑡) = 45

• maximum

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1. Match each description about the jump to a corresponding expression or equation and to a feature on the graph. One expression or equation does not have a matching verbal description. Its corresponding graphical feature is also not shown on the graph. Interpret that expression or equation in terms of the jump and in terms of the graph of the function. Record your interpretation in the last row of the table. Description of Jump

Expression or Equation

Feature of Graph

a. the greatest height that the jumper is from the river

b. the height from which the jumper was jumping

c. the time at which the jumper reached the highest point after the first bounce

d. the lowest point that the jumper reached in the entire jump

2. Use the graph to: a. estimate ℎ(0) and ℎ(4).

b. estimate the solutions to ℎ(𝑡) = 45 and ℎ(𝑡) = 0.

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Lesson Summary The graph of a function can offer useful information about the quantities in a situation. Some points and features of a graph are particularly informative. The graph of function ℎ is shown, which gives the height, in m, of a ball 𝑡 sec. after it is tossed up in the air. Several key features can be identified and interpreted. • The point (0, 20) is the vertical intercept, or 𝑦-intercept, of the graph. It is the point where the graph intersects the vertical axis. This point represents the initial height of the ball at 20 m because when 𝑡 is 0, the value of ℎ(𝑡) is 20. The statement ℎ(0) = 20 captures this information. • The point (1, 25) is the highest point on the graph, so it is a maximum of the graph. A maximum of a function is a value of the function that is greater than or equal to all the other values. The maximum of the graph of the function is the corresponding highest point on the graph.

The value 25 is also the maximum value of the function ℎ. It reveals that the highest point the ball reaches is 25 m, and that this happens 1 sec. after the ball is tossed.

• The point (3.2, 0) is a horizontal intercept, or 𝑥-intercept, of the graph. It is a point where the graph intersects the horizontal axis. This point is also the lowest point on the graph, so it represents a minimum of the graph. The 𝒙-intercept is the value of 𝑥 at the point where a line or graph intersects the 𝑥-axis. The value of 𝑦 is 0 at this point.

A minimum of a function is a value of the function that is less than or equal to all the other values. The minimum of the graph of the function is the corresponding lowest point on the graph. This point reveals that the ball hits the ground 3.2 sec. after being tossed up, so the height of the ball is 0 m when 𝑡 is 3.2, which is written as ℎ(3.2) = 0. © Accelerate Learning Inc. - All Rights Reserved

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• The height of the graph increases when 𝑡 is between 0 and 1. Then, the graph changes direction, and the height decreases when 𝑡 is between 1 and 3.2. Neither the increasing part nor the decreasing part is a straight line. This suggests that the ball increases in height in the first sec. after being tossed and then starts falling between 1 sec. and 3.2 sec. It also reveals that the height does not increase or decrease at a constant rate. Because the intercepts of a graph are points on an axis, at least 1 of their coordinates is 0. The 0 corresponds to the input or the output of a function, or both. • A vertical intercept (typically the 𝑦-intercept) is on the vertical axis, so its coordinates have the form (0,𝑏), where the first coordinate is 0 and 𝑏 can be any number. The 0 is the input. • A horizontal intercept (typically the 𝑥-intercept) is on the horizontal axis, so its coordinates have the form (𝑎,0), where 𝑎 can be any number and the second coordinate is 0. The 0 is an output. • A graph that passes through (0, 0) intersects both axes, so that point is both a horizontal intercept and a vertical intercept. Both the input and output are 0.

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Practice Problems 1. This graph represents Andre’s distance from his bicycle as he walks in a park. Decide whether the following statements are true or false. a. The graph has multiple horizontal intercepts. _______________ b. A horizontal intercept of the graph represents the time when Andre was with his bike. _______________ c. A minimum of the graph is (17, 1). _______________ d. The graph has two maximums. _______________

e. About 21 sec. after he left his bike, he was the farthest away from it, at about 8.3 feet (ft.). _______________ 2. The graph represents the temperature, in degrees Fahrenheit (℉), as a function of time. a. Tell the story of the temperature throughout the day.

b. Identify the maximum and minimum of the function and where the function is increasing and decreasing.

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3. The graph represents the height, in ft., as a function of time. Match each feature of the situation with a corresponding statement in function notation. Feature

Statement

maximum height

ℎ(0) = 7

minimum height

ℎ(1.5)

height staying the same

starting height

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ℎ(4) ℎ(𝑡) = 6 for 7 ≤ 𝑡 ≤ 8

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Review Problems 4. Here are the equations that define three functions. 𝑥

𝑓(𝑥) = 4𝑥 − 5 𝑔(𝑥) = 4(𝑥 − 5) ℎ(𝑥) = 4 − 5

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

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

1 1 1 c. Which function value is the largest: 𝑓�100 �, 𝑔� �, or ℎ� �? 100 100

5. Function 𝑓 is defined by the equation 𝑓(𝑥)= 𝑥2. a. What is 𝑓(2)?

b. What is 𝑓(3)?

c. Explain why 𝑓(2) + 𝑓(3)≠𝑓(5).

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Unit 3, Lesson 6: Using Graphs to Find Average Rate of Change

Warm-Up: Temperature Drop Here are the recorded temperatures, in degrees Fahrenheit (℉), at three different times on a winter evening. Time

4 p.m.

6 p.m.

10 p.m.

Temperature

25℉

17℉

8℉

• Tyler says the temperature dropped faster between 4 p.m. and 6 p.m. • Mai says the temperature dropped faster between 6 p.m. and 10 p.m.

Who do you agree with? Explain your reasoning.

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Exploration Activity: Drop Some More The table and graph show a more complete picture of the temperature changes on the same winter day. The function 𝑇 gives the temperature in degrees Fahrenheit, ℎ hours since noon.

𝒉

𝑻(𝒉)

0 2 3 4 5 6

1. Find the average rate of change for the following intervals. Explain or show your reasoning. a. between noon and 1 p.m.

b. between noon and 4 p.m.

c. between noon and midnight

7 8 9

10 11 12

18 20 20 25 23 17 15 11 11 8 6 7

2. Remember Mai and Tyler’s disagreement? Use average rate of change to show which time period (4 p.m. to 6 p.m. or 6 p.m. to 10 p.m.) experienced a faster temperature drop.

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Guided Activity: Average Rate of Change The graphs show the populations of California and Texas over time.

1. Estimate the average rate of change in the population in each state between 1970 and 2010. Show your reasoning.

2. In this situation, what does each rate of change mean?

3. Which state’s population grew more quickly between 1900 and 2000? Show your reasoning.

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Lesson Summary The graph of 1 winter day’s temperature in Greenville, South Carolina, as a function of time is shown. • The temperature was 35℉ at 9 a.m. and 45℉ at 2 p.m., an increase of 10℉ over those 5 hours (hr.).

• The change in temperature wasn’t constant, however. The temperature rose from 9 a.m. to 10 a.m., stayed steady for an hour, and then rose again until 2 p.m. To determine how fast the temperature rose between 9 a.m. to 2 p.m., on average, calculate the average rate of change. The average rate of change of a function 𝑓 between inputs 𝑎 and 𝑏 is the change in the outputs divided by the change in the inputs, 𝑓(𝑏)⁻𝑓(𝑎) . It is the slope of the line joining ((𝑎, 𝑓(𝑎)) and 𝑏⁻𝑎 (𝑏, 𝑓(𝑏)) on the graph. • The work to determine the average rate of change from 9 a.m. to 2 p.m. is shown. 45⁻35 = 10 = 2 5 5

On average, the temperature between 9 a.m. and 2 p.m. increased 2℉ per hr.

• The work to determine the average rate of change from 2 p.m. to 8 p.m. is shown. 30⁻45 = ⁻15 = 2.5 ⁻ 6 6

On average, the temperature between 2 p.m. and 8 p.m. decreased 2.5℉ per hr.

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Notice that the sign of the average rate of change impacts its interpretation in context. In this context, a positive average rate of change represents an average increase in temperature, and a negative average rate of change represents an average decrease, or drop, in temperature. If determining the average rate of change using 2 points on a graph, the average rate of change is the slope of the line that connects the 2 points.

Practice Problems 1. The temperature was recorded at several times during the day. Function 𝑇 gives the temperature in degrees Fahrenheit, 𝑛 hours since midnight. Here is a graph for this function.

For each time interval, decide if the average rate of change is positive, negative, or zero. a. From 𝑛 = 1 to 𝑛 = 5 ______________ b. From 𝑛 = 5 to 𝑛 = 7 ______________

c. From 𝑛 = 10 to 𝑛 = 20 ______________

d. From 𝑛 = 15 to 𝑛 = 18 ______________

e. From 𝑛 = 20 to 𝑛 = 24 ______________

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2. The graph shows the total distance, in feet (ft.), walked by a person as a function of time, in seconds (sec.). a. Was the person walking faster between 20 and 40 sec. or between 80 and 100 sec.?

b. Was the person walking faster between 0 and 40 sec. or between 40 and 100 sec.?

3. The percent of voters between the ages of 18 and 29 that participated in each United States presidential election between the years 1988 to 2016 are shown in the table. Year Percentage of Voters Ages 18-29

1988

1992

1996

2000

2004

2008

2012

2016

35.7

42.7

33.1

34.5

45.0

48.4

40.9

43.4

The function 𝑃 gives the percent of voters between 18 and 29 years old that participated in the election in year 𝑡. a. Determine the average rate of change for 𝑃 between 1992 and 2000.

b. Pick two different values of 𝑡 so that the function has a negative average rate of change between the two values. Determine the average rate of change.

c. Pick two values of 𝑡 so that the function has a positive average rate of change between the two values. Determine the average rate of change.

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Review Problem 4. 𝑊 gives the weight of a puppy, in pounds, as a function of its age, 𝑡, in months. Describe the meaning of each statement in function notation. a. 𝑊(2) = 5 b. 𝑊(6) > 𝑊(4) c. 𝑊(12) = 𝑊(15)

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Unit 3, Lesson 7: Interpreting and Creating Graphs

Warm-Up: Which One Doesn’t Belong: Temperature over Time Which graph doesn’t belong?

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Exploration Activity: Flag Raising A flag ceremony is held at an Independence Day event. The height of the flag is a function of time. Shown are graphs that are each a possible representation of the function.

1. Your teacher will assign some of or all of the graphs to your group. a. For each graph assigned to you, explain what it tells us about the flag.

b. Decide as a group which graph(s) appear to be most realistic and which ones appear to be least realistic.

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2. Another graph that relates time and height is shown. a. Can this graph represent the time and height of the flag? Explain your reasoning.

b. Is this a graph of a function? Explain your reasoning.

Your teacher will show a video of a flag being raised. Function 𝐻 gives the height of the flag over time. Height is measured in feet (ft.). Time is measured in seconds (sec.) since the flag is fully secured to the string, which is when the video clip begins. 3. On the coordinate plane, sketch a graph that could represent function 𝐻. Be sure to include a label and a scale for each axis.

4. Discuss with your group why the graphs in this activity only use quadrant I. Summarize your discussion.

Guided Activity: Two Pools To prepare for a backyard party, a parent uses two identical hoses to fill a small pool that is 15 inches (in.) deep and a large pool that is 27 in. deep.

The height of the water in each pool is a function of time since the water is turned on.

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Here are descriptions of three situations. For each situation, sketch the graphs of the two functions on the same coordinate plane, so that 𝑆(𝑡) is the height of the water in the small pool after 𝑡 minutes (min.), and 𝐿(𝑡) is the height of the water in the large pool after 𝑡 min. In both functions, the height of the water is measured in inches. • Situation 1: Each hose fills one pool at a constant rate. When the small pool is full, the water for that hose is shut off. The other hose keeps filling the larger pool until it is full.

• Situation 2: Each hose fills one pool at a constant rate. When the small pool is full, both hoses are shut off.

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• Situation 3: Each hose fills one pool at a constant rate. When the small pool is full, both hoses are used to fill the large pool until it is full.

Collaborative Activity: The Bouncing Ball Your teacher will show you 1 or more videos of a tennis ball being dropped. The height of the ball is a function of time. Suppose the height is ℎ ft., 𝑡 sec. after the ball is dropped.

1. Use the blank coordinate plane to sketch a graph of the height of the tennis ball as a function of time. To help you get started, some pictures and a table are provided. Complete the table with your estimates before sketching your graph.

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Time (sec.) 0 0.28 0.54 0.74 1.03 1.48 1.88 2.25

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Height (ft.)

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2. Identify the 𝑥- and 𝑦-intercepts of the graph. Interpret the coordinates of each in terms of the tennis ball.

3. Find the maximum and minimum values of the function. Interpret each in terms of the tennis ball.

Lesson Summary Graphs can be used to help visualize the relationship between quantities in a situation, even with only a general description. Consider the description of a hiker’s journey on a trail. A hiker walked briskly and steadily for about 30 min. and then took a 10 min. break. Afterward, she jogged all the way to the end of the trail, which took about 20 min. There, she took a 15 min. break and then started walking back leisurely, stopping twice to enjoy the scenery. Her return trip along the same trail took 105 min.

The sketch of a graph of the distance the hiker has traveled as a function of time based on this description is shown. Even though the specific distances she traveled and the length of the trail are unknown, the graph still shows some important features of the situation. For example:

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• the intervals over which the distance increased or stayed constant • how quickly the distance was increasing • the amount of time the hiker was hiking

If distance from the trailhead (the start of the trail) as a function of time were being considered, the graph of the function might appear as shown.

It shows the distance increasing as the hiker was walking away from the trailhead, and then decreasing as she was returning to the trailhead.

Practice Problems 1. Descriptions of 4 situations in which the volume of water in a tank is a function of time are given in the table. Match each description to a corresponding graph. Description

Graph Graph 1

a. An empty 20-gallon (gal.) water tank is filled at a constant rate for 3 min. until it is half full. Then, it is emptied at a constant rate for 3 min.

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Graph 2 b. A full 10-gal. water tank is drained for 30 sec., until it is half full. Afterward, it gets refilled.

Graph 3 c. A 2,000-gal. water tank starts out empty. It is being filled for 5 hr., slowly at first, and faster later. Graph 4 d. An empty 100-gal. water tank is filled in 50 min. Then, a dog jumps in and splashes around for 10 minutes, letting 7 gal. of water out. The tank is refilled afterward. 2. The graphs show the distance, 𝑑, traveled by two cars, A and B, over time, 𝑡. Distance is measured in miles (mi.) and time is measured in hours (hr.). Which car traveled slower? Explain how you know.

3. Clare describes her morning at school yesterday: “I entered the school on the first floor, then walked up to the third floor and stayed for my class for an hour. Afterwards, I had an hour-long class in the basement, and after that I went back to the ground level and sat outside to eat my lunch.” Sketch a possible graph of her height from the ground floor as a function of time. © Accelerate Learning Inc. - All Rights Reserved

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4. Tyler filled up his bathtub, took a bath, and then drained the tub. The function gives the depth of the water, in inches, 𝑡 min. after Tyler began to fill the bathtub. These statements describe how the water level in the tub was changing over time. Use the statements to sketch an approximate graph of function. a. 𝐵(0) = 0

b. 𝐵(1) < 𝐵(7) c. 𝐵(9) = 11

d. 𝐵(10) = 𝐵(23)

e. 𝐵(20) > 𝐵(40)

Review Problems 5. Two functions are defined by these equations: 𝑓(𝑥) = 5.1 + 0.8𝑥

𝑔(𝑥) = 3.4 + 1.2𝑥

Which function has a greater value when 𝑥 is 3.9? How much greater?

6. Function 𝑓 is defined by the equation 𝑓(𝑥) = 3𝑥 − 7. Find the value of 𝑐 so that 𝑓(𝑐) = 20 is true.

7. Function 𝑉 gives the volume of water (liters) in a water cooler as a function of time, 𝑡 (min.). This graph represents function 𝑉. 316 | Unit 3

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a. What is the greatest water volume in the cooler?

b. Find the value or values of 𝑡 that make 𝑉(𝑡) = 4 true. Explain what the value or values tell us about the volume of the water in the cooler.

c. Identify the horizontal intercept of the graph. What does it tell you about the situation?

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Unit 3, Lesson 8: Comparing Graphs

Warm-Up: Population Growth This graph shows the populations of Baltimore and Cleveland in the 20th century. 𝐵(𝑡) is the population of Baltimore in year 𝑡. 𝐶(𝑡) is the population of Cleveland in year 𝑡.

1. Estimate 𝐵(1930) and explain what it means in this situation. 2. Here are pairs of statements about the two populations. In each pair, which statement is true? Be prepared to explain how you know. a. 𝐵(2000) > 𝐶(2000) or 𝐵(2000) < 𝐶(2000) b. 𝐵(1900) = 𝐶(1900) or 𝐵(1900) > 𝐶(1900) 3. Were the two cities’ populations ever the same? If so, when?

Exploration Activity: Wired or Wireless? 𝐻(𝑡) is the percentage of homes in the United States that have a landline phone in year 𝑡. 𝐶(𝑡) is the percentage of homes with only a cell phone. Here are the graphs of 𝐻 and 𝐶. © Accelerate Learning Inc. - All Rights Reserved

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1. Estimate 𝐻(2006) and 𝐶(2006). Explain what each value tells us about the phones. 2. What is the approximate solution to 𝐶(𝑡) = 20? Explain what the solution means in this situation.

3. Determine if each equation is true. Be prepared to explain how you know. a. 𝐶(2011) = 𝐻(2011) b. 𝐶(2015) = 𝐻(2015) 4. Between 2004 and 2015, did the percentage of homes with landlines decrease at the same rate at which the percentage of cell-phones-only homes increased? Explain your reasoning.

Collaborative Activity: Audience of TV Shows The number of people who watched a TV episode is a function of that show’s episode number. Here are three graphs of three functions, 𝐴, 𝐵, and 𝐶, representing three different TV shows.

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1. Match each description with a graph that could represent the situation described. One of the descriptions has no corresponding graph. a. This show has a good core audience. They had a guest star in the fifth episode that brought in some new viewers, but most of them stopped watching after that.

b. This show is one of the most popular shows, and its audience keeps increasing.

c. This show has a small audience, but it’s improving, so more people are noticing.

d. This show started out huge. Even though it feels like it crashed, it still has more viewers than another show.

2. Which is greatest, 𝐴(7), 𝐵(7), or 𝐶(7)? Explain what the answer tells us about the shows.

3. Sketch a graph of the viewership of the fourth TV show that did not have a matching graph.

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Collaborative Activity: Functions 𝒇 and 𝒈

1. Here are graphs that represent two functions, 𝑓 and 𝑔.

Decide which function value is greater for each given input. Be prepared to explain your reasoning. a. 𝑓(2) or 𝑔(2) b. 𝑓(4) or 𝑔(4) c. 𝑓(6) or 𝑔(6) d. 𝑓(8) or 𝑔(8)

2. Is there a value of 𝑥 at which the equation 𝑓(𝑥) = 𝑔(𝑥) is true? Explain your reasoning.

3. Identify at least two values of 𝑥 at which the inequality 𝑓(𝑥) < 𝑔(𝑥) is true.

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Lesson Summary Graphs are useful for comparing 2 or more functions. The graphs of functions 𝐶 and 𝑇, which give the populations (in millions) of California and Texas in year 𝑥, are shown.

The graphs can be used to make multiple comparisons of the populations of the 2 states. Four examples are shown in the table with a description of how the answer can be determined from the graphs. The justifications are also conveyed using function notation. Comparison In the early 1900s, California had a smaller population than Texas. Around 1935, the 2 states had the same population of about 5 million people.

After 1935, California has had more people than Texas. Both populations have increased over time, with no periods of decline.

Description Using Graphs

Description Using Function Notation

The graph of 𝐶 is below the graph of 𝑇 when 𝑥 is 1900.

𝐶(1900) < 𝑇 (1900)

The graphs intersect at about (1935, 5)

When 𝑥 is greater than 1935, the graph of 𝐶(𝑥) is above that of 𝑇(𝑥).

Both graphs slant upward from left to right.

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𝐶(1935) = 5, 𝑇(1935) = 5, and 𝐶(1935) = 𝑇(1935) 𝐶(𝑥) > 𝑇 (𝑥) for 𝑥 > 1935 𝐶(2010) − 𝐶(1900) 𝑇(2010) − 𝑇(1900) > 2010 − 1900 2010 − 1900

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Practice Problems 1. Functions 𝑅 and 𝐷 give the height, in feet (ft.), of a toy rocket and a drone, 𝑡 seconds (sec.) after they are released. Here are the graphs of 𝑅 (for the rocket) and 𝐷 (for the drone). Write each statement in function notation.

a. At 3 sec., the toy rocket is higher than the drone.

b. At the start, the toy rocket is 25 ft. above the drone. 2. 𝐴(𝑡) is the average high temperature in Aspen, Colorado, 𝑡 months after the start of the year. 𝑀(𝑡) is the average high temperature in Minneapolis, Minnesota, 𝑡 months after the start of the year. Temperature is measured in degrees Fahrenheit.

Which function had the higher average rate of change between the beginning of January and middle of March? What does this mean about the temperature in the two cities?

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3. Here are two graphs representing functions 𝑓 and 𝑔.

Select all statements that are true about functions 𝑓 and 𝑔.

□ 𝑓(0) > 𝑔(0)

□ There are two values of 𝑥 where 𝑓(𝑥) = 𝑔(𝑥). □ 𝑓( − 1) < 𝑔( − 1) □ 𝑓( − 3) > 𝑔(4)

4. The three graphs represent the progress of three runners in a 400-meter (m) race.

The solid line represents runner A. The dotted line represents runner B. The dashed line represents runner C. a. One runner ran at a constant rate throughout the race. Which one?

b. A second runner stopped running for a while. Which one? During which interval of time did that happen?

c. Describe the third runner’s race. Be as specific as possible.

d. Who won the race? Explain how you know.

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Review Problems 5. Function 𝐴 gives the area, in square inches (sq. in.), of a square with side length 𝑥 inches (in.). a. Complete the table.

𝒙

𝑨(𝒙)

b. Represent function 𝐴 using an equation.

c. Sketch a graph of function 𝐴.

