STEMscopes Georgia Math Student Notebook Grade A1 by acceleratelearning

AL GE BR AІ Student Notebook

GEORGIA

Student Notebook – Algebra l ISBN: 978-1-64861-276-3 Published by Accelerate Learning Inc., 5177 Richmond Ave, Suite 800, Houston, TX 77056. Copyright © 2023 by Accelerate Learning Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior written consent of Accelerate Learning Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.

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GEORGIA

Student Notebook - Algebra l

Table of Contents Scope Name

Page Number

Properties of Functions

Linear Functions

Geometry on the Coordinate Plane

Linear Inequalities

Systems of Inequalities

Simplify Radicals

Polynomial Operations

113

Graphs of Quadratic Functions

137

Factors of Polynomials

165

Solve Quadratics

201

Transform Quadratic Functions

227

Exponential Functions

257

Exponential Extensions

275

Compare Function Types

295

iii

GEORGIA

Student Notebook - Algebra l

Table of Contents (Cont.) Scope Name

Page Number

Statistics

311

Model Data

331

Skills Quizzes

349

Glossary of Terms

435

Workspace

471

Properties of Functions

Explore 1

Name: _______________________ Date: ___________

Relations and Functions Part I Analyze the widget machines below to determine which two machines are working and what characteristics are required for a widget to be approved by factory standards.

Machine A

Machine B

INPUT

OUTPUT

Input 2 blobs 5 blobs 7 blobs

Output 1 widget 9 widgets 13 widgets

OUTPUT

Input 3 blobs 3 blobs 8 blobs

Machine C INPUT

Output 1 widget 5 widgets 2 widgets

OUTPUT

Input 1 blob 3 blobs 9 blobs

Output 1 widget 1 widget 14 widgets

1. Which two machines do you think are working correctly? What patterns did you ﬁnd between those two machines?

2. Which machine is malfunctioning and needs to be shut down? 3. What is different about the machine that is not functioning correctly?

Properties of Functions | 3

Explore 1

Properties of Functions

Part II Analyze each Widget Card, and determine whether each machine is functioning properly or not functioning properly. State your reasoning on the right side of each card. Cut out the cards, and glue or tape them in the corresponding columns below. Functioning Widget Machines

Nonfunctioning Widget Machines

Glue functioning Widget Cards here.

Glue nonfunctioning Widget Cards here.

4 | Properties of Functions

Explore 1

Properties of Functions

Reflect 1. Summarize your overall ﬁndings about the widget machines that were functioning correctly.

2. Summarize your overall ﬁndings about the widget machines that were not functioning correctly.

3. Is it possible for a functioning widget machine to have only one output value? Explain.

Properties of Functions | 5

Properties of Functions

Explore 2

Name: _______________________ Date: ___________

Evaluating Functions Part I Analyze the graph below showing the data collected by Dr. Angola’s team. Use the graph and the data to complete the report for Dr. Angola. Tracked Sea Turtle Migration y

Distance traveled (mi.)

300 F

250 E

200 D

150 100 50 0

Days

Report: Leatherback Sea Turtle Migration A

(5, 50)

ff(5) = 50

After 5 days, the turtles had traveled 50 miles.

(8, 80)

ff(___) = 80

After ____ days, the turtles had traveled 80 miles.

(10, 100)

ff(10) = ____

After 10 days, the turtles had traveled ____ miles.

After ____ days, the turtles had traveled ____ miles.

Properties of Functions | 7

Properties of Functions

Explore 2

1. Dr. Angola is most interested in the section of the graph containing points D, E, and F, as she believes the turtles will continue on this trend for the next several days. Complete the function below to represent the relationship between the number of days and the distance traveled for this section of the graph. f x) = _______ f( 2. Use your equation from question 1 to predict the distance traveled by the turtles after 36 days.

Part II Evaluate the functions for the given value, or ﬁnd the input for a given output. Complete the multispecies report for Dr. Angola. Report: Multispecies Migration Flatbacks f x) = 4(x f( x – 2)2

ff(14) = ____

Archelons a(x) = 7x

a(x) = 126 x = ____

Loggerheads l(x) = x2 + 5x

l(9) = ____

Hawksbills h(x) ) = 18 18x

h(x) = 54 x = ____

Greens g(x) = (x x + 4)(x – 2)

g(6) = ____

Kemp’s Ridleys k(x) ) = 16 16x x + 10

k(x) = 170 x = ____

8 | Properties of Functions

After _____ days, the Flatbacks had traveled _____ miles. After _____ days, the Archelons had traveled _____ miles. After _____ days, the Loggerheads had traveled _____ miles. After _____ days, the Hawksbills had traveled _____ miles. After _____ days, the Greens had traveled _____ miles. After _____ days, the Kemp’s Ridleys had traveled _____ miles.

Explore 2

Properties of Functions

Reflect 1. Consider the function t(x) = x2 + 8x x + 20. a. Find t(−4).

b. If t(x) represents the distance traveled by Terrapin sea turtles over time, does t(−4) make sense? Explain.

2. Dr. Angola explained that Archelons and Loggerheads often swim for 3 days and then take a break. She wants to know the total distance traveled by these two species after 3 days. Using the functions from the report on multispecies migration, evaluate a(3) + l(3).

3. Why might a student think that f( f x) means to multiply f times x?

4. Explain what “evaluate g(5)” means in your own words.

5. How is solving the equation f( f x) = 10 different from evaluating the expression f(10)? f

Properties of Functions | 9

Properties of Functions

Explore 3

Name: _______________________ Date: ___________

Domain and Range

Part I

Day

Number of Ticket Sales

140

Number of ticket sales

Ticket sales started slowly but increased by a larger margin each day. The data you collected was organized as a table and a graph. Analyze the data to answer the questions and make your recommendation to the committee.

160 140 120 100 80 60 40 20 0

4 5 Days

1. What are the possible inputs for this situation? 2. Do decimals make sense for this situation? Why or why not?

3. What are the values of the outputs of the situation? What do they represent?

4. Would negative values be possible outputs in this situation? Why or why not?

5. Based on the data collected, how many students should they anticipate at this event? What is your recommendation for food and prizes?

Properties of Functions | 11

Properties of Functions

Explore 3

Number Hour of Students 0

300

400

300

Number of students

Attendance The number of students at the end-of-year bash increased continuously until hour 2, when the number in attendance peaked. After hour 2, the number of students at the event decreased continuously until the end of the event at hour 4. The data you collected was organized as a table and a graph. Analyze the data to answer the questions, and make your recommendation to the committee.

400 300 200 100 0

2 3 Hour

6. Can the possible inputs be described by listing every possible value? Why or why not?

7. Describe the possible inputs using inequalities. 8. Is describing the possible outputs by listing every possible value the most efﬁcient way to describe the possible outputs? Why or why not? 9. Describe the possible outputs using inequalities.

10. Based on the data collected, would you recommend that next year’s end-of-year bash be extended to 5 hours in length? Explain.

12 | Properties of Functions

Properties of Functions

Explore 3 Part II

Cut out the Event Cards, look at the information on each card, and decide which event it belongs to. Once you have decided, glue or tape each card in the appropriate place below. Candy Grams The student event planning committee is selling candy grams the last week of school. The equation y = 2.5x is being used to calculate proﬁt. It has been determined that they need to sell between 55 and 60 candy grams to meet their proﬁt goal.

Glue the domain and range here.

Glue the graph here.

Egg Launch

The egg launch event resulted in the winner’s egg reaching a height of 150 feet and landing a distance of 8 feet from the starting platform.

Glue the domain and range here.

Glue the graph here.

Properties of Functions | 13

Properties of Functions

Explore 3 Backyard Bash

A dunk tank has been reserved for the backyard bash. After a hose is placed in the tank, the water rises at 18.75 inches per minute. It takes just under 8 minutes to ﬁll the tank.

Glue the domain and range here.

Glue the graph here.

Match each of the remaining graphs with its correct domain and range. Then, glue or tape the Event Cards in the spaces below.

Glue the graph here.

Glue the domain and range here.

14 | Properties of Functions

Properties of Functions

Explore 3 Reflect

1. What features of the graph did you look at to determine the domain and range?

2. What are the ways we can write domain and range to show that an endpoint is not included? 3. How is the domain of an algebraic representation different from the domain of a situation?

4. Is it possible to have a restricted domain and an unrestricted range or a restricted range and an unrestricted domain? Why or why not?

5. What types of situations would be discrete? List a few examples.

6. What types of situations would be continuous? List a few examples. 5

7. Draw a graph whose domain is [1, 5) and whose range is (−2, 2].

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

x 1

Properties of Functions | 15

Properties of Functions

Explore 4

Name: _______________________ Date: ___________

Linear vs. Nonlinear Functions Part I

Use the Graph Cards provided to complete the tables below. 1. Sort the graphs according to the criteria given below. List the name of the graphs in their respective categories. Criteria

Non-Curvy

Curvy

2. Complete the tables by sorting the graphs according to criteria of your choosing. List the name of the graphs in their respective categories. A. Criteria A

Properties of Functions | 17

Explore 4

Properties of Functions

B. Criteria B

C. Criteria C

18 | Properties of Functions

Explore 4

Properties of Functions

3. Which of the Parent Function graphs does not have a domain of all real numbers?

4. End behavior is what happens to the outputs of a function when we look farther and farther from 0 in both directions. Most of these graphs have end behavior of either very large positive or large negative numbers as the x values get farther from 0 in the positive and negative directions. Which functions do not? Describe their end behavior.

5. Two of the functions have increasing as well as decreasing parts. Which are they?

6. Only one Parent Function does not have an x-intercept. Which one is it?

7. Only one of these functions is considered linear, and the rest are not considered linear. Why do you think that is? What separates the linear graph from the rest?

8. Draw a linear graph that has an end behavior of output values getting larger and larger as the x values move away from 0 in the negative direction.

9. Explain why your graph in question 8 must have an x-intercept since it is a linear function with the given end behavior.

Properties of Functions | 19

Properties of Functions

Explore 4 Part II

1. Use the information provided to determine whether the graph being described is linear or nonlinear. Sketch the graph(s) that would satisfy the criteria. Criteria • As x moves in the positive direction, the y values get larger and larger. • One x-intercept • Range all real numbers

Sketch y

Circle the type of function that can satisfy the criteria: Linear Nonlinear Both • Restricted domain • Restricted range • Always increasing

Circle the type of function that can satisfy the criteria: Linear Nonlinear Both • Constant rate of change • One x-intercept • As x moves in the negative direction, the y values get larger and larger.

Circle the type of function that can satisfy the criteria: Linear Nonlinear Both • Domain all real numbers • One x-intercept • Always decreasing

Circle the type of function that can satisfy the criteria: Linear Nonlinear Both

20 | Properties of Functions

Properties of Functions

Explore 4

2. Describe a scenario that would lead to a nonlinear graph. Then, graph your scenario below. Be sure to label the axes and scale appropriately. y

3. Describe a scenario that would lead to a linear graph. Then, graph your scenario below. Be sure to label the axes and scale appropriately. y

Properties of Functions | 21

Explore 4

Properties of Functions

Reflect

1. How can you tell if a graph is linear or nonlinear by visual analysis alone?

2. Is it possible for a linear function to have two x-intercepts? Why or why not?

3. Could a linear graph have end behavior in the positive direction where the output values get closer and closer to 3? Why or why not?

4. What do the absolute value and quadratic Parent Functions have in common?

5. What is one unique feature of the square root function compared to the other Parent Functions?

6. Explain the similarities and differences between range and end behavior.

22 | Properties of Functions

Linear Functions

Explore 1

Name: _______________________ Date: ___________

Part I

Identify and Construct Arithmetic Sequences

Look for patterns in the Cup Stacking Cards. Use those patterns to help you answer the questions to determine which setups are allowed for each round of the competition. 1. Round 1 of the cup stacking tournament must use setups that increase or decrease by the same number of cups each time. Which setups can be used in round 1?

2. By how much do each of these stacking setups differ?

3. Round 2 of the cup stacking tournament must use setups that increase or decrease by a different number of cups each time. Which setups can be used in round 2?

4. Sketch a cup stacking setup that meets the criteria for round 1.

a. Identify the constant number of cups being added to or subtracted from each stack. b. How many cups would be in the 4th stack? c. Can you easily predict the number of cups needed in the 100th stack? Explain.

Linear Functions | 25

Linear Functions

Explore 1 Part II

1. Look for patterns in the Cup Stacking Charts to help you complete the ﬁrst four columns of the table, answer the questions, and determine the missing number of cups for the next terms. Do not attempt to ﬁll in the blank in the last column until after you have answered question 4. Competitor 1 Number of Cups in the First Term

Pattern/ Common Difference

Fifth Term

Fifth Term Work

100th Term Work

6 + 3(___)

2. What is the rule or pattern in competitor 1’s table that led you to the ﬁfth term? 3. Starting at the ﬁrst term, how many times would you need to add 3 to get to the 2nd term? How many times would you need to add 3 to get to the 5th term?

4. How many times do you think we would need to add 3 to the starting value of 6 to get to the 100th term? Why? Use your answer to ﬁll in the blank in the last column of the table.

5. How could you represent the number of times you would need to add 3 to the starting value of 6 in order to get the nth term?

6. Write an equation or rule that can be used to ﬁnd the nth term, and then use your equation to ﬁnd the 12th term.

26 | Linear Functions

Linear Functions

Explore 1

7. Look for patterns in the Cup Stacking Charts to help you complete the ﬁrst four columns of the table, answer the questions, and determine the missing number of cups for the next terms. Do not attempt to ﬁll in the blank in the last column until after you have answered question 10. Competitor 2 Number of Cups in the First Term

Pattern/ Common Difference

Fifth Term

Fifth Term Work

500th Term Work

7 + 5(___)

8. What is the rule or pattern in competitor 2’s table that led you to the ﬁfth term?

9. Starting at the ﬁrst term, how many times would you need to add 5 to get to the 2nd term? How many times would you need to add 5 to get to the 5th term?

10. How many times do you think we would need to add 3 to the starting value of 7 to get to the 500th term? Why? Use your answer to ﬁll in the blank in the last column of the table.

11. How could you represent the number of times you would need to add 5 to the starting value of 7 in order to get the nth term?

12. Write an equation or rule that can be used to ﬁnd the nth term, and then use your equation to ﬁnd the 12th term.

Linear Functions | 27

Linear Functions

Explore 1 Reflect

1. Why did the rules you generated for the nth term of each competitor both contain (n – 1)?

2. What are two ways to know if a sequence is arithmetic?

3. If the competition involves making 23 stacks, how many cups will each competitor have?

4. Is 9, 20, 31, 42 an arithmetic sequence? Why or why not?

5. Is 7, 13, 20, 26 an arithmetic sequence? Why or why not?

28 | Linear Functions

Linear Functions

Explore 2

Name: _______________________ Date: ___________

Create Recursive and Explicit Equations Part I Analyze the Vegetable Graphs and answer the questions to determine whether you’ll grow enough tomatoes and green beans for your grandma. 1. What common difference do you notice in each of the graphs?

2. What mathematical operations can you use to represent the tomato growth and the green bean growth?

3. Determine the number of tomatoes and green beans for days 5 and 6, and then record the data in the table. Day

Number of Tomatoes Number of Green Beans

4. How did you determine the number of tomatoes that will be available for harvest on day 6?

5. How did you determine the number of green beans that will be available for harvest on day 6?

Linear Functions | 29

Linear Functions

Explore 2 6. Complete the table below to help you answer the following questions. Tomatoes Day

Previous Value

Math Operation

Workspace

Output

7 7. How did you determine how many tomatoes will have grown by day 7?

8. Write an equation to ﬁnd the number of tomatoes, An, using the number of tomatoes grown on the previous day, An – 1. 9. Complete the table below to help you answer the following questions. Green Beans Day

Previous Value

Math Operation

Workspace

Output

6 7 10. How did you determine how many green beans will have grown by day 7?

11. Write an equation to ﬁnd the number of green beans, An, using the number of green beans grown on the previous day, An – 1. 12. If your grandma needs 40 tomatoes and 60 green beans on day 7, will you have enough tomatoes and green beans?

30 | Linear Functions

Linear Functions

Explore 2 Part II 1. Use the recursive equation below to complete the table, and determine whether you’ll have enough strawberries in time for your aunt. Strawberries Recursive equation: An = An – 1 + 7

Day (n)

Number of Strawberries (An)

2 3 4 2. Would it be efﬁcient to calculate the number of strawberries on day 10 using the recursive equation? Why or why not?

3. Complete the table below to efﬁciently calculate the number of strawberries on day 10.

Arithmetic or Non-Arithmetic (Circle one.)

Value of d

First Term

Explicit Equation

Day 10

Arithmetic Non-arithmetic

Linear Functions | 31

Linear Functions

Explore 2 4. Use the recursive equation below to complete the table, and determine whether you’ll have enough blueberries in time for your aunt. Blueberries Recursive equation: An = An – 1 + 9

Term (n)

Output

Day 1

3 blueberries

Day 2 Day 3 Day 4 5. What are the beneﬁts and drawbacks of using this equation compared to the explicit formula?

6. Complete the table below to efﬁciently calculate the number of blueberries on day 10. Arithmetic or Non-Arithmetic (Circle one.)

Value of d

First Term

Explicit Equation

Day 10

Arithmetic Non-arithmetic 7. Your aunt needs 75 strawberries and 80 blueberries to make her special jam. Will you have enough of each fruit on day 10? How do you know?

32 | Linear Functions

Explore 2

Linear Functions

Reflect 1. What are the ways you can tell if a sequence is arithmetic or non-arithmetic from a list and a graph?

2. Once you determine the recursive equation for a sequence, must you always know the value of the previous term (An – 1) before ﬁnding the next term? Why or why not?

3. What are the two types of equations used with sequences, and how do they differ?

4. If you were given an equation, how could you tell if it is recursive?

5. If you were given an equation, how could you tell if it is explicit?

Linear Functions | 33

Linear Functions

Explore 3

Name: _______________________ Date: ___________

Arithmetic Sequences and Linear Functions Part I Your parents called the ﬁrst shop, Bikes Galore. They organized the cost of bikes per full hour of rental in a table. Use the table to answer the questions below. Time (hours)

Cost (dollars)

1. Complete the table below to represent the cost of bikes at Bikes Galore. Arithmetic Sequence An = A1 + (n – 1)(d)

Linear Equation y = mx + b

Equation Domain Range 2. How is the domain of the arithmetic sequence different than the domain of the linear equation?

3. Your mom said that the most accurate type of function to represent this situation is arithmetic, and your dad said linear is better for this situation. Who is correct? Explain your thinking.

Linear Functions | 35

Linear Functions

Explore 3

4. Use both functions to ﬁnd the cost for 8 hours. Do they give the same value? Why or why not? Arithmetic: Linear:

5. Graph each function below to model the cost of renting a bike at the ﬁrst bike shop. Label and scale as appropriate. Arithmetic

Linear

6. Which graph is the most reasonable representation of the situation? Explain.

7. Your aunt is wondering if writing the equation in recursive notation would make it easier to determine the cost for 8 hours. Represent this situation using recursive notation, determine ff(8), and then explain whether this notation is easier than explicit when determining the cost for 8 hours.

36 | Linear Functions

Linear Functions

Explore 3 Part II

Complete the table to help you analyze and organize the data from your parents. Bikes n’ More

2 Wheels Only

Wheels Are Us

Pedals and More

Does this situation have a discrete or continuous domain? Explain. Is this situation best represented by an arithmetic or linear function? Explain. Function Cost for 8 hours Cost for 4.5 hours 1. Which shop would be the best option for 8 hours? Explain.

2. Which shop would be the best option for 4.5 hours? Explain.

Linear Functions | 37

Linear Functions

Explore 3 Reflect 1. How are arithmetic sequences and linear functions similar?

2. How are they different?

3. What is the linear word for common difference?

4. What is the linear word for initial amount?

5. Your friend was absent today. Explain to them the similarities and differences between arithmetic sequences and linear functions. Include how you know when to use which function.

38 | Linear Functions

Linear Functions

Explore 4

Name: _______________________ Date: ___________

Part I

Graph Linear Functions

1. The information Lashawn and Isaac have collected has come from different sources. It is given in a variety of formats. Review the route descriptions on the Driving Routes Cards, and answer the following questions. a. What are the starting values for each driving route? Give each as an ordered pair and as a sentence in context. Driving route A:

Driving route B:

Driving route C:

Driving route D:

b. What is the rate of change for each driving route? Give each as a unit rate and as a sentence in context. Driving route A:

Driving route B:

Driving route C:

Linear Functions | 39

Linear Functions

Explore 4

2. Review the routes on the Driving Routes Cards and the information in question 1.

a. Graph each driving route below. Label each line with A, B, C or D. Identify the x- and y-intercepts by writing the ordered pair near each point.

300

Distance (miles)

250 200 150 100 50

Time (hours)

b. Which driving route will get Lashawn and Isaac to their destination the fastest? How long will it take?

c. Driving routes A, D, and B all have the same y-intercept. Why do they have different arrival times?

40 | Linear Functions

Linear Functions

Explore 4 Part II

Lashawn and Isaac both own vehicles. Lashawn’s father also has offered to let them borrow his vehicle. 1. For each of the three vehicles, plot a line on the graph below showing the amount of gasoline in the vehicle, g (in gallons), as a function of distance traveled, d (in miles). 30

Isaac’s Vehicle

125

250

500

26 24 22

Lashawn’s Vehicle He has 20 gallons of gas at the beginning of the trip, and he can drive 15 miles for every 1 gallon of gas.

Gas (gallons)

Dad’s Vehicle The line passes through (300, 15), with a slope of − 1 . 20

20 18 16 14 12 10 8 6 4 2 0

2. Which vehicle could make the 600 mi. round trip without running out of gas?

100

200

300

400

500

600

Distance (miles)

3. Which vehicle has the worst gas mileage? Which has the best gas mileage?

4. Which vehicle has the most gas in the tank at the start of the trip?

Linear Functions | 41

Linear Functions

Explore 4 Reflect

1. Was it helpful to plot the different routes on the same graph? Why or why not?

2. Given the verbal description for a function, what pieces of information do you look for to plot the function on a graph?

3. Which of the following are equivalent descriptions of a line? Explain how you know which ones are equivalent. Select all that apply. A. B. C. D.

Start walking at three miles per hour, and stop after two hours. A line from (0, 0) to (2, 6) A line with a y-intercept of (0, 0) and a slope of 3 t

1.5

4.5

42 | Linear Functions

Linear Functions

Explore 5

Name: _______________________ Date: ___________

Graph from an Equation Part I Use the equations describing the salaries for the jobs below, where y is Lashawn’s income in dollars and t is the time in weeks, to analyze the situation. Technical Support Representative

y = 600t + 100

Math Camp Instructor

y = 500t

1. How much does Lashawn make as a technical support representative if he works 0 weeks? How much does he make if he works 4 weeks?

2. What are the two corresponding ordered pairs for the answers to question 1? Plot these ordered pairs on the graph, connect them with a line, and label the line.

3. How much does Lashawn make as an instructor at the math camp if he works 0 weeks? How much does he make if he works 4 weeks? Write your answer as two ordered pairs.

Income ($)

4. Plot the ordered pairs for the answers to question 3 on the graph, connect them with a line, and label the line.

3,000

2,000

1,000

5. How are the ordered pairs on the line and the equation related? 0

Linear Functions | 43

Explore 5

Linear Functions

6. Select another ordered pair that lies anywhere on the line that represents the technical support representative salary. How can you determine if this ordered pair is a solution to the equation for the technical support representative salary? Is it?

7. Since the ordered pair selected could be anywhere on the line, what does this tell you about all of the ordered pairs on the line?

8. Which job will provide Lashawn with the most summer income? How do you know?

44 | Linear Functions

Linear Functions

Explore 5

After weekly expenses, Lashawn will deposit the remainder of his paycheck. The ﬁnancial outlook for Lashawn over the ﬁrst four weeks can be described by 1,500tt – 3y = 3,000, where t is time in weeks and y is his net worth in dollars. 9. What is Lashawn’s net worth at the beginning of summer? Write this as an ordered pair.

10. What is the ordered pair that represents his net worth after 2 weeks?

11. Use these two ordered pairs to draw a line representing the given equation over the ﬁrst 4 weeks of summer.

Net worth (dollars)

1,000

500

Time (weeks)

–500

–1,000

12. What do you know about all of the ordered pairs on this line?

13. At what rate is Lashawn saving money?

14. Describe how Lashawn’s net worth changed over the ﬁrst four weeks of summer.

Linear Functions | 45

Linear Functions

Explore 5 Part II

The graph shows the loan repayment plans from two different lenders. Lashawn wants to ﬁnd an equation that represents each plan so he can see and model each one. 1. For an equation to represent the loan repayment plan from lender 1, what must be true of every ordered pair on that line?

3,000

(0, 3000)

2,800 2,600

2. What information do you need to know to write an equation in slope-intercept form?

Loan balance (dollars)

2,400 2,200 2,000 1,800

Lender 1

1,600 1,400 1,200 1,000

Lender 2

800 600 400

3. What is the y-intercept for the line that represents lender 1? What does this represent?

200 0

(12, 0) 2

(24, 0)

8 10 12 14 16 18 20 22 24 Time (months)

4. How do you ﬁnd the slope for the plan from lender 1? What is the slope?

5. Write an equation for the plan from lender 1 in slope-intercept form.

46 | Linear Functions

Explore 5

Linear Functions

6. For the plan from lender 2, identify the y-intercept, calculate the slope, and then write an equation in slope-intercept form.

7. Which lender will require that the loan be paid back more quickly? How do you know?

8. How long would it take Lashawn to pay off the loan from lender 1?

9. How long would it take Lashawn to pay off the loan from lender 2?

10. Which loan do you recommend, and why?

Linear Functions | 47

Linear Functions

Explore 5 Reflect

1. When given an equation for a line in a problem, do you ﬁnd it helpful to plot the equation? Why or why not?

2. How could you test an equation to see if it correctly represents a line plotted on a graph?

3. For a line plotted on a graph and its corresponding equation over some speciﬁed domain, which is a true statement? A. All of the ordered pairs on the line are solutions of the equation. B. All of the ordered pairs that satisfy the equation are on the line. C. Both statements are true.

48 | Linear Functions

Geometry on the Coordinate Plane

Explore 1

Name: _______________________ Date: ___________

Discover the Distance Formula Part I 1. How far is the high school from the town hall, which is located in the center of the city?

Arcade

Mall

7 6

Movie theater

5 Grocery store

2. Draw a straight line that connects the mall and the bowling alley. How far is the mall from the bowling alley? Write an expression that shows how to ﬁnd this answer.

4 3 2

1 Town hall -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 -1 Soccer field

High school 2

x 9 10

-2 -3 -4

Bowling alley

-5

3. Draw a straight line that connects the mall and the arcade. How far is the mall from the arcade? Write an expression that shows how to ﬁnd this answer.

4. Label the distance on the map for the mall to the bowling alley and the mall to the arcade. Then, draw a line to show the shortest distance from the arcade to the bowling alley. 5. Can you determine the distance from the arcade to the bowling alley? Why or why not? If possible, determine the distance and show your work.

Geometry on the Coordinate Plane | 51

Geometry on the Coordinate Plane

Explore 1

6. How was ﬁnding the distance between the arcade and bowling alley different from ﬁnding any of the other distances in questions 1–3?

7. What tools did you use to ﬁnd the distance between the arcade and bowling alley, and what additional information did you need?

8. Predict whether the soccer ﬁeld or movie theater is closer to the town hall. Explain.

Arcade

9. Calculate which is closer to the town hall, the soccer ﬁeld or the movie theater. Draw two triangles to show how you got your answer. Do the results match your prediction?

Mall

7 6

Movie Theater

5 Grocery Store

4 3 2

1 Town Hall -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 -1 Soccer Field

High School 2

x 9 10

-2 -3 -4

Bowling Alley

-5

52 | Geometry on the Coordinate Plane

Explore 1

Geometry on the Coordinate Plane

Part II 1. Which trip is longer? Tom’s dad drives a group of students from the high school to town hall to pick up something, then to the bowling alley, and then back to the high school to drop the other students off. Milagros’s mom drives from work at the town hall to the soccer ﬁeld to pick up her daughter, then to the grocery store, and then back to the town hall to grab her laptop that she forgot. Show or explain your work.

2. A chef is deciding where to place her new Cambodian restaurant and wants to calculate how far it will be from the mall, which is always crowded. Since the location is undecided, we will label it as (x1, y1). Write an expression that calculates the distance between the Cambodian restaurant and the mall.

3. The town is deciding where to place a new outdoor basketball court and wants to calculate how far it will be from the soccer ﬁeld, where athletes are often headed. Since the location is undecided, we will label it as (x2, y2). Write an expression that calculates the distance between the new basketball courts and the soccer ﬁeld.