6. Function 𝑓 is represented by 𝑓(𝑥) = 5(𝑥 + 11). a. Find 𝑓( − 2).

b. Find the value of 𝑥 such that 𝑓(𝑥) = 90 is true. 326 | Unit 3

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Unit 3, Lesson 9: Exploring Domain and Range

Warm-Up: Number of Barks Earlier, you saw a situation where the total number of times a dog has barked was a function of the time, in seconds (sec.), after its owner tied its leash to a post and left. Less than 3 minutes (min.) after he left, the owner returned, untied the leash, and walked away with the dog. 1. Could each value be an input of the function? Be prepared to explain your reasoning. 15 841 2

300

2. Could each value be an output of the function? Be prepared to explain your reasoning. 15 841 2

300

Exploration Activity: Cart Sort: Possible or Impossible? Your teacher will give you a set of cards that each contain a number. Decide whether each number is a possible input for the functions described here. Sort the cards into two groups—possible inputs and impossible inputs. Record your sorting decisions. 1. The area of a square, in square centimeters (sq. cm), is a function of its side length, 𝑠, in centimeters (cm). The equation 𝐴(𝑠) = 𝑠2 defines this function. a. Possible inputs:

b. Impossible inputs:

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2. A tennis camp charges $40 per student for a full-day camp. The camp runs only if at least 5 students sign up, and it limits the enrollment to 16 campers a day. The amount of revenue, in dollars, that the tennis camp collects is a function of the number of students that enroll. The equation 𝑅(𝑛) = 40𝑛 defines this function. a. Possible inputs:

b. Impossible inputs:

3. The relationship between temperature in Celsius and the temperature in Kelvin can be represented by a function 𝑘. The equation 𝑘(𝑐) = 𝑐 + 273.15 defines this function, where 𝑐 is the temperature in Celsius and 𝑘(𝑐) is the temperature in Kelvin. a. Possible inputs:

b. Impossible inputs:

Collaborative Activity: What about the Outputs? In an earlier activity, you saw a function representing the area of a square (function 𝐴) and another representing the revenue of a tennis camp (function 𝑅). Refer to the descriptions of those functions to answer these questions. 1. Here is a graph that represents function 𝐴, defined by 𝐴(𝑠) = 𝑠2, where 𝑠 is the side length of the square in centimeters. a. Name three possible input-output pairs of this function.

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b. Earlier we described the set of all possible input values of 𝐴 as “any number greater than or equal to 0.” How would you describe the set of all possible output values of 𝐴? 2. Function 𝑅 is defined by 𝑅(𝑛) = 40𝑛, where 𝑛 is the number of campers.

a. Is 20 a possible output value in this situation? What about 100? Explain your reasoning.

b. Here are two graphs that relate number of students and camp revenue in dollars. Which graph could represent function 𝑅? Explain why the other one could not represent the function.

c. Describe the set of all possible output values of 𝑅.

Exploration Activity: What Could Be the Trouble? Consider the function 𝑓(𝑥) =

6 . 𝑥 − 2

To find out the sets of possible input and output values of the function, Clare created a table and evaluated 𝑓 at some values of 𝑥. Along the way, she ran into some trouble. 1. Find 𝑓(𝑥) for each 𝑥-value Clare listed. Describe what Clare’s trouble might be. © Accelerate Learning Inc. - All Rights Reserved

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𝒙

𝒇(𝒙)

− 10

1 2

2. Use graphing technology to graph function 𝑓. What do you notice about the graph?

3. Use a calculator to compute the value you and Clare had trouble computing. What do you notice about the computation?

4. How would you describe the domain of function 𝑓?

Lesson Summary The domain is another key feature of functions. The domain is the complete set of possible values of the input of a function or relation.

Depending on the situation represented, a function may take all numbers as its input or only a limited set of numbers. Three examples are described. Function 𝐴 gives the area of a square, in sq. cm, as a function of its side length, 𝑠, in cm. 19

• The input of 𝐴 can be 0 or any positive number, such as 4, 7.5, or 3 . It cannot include negative numbers because lengths cannot be negative. • The domain of 𝐴 includes 0 and all positive numbers, or 𝑠 ≥ 0.

Function 𝑞 gives the number of buses needed for a school field trip as a function of the number of people, 𝑛, going on the trip. 330 | Unit 3

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• The input of 𝑞 can be 0 or positive whole numbers because a negative or fractional number of people doesn’t make sense. • The domain of 𝑞 includes 0 and all positive whole numbers. If the number of people at a school is 120, then the domain is limited to all nonnegative whole numbers up to 120, or 0 ≤ 𝑛 ≤ 120 where 𝑛 is a whole number.

Function 𝑣 gives the total number of visitors to a theme park as a function of days, 𝑑, since a new attraction was opened to the public.

• The input of 𝑣 can be positive or negative. A positive input represents days since the attraction was opened, and a negative input represents days before the attraction was opened. • The input can also be whole numbers or fractional. For example, the statement 𝑣(17.5) means 17.5 days after the attraction was opened.

• The domain of 𝑣 includes all numbers. If the theme park had been open for exactly 1 year before the new attraction was opened, then the domain would be all numbers greater than or equal to − 365, or 𝑑 ≥ − 365.

Functions can also be described using the range.

The range is the complete set of possible values of the output of a relation of function.

Once the domain of a function is known, the range can be determined to what makes sense in the situation. The range for each situation is described on the next page. • The output of Function 𝐴 is the area of a square, in sq. cm, which cannot be negative but can be 0 or greater and not limited to whole numbers. The range of 𝐴 is 0 and all positive numbers, or 𝐴≥0.

• The output of Function 𝑞 is the number of buses, which can only be 0 or positive whole numbers. If there are 120 people at the school, however, and if each bus could seat 30 people, then only up to 4 buses are needed. The range that makes sense in this situation would be any whole number that is at least 0 and at most 4, or 0≥𝑞≥4 where 𝑞 is a whole number.

• The output of Function 𝑣 is the number of visitors, which cannot be fractional or negative. The range of 𝑣, therefore, includes 0 and all positive whole numbers, or 𝑣≥0 where 𝑣 is a whole number. © Accelerate Learning Inc. - All Rights Reserved

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Practice Problems 1. The cost for an upcoming field trip is $30 per student. The cost of the field trip 𝐶, in dollars, is a function of the number of students 𝑥. Select all the possible outputs for the function defined by 𝐶(𝑥) = 30𝑥.

□ 20 □ 30 □ 50 □ 90

□ 100 2. A rectangle has an area of 24 sq. cm. Function 𝑓 gives the length of the rectangle, in centimeters, when the width is 𝑤 cm. Determine if each value, in cm, is a possible input of the function.

0.5 48

−6

3. Select all the possible input-output pairs for the function 𝑦 = 𝑥3.

□ ( − 1, − 1) □ ( − 2, 8) □ (3, 9)

□ (12 , 18 )

□ (4, 64)

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4. A small bus charges $3.50 per person for a ride from the train station to a concert. The bus will run if at least 3 people take it, and it cannot fit more than 10 people. Function 𝐵 gives the amount of money that the bus operator earns when 𝑛 people ride the bus. a. Identify all numbers that make sense as inputs and outputs for this function.

b. Sketch a graph of 𝐵.

Review Problems 5. Two functions are defined by the equations 𝑓(𝑥) = 5 − 0.2𝑥 and 𝑔(𝑥) = 0.2(𝑥 + 5). Select all statements that are true about the functions.

□ 𝑓(3)>0 □ 𝑓(3)>5

□ 𝑔( − 1) = 0.8

□ 𝑔( − 1)<𝑓( − 1) □ 𝑓(0) = 𝑔(0)

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6. The graph of function 𝑓 passes through the coordinate points (0, 3) and (4, 6).

Use function notation to write the information each point gives us about function 𝑓.

7. Match each feature of the graph with the corresponding coordinate point. If the feature does not exist, choose “none”.

Feature a. maximum b. minimum c. vertical intercept d. horizontal intercept

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Coordinate Point (0, 7)

(1.5, 2) (4, 16) none

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Unit 3, Lesson 10: Identifying Domain and Range from Different Representations Warm-Up: Notice and Wonder: Mapping Diagram A function is represented by the mapping diagram.

f (w)

–10

–1

–90

–597

What do you notice? What do you wonder?

Guided Activity: Identifying the Domain and Range of a Function 1. Consider the following statements. 12 ÷ 3 = 4 because 12 = 4 3. 6 ÷ 0 = 𝒙 because 6 = 𝒙 · 3.

In the second statement, what value can be substituted for 𝒙 so that the statement is true?

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2. Complete the table of values to represent the function described by the rule, and answer the following questions. a. Complete the table of values. Rule A

Table of Values

Divide 1 by the input

Input

Output

−3

−7

1 x

4 5

12 b. Complete the statements. In this function, _____ is not an allowable input. Therefore, the domain does not include ____. c. Work with your partner to determine if there is another input that is not allowable for 1𝑥 . d. The domain of rule A can be represented using inequalities for the input values defined as 𝒙. Complete the inequalities. e. Complete the statement.

𝑥 > ____ or 𝑥 < ____

These 2 inequalities can also be written as a single inequality, 𝑥 ≠ ___. 336 | Unit 3

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3. Use the given rule to complete the following. a. Complete the mapping diagram. Rule B

Mapping Diagram x

Square the input

–4

0 9

b. Based on the rule, what type of output values are unattainable for this function? c. Write an inequality to represent the range of this relation. y

4. For each function, write the domain and range as a list of numbers. Function A 5 4

Function B

Input

Output

–8

2 14

3 2 1

-5 -4 -3

-2

-1 0 -1

x 1

4 5

-2 -3 -4 -5

Domain: _______________

Domain: _____________

Range: ________________

Range: _______________

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5. A graph is shown. y

(–2, 4.5)

5 4 3 2 1

-4 - 3

-2

-1

0 -1

(3, 0.75) 1

-2 -3

a. Represent the domain and range of the graph using compound inequalities. Domain: _______________ Range: ________________ b. Compare this graph to function A in the previous problem. Discuss with your partner why this graph’s domain and range could not be represented by a list.

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Lesson Summary Functions can be represented in a variety of ways, including input-output rules and mapping diagrams, in addition to tables of values, graphs, and equations. • An input-output rule is a rule that takes an input value and uses it to determine an output value. • A mapping diagram shows how the inputs and outputs of a function are paired. The first column represents the domain of the function, and the second column represents the range. Arrows show how each input is mapped to each output. The domain of a function can be represented in several ways, including using lists, inequalities, or compound inequalities. A compound inequality is a conjunction of two or more inequalities.

For example, −4 ≤ 𝑥 ≤ 3 is a compound inequality that is read as “−4 is less than or 5 equal to 𝑥, which is less than or equal to 3 .” This inequality is true for any value of 𝑥 5 between −4 and 3 , including either of those values. 5

When the elements of the domain or range consist of a set number of values, the domain and range can be represented by a list.

When the elements of the domain or range consist of an infinite number of points between specified values, the domain and range are best represented by an inequality.

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Practice Problems 1. Identify the domain and range of each function. Represent each domain and range as a list or an inequality. Function 𝒙

𝒚

2.4

4.7

−4.7

−1.6

Domain

Range

3.9 Function

Domain

5 x

Divide 5 by the input

Function

Range

Domain

12 y 10 8

(1, 9)

(–1, 1) -10 -8

-6

4 2

-4 -2 0 -2

x 2

Range

10 12

-4 -6 -8

(4, –9)

-10 -12

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Review Problems 2. Jada walks to school. The function 𝐷 gives her distance from school, in meters (m), as a function of time, in minutes (min.), since she left home. What does 𝐷(10) = 0 represent in this situation?

3. Jada walks to school. The function 𝐷 gives her distance from school, in m, � min. since she left home. Which equation tells us “Jada is 600 m from school after 5 min.”? A. 𝐷(5) = 600 B. 𝐷(600) = 5 C. �(5) = 600 D. �(600) = 5

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Unit 3, Lesson 11: Introducing Set and Interval Notation

Warm-Up: Domain and Range of a Line Use the linear function 𝑓(𝑥) = −4𝑥 + 7 to answer the questions. 1. Which values are in the domain for 𝑓(𝑥)?

A. All values greater than −5 and less than 5

B. All values greater than −20 and less than 20 C. It cannot be determined. D. Any value

2. Which values are in the range for 𝑓(𝑥)?

A. All values greater than −5 and less than 5

B. All values greater than −20 and less than 20 C. It cannot be determined. D. Any value

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Guided Activity: Understanding Set Notation In the Warm-Up, you determined that any value, or all real numbers, are included the domain and range of 𝑓(𝑥) = −4𝑥 + 7. There are several ways to represent a domain and range of all real numbers shown in the table. Written Description

Inequalities

Set Notation

Domain

All real numbers

− ∞ < 𝑥 < ∞

{𝑥: 𝑥 ∈ ℝ} or {𝑥|𝑥 ∈ ℝ}

Range

All real numbers

−∞<𝑦<∞

{𝑦: 𝑦 ∈ ℝ} or {𝑦|𝑦 ∈ ℝ}

The representation of the domain using inequalities is read as, “negative infinity is less than 𝑥, which is less than infinity.”

The representation using set notation is often similar to an inequality, except when referring to a subset of the number system as in this example. The notation 𝑥 ∈ ℝ is read as, “𝑥 is an element of the real number system.” Refer to the Lesson Summary for other symbols that are used to represent sets of numbers in set notation.

Notice that the inequality sign used with infinity does not include an equal sign.

1. Discuss with your partner why the domain −∞ < 𝑥 < ∞ cannot be written as −∞ ≤ 𝑥 ≤ ∞ . Summarize your discussion.

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2. Complete the table for each written description. The first row has been completed. Written Description All numbers greater than or equal to negative three

Inequalities

Set Notation

𝑥 ≥ −3

{𝑥: 𝑥 ≥ −3} or {𝑥|𝑥 ≥ −3}

All numbers less than seven All numbers greater than or equal to negative two and less than or equal to eleven All numbers greater than zero and less than or equal to twenty

The set of integers

The set of rational numbers

3. Explain why writing the set of integers and the set of rational numbers using inequalities is not possible.

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Guided Activity: Understanding Interval Notation The graph of the function ℎ(𝑥) = 2𝑥2 + 4𝑥 − 6 is shown.

4 3 2 1 -5

-4

-3

-2

-1

0 -1

x 1

-2 -3 -4 -5 -6 -7 -8

Several representations of the domain and range of ℎ(𝑥) are shown, including an additional notation called interval notation.

Domain Range

-9 -10

Inequality

Set Notation

Interval Notation

−∞ < 𝑥 < ∞

{𝑥: 𝑥 ∈ ℝ} or {𝑥|𝑥 ∈ ℝ}

(−∞, ∞)

−8 ≤ 𝑦 < ∞

{𝑦| 𝑦≥ −8} or {𝑦:𝑦 ≥−8}

[−8, ∞)

1. Discuss with your partner what you think the symbols (, [, ), and ] mean in interval notation. Summarize your discussion.

2. Notice that −∞ and ∞ are always bound by parentheses when expressing an interval that goes infinitely in either direction. Explain why a bracket is not used for −∞ and ∞.

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Collaborative Activity: Using Set and Interval Notation 1. Complete the table by expressing each set of numbers given as an inequality using set and interval notation. Inequality

Set Notation

Interval Notation

𝑥 ≥ 0 −∞ < 𝑥 ≤ 9 −9 < 𝑦< 25 2

144 ≤ 𝑦 < 627 6 5

2. The graph of a function is shown on the coordinate plane. (−3, 3)

4 3 2 1

-6 -5 -4 -3 -2 -1 0 -1 -2 -3

Complete the table by expressing the domain and range of the function in set and interval notation. Set Notation

x 1

(5, −2.333)

-4 -5 -6

Interval Notation

Domain Range

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3. The graph of a function is shown on the coordinate plane.

Complete the table by expressing the domain and range of the function in set and interval notation. Set Notation

Interval Notation

Domain Range

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Lesson Summary In addition to inequalities, other ways to represent sets of numbers include set notation and interval notation. For a function, the domain, range, and intervals of either can be represented using both notations. • Set notation is a shorthand used to write sets—often sets with an infinite number of elements. The set {𝑥: 𝑥 >0} or {𝑥|𝑥 >0} is read as, ”the set of all 𝑥, such that 𝑥 is greater than 0.” • The table shows number sets within the real number system, examples of the types of numbers included in each set, and the symbol used when referring to the set in set notation if there is one.

Set of Numbers

Examples

Symbol

Natural Numbers

1, 2, 3, 4 …

ℕ

Whole Numbers

0,1, 2, 3,…

No symbol

Integers

…, −4, −3, −2, −1, 0, 1, 2, 3, 4…

ℤ

Rational Numbers

−207 ,−2 , −0.25, √9, 4.01, ∛125, 14 5 , 89

Irrational Numbers Real Numbers

6 −√17, 𝜋,√2 ∛28

All rational and irrational numbers

ℚ No symbol ℝ

• Interval notation is another way of expressing a set using an inequality. Parentheses, “(” or “)”, means “not included” or “open.” Brackets, “[”or “]”, means “included” or “closed.”

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Practice Problems 1. Express the set of natural numbers using set notation.

2. Express the domain and range of the function h(𝑥) = 2𝑥 − 5.3 using set and interval notation. Set Notation

Interval Notation

Domain Range

3. The graph of 𝑦 = 1.5𝑥2 + 3𝑥 − 4.7 is shown on the coordinate plane. Julissa incorrectly wrote the domain and range using interval notation. Her answer is shown. Domain: [−6.2, ∞.] Range: [−∞, ∞]

Explain the error(s) in Julissa’s work.

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Review Problems 4. The area 𝐴 of a rectangle is represented by the formula 𝐴 = 𝑙𝑤, where 𝑙 is the length and 𝑤 is the width. The length of the rectangle is 5. Write an equation that makes it easy to find the width of the rectangle if we know the area and the length.

5. The table shows the volume of water, in cubic meters (cu. m), 𝑉, in a tank after water has been pumped out for a certain number of minutes (min.). Time after Pumping Begins

Volume of Water (cu. m)

27.5

7.5

Which equation could represent the volume of water, in cu. m, after 𝑡 min. of water being pumped out? A. 𝑉 = 30 − 2.5𝑡 B. 𝑉 = 30 − 0.5𝑡

C. 𝑉 = 30 − 0.5𝑡2

D. 𝑉 = 30 − 0.1𝑡2

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Unit 3, Lesson 12: Digging Deeper into Domain and Range

Warm-Up: Which One Doesn’t Belong: Unlabeled Graphs Which one doesn’t belong?

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Exploration Activity: Time on the Swing A child gets on a swing in a playground, swings for 30 seconds (sec.), and then gets off the swing. 1. Here are descriptions of four functions in the situation and four graphs representing them. The independent variable in each function is time, measured in sec. Match each function with a graph that could represent it. Then, label the axes with the appropriate variables. Be prepared to explain how you make your matches. • Function 𝘩: The height of the swing, in feet (ft.), as a function of time since the child gets on the swing

• Function 𝑟: The amount of time left on the swing as a function of time since the child gets on the swing

• Function 𝑑: The distance, in ft., of the swing from the top beam (from which the swing is suspended) as a function of time since the child gets on the swing

• Function 𝑠: The total number of times an adult pushes the swing as a function of time since the child gets on the swing

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2. On each graph, mark one or two points that, if you have the coordinates, could help you determine the domain and range of the function. Be prepared to explain why you chose those points.

3. Once you receive the information you need from your teacher, describe the domain and range that would be reasonable for each function in this situation. Function 𝘩:

Function 𝑟:

Function 𝑑:

Function 𝑠:

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Collaborative Activity: Back to the Bouncing Ball A tennis ball was dropped from a certain height. It bounced several times, rolled along for a short period, and then stopped. Function 𝘏 gives its height over time. A partial graph of 𝘏 is shown. Height is measured in ft. Time is measured in sec.

Use the graph to help you answer each question. Be prepared to explain what each value or set of values means in this situation. 1. Find 𝘏 (0). 2. Solve 𝘏(𝑥) = 0. 3. Describe the domain of the function.

4. Express the range of the function using set or interval notation.

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Lesson Summary The graph of a function can provide information about its domain and range. The graphs of 2 functions explored previously in the unit are shown. • The first graph represents the best price of bagels as a function of the number of bagels bought. • The second graph represents the height of a bungee jumper as a function of seconds since the jump began. What are the domain and range of each function? The number of bagels cannot be negative but could include 0, which would mean no bagels were purchased. The domain of the function, therefore, includes 0 and positive whole numbers, or 𝑛 ≥ 0 where 𝑛 is whole numbers. In such cases, expressing the domain using set notation makes more sense because the domain includes only values from a specific subset of real numbers. Using set notation, the domain can be expressed as {𝑥: ∈ ℤ| 𝑥 ≥0}.

The lowest price can be $0 if no bagels are purchased, certain multiples of 1.25, certain multiples of 6, and so on. Therefore, the range includes 0 and certain positive values. In this case, it is easier to describe the domain verbally than any specific mathematical notation.

The domain of the height function would include any amount of time since the jump began, up until the jump is complete. The graph shows that this happened more than 70 sec. after the jump began, but the exact value of 𝑡 is not shown. The graph shows a maximum height of 80 meters (m) and a minimum height of 10 m. It can be concluded that the range of this function includes all values that are at least 10 and at most 80. Using an inequality, the range can be expressed as 10 ≤ 𝑦 ≤ 80. Using interval notation, it can be expressed as [10,80]. © Accelerate Learning Inc. - All Rights Reserved

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Practice Problems 1. A child tosses a baseball up into the air. On its way down, it gets caught in a tree for several seconds before falling to the ground.

Select the best description of the range of this function. A. {𝑥: 0 ≤ 𝑥 ≤ 8.5}

B. {𝑦: 5 < 𝑦 < 12.5}

C. {𝑦: 0 ≤ 𝑦 ≤ 12.5}

D. The range includes all integers between 0 and 12.5. 2. To raise funds for a trip, members of a high school math club are holding a game night in the gym. They sell tickets at $5 per person. The gym holds a maximum of 250 people. The amount of money raised is a function of the number of tickets sold. Which statement accurately describes the domain of the function? A. all numbers less than 250 B. all integers C. all positive integers D. all positive integers less than or equal to 250

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3. The graph of 𝘏 shows the height, in m., of a rocket 𝑡 sec. after it was launched. a. Find 𝘏(0). What does this value represent?

b. Describe the domain of this function.

c. Describe the range of this function.

d. Solve 𝘏(𝑥) = 0. What does this value represent? Review Problems 4. Mai has to decide between two cafeteria meal plans. Under plan A, each meal costs $2.50. Under plan B, one month of meals costs $30. a. Write an equation for function 𝐴, which gives the cost, in dollars, of buying 𝑛 meals under plan A.

b. Write an equation for function 𝘉, which gives the cost, in dollars, of buying 𝑛 meals under plan B.

c. Mai estimates that she’ll buy 15 meals per month. Which meal plan should she choose? Explain your reasoning.