Geometry on the Coordinate Plane | 53

Geometry on the Coordinate Plane

Explore 1 4. In this question, you will create a way to write the distance between the Cambodian restaurant at (x1, y1) and the basketball court at (x2, y2).

a. Draw a right triangle that would help you calculate this distance. b. Write an expression that represents the horizontal distance between the two points. This is the distance between (x1, y1) and (x2, y1).

y1 x1

c. Write an expression that represents the vertical distance between the two points. This is the distance between (x2, y2) and (x2, y1).

d. Create an equation using the expressions from parts b and c to ﬁnd the distance between the Cambodian restaurant at (x1, y1) and the basketball court at (x2, y2).

5. You have created a formula that can ﬁnd the distance between any two points! Use your formula to calculate the distance between the STEM City high school and the rival high school located at (19, 35).

54 | Geometry on the Coordinate Plane

Geometry on the Coordinate Plane

Explore 1 Reflect

1. How are the Pythagorean theorem and distance formula related?

2. Write down two real-life applications for using the distance formula.

3. Explain why the distance between A and B is the same as the distance between C and D without performing any calculations 5

y B

4 C

3 2 1

D -6

-5

-4

-3

-2

-1

A 1

x 6

-1

Geometry on the Coordinate Plane | 55

Explore 2

Geometry on the Coordinate Plane

Name: _______________________ Date: ___________

Discover the Midpoint Formula Part I The crew decides to try and dig halfway in between locations of different jewels on the map to see if they have any luck turning up more jewels along the way. 1. The crew wants to start digging halfway in between the purple circle gem and the blue teardrop gem on the map. Help them ﬁnd this location, and explain how you got there.

2. Where would the crew dig to ﬁnd the midpoint between the green triangular gem and the blue teardrop?

3. The light blue diamond is located at (−7, 4). What is the point halfway between this diamond and another off the map located at (15, 4)? Explain how you found the point.

4. What quadrant do you think the point exactly halfway in between the purple circle gem and the green triangle gem would lie in, and why?

Geometry on the Coordinate Plane | 57

Explore 2

Geometry on the Coordinate Plane

5. Use your answers from questions 1 and 2 to create a new point that could be directly in between the purple circle and the green triangle gems. Label the three halfway or midpoints on the map below.

6. See if your point is in the middle by ﬁnding the distance between the purple circle gem and your point as well as the distance between the green triangle and your point.

7. Looking at your midpoint between (−4, −2) and (6, 6), how does it help us to just look at the x-coordinates and then just look at the y-coordinates when ﬁnding the midpoint?

58 | Geometry on the Coordinate Plane

Explore 2

Geometry on the Coordinate Plane

8. Draw a line between the red oval and gold heart-shaped gems, identify the point you believe is halfway between the two gems, and explain why you picked it.

9. The red oval is at (−1, 9), and the gold heart is at (3, 1). How does the x-coordinate of the midpoint you found compare to the two original x values, and how does the y-coordinate of your midpoint compare to the original y values?

10. Find the midpoint between the green triangular gem and the following gems: a. The gold heart gem b. The green rectangular gem c. The gold triangle gem d. The point (24, −12) e. The point (−100, −50) f. The point (x1, y1)

Geometry on the Coordinate Plane | 59

Geometry on the Coordinate Plane

Explore 2 Part II

1. The last known locations of ship A and ship B are shown on the route map. Find the midpoint between the two ships where they would meet if these were their starting locations.

2. Use the space provided below to label the route map to show the coordinates of the locations of ship A and ship B, where we are using variables since we don’t know their exact location. y-axis

Ship A

Ship B

x-axis

3. Write down the coordinates of ship A and ship B in terms of x1, x2, y1, or y2.

4. Draw a dotted line to mark the distance between both ships. Then, mark the center of the distance with an X. 5. How could we write an expression that would tell us where the middle of the two x-coordinates lies without knowing the values of x1 and x2?

6. How could we write an expression that would tell us where the middle of the two y-coordinates lies without knowing the values of y1 and y2?

60 | Geometry on the Coordinate Plane

Explore 2

Geometry on the Coordinate Plane

7. Use your values from questions 5 and 6 to write a full coordinate point of the point halfway in between ship A and ship B.

8. You just created a formula that would allow you to ﬁnd the midpoint between any two starting locations. Explain what the x- and y-coordinates of your formula are calculating.

9. Where should the captain head if the ships are located at the following points: a. (200, 600) and (1800, 1000)? b. (150, 900) and (725, 852)? c. (57.3, 435.7) and (1236.8, 127.6)? d. (a, b) and (1400, 1000)?

10. If the captain hears the location of each ship from his spies and sets a course for (1000, 300), and ship B was located at (100, 600), determine where ship A was located when the captain set sail for the midpoint.

Geometry on the Coordinate Plane | 61

Explore 2

Geometry on the Coordinate Plane

Reflect 1. How is the midpoint formula similar to the average between two numbers?

2. How is the midpoint formula used in real life?

3. What is the midpoint between (a, 10) and (−5, b)?

4. If you are given a midpoint and only one endpoint, how could you ﬁnd the other endpoint?

5. How can you check that a midpoint you calculated is correct?

62 | Geometry on the Coordinate Plane

Geometry on the Coordinate Plane

Explore 3

Name: _______________________ Date: ___________

Shapes on the Coordinate Plane Part I 1. Determine the slopes of the sides for fence 1. Bottom side: Left side: Top side: Right side: 2. How are the slopes for each side related? 3. How is the slope of the top related to the slope of the bottom? Could these lines intersect if they were longer? Why or why not?

4. Determine the slopes of the sides for fence 2. Bottom side: Left side: Top side: Right side: 5. How is the slope of the top related to the slope of the right side? What type of angle does this relationship create?

6. Does the same pattern hold true for fences 3, 4, and 5? If not, explain the pattern or how it is different from fence 2. 7. Explain to farmer John how you decided which of the 5 fence areas are rectangles. In other words, explain how you determined which fence areas had 4 right angles or 1 set of parallel sides and 2 opposite angles that are right angles.

Geometry on the Coordinate Plane | 63

Geometry on the Coordinate Plane

Explore 3 Part II

1. What information is needed to determine the area of each rectangular fence region?

2. What is the formula to calculate distance on the coordinate grid? 3. Is there enough information to determine the length and width of each fence side? If yes, determine the length and width of each one. Round to the nearest tenth. Then, determine the area and perimeter. Round the area and perimeter to the nearest whole number. Only calculate length, width, area, and perimeter for the rectangular fences.

Fence 1

Fence 2

Fence 3

Fence 4

Fence 5

Is it a rectangle? Length (feet) Width (feet) Area (square feet) Perimeter (feet) 4. Which fenced-in area meets farmer John’s criteria? Explain.

64 | Geometry on the Coordinate Plane

Explore 3

Geometry on the Coordinate Plane

Reflect 1. What information is needed to determine if a shape on a coordinate grid is a square?

2. What information is needed to determine the area or perimeter of a rectangle on a coordinate grid? 3. How can the slope of two lines be used to determine if they are parallel or perpendicular?

Geometry on the Coordinate Plane | 65

Geometry on the Coordinate Plane

Explore 4

Name: _______________________ Date: ___________

Parallel and Perpendicular Lines Part I 1. The city planning committee determined there should be two streets parallel to Main Street. On the map below, draw three possible streets that are parallel to Main Street. 2. Write the equation of each street you drew. Equation 1: Equation 2: Equation 3: 3. What information might you need to clarify to determine which street is in the location the planning committee wanted?

y 10 9 8

ain

re St

5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1

x 1

9 10

-2 -3 -4 -5 -6 -7 -8 -9 -10

Geometry on the Coordinate Plane | 67

Geometry on the Coordinate Plane

Explore 4

4. The planning committee explained that one road parallel to Main Street should pass through the point 4 units east of the town’s center, the origin of the coordinate grid. The other street should pass through the point 9 units south of the town’s center. Draw these streets on the map below.

y 10 9 8

ain

re St

5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1

x 1

9 10

-2 -3 -4 -5 -6 -7 -8 -9 -10

5. Write the equation of each street you drew. Equation 1: Equation 2: 6. Conﬁrm you have drawn the same streets as at least one other person. Then, draw these streets on the Town Map Blueprint. 7. On the Town Map Blueprint, label the street passing through (0, –9) “Street 1.” Label the street passing through (4, 0) “Street 2.”

68 | Geometry on the Coordinate Plane

Explore 4

Geometry on the Coordinate Plane

The planning committee has given you the following information about existing and new streets. Existing street Street A: There is a street that extends from the northwest corner of town to the southeast corner of town. New streets Street B: There needs to be a street that is perpendicular to Main Street and goes through the point 5 units north of the town’s center. Street C: Another street needs to be built that is perpendicular to Main Street and goes through the point 6 units west and 1 unit north of the town’s center. 8. Explain how the slope of Main Street and street B are related.

9. Draw streets A, B, and C on the Town Map Blueprint. 10. Write the equation of each street you drew. Street A: Street B:

Street C:

11. Give the slope of street B and the point given by the planning committee. Use that information to write an equation in slope-intercept form, y = mx + b. Slope: Point: Equation: 12. Give the slope of street C and the point given by the planning committee. Use that information to write an equation in point-slope form, y – y1 = m(x x – x1). Slope: Point: Equation: 13. Rewrite the equation for street C in slope-intercept form, y = mx + b.

14. Do the equations derived from the graph in question 10 match the equations derived algebraically in questions 11 and 13? Why or why not?

Geometry on the Coordinate Plane | 69

Geometry on the Coordinate Plane

Explore 4 Part II

1. Determine which route from Town Hall to the hospital would be shorter. Town Hall to Hospital Using Previously Existing Roads

Using the Proposed New Road

Start on street A at (−1, 1). End on street A at (−8, 8).

Start on street B at (1.5, 3). End on street B at (−3, 9).

Distance:

2. Determine which route from the public library to Town Hall would be shorter. Public Library to Town Hall Using Previously Existing Roads

Using the Proposed New Road

Start on Main Street at (8, 6). Travel to (0, 0) on Main Street. Turn onto street A. End at (−1, 1).

Start on street 2 at (8, 3). Travel to (3.84, −0.12) on street 2. Turn onto street B. End at (1.5, 3).

Distance:

70 | Geometry on the Coordinate Plane

Geometry on the Coordinate Plane

Explore 4

3. Is there a route from the public library to the hospital using the existing streets that would be shorter than using the proposed new roads? Explain, and show all work. The public library has a driveway to the new street at (8, 3) and to the existing street at (8, 6). The hospital has a driveway to the new street at (−3, 9) and to the existing street at (−8, 8). Public Library to Hospital Using Previously Existing Roads

Using the Proposed New Road

Start on Main Street at ( ). Travel to ( ) on Main Street. Turn onto street A. End at ( ).

Start on street 2 at ( ). Travel to ( ) on street 2. Turn onto street B. End at ( ).

Distance:

4. Is there a route from the park to the public library using the existing streets that would be shorter than using the proposed new roads? Explain using mathematics. The park has a driveway to street C at (1.5, –9), to street 1 at (4, –6), and to street A at (9, −9). The public library has a driveway to street 2 at (8, 3) and to Main Street at (8, 6).

Geometry on the Coordinate Plane | 71

Explore 4

Geometry on the Coordinate Plane

Reflect 1. Compare and contrast the slopes of parallel and perpendicular lines.

2. Describe how an equation can be written for a line perpendicular to y = 1 x + 3 that 2 goes through (5, 1) using an algebraic process.

3. Explain how the distance formula was used to support creating the proposed new roads.

4. If you were a member of the city council, would you be satisﬁed with the additional streets, or would you be dissatisﬁed? Explain, and support your position with mathematics.

72 | Geometry on the Coordinate Plane

Linear Inequalities

Explore 1

Name: _______________________ Date: ___________

Solutions of Linear Inequalities Use the information listed below to determine if each ordered pair represents a scenario that is over budget, is under budget, or meets budget for the amount of money Jahzara has available for her start-up costs. Let x represent the number of bracelets and y represent the number of necklaces. • Jahzara has $500 to spend on start-up costs. • Materials for each bracelet cost $2. • Materials for each necklace cost $3. 1. Calculate the cost of supplies for each given situation. Determine if the costs of supplies are over budget, are under budget, or meet the budget. Then, select 5 different ordered pairs and complete the same calculations.

Equations and Calculations

Over Budget

Under Budget

Meets Budget

(20, 10) (120, 60) (220, 20) (30, 160) (40, 140) (200, 40)

Linear Inequalities | 75

Linear Inequalities

Explore 1

2. Write an inequality to represent the start-up costs. Let x represent the number of bracelets and y represent the number of necklaces. 3. Write a related equation to represent the start-up costs if Jahzara spends exactly $500. Let x represent the number of bracelets and y represent the number of necklaces. 4. Plot the ordered pairs from the previous page on the coordinate axis provided. • Use an to identify ordered pairs that are over budget. • Use a ● to identify ordered pairs that are under budget or meet the budget. y 180 170 160 150 140 130

Number of necklaces

120 110 100 90 80 70 60 50 40 30 20 10 0

90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270

Number of bracelets

5. Graph the related equation from question 3. 6. What do you notice about the

s and ●s?

7. On the coordinate axis, shade the half-plane that contains all of the points that make your inequality true.

76 | Linear Inequalities

Linear Inequalities

Explore 1

8. Why do you think this coordinate axis only shows positive x and y values?

9. Fill in the boxes to describe the constraints you identiﬁed using inequalities. x≥

y≥

10. What does the ordered pair (100, 50) mean in this situation?

11. What does the ordered pair (200, 100) mean in this situation?

Linear Inequalities | 77

Explore 1

Linear Inequalities

Reflect 1. Give three ordered pairs that are solutions. Are these in the shaded or unshaded half-plane?

2. What do you know about all of the coordinates in the shaded half-plane?

3. The ordered pair (54.2, 11.9) is in the shaded half-plane. Is this a reasonable solution? Why or why not?

4. What do you know about all of the coordinates in the unshaded half-plane?

5. Jahzara is selling the necklaces for $15 each and the bracelets for $8 each. How many bracelets and necklaces should she make? Her dad suggested the same number of each, maybe 100 and 100. Her aunt suggested making more necklaces because she is charging more for those. Jahzara was thinking she should make more bracelets because they take less time to make. How many bracelets and necklaces would you recommend Jahzara make? Support your answer with a math-based argument.

78 | Linear Inequalities

Graphing Linear Inequalities

40y

≤

2. Justify, using unit analysis, why 8 hours a week for 9 weeks is represented by 4,320.

Meaning

15x

15x x + 40y ≤ 4,320

1. Complete the table to explain the meanings of the terms and inequality symbol in context.

• Inequality from the app: 15x x + 40y ≤ 4,320 • Let x represent the number of bracelets made. • Let y represent the number of necklaces made.

Linear Inequalities | 79

4,320

Name: _______________________ Date: ___________

Use the given information to help Jahzara interpret the meaning of the inequality.

Explore 2

Linear Inequalities

8y y ≤ 864 – 3x

15x x + 40y ≤ 4,320

80 | Linear Inequalities

6. What additional constraints are needed so the region containing the solutions only contains reasonable solutions? Explain your reasoning.

5. Give an example of an ordered pair that contains integers and satisﬁes the inequality but is not a reasonable solution.

b. Do the inequalities have the same solution set? Why or why not?

a. Are the inequalities equivalent? Why or why not?

Aunt’s Inequality

Jahzara’s Inequality

4. Jahzara’s aunt ran the same data in a computer program she uses for work and was given a different inequality.

3. Explain how the app’s inequality, 15x x + 40y ≤ 4,320, differs from the related equation, 15x + 40y = 4,320.

Explore 2

Linear Inequalities

40 60

100

120

160

180

Number of bracelets

140

200

220

240

260

280

300

Linear Inequalities

Linear Inequalities | 81

9. Would the point (40, 93) be a reasonable test point to determine where to shade? Why or why not?

100

120

8. On the grid, shade the half-plane that contains all of the points that make your inequality true.

7. Graph the related equation from question 3.

Explore 2

Number of necklaces

82 | Linear Inequalities

3. How many of each type should Jahzara make? What recommendations would you give to Jahzara, and what math-based argument would you use to convince her that your recommendation is reasonable?

2. All of the points in the shaded region represent numbers of bracelets and necklaces that are within Jahzara’s budget. Are all of these options equally good and desirable? Give an example to support your position, and explain.

1. Both (8, 105) and (280, 3) are in the solution region. Both points mean a total of 4,320 minutes were spent on making jewelry. Does it make a difference which option is chosen? Why or why not?

Reflect

Explore 2

Linear Inequalities

Explore 3

Name: _______________________ Date: ___________

Writing Linear Inequalities Part I Analyze the graph to explain what the computer program has calculated for Jahzara’s jewelry income goal. Jahzara’s Jewelry Income Goal y 90

Number of necklaces

80 70 60 50 40 30 20 10 0

90 100 110 120 130 140 150 160 170

Number of bracelets

1. What do the x- and y-intercepts of the dashed line tell you within the context of this situation?

2. Why do you think the boundary line on the graph is dashed and not solid?

Linear Inequalities | 83

Linear Inequalities

Explore 3

Work through the problems below, and make a ﬁnal recommendation on the number of bracelets and necklaces Jahzara should make in order to meet her income goal. 3. Fill in the blanks to create an inequality that represents the situation on the graph. ___x x + ___y ___ 1,200 4. How did you determine the coefﬁcients to place in your equation?

5. How did you decide what inequality symbol to use?

6. Does the ordered pair (60, 48) satisfy the inequality?

7. Does the ordered pair (40.5, 70.5) satisfy the inequality? Is it a reasonable solution? Why or why not?

84 | Linear Inequalities

Linear Inequalities

Explore 3 Part II

Jahzara looked up other jewelry sellers on her computer program to compare their income goals. 1. Write an inequality for each of the graphs below. Each seller Jahzara researched has the same target of $1,200 that she has. Seller A y

Number of necklaces

Seller B

x 20

100

120

x 20

100

120

Number of bracelets

Inequality:

2. Which seller has a goal of more than $1,200? How do you know?

3. How would you describe the income goal of the other seller?

4. Which seller charges more for a necklace?

Linear Inequalities | 85

Linear Inequalities

Explore 3 Reflect 1. How can we tell if points along the boundary line are solutions?

2. Are all of the solutions in the shaded half-plane reasonable solutions?

3. How can the graph of an inequality be used to write an inequality and related equation?

4. How many of each type of jewelry should Jahzara make? What recommendations would you give to Jahzara, and what math-based argument would you use to convince her your recommendation is reasonable?

86 | Linear Inequalities

Systems of Inequalities

Explore 1

Name: _______________________ Date: ___________

Systems of Inequalities Part I Use the Graph Cards that represent the constraints for Jahzara’s start-up costs and the constraints for her projected income to determine some possible combinations of bracelets and necklaces she could make since she has $500 to spend, with bracelets costing $2 and necklaces costing $3 to make. 1. Write an inequality to represent the constraint for the start-up costs. 2. Determine which ordered pairs satisfy the constraint for the start-up costs. Ordered Pair

Yes

(80, 50) (105, 102) (140, 60) 3. Write an inequality to represent the constraint for the income given that she wants to earn at least $1,500 and sell each bracelet for $8 and necklace for $15.

4. Determine which ordered pairs satisfy the constraint for the income. Ordered Pair

Yes

(80, 50) (120, 70) (60, 68) 5. Why does (80, 50) satisfy the start-up constraint but not the income constraint?

6. Determine at least one ordered pair that satisﬁes both constraints. Explain how you know. 7. What does this pair of numbers represent in this situation?

Systems of Inequalities | 89

Systems of Inequalities

Explore 1 Part II The graph represents the constraints of Jahzara’s start-up costs.

Number of necklaces

1. On the graph, represent the constraints of the new income goal given that she wants to earn at least $2,000 and sell each bracelet for $8 and necklace for $15.

150 100 50

100

150

200

250

Number of bracelets 2. Determine at least three ordered pairs that satisfy both constraints. Explain how you know.

3. What do these ordered pairs represent in this situation?

4. The equation 6x x + 12y = P is used to determine Jahzara’s proﬁt. Which of the following combinations would have the greatest proﬁt? (250, 0) 90 | Systems of Inequalities

(0, 166)

Systems of Inequalities

Explore 1

Jahzara has changed her income goal once again. Now, she wants her income to be at least $2,600. She is still thinking she should sell the bracelets for $8 each and the necklaces for $15 each. Help her determine combinations of bracelets and necklaces to reach this new income goal. The graph represents the constraints of Jahzara’s start-up costs.

Number of necklaces

5. On the graph, represent the constraints of the new income goal.

150 100 50

100

150

200

250

300

350

Number of bracelets 6. Determine at least one ordered pair that satisﬁes both constraints. Explain how you know.

7. What does this represent in this situation?

Systems of Inequalities | 91

Explore 1

Systems of Inequalities

Reflect 1. If an ordered pair is in the shaded region of one linear inequality but not the other linear inequality, is it a solution to the system of linear inequalities?

2. What is the most efﬁcient way to determine all of the solutions to a system of linear inequalities?

3. What does it mean if there is not an overlapping region when a system of linear inequalities is graphed?

92 | Systems of Inequalities

Systems of Inequalities

Explore 2

Name: _______________________ Date: ___________

Part I

Systems of Inequalities Continued

Use the graphs that represent Mr. Khan’s puzzle to help Makena determine the two numbers. 1. Fill in the inequality symbol to represent Mr. Khan’s ﬁrst part of the puzzle. 2x x + 7 ___ y

3. Write an inequality to represent Mr. Khan’s second part of the puzzle. 5x x + 3 ___ y

2. Determine which pairs satisfy Mr. Khan’s ﬁrst part of the puzzle.

4. Determine which pairs satisfy Mr. Khan’s second part of the puzzle.

Yes

(10, 40)

(20, 30)

(17, 24)

(4, 25)

(3, 25)

5. Why does (20, 30) satisfy the second part of the puzzle but not the ﬁrst part of the puzzle?

6. Determine at least one ordered pair that satisﬁes both parts of the puzzle. Explain how you know.

7. What does this pair of numbers represent in this situation?

8. Is the ordered pair listed in question 6 the only possible set of numbers that could solve Mr. Khan’s puzzle? Why or why not?

Systems of Inequalities | 93

Systems of Inequalities

Explore 2 Part II The graph represents the ﬁrst part of Mr. Khan’s new puzzle.

1. On the same graph, represent the second part of Mr. Khan’s new puzzle. y 40 35

Second number

30 25 20 15 10 5

–5

0 –5

First number

2. Determine at least three ordered pairs that satisfy both parts of the puzzle. Explain how you know.

3. What do these ordered pairs represent in this situation?

4. Are the ordered pairs listed in question 2 the only possibilities that could solve Mr. Khan’s puzzle? Why or why not?

94 | Systems of Inequalities

Systems of Inequalities

Explore 2

Mr. Khan has changed the puzzle once again. Now, five times the first number plus 18 is less than the second number. Five times the first number minus 50 is more than the second number. What are the two numbers now? Help Makena determine the two numbers. The graph represents the first part of Mr. Khan’s puzzle. 5. On the same graph, represent the second part of Mr. Khan’s puzzle. y

40 35

Second number

30 25 20 15 10 5

–5

First number

6. Determine at least one ordered pair that satisﬁes both parts of the puzzle. Explain how you know.

7. What does this represent in this situation?

Systems of Inequalities | 95

Explore 2

Systems of Inequalities

Reflect 1. If an ordered pair is in the shaded region of one linear inequality but not the other linear inequality, is it a solution to the system of linear inequalities?

2. How can we make sure we shade the correct side of a linear inequality?

3. What is the most efﬁcient way to determine all of the solutions to a system of linear inequalities?

4. What does it mean if there is not an overlapping region when a system of linear inequalities is graphed?

96 | Systems of Inequalities

Simplify Radicals

Explore 1

Name: _______________________ Date: ___________

Simplify Square Roots Part I 1. Complete the table below using your knowledge of square roots and factoring to compute the length of each side of the two gardens.

Carrots

Tomatoes

Radishes

Area of Garden Length of Each Side As a Product of Prime Factors Common Terms Paired

√ √4 √(2 . 2) √22

Simpliﬁed

2. What is the difference between simplifying the square roots of 25 and simplifying the square root of 8?

Simplify Radicals | 99

Simplify Radicals

Explore 1 Part II

1. Use the Garden Signs Cards to complete the table. Determine the radical expression that represents the length and width of the garden plot. Write each radicand as its prime factors, and then give the expression for length and width in simplest radical form.

Lettuce

Radical

Prime Factorization

Simplest Radical Form

√ √45

√(3 . 3 . 5)

3√5 √

Potatoes

Green Beans

Strawberries

Cucumbers

Bell Peppers

2. In another garden, there is squash with a radical expression of 5√45 to represent the √ √45 width. Charlotte believes this simpliﬁes to 3√5 √ . Joshua believes the area simpliﬁes to 15√5 √ . Who is correct, and why?

100 | Simplify Radicals

Simplify Radicals

Explore 1

3. Fernando thought it took too long to write out the prime factorization of 72 and wants to consider a different way to simplify √72 √ . Which of the options below is more useful, and why? Option A: √72 =√ Option B: √72 =√ √ √ √6 . 12 √36 . 2

4. Fernando is trying to rework ﬁnding the simplest form for √200 . Which of the options √ below would be more efﬁcient? Option A: √200 =√ Option B: √200 =√ √ √ √50 . 4 √100 . 2

5. After looking at these more efﬁcient ways, with your guidance, Fernando tried to simplify √5 √ . √10 √ . Explain Fernando’s steps in each option, and decide which one is more efﬁcient in your opinion. Option A: √5 Option B: √5 √ . √10 √ √ . √10 √ √ √50

√5 . (√5 √ √ .√ √2)

√ 2.√ √5 √2

5√2 √

√25 . 2 √ 5√2 √

√ 2.√ √5 √2

Simplify Radicals | 101

Simplify Radicals

Explore 1 Reflect

1. In this Explore, we have been using radical notation. Rewrite the steps for the lettuce bed calculations using just rational exponents. √ √45 1

45 2

√(3 . 3 . 5) √

√(32 . 5) √

√ 2.√ √3 √5

3√5 √

2. Does writing the calculations in exponential form versus radical form make any real difference in performing the calculations?

3. After rewriting with fractional exponents, identify 2 places in question 1 where the laws of exponents were used.

4. We have been looking at areas that are always positive numbers, but would you get a real number if you try to take the square root of a negative number? Why or why not?

5. What does it mean when we say a number is a perfect square? How do perfect squares relate to the problems we have done in this Explore?

102 | Simplify Radicals

Simplify Radicals

Explore 2

Name: _______________________ Date: ___________

Simplify Square Root Expressions Part I 1. In determining how many bags of fertilizer we need, what is the ﬁrst calculation we need to make?

2. Complete the table to ﬁnd the area of each of the 3 different gardens.

Area Expression Expanded Form Coefﬁcients Multiplied Radicands Multiplied

Radish

Garden 1

Garden 2

2√2 √ . 2√2 √

5√6 √ . 10√2 √

10√2 √ . 4√6 √

2 . √2 √ . 2 . √2 √ 4 . √2 √ . √2 √

Radicals Simpliﬁed Area Expression Simpliﬁed

4 . √4 √ 4.2 8

3. What are the similarities and differences between the simpliﬁed area expressions?

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Simplify Radicals

Explore 2

4. The garden club needs the total combined area of all of the gardens. They need to identify the gardens that have like radicands. Which gardens have an expression for area with like radicands? What is the total combined area of garden 1 and garden 2?

5. Is the perimeter of the radishes larger or smaller than its area value?

6. Find the total combined area of the three gardens, give the answer in radical form, and then use technology to give an approximation to the nearest integer.

7. If they need one bag of fertilizer per 100 sq. ft. of garden, how many bags will they need?

8. The students wondered if they could generate a generic expression that could help with plans for next year. The groundskeeper suggested using variables. If they estimate an area of 5√x √ and a depth of 2√x √x √ , what would be the volume as a √x simpliﬁed radical expression?

9. Help the garden club simplify one more expression for volume: √ √4x . 5√x √x2y4 . √

104 | Simplify Radicals

Simplify Radicals

Explore 2 Part II

1. Help Agustin and the garden club. Use the space below to experiment with sums and products of rational and irrational numbers. Pick two numbers, ﬁnd the sum or product when possible, and then determine if the result is rational or irrational. Rational Numbers 7, 1 , 0, 1 2

Irrational Numbers √ , –√3 √3 √ ,

a. The sum of a rational number and a rational number ______ ______

+ +

______ ______

= =

______ ______

1 ,𝜋 (√3 √ )

Rational

Irrational

Rational

Irrational

b. The sum of a rational number and an irrational number ______ ______

+ +

______ ______

= =

______ ______

Rational

Irrational

Rational

Irrational

c. The sum of an irrational number and an irrational number ______ ______

+ +

______ ______

= =

______ ______

Rational

Irrational

Rational

Irrational

d. The product of a rational number and a rational number ______ ______

. .

______ ______

= =

______ ______

Rational

Irrational

Rational

Irrational

e. The product of a rational number and an irrational number ______ ______

. .

______ ______

= =

______ ______

Rational

Irrational

Rational

Irrational

f. The product of an irrational number and an irrational number ______ ______

. .