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5. Lin completes a 5K using a combination of walking and running. Here are four graphs that represent four possible situations. Each graph shows the distance, in m, as a function of time, in minutes (min.).

Match each description with a graph that could represent it.

Description

Graph

a. Lin starts out running but then slows down to a jog. After 10 min., she stops for a water break. She then runs the rest of the way.

Graph 1

b. Lin starts the race walking, gradually getting faster and faster.

Graph 2

c. Lin jogs at a steady pace for the entire race.

Graph 3

d. Lin starts out walking and then moves to a steady run. After 15 min., she stops to stretch a cramped leg. Then, she walks the rest of the way.

Graph 4

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Unit 4: Systems of Equations

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Unit 4, Lesson 1: Introduction to Systems of Linear Equations

Warm-Up: A System of Linear Equations 1. Consider the linear equations shown. 𝑦 = − 5𝑥 + 7 𝑦 = 2 − 3𝑥

a. Complete the statement.

If 𝑦 = − 5𝑥 + 7 and 𝑦 = 2 − 3𝑥, then − 5𝑥 + 7 = ___________________ . b. Solve the equation you wrote in part A for 𝑥.

c. Use your answer from part B to solve for 𝑦 using either equation. or

𝑦 = − 5𝑥 + 7

𝑦 = 2 − 3𝑥

d. Use the 𝑥- and 𝑦-values to create an ordered pair. (𝑥, 𝑦) = __________

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Exploration Activity: Exploring Systems of Equations Work with your partner to complete the following. 1. The graphs of 𝑦 = − 𝑥 + 5 and 𝑦 = 1 𝑥 + 2 are shown 2 on the coordinate plane. a. What do you notice about the graphs?

5 4 3 2 1 -2 -1 0 -1

x 1

-2

b. Use the definition found in the Lesson Summary to explain why the graph is a system of equations.

-3

c. Discuss with your partner what the quote from the definition, “The solution is the values for the variables that make all equations in the system true,” means. Summarize your discussion.

To find the values for the variables that make both 𝑦 = − 𝑥 + 5 and 𝑦 = 1 𝑥 + 2 true, the 2 equations can be set equal to each other. Since 𝑦 = − 𝑥 + 5 and 𝑦 = 1 𝑥 + 2, then − 𝑥 + 5 = 1 𝑥 + 2. 2

d. Solve the equation − 𝑥 + 5 = 1 𝑥 + 2 for 𝑥.

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The solution only gives the 𝑥-value. e. Decide who will be partner A and who will be partner B. Use the partially completed work to substitute the value of 𝑥 into 1 of the equations to find 𝑦.

Partner A 𝑦 = − 𝑥 + 5

𝑦 = − (

) + 5

𝑦 = ________ + 5 𝑦 = ________

Partner B 𝑦 = 1 𝑥 + 2 2

𝑦 = �1 ∙ � + 2 2

𝑦 = ________ + 2 𝑦 = ________

f. Verify the 𝑦-value with your partner.

g. Use the 𝑥- and 𝑦-values to create an ordered pair. (𝑥, 𝑦) = (________ , ________)

h. Plot your point on the original graph.

i. Discuss with your partner how to find the solution of the system using the graph.

2. Complete the statements. The solution of a system of equations is the

slope. 𝑦-intercept. point of intersection.

When 𝑥 = _______ and 𝑦 = _______, both 𝑦 = − 𝑥 + 5 and 𝑦 = 1 𝑥 + 2 are true. The point 2

(________ , ________) is a point on the graph of both lines and is therefore a solution to the system of equations.

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Guided Activity: Solving Linear Systems of Equations 1. A system of equations is shown. �

𝑦 = 0.8𝑥 𝑦 = 0.8𝑥 − 3.2

4 3

The system of equations is also shown on the coordinate plane.

2 1

Since 𝑦 = 0.8𝑥 and 𝑦 = 0.8𝑥 − 3.2, then 0.8𝑥 = 0.8𝑥 − 3.2.

-6 -5 -4 -3 -2 -1 0 -1

x 1

-2 -3 -4 -5

a. Solve 0.8𝑥 = 0.8𝑥 − 3.2 for 𝑥.

-6

b. Discuss with your partner how the solution to 0.8𝑥 = 0.8𝑥 − 3.2 and the graph of the system are related. Summarize your discussion.

2. Consider the following system of linear equations.

a. Complete the statement.

𝑦 = − 0.5𝑥 + 6 � 𝑦 = 6 − 1 𝑥 2

Since 𝑦 = − 0.5𝑥 + 6 and 𝑦 = 6 − 1 𝑥, then ___________ = ___________. 366 | Unit 4

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b. Solve the equation you wrote in part A for 𝑥.

c. Discuss with your partner what you noticed while solving the equation. Predict what the graph of the system of equations would look like. Make a sketch of your prediction.

d. Compare your predictions with another partner pair.

e. The graph of 𝑦 = 6 − 1 𝑥 is shown on the coordinate plane. 2

Graph 𝑦 = − 0.5𝑥 + 6 on the same coordinate plane. 9

8 7 6 5 4 3 2 1 -2 -1 0 -1

x 1

10 11 12 13 14

-2

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f. How many 𝑥- and 𝑦-value ordered pairs are the same for both lines?

g. How many solutions does this system have?

Lesson Summary This lesson explored systems of equations. A system of equations is a set of 2 or more equations. Each equation contains 2 or more variables. The solution is the values for the variables that make all equations in the system true. A system of linear equations can have 1 solution, no solutions, or infinitely many solutions. • For a system of linear equations with 1 solution, there is 1 ordered pair (𝑥, 𝑦) that satisfies all equations in the system. When graphed, this ordered pair will be the point where the graphs intersect. • For a system of linear equations with no solutions, there is not an ordered pair (𝑥, 𝑦) that satisfies all equations in the system. When graphed, there will be no point where the graphs intersect because the graphs are parallel lines. • For a system of linear equations with infinitely many solutions, there are infinite ordered pairs (𝑥, 𝑦) that satisfy all equations in the system. When graphed, the lines will lie on top of each other because the equations represent the same line. 368 | Unit 4

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Practice Problems 1. The graphs of 3 systems of equations are shown. Match each system of equations with its graphical representation. Graph

System of Equations 6

5 4

¡ �

3 2 1 -8 -7

-6 -5 -4 -3 -2 -1 0 -1

¡ �

-2 -3 -4

¡ �

-5 -6

6 4

¡ �

3 2 1

x 1

-2

¡ �

-3 -4 -5 -6

Book 1.indb 369

𝑦 = 1.25 + 0.25𝑥 𝑦 = 2 𝑥 +5

3 𝑦 = 1 𝑥 + 3 3

𝑦 = 1 𝑥 +1

6 𝑦 = 1 𝑥 − 1 6

-6 -5 -4 -3 -2 -1 0 -1

𝑦 = 1 𝑥 + 5

¡ �

𝑦 = 1 𝑥 + 5 4

𝑦 = 1.25 + 0.25𝑥 𝑦 = 2 𝑥 +5

3 𝑦 = 1 𝑥 + 3 3

𝑦 = − 1 𝑥 +1

6 𝑦 = − 1 𝑥 − 1 6

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5 4

¡ �

3 2 1 -6 -5 -4 -3 -2 -1 0 -1 -2

x 1

¡ �

-3 -4 -5 -6

¡ �

𝑦 = 1 𝑥 + 5 4

𝑦 = 1.25 + 0.25𝑥 𝑦 = 2 𝑥 +5

3 𝑦 = 1 𝑥 + 3 3

𝑦 = − 1 𝑥 +1

6 𝑦 = − 1 𝑥 − 1 6

Review Problem 2. Match each equation with a description of the function it represents. Function a. 𝑓�𝑥� = 2𝑥 + 4 b. 𝑔�𝑥� = 2(𝑥 + 4) c. ℎ�𝑥� = 4𝑥 + 2 d. 𝑘�𝑥� = 4(𝑥 + 2)

370 | Unit 4

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Description To get the output, add 4 to the input, and then multiply the result by 2.

To get the output, add 2 to the input, and then multiply the result by 4.

To get the output, multiply the input by 2, and then add 4 to the result.

To get the output, multiply the input by 4, and then add 2 to the result.

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3. Noah has a coin jar containing 𝑑 dimes and 𝑞 quarters worth a total of $5.00. Select all the equations that represent this situation.

□ 𝑑 + 𝑞 = 5

□ 𝑑 + 𝑞 = 500

□ 0.1𝑑 + 0.25𝑞 = 5

□ 10𝑑 + 25𝑞 = 500 □ 𝑑 = 50 □ 𝑞 = 20

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Unit 4, Lesson 2: Solving Systems by Graphing

Warm-Up: Solving and Verifying Systems from Graphs 1. The graph of the system � is shown.

𝑦 = 1 𝑥 + 2 3

𝑦 = 2𝑥 + 2

a. Mark the solution to the system on the graph. b. What are the coordinates of the solution to the system?

c. Using substitution, verify the solution satisfies both equations in the system. 𝒚 � 𝟏 𝒙 + 𝟐 𝟑

𝒚 = 𝟐𝒙 + 𝟐

d. Explain why your work in part C is sufficient to show that the coordinates are the solution to the system.

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Guided Activity: Approximating Solutions 𝑥 = 3.5

1. The graph of the system � 𝑦 = 2 𝑥 + 2 is shown.

a. Mark the solution to the system of equations.

6 4 2 -12 -10 -8

-6

-4

-2

x 2

-2

The solution to the system does not fall on the grid lines shown on the coordinate grid. Therefore, the solution can only be approximated from the graph.

-4 -6 -8 -10

b. What are the approximate coordinates of the solution?

c. Kyrie said, “I know the 𝑥-value of the solution is 3.5. I could use that to determine the 𝑦-value of the solution.” Discuss with your partner what Kyrie may have been thinking.

d. Test Kyrie’s conjecture to see if his strategy will work to determine the exact solution of the system.

e. Explain whether the solution determined in part D makes sense based on your approximation from part B.

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2. Use the coordinate plane to complete the following.

a. Graph the system �

𝑦 = − 3 𝑥 + 4

𝑦 = 3𝑥 − 4

on the coordinate plane.

4 2 -12 -10 -8

-6

-4

-2

x 2

-2

b. Mark the solution to the system on your graph.

-4 -6

c. What are the approximate coordinates of the solution?

-8 -10

Collaborative Activity: Solving Systems by Graphing 1. Diego bought some raisins and walnuts to make trail mix. Raisins cost $4 per pound (lb.) and walnuts cost $8 per lb.

a. Diego spent $15 on both ingredients. Complete the table with the amount of each ingredient Diego could have bought given the other. Be prepared to explain or show your reasoning. Raisins (lb.) Walnuts (lb.)

0.25

1.375

1.25

1.75

b. Diego bought a total of 2 lb. of raisins and walnuts combined. Complete the table with the amount of each ingredient Diego could have bought given the other. Be prepared to explain or show your reasoning. Raisins (lb.) Walnuts (lb.)

0.25

1.375

1.25

1.75

c. Diego spent $15 and bought exactly 2 lb. of raisins and walnuts combined. How many pounds of each item did he buy? Explain your reasoning. d. The system of equations � 4𝑥 + 8𝑦 = 15 represents this scenario. Use graphing 𝑥 + 𝑦 = 2 technology to graph the system of equations.

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e. Write the solution to the system of equations as an ordered pair.

f. Discuss with your partner how your answers to parts C and D relate. Summarize your discussion.

2. A family bought a total of 16 adult and child tickets to a magic show. Adult tickets, 𝑥, are $10.50 each and child tickets, 𝑦, are $7.50 each. The family paid a total of $141. The system of equations �

𝑥 + 𝑦 = 16

10.50𝑥 + 7.50𝑦 = 141

represents this scenario.

a. Use graphing technology to determine how many adult and child tickets were purchased. b. Complete the statement. The family purchased _____ adult tickets and ____ child tickets. 3. At a poster shop, Colin paid $16.80 for 2 large posters, 𝑥, and 3 small posters, 𝑦, of his favorite band. Daphne paid $14.15 for 1 large poster and 4 small posters of her favorite TV shows. Posters of the same size have the same price. The system of equations �

2𝑥 + 3𝑦 = 16.80 𝑥 + 4𝑦 = 14.15

represents this scenario.

a. Use graphing technology to determine price for a large poster and the price for a small poster. b. Complete the statement. A large poster costs _________, and a small poster costs _________.

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Lesson Summary A costume designer needs some silver and gold thread for the costumes for a school play. She needs a total of 240 yards (yd.). At a store that sells thread by the yd., silver thread costs $0.04 per yd., and gold thread costs $0.07 per yd. The designer has $15 to spend on the thread. How many yd. of each color should she get if she is buying exactly what is needed and spending her entire budget? This situation involves 2 quantities and 2 constraints, length and cost. Answering the question means finding a pair of values that meets both constraints simultaneously. To do so, graph the system of equations that represents the context on the same coordinate plane. Let 𝑥 represent the amount of silver thread and 𝑦 represent the amount of gold thread. • The length constraint can be represented by the equation 𝑥 + 𝑦 = 240.

• The cost constraint can be represented by the equation 0.04𝑥 + 0.07𝑦 = 15.

Every point on the graph of 𝑥 + 𝑦 = 240 is a pair of values that meets the length constraint. Every point on the graph of 0.04𝑥 + 0.07𝑦 = 15 is a pair of values that meets the cost constraint. The point where the 2 graphs intersect gives the pair of values that meets both constraints, which is the solution to the system of equations. A solution to a system of equations is a coordinate pair that makes both equations in the system true.

In this example, the solution is (60, 180), which represents 60 yd. of silver thread and 180 yd. of gold thread.

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The solution can be verified algebraically by substituting 60 for 𝑥 and 180 for 𝑦 in both equations. A solution to the system will make both equations true, as shown. 𝑥 + 𝑦 = 240 60 + 180 = 240 240 = 240

0.04𝑥 + 0.07𝑦 0.04(60) + 0.07(180) 2.40 + 12.60 15

= = = =

15 15 15 15

Practice Problems 1. Here are two equations. Equation 1: 6𝑥 + 4𝑦 = 34

Equation 2: 5𝑥 − 2𝑦 = 15

a. Decide whether each (𝑥,𝑦) pair is a solution to one equation, both equations, or neither of the equations. Solution to Equation 1 Solution to Equation 2 Solution to Neither Equation

(𝟑,𝟒)

(𝟒,𝟐.𝟓)

□ □ □

(𝟓,𝟓)

□ □ □

(𝟑,𝟐)

□ □ □

b. Is it possible to have more than 1 (𝑥,𝑦) pair that is a solution to both equations? Explain or show your reasoning.

2. Diego is thinking of 2 positive numbers. He says, “If we triple the first number and double the second number, then the sum is 34.” Diego then says, “If we take half of the first number and double the second, then the sum is 14.” The system of equations represents this scenario.

378 | Unit 4

Book 1.indb 378

3𝑥 + 2𝑦 = 34

� 1 𝑥 + 2𝑦 = 14 2

19/07/24 5:08 AM

a. Graph the system of equations on the coordinate plane. b. What are Diego’s 2 numbers?

3. Explain or show that the point (5, − 4) is a solution to this system of equations �

3𝑥 − 2𝑦 = 23 . 2𝑥 + 𝑦 = 6

Review Problems 4. Volunteer drivers are needed to bring 80 students to the championship baseball game. Drivers either have cars, which can seat 4 students, or vans, which can seat 6 students. The equation 4𝑐 + 6𝑣 = 80 describes the relationship between the number of cars 𝑐 and number of vans 𝑣 that can transport exactly 80 students. Explain how you know that this graph represents this equation.

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5. What is the 𝑥-intercept of the graph of 𝑦 = 3 − 5𝑥? A. �

3 , 0� 5

B. (− 5, 0) C. (0, 3)

D. �0, 5 � 3

380 | Unit 4

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Unit 4, Lesson 3: Solving Systems by Substitution

Warm-Up: Is It a Match? Here are graphs of two equations in a system.

Determine if each of these systems could be represented by the graphs. Be prepared to explain how you know. �

𝑥 + 2𝑦 = 8 𝑥 = − 5

�

𝑦 = − 7𝑥 + 13 𝑦 = − 1

Book 1.indb 381

�

3𝑥 = 8 3𝑥 + 𝑦 = 15

�

𝑦 = 2𝑥 − 7 4 + 𝑦 = 12

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Guided Activity: Solving Systems of Equations Using Substitution While systems of equations can be solved by graphing, they can also be solved algebraically. One method for solving systems of equations algebraically is using substitution. 𝑦 = 2𝑥 − 7 1. Use the system of equations � 𝑥 − 3𝑦 = 12 to complete the following.

a. Victor notices from the first equation of the system that the value of 𝑦 is equal to 2𝑥 − 7, so he substitutes this expression for 𝑦 in the second equation of the system, as shown. 𝑥 − 3(2𝑥 − 7) = 12

Discuss with your partner how Victor’s thought process applies the substitution property of equality.

b. Solve the equation Victor created using substitution. 𝑥 − 3(2𝑥 − 7) = 12 c. The solution to a system of equations is an ordered pair. Victor’s equation was used to find the 𝑥-value of the solution. Substitute the 𝑥-value into either equation in the system to find the 𝑦-value of the solution.

d. What is the solution to the system �

382 | Unit 4

Book 1.indb 382

𝑦 = 2𝑥 − 7 ? 𝑥 − 3𝑦 = 12

19/07/24 5:08 AM

Exploration Activity: Four Systems 1. Here are four systems of equations you saw earlier. Solve each system. Then, check your solutions by substituting them into the original equations to see if the equations are true. a. �

𝑥 + 2𝑦 = 8

b. �

𝑦 = − 7𝑥 + 13

c. �

3𝑥 = 8

d. �

𝑥 = − 5

𝑦 = − 1

3𝑥 + 𝑦 = 15

𝑦 = 2𝑥 − 7

4 + 𝑦 = 12

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Collaborative Activity: What about Now? Solve each system. 5𝑥 − 2𝑦 = 26

� 𝑦 + 4 = 𝑥

2𝑑 = 8𝑓

� 18 − 4𝑓 = 2𝑑

384 | Unit 4

Book 1.indb 384

2𝑚 − 2𝑝 = − 6

� 𝑝 = 2𝑚 + 10

�

𝑤 + 1 𝑧 = 4 7

𝑧 = 3𝑤 − 2

19/07/24 5:08 AM

Lesson Summary The solution to any system can be found by graphing, but graphing may not always be the most precise or the most efficient way to solve a system. A system of equations and its graph are shown. 3𝑝 + 𝑞 = 71

� 2𝑝 − 𝑞 = 30

The graph of the system shows an intersection at approximately (20, 10). Without using technology, it is not easy to tell what the exact values are. Instead of solving by graphing, the system of equations can be solved algebraically. One method of solving algebraically is by substitution. • Subtracting 3𝑝 from each side of the first equation, 3𝑝 + 𝑞 = 71, creates an equivalent equation, 𝑞 = 71 − 3𝑝. Rewriting the original equation this way allows the variable 𝑞 to be isolated. Because 𝑞 is equal to (71 − 3𝑝), the expression (71 − 3𝑝) can be substituted in place of 𝑞 in the second equation. Doing this results in an equation with only 1 variable, 𝑝, and makes it possible to find the value of 𝑝, as shown. 2𝑝 − 𝑞 2𝑝 − (71 − 3𝑝) 2𝑝 − 71 + 3𝑝 5𝑝 − 71 5𝑝

= = = = =

30 30 30 30 101

𝑝

20.2

= = = =

71 71 71 − 60.6 10.4

𝑝

101 5

Original equation

Substitute 71 − 3𝑝 for 𝑞. Apply the distributive property. Combine like terms. Add 71 to both sides.

Divide both sides by 5.

• Using the value of 𝑝, find the value of 𝑞 by substituting 20.2 for 𝑝 in either of the original equations and solving the equation. This process is shown for both equations. 3(20.2) + 𝑞 60.6 + 𝑞 𝑞 𝑞

Book 1.indb 385

2(20.2) − 𝑞 40.4 − 𝑞 −𝑞 −𝑞

= 30 = 30 = 30 − 40.4 = − 10.4

𝑞 = − 10.4 −1

𝑞 = 10.4

Unit 4 | 385

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• The solution to the system is the pair 𝑝 = 20.2 and 𝑞 = 10.4 or the point (20.2, 10.4) on the original graph. This method of solving a system of equations is called solving by substitution because an expression was substituted for 𝑞 in the second equation.

Practice Problems 1. Identify a solution to this system of equations: �

− 4𝑥 + 3𝑦 = 23

A. ( − 5, 2)

𝑥 − 𝑦 = − 7

B. ( − 2, 5) C. ( − 3, 4) D. (4, − 3)

2. Lin is solving this system of equations: �

6𝑥 − 5𝑦 = 34

3𝑥 + 2𝑦 = 8

She starts by rearranging the second equation to isolate the 𝑦-variable: 𝑦 = 4 − 1.5𝑥. She then substituted the expression 4 − 1.5𝑥 for 𝑦 in the first equation, as shown. 6𝑥 − 5(4 − 1.5𝑥) 6𝑥 − 20 − 7.5𝑥 − 1.5𝑥 𝑥

= = = =

34 34 54 − 36

𝑦 𝑦 𝑦

= = =

4 − 1.5𝑥 4 − 1.5 ⋅ ( − 36) 58

a. Check to see if Lin’s solution ( − 36, 58) makes both equations in the system true.

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b. If your answer to the previous question is “no,” find and explain her mistake. If your answer is “yes,” graph the equations to verify the solution of the system.

3. Solve each system of equations. a. �

2𝑥 − 4𝑦 = 20

𝑥 = 4

𝑦 = 6𝑥 + 11

b. � 2𝑥 − 3𝑦 = 7

𝑥 + 3𝑦 = − 5

4. Tyler and Han are trying to solve this system by substitution: � 9𝑥 + 3𝑦 = 3

Tyler’s first step is to isolate 𝑥 in the first equation to get 𝑥 = − 5 − 3𝑦. Han’s first step is to isolate 3𝑦 in the first equation to get 3𝑦 = − 5 − 𝑥.

Show that both first steps can be used to solve the system and will yield the same solution.

Book 1.indb 387

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Review Problems 5. Kiran buys supplies for the school’s greenhouse. He buys 𝑓 bags of fertilizer and 𝑝 packages of soil. He pays $5 for each bag of fertilizer and $2 for each package of soil, and spends a total of $90. The equation 5𝑓 + 2𝑝 = 90 describes this relationship. If Kiran solves the equation for 𝑝, which equation would result? A. 2𝑝 = 90 − 5𝑓 B. 𝑝 = 5𝑓 − 90 2

C. 𝑝 = 45 − 2.5𝑓 D. 𝑝 = 85𝑓 2

6. Elena wanted to find the slope and 𝑦-intercept of the graph of 25𝑥 − 20𝑦 = 100. She decided to put the equation in slope-intercept form first. Here is her work. 25𝑥 − 20𝑦 = 100 20𝑦 = 100 − 25𝑥 𝑦 = 5 − 5𝑥 4

She concluded that the slope is − 5 and the 𝑦-intercept is (0, 5). a. What was Elena’s mistake?

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b. What are the slope and 𝑦-intercept of the line? Explain or show your reasoning.

7. Here are two equations: Equation 1: 𝑦 = 3𝑥 + 8

Equation 2: 2𝑥 − 𝑦 = − 6 Without using graphing technology: a. Find a point that is a solution to Equation 1 but not a solution to Equation 2.

b. Find a point that is a solution to Equation 2 but not a solution to Equation 1.

Book 1.indb 389

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c. Graph the two equations.

d. Find a point that is a solution to both equations.

390 | Unit 4

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Unit 4, Lesson 4: Solving Systems by Elimination – Part 1

Warm-Up: Multiplying Equations 1. The system of equations � 4𝑥 + 𝑦 = 1 is graphed. 𝑥 + 2𝑦 = 9 a. Identify the solution to the system.

b. Multiply the equation 4𝑥 + 𝑦 = 1 by each multiple indicated in the table to generate equivalent equations. Multiple

Equivalent Equation

−5 1 3

Book 1.indb 391

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Exploration Activity: Adding Equations Diego is solving this system of equations. �

Here is his work: 4𝑥 + 3𝑦 = + − 4𝑥 + 5𝑦 = 0 + 8𝑦 = 𝑦 =

4𝑥 + 3𝑦 = 10 − 4𝑥 + 5𝑦 = 6

10 6 16 2

4𝑥 + 3(2) = 4𝑥 + 6 = 4𝑥 = 𝑥 =

1. Make sense of Diego’s work and discuss with a partner.

10 10 4 1

a. What did Diego do to solve the system?

b. Is the pair of 𝑥- and 𝑦-values that Diego found actually a solution to the system? How do you know?

2. Does Diego’s method work for solving these systems? Be prepared to explain or show your reasoning. a. �

392 | Unit 4

Book 1.indb 392

2𝑥 + 𝑦 = 4 𝑥 − 𝑦 = 11

b. �

8𝑥 + 11𝑦 = 37 8𝑥 + 𝑦 = 7

19/07/24 5:08 AM

Guided Activity: Solving Systems of Linear Equations Using Elimination In addition to solving systems of equations by graphing and using substitution, systems of equations can also be solved algebraically using elimination. 1. Consider the system of equations, � a. Add the two equations together.

2𝑥 + 4𝑦 = 13

. −2𝑥 + 4𝑦 = 11 2𝑥 + 4𝑦 = 13

+ ( − 2𝑥 + 4𝑦 = 11)

b. Subtract one equation from the other.

2𝑥 + 4𝑦 = 13

− ( − 2𝑥 + 4𝑦 = 11)

c. What do you notice about the resulting equations in parts A and B?

d. The graphs of the original equations in the system and the resulting equations from parts A and B are shown on the same coordinate grid.

2x + 4y = 13

–2x + 4y = 11

What do you notice?

8y = 24 x

-5

e. Use the resulting equation from part A to solve the original system of equations algebraically.

-5

4x = 2

Book 1.indb 393

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f. Verify your solution using the graph. When equations in a system are added or subtracted, the resulting equation shares the solution of the original system.

2. Use the system of equations � a. Complete the statement. Adding Subtracting

5𝑎 − 𝑏 = 9 to complete the following. 2.5𝑎 + 𝑏 = 6

the original equations in this system would result in eliminating 𝑎. 𝑏.

one of the variables,

b. Perform the operation selected in part A. 5𝑎 − 𝑏 = 9

(2.5𝑎 + 𝑏 = 6) c. Solve the system using your equation from part B.

𝑎 = _____

394 | Unit 4

Book 1.indb 394

𝑏 = _____

19/07/24 5:08 AM

Collaborative Activity: Adding and Subtracting Equations to Solve Systems Three systems of equations are shown. System A: �

4𝑥 + 3𝑦 = 10 − 4𝑥 + 5𝑦 = 6

System B: �

2𝑥 + 𝑦 = 4 𝑥 − 𝑦 = 11

System C: �

8𝑥 + 11𝑦 = 37 8𝑥 + 𝑦 = 7

For each system: 1. Use graphing technology to graph the original two equations in the system. Then, identify the coordinates of the solution. System A: __________

System B: __________

System C: __________

2. Find the sum or difference of the two original equations that would enable the system to be solved.

3. Graph the third equation on the same coordinate plane. Make an observation about the graph.

Book 1.indb 395

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Lesson Summary Another way to solve systems of equations algebraically is by elimination. Elimination is a method of solving a system of two equations in two variables where you add or subtract a multiple of one equation to another in order to get an equation with only one of the variables. Just like in substitution, the goal is to eliminate 1 variable to solve for the other variable. This is done by adding or subtracting equations in the system. Consider the system �

5𝑥 + 7𝑦 = 64 . 0.5𝑥 − 7𝑦 = − 9

• Notice that 1 equation has 7𝑦, and the other has − 7𝑦.

• If the second equation is added to the first, the 7𝑦 and − 7𝑦 add up to 0, which eliminates the 𝑦-variable so the equation can be solved for 𝑥, as shown. • Substitute 10 for 𝑥 in either of the equations, and find 𝑦. 5𝑥 + 7𝑦 = 5(10) + 7𝑦 = 50 + 7𝑦 = 7𝑦 = 𝑦 =

64 64 64 14 2

• The solution to the system is (10, 2).

0.5𝑥 − 7𝑦 0.5(10) − 7𝑦 5 − 7𝑦 − 7𝑦 𝑦

= −9 = −9 = −9 = − 14 = 2

5𝑥 + 7𝑦 = 64

+ (0.5𝑥 − 7𝑦 = − 9)

5.5𝑥 + 0 = 55 5.5𝑥 = 55 𝑥 = 10

In this system, the coefficient of 𝑦 in the first equation happens to be the opposite of the coefficient of 𝑦 in the second equation. The sum of the terms with 𝑦-variables is 0. What if the equations don’t have opposite coefficients for the same variable, like in the system shown?

396 | Unit 4

Book 1.indb 396

�

8𝑟 + 4𝑠 = 12 8𝑟 + 𝑠 = − 3

19/07/24 5:08 AM

• Notice that both equations include the term 8𝑟. If the second equation is subtracted from the first, the 𝑟-term will be eliminated because 8𝑟 − 8𝑟 is 0, as shown.

8𝑟 + 4𝑠 = 12

− (8𝑟 + 𝑠 = − 3)

0 + 3𝑠 = 15 3𝑠 = 15

• Substitute 5 for 𝑠 in either equation to find 𝑟 as shown.

𝑠 = 5

8𝑟 + 4𝑠 = 12

8𝑟 + 4(5) = 12 8𝑟 + 20 = 12

8𝑟 =− 8𝑟

𝑟 = − 1 • The solution to the system is 𝑟 = − 1, 𝑠 = 5.

Adding or subtracting the equations in a system creates a new equation. To prove the new equation shares a solution with the original system, graph the original equations in the system and the new equation on the same coordinate plane. All 3 lines intersect at the same point.

Future lessons will investigate why this strategy works.

Book 1.indb 397

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Practice Problems 1. Which equation is the result of adding these two equations?

A. − 5𝑥 − 6𝑦 = 14

�

− 2𝑥 + 4𝑦 = 17 3𝑥 − 10𝑦 = − 3

B. − 𝑥 − 6𝑦 = 14 C. 𝑥 − 6𝑦 = 14

D. 5𝑥 + 14𝑦 = 20

2. Which equation is the result of subtracting the second equation from the first?

A. − 9𝑥 − 4𝑦 = 8

�

4𝑥 − 6𝑦 = 13 − 5𝑥 + 2𝑦 = 5

B. − 𝑥 + 4𝑦 = 8 C. 𝑥 − 4𝑦 = 8

D. 9𝑥 − 8𝑦 = 8

3. Solve this system of equations without graphing: �

5𝑥 + 2𝑦 = 29 5𝑥 − 2𝑦 = 41

�

6𝑥 + 21𝑦 = 103 − 6𝑥 + 23𝑦 = 51

4. Here is a system of linear equations:

Would you rather use subtraction or addition to solve the system? Explain your reasoning. 398 | Unit 4

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Review Problems 5. Kiran sells 𝑓 full boxes and ℎ half-boxes of fruit to raise money for a band trip. He earns $5 for each full box and $2 for each half-box of fruit he sells and earns a total of $100 toward the cost of his band trip. The equation 5𝑓 + 2ℎ = 100 describes this relationship. Solve the equation for 𝑓.

6. The volume of a cylinder is represented by the formula 𝑉 = 𝜋𝑟2ℎ.

Volume (cubic inches)

Radius (inches)

96𝜋

Find each missing height and show your reasoning.

31.25𝜋 𝑉

Height (inches)

2.5 𝑟

7. Match each equation with the slope, 𝑚, and 𝑦-intercept of its graph. Slope and 𝒚-intercept 𝑚 = − 6 , 𝑦-int. = (0, 12)

5𝑥 − 6𝑦 = 30

𝑚 = − 5 , 𝑦-int. = (0, 1)

𝑦 = 5𝑥 + 1

𝑚 = − 6 , 𝑦-int. = (0, 5) 6

𝑚 = 5 , 𝑦-int. = (0, 1) 6

𝑚 = 5 , 𝑦-int. = (0, − 1) 6

𝑚 = 5 , 𝑦-int. = (0, − 5) 6

Book 1.indb 399

Equation

𝑦 = 5 − 6𝑥 6

5𝑥 − 6𝑦 = 6

5𝑥 + 6𝑦 = 6

6𝑥 + 𝑦 = 12 Unit 4 | 399

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Solve each system of equations. a. �

2𝑥 + 3𝑦 = 4 2𝑥 = 7𝑦 + 24

b. �

5𝑥 + 3𝑦 = 23 3𝑦 = 15𝑥 − 21

400 | Unit 4

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Unit 4, Lesson 5: Solving Systems by Elimination – Part 2

Warm-Up: Is It Still True? Here is an equation: 50 + 1 = 51.

1. Perform each of the following operations and answer these questions: What does each resulting equation look like? Is it still a true equation? a. Add 12 to each side of the equation.

b. Add 10 + 2 to the left side of the equation and 12 to the right side.

c. Add the equation 4 + 3 = 7 to the equation 50 + 1 = 51.

2. Write a new equation that, when added to 50 + 1 = 51, gives a sum that is also a true equation.

3. Write a new equation that, when added to 50 + 1 = 51, gives a sum that is a false equation.

Book 1.indb 401

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Collaborative Activity: Classroom Supplies A teacher purchased 20 calculators and 10 measuring tapes for her class and paid $495. Later, she realized that she didn’t order enough supplies. She placed another order of 8 of the same calculators and 1 more of the same measuring tape and paid $178.50. This system represents the constraints in this situation. �

20𝑐 + 10𝑚 = 495 8𝑐 + 𝑚 = 178.50

1. Discuss with a partner:

a. In this situation, what do the solutions to the first equation mean?

b. What do the solutions to the second equation mean?

c. For each equation, how many possible solutions are there? Explain how you know.

d. In this situation, what does the solution to the system mean?

2. Find the solution to the system. Explain or show your reasoning.

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3. To be reimbursed for the cost of the supplies, the teacher recorded: “Items purchased: 28 calculators and 11 measuring tapes. Amount: $673.50.”

a. Write an equation to represent the relationship between the numbers of calculators and measuring tapes, the prices of those supplies, and the total amount spent.

b. How is this equation related to the first two equations?

c. In this situation, what do the solutions of this equation mean?

d. How many possible solutions does this equation have? How many solutions make sense in this situation? Explain your reasoning.

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Guided Activity: Solving Systems of Linear Equations Using Elimination In the previous lesson, you solved the system of equations �

5𝑎 − 𝑏 = 9 by eliminating 𝑏. 2.5𝑎 + 𝑏 = 6

1. Riley wondered whether it is possible to solve the system by eliminating the 𝑎 terms instead of the 𝑏 terms first. He suggested multiplying the second equation by 2 and then subtracting it from the first equation. The first steps Riley suggests are shown. Complete Riley’s work and use the results to solve the system.

2(2.5a + b = 6) a+

–(

5a – b = 9

)

When solving a system of equations using elimination, one or both of the equations can be multiplied by a factor before adding or subtracting the equations. The resulting equation shares the same solution with the original system. 2. Solve each system of equations algebraically. Show your work. Then, check your solutions.

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a. �

2𝑥 + 3𝑦 = 7 − 2𝑥 + 4𝑦 = 14

b. �

2𝑥 + 3𝑦 = 7 3𝑥 − 3𝑦 = 3

c. �

2𝑥 + 3𝑦 = 5 2𝑥 + 4𝑦 = 9

d. �

2𝑥 + 3𝑦 = 16 6𝑥 − 5𝑦 = 20

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Lesson Summary When solving a system with 2 equations, why is it acceptable to add the 2 equations or to subtract 1 equation from the other? Remember that an equation is a statement that says 2 quantities are equal. For example, the equation 𝑎 = 𝑏 says a number 𝑎 has the same value as another number 𝑏. The equation 10 + 2 = 12 says that 10 + 2 has the same value as 12. If 𝑎 = 𝑏 and 10 + 2 = 12 are true statements, then adding 10 + 2 to 𝑎 and adding 12 to 𝑏 means adding the same amount to each side of 𝑎 = 𝑏. The result, 𝑎 + 10 + 2 = 𝑏 + 12, is also a true statement. As long as an equal amount is added to each side of a true equation, both sides of the resulting equation will remain equal. The same reasoning applies when adding variable equations in a system. �

𝑒 + 𝑓 = 17 − 2𝑒 + 𝑓 = − 1

In each equation, if (𝑒,𝑓) is a solution, the expression on the left of the equal sign and the number on the right are equal. Because − 2𝑒 + 𝑓 is equal to − 1:

• Adding − 2𝑒 + 𝑓 to 𝑒 + 𝑓 and adding − 1 to 17 means adding an equal amount to each side of 𝑒 + 𝑓 = 17. Both sides of the new equation, − 𝑒 + 2𝑓 = 16, remain equal. 𝑒 + 𝑓 = 17

+ (− 2𝑒 + 𝑓 = − 1) − 𝑒 + 2𝑓 = 16

The 𝑒- and 𝑓-values that make the original equations true also make this equation true.

• Subtracting − 2𝑒 + 𝑓 from 𝑒 + 𝑓 and subtracting − 1 from 17 means subtracting an equal amount from each side of 𝑒 + 𝑓 = 17. Both sides of the new equation, 3𝑒 = 18, remain equal. 𝑒 + 𝑓 = 17

− ( − 2𝑒 + 𝑓 = − 1) 406 | Unit 4

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The 𝑓-terms are eliminated, but the 𝑒-value that makes both of the original equations true also makes this equation true. From 3𝑒 = 18, it can be determined that 𝑒 = 6. Because 6 is also the 𝑒-value that makes the original equations true, it can be substituted into either of the equations to find the 𝑓-value. The solution to the system is 𝑒 = 6, 𝑓 = 11, or the point (6, 11) on the graphs representing the system. Substituting 6 and 11 for 𝑒 and 𝑓 in any of the equations results in true equations.

Practice Problems 1. Solve this system of linear equations without graphing. �

5𝑥 + 4𝑦 = 8 10𝑥 − 4𝑦 = 46

2. Select all the equations that share a solution with this system of equations.

□ 7𝑥 + 3𝑦 = 50

�

5𝑥 + 4𝑦 = 24 2𝑥 − 7𝑦 = 26

□ 7𝑥 − 3𝑦 = 50

□ 5𝑥 + 4𝑦 = 2𝑥 − 7𝑦 □ 3𝑥 − 11𝑦 = − 2 □ 3𝑥 + 11𝑦 = − 2 3. Students performed in a play on a Friday and a Saturday. For both performances, adult tickets cost dollars each and student tickets cost dollars each. On Friday, they sold 125 adult tickets and 65 student tickets and collected $1,200. On Saturday, they sold 140 adult tickets and 50 student tickets and collected $1,230. 125𝑎 + 65𝑠 = 1,200 This situation is represented by this system of equations: � 140𝑎 + 50𝑠 = 1,230. © Accelerate Learning Inc. - All Rights Reserved

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a. What could the equation 265𝑎 + 115𝑠 = 2,430 mean in this situation?

b. The solution to the original system is the pair 𝑎 = 7 and 𝑠 = 5. Explain why it makes sense that this pair of values is also a solution to the equation 265𝑎 + 115𝑠 = 2,430.

4. Which statement explains why 13𝑥 − 13𝑦 = − 26 shares a solution with this system of 10𝑥 − 3𝑦 = 29 equations, � . − 3𝑥 + 10𝑦 = 55 A. Because 13𝑥 − 13𝑦 = − 26 is the product of the two equations in the system of equations, it must share a solution with the system of equations.

B. The three equations all have the same slope but different 𝑦-intercepts. Equations with the same slope but different 𝑦-intercepts always share a solution. C. Because 10𝑥 − 3𝑦 is equal to 29, I can add 10𝑥 − 3𝑦 to the left side of − 3𝑥 + 10𝑦 = 55 and add 29 to the right side of the same equation. Adding equivalent expressions to each side of an equation does not change the solution to the equation. D. Because − 3𝑥 + 10𝑦 is equal to 55, I can subtract − 3𝑥 + 10𝑦 from the left side of 10𝑥 − 3𝑦 = 29 and subtract 55 from its right side. Subtracting equivalent expressions from each side of an equation does not change the solution to the equation.

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Review Problems 5. Select all equations that can result from adding these two equations or subtracting one from the other. �

□ − 2𝑥 − 4𝑦 = 8

𝑥 + 𝑦 = 12 3𝑥 − 5𝑦 = 4

□ − 2𝑥 + 6𝑦 = 8 □ 4𝑥 − 4𝑦 = 16 □ 4𝑥 + 4𝑦 = 16

□ 2𝑥 − 6𝑦 = − 8 □ 5𝑥 − 4𝑦 = 28

6. Solve each system of equations. a. �

7𝑥 − 12𝑦 = 180 7𝑥 = 84

b. �

− 16𝑦 = 4𝑥 4𝑥 + 27𝑦 = 11

7. Here is a system of equations: �

7𝑥 − 4𝑦 = − 11 . 7𝑥 + 4𝑦 = − 59

Would you rather use subtraction or addition to solve the system? Explain your reasoning.

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8. In places where there are crickets, the outdoor temperature can be predicted by the rate at which crickets chirp. One equation that models the relationship between 1 chirps and outdoor temperature is 𝑓 = 4 𝑐 + 40, where 𝑐 is the number of chirps per minute (min.) and 𝑓 is the temperature in degrees Fahrenheit (℉). a. Suppose 110 chirps are heard in a min. According to this model, what is the outdoor temperature?

b. If it is 75℉ outside, about how many chirps can we expect to hear in 1 min.? c. The equation is only a good model of the relationship when the outdoor temperature is at least 55℉. (Below that temperature, crickets aren’t around or inclined to chirp.) How many chirps can we expect to hear in a min. at that temperature?

d. On the coordinate plane, draw a graph that represents the relationship between the number of chirps and the temperature.

e. Explain what the coefficient 4 in the equation tells us about the relationship.

f. Explain what the 40 in the equation tells us about the relationship.

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Unit 4, Lesson 6: Solving Systems Algebraically

Warm-Up: Multiplying Equations by a Number Consider two equations in a system: �

4𝑥 + 𝑦

𝑥 + 2𝑦

Equation A Equation B

1. Use graphing technology to graph the equations. Then, identify the coordinates of the solution.

2. Write a few equations that are equivalent to equation A by multiplying both sides of it by the same number, for example, 2, − 5, or 1 . Let’s call the resulting equations A1, A2, and A3. Record your equations here: 2 a. Equation A1:

b. Equation A2:

c. Equation A3:

3. Graph the equations you generated. Make a couple of observations about the graphs.

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Exploration Activity: Writing a New System to Solve a Given System Here is a system you solved by graphing earlier. �

4𝑥 + 𝑦

𝑥 + 2𝑦

To start solving the system, Elena wrote:

And then she wrote:

Equation A Equation B

4𝑥 + 𝑦 = 1 4𝑥 + 8𝑦 = 36 4𝑥 + 𝑦 = 1 − (4𝑥 + 8𝑦 = 36) − 7𝑦 = − 35

1. What were Elena’s first two moves? What might be possible reasons for those moves?

2. Complete the solving process algebraically. Show that the solution is indeed 𝑥 = − 1, 𝑦 = 5.

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Collaborative Activity: What Comes Next? Your teacher will give you some slips of paper with systems of equations written on them. Each system represents a step in solving this system. 4 𝑥 + 6𝑦 = 15 � 5

− 𝑥 + 18𝑦 = 11

Arrange the slips in the order that would lead to a solution. Be prepared to: • Describe what move takes one system to the next system. • Explain why each system is equivalent to the one before it.

Exploration Activity: Build Some Equivalent Systems Here is a system of equations: �

12𝑎 + 5𝑏 = − 15 8𝑎 + 𝑏

= 11

1. To solve this system, Diego wrote these equivalent systems for his first two steps.

Step 1: �

12𝑎 + 5𝑏

= − 15

− 40𝑎 + − 5𝑏 = − 55

Step 2: �

12𝑎 + 5𝑏 = − 28𝑎

− 15 − 70

Describe the move that Diego made to get each equivalent system. Be prepared to explain how you know the systems in Step 1 and Step 2 have the same solution as the original system.

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2. Write another set of equivalent systems (different than Diego’s first two steps) that will allow one variable to be eliminated and enable you to solve the original system. Be prepared to describe the moves you make to create each new system and to explain why each one has the same solution as the original system.

3. Use your equivalent systems to solve the original system. Then, check your solution by substituting the pair of values into the original system.