______ ______

= =

______ ______

Rational

Irrational

Rational

Irrational Simplify Radicals | 105

Explore 2

Simplify Radicals

2. Based on your experiments, decide if each statement is sometimes true, always true, or never true. a. The sum of a rational number and a rational number is rational. Sometimes true Always true Never true b. The sum of a rational number and an irrational number is irrational. Sometimes true Always true Never true c. The sum of an irrational number and an irrational number is irrational. Sometimes true Always true Never true d. The product of a rational number and a rational number is rational. Sometimes true Always true Never true e. The product of a rational number and an irrational number is irrational. Sometimes true Always true Never true f. The product of an irrational number and an irrational number is irrational. Sometimes true Always true Never true

3. For the 3 statements that you selected “sometimes true,” explain to Agustin using a counterexample why this doesn’t always work.

106 | Simplify Radicals

Simplify Radicals

Explore 2 Reflect

1. When simplifying a square root, if the radicand is a variable that has a power that is an even whole number, will it simplify to an expression with or without a radical? Explain.

2. When simplifying a square root, if the radicand is a variable that has a power that is an odd whole number, will it simplify to an expression with or without a radical? Explain.

3. Consider the product of rational and irrational factors. Decide if the product will sometimes or always be rational or irrational. Justify your answer with an example.

Sometimes or Always True

Example

Irrational . Irrational = Irrational Irrational . Rational = Irrational Rational . Rational = Rational

Simplify Radicals | 107

Simplify Radicals

Explore 3

Name: _______________________ Date: ___________

Simplify Cube Root Expressions Part I 1. Fill out the table below by using the information on the Storage Cube Cards, along with your knowledge of cube roots and factoring, to compute the length of each side of the storage containers, and then answer the question that follows.

Seeds

Volume of Container

Length of Each Side

Write as a product of prime factors.

Group trios of common terms.

∛ ∛8

∛2 . 2 . 2 ∛

∛ 3 ∛2

Simplify.

Topsoil

Fertilizer

Equipment

2. How do simplifying a square root and simplifying a cube root differ?

Simplify Radicals | 109

Simplify Radicals

Explore 3 Part II

1. Fill in the chart to determine the size of the cube-shaped transport containers, and then answer the questions that follow. Volume Lettuce

405

Potatoes

200

Green Beans

Bell Peppers

250x3

Tomatoes

16x6y3

Length of Side

Prime Factorization

Simpliﬁed

∛ ∛405

∛(3 . 3 . 3 . 3 . 5) ∛

3∛15 ∛

2. If you were wanting to add together the lengths of the tomato and bell pepper bins, would it be easier to use the original cube root form or the simpliﬁed form? Why?

3. If a cube root has a radicand with a variable exponent divisible by 3, will the variable simplify to an expression with or without a radical? Explain.

110 | Simplify Radicals

Explore 3

Simplify Radicals

Reflect 1. Emma and Amal are looking over their calculations for the garden project. Emma 1 uses the equation L = ∛V , and Amal wrote L =V 3 . Which student is correct? Why?

2. When we have an exponent to an exponent, does the order matter? Does 1 1 (x 3 )2 = (x2) 3 ? Why or why not? What would this expression simplify to?

3. When taking the square root of a number, we ﬁnd groups of 2, and for cube roots, we ﬁnd groups of 3. How might we go about simplifying the 5th root of a number?

4. If you try to take a square root of a negative number, you will not get a real solution. Do you think it is possible to get a real number if you take the cube root of a negative number? Give an example.

Simplify Radicals | 111

Polynomial Operations

113

Polynomial Operations

Explore 1

Name: _______________________ Date: ___________

Adding and Subtracting Polynomials Part I Algebra Mathematica has given each mosaic art piece a title that includes the terms below, and they used the pieces shown below to make the art. The artist did not indicate which term goes with which tile piece. Match each tile piece to a term, and justify your choice.

1 unit

A units

Term

Yellow Square

Green Rectangle

Blue Square

Justiﬁcation

Polynomial Operations | 115

Polynomial Operations

Explore 1 Part II

Complete the table by writing the letter that corresponds to each crate to organize the art.

Before Mosaic

Set Name

Before Title

Thrice Duce

Crate:

Missing Middle

Crate:

Opposites 7 to 1

Crate:

So Odd

Crate:

116 | Polynomial Operations

After Title

After Mosaic

Explore 1

Polynomial Operations

Part III Create your own piece for the Before and After Collection. Before Title

After Title

Before Mosaic

After Mosaic

Polynomial Operations | 117

Explore 1

Polynomial Operations

Reflect 1. Compare and contrast simplifying the expressions in the diagram below. (2x x + 3y) – 4(x x + y)

(2a2 + 3a) – 4(a2 + a)

2. How would you explain subtracting polynomials to a peer who was absent today?

118 | Polynomial Operations

Polynomial Operations

Explore 2

Name: _______________________ Date: ___________

Multiplying Polynomials with Models Part I

1. Complete the product and image for set B in the table below. Set A

Set B

Multiplicand (factor)

2A2 + 2A + 3

A2 + 6A – 1

Multiplier (factor)

Product

6A2 + 6A + 9

Image for the Series

2. Multi-term multiplication can be done using an array. Fill in the product below, and explain how you know where to write each part of your answer.

−1

2 3. What would the algebraic representation for the product be if set A had been lost in transit? Explain your reasoning.

Polynomial Operations | 119

Polynomial Operations

Explore 2 Part II

1. Complete the table to help organize sets C and D of the Times Up Collection. Set C

Set D

Multiplicand (factor)

2x x–7

4x x+5

Multiplier (factor)

Product

Image for the Series

Array

2. How does the array area model compare to the image for the series?

3. How can the method of vertical multiplication with multi-digit numbers be used to multiply polynomials? For example, 45 times 2 can be written as shown. Show or explain how this method can be used to multiply 4x x + 5 times 2x. 45 x 2 90 1

120 | Polynomial Operations

Explore 2

Polynomial Operations

Reflect 1. Show or explain how an array could be used to multiply 3A2 + 7A – 10 times 4A.

2. Could vertical multiplication be used to multiply 3A2 + 7A – 10 times 4A? Why or why not?

3. Explain what is different about ﬁnding the sum of 3A2 + 7A – 10 plus 4A compared to ﬁnding the product of 3A2 + 7A – 10 times 4A.

4. Why do you think an area model might be better than algebra tiles in situations where you are multiplying large quantities?

5. How would you explain the process of multiplying a monomial by a polynomial to someone who had never done it before?

Polynomial Operations | 121

Polynomial Operations

Explore 3

Name: _______________________ Date: ___________

Multiplying Polynomials Part I Use the Family Backyard Design Cards to answer the questions below. 1. Use the tiles sketched on the architect’s plans to determine the algebraic expressions for the width factor, length factor, and area. Complete the table below to demonstrate to the lead architect that you understand their diagram. Project

Width Factor

x–4

Length Factor

Area x2 – 4x x + 2x – 8

x+3

2. Another intern is having trouble with the project sketches. The lead architect asks you to explain how the tiles sketched on the plans represent the algebraic expressions. What do you tell the other intern?

3. The lead architect has now asked you to demonstrate how to ﬁnd the algebraic expressions using area models or arrays. Complete the arrays below, and circle any boxes that contain like terms.

Project A

Project B

Polynomial Operations | 123

Polynomial Operations

Explore 3 Part II

Use the Summer Project Cards to complete the steps below. 1. Draw a sketch of each design proposal with the dimensions labeled. 2. Calculate the area of each project using an area model. 3. Circle any entries in the area models that contain like terms. 4. Write the terms from the area models. 5. Write the simpliﬁed products. Formula: A = bh

Family Pool

Sketch:

Area model:

Write the terms from the area model. Simpliﬁed product for the area: Formula: A = 1 bh 2

Vegetable Garden

Sketch:

Area model:

Write the terms from the area model. Simpliﬁed product for the area: 124 | Polynomial Operations

Explore 3

Polynomial Operations

Reflect 1. If needed, would you be able to draw representations using algebra tiles for both the family pool and the vegetable garden? Explain your reasoning.

2. When using an area model, can you identify any similarities to the distributive property? Use the factors 2x x + 3 and 3x + 5 and an area model to help explain.

3. Use an area model or written description to explain why (x x + 1) times (x + 1) is not 2 x + 1.

Polynomial Operations | 125

Polynomial Operations

Explore 4

Name: _______________________ Date: ___________

Special Products Part I 1. Use the Gate Code Cards to unlock the gates. Glue the correct matches in the appropriate boxes below. Use the workspace as needed. Note that not all of the Gate Code Cards will be used. Gate

(

)(

Workspace

)

Gate

(

−

)(

Code 2

Workspace

−

)

Code 2

−

Polynomial Operations | 127

Polynomial Operations

Explore 4 Gate

)(2

Workspace

)

Gate

−

)(3

Code (2

)2 + (2

Workspace

−

)

)(

) + (2

)(

Code (3

)2 − (3

)(

) − (3

2. What series of steps did your group come up with for cracking each gate code?

128 | Polynomial Operations

Explore 4

Polynomial Operations

3. What similarities and differences did you observe between simplifying each gate and its corresponding code and simplifying expressions?

+ )( + ) , and replace the square with the 4. If we look at the ﬁrst gate, ( variable x and the circle with the number 3, what is the new solution? Show your work in the space below.

5. Did your process change from the ﬁrst time you simpliﬁed this expression when it was only shapes?

− )(3 − ) , and replace the square with the 6. If we look at the last gate, (3 variable x and the circle with the number 2, what is the new solution? Show your work in the space below.

Polynomial Operations | 129

Polynomial Operations

Explore 4 Part II

1. Complete the table by breaking the codes for the remaining four gates listed. Gate

Workspace

Code

(x x + 5)(x – 5)

(3w w + 7)(3w – 7)

(2y y – z)(2y y + z)

(4x x + 9)(4x – 9)

2. What series of steps did your group come up with for cracking each gate code?

130 | Polynomial Operations

Explore 4

Polynomial Operations

Reflect 1. Explain how you would use patterns to multiply (3x x + 2y)(3x x – 2y) without having to use an area model or write out all of the steps.

2. Explain why there is no middle term when multiplying expressions in the form (a + b)(a – b).

3. When squaring a binomial, how is the middle term of the product related to the terms in the binomial? You may use the expansion of (x x + 7)2 to help explain your answer.

Polynomial Operations | 131

Polynomial Operations

Explore 5

Name: _______________________ Date: ___________

Standard Form of a Quadratic Function Part I 1. Use the Blueprint of the neighbor’s home to illustrate each student’s equation. Then, give a brief explanation of your illustration. Sai

Maria

James

2. Simplify each equation to verify the equations are equivalent algebraically. Write each equation in standard form with the terms in order of descending degree. What do the simpliﬁed equations have in common?

3. If the shaded area of the neighbor’s house is 212 square feet, determine the length and width of the whole house.

Polynomial Operations | 133

Polynomial Operations

Explore 5 Part II 1. How does each equation compare with the data they recorded?

2. Is there anything all the students’ equations have in common?

3. Could all three students have the correct equation? Why or why not?

4. Simplify each equation to verify the equations are equivalent algebraically. Write each equation with the terms in order of descending degree.

134 | Polynomial Operations

Explore 5

Polynomial Operations

5. Use substitution with James’s equation to determine ff(0). Interpret this ordered pair in terms of the situation.

6. What ordered pair could be used to verify James’s equation represents the rocket landing after 11 seconds? Does this ordered pair satisfy James’s equation? Why or why not?

7. What ordered pair could be used to verify James’s equation represents the maximum height of the rocket occurred 5 seconds after launch at a height of 90 meters? Does this ordered pair satisfy James’s equation? Why or why not?

Polynomial Operations | 135

Polynomial Operations

Explore 5 Reflect

1. Another student, George, created an equation g(x) for the neighbor’s house. Describe how it could be determined if g(x) is equivalent to the other students’ equations.

2. Determine if George’s equation, g(x), is equivalent to Sai’s equation, f( f x). Are they equivalent? Why or why not? g(x) = (2x x – 4)(3) + 4x f x) = 3(2x) + 3(4) + 4(x) f(

3. Which of the following equations are written in standard form? If an equation is not written in standard form, then rewrite it in standard form. Equation

Is it in standard form?

p(x) = x2 + 5x x+2

Yes No

q(x) = 9x x – x2 + 13

Yes No

r(x) = 4(x x + 1)2 + 6

Yes No

s(x) = x2 + 5

Yes No

t(x) = (2x x + 3)(x – 7)

Yes No

136 | Polynomial Operations

Rewrite if needed.

Graphs of Quadratic Functions

137

Graphs of Quadratic Functions

Explore 1

Name: _______________________ Date: ___________

Key Features and Attributes Part I 1. Use the sketch of the jump and the judges’ notes to complete the contestant card. Identify the ordered pairs of points B–E and the hang time. Amari, Skateboard Finalist (0, 0) A: ________ B: ________ B

C: ________

Height (feet)

D: ________ E: ________ Time spent in the air:

______ seconds

Time (seconds)

Judges’ Notes • Amari was in the air for 8 seconds. Use this information to determine the coordinates of point E. • The peak of Amari’s jump was timed to be exactly halfway through his jump and was 16 feet high. Use this information to determine the coordinates of point C. • One second before his maximum height, he was 15 feet above the ground. Use this information to determine the coordinates of point B. • Amari’s jump was perfectly symmetrical when measuring time vs. height. Use this information to determine the coordinates of point D. © Accelerate Learning Inc. – All Rights Reserved

Graphs of Quadratic Functions | 139

Graphs of Quadratic Functions

Explore 1

2. Use the sketch of the jump and the judges’ notes to complete the contestant card. Identify the ordered pairs of points A–E and the hang time. Harry, Skateboard Finalist A: ________ B: ________ C: ________ D: ________ Height (feet)

C B

E: ________ Time spent in the air:

______ seconds

A Time (seconds)

Judges’ Notes Use the information below to identify the coordinates above. • Harry reached a maximum height of 9 feet after 3 seconds. • Harry’s jump was perfectly symmetrical when measuring time vs. height. • Harry crossed point D after 5 seconds at a height of 5 feet.

3. Which skateboard ﬁnalist won the hang-time award for being in the air the longest?

140 | Graphs of Quadratic Functions

Graphs of Quadratic Functions

Explore 1 Part II

1. Use the data shown in the picture that was collected by the analyst to complete each contestant card. Aalia, Dirt Bike Finalist

Manuela, Dirt Bike Finalist

Height (meters)

( _, 20)

(0, 10)

(3, 16) (1, 12) (0, 7)

(-2, 0)

(12, 0) Time (seconds)

(-1, 0) Time (seconds)

Reasonable root(s): ________

y-intercept: ________

Vertex: ________

Reasonable root(s): ________

Axis of symmetry: ________

Minimum or maximum (Circle one.)

What does the point (0, 10) represent in regard to the jump?

Manuela’s jump was perfectly symmetrical. Use the property of symmetry to determine Manuela’s height at 5 seconds. _______ meters

2. Which dirt bike ﬁnalist won the hang-time award for being in the air the longest?

Graphs of Quadratic Functions | 141

Graphs of Quadratic Functions

Explore 1

3. Use the data shown in the picture that was collected by the analyst to complete each contestant card. Aditi, Cliff Diving Finalist

Yume, Cliff Diving Finalist

(0, 30)

Height (feet)

(0, 30)

(2, 0)

Time (seconds)

(15, 0)

(__, 0)

( __, -42.25)

Time (seconds)

(10, 0)

( 6.5, -12.25)

Vertex: ______________

x-intercept(s): ________________

Minimum or maximum (Circle one.)

y-intercept: ________

Axis of symmetry: ________

Minimum or maximum (Circle one.)

How long was Aditi under water?

What depth did Yume reach?

________ seconds

________ feet

4. Which cliff diving ﬁnalist spent the most time underwater?

142 | Graphs of Quadratic Functions

Graphs of Quadratic Functions

Explore 1 Reflect

1. What shape is the graph of a quadratic function, and what are some of the graph’s key features?

2. Describe the relationship between the vertex and the axis of symmetry.

3. If the parabola opens upward, what can you say about the vertex? If it opens downward, what can you say?

4. The point (3, 4) lies on a parabola whose axis of symmetry is x = 1. What other point must also lie on the parabola, and why? You may want to draw a sketch.

5. How many x-intercepts could a quadratic function plotted on the xy plane (a parabola) have?

6. Complete the table to give possible ordered pairs for the following key features. Then, describe how the vertex could be identiﬁed if only the table were given. y

Key Feature x

x-intercept vertex x-intercept

Graphs of Quadratic Functions | 143

Graphs of Quadratic Functions

Explore 2

Name: _______________________ Date: ___________

Roots, Zeros, x-intercepts, and Solutions Part I Use the Results Cards to complete Part I. 1. Analyze the data on the Results Cards, and complete the summary table below.

Finalist

Initial Time, x x, and Height, y (x, x y) x,

Maximum Time, x x, and Height, y (x, x y) x,

Ending Time, x x, and Height, y (x, x y) x,

Equation

A B C 2. Are any of the ordered pairs in the table x-intercepts? If so, which ones?

3. Compare and contrast the ordered pairs for the initial values and the ending values with the equations for ﬁnalists A and B.

4. Compare and contrast the ordered pairs for the x-intercepts with the equation for ﬁnalist C.

5. What generalization can you make about the x-intercepts and the equation?

Graphs of Quadratic Functions | 145

Graphs of Quadratic Functions

Explore 2 Part II

Analyze each Results Card. Determine whether there is an error and how to ﬁx it. Contestant A Is there an error? If so, what is the error?

Write a correct equation, and sketch a corresponding graph with labels for that equation.

Correct start time: Correct end time:

Contestant B Is there an error? If so, what is the error?

Write a correct equation, and sketch a corresponding graph with labels for that equation.

Correct start time: Correct end time:

Contestant C Is there an error? If so, what is the error?

Write a correct equation, and sketch a corresponding graph with labels for that equation.

Correct start time: Correct end time:

146 | Graphs of Quadratic Functions

Explore 2

Graphs of Quadratic Functions

Reflect 1. Agustin and Jamere are arguing about which equation corresponds with the graph below. According to Agustin, the correct equation is y = (x + 2)(x + 5), and Jamere argues that the correct equation is y = (x – 2)(x – 5). Who is correct, and why? y

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

-3 -2

-1

0 -1

x 1

-2 -3

2. Why might the x-intercepts be called zeros?

3. According to Chole, y = x(x x + 1) only has one root, and it is −1. a. Is she correct? b. If she is not correct, rewrite the equation to help Chole more clearly see the other root.

4. Would the equations y = 8x(x x – 4) and y = x(x x + 1) both have roots at the origin? Why or why not?

5. Would y = (x + 9)(x + 9) have 1 or 2 x-intercepts? Explain your reasoning.

Graphs of Quadratic Functions | 147

Graphs of Quadratic Functions

Explore 3

Name: _______________________ Date: ___________

Equations in Vertex Form Part I 1. Use the piece of string to model the suspension wire for bridges A, B, and C. Then, in the table below, record the coordinate that describes the top of each tower and the coordinate that describes the distance when the wire is 0 meters above the road.

Bridge

Top of West Tower

(0, 9)

Point Where Wire Touches Road

B C

Top of East Tower

(80, 8) (0, 4.5)

2. Describe the relationship between the towers and where the support wire touches the road.

3. The following equations are written in vertex form. Use substitution to determine which equation represents the height of the support wire as a function of the distance from the west tower for each bridge. Equation 1 y = 0.01(x – 30)2

Equation 2 y = 0.005(x – 30)2 + 0

Equation 3 y = 0.005(x – 40)2

4. Explain your method for at least one equation.

Graphs of Quadratic Functions | 149

Graphs of Quadratic Functions

Explore 3

5. Use the equations to determine the height of the support wire 40 meters from the west tower. Bridge A:

Bridge B:

Bridge C:

6. Do any of the bridge support wires have a minimum height 40 meters from the west tower? If so, explain.

7. Which, if any, part of the equation can be used to determine the distance from the west tower the minimum height will occur? Does this hold true for all three equations? Explain.

8. Describe how the minimum height of the support wire, the distance from the west tower, and the equation are related.

150 | Graphs of Quadratic Functions

Graphs of Quadratic Functions

Explore 3 Part II

Use information about each bridge to answer the questions that follow. Bridge 1

Distance across = 130 m

1. If the vertex of the wire in the picture was at (10, 0), how long would the bridge be? Explain, and label the distance.

2. Would the total distance between towers be different if the vertex was at (10, 2) instead? Why or why not?

Graphs of Quadratic Functions | 151

Graphs of Quadratic Functions

Explore 3 Bridge 2

The equation h = 0.05(d d – 10)2 + 2 models the height of the support wire as a function of the distance from the west tower of bridge 2. 3. What value of d would give the minimum height of the support wire? Explain.

4. How can you use the value you determined in the previous question to determine the minimum support height of the wire above ground?

Bridge 3 5. Based on the following details, complete the missing parts of the equation for bridge 3.

Height of Lowest Point

Distance from a Tower to the Middle

4 meters

20 meters

h = 0.025(d d – ___)2 + ___

152 | Graphs of Quadratic Functions

Explore 3

Graphs of Quadratic Functions

Reflect 1. Compare and contrast the location of the minimum for the quadratic functions f( f x) and h(x). f x) = (d f( d – 40)2 + 5 and h(x) = (d d + 40)2 + 5

2. Explain why substituting in x = −2 will give the function f( f x) = (x x + 2)2 – 5 its lowest value.

3. Why did knowing the vertex of the bridge’s suspension wire tell us how long the bridge was?

4. How can you ﬁnd and explain the vertex of an equation in the form y = a(x x – h)2 + k?

Graphs of Quadratic Functions | 153

Graphs of Quadratic Functions

Explore 4

Name: _______________________ Date: ___________

Part I

Solving for a in Quadratic Functions

Use the parabola below to model the tunnel opening, and answer the questions that follow. y

x -10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1. The ﬁrm’s intern starts by creating the equation y = x2 + 8. What about the intern’s model is successful, and what features need to change?

2. An associate at the architecture ﬁrm adjusted the intern’s equation to y = −x2 + 8. Does this equation match the points labeled on your graph?

3. The lead architect knows that the first two attempts were close and suggests that the team find a coefficient to put in front of the equation so it goes through each of the points since −1 did not quite work. Find the value of a in the equation y = ax2 + 8 so it passes through the x-intercepts and the vertex.

Graphs of Quadratic Functions | 155

Graphs of Quadratic Functions

Explore 4

Use the parabola below to model the tunnel opening, where the x- and y-axes measure distance in meters, and answer the questions that follow. y

(6, 4)

(–6, 4)

(–8, 0)

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

(8, 0) 9

x 10

4. Since we know the two x-intercepts and don’t know the vertex, with what form would it be easiest to create the equation? 5. Write an equation that passes through both x-intercepts. 6. Solve for the coefﬁcient in your equation so your model also has a height of 4 m when it is 6 m away from the center.

7. Use your equation to determine the maximum height of this tunnel.

156 | Graphs of Quadratic Functions

Graphs of Quadratic Functions

Explore 4 Part II

The architecture ﬁrm needs a graph and an equation to model a tunnel that has an opening that goes through the points (7, 0), (−7, 0), and (3, 8). 1. Sketch the parabola on the coordinate grid to represent the opening of the tunnel. Label the x-intercepts. y

x -10

-9

-8

-7

-6

-5

-4

-3

-2

-1

2. Write an equation in factored form, and then determine the value of a that would model the opening of the tunnel shown.

3. Use your equation to determine the vertex of the parabola. Adjust your sketch if needed.

Graphs of Quadratic Functions | 157

Explore 4

Graphs of Quadratic Functions

Reflect 1. Compare and contrast the methods to determine the coefﬁcient when given the x-intercepts and when given the vertex.

2. How did substitution help us solve for the full equation of a parabola when given the graph?

3. Can an exact equation be determined to represent the parabola if only the roots are known? Why or why not?

158 | Graphs of Quadratic Functions

Graphs of Quadratic Functions

Explore 5

Name: _______________________ Date: ___________

Identifying Key Features and Equivalent Quadratic Equations Part I For each statement below, select agree or disagree. Explain why you agree or disagree using data from the witness statements, and if you disagree, rewrite the statement so it is true. 1. The water balloon was thrown from the ground level. Agree

Disagree

2. The maximum height of the water balloon was below the 10th ﬂoor. Agree

Disagree

3. The water balloon was in the air for a total of 3 seconds. Agree

Disagree

Graphs of Quadratic Functions | 159

Graphs of Quadratic Functions

Explore 5

Mrs. Evelyn and Andres write several possible equations to describe the water balloon’s height with respect to the ground as a function of time. Mrs. Evelyn’s Equations

Andres’s Equations

a(x) = −(x x + 1)2 + 9 b(x) = −(x x – 2)2 + 9 c(x) = −(x x – 3)2 + 9

d(x) = −x(x x + 6) e(x) = −(x x + 2)(x – 4) f x) = −(x f( x + 1)(x – 5)

4. Identify the vertex for each of Mrs. Evelyn’s equations. Explain how you identiﬁed the vertex from the equation. Equation A

a(x) = −(x x + 1)2 + 9

b(x) = −(x x – 2)2 + 9

c(x) = −(x x – 3)2 + 9

Vertex

5. Use information from the Eyewitness Accounts to explain how Mrs. Evelyn may have derived her equations.

6. Can any of Mrs. Evelyn’s equations be eliminated based on the location of the vertex? Why or why not?

160 | Graphs of Quadratic Functions

Graphs of Quadratic Functions

Explore 5

7. Identify the roots and vertex for each of Andres’s equations. Explain how you identiﬁed the vertex from the equation. Equation D

d(x) = −x(x x + 6)

e(x) = −(x x + 2)(x – 4)

f x) = −(x f( x + 1)(x – 5)

Roots

Vertex

8. Use information from the Eyewitness Accounts to explain how Andres may have derived his equations.

9. Can any of Andres’s equations be eliminated based on the location of the roots or vertex? Why or why not?

Graphs of Quadratic Functions | 161

Graphs of Quadratic Functions

Explore 5 Part II

1. List the remaining equations from both Mrs. Evelyn and Andres that have a reasonable maximum. Then, use substitution to determine the y-intercept for the remaining equations. Equation

Substitution

y-intercept

2. Based on the y-intercept for the equations remaining for Mrs. Evelyn and Andres, what ﬂoor(s) would the water balloon likely come from? Support why the ﬂoor choice is reasonable or unreasonable given the eyewitness accounts.

3. Using evidence from the eyewitness accounts as well as mathematical reasoning, explain which, if any, should be eliminated.

162 | Graphs of Quadratic Functions

Graphs of Quadratic Functions

Explore 5

4. What are the y-intercept and vertex for the remaining equations from both Mrs. Evelyn and Andres? What do you think will happen if they are both rewritten in standard form? Explain.

Equation

Mrs. Evelyn

Andres

b(x) = −(x x – 2)2 + 9

f x) = −(x f( x + 1)(x – 5)

y-intercept Vertex

5. Write both Mrs. Evelyn’s and Andres’s equations in standard form. Mrs. Evelyn’s Equation

Andres’s Equation

6. Are the equations equivalent? Explain.

7. Use substitution and the standard form of their equation to determine the y-intercept.

8. What relationship do you notice about the y-intercept and the standard form: y = Ax2 + Bx + C?

Graphs of Quadratic Functions | 163

Graphs of Quadratic Functions

Explore 5 Reflect

1. How can you ﬁnd the vertex of a function that is written in factored form?

2. Which quadratic form makes it easiest to ﬁnd the y-intercept, and why?

3. If you were asked to identify the maximum or minimum of a quadratic equation, which form would you want to use, and why?

4. How can you tell if two equations are equivalent when they are written in different forms?

5. If two equations have the same vertex and the same roots, will they have equivalent equations? Why or why not?

6. Sketch as many quadratic functions as you can that all have the following attributes. Then, write the equation(s) to represent the function(s) graphed. y

Vertex

x-intercepts

4 3

(1, −4)

(−1, 0)

(3, 0)

2 (–1, 0)

–4 –3 –2 –1 0 –1

(3, 0) 1

4 x

–2 –3 –4

164 | Graphs of Quadratic Functions

(1, –4)

Factors of Polynomials

165

Factors of Polynomials

Explore 1

Name: _______________________ Date: ___________ GLUE

Factoring Using Models

Part I Use the Art Pieces Cards to complete the following steps. 1. Arrange the algebra tiles to form a mosaic. The algebra tiles must be arranged in the shape of a rectangle. The large square(s) must be in the upper left corner of the rectangle, and the small square(s) must be in the lower right corner of the rectangle. 2. Use the algebra tiles and Algebra Tiles Factoring Mat to determine the correct arrangement of tiles. Then, complete the table.

Art

Total Area in Standard Form

Sketch of Assembled Mosaic

Dimensions in Factored Form

Checked Answer (Multiply.)

Factors of Polynomials | 167

Factors of Polynomials

Explore 1

3. Can every quadratic trinomial expression be factored? If yes, explain. If no, provide an example.

4. The art below fell off of the display, and there are two pieces missing. The total area was x2 + x – 2.

a. What are the two missing pieces? What do they represent?

b. What are the dimensions in factored form?

168 | Factors of Polynomials

Factors of Polynomials

Explore 1 Part II

1. The artist orders a frame for a work of art with an area of 2x2 + 7x x + 6. Based on the rules from the artist, you know where to place the large and small squares. Sketch where the remaining algebra tiles should be placed.

2. Use the sketch to complete the area model below. Write the dimensions on the area model. Check your answer using multiplication with the blank area model. Area: 2x2 + 7x x+6

Check using multiplication.

Width

Length

2x2

Look for relationships to explain how to use the area model. 3. What is the relationship between the ﬁrst degree term, 7x, and the ﬁrst degree terms you wrote inside the area model?

4. How are the second degree term and the constant, 2x2 and 6, related to the ﬁrst degree terms you wrote in the area model?

Factors of Polynomials | 169

Factors of Polynomials

Explore 1

5. Complete the area models below to determine the dimensions of the art with the given areas. Check your answers using multiplication with the blank area models. Area: 4x2 + 8x x+3

Check using multiplication.

Width

Length

4x2

Area: 2x2 + x – 6

Width 170 | Factors of Polynomials

Check using multiplication.

Length

Factors of Polynomials

Explore 1 Reflect

1. Tomas and Paola each factored the expression 2x2 + 9x x + 10 using an area model. Which student is correct? Explain your reasoning. Tomas

2x2

+ 2 Length

2x2

+ 5 Length

Width 2x

Width x

Paola

2. Nyah is struggling to calculate the dimensions of x2 + 2x x – 3 using algebra tiles. She has arranged the algebra tiles, but she cannot make a rectangle.

a. Show Nyah how to factor x2 + 2x x – 3 using an area model.

b. How can Nyah factor the expression using algebra tiles?

3. Describe when algebra tiles or an area model is more useful for factoring.

Factors of Polynomials | 171

Factors of Polynomials

Explore 2

Name: _______________________ Date: ___________

Factoring Using Patterns Part I Your boss has separated the calling cards into 3 pods as listed below. Positive – Positive Pod 1. Determine the pattern to complete the missing values on the calling cards. 12 3

6 4

2. Describe the pattern in the positive – positive pod.

Positive – Negative Pod 3. Determine the pattern to complete the missing values on the calling cards. −5 1

−5

−8

−15

−3

−10

−2

−4

4. Describe the pattern in the positive – negative pod.

Negative – Negative Pod 5. Determine the pattern to complete the missing values on the calling cards. 6

3 −1

−3 −4

−8

−6

−7

−1

−2 −5

6. Describe the pattern in the negative – negative pod.

Factors of Polynomials | 173

Factors of Polynomials

Explore 2 a = 1 Trinomial House ax2 + bx + c

7. Now that you have decoded the calling cards, use the information your boss compiled below to decode and apprehend members of the a = 1 trinomial family. Complete the table, and answer the questions. Bandit in Standard Form

x2 + 7x x + 10

x2 + 1x x–6

x2 – 5x x+4

−6

Target Product Target Sum

Calling Card

Bandit in Factored Form

(x x + )(x x + 5)

(x x – )(x

)

8. Which term(s) is(are) used to determine the target product? 9. Which term(s) is(are) used to determine the target sum? 10. Describe how standard form and factored form are related.

11. Will this work for all quadratics where a = 1? Why or why not?

Congratulations! You have discovered a pattern for the a = 1 trinomial family. They are out of business and ofﬁcially apprehended! 174 | Factors of Polynomials

Factors of Polynomials

Explore 2 Part II

Perfect Square Trinomial House ax2 + bx + c

1. Use the information your boss compiled below to decode and apprehend members of the perfect square trinomial house. Complete the table, and answer the questions. Bandit in Standard Form

x2 + 10x x + 25

x2 + 14x x + 49

x2 – 6x x+9

−6

Bandit in Factored Form

(x x + )(x x + 5)

(x x + )(x x+ )

Bandit as a Perfect Square

(x x + 5)2

Target Product Target Sum

Calling Card

2. How is the name perfect square trinomial related to the pattern of a factored perfect square trinomial?

3. Which of the following statements best describes the pattern of factoring a perfect square trinomial? A. If half of c squared is b, then the constant in the factors is half of c. B. If half of b squared is c, then the constant in the factors is half of b. C. If the square root of c is 2 more than b, then the constant in the factors is the square root of c.

Factors of Polynomials | 175

Factors of Polynomials

Explore 2 Difference of Squares House ax2 + bx + c or ax2 – c

4. Use the information your boss compiled below to decode and apprehend members of the difference of squares house. Complete the table, and answer the questions. Bandit in Standard Form

x2 + 0x x – 25

x2 – 16

4x2 – 9

−25

−16

−36

Target Product Target Sum

Calling Card

Bandit in Factored Form

(2x x + 3)

5. How is the name difference of squares related to the pattern of the quadratic in standard form?

6. Would a2 – b2 be considered a difference of squares quadratic? Why or why not?

176 | Factors of Polynomials

Factors of Polynomials

Explore 2 Part III

To apprehend the remaining quadratic bandits, paste each of the Quadratic Bandits Wanted Posters into the correct family in the table below. a = 1, Trinomials in Standard Form House

Difference of Squares House

Perfect Square Trinomial House

Factors of Polynomials | 177

Factors of Polynomials

Explore 2 Reflect 1. What patterns did you notice? How is this helpful?

2. Explain why the two constant values inside factors of x2 – 10x x + 24 must both be negative.

3. How did you decide which quadratic expressions were perfect square trinomials and which were a difference of squares?

4. How can you check that a quadratic has been factored correctly?

5. Complete the sentences to describe the process of transforming a quadratic equation or expression from standard form to factored form. To determine possible factors, I ﬁrst try to– Next, I think about– I know I have an equivalent expression in factored form if– To verify that the expression in factored form is equivalent to the original expression in standard form, I can–

178 | Factors of Polynomials

Explore 3

Factors of Polynomials

Name: _______________________ Date: ___________

Completing the Square – Introduction Part I 1. Write an expression in terms of x, the length, for the length and width of Miranda’s square block pattern that is 5 inches longer than the standard square block pattern.

2. Use the linear factors to write an equation in terms of x for the area of the square block pattern Miranda will use.

3. When Miranda draws her square block pattern, it looks like this. How many rectangles are on each side of the large square?

4. How is this diagram related to the expressions for length and width from question 1?

Factors of Polynomials | 179

Factors of Polynomials

Explore 3

5. Use the questions to complete the table by describing how the area equation is related to the square block pattern. Miranda’s square block pattern is shown below. Question

+10x

+25

Describe the portion of the area of the square block pattern this term represents.

Explain why all three terms are needed to describe the area of the entire square block pattern. How does the equation in question two on the previous page relate to the equation A(x) = x2 + 10x x + 25?

180 | Factors of Polynomials

Factors of Polynomials

Explore 3 Part II

Use the Quilt Orders Cards to complete the tables and answer the questions. 1. Help Miranda put the ripped quilt orders back together using the information in each order. Glue the pieces of the order in the correct locations. You will have extra pieces.

Myra ordered a quilt for her daughter. She wants a quilt that has squares 4 inches larger than the standard square.

Koﬁ ordered a quilt for himself. He wants a quilt that has squares 3 inches larger than the standard square.

Diagram

Area

Dimensions

Order

Orders for Abuela Florencia’s Quilts

Factors of Polynomials | 181

Factors of Polynomials

Explore 3

2. It appears there were 4 orders because Miranda found some partial expressions for area. Read each order carefully, and glue the diagrams in the correct locations. Then, complete the table by ﬁnishing the equation and writing the expression in two forms. Order 3

Order 4

x2 – 10x x + ___

x2 + 4 4x x + ___

Diagram

Area

Dimensions

182 | Factors of Polynomials

Factors of Polynomials

Explore 3

Miranda organized some of the expressions used so far. She has lined up each area expression, ax2 + bx + c, with the corresponding dimension expression, (x x + h)2. Area Expression

x2 + 8x x + 16

x2 + 6x x+9

x2 – 10x x + 25

Dimension Expression

(x x + 4)2

(x x + 3)2

(x x – 5)2

3. Miranda believes she has noticed a pattern and can determine h using the value of b. Is there a pattern? If there is a pattern, describe the pattern and show how the value of b can be used to determine the value of h.

4. Miranda believes there is another pattern. She thinks the value of h can be used to ﬁnd the value of c. Is there a pattern? If there is a pattern, describe the pattern and show how the value of h can be used to determine the value of c.

5. Miranda begins to wonder if b can be used to determine the value of c. Describe how b can be used to determine the value of c, and give one example to show how the value of b can be used to determine the value of c.

Factors of Polynomials | 183

Factors of Polynomials

Explore 3

6. Miranda thought she could use an array with the algebraic terms to show the square block diagram because drawing so many little squares and rectangles is tedious. Is the array an equivalent representation? Why or why not? Dimension Equation

Area Equation

A(x) = x + 14x x + 49 2

Square Block Pattern

Array

A(x) = (x x + 7)

7. Help Miranda complete the arrays and dimension equations in the table below. Area Equation

A(x) = x2 + 16x x + 64

A(x) = x2 – 8x x + 16

–4x

A(x) = x2 – 6x x+9

Array

Dimension Equation

184 | Factors of Polynomials

A(x) = _______

A(x) = (x x – 4)2

A(x) = _______

Factors of Polynomials

Explore 3 Reflect

1. Abuela gave Miranda 2 more orders. She only gave her the area expression. Help Miranda decide if these are going to make a perfect square block pattern, and write your answers in the table. Justify why or why not. x2 + 20x x + 100

x2 + 16 16x x+8

2. When is using a diagram to complete the square unreasonable? Explain.

3. How can you use the value of b to determine the value of c?

4. How are the given polynomial and the diagram related? What would need to happen to the polynomial before being able to complete the square using the equation? A(x) = 2x2 + 16x x + 32

Factors of Polynomials | 185

Completing the Square – Advanced

A(x) = (x x + 3)2 + 4

Dimension Equation

A(x) = x2 + 6x x+9+4

A(x) = x2 + 6x x + 13

Underline the term that represents the extra fabric squares.

Area Equation

Number of Extra Fabric Squares

Diagram of Square Block Pattern

Area Equation

Miranda’s Bag

Bag 1

Square Block Patterns

Factors of Polynomials | 187

Bag 2

Name: _______________________ Date: ___________

1. Use the Bags of Fabric Cards to complete the table and help Miranda.

Part I

Explore 4

Factors of Polynomials

188 | Factors of Polynomials

Dimension Equation

Array

Area Equation

A(x) = (x x + 12)2 + __

12x

x²

A(x) = x2 + 24x x + 146

–9x

A(x) = (x ___)2 + __

x²

A(x) = x2 – 18x x + 91

4. Miranda wants to use arrays for these patterns that have larger numbers. Help her complete the missing parts in the table.

3. How is the number of extra fabric squares related to the number after the parentheses that is added or subtracted in the dimension equation?

2. Miranda showed there were 4 extra fabric squares by underlining the + 4. How are the + 9 and + 4 in the area equation A(x) = x2 + 6x x + 9 + 4 related to the + 13 in the previous area equation of A(x) = x2 + 6x x + 13?

Explore 4

Factors of Polynomials

Dimension Equation

Diagram of Complete Square Block Pattern

A(x) = (x x + 4)2 – 6

A(x) = x2 + 8x x + 16 – 6

A(x) = x2 + 8x x + 10 + 6 – 6

6 small squares

How many small squares will Miranda take out of the supply closet?

Area Equation Showing the Number Taken Out of the Supply Closet

6 small squares

A(x) = x2 + 8x x + 10

How many small squares does the square block pattern need?

Area Equation

Diagram of Square Block Pattern

1. Complete the table to help Miranda.

Part II

Explore 4

Square Block Patterns

Factors of Polynomials| 189

Factors of Polynomials

190 | Factors of Polynomials

Dimension Equation

Array

Workspace

Area Equation

225

A(x) = (x x + 15)2 – __

15x

x²

–8x

A(x) = (x x ___)2 – __

x²

We have ____, so we have 34 less than what we need.

−8 · ___ = 64

15 · 15 = 225 We have 200, so we have 25 less than what we need.

A(x) = x2 – 16x x + 30

A(x) = x2 + 30x x + 200

4. Miranda wants to use arrays again for these patterns that have larger numbers. Help her complete the missing parts in the table.

3. How is the number of fabric squares taken from the supply closet related to the number after the parentheses that is added or subtracted in the dimension equation?

2. Miranda showed there were 6 fabric squares she needed from the supply closet by underlining the – 6. In the area equation A(x) = x2 + 8x x + 10 + 6 – 6, why did Miranda add 6 and subtract 6?

Explore 4

Factors of Polynomials

(x x + ___)2 – ___

(x2 + 8x x + ___) – ___

(x2 + 8x x + ___) + 10 – ___

(x2 + 8x) + 10

x2 + 8x x + 10

Algebraic Representation

How are the representations related?

Factors of Polynomials| 191

Model

5. Miranda wants to represent this pattern algebraically. Help her complete the missing parts of the table.

Explore 4

Factors of Polynomials

192 | Factors of Polynomials

2(x x + ___)2 – ___

2(x2 + 6x x + ___) – ___

2(x2 + 6x x + ___) + 7 – ___

2(x2 + 6x) + 7

(2x2 + 12x) + 7

2x2 + 12x x+7

Algebraic Representation

How are the representations related?

Model

6. Miranda wants to represent this pattern algebraically. Help her complete the missing parts of the table.

Explore 4

Factors of Polynomials

3. Two students were trying to complete the square. Who did it incorrectly? Explain their mistake.

= 2(x2 + 2x x+_)+7–_ = 2(x2 + 2x x + 1) – 1 + 7 = 2(x2 + 2x x + 1) + 6 = 2(x x + 1)2 + 6

= 2(x2 + 2x x+_)+7–_ = 2(x2 + 2x x + 1) – 2 + 7 = 2(x2 + 2x x + 1) + 5 = 2(x x + 1)2 + 5

Factors of Polynomials| 193

= 2(x2 + 2x) + 7

f x) = 2x2 + 4x f( x+7

Student B

= 2(x2 + 2x) + 7

f x) = 2x2 + 4x f( x+7

Student A

2. How do you determine what number needs to be added to complete the square for x2 + bx? When you add a number, do you always have to subtract the same number? Why or why not?

1. Orders can be written in either standard form, x2 + bx + c, or vertex form, (x x + h)2 + k. Describe how c and h can be used to determine k, the number of blocks that will be left over.

Reflect

Explore 4

Factors of Polynomials

Explore 5

Name: _______________________ Date: ___________

Quadratic Equations – Key Features Part I 1. Taio and Myra started by graphing f( f x) = 2x2 – 3 and g(x) = −(x x – 1)2 + 4 on their graphing calculators. Myra noticed that one parabola opened upward and one opened downward. Explain to Taio and Myra what caused this, and give examples to illustrate your explanation. Include how you know from an equation whether a parabola has a maximum or a minimum. Y1 = -(X - 1)2 + 4

Y1 = 2X2 - 3

X=0

Y = -3

X=0

Y=3

Factors of Polynomials | 195

Factors of Polynomials

Explore 5

2. Next, Taio graphed m(x) = −(x x – 1)2 + 4 and h(x) = −(x x + 3)2 + 4 on the same screen and noticed that the graphs were very similar. Explain to Taio and Myra how the values in the equation connect to the graph. y

(–3, 4)

5 4

(1, 4)

3 2 1 –5 –4 –3 –2 –1 0

3. The next equation Taio and Myra review is j(x) = (x x – 2)(x – 8). Before they use their calculators to see a graph of this equation, what points do you already know will be on the graph?

4. Taio and Myra realized that they could quickly ﬁnd the vertices or x-intercepts from the functions m(x), h(x), and j(x). They wondered what strategies they could use to ﬁnd this point from an equation in a different form, such as k(x) = x2 – 6x x + 8. Explain the strategies.

196 | Factors of Polynomials

Factors of Polynomials

Explore 5 Part II

Use the questions to help you organize the Note Cards and help Taio and Myra ﬁnd a pattern so they don’t fail their test! 1. Circle the y-intercept on the graph. 2. Find the cards that show this y-intercept as an ordered pair and in the equation written in standard form. y-intercept

Standard Form

Graph A y

1 –3 –2 –1 0 –1

–2 –3 –4 –5 –6 –7

3. How does the equation in standard form show the y-intercept?

–8 –9

4. Complete the table by factoring the quadratic function and determining the zeros. Workspace

Factored Form

Zeros

5. How are the factors related to the zeros?

6. How can the zeros be used to determine the x value of the vertex?

Factors of Polynomials | 197

Factors of Polynomials

Explore 5 7. Circle the vertex on the graph. 8. Find the cards that show this vertex and the vertex of the function in an equation in vertex form, and attach the cards in the appropriate columns below. Vertex Form

Graph B y

8 7 6

Vertex

5 4 3 2 1

9. Find the card that shows the axis of symmetry for this function, and attach it below.

–5 –4 –3 –2 –1 0

Axis of Symmetry

10. How can the vertex be used to determine the axis of symmetry?

11. Find the card that rewrites this function in standard form, and attach it below. Standard Form

198 | Factors of Polynomials

Factors of Polynomials

Explore 5

12. Help Taio and Myra ﬁnd a pattern so they don’t fail their test! Find the remaining Note Card with an equation in standard form, and attach it below. Standard Form

13. Complete the table by completing the square, and ﬁnd the card with the equivalent equation written in vertex form. Attach the appropriate card in the Vertex Form column below. Workspace

Vertex Form

Compare and contrast the equations in standard form and vertex form.

14. Complete the table by determining the axis of symmetry and attaching the appropriate card in the Axis of Symmetry column below. Explain how the vertex can be used to determine the axis of symmetry. Explanation

Axis of Symmetry

Factors of Polynomials | 199

Explore 5

Factors of Polynomials

Reflect 1. How is the y-intercept on the graph related to the equation y = ax2 + bx + c?

2. How is the vertex on the graph related to the equation y = (x – h)2 + k?

3. How are the roots on the graph related to the equation y = (x – p)(x x – q)?

4. How can the vertex, roots, y-intercept, and axis of symmetry be determined from a quadratic equation in standard form?

200 | Factors of Polynomials

Solve Quadratics

201

Solve Quadratics by Taking Square Roots

Name: _______________________ Date: ___________

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5. Do both values represent a reasonable side length for the print? Explain.

4. Give 2 values that satisfy the equation from question 1.

3. Calculate all of the values for s that make the equation true.

Use the questions provided to guide you through the mathematical calculations necessary to complete the art assignments. 1. Write an equation to represent the area of the print. The area of a square can be found using the formula A = s2, where s is side length. Assignment 1 Take photos of interesting patterns in objects either made by people or 2. What is the ﬁrst step to solving this equation? Explain how found in nature. Use a square image that “undoes” the second power. format. Adjust the camera settings so the image detail is clear when printed with an area of 121 square inches.

Part I

Explore 1

Solve Quadratics

204 | Solve Quadratics

Assignment 2 Convert one of your square photographs into the background for a square poster advertisement. Extend the length and width of your photograph by 22 inches. The area of the poster should be 900 square inches.

Explore 1

11. What is a reasonable side length of the original photograph? Explain.

10. Calculate all of the values for x that make the equation true.

9. Why is the order important when solving?

8. What is the ﬁrst step to solving this equation? What is the second step?

7. Write an equation to represent the area of the poster. The area of a square can be found using the formula A = s2, where s is side length.

6. Write an expression to represent the side length of the poster. Let x represent the side of the photograph measured in inches.

Solve Quadratics

1. What are the maximum dimensions of the art installation Mateo can create for the show?

Art Installation

Buffer of empty space

Exhibition guideline: Each participant will be allocated a square of wall space that is 20 square feet, but there must be a buffer of empty space surrounding the square art installation that is at least 20% of the total allocated space.

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2. If one pint of paint covers 54 square feet and Mateo shares the paint with two other students equally, what are the dimensions of the largest colored background he can use?

Exhibition guideline: Each participant may set a colored background to their work by painting large sheets of paper and pinning them to the wall behind their photos, ﬁlling as much of the 20 square feet as they wish.

Read each guideline, and answer the question that follows. Leave answers in simplest radical form if necessary.

Part II

Explore 1

Solve Quadratics

206 | Solve Quadratics

3. Could the method used to solve the problems in this Explore be used to solve 4x2 + 3x x = 0?

2. When you take the square root, how many solutions to the original equation do you get, and how do you decide if they are reasonable?

1. What method or principle are you using to solve for the unknown value?

Reflect

Explore 1

Solve Quadratics

Explore 2

Name: _______________________ Date: ___________

Solve Quadratics by Completing the Square Part I 1. To prepare for the pumpkin launch, Alexander and Maria Jose were discussing how to solve quadratic equations using square roots. Alexander was pretty certain he knew the 3 steps needed, but he wasn’t sure in what order to do those steps. Help Alexander by writing the 3 steps in the correct order in the table. Alexander’s mixed-up steps are as follows: • Solve for both solutions. • Isolate the term with the square on one side of the equation. • Take the positive and negative square roots of both sides.

Steps 1. 2. 3. 2. Use the ordered steps to show Alexander and Maria Jose how to solve the equation x2 – 9 = 0. Solve x2 – 9 = 0.

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Solve Quadratics

Explore 2

3. Identify the equation and measured distance for each team on the Pumpkin Launch Notebooks, and write them in the space provided. Team 1 Equation: Measured distance:

Team 2 Equation: Measured distance:

4. In the tables below, complete the steps to solve the equations from the Pumpkin Launch Notebooks that represent the distance each catapult is expected to launch the pumpkin. Round answers to the nearest tenth if needed. Team 1

Team 2

x≈

5. Are both solutions for team 1 reasonable projections? Why or why not?

6. Are both solutions for team 2 reasonable projections? Why or why not?

7. How far was each team from their projection? Which team was closest to their projection and won this round?

208 | Solve Quadratics

Solve Quadratics

Explore 2 Part II 1. Identify the equation for each team on the Pumpkin Launch Notebooks, and write it in the space provided. Equation:

Team 1

Equation:

Team 2

2. Maria Jose suggested they complete the square to rewrite each team’s equation in vertex form. She thinks the equation will be easier to solve if it is rewritten. Team 1 Equation in vertex form:

Team 2 Equation in vertex form:

3. Use the vertex form of each equation to solve and determine the expected distance each team will launch their pumpkin. Write the steps to solve in the tables below. Team 1

Team 2

4. Are both solutions for each team reasonable projections? Why or why not?

5. How far was each team from their projection? Which team was closest to their projection and won this round?

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Explore 2

Solve Quadratics

Reflect 1. When calculating the solution to a quadratic equation like x2 = 4, how many solutions are there? Explain.

2. When solving a quadratic of the form x2 = 0, how many solutions are there? Explain.

3. If you were looking at the quadratic x2 = −4, how many real solutions can you ﬁnd? Explain.

4. If you have a quadratic equation in vertex form, (x x – h)2 + k = 0, what criteria do you think need to be met for there to be 0, 1, or 2 solutions?

5. What is the vertex of the equation y = (x – 3)2 – 4? Set the y value equal to 0, and ﬁnd the x-intercepts. Explain why these x-intercepts make sense given your vertex.

210 | Solve Quadratics

Solve Quadratics

Explore 3

Name: _______________________ Date: ___________

Solve Quadratics by Factoring Part I 1. Determine an expression for the length and width of the pool in terms of w, and write it on the diagram. Then, determine an expression for the area of the pool in terms of w, and complete the table.

Width

Length

In Sq. Yards

Area of the Pool 24

In Terms of w

2. Write an equation for the area of the pool in terms of w.

3. Rewrite your equation so it is in standard form.

4. Factor the equation.

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Solve Quadratics

Explore 3

5. Use the zero product rule outlined below to answer the following questions. If ab = 0, then either a = 0 or b = 0. a. How does this illustrate the zero product rule?

0 0=0 0 1=0 0 9=0 0 58 = 0 0 2,352 = 0

b. If 3b = 0, then what is the value of b? How do you know?

c. If 3(b + 5) = 0, then what is the value of (b + 5)? How do you know?

d. If a(b + 5) = 0, then what must be true of either a or (b + 5)?

6. If (w w + 6)(w – 4) = 0, then what must be true of w, and how do you know?

7. Are both of these solutions a reasonable width for the pool? Explain.

8. What are the dimensions of the pool?

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Solve Quadratics

Explore 3 Part II

1. Complete the tables below, and then answer the questions to determine how wide the pool deck needs to be.

6 yd.

4 yd.

x Pool

Pool and Deck in Terms of x

Length

6 + 5x

Width

Area

(6 + 5x)(

)

Area in Square Yards of Pool Plus Deck 2. What is the equation for the area of the pool and deck in terms of x?

3. Simplify the equation by putting it in standard form.

4. What is the factored form of the equation?

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Solve Quadratics

Explore 3

5. If 2(5x x + 21)(x – 1) = 0, then what must be true of either (5x + 21) or (x – 1)?

6. Use the zero product rule to solve 2(5x x + 21)(x – 1) = 0. Give the equations that will be used to solve, and then solve the equations.

7. Are both of these solutions a reasonable width for the pool decking? Explain.

8. How wide should the pool decking be?

9. What are the total dimensions of the pool and decking in yards?

214 | Solve Quadratics

Solve Quadratics

Explore 3 Reflect

1. Do you think there is any limit to the number of factors that can be used when applying the zero product rule?

2. What would be the solutions to the factored equation (x x + 2 )(x x – 3 ) = 0?

3. Why is the equation 2x(x x – 1) + 3(x – 1) = 0 not in the proper form to use the zero product rule? What would you need to do in order to use the zero product rule?

4. Why are the solutions to (x x + 2)(x – 5) = 7 not 5 and −2?

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Solve Quadratics

Explore 4

Name: _______________________ Date: ___________

Using the Quadratic Formula Part I: Analysis and Development Phases

Developer’s Notebook

* The 󰇧󰇴󰇵 n󰈥e󰈧󰈤 t󰈢 󰇷󰈣󰇱ve AN󰈘 qu󰇧󰈧󰈦󰈜ti󰈛! * Notes to self:

Standard form of a quadratic function: y = ax2 + bx + c Standard form of a quadratic equation: ax2 + bx + c = 0

1. The ﬁrst thought for a way to solve any quadratic was to graph. The team was trying to solve the equation 0 = x2 – 4x x + 3, so they 2 graphed the function g(x) = x – 4x x + 3. Based on the graph, what are the solutions to the equation 0 = x2 – 4x x + 3?

2 g(x)

–2

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Solve Quadratics

Explore 4

The development team wants to see which algebraic tools are at their disposal since they will not always be able to graph to solve their quadratics. Two teams focus on different aspects of solving quadratic equations in the form ax2 + bx + c = 0. Team A

Team B

Factoring

Completing the square

2. You are trying to solve the equation 0 = x2 + 6x x + 8. Use team A’s strategy.

3. Explain why team A’s strategy would not help you solve the similar equation of 0 = x2 + 6x x + 7.

4. Use team B’s strategy to solve the equation 0 = x2 + 6x x + 7.

5. What makes both teams’ strategies difﬁcult and not efﬁcient when solving the equation 0 = 2x2 + 5x x + 1?

218 | Solve Quadratics

Solve Quadratics

Explore 4 Part II: Test Phase Use the new app to ﬁnd the zeros of the quadratic functions in the table.

Ultimately, after much debate, the teams come together and develop a new method they believe will work every time. Your job is to test their solution equation below.

INPUT a, b, c

Solver:

−b ± √b2 − 4ac 2a

OUTPUT x = 1. Complete the table by solving the quadratic equations using the new app. Begin by ﬁrst identifying the inputs. Solutions should be written in simplest radical form. Quadratic Equation

Inputs a

0 = 1x2 + 6x x+8

Solver

–6 + √6 – 4(1)(8) 2(1) 2

Outputs

–6 + 2 2

x = −4 or −2

a 0 = x2 + 6x x+7

b c a

0 = 2x2 + 5x x+1

b c

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Solve Quadratics

Explore 4

2. The teams verified that their new app’s first two solutions matched the solutions from their initial attempts to factor and complete the square. Verify the solutions to the final quadratic equation by graphing or substitution, and complete the table below.

Quadratic Equation

Solutions

Verify by substitution or graphing.

0 = 2x2 + 3x x–5

3. What is the beneﬁt of the app compared to the factoring method?

4. What is the beneﬁt of the app compared to the completing the square method?

5. The team tried to solve the equation 0 = x2 + 2x x + 5 and got x =

–2 + √–16 , 2

which led them to say there are no solutions. Did this equation break the app?

220 | Solve Quadratics

Explore 4

Solve Quadratics

Reflect 1. What would be the ﬁrst step to solving this quadratic equation using the quadratic formula? (x x – 3)2 + 4x x–3=0

2. What information do we need from the standard form equation to use the quadratic formula?

3. For the quadratic equation x2 + 2x x + 1 = 0, the app only gives one solution, x = −1. Is it correct or malfunctioning? Check by graphing or factoring.