Lesson Summary Two algebraic strategies for solving systems of equations have been explored, solving by substitution and solving by elimination. In some systems, the equations may provide a clue as to which strategy may be more efficient based on their structure. Consider the system of equations �

𝑦 = 2𝑥 − 11

3𝑥 + 2𝑦 = 18

• In this system, 𝑦 is already isolated in the first equation. The system can be solved by substituting 2𝑥 − 11 for 𝑦 in the second equation and finding the value of 𝑥. Consider the system of equations �

3𝑥 − 𝑦 = − 17

. − 3𝑥 + 4𝑦 = 23

• This system is also set up nicely for elimination because of the opposite coefficients of the 𝑥-variable. Adding the equations eliminates 𝑥, so the resulting equation can be solved for 𝑦. In other systems, which strategy to use is less straightforward, either because no variables are isolated or because no variables have equal or opposite coefficients.

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Consider the system of equations �

2𝑥 + 3𝑦 3𝑥 − 9𝑦

= 15

= 18

Equation A Equation B

To solve this system by elimination, first rewrite 1 or both equations so that a variable can be eliminated. To do that, multiply both sides of an equation by the same factor. Remember that doing this doesn’t change the equality of the 2 sides of the equation, so the 𝑥- and 𝑦-values that make the first equation true also make the new equation true. There are different ways to eliminate a variable with this approach, as shown in the table. Strategy

Resulting System

Multiply Equation A by 3 to get 6𝑥 + 9𝑦 = 45. Adding this equation to Equation B eliminates 𝑦.

Multiply Equation B by 2 to get 2𝑥 − 6𝑦 = 12. 3

Subtracting this equation from Equation A eliminates 𝑥.

Multiply Equation A by 1 to get 𝑥 + 3 2

𝑦 = 7 1 , and multiply Equation B by 1 to 2

get 𝑥 − 3𝑦 = 6. Subtracting 1 equation from the other eliminates 𝑥.

� �

6𝑥 + 9𝑦

= 45

Equation A1

2𝑥 + 3𝑦

= 15

Equation A

3𝑥 − 9𝑦 = 18

2𝑥 − 6𝑦 �

𝑥 + 3 𝑦 2

𝑥 − 3𝑦

Equation B

= 12

Equation B1

=7 1

Equation A2 Equation B2

Each multiple of an original equation is equivalent to the original equation, so each new pair of equations is equivalent to the original system and has the same solution. The steps for solving the original system using the first equivalent system generated in the table are shown. • Add the 2 equations to eliminate 𝑦, leaving a new equation 9𝑥 = 63, or 𝑥 = 7. 6𝑥 + 9𝑦 = 45

+ (3𝑥 − 9𝑦 = 18) 9𝑥 + 0 = 63 𝑥 = 7

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• Combine 𝑥 = 7 and the original 3𝑥 − 9𝑦 = 18 to generate another equivalent system. �

𝑥

3𝑥 − 9𝑦

= 18

• Substitute 7 for 𝑥 in the second equation to solve for 𝑦. 3(7) − 9𝑦 = 18

21 − 9𝑦 = 18 − 9𝑦 = − 3 1 𝑦 = 3

When solving a system by elimination, a series of equivalent systems, or systems with the same solution, is created.

Two systems are equivalent systems if they share the exact same solution set.

Each equivalent system gets closer and closer to the solution of the original system. �

2𝑥 + 3𝑦 3𝑥 − 9𝑦

= 15 = 18

�

6𝑥 + 9𝑦 3𝑥 − 9𝑦

= 45 = 18

�

𝑥

3𝑥 − 9𝑦

= 18

�

𝑥

𝑦

= 1 3

Practice Problems 1. Solve each system of equations. a. �

2𝑥 − 4𝑦 = 10

𝑥 + 5𝑦 = 40

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b. �

3𝑥 − 5𝑦 = 4

− 2𝑥 + 6𝑦 = 18

2. Tyler is solving this system of equations. �

4𝑝 + 2𝑞 = 62 8𝑝 − 𝑞 = 59

He can think of two ways to eliminate a variable and solve the system. • Multiply 4𝑝 + 2𝑞 = 62 by 2, and then subtract 8𝑝 − 𝑞 = 59 from the result.

• Multiply 8𝑝 − 𝑞 = 59 by 2, and then add the result to 4𝑝 + 2𝑞 = 62.

Do both strategies work for solving the system? Explain or show your reasoning.

3. Andre and Elena are solving this system of equations. � • Andre’s first step is to write 3𝑥 = 9𝑥 − 30.

• Elena’s first step is to create a new system, �

𝑦 = 3𝑥

𝑦 = 9𝑥 − 30

3𝑦 = 9𝑥

𝑦 = 9𝑥 − 30

Do you agree with either first step? Explain your reasoning.

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4. Select all systems that are equivalent to this system.

□ � 6𝑑 + 4.5𝑒

= 16.5

�

6𝑑 + 4.5𝑒 5𝑑 + 0.5𝑒

= 16.5 =4

45𝑑 + 4.5𝑒 = 4

□ � 30𝑑 + 22.5𝑒 = 82.5 5𝑑 + 0.5𝑒

30𝑑 + 3𝑒

= 24

6𝑑 + 0.6𝑒

= 4.8

□ � 30𝑑 + 22.5𝑒 = 82.5 □ � 6𝑑 + 4.5𝑒 □ � 12𝑑 + 9𝑒

= 16.5 = 33

10𝑑 + 0.5𝑒 = 8

□ � 6𝑑 + 4.5𝑒 11𝑑 + 5𝑒

= 16.5 = 20.5

𝑝 + 8𝑞 = − 8 5. Here is a system of equations with a solution: � 1 𝑝 + 5𝑞 = − 5 . 2

a. Write a system of equations that is equivalent to this system. Describe what you did to the original system to get the new system.

b. Explain how you know the new system has the same solution as the original system.

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Review Problems 6. The cost to mail a package is $5.00. Noah has postcard stamps that are worth $0.34 each and first-class stamps that are worth $0.49 each. a. Write an equation that relates the number of postcard stamps 𝑝, the number of first-class stamps 𝑓, and the cost of mailing the package. b. Solve the equation for 𝑓. c. Solve the equation for 𝑝. d. If Noah puts 7 first-class stamps on the package, how many postcard stamps will he need?

2𝑥 + 7𝑦 = 8

− 7𝑥 + 3𝑦 = − 65

7. Here is a system of linear equations: �

𝑦 + 2𝑥 = 14

Find at least one way to solve the system by substitution and show your reasoning. How many ways can you find? (Regardless of the substitution that you do, the solution should be the same.)

8. Here is a system of equations: �

− 7𝑥 + 10𝑦 = − 135

Write an equation that results from subtracting the two equations.

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9. A grocery store sells bananas for 𝑏 dollars a pound (lb.) and grapes for 𝑔 dollars a lb. Priya buys 2.2 lb. of bananas and 3.6 lb. of grapes for $9.35. Andre buys 1.6 lb. of bananas and 1.2 lb. of grapes for $3.68. This situation is represented by the system of equations, �

2.2𝑏 + 3.6𝑔 = 9.35 1.6𝑏 + 1.2𝑔 = 3.68

Explain why it makes sense in this situation that the solution of this system is also a solution to 3.8𝑏 + 4.8𝑔 = 13.03.

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Unit 4, Lesson 7: Systems of Linear Equations and Their Solutions

Warm-Up: A Curious System Andre is trying to solve this system of equations, �

𝑥 + 𝑦 = 3

4𝑥 = 12 − 4𝑦

Looking at the first equation, he thought, “The solution to the system is a pair of numbers that add up to 3. I wonder which two numbers they are.”

1. Choose any two numbers that add up to 3. Let the first one be the 𝑦-value and the second one be the 𝑥-value.

2. The pair of values you chose is a solution to the first equation. Check if it is also a solution to the second equation. Then, pause for a brief discussion with your group.

3. How many solutions does the system have? Use what you know about equations or about solving systems to show that you are right.

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Exploration Activity: What’s the Deal? A recreation center is offering special prices on its pool passes and gym memberships for the summer. On the first day of the offering, a family paid $96 for 4 pool passes and 2 gym memberships. Later that day, an individual bought a pool pass for herself, a pool pass for a friend, and 1 gym membership. She paid $72. The system of equations that represents this situation is �

4𝑝 + 2𝑔 = 96 2𝑝 + 𝑔 = 72

price of each pool pass and 𝑔 is the price of each gym membership.

, where 𝑝 is the

1. Find the price of a pool pass and the price of a gym membership by solving the system algebraically. Show your work.

2. Use graphing technology to graph the equations in the system. Make 1–2 observations about your graphs.

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Collaborative Activity: Card Sort: Sorting Systems Your teacher will give you a set of cards. Each card contains a system of equations. Sort the systems into three groups based on the number of solutions each system has. Be prepared to explain how you know where each system belongs.

Collaborative Activity: One, Zero, Infinitely Many Here is an equation: 5𝑥 − 2𝑦 = 10.

Create a second equation that would make a system of equations with: 1. One solution

2. No solutions

3. Infinitely many solutions

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Lesson Summary Most of the systems explored in this unit have 1 pair of values that satisfies both equations. However, recall from previously in this unit that not all systems have 1 solution. Some systems have infinitely many solutions, and others have no solution. Consider 3 systems of equations and their graphs.

System 1: �

3𝑥 + 4𝑦 = 8 3𝑥 − 4𝑦 = 8

The graphs of the equations in System 1 intersect at 1 point. The coordinates of the point are the 1 pair of values that are simultaneously true for both equations. When solving the system of equations, there is exactly 1 solution.

System 2: �

3𝑥 + 4𝑦 = 8

6𝑥 + 8𝑦 = 16

The graphs of the equations in System 2 appear to be the same line. This suggests that every point on the line is a solution to both equations. In other words, the system has infinitely many solutions.

System 3: �

3𝑥 + 4𝑦 = 8

3𝑥 + 4𝑦 = − 4

The graphs of the equations in System 3 are parallel. If the lines never intersect, then there is no common point that is a solution to both equations. Therefore, the system has no solutions.

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Without graphing, how can it be determined algebraically that System 2 has infinitely many solutions? Two methods are described. • Notice that 3𝑥 + 4𝑦 = 8 and 6𝑥 + 8𝑦 = 16 are equivalent equations. Multiplying the first equation by 2 produces the second equation. Multiplying the second equation by 1 2

results in the first equation. This means that any solution to the first equation is a solution to the second.

• Rearranging 3𝑥 + 4𝑦 = 8 into slope-intercept form produces 𝑦 = 8 − 3𝑥 , or

4 3 16 − 6𝑥 𝑥. Rearranging 6𝑥 + 8𝑦 = 16 gives 𝑦 = , which is also 𝑦 = 2 − 3 𝑥. 𝑦 = 2 − 4 4 8

Both lines have the same slope and the same 𝑦-intercept.

Without graphing, how can it be determined algebraically that System 3 has no solutions? Two methods are described. • Notice that in first equation 3𝑥 + 4𝑦 equals 8, but in the other equation, it equals − 4. Because it is impossible for the same expression to equal 8 and − 4, there must not be a pair of 𝑥- and 𝑦-values that are simultaneously true for both equations. This means that the system has no solutions. Rearranging each equation into slope-intercept form results in 𝑦 = − 3 𝑥 + 2 and 4

𝑦 = − 3 𝑥 − 1. Both graphs have the same slope, but their 𝑦-intercepts are different. 4

This proves that the lines are parallel and will never cross.

Practice Problems 1. Here is a system of equations: �

3𝑥 − 𝑦 = 17 𝑥 + 4𝑦 = 10

a. Solve the system by graphing the equations (by hand or using technology).

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b. Explain how you could tell, without graphing, that there is only one solution to the system.

2. Consider this system of linear equations: �

𝑦 =

4 𝑥 − 3 5

4 𝑥 + 1 𝑦 = 5

a. Without graphing, determine how many solutions you would expect this system of equations to have. Explain your reasoning.

b. Try solving the system of equations algebraically and describe the result that you get. Does it match your prediction?

3. How many solutions does this system of equations have? Explain how you know.

�

9𝑥 − 3𝑦 = − 6

5𝑦 = 15𝑥 + 10

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4. Select all systems of equations that have no solutions.

□� □� □� □� □�

𝑦 = 5 − 3𝑥

𝑦 = − 3𝑥 + 4 𝑦 = 4𝑥 − 1

4𝑦 = 16𝑥 − 4 5𝑥 − 2𝑦 = 3

10𝑥 − 4𝑦 = 6

3𝑥 + 7𝑦 = 42

6𝑥 + 14𝑦 = 50 𝑦 = 5 + 2𝑥

𝑦 = 5𝑥 + 2

Review Problems 5. Solve each system of equations without graphing. a. �

b. �

2𝑣 + 6𝑤 = − 36 5𝑣 + 2𝑤 = 1

6𝑡 − 9𝑢 = 10 2𝑡 + 3𝑢 = 4

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6. Here is a system of linear equations: �

6𝑥 − 𝑦 = 18

. 4𝑥 + 2𝑦 = 26

Select all the steps that would help to eliminate a variable and enable solving.

□ Multiply the first equation by 2, and then subtract the second equation from the result.

□ Multiply the first equation by 4 and the second equation by 6, and then subtract the resulting equations.

□ Multiply the first equation by 2, and then add the result to the second equation. □ Divide the second equation by 2, and then add the result to the first equation. □ Multiply the second equation by 6, and then subtract the result from the first equation.

7. Here is a system of equations: �

− 𝑥 + 6𝑦 = 9 𝑥 + 6𝑦 = − 3

Would you rather use subtraction or addition to solve the system? Explain your reasoning.

8. Consider this system of equations, which has one solution: �

2𝑥 + 2𝑦 = 180

0.1𝑥 + 7𝑦 = 78

Here are some equivalent systems. Each one is a step in solving the original system.

Step 1: �

7𝑥 + 7𝑦 = 630

0.1𝑥 + 7𝑦 = 78

Step 2: �

6.9𝑥 = 552

0.1𝑥 + 7𝑦 = 78

a. Look at the original system and the system in Step 1.

Step 3: �

𝑥 = 80

0.1𝑥 + 7𝑦 = 78

i. What was done to the original system to get the system in Step 1?

ii. Explain why the system in Step 1 shares a solution with the original system.

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b. Look at the system in Step 1 and the system in Step 2. i. What was done to the system in Step 1 to get the system in Step 2?

ii. Explain why the system in Step 2 shares a solution with that in Step 1.

c. What is the solution to the original system?

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Unit 4, Lesson 8: Solving Equations in the Form 𝒇(𝒙) = 𝒈(𝒙) Warm-Up: Pattern The mathematical statements shown express patterns. 𝑎 = 15(104)11 𝑐 = 13(88)9

1. Determine what value completes 𝑏 = 14( )10. 2. What strategies did you use to try to determine the pattern?

3. Continue the pattern. 𝑑 = 12( )8 𝑒 = 11( )7 ℎ = 8( )4

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Guided Activity: Relating Intersections of Graphs to an Equation 1. The graph of the function 𝑦 = 𝑓(𝑥), where 𝑓(𝑥) = 3𝑥 + 1, is shown on the coordinate plane.

y 10

a. Discuss with your partner whether each point on the graph of 𝑦 = 3𝑥 + 1 is a solution to the equation.

b. Work with your partner to complete the table for each 𝑥-value given. Then, locate each ordered pair on the graph, and label it with the label indicated in the last column.

𝒙-value −1 1 2.5 3

x -5

Ordered Pair (𝒙, 𝟑𝒙 + 𝟏)

𝒚-value of 𝒇(𝒙)

Label P M K L

c. Discuss with your partner how the 𝑥-value of each ordered pair found relates to the equation 𝑦 = 3𝑥 + 1 for the 𝑦-value of the ordered pair. y

The set of points that includes all the solutions to 𝑦 = 3𝑥 + 1 is written as (𝑥, 3𝑥 + 1), where 𝑥 is a real number. When plotted on the coordinate grid, this set of points results in the graph of 𝑦 = 3𝑥 + 1.

2. The graph of the function 𝑦 = 𝑔(𝑥), where 𝑔(𝑥) = 𝑥2 − 2𝑥 + 5, is shown on the coordinate plane. a. Work with your partner to complete the table for each 𝑥-value given. Then, locate each ordered pair on the graph, and label it with the label indicated in the last column. 432 | Unit 4

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x -5

19/07/24 5:08 AM

𝒙-value −1 1 2.5 4

Ordered Pair (𝒙, 𝒙𝟐 − 𝟐𝒙 + 𝟓)

𝒚-value of 𝒈(𝒙)

Label H M D L

b. Complete the statement. The set of points that includes all the solutions to 𝑦 = 𝑥2 − 2𝑥 + 5 is written as ___________________________, where 𝑥 is a real number.

3. Graph the function 𝑓(𝑥) = 3𝑥 + 1 on the same coordinate plane as 𝑔(𝑥) = 𝑥2 − 2𝑥 + 5 in problem 2.

4. Discuss with your partner what you notice about the graphs of the 2 functions.

5. Complete the statements. The graphs of 𝑓(𝑥) and 𝑔(𝑥) share the points labeled

means that for the 𝑥-values

−1 1 2.5 4

and

functions are the same. Therefore, there

𝑓(𝑥) = 𝑔(𝑥), the

𝑥-value(s) 𝑦-value(s)

and

H. M. D. L.

This

the 𝑦-values of both

is 1 solution are 2 solutions are no solutions

to the equation

where the graphs intersect.

Book 1.indb 433

− 1, 1, 2.5, 4,

H M D L

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6. In problem 3, you completed the graph of the equation 3𝑥 + 1 = 𝑥2 − 2𝑥 + 5. Discuss with your partner what you notice about the relationship between the equation 3𝑥 + 1 = 𝑥2 − 2𝑥 + 5 and the concepts explored in problems 1–5. Summarize your discussion.

Collaborative Activity: Solutions of Functions Three pairs of functions and their graphs are shown in the table. 1. Complete the table for each pair of functions. Functions

𝑓(𝑥) = 𝑥 − 3

𝑔(𝑥) = 𝑥 − 2𝑥 − 7 2

434 | Unit 4

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Graphs

Number of Solutions

Solutions to 𝒇(𝒙) = 𝒈(𝒙)

¡ 1 solution ¡ 2 solutions ¡ No solutions

19/07/24 5:08 AM

Functions

Number of Solutions

Graphs

Solutions to 𝒇(𝒙) = 𝒈(𝒙)

𝑓(𝑥) = − 2|𝑥 + 3|

¡ 1 solution

¡ 2 solutions

𝑔(𝑥) = − 2𝑥

¡ No solutions

¡ 1 solution

𝑓(𝑥)= 1 𝑥 + 7 3 3 1

𝑔(𝑥) = �2 �

¡ 2 solutions

𝑥

¡ No solutions

2. The graphs of 𝑦 = ℎ(𝑥) and 𝑦 = 𝑚(𝑥) are shown on the coordinate plane, where ℎ(𝑥) = − 2𝑥 and 𝑚(𝑥) = 3(𝑥 + 1).

a. How many solutions are there to the equation ℎ(𝑥) = 𝑚(𝑥)?

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A table of values for both functions is shown. 𝒙

𝒉(𝒙)

𝒎(𝒙)

−2

−

1 4

−3

−1 0 1 2

−

1 8

−

1 2

−1 −2 −4

−6

3 6 9

b. Discuss with your partner what you know about the solution(s) to the equation ℎ(𝑥) = 𝑚(𝑥) based on the graphs and table of values. c. Complete the statements. Based on the graphs of the functions, the values of 𝑚(𝑥) are

less than greater than

the values of ℎ(𝑥) for 𝑥-values less than the 𝑥-value where the graphs intersect. After the point of intersection, the values of 𝑚(𝑥) are

less than greater than

the

values of ℎ(𝑥) for each 𝑥-value. Based on the table of values, there is a solution to the equation ℎ(𝑥) = 𝑚(𝑥) between 𝑥 = ______ and 𝑥 = ______. d. Discuss with your partner how you could determine a more specific approximation for the solution to the equation ℎ(𝑥) = 𝑚(𝑥). Summarize your discussion.

e. Use a graphing tool to approximate the solution to the equation ℎ(𝑥) = 𝑚(𝑥). 436 | Unit 4

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Lesson Summary When solving equations in the form 𝑓(𝑥) = 𝑔(𝑥), the solutions can be determined by identifying the 𝑥-values for which the values of the function are the same. These points represent 𝑥-values where the graphs of the functions intersect. Equations in the form 𝑓(𝑥) = 𝑔(𝑥) can have 1 solution, no solutions, or multiple solutions depending on how many times the functions 𝑓(𝑥) and 𝑔(𝑥) intersect. Any function type can be included in a system of equations as explored in the lesson. Note that the solution to a system of equations is a coordinate pair (𝑥, 𝑦), and any solution to the equation 𝑓(𝑥) = 𝑔(𝑥) is written in the form 𝑥 = ____. The solution is not a point that satisfies the system, but the 𝑥-value that makes the equation true.

Absolute value, exponential, and quadratic functions like those included in problems in this lesson will be explored more deeply later in this course. y

Practice Problems 5

1. The graphs of 𝑓(𝑥) and 𝑔(𝑥) are shown on the coordinate plane. What is (are) the solution(s) to the equation 𝑓(𝑥) = 𝑔(𝑥)?

x -5

-5

2. The graph of the absolute value function 𝑝(𝑥) = 3(𝑥) − 1 is shown on the coordinate plane.

a. Graph the function 𝑘(𝑥) = − 3𝑥 + 5 on the same coordinate plane. b. How many solutions are there to the equation 𝑝(𝑥) = 𝑘(𝑥)?

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c. What is (are) the solution(s) to the equation 𝑝(𝑥) = 𝑘(𝑥)?

3. The graph of the quadratic function ℎ(𝑥) = 2(𝑥 − 3)2 is shown on the coordinate plane. a. Graph the function 𝑑(𝑥) = 𝑥 − 5 on the same 2 coordinate plane.

b. How many solutions are there to the equation ℎ(𝑥) = 𝑑(𝑥)?

c. What is (are) the solution(s) to the equation ℎ(𝑥) = 𝑑(𝑥)?

438 | Unit 4

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Review Problem 4. The cell phone plan from Company C costs $10 per month, plus $15 per gigabyte for data used. The plan from Company D costs $80 per month, with unlimited data. Rule 𝐶 gives the monthly cost, in dollars, of using 𝑔 gigabytes of data on Company C’s plan. Rule 𝐷 gives the monthly cost, in dollars, of using 𝑔 gigabytes of data on Company D’s plan. a. Write a sentence describing the meaning of the statement 𝐶(2) = 40. b. Which is less, 𝐶(4) or 𝐷(4)? What does this mean for the two phone plans? c. Which is less, 𝐶(5) or 𝐷(5)? Explain how you know. d. For what number 𝑔 is 𝐶(𝑔) = 130? e. Draw the graph of each function.