4. What is the maximum number of solutions for a quadratic equation, and how do you know that from looking at the quadratic formula?

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Solve Quadratics

Explore 5

Name: _______________________ Date: ___________

Choosing the Best Method Part I Given the variety of methods for solving quadratic equations that you have available in your toolbox, compare different pairs of quadratic equations, and determine what type of equation is best solved by a particular method.

Graphing Factoring Taking the square root completing the square Quadratic Formula

Equation A

Equation B

x2 – 2x x – 15 = 0

(x x – 1)2 = 20

1. Which of the two quadratic equations above could be easily solved by taking the square root? Why?

2. If you were to solve that same problem using the quadratic formula or by completing the square, what would you ﬁrst need to do to the equation?

Equation C

Equation D

x2 + 6x x – 25 = 0

2x2 – 5x x–4=0

3. Now consider completing the square on the equations above. Which equation would be easier to solve using that method? Explain or demonstrate.

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Solve Quadratics

Explore 5 Equation E

Equation F

x2 – 2x x – 15 = 0

x2 + 6x x – 25 = 0

4. Which of the two quadratic equations above would be easier to factor? Why?

5. Write that equation in factored form, and give the real solutions.

6. Try solving the same problem using the quadratic formula: x =

–b + √b2 – 4ac 2a

7. Which method was more efﬁcient in this case?

Equation G

Equation H

6.02x2 – 5.99x x – 0.4 = 0

2x2 – 5x x–4=0

8. Could you use the quadratic formula on both of the equations above? 9. Which would be easier to evaluate by hand? Why?

10. What other method remaining in your toolbox could you use? 11. Does graphing always ﬁnd the exact real solutions of a quadratic equation, or are there limitations? Explain.

224 | Solve Quadratics

Solve Quadratics

Explore 5 Part II

Create a set of quadratic equation examples that the new hire can use as a reference. For each example problem, place a check mark by the most efﬁcient method to use. Justify your choice at the bottom of each table. x2 + 5x x – 14 = 0 Graph

Factor

Taking the Square Root

Completing the Square

Quadratic Formula

Completing the Square

Quadratic Formula

Completing the Square

Quadratic Formula

Completing the Square

Quadratic Formula

Justiﬁcation:

(x x – 3)2 – 25 = 0 Graph

Factor

Taking the Square Root

Justiﬁcation:

−4 2 + 5x −4x x+7=0 Graph

Factor

Taking the Square Root

Justiﬁcation:

x2 – 10x x=5 Graph

Factor

Taking the Square Root

Justiﬁcation:

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Explore 5

Solve Quadratics

Reflect 1. Can an equation have different solutions depending on the method you use to solve it?

2. What methods would you consider ﬁrst to solve a quadratic equation? Why?

3. Does using the quadratic formula or completing the square work to solve any quadratic equation?

4. How can you check your solutions?

226 | Solve Quadratics

Transform Quadratic Functions

227

Transform Quadratic Functions

Explore 1

Name: _______________________ Date: ___________

Translations of Quadratics Part I: Ground Squirrels 1. The basic path is modeled by f( f x) = x2. You are given three equations to choose from. Use technology to graph the three functions to determine which path will get the squirrel to his acorn.

function selection

LEVEL 1

x² −6 option 1

y 16

x² − 4 option 2

15 14 13 12

x² −2 option 3

11 10 9 8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3

2. How does subtracting 2 from the function change each ordered pair of the basic path?

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Transform Quadratic Functions

Explore 1

3. The basic path is modeled by f( f x) = x2. You are given three equations to choose from. Use technology to graph the three functions to determine which path will get the squirrel to his acorn.

function selection

LEVEL 1

x² + 6 option 1

y 16

x² + 4 option 2

15 14 13 12

x² +2 option 3

11 10 9 8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3

4. Describe how k changes the function when f( f x) = x2 is transformed to g(x) = x2 + k. Include an explanation for positive k and negative k values.

230 | Transform Quadratic Functions

Transform Quadratic Functions

Explore 1

5. The basic path is modeled by f( f x) = x2. You are given three equations to choose from. Use technology to graph the three functions to determine which path will get the squirrel to his acorn.

function selection

LEVEL 1

(x − 1)²

option 1

(x − 2)²

15 14 13

option 2

(x − 3)²

11 10 9

option 3

8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3

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Transform Quadratic Functions

Explore 1

6. The basic path is modeled by f( f x) = x2. You are given three equations to choose from at the beginning of level 1. Use technology to graph the three functions to determine which path will get the squirrel to his acorn.

function selection

LEVEL 1

(x + 1)²

option 1

(x + 2)²

15 14 13

option 2

(x + 3)²

11 10 9

option 3

8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3

7. Describe how h changes the function when f( f x) = x2 is transformed to g(x) = (x x – h)2. Include an explanation for positive h and negative h values.

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Transform Quadratic Functions

Explore 1 Hidden Level 1.1 8. The paths for squirrel A and squirrel B are described as transformations of the quadratic parent function, f( f x). Describe the transformations of each squirrel’s path.

LEVEL 1.1 y

Squirrel A

Squirrel B

16 15 14

f x) – 5 f(

f x) + 5 f(

12 11 10 9 8 7 6 5 4 3

9. If the acorn is located at a y-intercept of 5, which squirrel has a path that would get the acorn?

2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3

10. Explain how the y-intercept of the parent function compares to the y-intercept of the path taken by the squirrel that gets the acorn.

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Transform Quadratic Functions

Explore 1 Part II: Level Up

1. The basic path is modeled by f( f x) = x2. Your hungry squirrel can only reach the acorn using horizontal and vertical shifts. Which action should you take? A. Perform a vertical shift 3 units down and a horizontal shift 2 units to the left. B. Perform a vertical shift 4 units up and a horizontal shift 3 units to the right.

y 16 15 14

C. Perform a vertical shift 4 units down and a horizontal shift 3 units to the left.

13 12 11 10 9 8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3

2. Your squirrel needs an equation for the correct path in the form of g(x) = (x x – h)2 + k. Which equation should you use? A. g(x) = (x x + 3)2 – 4 B. g(x) = (x x – 3)2 + 4 C. g(x) = (x x + 3)2 + 4 D. g(x) = (x x – 3)2 – 4

234 | Transform Quadratic Functions

Transform Quadratic Functions

Explore 1

3. The basic path is modeled by f( f x) = x2. Your hungry squirrel can only reach the acorn using horizontal and vertical shifts. Which action should you take? A. Perform a vertical shift 3 units down and a horizontal shift 2 units to the left. B. Perform a vertical shift 2 units up and a horizontal shift 3 units to the right.

y 16 15 14

C. Perform a vertical shift 2 units down and a horizontal shift 3 units to the left.

13 12 11 10 9 8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3

4. Your squirrel needs an equation for the correct path in the form of g(x) = (x x – h)2 + k. Which equation should you use? A. g(x) = (x x + 3)2 – 2 B. g(x) = (x x – 3)2 + 2 C. g(x) = (x x + 3)2 + 2 D. g(x) = (x x – 3)2 – 2

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Transform Quadratic Functions

Explore 1

5. The basic path is modeled by f( f x) = (x x + 3)2 – 2. Your hungry squirrel can only reach the acorn using horizontal and vertical shifts. Which action should you take? A. Perform a vertical shift 2 units up and a horizontal shift 2 units to the left. B. Perform a vertical shift 2 units up and a horizontal shift 2 units to the right.

y 16 15 14

C. Perform a vertical shift 2 units down and a horizontal shift 2 units to the right.

13 12 11 10 9 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

0 -1

-2 -3

6. Your squirrel needs an equation for the correct path in the form of g(x) = (x x – h)2 + k. Which equation should you use? A. g(x) = (x x + 2)2 + 2 B. g(x) = (x x – 2)2 + 2 C. g(x) = (x x + 1)2 + 0 D. g(x) = (x x – 1)2 + 0

236 | Transform Quadratic Functions

Transform Quadratic Functions

Explore 1

7. The basic path is modeled by f( f x) = (x x – 5)2 + 4. Your hungry squirrel can only reach the acorn using horizontal and vertical shifts. Which action should you take? A. Perform a vertical shift 3 units up and a horizontal shift 4 units to the left. B. Perform a vertical shift 3 units up and a horizontal shift 4 units to the right.

y 20 19 18

C. Perform a vertical shift 3 units down and a horizontal shift 4 units to the right.

17 16 15 14 13 12 11 10

8. Your squirrel needs an equation for the correct path. Which equation should you use?

9 8 7 6 5 4

A. g(x) = (x x + 1) + 7 2

B. g(x) = (x x – 4)2 + 3

3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

0 -1

C. g(x) = (x x + 4)2 + 3 D. g(x) = (x x – 1)2 + 7 9. This is the ﬁnal challenge! Master level 2 by writing g(x) as a function of f( f x). Use the transformations to write g(x) in the form of g(x) = f( f x – h) + k, where h represents horizontal translation from f( f x), and k represents the vertical translation from f( f x). Which equation is correct? A. g(x) = f( f x + 4) + 3 B. g(x) = f( f x – 4) + 3 C. g(x) = f( f x – 4) – 3

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Transform Quadratic Functions

Explore 1 Hidden Level 2.1 10. The paths for squirrel A and squirrel B are transformations of the quadratic parent function, f( f x). Identify the vertex of each path.

LEVEL 2.1

Squirrel A

y 16

x y

1 12

2 5

5 −4

8 5

14 13 12

11 10 9 8

Squirrel B

7 6 5

g(x) = f( f x + 3) + 7

4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0

-1

-2 -3

11. Describe the transformation from the parent function, and compare the x-intercepts of each function.

12. Which squirrel has a path that would get the acorn?

238 | Transform Quadratic Functions

Explore 1

Transform Quadratic Functions

Reflect 1. Describe what the letters h and k do for the function f( f x) = (x x – h)2 + k when h and k are positive.

2. When performing a translation involving h and k values, does it matter in which order you move the vertex?

3. Write a new equation that will move the parent function to the left and up.

4. Write a new equation that will move the parent function to the right and down.

5. Explain how you can identify the vertex on a table, a graph, and an equation. Table: Graph: Equation:

Transform Quadratic Functions | 239

Transform Quadratic Functions

Explore 2

Name: _______________________ Date: ___________

Dilations of Quadratics Part I: Ground Squirrels and Flying Squirrels 1. The basic path is modeled by f( f x) = x2. You are given equations to choose from. Use technology to graph the functions to determine which path will get the squirrel to his acorn.

function selection

LEVEL 3

2x² option 1

y 16

3x² option 2

15 14 13 12

5x² option 3

11 10 9 8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3

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Transform Quadratic Functions

Explore 2

2. The basic path is modeled by f( f x) = x2. You are given equations to choose from. Use technology to graph the functions to determine which path will get the squirrel to his acorn.

function selection

LEVEL 3

1 2 x² option 1

y 16

1 3 x² option 2

1 5 x² option 3

14 13 12 10 9 8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0

-1

-2 -3

3. Describe how a changes the function when f( f x) = x2 is transformed to g(x) = ax2. Include descriptions for numbers greater than 1 and less than 1.

242 | Transform Quadratic Functions

Transform Quadratic Functions

Explore 2

4. The basic path is modeled by f( f x) = x2. You are given equations to choose from. Use technology to graph the functions to determine which path will get the squirrel to his acorn.

function selection (12 x)²

LEVEL 3 y

option 1

(2x)²

15 14 13

option 2

(3x)²

11 10 9

option 3

8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3

5. Describe how b changes the function when f( f x) = x2 is transformed to g(x) = (bx)2. Explain using the horizontal changes.

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Transform Quadratic Functions

Explore 2

6. The ﬂying squirrels take a different path. Instead of using the function y = x2, they follow the path of y = −x2. Graph the new path below to determine if the squirrel will reach the acorn. y

LEVEL 3 15 14 13

12 11 10 9 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

0 -1

-2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12

7. Describe how a negative coefﬁcient changes the function when f( f x) = x2 is transformed to g(x) = −x2.

244 | Transform Quadratic Functions

Transform Quadratic Functions

Explore 2 Hidden Level 3.1 8. The paths for squirrel A and squirrel B are transformations of the quadratic parent function, f( f x). Which squirrel has a path that would get the acorn? Squirrel A

LEVEL 3.1 y

Squirrel B

16 15

The quadratic function, g(x), passing through the points (−2, −8) (0, 0) (2, −8)

14 13 12 11 10

h(x) = −0.5x2

9 8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0

-1

-2 -3

9. Describe the transformation from the parent function for each path.

10. To get to level 4, select all of the true statements. The functions g(x) and h(x) have the same maximum value because the vertex doesn’t shift when dilated about the origin. For all values of x, g(x) < h(x). For all nonzero values of x, g(x) > h(x). g(x) = h(x) when x = 0. The function g(x) has 2 roots, and the function h(x) has no real roots.

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Transform Quadratic Functions

Explore 2 Part II: Level Up

1. The basic path is modeled by f( f x) = x2. Your hungry squirrel can only reach the acorn using dilations and/or reﬂections. Which action should you take? A. Perform a reﬂection over the y-axis and a vertical stretch by a factor of 2.

Leve l 4 15 14 13

12 11

B. Perform a reﬂection over the x-axis and a vertical stretch by a factor of 4.

10 9 8 7 6 5

C. Perform a reﬂection over the x-axis and a vertical stretch by a factor of 6.

4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

0 -1

-2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12

2. Your squirrel needs an equation for the correct path in the form of g(x) = a · f( f x). Which equation should you use? A. g(x) = −2 · f( f x) B. g(x) = 4 · f( f x) C. g(x) = −4 · f( f x) D. g(x) = 6 · f( f x) 3. Write an equation in the form of g(x) = ax2 to represent the squirrel’s path.

246 | Transform Quadratic Functions

Explore 2

Transform Quadratic Functions

4. The basic path is modeled by f( f x) = 2x2. Your hungry squirrel can only reach the acorn using dilations and/or reflections. Which action should you take? A. Perform a reflection over the x-axis and a vertical compression by a factor of 0.25. B. Perform a reflection over the y-axis and a vertical compression by a factor of 0.5.

Leve l 4 1515 1414 1313

1212 1111 1010 9 9 8 8 7 7 6 6

C. Perform a reﬂection over the x-axis and a vertical compression by a factor of 1 . 2

5 5 4 4 3 3 2 2 1 1 0 0 -10-10-9 -9-8 -8-7 -7-6 -6-5 -5-4 -4-3 -3-2 -2-1 -1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 91010 -1 -1

-2 -2 -3 -3 -4 -4 -5 -5 -6 -6 -7 -7 -8 -8 -9 -9 -10-10 -11-11 -12-12

5. Your squirrel needs an equation for the correct path in the form of g(x) = a · f( f x). Which equation should you use? A. g(x) = −0.5 · f( f x) B. g(x) = 0.5 · f( f x) C. g(x) = −0.25 · f( f x) 6. Write an equation in the form of g(x) = ax2 to represent the squirrel’s path.

Transform Quadratic Functions | 247

Transform Quadratic Functions

Explore 2

7. The basic path is modeled by f( f x) = −5x2. Your hungry squirrel can only reach the acorn using dilations and/or reﬂections. Describe the transformation.

Leve l 4 y 11 10 9 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

0 1

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

8. Your squirrel needs an equation for the correct path in the form of g(x) = a · f( f x). Write the equation.

9. Write an equation in the form of g(x) = ax2 to represent the squirrel’s path.

248 | Transform Quadratic Functions

Transform Quadratic Functions

Explore 2 Hidden Level 4.1 10. The paths for squirrel A and squirrel B are transformations of the quadratic parent function, f( f x). Which squirrel has a path that would get the acorn? Squirrel A

LEVEL 4.1 y

Squirrel B

16 15 14

x −2

g(x)

12 11 10

−9

9 8

7 6

h(x) = −(x x – 1)2 + 6

5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

−9

-2 -3

Which statement is true? Select all that apply. g(−2) is less than h(−2). g(0) is less than h(0). g(0) is equal to h(0). g(2) is less than h(2). g(4) is greater than h(4).

Transform Quadratic Functions | 249

Explore 2

Transform Quadratic Functions

Reflect 1. Describe how the coefﬁcients a and b transform the function f( f x) = a(bx)2. Explain positives and negatives as well as numbers larger and smaller than 1.

2. Describe the transformation to a quadratic function if a is exactly one. Justify your position using a property of multiplication such as the zero property, the identity property, or the associative property.

3. Describe the transformation to a quadratic function if a is exactly zero. Justify your position using a property of multiplication such as the zero property, the identity property, or the associative property.

250 | Transform Quadratic Functions

Transform Quadratic Functions

Explore 3

Name: _______________________ Date: ___________

Transformations of Quadratics Part I: Choose a Squirrel 1. The basic path is modeled by f( f x) = x2. You are given equations to choose from. Use technology to graph the functions to determine which path will get the squirrel to both acorns. The orange dot must pass through one acorn. y

LEVEL 5 15

function selection

14 13

12 11

−(x − 2)² + 1

10 9

option 1

8 7

x² + 5

6 5

option 2

4 3

−x² + 5 option 3

2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

0 -1

-2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12

2. Describe the transformations from f to g.

3. There is another function, h(x), that uses translations and will result in the parabola going through the points (0, 5) and (2, 1). Write the function, and describe the translations.

Transform Quadratic Functions | 251

Transform Quadratic Functions

Explore 3

4. From the basic function f( f x) = x2, write a function that can be used to get the two acorns. The orange dot must pass through one acorn. Give the transformed function, g(x), as a function of f( f x).

LEVEL 5 15 14 13

12 11 10 9 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

0 -1

-2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12

252 | Transform Quadratic Functions

Transform Quadratic Functions

Explore 3 Part II: Level Up

1. The basic path is modeled by f( f x) = 2(x x + 3)2 – 4. Write a function that can be used to get the two acorns. The orange dot must pass through one acorn. Give the transformed function, g(x), as a function of f( f x).

LEVEL 6 y 15 14 13 12 11 10 9 8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3 -4

Transform Quadratic Functions | 253

Transform Quadratic Functions

Explore 3

2. The basic path is modeled by f( f x) = −(x x – 2)2 + 3. Write a function that can be used to get the two acorns. The orange dot must pass through one acorn. Give the transformed function, g(x), as a function of f( f x).

LEVEL 6 15 14 13

12 11 10 9 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

0 -1

-2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12

254 | Transform Quadratic Functions

Transform Quadratic Functions

Explore 3

3. From the basic function f( f x) = x2, write a function that can be used to get the two acorns. The orange dot must pass through one acorn. Give the transformed function, g(x), as a function of f( f x).

LEVEL 6 y 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

1 0 -1

-2 -3

Transform Quadratic Functions | 255

Transform Quadratic Functions

Explore 3 Reflect

1. Write your own quadratic function with a vertical stretch, a shift to the right, and a shift down from the parent function.

2. Write your own quadratic function that contains a reﬂection over the x-axis, a shift to the left, and a shift up from the parent function.

3. Draw a graph that reﬂects over the x-axis, shifts to the right, and shifts up from the parent function.

256 | Transform Quadratic Functions

Exponential Functions

257

Attributes of Exponential Functions

Name: _______________________ Date: ___________

Exponential Functions | 259

4. Look at the x-axis on each graph. Do both functions have an x-intercept? If not, describe the graph of the function as it relates to the x-axis. Is it getting closer, getting farther, touching, not touching, crossing, or curving back upward?

3. What does the y-intercept represent in this situation?

2. What is the y-intercept for each microbe population? Write it as an ordered pair.

1. Which microbe seems to be growing faster? How do you know this?

Use the Microbe Games Cards to review the information you have collected about your ﬁrst two contestants, microbe A and microbe B, and answer the following questions.

Part I

Explore 1

Exponential Functions

1 2 3

260 | Exponential Functions

−2

1 16

−1

4−2

−2

c(x)

−1

c(x) = 4x

Microbe C

5(−2) + 4

d(x) = 5x x+4

Microbe D

3. Circle the y-intercept on the tables and graph.

−6

d(x)

2. Graph each function on the graph, and connect the points.

1. Complete the tables of values for these functions.

Part II

Explore 1

-4

-2

Exponential Functions

Microbe D d(x) = 5x x+4

Microbe C c(x) = 4x

Exponential Functions | 261

8. Considering the function that represents the growth of the function, why do you think one microbe is growing faster than the other?

7. Which microbe is increasing most rapidly, microbe C or microbe D?

6. What is happening to c(x) and d(x) as x approaches negative inﬁnity on the graph?

5. What is happening to c(x) and d(x) as x approaches inﬁnity on the graph?

4. What is the y-intercept of each function?

Explore 1

Exponential Functions

262 | Exponential Functions

5. What can we conclude about the growth of a function when we see a variable being used as an exponent?

4. Did these exponential functions have x-intercepts? Why or why not?

3. Why did microbe D not grow as quickly as microbe C, even though the numbers in the equation were just as large?

2. Considering the mathematical representations, why did this microbe increase most rapidly and win?

1. Which microbe is increasing most rapidly and would be crowned the winner of the Microbe Games? Explain how you know.

Reflect

Explore 1

Exponential Functions

Explore 2

Name: _______________________ Date: ___________

Domain and Range Part I Students spray their desks with an antibacterial solution at the end of each day. The average desk has 100 microbes before the solution is sprayed, and the solution kills 70% of the microbes every minute. y

Number of bacteria

100

f(x) = 100(0.3)x f(x f( f(x)

x 2

Minutes since spray

1. Describe the graph as time elapses.

2. Does the graph appear to ever reach exactly 0 or negative values? Find ff(15) to check your hypothesis.

3. If the domain is restricted to the ﬁrst three minutes (0 ≤ x ≤ 3), what is the range of this graph? What about when the domain is all values greater than or equal to 0 (0 ≤ x)?

Exponential Functions | 263

Exponential Functions

Explore 2 Part II

You have strategically collected samples from four different locations, and now you must review your data and establish a reasonable domain and range to be able to complete your lab report. • Determine whether the bacteria represents a growth function or a decay function. • Establish a reasonable domain and a reasonable range for each scenario.

Location

Growth or Decay

Domain

Range

Restroom sink Library computer keyboard Lunch table Hallway water fountain 1. What did you have to take into consideration to determine a reasonable domain and a reasonable range?

2. Would you use negative input or output values in any of these scenarios? Explain.

264 | Exponential Functions

Explore 2

Exponential Functions

Reflect 1. How can you tell where an exponential graph has a horizontal asymptote?

2. When is it beneﬁcial to look at unrestricted domains?

3. When is it beneﬁcial to look at restricted domains?

4. Does the initial value or constant multiplier have any role in determining the restrictions on the domain and range of an exponential function?

Exponential Functions | 265

Initial Value and Constant Multiplier

Name: _______________________ Date: ___________

Elk

Wolves

25 – 20 = 5 31 – 25 = 6 39 – 31 = 8

Difference of Change for Each Output

Yes No

Constant Rate of Change?

9,000 ÷ 12,000 = 0.75 6,750 ÷ 9,000 = 0.75 5,063 ÷ 6,750 ≈ 0.75

Factor of Change for Each Output

Exponential Functions | 267

Yes No

Constant Factor of Change?

2. The tables from the Ranger’s Notebook Cards show equal units of growth in the input columns. Determine the difference of change and if it is a constant rate of change for each output. Determine the factor of change and if it is a constant factor of change for each output. Complete the table below.

1. Identify the y-intercept of each graph. What does each represent in the context of this scenario?

Use the questions below to help you understand the information on the Ranger’s Notebook Cards about the wolves and elk in Yellowstone. Then, ﬁll in the ranger log, and give your insight.

Part I

Explore 3

Exponential Functions

268 | Exponential Functions

Percent Increase or Decrease

Constant Multiplier

Population Increasing or Decreasing?

Initial Value

Wolves

Elk

6. Complete the ranger log, and give the head ranger insight about the impact the wolves may have had on other mammal populations in the years following their arrival.

5. The base, b, of an exponential function of the form f( f x) = a · bx is the ratio, or factor of change. How is the constant multiplier (factor of change) related to the common ratio of a geometric sequence?

4. Where do you see this ratio (factor of change) represented in each equation? What must this mean about that value?

3. Linear functions grow by the same rate of change per unit of horizontal growth. Exponential functions grow by the same factor per unit of horizontal growth. Are the functions that represent the wolf and elk populations linear or exponential? Explain.

Explore 3

Exponential Functions

P(x) = 52(1.04)x G(x) = 14(0.94)x O(x) = 65(1.08)x L(x) = 100(0.64)x

Peregrine falcon

Golden eagle Osprey Loon

Raven

Chickadee

Sandhill crane

Woodpecker

Bird

R(x) = 580(1.24)x

C(x) = 800(1.12)x

S(x) = 52(0.98)x

W(x) = 135(1.02)x

Model

Exponential Functions | 269

6. Which bird population is decreasing the fastest and may need special environmental protections?

5. Which bird has the largest starting population? What is that population?

4. Choose one of the decreasing models, and state the percent decrease each year.

3. Choose one of the increasing models, and state the percent increase each year.

2. Identify all of the bird populations that are decreasing. How do you know?

1. Identify all of the bird populations that are increasing. How do you know?

Model

Bird

There are many birds that also call Yellowstone their home. Use the population models below for each of the birds to answer the questions that follow.

Part II

Explore 3

Exponential Functions

270 | Exponential Functions

5. What conclusions can be made about the constant multiplier in any exponential function?

4. Would it make sense for the wolf population to keep increasing exponentially for an undeﬁned amount of time? Explain.

3. In your ranger log, you also noted that the wolf population was increasing by 25% each year. Explain how you calculated that value.

2. In your ranger log, you noted that the elk population was decreasing by 25% each year. If the value of the constant multiplier is 0.75, how did you calculate the 25%?

1. Based on your analysis of the information provided by the park rangers, what can you conclude about the reintroduction of wolves to the park?

Reflect

Explore 3

Exponential Functions

Write Exponential Functions

Name: _______________________ Date: ___________

Years since Reintroduction

> ___________________ > ___________________ 44.18 ≈ 44 > ___________________ 41.53 ≈ 42 > ___________________ 39.04 ≈ 39 47

Population

Coyote

Exponential Functions | 271

4. How can the constant multiplier be used to determine whether the exponential function is characterized by growth or decay?

3. How is the constant multiplier related to the percent increase or decrease?

2. What are the constant multipliers and initial values for the cougar and coyote populations?

> ___________________ 19.2 ≈ 19 > ___________________ 24.58 ≈ 25 > ___________________ 31.46 ≈ 31 > ___________________ 40.26 ≈ 40

Population

Years since Reintroduction

Cougar

1. In the tables below, show the process for calculating the constant multiplier.

Use the Ranger Hint Cards and tables below to help you calculate the initial value and constant multiplier for each animal. Answer the questions to help you write an exponential function equation for each animal. Round all decimals to the nearest hundredth, if necessary.

Part I

Explore 4

Exponential Functions

Coyote

Cougar

Initial Value (a)

Percent Increase or Decrease Constant Multiplier (b) Growth or Decay

Exponential Function Equation f x) = abx f(

272 | Exponential Functions

7. How can the constant multiplier be used to determine the percent increase or percent decrease?

6. If the output values have an increasing or decreasing pattern, what can that tell you about the constant multiplier?

5. Complete the table.

Explore 4

Exponential Functions

Graph Card

Table Card

Equation Card

Constant Multiplier

Initial Value

Equation

Exponential Functions | 273

4. Does the initial value have any inﬂuence on whether a function will be increasing or decreasing? Why or why not?

3. What is the percent decrease in each of the decreasing functions?

2. Once your table is ﬁlled in, circle all of the increasing functions.

F – Bat

E – Sheep

D – Otter

C – Jackrabbit

B – Grizzly

A – Bison

Animal Card

1. Reorganize your Animal Population Cards so you can give your ﬁnal report to the head ranger. Fill in each column of the table below with the letter of the card that corresponds to each written description. Then, ﬁll in the constant multiplier, initial value, and equation for each row.

Part II

Explore 4

Exponential Functions

274 | Exponential Functions

4. If you know an exponential function has a y-intercept of (0, 40) and passes through the point (2, 90), how could you write an equation in the form y = a(bx)?

3. If a peer were absent today, how would you explain to them how to use a table or description of a scenario to write an exponential equation to represent the data?

2. If the population size is increasing by 50%, why is the growth factor written as 1.5 in the equation?

1. When thinking about writing exponential functions in an equation, which value is considered the base?

Reflect

Explore 4

Exponential Functions

Exponential Extensions

275

Exponential Extensions

Explore 1

Name: _______________________ Date: ___________

Solve Exponential Equations Part I 1. Gus the German shepherd puppy weighed 1.5 pounds at birth and is growing exponentially! Fill out the table below to track his weight over time if his weight doubles every month.

Time (months)

Weight (lb.)

1.5

2. Write an equation to model Gus’s weight over these ﬁve months. 3. If Gus continued to grow at this rate, how much would he weigh after seven months? Does this value seem reasonable or not?

4. Gus’s sibling, Duke, only weighed one pound at birth, but his weight tripled every month for the ﬁrst few months. Complete the table for Duke, and write an equation that models this growth. Time (months)

Weight (lb.)

Exponential Extensions | 277

Exponential Extensions

Explore 1

5. How could you use the tables you created in questions 1 and 4 to solve the equation 1.5(2x) = 3x?

6. What does the solution mean in the context of the German shepherd siblings?

7. Peaches is a morkie, a small teacup dog who will only grow to about 10 pounds. For small puppies, it is easier to measure their weight in grams. Peaches weighed 25 grams at birth, and her weight doubles every month for six months. Write an equation to determine when Peaches weighs 100 grams.

8. Isolate the base and exponent in your equation, and determine when Peaches weighs 100 grams.

9. To determine when Peaches weighs 800 grams, we need to solve the equation 25(2x) = 800. Graph f( f x) = 25(2x) on the coordinate grid below, and use your graph to solve this equation. 900

800

Weight (grams)

700 600 500 400 300 200 100 0

278 | Exponential Extensions

4 5 6 Time (months)

x 10

Explore 1

Exponential Extensions

Part II Chelsea’s 160 followers started to triple every month for the next ﬁve months as Gus and Duke grew. 1. Write an equation to model Chelsea’s followers during this period of exponential growth. Deﬁne your variables.

2. Determine when Chelsea has 4,320 followers. Show the equation you solved, and explain your work.

3. Peaches’s owner, Alejandro, also has experienced a steep gain in followers. He started with 810 followers and doubles his total each month. Write and solve an equation to determine when Alejandro has 51,840 followers.

4. What equation could you solve to determine when Chelsea and Alejandro have the same number of followers?

5. Since Alejandro initially had more followers than Chelsea, why would we think they could have the same number of followers at some point in time?

Exponential Extensions | 279

Exponential Extensions

Explore 1

6. Graph f( f t) = 160(3t) and g(t) = 810(2t) using technology, and sketch the results below. Use the graph to solve the equation you created in question 4. 30,000

20,000

10,000

-1

x 1

Reflect 1. How can we use tables and graphs to help solve exponential equations?

2. Solve each equation below for x. a. 2x = 32

b. 4(2x) = 32

c. 4(2x) + 16 = 32

280 | Exponential Extensions

Exponential Extensions

Explore 2

Name: _______________________ Date: ___________

Exponentials in the Form y = a(b bx) + k Part I Analyze the Cooling Curves Card for tea, coffee, and hot chocolate. The detective’s notes below provide additional information about each beverage and the suspect who drank it.

Det󰇪󰈛󰇺󰇯ve’s No󰇹󰇪󰈤 Sus󰇴󰇪󰈝t

Bev󰇪󰇶󰈜󰈪e

Cur󰇶󰇪󰈡t T󰈥󰇲󰇵er󰇧󰇹󰇼󰈦e (°F)

Tea

100

Coﬀee

Hot chocolate

1. Use the cooling curve to determine the temperature of each beverage when each suspect received it.

2. Consider each equation and its graph. If there was a fourth beverage and its equation was I(t) = 130(0.37)t + 70, where would that curve be located on the graph with respect to the other curves? Why?

3. Based on the cooling curves and the detective’s notes, how long was each suspect at the coffee shop?

4. Which suspect was caught in a lie? Why?

Exponential Extensions | 281

Exponential Extensions

Explore 2 Part II

Analyze the Cooling Curves Card for the engines of the suspects’ cars. The detective’s notes provide additional information about the engines.

Det󰇪󰈛󰇺󰇯ve’s No󰇹󰇪󰈤 Sus󰇴󰇪󰈝t

Cur󰇶󰇪󰈡t T󰈥󰇲󰇵er󰇧󰇹󰇼󰈦e (°F)

100

1. Since all of the functions have the same constant on the end, what must be the same about each graph? Explain what this means in context.

2. All of the functions are written in the form y = a(bx) + k and have identical a values. What does this commonality lead to on the graph?

3. What does reducing the value of b do to the graphs of the functions?

4. Which engine is cooling at the fastest rate? 5. Based on the current temperature of each engine, for how long has each car been cooling? 6. Which suspect was caught in a lie? Why?

282 | Exponential Extensions

Explore 2

Exponential Extensions

Reflect 1. What are the horizontal asymptote and y-intercept of a function written in the form y = a(bx) + k?

2. How would the graph of f( f x) = 5(0.85)x + 2 compare to the graph of g(x) = 5(0.25)x + 2?

3. For an equation written in the form y = a(bx) + k, why does the value of b not impact the horizontal asymptote or the y-intercept?

4. Create a function in the form y = a(bx) + k, where the value of b is 5, the horizontal asymptote is y = −3, and the y-intercept is at (0, 10).

5. Use the functions f( f x) = 2x and g(x) = 8(2x) to answer the questions below. a. What kind of transformation took place to turn f( f x) into g(x) = 8(2x)?

b. Explain why the expression 2x+3 is equivalent to 2x(23).

c. Evaluate the part of the expression 2x(23) inside the parentheses, and use your work to explain why h(x) = 2x+3 is equivalent to g(x) = 8(2x). You can graph g(x) and h(x) using technology to prove they are equivalent and that the two transformations are the same for this exponential graph, f( f x). © Accelerate Learning Inc. – All Rights Reserved

Exponential Extensions | 283

Exponential Extensions

Explore 3

Name: _______________________ Date: ___________

Geometric Sequences Part I Analyze the Green Beans Graph and answer the questions to determine if you’ll grow enough green beans for your grandma. 1. What constant ratio do you notice in the graph?

2. What mathematical operation can you use to represent the green bean growth?

3. Look for patterns in the Green Beans Graph to help you complete the ﬁrst four columns of the table, answer the questions, and determine the missing number for the next terms. Do not attempt to ﬁll in the blank in the last two columns until after you have answered question 6. Number of Green Beans on Day 1

Pattern/ Common Ratio

Green Beans on Day 5

Green Beans on Day 6

6th Term Work

100th Term Work

5(____________)

5 · 2___

4. Starting at the ﬁrst term, how many times would you need to multiply by 2 to get to the 2nd term? How many times would you need to multiply by 2 to get to the 6th term?

5. What is another way to write the multiplication you expressed for the sixth term in question 4?

Exponential Extensions | 285

Explore 3

Exponential Extensions

6. How many times do you think we would need to double the starting value of 5 to get to the 100th term? Why? Use your answer to ﬁll in the blank in the last column of the table.

7. How could you represent the number of times you would need to double the starting value of 5 in order to get the nth term?

8. Write an equation or rule that will determine the number of days to grow green beans, and use your equation to ﬁnd the 12th term.

9. If your grandma needs 300 green beans on day 7, will you have enough green beans?

10. How does developing a formula for a geometric sequence compare to developing a formula or general rule for an arithmetic sequence?

286 | Exponential Extensions

Exponential Extensions

Explore 3 Part II Use the recursive equation below to complete the table and determine if you’ll have enough blueberries in time for your aunt. Blueberries Recursive equation: An = 2 · An – 1 Term (n)

Output

Day 1

3 blueberries

Day 2 Day 3 Day 4 1. What are the beneﬁts and drawbacks of using this equation compared to the explicit formula?

2. Complete the table below to efﬁciently calculate the number of blueberries on day 10. Arithmetic or Geometric (Circle one.)

Value of d or r

First Term

Explicit Equation

Day 10

Arithmetic Geometric 3. Your aunt needs 1,000 blueberries to make her special jam. Will you have enough on day 10? How do you know?

Exponential Extensions | 287

Exponential Extensions

Explore 3 Use the recursive equation below to complete the table and determine if you’ll have enough blackberries in time for your aunt. Blackberries Recursive equation: An = 1.5 · An – 1 Term (n)

Output

Day 1

32 blackberries

Day 2 Day 3 Day 4 4. Complete the table below to efﬁciently calculate the number of blueberries on day 10. Arithmetic or Geometric (Circle one.)

Value of d or r

First Term

Explicit Equation

Day 10

Arithmetic Geometric 5. Your aunt needs 1,000 blackberries to make her special jam. Will you have enough on day 10? How do you know?

288 | Exponential Extensions

Explore 3

Exponential Extensions

Reflect 1. What are the ways you can tell if a sequence is geometric from a list and a graph?

2. Once you determine the recursive equation for a sequence, must you always know the value of the previous term (An – 1) before ﬁnding the next term? Why or why not?

3. What are the two types of equations used with sequences, and how do they differ?

4. If you were given an equation, how could you tell if it is recursive?

5. If you were given an equation, how could you tell if it is explicit?

Exponential Extensions | 289

Exponential Extensions

Explore 4

Name: _______________________ Date: ___________

Geometric Sequences and Exponential Functions Part I Refer to the graphs to answer the questions that follow. 500 450 400 350 300 250 200 150 100 50 1

9 10

Number of refurbished chairs

Sia’s Graph

Emma’s Graph

500 450 400 350 300 250 200 150 100 50

Time (years)

9 10

Time (years)

1. What type of function does Sia’s graph represent? 2. Why might Sia use this type of function to depict the scenario?

3. How do the domains of both graphs compare?

4. Is Emma’s graph continuous or discrete? 5. Why might Emma represent the situation graphically this way?

6. Which graph best represents the situation? Why?

Exponential Extensions | 291

Exponential Extensions

Explore 4

Ishaan asks his assistants to create an equation he can use to determine the number of chairs he can refurbish over time. Both assistants create equations based on their graphs. Sia’s Equation

Emma’s Equation

7. Complete the table above by writing the function that represents Sia’s graph and the equation for the geometric sequence that represents Emma’s graph. 8. How are Sia’s and Emma’s equations similar?

9. How are Sia’s and Emma’s equations different?

10. Use both equations to determine the number of chairs refurbished after 3 years. Sia’s Equation

Emma’s Equation

11. How do your answers compare? 12. Consider your answers to questions 10 and 11 as well as Sia’s and Emma’s graphs. If I wanted to determine the number of chairs refurbished at 2.5 years, could I use either equation? Why or why not?

292 | Exponential Extensions

Exponential Extensions

Explore 4 Part II

Complete the table to help Ishaan determine what type of function he should use for each scenario. Movie Tickets

Overtime

Repairs

Lateness

Does this situation have a discrete or continuous domain? Explain. Is this situation best represented by an exponential or geometric function? Explain. Function

Exponential Extensions | 293

Exponential Extensions

Explore 4 Reflect

1. What is the difference between geometric growth and exponential growth?

2. Refer to the table of values that represents the weight of a young orangutan. Age (years)

Weight (lb.)

a. Does the table of values likely represent an exponential function or a geometric sequence? Explain.

b. Write a possible function for the table of values.

c. How does this function you created in part b differ from a geometric sequence that would match the table of values starting at an age of 1?

3. Write a scenario for which the following geometric sequence could apply: An = 3( 1 )n – 1 2

294 | Exponential Extensions

Compare Function Types

295

Compare Function Types

Explore 1

Name: _______________________ Date: ___________

Average Rate of Change Introduction Part I Analyze the hamburger sales data and pizza slice sales data. Use the data to complete the tables and answer the questions that follow.

Number of sales

Hamburger Sales 600 500 400 300 200 100 0

Month Hamburger Sales Time Period

Change in Sales

Change in Months

Average Change in Sales per Month

From month 1 to month 2

200

200 hamburgers per month

From month 2 to month 3

______ hamburgers per month

From month 3 to month 4

______ hamburgers per month

From month 4 to month 7

______ hamburgers per month

From month 5 to month 9

______ hamburgers per month

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Compare Function Types

Explore 1

Pizza Slice Sales Chart

Month

Number of Sales

400

550

600

500

350

400

300

400

Pizza Slice Sales Time Period

Change in Sales

Change in Months

Average Change in Sales per Month

From month 2 to month 3

______ pizza slices per month

From month 3 to month 5

______ pizza slices per month

From month 5 to month 9

______ pizza slices per month

1. The ﬁrst Student Celebration Day will occur after the third month of school. Which food item had the larger average change in sales per month from month 2 to month 3?

2. The last Student Celebration Day will occur toward the end of the school year and will be based off of the average change in sales per month from month 5 to month 9. Which food item should be purchased? Explain.

298 | Compare Function Types

Compare Function Types

Explore 1 Part II

Use the Food and Beverage Options Cards to analyze the previous year’s sales for the options below. Make the corresponding calculations, and then select the option for your recommendation. Healthy Food Options Base your recommendation on the higher average rate of change from month 1 to month 5. Average rate of change during the speciﬁed time period: a. Salad sales:

b. Chicken wrap sales:

Recommendation: Dessert Options Base your recommendation on the higher average rate of change from the interval 2 ≤ x ≤ 9. Average rate of change during speciﬁed time period: a. Chocolate brownie sales:

b. Ice cream pop sales: Recommendation: Beverage Options

Base your recommendation on the lower total cost for 50 gallons of beverage. Use the average cost per gallon to ﬁll in the missing values in the table. Gallons of Lemonade Needed

Total Cost ($)

Gallons of Limeade Needed

Total Cost ($)

$32.50 $175.00

50 Recommendation:

Compare Function Types | 299

Compare Function Types

Explore 1 Reflect

1. How are slope and average rate of change similar? How are they different?

Number of sodas sold

2. Kamille was analyzing the sales data for sodas sold during the previous year. The sales data is displayed on the graph below. 300 250 200 150 100

Month

Kamille claimed that the average rate of change from month 8 to month 10 was 50 sodas per month. Is her claim accurate? Justify your answer.

3. Draw in the line segment that shows the average rate of change for months 2 through 10, and then ﬁnd the value and explain it in context.

300 | Compare Function Types

Compare Function Types

Explore 2

Name: _______________________ Date: ___________

Compare Linear and Exponential Representations Part I The amount of caffeine in Lucy’s body over time is modeled by the table below. The amount of caffeine in Hannah’s body over time is modeled by the graph below. Hannah y

Lucy Amount of Caffeine (mg)

100

125 Amount of caffeine in body (mg)

Time (hours)

100

25 x

Time (hours)

1. Who consumed more caffeine initially, Lucy or Hannah? How much more? Explain.

2. How did Lucy’s amount of caffeine change compared to Hannah’s?

3. Both Lucy and Hannah start to lose energy when their levels of caffeine dip below 25 mg. What’s the maximum amount of time they can optimally study together? Why?

4. Lucy and Hannah both have the same amount of caffeine in their systems at 0 and 10 hours. Who has more caffeine in between those times?

Compare Function Types | 301

Explore 2

Compare Function Types

Lucy and Hannah’s classmates, Henry and Charlotte, are also in the library studying and decide to drink coffee as well. 5. Henry senses a crash coming when he has 80 mg of caffeine in his system, so he starts to drink more caffeine. After two hours, he has 125 mg in his system. a. If Henry drinks gradually for two hours, the amount of caffeine will increase linearly. Calculate the slope for your equation by using the amount of caffeine in Henry’s system for the ﬁrst two hours.

b. What would the y-intercept of your equation be?

c. Write the equation to model Henry’s caffeine intake if it increases linearly.

d. If Henry slowly drinks more and more coffee, the amount of caffeine in his system will increase exponentially. What would the initial amount of caffeine be?

e. Substitute the amount of caffeine in Henry’s system after two hours to solve for b, and write a model in the form y = a(bx).

6. Hannah’s and Lucy’s friend Charlotte started off with 150 mg. After 2 hours, Charlotte had 96 mg in her system. a. If the amount of caffeine in Charlotte’s body decreases at a linear rate, write the equation that models her caffeine levels in the form y = mx + b, where y is the number of mg of caffeine in Charlotte’s body and x is time in hours.

b. If the amount of caffeine in Charlotte’s body decreases at an exponential rate, write the equation that models her caffeine levels in the form y = a(bx), where y is the number of mg of caffeine in Charlotte’s body and x is the time in hours.

302 | Compare Function Types

Compare Function Types

Explore 2 Part II Use the questions that follow to analyze the Coffee Shop Cards. 1. Complete the table below. Coffee Shop

Linear or Exponential Growth?

y-intercept

A B C D 2. How could you tell which functions were linear and which were exponential?

3. Compute the predicted output at 5 years for each coffee shop.

4. Determine the average rate of change over the ﬁrst three years, on the interval [1, 3], for each coffee shop.

5. Write equations to match the growth of sales for coffee shop A and coffee shop B.

6. Which coffee shops do you think are larger chains that are growing faster than the local shops? Explain.

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Compare Function Types

Explore 2 Reflect

1. The table below contains points found on a continuous function f( f x). The function g is deﬁned by g(x) = 0.5x. x

−4

−3

−2

f x) f(

−2

−4

−8

a. Which function is linear, and which is exponential? Explain.

b. Which function has a larger y-intercept?

c. What is the end behavior of each function as x moves farther and farther from 0 in the positive direction?

2. Use the table to answer the questions below. a. Write a linear equation that could model the points in this table.

f x) f(

b. Write an exponential equation in the form ff(x) = a(bx) that models the points in this table.

c. What is the output when x = 5 for each of the models you created?

304 | Compare Function Types

Compare Function Types

Explore 3

Name: _______________________ Date: ___________

Compare Linear, Exponential, and Quadratic Relationships Part I Function A x

a(x)

−4

−1

−3

−10

Function B

Function C

b(x) = 2x2 + 1

1. Which function is linear? Quadratic? Exponential? How do you know?

2. Which function has the greatest y-intercept? What is it?

3. Which function does not have an x-intercept? How do you know?

4. As x gets farther from 0 in the negative direction, describe the end behavior of each of the functions.

5. When x = 10, which function will have the greatest output?

Compare Function Types | 305

Compare Function Types

Explore 3

6. Explain how to use the tables below to help distinguish between the linear and exponential relationship quickly. x

200

2,000

20,000

7. The tables below both show quadratic relationships. Calculate the ﬁrst difference and then the difference of these differences, known as the second difference. x

1st Difference

2nd Difference

125

180

306 | Compare Function Types

1st Difference

2nd Difference

Compare Function Types

Explore 3

8. What information in the table lets you know it is a quadratic relationship? What do they both have in common?

9. Fill in the missing values in the table of the quadratic relationship. x

1st Difference

2nd Difference

4 5 6

10. Explain how completing a table with a quadratic relationship compares to completing a table with a linear relationship.

Compare Function Types | 307

Compare Function Types

Explore 3 Part II

1. In the first year that the school recorded the first language of its students, 4% spoke Spanish. In year 3, 16% of students spoke Spanish as a first language. The school wants to predict how many students enrolled will have learned Spanish as their first language in year 5. Fill out the table below with the % of students who speak Spanish as their first language based on each potential model. Year

Linear Growth

Quadratic Growth

Exponential Growth

2 3

9 16

4 5

2. What is different about the growth of the three models? Which ends up with larger values the fastest?

3. What are some drawbacks to each of these models? Why might they not be accurate going forward?

308 | Compare Function Types

Compare Function Types

Explore 3

4. The district has three schools that started offering courses at a local community college for the first time. Below is the data for the first five years from the three schools represented in different ways. School A 70

School B

Year

Students Enrolled

Students enrolled

30 1

Year

School C c(x) = 2x, where x is measured in years and c(x) is the number of students enrolled.

5. Describe the differences in the types of growth for each of the schools.

6. In year 5, which school has the most students enrolled in a course at the community college? Which school has the least?

7. If the trends for each school continue, which school will have the most students enrolled in year 10? How do you know?

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Compare Function Types

Explore 3 Reflect

1. How can we determine whether a table represents a linear, quadratic, or exponential relationship?

2. How do we ﬁnd the second difference from a table?

3. If I have two functions, f( f x) = 10x2 and g(x) = 3x, as the x values get larger and larger, which function will have a greater value, and why?

4. Which function has a larger y-intercept, the quadratic function represented in the table below or the exponential function modeled by the equation g(x) = 3(2x) – 1?

f x) f(

1st Difference

2nd Difference

310 | Compare Function Types

Statistics

311

Statistics

Explore 1

Name: _______________________ Date: ___________

Shape of Data Part I: Mrs. Ortega’s Class Data Mrs. Ortega showed her class many different representations of data that she took from class surveys and assessments. Your job is to describe the data in each set as best you can. Description of the Shape

Graph

Prediction of What Caused the Shape Two peaks could be caused by piglets and adult pigs being weighed at the same time.

Weight of pigs

Grades

90 100

Percentages

Statistics | 313

Statistics

Explore 1 The mathematical descriptor for each data set is below.

Weight of pigs

Grades

Bimodal

Bell-shaped/symmetric

90 100

Percentages

Winning lottery numbers

Skewed left

Uniform

1. In the data that is labeled “skewed left,” is most of the data on the left or right side?

2. Describe the shape you think the data would take in each of the scenarios below. a. Ages of people who post on social media daily b. Number of siblings of the people in the class c. Heights of the girls in your school d. Ages of people at a Little League baseball game e. Results when rolling a die f. Points the football team scores in each game

314 | Statistics

Statistics

Explore 1 Part II: Mrs. Ortega’s Class Research Continued

The dot plot and box plot represent the same data for students in Mrs. Ortega’s class. Analyze Mrs. Ortega’s class data to make decisions about the center and spread of the data. Time per Day on Phones

Time (hours)

4 5 6 Time (hours)

1. Complete the table to help you analyze Mrs. Ortega’s class data. Mrs. Ortega’s Class Mean Fill in the blanks using the words mean and median.

Median The _______ is larger than the _______.

2. Is the data skewed left, skewed right, or symmetrical? Explain how the shape is related to the mean and median.

3. Do you think it is easier to see the shape of the data from the dot plot or box plot?

Statistics | 315

Statistics

Explore 1

Mrs. Ortega’s class collected and analyzed more data on two different topics. Use the data to conﬁrm or revise your conclusion about the relationship between shape and mean or median. Lunch Data

How many miles do you drive to school each day?

How much do you spend on lunch each day?

Frequency

Driving Data

3 2 1 0

Miles driven

3 2 1 0

Money spent

Shape: skewed right

Shape: symmetrical

Mean: 6.6

Mean: 5

Median: 5

Median: 5.1

4. Is the mean or median a better representation of the center of the lunch data? Explain using the shape of the graph to justify your position.

5. How does the relationship between the mean and median relate to the shape of the data when the mean and median are similar? How does it relate when the mean is larger than the median? How does it relate when the mean is less than the median?

316 | Statistics

Explore 1

Statistics

Reflect 1. How does the shape of the data affect the mean and median?

2. Explain the shape of data you would expect for the following types of mean and median. a. The mean is less than the median. b. The mean and median are similar. c. The mean is greater than the median.

3. If you knew the shape of a data set was symmetric, what would you know about the mean and median of the data set?

4. If you knew the shape of a data set was skewed right, what would you know about the mean and median?

5. Create a scenario that would be bimodal.

Statistics | 317

Statistics

Explore 2

Name: _______________________ Date: ___________

Standard Deviation Part I Use the Trashketball Tourney Player Data Cards to analyze the players and justify your draft selection using mean and standard deviation. 1. Which player do you think is the best shooter, and why?

2. Which player has the highest average number of makes?

3. How would you describe the differences in the cards?

4. If Mr. Williams offered no homework for the night if someone could make 4 out of their 6 shots, who would you pick to go shoot, and why?

5. If Mr. Williams offered no homework for as many days as the number of shots made out of 6, who would you pick to shoot, and why?

6. Why does just ﬁnding the mean not give us enough information about each of the shooters?

Statistics | 319

Statistics

Explore 2 7. Complete the table to organize your thoughts. Least Consistent

In Between

Most Consistent

Player How far are the data points from center? How far are the data points from each other? 8. We are going to look at a new measure of spread called standard deviation to see how it differs for each shooter. Shooter

Mean

Standard Deviation

Joaquin

1.63

Reyansh

Amari

2.11

Grace

1.89

a. How do the standard deviation values relate to your answers in the table for question 7?

b. There is another player in the class, Charles, who also had a mean of 4 and a standard deviation of 2.83. What can you say about Charles’s shooting performance?

320 | Statistics

Statistics

Explore 2 Part II

Trashketball Tryouts Shooting Results

Mark your makes with a . Mark your misses with an .

Total Made Baskets Group Results

Player

Number of Makes

Statistics | 321

Statistics

Explore 2 Trashketball Tryouts Calculation Results

Group ___ Stat Card Mean:

Standard deviation:

0 1 2 3 4 5 6 7 8 9 1. Using the class data, which group had the most clustered data? How do you know?

2. Which group had the most spread in their data? How do you know based on the stat card?

3. Mr. Williams wants to give awards to groups as prizes for their games. What awards would you suggest he give based on the statistics?

322 | Statistics

Statistics

Explore 2 Reflect 1. How does the standard deviation relate to the mean?

2. Sketch a dot plot that also has 6 data points and has a greater standard deviation than the plot below.

3. Would it be possible to have a standard deviation less than 0? Why or why not?

Statistics | 323

Statistics

Explore 3

Name: _______________________ Date: ___________

Outliers Part I: Understanding Outliers 1. Mr. Thomas gives Ishaan the class data, which he turns into a dot plot and a box plot. Do you believe there are any outliers? If so, what would you identify as an outlier, and why?

History Test Scores

100

History Test Scores

2. Identify the minimum, quartile one, median, quartile three, and maximum.

3. Which representation made it easiest to ﬁnd this information, and how do the potential outliers look in this plot of the data?

4. How much longer is the ﬁrst whisker than the inner box?

Statistics | 325

Statistics

Explore 3

5. One way to determine whether there are outliers that should not be included in the data set is to multiply the interquartile range (IQR), which is the length of the box, by 1.5 and exclude any values of this distance less than quartile 1 or greater than quartile 3. Complete the table below to determine whether there are any outliers. Data 1.5 Times IQR Less Than Q1 Calculations

Data 1.5 Times IQR Greater Than Q3

71 – 1.5(27) = 30.5

Outliers 6. Why is the score of 30 not an outlier using this method?

7. The student handbook says, “Whenever a student demonstrates a deﬁciency in mastery of the material, being an outlier, the individual student may request a retake at the teacher’s availability. If there are 3 or more students showing a learning deﬁciency, the class may request a retake at the teacher’s availability.” Are individual students or the class owed a retake? How are outliers a good mathematical way to determine retake status?

326 | Statistics

Statistics

Explore 3 Part II: The Impact of Outliers Analyze the two dot plots below from Mr. Fynn’s science classes.

Period 1

Period 2

Median = 55 Mean = 55

90 100

Median = 52.5 Mean = 47.2

1. How do the outliers 10, 20, and 25 in period 2 affect the measures of center for the data from period 1?

Use the following information to answer questions 2 and 3. In period 1, there were partners who won the review game and could apply extra credit. 2. Describe what would happen to the mean and median in period 1 if scores of 200 and 150 were added to the data.

3. What conclusion can you draw about the shape of the data if the median is smaller than the mean? What conclusion can you draw if the mean and median are similar? What conclusion can you draw if the median is larger than the mean?

Statistics | 327

Statistics

Explore 3 Analyze the two box plots below from Mrs. Paola’s 5th period English class.

Period 5 Version A

Period 5 Version B

IQR 15

IQR 10

Standard deviation: 20.849

90 100

Standard deviation: 7.071

4. How do the outliers 10, 95, and 100 affect the measures of variability?

5. Suppose Mrs. Paola decided only to include the outlier of 10. She did not include 95 and 100 because their scores were not valid. How would the IQR and standard deviation be affected?

6. Mrs. Paola believes that because both the IQR and standard deviation change, it wouldn’t matter which she would consider when there are outliers. Explain whether you agree or disagree.

328 | Statistics

Statistics

Explore 3 Part III: Analyzing the Source and Inclusion of Outliers Tackles Made This Season by Individual Players 3, 50, 57, 60, 70, 75, 84, 85, 88, 90, 120

Mean

Standard Deviation

71.091

28.308

Five-Number Summary Minimum

Median

Maximum

120

1. Identify the outlier(s), and hypothesize possible causes for the outlier(s).

2. Brainstorm with your group various scenarios where you believe the outlier(s) should or should not be included. Scenarios to Include Outliers

Scenarios That Are Unclear

Scenarios to Exclude Outliers

3. Assuming you don’t know the reason behind any of the data, should the coach use the outlier(s) when analyzing the data?

Statistics | 329

Statistics

Explore 3 Reflect 1. How do outliers affect the measures of center?

2. How do outliers affect the measures of variability?

3. Circle which measure you would choose in each of the following scenarios. a. Skewed data Mean or Median b. Symmetric data Mean or Median c. Data with an outlier Mean or Median

330 | Statistics

Model Data

331

Caffeine intake (milligrams)

Is this headline an accurate representation of the data? Explain your reasoning.

Model Data

Name: _______________________ Date: ___________

Correlation and Causation

1. There is a negative correlation between the amount of caffeine consumed and the number of hours a person sleeps.

Hours spent sleeping

Model Data | 333

Is this headline an accurate representation of the data? Explain your reasoning.

2. There is a positive correlation between ice cream sales at the beach and sunglasses sales.

Ice cream sales

Analyze each scatterplot and corresponding headline to determine whether the headline is an accurate representation of the data.

Part I

Explore 1

Sunglasses sales

If the number of bus stops increases, then the amount of time to complete the route increases.