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Unit 5: Linear and Absolute Value Functions

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Unit 5, Lesson 1: Identifying Key Features of Lines from Different Representations

Warm-Up: Pairs of Equations and Functions 1. Two pairs of equations and functions are shown. Circle the pair(s) in which you think the equation and function represent the same linear relationship. Pair A

Pair B

𝑦 + 7 = 5 (𝑥 − 8) and 𝑓(𝑥) = 5 𝑥 − 1 4

𝑦 − 3 = − 6(𝑥 + 5) and 𝑓(𝑥) = − 6𝑥 − 27

2. Substitute the 𝑥-value indicated into the equation and the function to determine if the output is the same. 𝒙- value

Equation

Function

𝑦 + 7 = 5 (𝑥 − 8) 4

𝑓(𝑥) = 5 𝑥 − 1

𝑦 − 3 = − 6(𝑥 + 5)

𝑓(𝑥) = − 6𝑥 − 27

−𝟐

𝟗

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Guided Activity: Key Features from a Graph 1. A line segment that represents a function is shown on the coordinate plane. a. Draw a vertical line from the line segment’s left endpoint to the 𝑥-axis. What is the 𝑥-value?

6 y 5 4

(–3, 3)

3 2 1

-6 -5

-4

-3

b. Draw a vertical line from the line segment’s right endpoint to the 𝑥-axis. What is the 𝑥-value?

-2

-1

0 -1

(0, 1) (1.5, 0) 1

x 4

-2 -3

(5, –2.333)

-4 -5 -6

c. Discuss with your partner how finding the 𝑥-values of the line segment’s left and right endpoints helps to determine the domain of the function that the line segment represents.

d. Use the 𝑥-values of the line segment’s left endpoint and right endpoint to write the domain as an inequality. __________ ≤ 𝑥 ≤ ___________

e. Work with your partner to determine the range of the function that is represented by the line segment. Use the 𝑦-values of the line segment’s left endpoint and right endpoint to write the range as an inequality. __________≤ 𝑦 ≤ ___________

f. Determine the other key features of the line segment. • The slope of the line segment is __________. • As the 𝑥-values are increasing, the 𝑦-values are ____________________. 444 | Unit 5

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• The coordinates of the point where the line crosses the 𝑦-axis are ____________. The 𝑦-value of the point is ________. • The coordinates of the point where the line crosses the 𝑥-axis are _____________. The 𝑥-value of the point is ________.

g. Determine the intercepts.

𝒙-intercept

𝒚-intercept

2. A line that represents a function is shown on the coordinate plane. The line is similar to the line segment represented in the previous problem, except that it does not have endpoints.

6 y 5 4 3 2 1

a. Complete the statement.

x -6 -5

-4

As the 𝑥-values increase, the 𝑦-values

-3

-2

-1

0 -1

-2

O increase. O decrease.

-3 -4 -5 -6

b. Work with your partner to determine the key features of the line. Key Features 𝑥-intercept

𝑦-intercept

Domain Range

Slope

Book 1.indb 445

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3. Which key features were different for the line than the line segment?

Collaborative Activity: Key Features of a Linear Function from a Table 1. A table of values that represents a function is shown. 𝒙

a. With your partner, discuss which key features of this function, if any, can be determined from the table of values.

−2

□ Domain □ Range □ Slope □ Increasing/Decreasing □ 𝑥-intercept □ 𝑦-intercept

b. Use the table of values to graph the function it represents, 𝑓(𝑥) = 5 𝑥 − 4. 2

c. Complete the statement.

𝒚

−9

−1

− 6.5

− 1.5

3.5

−4

11 10 9 8 7 6 5

As the 𝑥-values increase, the 𝑦-values

O increase. O decrease.

4 3 2 1 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1

1 2

3 4

6 7

9 10 11 12

-2 -3 -4 -5 -6

d. Find the key features of the function 𝑓(𝑥) = 5 𝑥 − 4. 2

446 | Unit 5

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-7 -8 -9 -10 -11 -12

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Key Features Domain

𝑥-intercept

Range

𝑦-intercept Slope

Collaborative Activity: Different Forms of a Line 1. Consider the equation 𝑦 + 3 = − 3 (𝑥 + 8). 4

11 10

a. Graph 𝑦 + 3 = − 3 (𝑥 + 8).

8 7 6 5 4 3 2 1 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1

1 2

3 4

6 7

9 10 11 12

-2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12

b. Complete the statement. As the 𝑥-values increase, the 𝑦-values

Book 1.indb 447

O increase. O decrease.

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c. Find the key features of 𝑦 + 3 = − 3 (𝑥 + 8). 4

Key Features Domain

𝑥-intercept

Range

𝑦-intercept Slope

2. The equation 𝑦 + 3 = − 3 (𝑥 + 8) is one form of an equation or function that represents 4

the line graphed in question 1. The table shows 4 common representations of the same line. Point-Slope Form

Slope-Intercept Form

𝑦 + 3 = − 3 (𝑥 + 8)

𝑦 = − 3 𝑥 − 9

Standard Form

Function Notation

3𝑥 + 4𝑦 = − 36

𝑓(𝑥) = − 3 𝑥 − 9

a. Which equation forms make it easiest to find the line’s slope?

b. Which equation forms make it easiest to find the line’s 𝑦-intercept?

448 | Unit 5

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Lesson Summary A function that models a relationship between 2 quantities includes several key features that convey information about the relationship. Knowing how to identify key features from graphs, tables, and equations helps with interpreting the relationship between 2 quantities. Interpreting these key features provides a means of picturing how quantities relate to each other within a real-world context as well, a skill that will be practiced in the next lesson. For linear functions, key features include 𝑥- and 𝑦-intercepts, domain, range, and slope. Different representations more readily reveal different key features. Any given representation can be translated among other representations as needed to help identify all key features of interest. For a linear function, key features are defined as follows. • The 𝑥- and 𝑦-intercepts are the locations where the graph of the function intersects the 𝑥- and 𝑦-axis, respectively. Using a table or equation, the 𝑥-intercept is the 𝑥-value where 𝑦 = 0, and the 𝑦-intercept is the 𝑦-value where 𝑥 = 0.

• The domain is the set of all possible input values, or 𝑥-values, for the function. • The range is the set of all possible output values, or 𝑦-values, for the function. • The slope is the ratio of the change in 𝑦-values to the change in 𝑥-values. This describes the steepness and direction of the function.

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Practice Problems

1. The graph of a line is shown on the coordinate plane.

3 2 1 x

a. Complete the statement.

-5

-4

-3

-2

-1

-1 -2

As the 𝑥-values increase, the 𝑦-values

-3

O increase. O decrease.

-4 -5

b. Determine the key features of the line. Key Features 𝑥-intercept

𝑦-intercept

Domain Range

Slope

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2. A group of students are discussing how to find the slope and 𝑦-intercept of the line with the equation 9𝑥 − 15𝑦 = − 75. a. Complete each student’s work according to their description.

Student

Description

Work 9𝑥 − 15𝑦 = − 75

Malcolm

Olivia

William

To find the slope and 𝑦-intercept, I rewrite the equation in slope-intercept form.

To find the slope, I mentally move the 𝑥-term to the other side and divide by the coefficient of the 𝑦-term. To find the 𝑦- intercept, I divide the value that the equation is equal to by the coefficient of the 𝑦-term.

To find the slope, I write the coefficient of the 𝑥-term divided by the coefficient of the 𝑦-term. To find the 𝑦-intercept, I write the value that the equation is equal to, divided by the coefficient of the 𝑦-term.

− 15𝑦 = − 9𝑥 − 75 −15 𝑦 = −9 𝑥 − 75 −15 −15 −15

𝑦 = ________ 𝑥 + _______

Slope = –15 or y-intercept =

–15

Slope = –15 or y-intercept =

–15

b. Which student would not find the correct slope and/or 𝑦-intercept?

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Review Problems 3. Solve each system of equations without graphing. a.

� 4𝑑 + 7𝑒 = 68 − 4𝑑 − 6𝑒 = −72

1 𝑥 + 𝑦 = 1 4 � 3 𝑥 − 𝑦 = 4 2 3

4. Mai and Tyler are selling items to earn money for their elementary school. The school earns 𝑤 dollars for every wreath sold and 𝑝 dollars for every potted plant sold. Mai sells 14 wreaths and 3 potted plants and the school earns $70.50. Tyler sells 10 wreaths and 7 potted plants and the school earns $62.50. This situation is represented by this system of equations:

+ 3𝑝 = 70.50 {14𝑤 10𝑤 + 7𝑝 = 62.5

Explain why it makes sense in this situation that the solution of this system is also a solution to 4𝑤 + (−4𝑝) = 8.00.

452 | Unit 5

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Unit 5, Lesson 2: Relating Key Features of Lines to Real-World Contexts

Warm-Up: Equations of Lines 1. Place a digit in each blue box to make the statement true. Use only the digits 1 to 9, no more than 1 time each.

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Collaborative Activity: Determining Key Features of Linear Functions from a Graph 1. Complete the table by matching the equations, tables of values, and graphs. In the last column, note any key features that you use to determine the matches. Equation

Table

Graph

Key Features Used

O Green O Purple O Orange

4𝑥 + 𝑦 = − 3

𝑦 = 1 𝑥 − 3 2

𝑦 − 4 = 1 (𝑥 − 6) 2

O Green O Purple O Orange O Green O Purple O Orange

Tables of Values 1

452 | Unit 5

Book 1.indb 452

Graphs y

𝒙 𝒚

−3 −4.5

−2 −4

−1 −3.5

0 −3

1 −2.5

2 −2

𝒙 𝒚

−3 9

−2 5

−1 1

0 −3

1 −7

2 −11

𝒙 𝒚

−3 −0.5

−2 0

−1 0.5

0 1

1 1.5

2 2

x -20

-10

-20

19/07/24 5:09 AM

2. With your partner, discuss which key features they used for each graph, and discuss why they used those key features to determine the graph’s matching equation and table. Record your summary of their key features and their reasoning. You are the only person who should write in the middle column of the table. Have your partner initial next to your summary, indicating that your summary is correct. Graph

My Summary of My Partner’s Key Features

Initials

Green

Purple

Orange

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Guided Activity: Determining Key Features of Linear Functions in Context Bella is using a gift card to buy coffee at a coffee shop. The initial value of the gift card is $20. The function 𝐴(𝑐) = 20 − 2.75𝑐 represents the amount of money, 𝐴, in dollars, that is still left on the gift card after purchasing 𝑐 cups of coffee. A graph and a table of values representing the function are shown.

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5

4 3

𝒄 0 1 2 3 4 5 6 7 7. 27

𝑨 20 17.25 14.5 11.75 9 6.25 3.5 0.75 0

2 1 -1

0 -1

454 | Unit 5

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1. Work with your partner to identify the key features of the function 𝐴(𝑐) = 20 − 2.75𝑐. Key Features Domain

𝑥-intercept

Range

𝑦-intercept Slope

2. Complete the statement. As the 𝑐-values increase, the 𝐴-values

O increase. O decrease.

The definition of domain states, “The domain may vary depending on the context.” In this context, the domain of the function 𝐴(𝑐) = 20 − 2.75𝑐 should only be whole numbers 7 or less. 3. Work with your partner to write an explanation for why the domain should only be whole numbers 7 or less. Then, if necessary, fix the domain you wrote for 𝐴(𝑐) = 20 − 2.75𝑐 in problem 1. 4. Complete the interpretation of the slope of the function.

For every

the gift card

O 1 cup O 2.75 cups

of coffee purchased, the amount of money still left on

O decreases O increases

Book 1.indb 455

O $1.00. O 2.75.

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5. Work with your partner to explain what the 𝑦-intercept indicates about the gift card.

A key feature is viable if the value(s) are feasible for the real-world context. 6. Explain why the 𝑥-intercept for the function 𝐴(𝑐) = 20 − 2.75𝑐 is not viable for the context.

Lesson Summary When real-world contexts are modeled with linear functions, key features have specific meanings within the context. • The 𝑦-intercept is typically interpreted as the starting, or initial value.

• The 𝑥-intercept has different interpretations based on what it would mean for the function to have a value of 0. In a context where negative 𝑦-values are not viable in the range, the 𝑥-intercept would represent the ending, or final value. • The slope describes how the 𝑦-value changes in relation to the 𝑥-value. Slope is generally interpreted as an increase or decrease by some quantity per 1 unit of the other quantity.

Real-world contexts often include restrictions, or constraints, on what values are possible in the domain and/or range of the function. When interpreting any function, it is important to keep these constraints in mind to determine which values are viable, or feasible, in the real-world context. For example, there are situations where only integers, only positive values, or only values within a certain interval make sense. In cases where there are constraints due to the context, the domain and range should reflect this.

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Practice Problems 1. Use the description of the context and the representations of the function to determine each key feature and whether or not it is viable in context. Domain

Is it viable in context? O Yes O No

Range

Is it viable in context?

Description 𝑃(𝑥) represents the percentage, 𝑃, of US companies who performed drug tests on employees or job applicants from 1998 to 2003, where 𝑥 is the number of years since 1998. Representations

O Yes O No 𝒙-intercept

𝑃(𝑥) = − 1.7𝑥 + 74.5

Is it viable in context?

200

O Yes O No 𝒚-intercept

100

Is it viable in context? O Yes O No

(0, 74.5) (43.824, 0)

-100

100

200

100

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2. Use the description of the context and the representations of the function to determine each key feature and whether or not it is viable in context. Domain

Is it viable in context? O Yes O No

Range

Is it viable in context? O Yes O No

𝒙-intercept

Description 𝑓(ℎ) represents the length of a woman’s femur, 𝑓, in inches (in.), as a function of her height (in.), ℎ. Representations

𝑓(ℎ) = 0.432ℎ − 10.44 50

Is it viable in context? O Yes O No

𝒚-intercept

Is it viable in context?

(24.167, 0) 0

h 50

(0, –10.44)

O Yes O No

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3. Use the description of the context and the representations of the function to determine each key feature and whether or not it is viable in context. Domain

Is it viable in context? O Yes O No

Range

Is it viable in context? O Yes O No

𝒙-intercept

Is it viable in context? O Yes O No

𝒚-intercept

Is it viable in context? O Yes O No

Book 1.indb 459

Description A construction company charges $8 per hour for trash removal, plus a one-time fee of $150 for the use of a dumpster. The total cost, 𝑦, in dollars ($), is a function of the number of hours, 𝑥, that the company takes to remove the trash. Representations 8𝑥 − 𝑦 = − 150

Number of Hours

Fee ($)

−10

$70

−4

$118

$190

$150

$342

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Review Problem 4. The graph of a line is shown on the coordinate plane.

6 y 5 4 3

a. Complete the statement.

2 1

As the 𝑥-values increase, the 𝑦-values

-6 -5

-4

-3

-2

-1

0 -1

-2

O increase. O decrease.

-3 -4 -5

b. Determine the key features of the line.

-6

Key Features 𝑥-intercept

𝑦-intercept

Domain Range

Slope

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Unit 5, Lesson 3: Graphing and Interpreting Key Features of Lines

Warm-Up: Finding Slope 1. Find the slope of each line using the given information. Line

Slope

The line that passes through (4, 1) and (10, 4)

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Guided Activity: Graphing a Line and Interpreting Key Features in Context A local community theater holds 140 seats. The theater uses the equation 16𝑥 + 8𝑦 = 1120 to check that each performance makes enough ticket sales to earn the $1,120 needed to cover production costs. A table of values representing this situation is shown. The variable 𝑥 represents the number of adult tickets, and 𝑦 represents the number of child tickets.

Adult Tickets, 𝒙

Child Tickets, 𝒚

104

140

1. Rewrite 16𝑥 + 8𝑦 = 1,120 in slope-intercept form.

2. Graph the equation 16𝑥 + 8𝑦 = 1,120.

200

3. Circle the 𝑥-intercept in the table, and place the point on the graph. 100

4. Complete the interpretation of the 𝑥- intercept.

-100

100

200

The community theater will meet O 0 production costs when it sells O 70 O 140 O adult O child

-100

tickets.

5. Circle the 𝑦-intercept in the table, and place the point on the graph.

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6. Complete the interpretation of the 𝑦-intercept.

O 0 The community theater will meet production costs when it sells O 70 O 140 O adult O child

tickets.

7. Complete the interpretation of the slope. For every ______ child ticket(s) sold, the community theater needs to sell ______ fewer adult ticket(s).

8. Determine the domain and range of the function in context, and explain any limitations on each. Key Feature

Domain

Range

Key Feature in Context

Explanation

9. Complete the statements. As the number of adult tickets sold, 𝑥, increases, the number of child tickets, 𝑦, that

need to be sold to make a profit

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O increases. O decreases.

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Collaborative Activity: Graphing a Line and Interpreting Key Features in Context The monarch butterfly is essential to the pollination process of many native and farmed North American plants. Every year, these butterflies migrate thousands of miles over the course of several months. A South Carolina Wildlife Federation scientist has been tracking their population since the 2005 migration cycle. The scientist uses the function 𝐵(𝑥) = −35𝑥 + 325 to estimate the number of butterflies, where 𝑥 is the number of years since 2005 and 𝐵 is the number of butterflies, in thousands. 1. Graph the function.

400

300

200

100

-6 -4

-2

14 16

-100

2. Solve − 35𝑥 + 325 = 0. 3. Which intercept is the solution to − 35𝑥 + 325 = 0? 4. Interpret the intercept found in question 3 in context.

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5. Find 𝐵(0). 6. Which intercept is the value of 𝐵(0)? 7. Interpret the intercept found in question 5 in context.

8. Determine the domain and range of the function in context, and explain any limitations on each. Key Feature

Domain

Range

Key Feature in Context

Explanation

9. Complete the statements.

As the

O number of years since 2005 O number of butterflies, in thousands,

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increases by 1, the

O increases. O decreases.

O O O O

35. 325. 35,000. 325,000.

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Lesson Summary For each context in this lesson, different information was used to graph the linear relationships. To graph a linear relationship given a real-world context, either of the following is needed: • The slope and 𝑦-intercept • Two ordered pairs

Use the given information to translate among the different representations of the relationship to determine what is needed to graph the function. Then, sketch the graph of the line, or use technology to graph the relationship. Use the context and the graph to interpret the real-world meaning of the line’s key features.

Practice Problems 1. At a gym in Aiken, South Carolina, the cost for a membership is given by the equation 𝑦 = 32.50𝑥 + 65, where 𝑦 is the total cost, in dollars, for a membership for a certain number of months, 𝑥. The gym charges a one-time initial membership fee and a recurring monthly fee. Graph the equation. 150

125 100 75 50 25 x

-10

-5

-25 -50

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2. Determine the key features of the function from problem 1. Key Features 𝑥-intercept

𝑦-intercept

Domain Range

Slope

3. Interpret the 𝑦-intercept in the context of problem 1.

4. Interpret the slope in the context of problem 1.

Review Problems 5. Which equation has the same solution as 10𝑥 − 𝑥 + 5 = 41? A. 10𝑥 + 5 = 41

B. 10𝑥 − 5 + 𝑥 = 41 C. 9𝑥 = 46

D. 9𝑥 + 5 = 41

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6. Elena is solving this system of equations: � 10𝑥 − 6𝑦 = 16 5𝑥 − 3𝑦 = 8

She multiplies the second equation by 2, and then subtracts the resulting equation from the first. To her surprise, she gets the equation 0 = 0. What is special about this system of equations? Why does she get this result and what does it mean about the solutions? (If you are not sure, try graphing them.)

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Unit 5, Lesson 4: Solving Problems Modeled with Linear Functions

Warm-Up: Air Pressure 1. As you watch the video of a football being inflated, use the coordinate grid to plot the pressure, in pounds per square inch (psi), over time.

Air pressure (psi)

x 0

10 11 12 13 14 15

Time (seconds) 2. The person inflating the football paused on occasion. Explain how the graph would look different if the person did not pause.

3. Explain why it is possible that the 𝑦-intercept is greater than 0.

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Collaborative Activity: Finding Solutions For weight management exercise and improving cardio fitness, the function 𝐻(𝑎) = 0.7(220 − 𝑎) models the recommended heart rate, 𝐻, in beats per minute (BPM) based on a person’s age, 𝑎, in years. 1. Work with your partner to determine if the variable for age, 𝑎, should have a constraint. Explain your choice.

2. Write the constraints.

300 250 200 150

3. Graph the function in terms of the context.

100 50 x

-150

-100

-50

100

150

200

250

300

-50 -100

4. Use the graph to find the solution for 0.7(220 − 𝑎) = 70. Label the point.

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-150 -200

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5. Dontrell was working out. He wondered if his heart rate of 125 BPM was close to the recommended BPM for his age. He wrote the equation 125 = 0.7(220 − 𝑎). Solve the equation to determine the age associated with a recommended BPM of 125.

6. Determine the age associated with a recommended heart rate of 194.

7. Find the solution for 𝐻(𝑎) = 0.

8. Determine if the solution to 𝐻(𝑎) = 0 is viable. Justify your answer.

9. What key feature is found when solving 𝐻(𝑎) = 0?

10. Find 𝐻� 1 �. 2

11. Determine if the value for 𝐻� 1 � is reasonable for the context. Explain why or why 2 not.

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Lesson Summary To graph a linear function, use the form of the function given to identify at least 2 points or 1 point and the slope. Use the point(s) and slope to graph a line on the coordinate plane. If there are constraints on the context, then consider those when graphing and when finding and interpreting solutions to the function. To solve a problem using the graph of the function, find a value along the 𝑥-axis, and use the graph to read the corresponding 𝑦-value, or vice-versa, depending on the question being asked. Consider the context to determine whether all solutions represented by the graph are viable in the context. For example, a person cannot be a negative age or an age beyond what is realistic for a human. The function can also be used to solve problems algebraically. Substitute a value for the output, and solve to find the corresponding input value, or vice versa, depending on the question being asked. Again, the context needs to be carefully considered to determine if all values calculated using the function are viable. For example, it is not always possible to purchase partial quantities of something, and purchases are often limited by the budget available.

Practice Problems 1. Yasmani uses a gift card to buy video games. The initial value of the gift card is $250. The function 𝑉(𝑥) = 250 − 15𝑥 represents the amount of money, 𝑉, y in dollars, that is still left on the gift 250 card after purchasing 𝑥 video games that cost $15 each. 200

a. Describe the constraints.