Proposal A

334 | Model Data

Proposal E

Proposal D

Proposal C

Proposal B

If-Then Statement

Proposal Card

Causation?

Correlation? (If so, what type?)

Analyze each Proposal Card to determine if there is a correlation between the variables. Write each headline in ifthen form. Consider whether or not the if variable is a cause and the then variable is an effect to determine if the headline represents causation.

Part II

Explore 1

Model Data

4. Can two variables show causation but not association?

d. Negative correlation and causal

c. Negative correlation but not causal

b. Positive correlation and causal

a. Positive correlation but not causal

3. Create a new scenario that would represent the criteria described below.

2. Name the possible lurking variables for Proposal E.

1. Which proposals would you approve? Explain your reasoning.

Reflect

Explore 1

Model Data | 335

Model Data

Explore 2

Name: _______________________ Date: ___________

Linear Regression Part I Measure your height and wingspan in inches. Record your measurements on the class Data Collection Tables, and graph the points below.

Height vs. Wingspan 75 70

Wingspan (inches)

65 60 55 50 45 40 35 30 25 20 15 10 5 0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

Height (inches)

1. What is the relationship between height and wingspan?

2. How could we determine the approximate wingspan of someone with a height of 75 inches?

3. Draw a straight line through the data, and write a linear equation to represent the line. 4. Since our comics need to be accurate, predict the wingspan of a cartoon with a height of 6 inches. 5. What does the slope of the line mean in terms of the height and wingspan of our comics?

Model Data | 337

Model Data

Explore 2 Part II

Analyze each Data Card to determine if there is a correlation between the variables. Use your calculator to view the scatterplot of the data, calculate the line of best ﬁt, and calculate the correlation coefﬁcient.

Data Card

Equation for the Line of Best Fit

Correlation Coefﬁcient (rr value)

Description of the Shape and Correlation of the Data

Data A

Data B

Data C

Data D

338 | Model Data

Explore 2

Model Data

Reflect 1. Which set(s) of data has(have) a positive correlation? What do you notice about the correlation coefﬁcient of the data?

2. Which set(s) of data has(have) a negative correlation? What do you notice about the correlation coefﬁcient of the data?

3. Which set of data has the weakest correlation? What do you notice about the correlation coefﬁcient of the data?

4. Predict: If you have a correlation coefﬁcient of 0, what does that tell you about the strength of the correlation of the data?

5. Which sets of data are useful for drawing accurate people in a comic? Explain your reasoning.

6. A cartoon that has a height of 3 inches was submitted to the newspaper. Use the equations you created in Part I and Part II to calculate the measurements below. Wingspan: Femur bone length: Hair length:

Model Data | 339

Model Data

Explore 3

Name: _______________________ Date: ___________

Residuals Part I Use the data on the Sales Report to calculate the differences between the predicted and actual values. Then, plot those differences on the graphs below. Pens x

Actual y

Fidget Spinners

Predicted Value ŷ

Difference (y y − ŷ)

Actual y

107

115

–5

–10

1. What patterns do you notice in the Difference column of the table or in the graph? What does this tell you?

Predicted Value ŷ

Difference (y y − ŷ)

2. What patterns do you notice in the Difference column of the table or in the graph? What does this tell you?

Model Data | 341

Explore 3

Model Data

Part II Analyze each scatterplot and residual plot on the Graph Cards to determine if a linear model is the best representation of the data.

Graph Card

Does a linear model ﬁt the data?

If yes, justify by describing the shape of the data and the residual plot. If no, justify by describing the shape of the data and the residual plot. Then, suggest a better model and scenario for that model.

342 | Model Data

Explore 3

Model Data

Reflect 1. Which graphs are best represented by a linear model? Describe the residual plots.

2. Which graphs are not well represented by a linear model? Describe the residual plots.

3. In general, what does the residual plot of a strong linear ﬁt look like?

4. Compare and contrast scatterplots and residual plots.

5. Why is a residual plot useful for students ordering inventory for a school store?

Model Data | 343

Model Data

Explore 4

Name: _______________________ Date: ___________

Quadratic and Exponential Regression Part I Analyze the animal populations, and describe any trends in the changes in populations. Determine which function type best models each change in population. Years since 2015

Population Black Bear

Coyote

Bobcat

1. What trends do you notice with the black bear population? What type of function best models the change in the population of black bears?

2. What trends do you notice with the coyote population? What type of function best models the change in the population of coyotes?

3. A linear function best represents the change in the bobcat population. Calculate the line of best ﬁt, and explain why a linear model best represents the change in the bobcat population.

Model Data | 345

Model Data

Explore 4 Part II

Analyze the animal population data on the Population Cards to determine which function type best models the change over time. Use the model to make predictions. Animal Species

Type of Function Model

Regression Equation

Predicted Population in 2026

Description of the Shape of the Graph

Predicted Population in 2026

(Round to the nearest tenth.)

Sierra Nevada bighorn sheep

Mule deer

Northern Paciﬁc rattlesnake

Western pond turtle

California ground squirrel

Animal Species

Type of Function Model

Little brown bat

Great gray owl

346 | Model Data

Explore 4

Model Data

Reflect 1. Based on the Population Cards, which animal populations are best represented by quadratic functions?

2. What are the patterns or trends in the animal population data that might indicate that a quadratic function best models a data set?

3. Based on the Population Cards, which animal populations are best represented by exponential functions?

4. What are the patterns or trends in the animal populations data that might indicate that an exponential function best models a data set?

5. How can you tell by looking at the data if an animal species is in danger of becoming extinct?

6. Are these models useful for making predictions of animal populations in 10 years? How about 20 years? How about 100 years? Why or why not?

Model Data | 347

Skills Quizzes

349

Properties of Functions

Skills Quiz

Name: _______________________ Date: ___________

Properties of Functions Directions: Solve each problem. Show or explain your mathematical thinking.

1. What kind of function does the ﬁrst graph depict? What kind of function does the second graph depict? How do they differ in terms of slope? y

x x

Properties of Functions | 351

Properties of Functions

Skills Quiz

2. Draw the parent linear function and parent square root function on the grid below. How do the two graphs differ in terms of domain? 6

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

x 9 10

-2 -3 -4

3. The function f( f x) = 3x x + 9 is a continuous function whose domain is all real numbers. What is the value of x when f( f x) = 15?

4. Given g(x) = 4x x + d and g(4) = 15, determine the value of d.

5. For the function f( f x) = −5x x – 5, evaluate the following expression. ff(0) = __________

352 | Properties of Functions

Properties of Functions

Skills Quiz 6. For the function g(x) = 3x x + 2, evaluate the following expression. g(2) = __________

7. For the function h(x) = 7x x + 0.25, evaluate the following expression. h(−2) = __________ 6x

8. For the function j(x)) = x - 4 , evaluate the following expression. j(−4) = __________

9. For the function k(x) = −3x x + 1, which is greater: k(2) + 1 or k(2 + 1)?

10. Draw the parent quadratic function and parent linear function on the grid below. How do the two graphs differ in terms of range?

5 4 3 2 1 -5

-4

-3

-2

-1 0 -1

x 1

-2 -3

Properties of Functions | 353

Linear Functions

Skills Quiz

Name: _______________________ Date: ___________

Linear Functions Directions: Solve each problem. Show or explain your mathematical thinking.

1. Use this graph of h(x) to answer the following questions. a. Name the function family, and write a function in slope-intercept form that matches the graph. y 8 6 4 2

x -8

-6

-4

-2

-2 -4

h(x)

-6 -8

b. Complete the table for h(x). x

h(x)

−10 10 −2 2 6

Linear Functions | 355

Linear Functions

Skills Quiz

2. Which choice gives a linear function that has solutions (4, −7) and (−2, −4)?

A. y = 2x – 15 B. y = 1 x – 3 2

C. y = 7x – 2 D. y = – 1 x – 5 2

3. Compare the two functions f( f x) and g(x). f x) = −4(x f( x – 2) x

g(x)

−9

−7

−3

4.5

a. Which function has a greater slope? b. Which function has a greater x-intercept? c. Which function has a greater y-intercept?

356 | Linear Functions

Linear Functions

Skills Quiz For questions 4 and 5, use the following information.

Marcos orders a chocolate milkshake at his favorite restaurant. The milkshake is 300 mL. He ﬁnishes his drink in 10 minutes by drinking it at a constant rate. 4. Write a function to represent this situation where f( f x) represents the volume in mL and x represents the number of minutes since Marcos received the milkshake. Graph the function representing the volume of the milkshake over time. y

350 300 250 200 150 100 50

x -2

5. What is a reasonable domain and what is a reasonable range for this scenario? The domain is __________. The range is __________.

Linear Functions | 357

Linear Functions

Skills Quiz Use the following information for questions 6–8. The following information describes daily bike rental fees for Bob’s Bikes. • 1 hour costs $5.00. • 2 hours costs $9.00. • 3 hours costs $13.00.

6. Answer the questions below for the information modeled as an arithmetic sequence. a. Write an arithmetic sequence to model the rental fees.

b. Determine the domain and the range.

7. Answer the questions below for the information modeled as a linear equation. a. Write a linear equation in slope-intercept form to model the rental fees.

b. Determine the domain and the range.

8. Would an arithmetic sequence or linear function best model the situation if the bike could be rented for only full hours? Justify your answer.

358 | Linear Functions

Linear Functions

Skills Quiz

9. Melissa starts reading a book for school by starting on page 32. If she reads it at a rate of 50 pages per hour, how much time, t, in hours will it take her to be on page p? Determine the common difference, and write an explicit formula to determine the page number that she is on.

10. The explicit formula of an arithmetic sequence is An = −11 + 22(n – 1). Complete the missing values in the recursive formula of the sequence. A1 = __________ An = __________

Linear Functions | 359

Geometry on the Coordinate Plane

Skills Quiz

Name: _______________________ Date: ___________

Geometry on the Coordinate Plane Directions: Solve each problem. Show or explain your mathematical thinking.

Use the following information to answer questions 1 and 2. The line y = 5 x + 11 that passes through (−1, 3) and (1, 8) is represented on the graph 2

below. y 10

-10

-5

-10

1. What is the distance between the points (−1, 3) and (1, 8) on the line?

2. What is the midpoint between the points (−1, 3) and (1, 8) on the line?

Geometry on the Coordinate Plane | 361

Geometry on the Coordinate Plane

Skills Quiz

3. Two roads meet at a right angle. One of the roads can be modeled by the equation y = 2 x + 10, and they meet at (−6, 6). Write an equation that models the other road. 3

4. What is the perimeter of the triangle below?

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

x 1 2 3 4 5

-3

5. The face of Mount Snow can be modeled by the equation y = −4.8x + 3,500, where y is the height of the mountain in feet and x is the horizontal distance from the point directly below the mountaintop at its base, also in feet. Mount Blizzard has a peak of 4,500 feet and a face that is just as steep as Mount Snow. Write an equation for the line that matches the faces of Mount Blizzard.

362 | Geometry on the Coordinate Plane

Geometry on the Coordinate Plane

Skills Quiz

6. Two top-rated local coffee shops are located at (2, −1) and (0, 5) on a coordinate grid, where each unit represents 0.1 miles. If someone wanted to try coffee at both shops to determine their favorite, how far would they have to travel in miles?

A. 2√10 miles B. √10 miles 2

C. √10 miles 5

√10

D. 10 miles

7. The right triangle below models a garden where each unit is equal to 1 foot. What is the area of the garden? 8

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

x 1 2 3 4 5

-2

Geometry on the Coordinate Plane | 363

Skills Quiz

Geometry on the Coordinate Plane

8. If one stretch of railing on a railroad track can be modeled by the equation 8x x – 4y = −32, what is the slope of a parallel railing?

A. 8 B. −4 C. −2 D. 2

364 | Geometry on the Coordinate Plane

Geometry on the Coordinate Plane

Skills Quiz Use the following information to answer questions 9 and 10.

The line y = − 1 x – 1 is graphed below. 2

y 10

-10

-5

-10

9. Johnny lives at the point (−2, 0), and Marco lives at the point (0, −1). Who lives closer to the school located at (0, 3)? A. Marco lives closer to the school. B. Johnny lives closer to the school. C. They live the same distance away from the school.

10. What is the midpoint between Johnny’s and Marco’s houses where they could meet up and walk together to school if they wanted to? A. (−1, −0.5) B. (−0.5, −1) C. (−1.5, −0.5) D. (−1, −1.5)

Geometry on the Coordinate Plane | 365

Linear Inequalities

Skills Quiz

Name: _______________________ Date: ___________

Linear Inequalities Directions: Solve each problem. Show or explain your mathematical thinking. 1. Which of the following inequalities is represented by the graph below? 10

x -10

-5

-10

A. y < x + 3 B. y > x + 3 C. y ≤ x + 3 D. y ≥ x + 3

Linear Inequalities | 367

Linear Inequalities

Skills Quiz

2. The Hargrove family owns a farm that grows cherries. There are 11 rows of cherry trees, and in the past, the farm has yielded no less than t total pounds of cherries. Each employee picks cherries from one row of trees. Which inequality can be used to determine p, the approximate weight in pounds of the cherries each employee will pick? A. 11p ≤ t B. 11tt ≤ p C. 11p ≥ t D. 11tt ≥ p Use the following information for questions 3 and 4. Janet works part time at the yogurt shop to earn extra money, m, to buy a new scooter. She currently has $750 saved and wants to add $35 each week from her paycheck. 3. Write an inequality that could be used to determine w, the number of weeks she will have to save to have enough money, m, to buy the scooter.

4. If m = $1,250, how many weeks will Janet have to save in order to reach her goal? Explain your reasoning.

368 | Linear Inequalities

Linear Inequalities

Skills Quiz 5. Which of the following graphs represents the inequality? y – 4x < 4 A.

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

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

6 5 4 3 2 1

1 2 3 4 5 6

6 5 4 3 2 1

x 1 2 3 4 5 6

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

x 1 2 3 4 5 6

Linear Inequalities | 369

Linear Inequalities

Skills Quiz 6. Graph the inequality x > 2 on the grid below.

x -10

-5

-10

370 | Linear Inequalities

Linear Inequalities

Skills Quiz 7. Which inequality is represented by the graph below? 10

x -10

-5

-10

A. y ≥ 2x – 2 B. y > 2x – 2 C. y < 2x – 2 D. y ≤ 2x – 2

Linear Inequalities | 371

Linear Inequalities

Skills Quiz 8. Which graph represents the inequality y < 4?

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

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

372 | Linear Inequalities

6 5 4 3 2 1

1 2 3 4 5 6

6 5 4 3 2 1

x 1 2 3 4 5 6

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

x 1 2 3 4 5 6

Linear Inequalities

Skills Quiz

9. Given the inequality 7y y < 2x + 8, which is represented on the graph below, determine whether (−4, 0) is included in the solution set.

x -10

-5

-10

Linear Inequalities | 373

Linear Inequalities

Skills Quiz

10. Graph the inequality 3y − 15 ≤ 2x − 8 to determine whether the point (4, 5) is a solution.

x -10

-5

-10

374 | Linear Inequalities

Systems of Inequalities

Skills Quiz

Name: _______________________ Date: ___________

Systems of Inequalities Directions: Solve each problem. Show or explain your mathematical thinking. For questions 1 through 3, graph both inequalities on the same graph. Clearly show the solution set of the system of inequalities. 1. 2x x+3>y 4x x – 8y ≤ 0 10

9 8 7 6 5 4 3 2 1 -10 -9

-8 -7 -6 -5 -4 -3

-2 -1 0 -1

x 1

-2 -3 -4 -5 -6 -7 -8 -9 -10

Systems of Inequalities | 375

Systems of Inequalities

Skills Quiz

2. y ≥ 2(x – 1) y < 2x + 1

4 3 2 1

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

x 10

-2 -3 -4 -5

3. −3x x – 7y > 35 4x x–y>1 10

8 6 4 2 -10

-8

-6

-4

-2

0 -2 -4 -6 -8 -10

376 | Systems of Inequalities

Skills Quiz

Systems of Inequalities

For questions 4 through 10, refer to the following information. Julius works as a cashier and as a babysitter after school on different days. He only has time to work, at most, 10 hours per week between the two jobs. As a babysitter, he earns $10 per hour, and as a cashier, he earns $7.25 per hour. In order to buy a new computer for himself in several months, he needs to make at least $80 per week. 4. Write a system of inequalities describing Julius’s situation by using b, hours worked as a babysitter, and c, hours worked as a cashier.

5. Which of the following represents constraints in this situation? Select all that apply. A. c < b B. b ≥ 0 C. c = b D. c ≥ 0

Systems of Inequalities | 377

Systems of Inequalities

Skills Quiz

6. Graph the system of inequalities, including the constraints. Use b as the horizontal axis and c as the vertical axis. 12

11 10 9

Hours spent as a cashier (c)

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

x 1

9 10 11 12 13 14 15

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

Hours spent babysitting (b)

7. What does the ordered pair (7, 2) mean in terms of this situation?

8. How much money per week would Julius make according to the point that you found?

9. Does a combination of 5 hours in each job satisfy your equations? Justify your answer.

10. Is (−1, 10) a viable solution for this situation? Justify your answer.

378 | Systems of Inequalities

Simplify Radicals

Skills Quiz

Name: _______________________ Date: ___________

Simplify Radicals Directions: Solve each problem. Show or explain your mathematical thinking. For questions 1 through 6, simplify the radicals. 1. √81 √

2. √40 √

3. √384 √

4. √96 √

5. 3 √504 √

√ 200 49

Simplify Radicals | 379

Skills Quiz

Simplify Radicals

7. A gardener is looking at mock-ups of his rectangular garden that have adjustable lengths and widths. Answer the following questions based on the changing parameters. a. If the length is 3 units and the width is √12 units, is the perimeter of the garden a rational number or an irrational number?

b. If the length is 3 units and the width is √12 units, is the area of the garden a rational number or an irrational number?

c. Adjust either the length or width value from part b so that the area is a rational number.

8. Niko was simplifying the radical expression √675 . His work is shown below. √ Step 1: √675

Step 2: √ √25 x 27

Step 3: √ √(5 x 5) x (9 x 3)

Step 4: √ √(5 x 5) x (3 x 3 x 3) Step 5: 8√3 √

In what step, if any, did Niko make a mistake in simplifying the radical expression?

380 | Simplify Radicals

Skills Quiz

Simplify Radicals

9. Which of the following is equivalent to the expression √9 . √100? √ A. √39 B. 30

√ C. 3√10

√ D. 10 √9

10. The sum of a rational number and an irrational number is guaranteed to be which of the following?

A. Positive B. Rational C. Irrational D. Not enough information to decide

Simplify Radicals | 381

Polynomial Operations

Skills Quiz

Name: _______________________ Date: ___________

Polynomial Operations Directions: Solve each problem. Show or explain your mathematical thinking.

1. The function f( f x) = 7x2 – 4x x + 1 is a continuous function whose domain is all real numbers. Evaluate f(−9). f

Use the following polynomial to answer questions 2–4.

h(x) = 5x x + 7x2 – 1

2. Write h(x) in standard form.

3. How many terms does h(x) have? Identify the terms.

4. Identify the coefﬁcients.

5. For the function f( f x) = 7x2 + x – 3, evaluate the following expression. ff(6) = __________

Polynomial Operations | 383

Polynomial Operations

Skills Quiz

6. This algebra tiles model demonstrates multiplying polynomials. What multiplication problem does it represent? Write the product as a polynomial in its simplest form.

7. Write the two functions below in standard form, and write the y-intercept of each.

f x) = (2x f( x – 3)2

g(x) = (5x x + 4)(5x – 4)

8. Simplify the expression, and give the answer in standard form. (5x2 – 2x x + 1) + (−x – 4)

9. Which expression below shows (4x2 – 7) – (3x2 + 5x x – 6) simpliﬁed correctly?

A. 7x2 – 5x x – 13 B. x2 – 5x x–1 C. 7x2 + 5x x–1 D. x2 + 5x x – 13

384 | Polynomial Operations

Skills Quiz

Polynomial Operations

10. The function h(t) = −8t2 + 80tt + 14 models the height of a performer when shot from a cannon, where h(t) is the height of the performer in feet and t is measured in seconds. Explain what h(7) means in context.

Polynomial Operations | 385

Graphs of Quadratic Functions

Skills Quiz

Name: _______________________ Date: ___________

Graphs of Quadratic Functions Directions: Solve each problem. Show or explain your mathematical thinking. For questions 1 through 6, use the following information. The height, in meters, of a projectile in the air can be represented by the equation h(t) = −4.9t2 + 9.8tt + 2, where t is time in seconds. This table represents this function and is symmetric about x = 1. 1. Complete the table, and sketch a graph on the grid provided. t

h(t)

−0.187 0 1

6.9 2

2.187

0 y 10

x -10

-5

-10

Graphs of Quadratic Functions | 387

Graphs of Quadratic Functions

Skills Quiz

2. How did the symmetry of the graph help you ﬁll out the table?

3. What is the initial height of the projectile? How is this depicted in the equation? How is this depicted in the graph? How is this depicted in the table?

4. What is the maximum height? When does it occur? Rewrite h(x) so that the maximum is more apparent.

5. What is a reasonable domain and range for this context?

6. A ball travels according to the path h(t) = −16(tt – 3.5)2 + 22, where t is time in seconds and h(t) is the ball’s height in meters. Does this ball reach a height of 25 meters at any point? How do you know?

388 | Graphs of Quadratic Functions

Graphs of Quadratic Functions

Skills Quiz

A quadratic function, f( f x) = 2(x x – 2)(x + 5), has zeros at x = 2 and x = −5.

7. Write f( f x) in standard form, and state the y-intercept.

8. Which of the quadratic functions below opens downward? f x) = 3x2 – 3x f( x – 100

g(x) = −(x x + 4)2 + 100

h(x)

100

150

175

150

9. Graph the function f( f x) = (x x + 1)2 – 4 below. Then, label the x-intercept(s), y-intercept, and vertex of the graph.

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

x 1

-2 -3 -4 -5

Graphs of Quadratic Functions | 389

Graphs of Quadratic Functions

Skills Quiz 10. Identify the features of your graph in question 9. Over what interval is f( f x) increasing? Over what interval is f( f x) decreasing? What is the equation for the line of symmetry for f( f x)?

390 | Graphs of Quadratic Functions

Factors of Polynomials

Skills Quiz

Name: _______________________ Date: ___________

Factors of Polynomials Directions: Solve each problem. Show or explain your mathematical thinking. For questions 1 through 3, factor the polynomial functions. Then, determine their zero(s) and x-intercept(s). 1. f( f x) = x2 – 10x x + 16 The factored form is _____________________. The zero(s) is(are) _____________. The x-intercept(s) is(are) ________________.

2. g(x) = 4x2 – 81 The factored form is ____________________. The zero(s) is(are) ______________. The x-intercept(s) is(are) ________________.

3. h(x) = x2 + 6x x+9 The factored form is ___________________. The zero(s) is(are) _____________. The x-intercept(s) is(are) _______________.

Factors of Polynomials | 391

Factors of Polynomials

Skills Quiz

4. Joe uses an area model to factor the quadratic j(x). What is the factored form of j(x)?

j(x) = 2x2 + 7x x – 15

x x

5. A skateboard ramp is modeled by the equation k(x) = x2 – 6x x + 5. Complete the square to write the function in vertex form, and then determine the vertex.

The vertex is __________. 6. Is the vertex you found in question 5 a minimum or maximum value of your graph? Explain what it describes in context.

392 | Factors of Polynomials

Factors of Polynomials

Skills Quiz 7. Consider the following functions to answer the questions below. m(x) = 7(x x – 1)2 and n(x) = (x x – 4)(x + 6)

a. Which function has a vertex with a greater y-coordinate, m(x) or n(x)? b. Which function has a vertex with a lesser x-coordinate, m(x) or n(x)? c. Which function has two zeros, and what are the values of the zeros?

8. If the ﬂight of a ball is modeled by y = −8(x – 9)(x + 2), where y is the height in yards and x is the time in seconds, when does the ball hit the ground?

Use the information below for questions 9 and 10. A farmer wants to fence in a section of his property. He has 100 feet of fencing available. This is the equation for the area, A, in square feet that the fence will enclose based on the width, w, in feet. A = w(50 – w)

9. Since the equation is given in factored form, explain what the two factors reveal about the context.

10. Rewrite the equation in vertex form. Then, identify the vertex, and explain what the ordered pair means in the context of this situation.

Factors of Polynomials | 393

Solve Quadratics

Skills Quiz

Name: _______________________ Date: ___________

Solve Quadratics Directions: Solve each problem. Show or explain your mathematical thinking. 1. Fill in the table below, and then graph the function f( f x) = x2 – 1 with the domain −3 ≤ x ≤ 3.

f x) f(

−3 −2 −1 0 1 2 3

Solve Quadratics | 395

Solve Quadratics

Skills Quiz 2. What is the solution set for the function below? x2 + 5x x – 24 = 0

3. What values of x satisfy the equation x2 + x = 20?

4. The equation y = x2 – 4x x + 3 models the revenue for a company, y, in millions of dollars after x years. Solve for x when y = 15, and explain what your solution means in the context.

5. Solve the following equation for t. 8 = t2 + 2t

396 | Solve Quadratics

Solve Quadratics

Skills Quiz

6. The equation y = x2 – 2x x – 3 represents the profits made by Erin’s new printing business, where x is the number of months since opening and y is the net profit in dollars. Negative y values indicate that the business is losing money, which is expected in the beginning. Erin really needs this business to be profitable after 4 months, at the latest. Graph the equation below, find the solutions, and determine whether she will meet her 4-month requirement.

x 0

-5

- 10

Solve Quadratics | 397

Solve Quadratics

Skills Quiz

7. The height of a water balloon is modeled by the function h(t) = −4(tt – 1.5)2 + 12, where h(t) is measured in feet and t is measured in seconds. When is the height of the balloon 8 feet in the air?

For questions 8 and 9, solve each quadratic equation by factoring or completing the square, and be sure to express each root in the simplest radical form.

8. Solve 2x2 – 12x x + 4 = 0.

9. Solve 3x2 + 6x x – 24 = 0.

398 | Solve Quadratics

Skills Quiz

Solve Quadratics

10. Andres and Vivann are competing as a team in a catapult competition where they both build the catapult, model the trajectory they predict it will take, and predict where the projectile will land. They have their catapult built and have come up with the equation y = −x2 + 9x x + 10 for their trajectory model, where y is the height of the projectile in meters and x is the horizontal distance traveled in meters.

a. What are the x values that make the model equation equal 0?

b. Are both solutions reasonable? Explain.

c. What is the predicted distance that Andres and Vivann’s catapult will launch the projectile?

Solve Quadratics | 399

Transform Quadratic Functions

Skills Quiz

Name: _______________________ Date: ___________

Transform Quadratic Functions Directions: Solve each problem. Show or explain your mathematical thinking. 1. The graph shows the plot for f( f x) = x2. Plot the function h(x) = (x x – 3)2 – 6. Label the vertex of f( f x) and h(x). y

7 6 5 f(x)

4 3 2 1

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

0 -1

-2 -3 -4 -5 -6 -7

How did the vertex move from the original plot to achieve the new plot?

Transform Quadratic Functions | 401

Transform Quadratic Functions

Skills Quiz 2. If f( f x) = 3x2, graph the function g(x) = f( f x + 3) below. y

3. Which function has a lesser y-intercept value? y

11 10 9 8 7 6 5

g(x)

4 3 2 1 -3 -2 -1

0 -1

9 10 11

-2 -3

f x) = (x f( x – 3)2 + 8

402 | Transform Quadratic Functions

Transform Quadratic Functions

Skills Quiz 4. Which function has a greater maximum y value? y

11 10 9 8 7 6

j(x)

5 4 3 2 1

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

0 -1

9 10 11

-2 -3 -4 -5 -6 -7 -8 -9 -10 -11

h(x) = −(7x x – 1)2 + 10

5. Given the graph of m(x) and the table for quadratic function p(x), determine which has the smallest root. m(x))

p(x)

-10

-5

p(x)

−4

−3

−1

−2

-10

Transform Quadratic Functions | 403

Transform Quadratic Functions

Skills Quiz

6. The function h(t) = −16t2 + 80tt + 5 models the height of a ball when thrown by Henry. The value t is time in seconds, and the value h(t) is height in feet. When William threw the same ball, the following data was recorded. • At 0 seconds, the ball was at a height of 6.5 feet. • After 1.25 seconds, the ball was at a height of 31.5 feet. • After 2.6 seconds, the ball was at a height of 2.24 feet. Which ball was initially thrown from a higher height?