150 100

b. Write the constraints using inequalities.

50 x

-10

c. Interpret the slope in terms of the context.

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-5

-50 -100

19/07/24 5:09 AM

d. Graph the function in terms of the context.

e. Use the graph to find the solution to 𝑉(𝑥) = 70. Label the point.

f. Find the solution of 𝑉(𝑥) = 305.

g. Determine if the solution to 𝑉(𝑥) = 305 is viable. Justify your answer.

h. Solve 250 − 15𝑥 = 125.

i. Explain why the solution to 250 − 15𝑥 = 125 does not make sense for the context.

j. How many games could Yasmani buy if the gift card had $55 left on it?

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2. Consider the function 𝑓(𝑥) = −2 𝑥 + 4. 5

y 8 7 6

a. Graph 𝑓(𝑥) = −2 𝑥 + 4.

4 3 2 1

b. Use the graph to find the solution to 𝑓(𝑥) = 8. Label the point.

-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 -2

x 1 2 3

4 5 6

7 8 9 10 11 12

-3 -4 -5 -6 -7 -8

c. Solve − 2 𝑥 + 4 = 8. 5

d. How does the solution to the equation relate to the point you labeled on the graph?

Review Problems 3. Andre does not understand why a solution to the equation 3 − 𝑥 = 4 must also be a solution to the equation 12 = 9 − 3𝑥. Write a convincing explanation as to why this is true.

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4. Three siblings are participating in a family-friendly running event. • The oldest sibling begins at the start line of the race and runs 7 miles per hour (mph) the entire time. • The middle sibling begins at the start line and walks at 3.5 mph throughout the race. • The youngest sibling joins the race 4 miles (mi.) from the start line and runs 5 mph the rest of the way.

Match each graph to the sibling whose running is represented by the graph.

Oldest Sibling

Graph A

Middle Sibling

Graph B

Youngest Sibling

Graph C

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Unit 5, Lesson 5: Introducing Absolute Value Functions

Warm-Up: How Good Were the Guesses? Before this lesson, you were asked to guess the number of objects in the jar. The guesses of all students have been collected. Your teacher will share the data and reveal the actual number of objects in the jar. Use that number to calculate the absolute guessing error of each guess, or how far the guess is from the actual number. Suppose the actual number of objects is 100. • If your guess is 75, then the absolute guessing error is 25.

• If your guess is 110, then the absolute guessing error is 10.

Record the absolute guessing error of at least 12 guesses in Table A of the handout from your teacher (or elsewhere as directed).

Exploration Activity: Plotting the Guesses Refer to the table you completed in the warm-up, which shows your class’s guesses and absolute guessing errors. 1. Plot at least 12 pairs of values from your table on the coordinate plane on the handout (or elsewhere as directed by your teacher). 2. Write down 1–2 other observations about the completed scatter plot.

3. Is the absolute guessing error a function of the guess? Explain how you know.

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Exploration Activity: Oops, Try Again! Earlier, you guessed the number of objects in a container and then your teacher told you the actual number. Suppose your teacher made a mistake about the number of objects in the jar and would like to correct it. The actual number of objects in the jar is ____. 1. Find the new absolute guessing errors based on this new information. Record the errors in Table B of the handout (or elsewhere as directed by your teacher). 2. Make 1–2 observations about the new set of absolute guessing errors.

3. a. Predict how the scatter plot would change given the new actual number of objects. (Would it have the same shape as in the first scatter plot? If so, what would be different about it? If not, what would it look like?)

b. Use technology to plot the points and test your prediction.

4. Can you write a rule for finding the output (absolute guessing error) given the input (a guess)?

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Lesson Summary In a number guessing game where the guess that is closest to a target number wins, it doesn’t matter if the guess is above or below the target number. What matters is how far off the guess is from the target number, or the absolute guessing error. The smaller the absolute guessing error, or the closer it is to 0, the better.

Suppose 8 people made these guesses for the number of pretzels in a jar: 14, 15, 19, 21, 23, 24, 26, and 28. If the actual number of pretzels is 22, the absolute guessing error of each number is as shown in the table. Guess Absolute Guessing Error

In this case, 21 and 23 are both winning guesses. Even though one number is an underestimate and the other an overestimate, 21 and 23 are both 1 away from 22. Of all the possible absolute guessing errors, 1 is the smallest one.

If the guesses and the guessing errors were plotted on a coordinate plane, the points would form a V shape. Notice that the V shape is above the horizontal axis, suggesting that all the vertical values are positive.

Suppose the actual number of pretzels is 19. The absolute guessing errors of the same 8 guesses are shown in this table. Guess Absolute Guessing Error

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14 5

15 4

19 0

21 2

23 4

24 5

26 7

28 9

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Notice that all the errors are still nonnegative. When these points are plotted on a coordinate plane, they still form a V shape, and they all land on or above the horizontal line. Why does the relationship between guesses and absolute guessing errors always have this kind of graph? This will be explored further in upcoming lessons.

Practice Problems 1. A group of ten friends played a number guessing game. They were asked to pick a number between 1 and 20. The person closest to the target number wins. The ten people made these guesses. Guess Absolute Guessing Error

a. The actual number was 14. Complete the table with the absolute guessing errors. b. Graph the guess and absolute guessing errors.

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c. Is the absolute guessing error a function of the guess? Explain how you know.

2. Bags of walnuts from a food producer are advertised to weigh 500 grams (g) each. In a certain batch of 20 bags, most bags have an absolute error that is less than 4 g.

Could this scatter plot represent those 20 bags and their absolute errors? Explain your reasoning.

3. The class guessed how many objects were placed in a mason jar. The graph displays the class results, with an actual number of 47. Suppose a mistake was made, and the actual number is 45.

Explain how the graph would change, given the new actual number.

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Review Problems 4. Function 𝐷 gives the height of a drone 𝑡 seconds after it lifts off.

Sketch a possible graph for this function given that: • 𝐷(3) = 4

• 𝐷(10) = 0

• 𝐷(5)>𝐷(3)

5. The population of a city grew from 23,000 in 2010 to 25,000 in 2015. a. What was the average rate of change during this time interval?

b. What does the average rate of change tell us about the population growth?

6. Select all systems that are equivalent to this system of equations: � 4𝑥 + 5𝑦 = 1 □ � 4𝑥 − 4𝑦 = 3 5

4𝑥 + 5𝑦 = 1 𝑥 − 𝑦 = 3 8

□ � 𝑥 𝑥 − + 𝑦 4 =𝑦 = 3 4 8

4𝑥 + 5𝑦 = 1 □ � 5𝑥 − 5𝑦 = 3

8𝑥 + 10𝑦 = 2 □ � 8𝑥 − 8𝑦 = 3

□�

𝑥 + 𝑦 = 1

5 3 𝑥 − 𝑦 = 8

482 | Unit 5

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Unit 5, Lesson 6: Exploring Absolute Value Functions

Warm-Up: Temperature in Toronto Toronto is a city at the border of the United States and Canada, just north of Buffalo, New York. Here are twelve guesses of the average temperature of Toronto, in degrees Celsius (℃), in February 2017.

−5

−1

1.5

− 2.5

− 0.5

1. The actual average temperature of Toronto in February 2017 is 0℃.

Use this information to sketch a scatter plot representing the guesses, 𝑥, and the corresponding absolute guessing errors, 𝑦.

2. What rule can you write to find the output given the input?

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Collaborative Activity: The Distance Formula The function 𝐴 gives the distance of 𝑥 from 0 on the number line. 1. Complete the table, and sketch a graph of function 𝐴. 𝒙 8

𝜋

1 2

𝑨(𝒙) 5.6

−1 2

−1

− 5.6

2. Armando and Elise are trying to write a rule for this function. • Armando says that the function looks like parts of 2 linear functions, where the slope is −1 up until a certain point and then changes to a slope of 1, so he writes 𝐴(𝑥) = � 𝑥, for 𝑥 ≥ 0 . −𝑥, for 𝑥 < 0

• Elise says the absolute value is the distance of a number from 0, so she writes 𝐴(𝑥) = |𝑥|. Discuss with your partner why both equations correctly represent function 𝐴. Summarize your discussion. 484 | Unit 5

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Exploration Activity: Moving Graphs Around Graph the functions 𝑓(𝑥) = |𝑥 + 𝑎| and 𝑔(𝑥) = |𝑥| + 𝑏. Experiment with different values of 𝑎 and 𝑏, and observe what happens to the graphs. 1. How does changing 𝑎 change the graph?

2. How does changing 𝑏 change the graph?

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Collaborative Activity: More Moving Graphs Around Five equations and 4 graphs are shown. 1. Match each equation with the graph that represents it. One equation will have no match. Equation

Graph

a. 𝑦 = |𝑥 − 3|

b. 𝑦 = |𝑥 − 9| + 3

c. 𝑦 = |𝑥| − 6

d. 𝑦 = |𝑥 + 3|

e. 𝑦 = |𝑥 + 3| − 6

486 | Unit 5

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2. For the equation without a match, sketch a graph on the blank coordinate plane. Equation:_____

3. Use graphing technology to check your matches and your graph. Revise your matches and graphs as needed.

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Lesson Summary In a guessing game, each guess can be seen as an input of a function and each absolute guessing error as an output. Because absolute guessing error tells how far a guess is from a target number, the output is distance. Suppose the target number is 0. The distance of a guess, 𝑥, from 0 is found by calculating 𝑥 − 0. Because distance cannot be negative, find |𝑥 − 0| or simply |𝑥|. If function 𝑓 gives the distance of 𝑥 from 0, then it can be defined with the equation 𝑓(𝑥) = |𝑥|.

Function 𝑓 is an absolute value function. It gives the distance of 𝑥 from 0 by finding the absolute value of 𝑥.

The graph of function 𝑓 is a V shape with the 2 lines converging at (0, 0). This point is called the vertex of the graph. It is the point where a graph changes direction, from going down to going up or the other way around. The vertex of the graph of an absolute value function is the point where the graph changes from increasing to decreasing, or vice versa. Two additional examples are shown. • The distance between 𝑥 and 4 is represented as 𝑥 − 4. Distance cannot be negative, so consider the absolute value of that difference, |𝑥 − 4|. If function 𝑝 gives the distance of 𝑥 from 4, then it is defined by the equation 𝑝(𝑥) = |𝑥 − 4|. The graph of 𝑝 is shown.

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• The distance between 𝑥 and 4 is represented as 𝑥 − (−4), which is equal to 𝑥 + 4. Distance cannot be negative, so consider the absolute value of that difference, |𝑥 + 4|. If function 𝑞 gives the distance of 𝑥 from −4, then it is defined by the equation 𝑞(𝑥) = |𝑥 + 4|. The graph of 𝑞 is shown. Notice that the graphs of 𝑝 and 𝑞 are like that of 𝑓, but they have shifted horizontally.

Practice Problems 1. Shown are 4 absolute value functions and 3 coordinate pairs. Each coordinate pair represents the vertex of the graph of an absolute value function. Match each function with the coordinates of the vertex of its graph. The vertex coordinates of the graph of 1 function are not shown. Function

Vertex

a. 𝑝(𝑥) = |𝑥 − 9|

(−9, 0)

c. 𝑟(𝑥) = |𝑥 + 9|

(0, −9)

b. 𝑞(𝑥) = |𝑥| + 9 d. 𝑡(𝑥) = |𝑥| − 9

(9, 0)

2. Function 𝐺 is defined by the equation 𝐺(𝑥) = |𝑥|.

Function 𝑅 is defined by the equation 𝑅(𝑥) = |𝑥| + 2.

Describe how the graph of function 𝑅 relates to the graph of 𝐺, or sketch the graphs of the two functions to show their relationship.

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3. Here is the graph of a function. Select the equation for the function represented by the graph. A. 𝑦 = |𝑥| − 5 B. 𝑦 = |𝑥| + 5 C. 𝑦 = |𝑥 − 5| D. 𝑦 = |𝑥 + 5|

Review Problems 4. The temperature was recorded at several times during the day. Function 𝑇 gives the temperature in degrees Fahrenheit, ℉, 𝑛 hours (hr.) since midnight. Here is a graph for this function.

a. Pick two consecutive points and connect them with a line segment. Estimate the slope of that line. Explain what that estimated value means in this situation.

b. Pick two non-consecutive points and connect them with a line segment. Estimate the slope of that line. Explain what that estimated value means in this situation.

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5. A tennis ball is dropped from an initial height of 30 feet (ft.). It bounces 5 times, with each bounce height being about 2 of the height of the previous bounce. 3

Sketch a graph that models the height of the ball over time. Be sure to label the axes.

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Unit 5, Lesson 7: Solving One-Variable Absolute Value Equations

Warm-Up: Critical Thinking 1. Place a digit in each blue box to make the statement true. Use digits 1 to 9 only no more than once each.

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Exploration Activity: Graphs of Absolute Value Equations 1. The graph of the absolute value equation 𝒚 = |𝒙| − 𝟐 is shown.

The graph of 𝒚 = |𝒙| − 𝟐 can be used to find the solution to 1 = |𝑥| − 2. a. What is the equation of the green line?

b. What are the points of intersection for the 2 equations?

c. Substitute the 𝑥-coordinates of the points of intersection into the equation 1 = |𝑥| − 2. Show your work in the table. 1 = |𝑥| − 2

1 = |𝑥| − 2

d. Discuss with your partner what you noticed when the 𝑥-coordinates were substituted into the equation. Summarize your discussion.

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2. The graphs of 𝑦 = |𝑥| and 𝑦 = −|𝑥|+ 2 are shown on the coordinate plane. a. Decide who will be partner A and who will be partner B, and then complete the table. Partner A

Partner B

What is (are) the value(s) of 𝑦 when 𝑥 = 2 on the blue graph?

What is (are) the value(s) of 𝑦 when 𝑥 = 0 on the orange graph?

What is (are) the value(s) of 𝑥 when 𝑦 = 1 on the blue graph?

What is (are) the value(s) of 𝑥 when 𝑦 = 0 on the orange graph?

What is (are) the value(s) of 𝑥 when 𝑦 = 3 on the blue graph?

What is (are) the value(s) of 𝑥 when 𝑦 = −3 on the orange graph?

b. What was true when there was only 1 value of 𝑥 for the given 𝑦-value? c. What was true when there were 2 values of 𝑥 for the given 𝑦-value? d. What was true when there were no values of 𝑥 for the given 𝑦-value? e. Discuss with your partner the circumstances under which an absolute value function would have each number of solutions.

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3. Four absolute value equations are shown. a. Work with your partner to determine the 𝑥-values that make each equation true. 𝒙 = |𝟮|

𝒙 = |−𝟮|

|𝒙| = 𝟯

|𝒙| = −𝟯

b. What did you notice about the 4 equations and their solutions?

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Guided Activity: Solving Absolute Value Equations 1. The graphs of 𝑦 = |𝑥| + 1 and 𝑦 = 6 are shown on the coordinate plane. a. Use the graph to determine the solution(s) to |𝑥| + 1 = 6. b. Solve |𝑥| + 1 = 6 algebraically.

2. The graphs of 𝑦 = |𝑥 + 1| and 𝑦 = 6 are shown on the coordinate plane. a. Use the graph to determine the solution(s) to |𝑥 + 1| = 6. b. Solve |𝑥 + 1| = 6 algebraically. 3. Solve 16 = |𝑦| + 5 algebraically.

4. Solve |9𝑧| = 15 algebraically.

Book 1.indb 497

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Collaborative Activity: Solving Multistep Absolute Value Equations 1. The mass of a regulation baseball, 𝑚, should be 145.5 grams (g). The manufacturing tolerance is within 3.5 g. a. Use the equation |𝑚 − 145.5| = 3.5 to determine the minimum and maximum values of the mass of a baseball.

b. Express all values of 𝑚 according to the manufacturing tolerance. Use set notation.

2. Solve each absolute value equation. a. |𝑦 − 7| − 4 = 3 b. 0 = |5𝑦 − 18| − 42 c. 64 = − 4|2𝑦 + 9| d. |5 − 8𝑥| + 11 = 11

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3. Discuss with your partner which of the equations from problem 2, if any, will have an intersection at the vertex of the graph. Summarize your discussion.

Lesson Summary To solve an absolute value equation algebraically, use the following steps. 1. Use inverse operations to isolate the absolute value term or expression to 1 side of the equation. 2. Write 2 equivalent equations without using absolute value, if possible. 3. Solve the equation(s), if possible.

Absolute value equations can have 1, 2, or no solutions. Three examples are shown. • Example 1: −3 + |2𝑥 − 1| = 4 |2𝑥 − 1| = 7

2𝑥 − 1 = 7 2𝑥 = 8 𝑥 = 4

2𝑥 − 1 = 2𝑥 = 𝑥 =

The solutions to −3 + |2𝑥 − 1| = 4 are 𝑥 = 4 and 𝑥 = −3.

Book 1.indb 499

−7 −6 −3

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• Example 2: |𝑥 + 1| − 2 |𝑥 + 1| 𝑥 + 1 𝑥

= = = =

The only solution to |𝑥 + 1| − 2 = −2 is 𝑥 = −1.

−2 0 0 −1

• Example 3: |𝑥 + 2| + 7 = 4 |𝑥 + 2| = −3

The equation |𝑥 + 2| = −3 is a false equation since there is not a number whose absolute value is equal to −3. Therefore, there are no solutions to |𝑥 + 2| + 7 = 4.

Practice Problems 1. Solve |𝑤| − 42 = −37.

2. Solve −|𝑥 − 1| − 1 = 0.

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3. Solve |𝑥 − 2| + 6 = 6.

4. Solve 6|𝑥 + 1| − 18 = 0.

5. The width of a machine part, 𝑤, should be 4.75 centimeters (cm). The manufacturing tolerance is within 0.05 cm. The equation |𝑤 − 4.75| = 0.05 can be used to determine the minimum and maximum widths of the machine part. a. Determine the minimum and maximum widths of the machine part.

b. Express all values of 𝑤 according to the manufacturing tolerance. Use set notation.

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Review Problems 6. Imagine a situation where a person is using a garden hose to fill a child’s pool. Think of two quantities that are related in this situation and that can be seen as a function. a. Define the function using a statement of the form “____ is a function of _____.” Be sure to consider the units of measurement.

b. Sketch a possible graph of the function. Be sure to label the axes. Then, identify the coordinates of one point on the graph and explain its meaning.

7. 𝑊 gives the weight of a puppy, in pounds, as a function of its age, 𝑡, in months. Describe the meaning of each statement in function notation. a. 𝑊(2) = 5 b. 𝑊(6) > 𝑊(4) c. 𝑊(12) = 𝑊(15)

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Unit 5, Lesson 8: Graphing Absolute Value Functions

Warm-Up: Error Analysis Justin’s work for solving 10 − 5|𝑥 − 4| = 45 is shown.

10 — 5|x — 4| = 45 5|x — 4| = 45 5|x — 4| = 45 5 5 |x — 4| = 9

x — 4= 9 +4 +4

x — 4 = —9

x = 13

+4 +4

x = —5

1. Circle the mistake Justin made.

2. Find the correct solution to 10 − 5|𝑥 − 4| = 45.

Book 1.indb 503

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Guided Activity: Graphing Absolute Value Functions 1. The table of values represents the absolute value function 𝑓(𝑥) = |𝑥|. Use the table of values to graph 𝑓(𝑥) = |𝑥|. 𝒙

𝒇(𝒙)

−1

−2 0 1 2

2 0 1 2

5 4 3 2 1

-6 -5 -4 -3 -2 -1 0 -1

-2 -3 -4 -5 -6

2. The table of values represents the absolute value function ℎ(𝑥) = |𝑥 + 1| − 2. Use the table of values to graph ℎ(𝑥) = |𝑥 + 1| − 2. 𝒙

−3 −2 −1 0 1

𝒉(𝒙) 0

−1 −2 −1 0

5 4 3 2 1 -6 -5 -4 -3 -2 -1 0 -1

x 1

-2 -3 -4 -5 -6

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3. A partial table of values for the function 𝑡(𝑥) = |𝑥 − 1| − 1 is shown. Complete the table of values, and graph the function 𝑡(𝑥) = |𝑥 − 1| − 1. 𝒙

−3

𝒕�𝒙�

5 4 3

−2

2 1

−1

-6 -5 -4 -3 -2 -1 0 -1

x 1

-2 -3

-4

-5 -6

a. Discuss the key features shown in the table for the function 𝑡(𝑥) = |𝑥 − 1| − 1. Then, complete the table. 𝒙-intercept(s)

End Behavior

𝒚-intercept

As 𝑥 → ∞, 𝑦 → ______.

As 𝑥 → −∞, 𝑦 → ______.

b. Complete the table of additional key features for the function 𝑡(𝑥) = |𝑥 − 1| − 1. Key Features

Domain

Increasing interval(s)

Range

Decreasing interval(s)

Vertex

Maximum/minimum?

Book 1.indb 505

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Collaborative Activity: Identifying Key Features of Absolute Value Functions Absolute value functions have an axis of symmetry, which is a vertical line in the form 𝑥 = ℎ that contains the vertex and divides the graph of the function into 2 halves. Two absolute value functions are shown on the coordinate grid. 6

5 4 3 2 1 -6 -5 -4 -3 -2 -1 0 -1

x 1

-2 -3 -4 -5 -6

Work with your partner to complete the following. 1. Complete the table for each function. Function

Vertex

Equation of the Axis of Symmetry

Blue Orange

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2. List 2 points on the orange graph that are equidistant from the axis of symmetry.

3. List 2 points on the blue graph that are equidistant from the axis of symmetry.

4. Discuss the additional key features shown for each function. Then, complete the table. Function

𝒙-intercept(s)

End Behavior

𝒚-intercept

As 𝑥 → ∞, 𝑦 →______.

Blue

As 𝑥 → −∞, 𝑦 →______. As 𝑥 → ∞, 𝑦 →______.

Orange

As 𝑥 → −∞, 𝑦 →______.

5. Complete the table of additional key features for the orange function. Key Features Domain

Increasing interval(s)

Range

Decreasing interval(s)

Maximum

Minimum

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6. Complete the table of additional key features for the blue function. Key Features Domain

Increasing interval(s)

Range

Decreasing interval(s)

Maximum

Minimum

Lesson Summary One way to graph an absolute value function is to create a table of values to find ordered pairs (𝑥, 𝑦) that satisfy the function. Use the table and equation to identify key features of the absolute value function that can be used to graph the function. In addition to domain, range, and 𝑥- and 𝑦-intercepts, absolute value functions include other key features of interest. • The axis of symmetry is a vertical line in the form 𝑥 = ℎ that divides the graph of the function into 2 halves. • An increasing interval is an interval of the domain over which the function increases as the value of 𝑥 increases. • A decreasing interval is an interval of the domain over which the function decreases as the value of 𝑥 increases. • The vertex is the point where the function changes from increasing to decreasing. This point is the maximum if the graph opens downward or the minimum if the graph opens upward. • The end behavior describes how the function behaves “at the ends” of the graph. How the outputs of a function change as we look at input values farther and farther from 0 is called end behavior.

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The function 𝑓(𝑥) = −⃒𝑥 + 2 ⃒ − 1 is shown on the coordinate plane with several key features labeled. • Notice that at the left end of the graph, as 𝑥 approaches −∞, 𝑓(𝑥) approaches −∞, and at the right end of the graph, as 𝑥 approaches ∞, 𝑓(𝑥) approaches −∞. This describes the end behavior of the function.

• From the left end of the function to 𝑥 = − 3 , the function is 2

increasing. From the 𝑥 = − 3 2

to the right end of the function, it is decreasing. Intervals can be described using inequalities, set notation, or interval notation. Unless specified, any form is acceptable. Examples of each notation for this function are shown.

Increasing Interval Decreasing Interval

Inequality

Set Notation

Interval Notation

−∞ < 𝑥 < −3

� 𝑥⃒ − ∞< 𝑥 < − 3 �

�− ∞, − 3 �

− 3 < 𝑥 < ∞ 2

� 𝑥⃒ − 3 < 𝑥 < ∞� 2

�− 3 , ∞� 2

Be careful not to confuse intervals where the function is increasing or decreasing with intervals where the function is positive or negative. This will be explored more in future courses. However, in this example, there is no positive interval since all values of 𝑓(𝑥) are negative for any value of 𝑥. The negative interval for the function is −∞ < 𝑥 < ∞.

Book 1.indb 509

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Practice Problems 1. Consider the function 𝑔(𝑥) = |𝑥 + 2| − 3.

a. Complete the table of values, and then graph 𝑔(𝑥) = |𝑥 + 2| − 3. 𝒙

−5 −4 −3

−2 −1 0 1

𝒈�𝒙�

5 4 3 2 1 -6 -5 -4 -3 -2 -1 0 -1

x 1

-2 -3 -4 -5 -6

b. Label the vertex, and draw the axis of symmetry on the graph.

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2. Complete the table of key features for the function in problem 1. Key Features Increasing interval(s)

𝑥-intercept(s)

Decreasing interval(s)

𝑦-intercept Domain

Vertex

Range

Maximum/ minimum?

Left end behavior

As 𝑥 → ____, 𝑦 → ____. Right end behavior As 𝑥 → ____, 𝑦 → ____.

Review Problems 3. An ecologist is studying a forest with a mixture of tree types. Since the average tree height in the area is 40 feet (ft.), he measures the height of the tree against that. He also records the type of tree. The results are shown in the table and segmented bar graph. Is there evidence of an association between tree height and tree type? Explain your reasoning.

Book 1.indb 511

Deciduous Evergreen Total

Under 40 ft.

40 ft. or taller

Total

14 59

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4. Solve each system of equations without graphing. Show your reasoning. a. �

−5𝑥 + 3𝑦 = −8 3𝑥 − 7𝑦 = −3

b. �

−8𝑥 − 2𝑦 = 24 5𝑥 − 3𝑦 = 2

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Unit 5, Lesson 9: Interpreting Key Features of Absolute Value Functions

Warm-Up: Key Features of Absolute Value Functions The graph of ℎ(𝑥) = − ⃒𝑥 + 1 ⃒ + 2 is

shown on the coordinate plane.

Book 1.indb 513

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1. Complete the table of key features for ℎ(𝑥). Key Features Increasing interval(s)

𝑥-intercept(s)

Decreasing interval(s)

𝑦-intercept Domain

Vertex

Range

Maximum/ Minimum?

Left end behavior

As 𝑥→ ____, 𝑦→ ____.

Right end behavior

As 𝑥→ ____, 𝑦→ ____.

2. What is the equation for the axis of symmetry?

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Collaborative Activity: Representing Real-World Applications of Absolute Value Functions The clearance below the Sunshine Skyway Bridge over Tampa Bay, Florida, is 180.5 feet (ft.) above sea level. According to the distribution of tidal phases, the water under the bridge can fluctuate up to 4 ft. above or below its normal depth. If 𝑥 is the amount of clearance, in ft., that the bridge has above the water, and 𝑦 is the variation in water depth, in ft., compared to sea level, the equation to find the variation in water depth is 𝑦 = |𝑥 − 180.5|. 1. Discuss with your partner the maximum and minimum clearance between the bridge and the water. Summarize your discussion in the table. Minimum Clearance (ft.)

Book 1.indb 515

Maximum Clearance (ft.)

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2. Complete the table, and graph the absolute value equation to find the variation from the normal depth. 𝒙

𝒚

Too much variation

Variation

176.5 177.5 178.5 179.5

180.5 181.5

x 176

182.5

177

178

180

181

182

183

184

185

Feet above water

183.5 184.5

179

Too much variation

3. Interpret the vertex of 𝑦 = |𝑥 − 180.5|.

4. The graphs of the equations 𝑦 = |𝑥 − 180.5| and 𝑦 = 4 are shown.

(176.5, 4) 160

(184.5, 4) 180

200

Work with your partner to explain how the graphs show the minimum and maximum clearance.

-20

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Guided Activity: Representing Real-World Applications of Absolute Value Functions 5

1. The average length of a football used at the college level is 1132 inches (in.), with a 9

tolerance of 32 of an in. If 𝑥 is the length of the football, and 𝑉 is the variation from 5

1132 in., the function 𝑉(𝑥) = ⃒𝑥 − 1132 ⃒ can be used to find the amount of variation. a. Determine the minimum and maximum 𝑥-values.

b. Complete the table for 𝑉(𝑥) = ⃒𝑥 − 1132 ⃒. Use the minimum 𝑥-value in the first 3

column and the maximum 𝑥-value in the last column. Add 32 to each 𝑥-value,

starting with the minimum and ending with the maximum. 𝒙

𝒚

c. Graph the absolute value function.

0.3

d. Interpret the vertex of 5 𝑉(𝑥) = ⃒𝑥 − 1132 ⃒.

Variation

0.25

e. Interpret the axis of symmetry of 5

𝑉(𝑥) = ⃒𝑥 − 1132 ⃒.

0.2 0.15 0.1 0.05 x 10.75

11.25

11.5

11.75

Book 1.indb 517

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2. For centuries, the average internal body temperature of humans has been designated as 98.6 degrees Fahrenheit (℉). The temperature can vary by as much as 1.8° and still be considered normal. If 𝑥 is the body temperature, and 𝑦 is the variation from the average, the equation to find the amount of variation is 𝑦 = |𝑥 − 98.6|. The table of values and the graph for the context are shown. 𝒙

96.5

Too much variation

96.8 97.1

1.8

97.4

1.5 1.2

97.7 98.0 98.3 98.6

0.9 0.6 0.3 0

98.9

0.3

99.5

0.9

99.2 99.8

100.1 100.4 100.7

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𝒚

0.6 1.2 1.5 1.8

Too much variation

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Complete the table of key features. Key Features Increasing interval(s)

𝑥-intercept

Decreasing interval(s)

Domain

Vertex

Range

Maximum/ Minimum?

𝑦-intercept

Axis of symmetry

Left end behavior Right end behavior

As 𝑥→______, 𝑦→_______.

Lesson Summary Absolute value functions can be used to represent real-world contexts. They are often used to model allowable or acceptable variances from some key value. In such contexts, there may be restrictions on the domain and/or range, which can put limits on the viability of key features of the function in context. For example, in this lesson, domain and range are often limited to ranges that are within an acceptable variance. Anything beyond these values is no longer acceptable. These limitations then affect the limits of the end behavior of the absolute value function. In context, the vertex and axis of symmetry each have a real-world interpretation. They do have a similar meaning because the vertex lies on the axis of symmetry, but their interpretations are not exactly the same. For example, a manufacturing company builds components. A particular component is specified to be 19.8 millimeters (mm) in width. For the components produced to be useable, their widths must be within an acceptable range. To test whether each component produced can be used or must be discarded, the manufacturing company uses the function 𝐸(𝑥) = |𝑥 − 19.8|, where 𝑥 is the width of the manufactured component, in mm, and 𝐸 is the amount of acceptable variation in width for the component to function properly. The vertex of the function represents a component that does not vary from the specified width of 19.8 mm, whereas the axis of symmetry represents the specified width. © Accelerate Learning Inc. - All Rights Reserved

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Practice Problems 1. A collectible 1982 sports car has an average fuel economy of 16.27 miles per gallon (mpg). The margin of error is 1.12 mpg. If 𝑥 is the mileage and 𝐸 is the amount of error, the function 𝐸(𝑥) = |𝑥 − 16.27| can be used to find the amount of variation. Graph the absolute value function for the expected range of the sport car’s fuel economy.

2. Complete the table of key features for 𝐸(𝑥) = |𝑥 − 16.27| in problem 1. Key Features

𝑦-intercept

Decreasing interval(s)

Domain

Vertex

Range

Maximum/Minimum?

Axis of symmetry

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Increasing interval(s)

𝑥-intercept

Left end behavior Right end behavior

As 𝑥→______, 𝑦→_______.

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3. Interpret the vertex of 𝐸(𝑥) = |𝑥 − 16.27| in problem 1.

4. Interpret the axis of symmetry of 𝐸(𝑥) = |𝑥 − 16.27| in problem 1.

Review Problems 5. Diego is buying shrimp and rice to make dinner. Shrimp costs $6.20 per pound (lb.) and rice costs $1.25 per lb. Diego spent $10.55 buying shrimp and rice. The relationship between pounds of shrimp, 𝑠, pounds of rice, 𝑟, and the total cost is represented by the equation 6.20𝑠 + 1.25𝑟 = 10.55. Write an equation that makes it easy to find the number of pounds of rice if we know the number of pounds of shrimp purchased.

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6. To raise funds for uniforms and travel expenses, the soccer team is holding a car wash in a part of town with a lot of car and truck traffic. The team spent $90 on supplies like sponges and soap. They plan to charge $10 per car and $20 per truck. Their goal is to raise $460.

How many cars do they have to wash if they wash the following numbers of trucks? a. 4 trucks

b. 15 trucks c. 21 trucks d. 27 trucks e. 𝑡 trucks

522 | Unit 5

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Unit 5, Lesson 10: Digging Deeper into Absolute Value Functions

Warm-Up: Verifying Absolute Value Solutions A graph representing the equation |2𝑥 − 3| = 5 is shown on the coordinate plane.

1. Use the graph to determine the solution(s) to |2𝑥 − 3| = 5, if there are any. 2. Verify the solution to |2𝑥 − 3| = 5 by solving algebraically.

Book 1.indb 523

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Collaborative Activity: Interpreting Key Features in Context According to a study, the average fuel economy of a popular pick-up truck is 15 miles per gallon (mpg). The function 𝑇(𝑚) = |𝑚 − 15| models the study’s results. 1. Graph 𝑇(𝑚) = |𝑚 − 15|.

2. The survey showed that the responses were between 9 mpg and 23 mpg. Based on this information, what constraints should be placed on 𝑇(𝑚)?

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3. Over which interval is the function increasing?

4. Which key feature gives the average fuel economy?

5. Explain why the 𝑦-intercept does not make sense for the context.

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Exploration Activity: The Impact of the Leading Coefficient 1. The table shows several graphs of absolute values functions in which the value of 𝑎 in the equation 𝑦 = 𝑎|𝑥| keeps changing. 𝒚 = |𝒙|

𝒚 = −|𝒙|

𝟏 𝒚 = 𝟑 |𝒙|

𝒚 = −𝟑|𝒙|

a. What do you notice about the graphs?

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b. Describe similarities and differences between the graphs in which 𝑎>0 and the graphs in which 𝑎<0. c. Describe similarities and differences among the graph in which |𝑎|>1, the graph in which 0<|𝑎|<1, and the parent function 𝑦 = |𝑥|. For linear functions in the form 𝑦 = 𝑚𝑥 + 𝑏, 𝑚 is considered the leading coefficient. For absolute value functions in the form 𝑦 = 𝑎|𝑥| + ℎ, 𝑎 is considered the leading coefficient. 2. Complete the statements. Function Type

Description of the Leading Coefficient 𝑚 > 0

Linear

𝑚 < 0 |𝑚| > 1 0 < |𝑚| < 1 𝑎 > 0

Absolute Value

𝑎 < 0 |𝑎| > 1 0 < |𝑎| < 1

Book 1.indb 527

Impact on the Shape of the Graph Increases from left to right Decreases from left to right Increases from left to right Decreases from left to right Flatter slope than 𝑦 = 𝑥

Steeper slope than 𝑦 = 𝑥 Flatter slope than 𝑦 = 𝑥

Steeper slope than 𝑦 = 𝑥 Opens upward

Opens downward Opens upward Opens downward Opens wider than 𝑦 = |𝑥|

Opens more narrowly than 𝑦 = |𝑥| Opens wider than 𝑦 = |𝑥|

Opens more narrowly than 𝑦 = |𝑥|

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3. Without graphing, predict the impact each value of 𝑎 will have on the shape of the function 𝑓(𝑥) = |𝑥 + 2| by completing each statement. a. 𝑓(𝑥) = − |𝑥 + 2|

Compared to the graph of 𝑓(𝑥) = |𝑥 + 2|, the graph of 𝑓(𝑥) = − |𝑥 + 2| will . . .

b. 𝑓(𝑥) = 2 |𝑥 + 2|

Compared to the graph of 𝑓(𝑥) = |𝑥 + 2|, the graph of 𝑓(𝑥) = 2 |𝑥 + 2| will . . .

c. 𝑓(𝑥) = − 4|𝑥 + 2|

Compared to the graph of 𝑓(𝑥) = |𝑥 + 2|, the graph of 𝑓(𝑥) = − 4|𝑥 + 2| will . . .

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Guided Activity: Graphing Absolute Value Functions in the Form 𝒚 = 𝒂|𝒙 − 𝒉| + 𝒌

Absolute value functions in the form 𝑓(𝑥) = 𝑎|𝑥 − ℎ| + 𝑘 have a vertex at (ℎ, 𝑘). 1. The graph of 𝑓(𝑥) = |𝑥| is shown on the coordinate plane. Use the graph of 𝑓(𝑥) = |𝑥| to graph 𝑔(𝑥) = |𝑥 + 7| − 9.

2. The graph of 𝑓(𝑥) = − |𝑥| is shown on the coordinate plane. Use the graph of 𝑓(𝑥) = − |𝑥| to graph ℎ(𝑥) = −|𝑥 − 3| + 5.

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Lesson Summary The parent function of an absolute value function is 𝑦 = |𝑥|. A parent function is the most basic form from which a family of similar functions is derived. Similar to transformations of figures in geometry, functions can also be transformed. Transformations of functions will be explored in more depth in future units. For now, when graphing absolute value functions, it can be helpful to understand how the value of 𝑎 impacts the shape of the graph. This knowledge, combined with other strategies learned in the last few lessons, will help ensure your graphs are accurate. For absolute value functions, the parent function can be transformed into a function of the form 𝑦 = 𝑎|𝑥 − ℎ| + 𝑘, where 𝑎 represents a vertical stretch or compression and (ℎ, 𝑘) reveals the location of the vertex. The value of 𝑎 determines whether the absolute value function opens up or down and whether it is vertically stretched or compressed as compared to the graph of 𝑦 = |𝑥|. • When 𝑎 is positive, the function opens up. When 𝑎 is negative, this acts as a reflection across the 𝑥-axis, and so the function opens down.

• When |𝑎| > 1, the function is stretched vertically. When 0 < |𝑎| < 1, the function is compressed vertically. Both these are dilations on the 𝑦-values.

The values of ℎ and 𝑘 indicate the location of the vertex. The vertex of 𝑦 = |𝑥| is at (0, 0) because the values of ℎ and 𝑘 both equal 0. When ℎ≠0, this indicates a horizontal translation, and when 𝑘≠0, it indicates a vertical translation.

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Practice Problems 1. The average range of the two-way radios that Madison and Giovanni are using can be modeled by function 𝑅(𝑚) = |𝑚 − 5.5|, where 𝑚 is the distance in miles (mi.) between the radios. a. Madison and Giovanni determined that the distance between them while using the radios could vary by up to 1 of a mi. before the radios would no longer 2

communicate. What constraints should be put on 𝑅(𝑚)?

b. Over which interval is the function decreasing?

c. What is the interpretation of the vertex in this context?

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2. Use key features to match each function with its graph. Function

Graph

𝐴(𝑥) = 4|𝑥 + 3| − 5

𝐵(𝑥) = − 0.5|𝑥 + 3| − 5

𝐶(𝑥) = − 2|𝑥 − 2| + 8

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Review Problem 3. Noah is solving an equation and one of his moves is unacceptable. Here are the moves he made. 2(𝑥 + 6) − 4 2𝑥 + 12 − 4 2𝑥 + 8 2𝑥 2

= = = = =

8 + 6𝑥 8 + 6𝑥 8 + 6𝑥 6𝑥 6

Original equation Apply the distributive property. Combine like terms. Subtract 8 from both sides. Divide each side by 𝑥.

Which answer best explains why the “divide each side by 𝑥 step” is unacceptable? A. When you divide both sides of 2𝑥 = 6𝑥 by 𝑥 you get 2𝑥2 = 6𝑥2.

B. When you divide both sides of 2𝑥 = 6𝑥 by 𝑥 it could lead us to think that there is no solution while in fact the solution is 𝑥 = 0. C. When you divide both sides of 2𝑥 = 6𝑥 by 𝑥 you get 2 = 6𝑥.

D. When you divide both sides of 2𝑥 = 6𝑥 by 𝑥 it could lead us to think that there is no solution while in fact the solution is 𝑥 = 3.

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Unit 5, Lesson 11: Comparing Key Features of Functions

Warm-Up: Which One Doesn’t Belong? The graphs of 4 functions are shown. Function A

Function B

Function C

Function D

1. Which function doesn’t belong?

2. Explain your reasoning.

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Collaborative Activity: Comparing Key Features of Functions Work with your partner to complete the following. 1. The table shows 2 different absolute value functions. Function F

Function P 𝒙

𝒚

−5 2

−5

−1

𝑦 = − |𝑥 + 2| − 3 4

−2

1 2

−1

−4

a. Which function has no 𝑥-intercept(s)? b. Complete the table by selecting the function(s) described in each statement. Statement O O O O

Function F Function P Neither Both

The right end behavior is as 𝑥→∞, 𝑦→ − ∞.

O O O O

Function F Function P Neither Both

The function has no positive 𝑦-values.

O O O O

Function F Function P Neither Both

The function opens up.

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2. Two functions are shown on the coordinate plane.

a. Which function has the larger 𝑥-intercept?

b. Which function has the smaller 𝑦-intercept?

c. Complete the statements. Function A is Function B is Both functions are Neither function is Function A Function B Both functions Neither function

decreasing over the interval ( − ∞, ∞).

cross(es) the 𝑥-axis.

d. Which function’s left end behavior could be described by the phrase, “As 𝑥→ − ∞, 𝑓(𝑥)→ − ∞”?

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3. The table shows 3 different functions. Function H

Function M

𝑦 = 5𝑥 − 11.5

Function K

𝒙

𝒚

−2

−5

−1 1 2

−1 7

a. Determine the type of function that each could represent. Function H

Function K

Function M

O Linear O Absolute value

b. Which key features are the same for all 3 functions?

□ Domain □ Range □ 𝑥-intercepts □ 𝑦-intercepts □ Right end behavior □ Left end behavior □ Intervals where the function is increasing □ Intervals where the function is decreasing 538 | Unit 11

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c. Assuming Function K represents a linear function, which function(s) include(s) all real numbers in the range?

d. Complete the statement. The 𝑥-intercept of Function

of Function

H K M

is less than is in between is greater than

the 𝑥-intercepts

H. K. M.

Lesson Summary When comparing the key features of 2 different functions, use the representations to determine what the comparable key features are. To make the best comparison, it may be necessary to use the given representations to create additional representations in other forms to make sure all key features are revealed. Keep in mind, not all function types have all the same key features. When comparing 2 different types of functions, only the common key features can be compared. Common key features include domain, range, 𝑥- and 𝑦-intercepts, leading coefficient, increasing and decreasing intervals, and end behavior.

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Practice Problems 1. Two linear functions are shown.

Function D

Function H

a. Which function has a positive slope?

b. Which function has the steeper slope?

𝑦 − 1 = 11(𝑥 + 1.2)

c. Which function has the greater 𝑥-intercept?

2. The table shows 2 functions.

Function A

a. Which function has the smaller 𝑦-intercept?

Function B

O Function A O Function B

8 7 6 5 4 3

b. As 𝑥→∞, which function has larger values of 𝑦? O Function A O Function B

2 1 -5 -4 -3 -2 -1 0 -1

x 1

𝑥 − 2𝑦 = − 2

-2 -3 -4 -5

c. Compare the ranges of the functions.

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3. Two functions are shown.

Function L

𝑓(𝑥) = |𝑥 + 2| + 5

Function Y 𝒙

𝒚

a. Describe how the intercepts of the functions compare in the table. Key Feature

How do they compare?

𝑥-intercept(s) 𝑦-intercept b. Compare the domains of the functions.

c. Compare the interval(s) over which Function L and Function Y are increasing.

Review Problems 4. Noah has a coin jar containing 𝑑 dimes and 𝑞 quarters worth a total of $5.00. Select all the equations that represent this situation.

□ 𝑑 + 𝑞 = 5 □ 𝑑 + 𝑞 = 500 □ 0.1𝑑 + 0.25𝑞 = 5 □ 10𝑑 + 25𝑞 = 500 □ 𝑑 = 50 □ 𝑞 = 20

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5. The table shows the volume of water in a tank after it has been filled to a certain height. Which equation could represent the volume of water in cubic inches (cu.in.), 𝑉, when the height is ℎ inches (in.)? Water Height (in.)

Water Volume (cu. in.)

1.05

28.35

8.40

A. ℎ = 𝑉

𝑉

B. ℎ = 4

C. 𝑉 = ℎ2 + 0.05 D. 𝑉 = 1.05ℎ3

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