7. Describe the transformations from the quadratic parent function to f( f x). f x) = (5x)2 – 4 f(

8. Describe the transformations from the quadratic parent function to g(x). g(x) = −3(x x – 1)2 + 10

404 | Transform Quadratic Functions

Transform Quadratic Functions

Skills Quiz

9. The function h(x) is graphed. Describe the transformations from the quadratic parent function to h(x). y

11 10 9 8 7 6 5 4 3 2 1 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

0 -1

9 10 11

-2 -3 -4 -5 -6

h(x)

-7 -8 -9 -10 -11

10. The function f( f x) = −2(x x – 2)2 + 8 models the path of a bouncy ball thrown against the ground where f( f x) is height measured in feet and x is the horizontal distance traveled in feet. When the bouncy ball is thrown again, it bounces according to the model g(x) = 2f( f x + 1) – 4. How does the bounce of g(x) differ from the bounce of f( f x)? f(

Transform Quadratic Functions | 405

Exponential Functions

Skills Quiz

Name: _______________________ Date: ___________

Exponential Functions Directions: Solve each problem. Show or explain your mathematical thinking. For questions 1 and 2, choose the equation that matches the graph. 1. y 10

x -10

-5

-10

A. y = 3x B. y = 3(1)x 1

C. y = 3 (3)x D. y = –(3)x

Exponential Functions | 407

Exponential Functions

Skills Quiz 2. y 10

x -10

-5

-10

A. y = 2x B. y = 3(2)x C. y = 2(3)x D. y = −(2)x

3. A scientist is experimenting with bacteria in his laboratory. He starts out with 3 bacteria. The number of bacteria quadruples every hour. Write the equation that represents the number of bacteria, f( f x), of this bacteria after x hours.

408 | Exponential Functions

Exponential Functions

Skills Quiz

4. This is the graph of the function f( f x)) = 6( 1 )x. Identify the key features listed below. 2

8 7 6 5 4 3 2 1 -2

-1

0 -1

x 9

-2

The y-intercept is __________. The domain is __________. The range is __________. The growth factor is __________. Can a data point in this plot have an x-coordinate of −1? Why or why not?

5. Does this function represent exponential growth or decay? g(x)) = 5( 1 )x 3

A. Growth B. Decay

Exponential Functions | 409

Exponential Functions

Skills Quiz

The table below models the number of hours Michael spends doing homework based on the number of after-school activities he has. Use the table to answer questions 6 and 7.

x (number of after-school activities)

f x) (hours f( of homework completed)

3 2

3 4

6. Evaluate ff(2), and explain what it means in the context.

7. Write an equation for f( f x) in the form f( f x) = a(bx).

8. Use your equation to ﬁnd ff(10). Does this value make sense in the context?

410 | Exponential Functions

Exponential Functions

Skills Quiz 9. Graph the exponential function f( f x). f x)) = 3( 1 )x f( 2

-10

-5

-10

Exponential Functions | 411

Skills Quiz

Exponential Functions

10. Identify the following key features for the function f( f x) given in question 9 above.

a. The x-intercepts are __________. b. The y-intercepts are __________. c. As x → ∞, y is __________.

d. As x → –∞, y is __________. e. What is the domain of f( f x)?

f. Can a data point in this plot have a y-coordinate of −1? Why or why not?

412 | Exponential Functions

Exponential Extensions

Skills Quiz

Name: _______________________ Date: ___________

Exponential Extensions Directions: Solve each problem. Show or explain your mathematical thinking. For questions 1 through 8, use the graph of f( f x) = 2(5x). 10

-10

-5

-10

1. Describe the transformations from f( f x) to g(x) if g(x) = f( f x) – 2.

2. Which function, f( f x) or g(x), has the greater x-intercept?

3. Describe the transformations from f( f x) to h(x) if h(x) = −f( f x). f(

4. Which function, f( f x) or h(x), is increasing?

Exponential Extensions | 413

Exponential Extensions

Skills Quiz 5. Describe the transformations from f( f x) to k(x) if k(x) = 2f( f x). f(

6. Which function, f( f x) or k(x), grows at a faster rate?

414 | Exponential Extensions

Exponential Extensions

Skills Quiz For questions 7 and 8, refer to the following information. The functions f( f x), g(x), and h(x) are graphed. y

g(x) g(x) ff(x) f( f(x (x) -10

-5

h(x) h(x) 0

-5

-10

7. Which choice gives g(x) in terms of f( f x)? A. g(x) = f( f x + 2) B. g(x) = f( f x – 2) C. g(x) = f( f x) + 2 D. g(x) = 2g(x)

8. Could the function h(x) be written as h(x) = f( f x) – 2? Explain.

Exponential Extensions | 415

Exponential Extensions

Skills Quiz

9. A certain bacterial population triples every hour. After the ﬁrst hour, there are 1,500 cells. How many bacteria will be present after t hours? a. Write an explicit formula for the geometric sequence. Explain.

b. Write as an exponential function.

c. How does a geometric sequence differ from an exponential function?

416 | Exponential Extensions

Exponential Extensions

Skills Quiz

10. Determine the common difference or common ratio. Then, write an explicit formula for the sequence.

250

(5, 243)

200

150

100 (4, 81)

50 (3, 27)

(0, 1)

(1, 3)

(2, 9) x

Exponential Extensions | 417

Compare Function Types

Skills Quiz

Name: _______________________ Date: ___________

Compare Function Types Directions: Solve each problem. Show or explain your mathematical thinking.

1. Which function has a larger y-intercept: f( f x) = x2 + 2 or g(x) as pictured below?

7 6 5 4 3

g(x)) g(x)

2 1 -9

-8

-7

-6

-5

-4

-3

-2

-1

-1 -2

2. Which exponential function has a larger horizontal asymptote: f( f x) = 5x or g(x), as modeled by the table below? x

–10

–8

–6

–4

–2

g(x)

1.00002

1.0002

1.0014

1.0123

1.1111

Compare Function Types | 419

Skills Quiz

Compare Function Types

3. Find the average rate of change for the function f( f x) = 9x2 + 1 from x = 0 to x = 2. How does the answer compare with the average rate of change for the function g(x) = 9x x + 1 over the same interval?

4. Find the average rate of change for the functions f( f x) = 3x2 – 1 and g(x) = 3x – 1 on the interval [–2, 0]. Which function has a greater rate of change over this interval?

5. Explain how to decide whether the average rate of change on the interval 0 ≤ x ≤ 3 is greater for f( f x) = 5(2x) or for g(x) = (x x – 3)2 + 10 without making any calculations.

420 | Compare Function Types

Compare Function Types

Skills Quiz

6. Find the average rate of change for the function h(x)) = 1 x2 + 6 from x = 0 to x = 1. 8

How does the answer compare with the average rate of change for the function g(x)) = 1 x + 6 from x = 0 to x = 1? 8

7. A student takes the functions y = 7x2 + 4 and y = 7x + 4. Without solving for an average rate of change, how can they prove that the two functions have the same average rate of change from x = 0 to x = 1?

8. Write the equation of a line and an exponential function in the form y = a(bx) that passes through (0, 4) and (2, 16).

9. A student takes a quadratic function and a linear function. What are the conditions for the functions to have the same average rate of change from x = −1 to x = 0?

Compare Function Types | 421

Skills Quiz

Compare Function Types

10. Which function has a greater maximum output: f( f x) = −(x x + 6)(x – 2) or g(x) = −(x x – 4)2 + 15?

422 | Compare Function Types

Statistics

Skills Quiz

Name: _______________________ Date: ___________

Statistics Directions: Solve each problem. Show or explain your mathematical thinking. 1. Determine the outliers in the data set of values of ages for a chess tournament entered into the competition’s database. Data values: 3, 15, 18, 18, 19, 22, 22, 30

Use the following information for questions 2 and 3. The shopping times for two groups at City Center Mall were recorded and displayed on the box plots below.

Group A Group B 0

Shopping Time (min.) 2. Which group had a larger median shopping time?

3. Which group had a larger IQR?

Statistics | 423

Statistics

Skills Quiz Use the following information for questions 4 and 5.

The table below displays the test scores on the Algebra I ﬁnal exam for Mr. Alvarez’s class. Ms. Brown’s class took the same ﬁnal exam; her class had a mean score of 80 and a median score of 82. Student

Score

Student

Score

Student

Score

100

4. Which class had a higher mean score?

5. Which class had a higher median score?

424 | Statistics

Statistics

Skills Quiz 6. Which choice best describes this dot plot?

Ages of Employees at Caroline’s Confections

A. Symmetrical

B. Skewed right

C. Skewed left

D. Center at 30

Statistics | 425

Statistics

Skills Quiz Refer to the following information and table to answer questions 7–9.

Two rival basketball teams, the Marvels and Spacers, each measure the vertical jumps of 10 of their players. The following table shows the measurements. Marvels

Spacers

40 in.

38 in.

37.5 in.

37 in.

37.4 in.

37 in.

34 in.

36 in.

30 in.

29 in.

30 in.

26 in.

24 in.

23 in.

22 in.

7. What is the median vertical jump measurement for the Marvels?

8. What is the difference in the mean between the Marvels and the Spacers?

9. Which team’s data has a smaller range?

426 | Statistics

Statistics

Skills Quiz

10. A recreational center begins hosting shuffleboard games. The first month, the center tracks how many visitors of different age groups come to play shuffleboard. The following bar plot shows the data recorded.

Number of players

Shuffleboard Players by Age 80 60 40 20 0

10–19

20–29

30–39

40–49

50–59

60–69

70–79

Age

Based on the data, what type of spread does the data exhibit, and where would the center most likely advertise their shufﬂeboard games ﬁrst? A. Uniform distribution; a coffee shop B. Skewed left; a retirement home C. Skewed right; a museum D. Bimodal; a community college

Statistics | 427

Model Data

Skills Quiz

Name: _______________________ Date: ___________

Model Data Directions: Solve each problem. Show or explain your mathematical thinking. 1. Each of the following values represents the correlation coefﬁcient calculated for a regression equation. −0.965, 0.459, 0.9468, −0.834 Rank them in order from best to worst ﬁt.

2. The strength of the linear association between two variables x and y is being determined by calculating their correlation coefﬁcient, r. Which of the following results would indicate the strongest linear association between the variables?

A. r = −0.9244 B. r = −0.8491 C. r = 0.8742 D. r = 0.9062

Model Data | 429

Model Data

Skills Quiz

3. The two graphs below represent the same set of data. Which of the graphs demonstrates the line of best ﬁt? Circle your answer.

4. In a study performed on lab rats, two conditions were identiﬁed and compared.

a. 98% of the rats with condition A also showed evidence of condition B. b. 28% of rats with condition B also showed evidence of condition A. What can be inferred about the relationship between condition A and condition B? A. Condition B is caused by condition A. B. Condition A is caused by condition B. C. Condition A is not associated with condition B. D. Condition A is associated with condition B, but it is not a causal relationship.

430 | Model Data

Model Data

Skills Quiz

5. What type of correlation does the graph below show? 10

x 0

A. Positive correlation B. Negative correlation C. No correlation D. Vertical correlation

6. What type of function is most likely going to ﬁt the set of data on the graph below?

120

A. Exponential 90

B. Linear 60

C. Quadratic 30

D. None 0

Model Data | 431

Model Data

Skills Quiz

7. The scatterplot displays the money Syad earned from dog walking last week. Draw a line of best ﬁt on your graph. Then, write the equation for the line of best ﬁt.

Money Earned from Dog Walking y

Money earned (dollars)

11 10 9 8 7 6 5 4 3 2 1 0

x 1

9 10

Time (hours)

8. Does the data from question 7 show association and/or causation?

432 | Model Data

Model Data

Skills Quiz

9. Sarai was examining the average high temperature, f( f x), in degrees Fahrenheit for each month, x, in her hometown. She used her calculator’s quadratic regression feature to display the information below. Use the information to generate a quadratic regression equation. Round to the nearest hundredth.

QuadReg y = ax2 + bx + c a = –1.26023976 b = 16.83066933 c = 36.36363636 R2 = 0.9418529739

10. A test preparation company plots the SAT scores of 1,000 students, y, with respect to their time spent studying in hours, x. The company creates the line of best ﬁt: y = 45.3x + 720. Which of the following is true about the line of best ﬁt?

A. The y-intercept predicts that every hour spent studying improves SAT scores by 45.3 points. B. The slope predicts that every hour spent studying improves SAT scores by 45.3 points. C. The average score on the SAT was 720. D. The y-intercept means that everyone who did not study received at least a 720.

Model Data | 433

GLOSSARY OF TERMS absolute value

angle-angle criterion

absolute value: the distance a number

additive inverse: what must be added to

is from zero on a number line; also

a number in order for the sum of the two

called the magnitude of a number; never

numbers to be zero

negative adjacent angles: two angles that have absolute value equation: an equation

the same vertex and a common ray but

in which x is c units from b in either

no interior common points

direction algebraic expression: numbers, absolute value function: a function that

variables, and symbols grouped together

contains an algebraic expression within

without an equal sign to show a

absolute value symbols

relationship

absolute value inequality: an inequality

algorithm: a step-by-step method for a

in which the distance from x to b is less

solution

than/greater than c altitude: the height of a polygon acute angle: an angle that measures less than 90°

amplitude: the height from the center line to the peak (or to the trough)

acute triangle: a triangle where every angle measures less than 90°

angle: a geometric figure formed by two rays with the same endpoint (vertex)

addends: the numbers added together to form a sum; any numbers being added

angle-angle criterion: the criterion which states that if two triangles have

addition property of equality: the

two pairs of congruent angles, then the

mathematical property which states that

triangles are similar

435

GLOSSARY OF TERMS angle measure

bar graph

angle measure: the measure of the

association: the form (linear/nonlinear),

angle formed by the two rays from a

direction (positive/negative/none), and

common vertex

strength (weak/moderate/strong) seen between two variables in a scatterplot

angle sum theorem: the theorem which states that the sum of the three interior

associative property of addition: the

angles of a triangle is equal to 180°

mathematical property which states that when adding three or more numbers, the

approximate: to find a number that is

placement of the grouping symbols does

close to the given number on a number line

not affect the sum, e.g., (a + b) + c = a + (b + c)

arc: a part of the circumference of a circle or a section of a curve

associative property of multiplication: the mathematical property which states

area: the number of square units it takes

that when multiplying three or more

to cover the two-dimensional surface of

numbers, the placement of the grouping

an object

symbols does not affect the product, e.g.,

area model: a model where the length

(a × b) × c = a × (b × c)

and width represent the factors and

asymptote: a line that a graph

are configured through the operation of

approaches but never crosses as the

multiplication

value of a variable becomes extremely

arithmetic pattern: a number pattern

large or small

that changes at the same rate, either

axis of symmetry: the line that divides

increasing or decreasing

a figure into two identical parts that are

arithmetic sequence: a sequence where

mirror images of each other

the successive terms differ by the same

bar graph: a graph that uses horizontal

number d, called the common difference,

or vertical rectangular bars to show each

where d ≠ 0

category of qualitative data

436

GLOSSARY OF TERMS base

chance

base: (1) the lower number of an

budget: a financial plan that estimates

exponent that is multiplied by itself;

expenditure for a certain period of time

(2) the surface that a solid object stands on categorical data: a type of data that can base of a polygon: the polygon side that

be divided into groups

is perpendicular to the altitude categorical variable: nonnumerical data base of a triangle: the triangle side that

represented by a letter or symbol

is perpendicular to the altitude category: a collection of objects with benchmark fraction: a familiar fraction

shared attributes

used as a reference point in order to measure, compare, and assess the

causation: the action of one event

reasonableness of a fractional value

causing another event to occur

binomial: a polynomial expression

center: referring to measures of center in

containing two terms

data collection

bivariate categorical data: data for two

center of a circle: the point that is an

nonnumerical variables

equal distance from any point on the circle

bivariate data: data for two variables that are paired to each other

center of a data set: a value in the middle of a distribution that represents a

boundary line: a line that corresponds

typical value of the data set

to the function that divides the coordinate plane into two halves

central angle: an angle in a circle with its corner in the circle’s center

box plot: a diagram that shows the fivenumber summary of a distribution

chance: the possibility of something happening

437

GLOSSARY OF TERMS circle

commutative property of addition

circle: a closed round figure in which

commission: money earned for selling a

every point on the boundary is equidistant

product, usually earned as a percentage

from the center

of the sales

circumference: the distance around a

common coefficient: when a variable

circle

has the same coefficient in two or more equations regardless of the sign

classify: to arrange into groups according to shared characteristics

common denominator: a denominator that is the same in two or more fractions

clockwise rotation: rotating in the direction in which the hands of a clock

common difference: the nonzero

normally move

constant difference, d, of any term and the previous term in an arithmetic

cluster: a group of data occurring closely

sequence

together on a graph common factor: a factor that two or coefficient: the number placed directly

more numbers share

before a variable that tells you to multiply that number by the variable

common multiple: a multiple that two or more numbers share

coinciding lines: lines that lie one on top of the other; the same line with the

common ratio: the ratio of each term

equations expressed in different forms

of a geometric progression to the term preceding it

combine like terms: to add together terms that have the same variable(s),

commutative property of addition: the

including their exponent

mathematical property which states that when adding two or more numbers, the order of the addends does not affect the sum; a + b = b + a

438

GLOSSARY OF TERMS commutative property of multiplication

constant

commutative property of

compound event: a combination of two

multiplication: the mathematical

or more simple events (with two or more

property which states that when

outcomes)

multiplying two or more numbers, the order of the factors does not affect the

compound interest: interest calculated

product; a × b = b × a

multiple times in a given time period so that interest is calculated on the original

complementary angles: two acute

amount and previous interest

angles that, when added, make 90°; two angles whose sum is 90°

conditional relative frequency: the fraction used to express the ratio of the

complete the square: the process used

number of participants in a group that

to form a perfect square trinomial for

meet a certain qualification

the purpose of finding the solution(s) by taking the square root

cone: a solid (three-dimensional) shape that has a flat, circular base joined to a

complex fraction: a fraction where the

point (vertex) by a curved side

numerator and/or the denominator are fractions

congruent: having exactly the same shape and size; being identical; congruent

complex solutions of a quadratic

objects coincide when they overlap.

equation: in the form a + bi; solutions that occur when the value under the

congruent angles: angles that have the

radical of the quadratic formula is less

same measure

than zero congruent figures: figures with the composite figure: a figure that consists

same size and shape

of two or more geometric shapes constant: a fixed number that stands composite number: a number with more

alone in an equation or expression

439

GLOSSARY OF TERMS constant of proportionality

correlation coefficient

constant of proportionality: the

convert: to change the form of a

positive constant, usually denoted k, that

measurement using different units

relates two quantities in the form y = kx

without changing the size or amount of the quantity being measured

constant of variation: the constant (unchanged) ratio of two quantities; in

coordinates: a pair of numbers that

direct variation, it is usually denoted as k.

provides the location of a point along the coordinate plane using the values of the

constant rate of change: a rate of

x-axis and y-axis

change that does not vary coordinate pair: the location of a single constant speed: the rate of fixed speed

point on a coordinate plane where the

per time

first and second values represent the position relative to the x-axis and y-axis,

constraint: a condition that the solution

respectively (x, y); also known as ordered

must satisfy

pair

continuous: data that can contain any

coordinate plane: two perpendicular

real number value between data points;

number lines, called the x-axis and the

data points can be connected.

y-axis, that intersect at the point (0, 0)

converse of the Pythagorean theorem: the theorem which states that

and create four quadrants; also called a graph, coordinate grid, or Cartesian plane

if the square of the length of the longest

correlation: the relationship between two

side of a triangle is equal to the sum of

variables that vary together

the squares of the other two sides, then the triangle is a right triangle; if c² = a²

correlation coefficient: a number r

+ b², then it is a right triangle.

that describes how closely the points in a scatterplot are related, where −1 ≤ r ≤ 1

440

GLOSSARY OF TERMS corresponding angles

data point

corresponding angles: angles in the

cube root: a number that, when

same position in different plane figures

multiplied by itself three times, produces the given number

corresponding congruent angles: angles in identical positions formed by a

cube root function: a function of the

transversal line cutting through two lines

form f( f x) =

corresponding sides: two sides that are

cubic number: a number to the power

in the same position in different plane

of three, i.e., 2³ represents the cubic

figures; in scale drawings, these sides will

number 8 and can be read as two cubed

have a proportional relationship.

or two to the power of three.

corresponding similar sides: sides in

cylinder: a solid (three-dimensional)

matching positions of similar figures that

shape that has two flat, circular, parallel

have a proportional relationship

bases joined by a curved surface at a

fixed distance counterclockwise rotation: rotating in the opposite direction in which hands of a

data: a collection of organized facts,

clock normally move

usually in numerical form, words, measurements, or descriptions

credit: a positive money value data distribution: a function or a listing cross-section: a two-dimensional shape

that shows all the possible values (or

that is created when a three-dimensional

intervals) of the data

shape is sliced data point: a point on a scatterplot that cube: a solid figure with six congruent

represents the data

square faces

441

GLOSSARY OF TERMS data set

difference

data set: a collection of organized facts,

degree (°): the unit of measure for an

usually in numerical form, but can also

angle

be given in words, measurements, or descriptions

degree of a polynomial: the largest exponent or the largest sum of exponents

debt: money that is owed; describes a

of a term within a polynomial

person’s bank account balance when it is less than zero

denominator: the bottom number within a fraction; the number that represents

decimal: a number that uses a decimal

the whole and how many parts total are

point followed by digits that show a value

in the whole

smaller than one, in powers of ten that decrease; a number with one or more

dependent variable: a variable,

digits to the right of the decimal point

often y, that relies on the value of the independent variable

decimal expansion: the decimal form of a number

deposit: a sum of money that is put into a bank account

decimal notation/decimal form: a number that uses a decimal point followed

deviation: the amount by which a single

by digits showing values less than one

measurement differs from a fixed value

decompose: to separate into parts or

diameter: any straight line segment that

elements (e.g., geometric figures or

passes through the center of the circle

numbers)

and has endpoints that lie on the circle

decreasing: the measure of the

difference: a number that is the result of

steepness of a line that shows the slant

subtraction

downward from left to right

442

GLOSSARY OF TERMS difference of two squares

dividend

difference of two squares: the

distance: a measurement of the length

difference of two squares, such as a² – b²

between two points

being factored into (a + b)(a – b) distance formula: the formula used to digit: any one of the numbers 0–9 dilation: a type of transformation where

find the distance, d, between two points (x1, y1) and (x2, y2) on the coordinate plane;

a scale factor is used to enlarge or reduce the distances in the original image

distance-time graph: a graph that shows the distance traveled by an object

dimension: something measurable (such

against the time it takes; any given

as length, width, and height)

point represents the speed of the object

direct variation: a relationship between two variables including a constant (k) discount: the amount subtracted from the original cost of an item discrepancy: a lack of compatibility or similarity between two or more things discrete: data that cannot contain the real number values between data points; data points are not connected. discriminant: the expression under the square root of the quadratic formula that determines the types of solutions of a quadratic equation

(distance per time). distribution: a list of all the possible values of the data and how often they occur distributive property: the mathematical property which states that multiplying the sum or difference of a group of terms by a number or variable is the same as multiplying each term by a number or variable and then adding or subtracting the products dividend: the number you divide into; a quantity that is to be divided by another quantity; a number that shows the amount of equal parts of a whole; the numerator (top number) that tells the number or quantity; a quantity to be divided

443

GLOSSARY OF TERMS division property of equality

equivalent

division property of equality: the

elimination method: a method of

mathematical property that states that

solving systems by adding or subtracting

dividing both sides of an equation by

equations to eliminate a variable

the same number gives us an equivalent equation

end behavior: the trend the graph follows as x approaches infinity in the

divisor: the quantity by which another

negative and positive directions

quantity is to be divided endpoint: the point at the end of a line domain: the set of all possible input

segment or ray

(x x values) of a function enlarge: to create a similar image that is dot plot: a method of visually displaying

now larger than the original image

a distribution of data values where each data value is shown as a dot or mark

equal sign: the symbol used to show

above a number line

that two quantities or expressions are the same

double number line diagram: a pair of parallel number lines used to represent

equal to (=): having exactly the same

equivalent ratios

amount or value

downward: the direction a parabola

equation: a mathematical statement that

opens when the value of a < 0

shows that two expressions are equal to each other

edge: a line at which a space or shape terminates, where two faces of a 3-D solid

equilateral triangle: a triangle with

intersect

three congruent sides and three congruent angles

element: any distinct number or value that is part of a set

444

equivalent: equal in value or amount

GLOSSARY OF TERMS equivalent expressions

expression

equivalent expressions: expressions

exponent: a mathematical notation that

that name the same number no matter

indicates the number of times the base

what value is substituted for the variable

number is multiplied by itself; also called power

equivalent ratios: two or more ratios that are equal; two different ratios

exponential decay: the process of

representing the same value

reducing an amount by a consistent percentage rate over a period of time

estimate: an approximation of an overall amount or value

exponential expression: an expression involving a term with a variable as an

evaluate: to determine or calculate the

exponent; 2x for example

numerical value of something exponential function: a function in the even function: when x is replaced with

form of f( f x) = abx where a and b are real

−x x in a function and the function is

numbers and a ≠ 0, b ≠ 1, and b > 0

simplified, the resulting function will be identical to the original function.

exponential growth: the change that occurs when an original amount is

event: one (or more) outcome(s) of an

increased by a consistent rate over a

experiment

period of time

experimental probability: the ratio that

exponential notation: an expression

compares the number of occurrences to

that takes the form aⁿ, where a is

the number of trials

multiplied by itself n times

explicit formula: a formula to find the

expression: numbers, variables, and

nth term of a sequence

symbols grouped together without an equal sign to show a relationship

445

GLOSSARY OF TERMS exterior angle of triangles theorem

gap

exterior angle of triangles theorem: the

force of gravity: the universal force of

mathematical theorem which states that

attraction acting between all matter

an exterior angle is equal to the sum of the two opposite interior angles of a triangle

formula: a mathematical statement or rule written with symbols

factor: A number or algebraic expression that another number or algebraic

fraction: a number that shows a part of a

expression can be divided by without

whole or part of a set

having a remainder frequency: how often a number occurs in factors: expressions that are multiplied

a data set

together to get a polynomial; factors that appear in the form of ax + b and cannot

frequency table: a table that lists

be factored further

outcomes and the number of times that they occur

factor pair: a set of two factors that multiply to give a particular product;

function: a special relationship between

listing factor pairs is a strategy used to

values; each input value gives back

determine all the factors of a number.

exactly one output value.

factor tree: a mathematical tool to help

function notation: a way of representing

break down a number into its prime

y, the dependent value in a relationship,

factorization

as f( f x), read “ff of x” where f names the function

figure: a two-dimensional shape function rule: the dependent variable five-number summary: the five values

(range, output, y value) expressed

used to make a box plot, including the

in terms of the independent variable

lowest value, lower quartile, median,

(domain, input, x value)

upper quartile, and highest value gap: a missing range of values in a data set 446

GLOSSARY OF TERMS geometric sequence

horizontal reflection

geometric sequence: a sequence in

half-plane: a planar region consisting of

which the ratio of successive terms is

all points on one side of an infinite straight

a constant r, called the common ratio,

line, and no points on the other side

where r ≠ 0 and r ≠ 1 height: the perpendicular distance from a graph: a visual representation of data

vertex to the opposite side of a figure

graph of a quadratic function:

height (3-D figure): the vertical

the attributes of a quadratic function

distance from the top of an object or

including the vertex, the y-intercept, the

figure to its base

x-intercepts, and the axis of symmetry histogram: a special type of bar graph graphing method: a method of solving

with numerical intervals as its labels

systems by graphing horizontal: describes the direction of gratuity: money given above the amount

a line that travels from left to right,

charged for a service; tip

perpendicular to a corresponding vertical line; from left to right; parallel to the

greater than (>): more than another

horizon

(e.g., 49 > 12) horizontal dilation: the act of expanding greater than or equal to (≥): more

or contracting in the horizontal direction

than or the same as another horizontal number line: describes the greatest common factor: the largest

direction of a horizontal number line that

same factor of two or more numbers

travels from left to right, perpendicular to a corresponding vertical line; from left to

grouping symbols: symbols that help to

right: parallel to the horizon

organize mathematical expressions; braces { }, brackets [ ], and parentheses ( )

horizontal reflection: a reflection over a vertical line such as the y-axis

447

GLOSSARY OF TERMS horizontal shift

inequality phrase

horizontal shift: a change in a function

increasing slope: the measure of the

that moves the function left or right

steepness of a line that shows the slant upward from left to right

horizontal translation: a shift in the base of the graph to the left or right

increasing/decreasing: a function is increasing if f( f b) > f( f a) and decreasing

hundredths: the second digit to the right

if f( f b) < f( f a) for any two input values a

of the decimal point; a hundredth is one

and b.

out of 100 equal parts of a whole. increments: the evenly spaced and scaled hypotenuse: the longest side of the right

markings used to locate and plot points

triangle, the side opposite of the right angle independent variable: a variable, often identity property of addition: the

x, that does not rely on the value of

mathematical property which states that

another variable

adding zero to a number does not change the value

index: a number indicating how many of a kind you need to put together to be able

identity property of multiplication: the

to move that number or variable from

mathematical property which states that

inside the radical to outside the radical

multiplying 1 by any number does not change the value

inequality: a mathematical sentence that uses symbols such as <, ≤, >, or ≥ to

image: the new figure in a transformation

compare two quantities

improper fraction: a fraction that has a

inequality notation: notation in

numerator that is greater than or equal to

which the solution is represented by an

the denominator

inequality statement

increasing: when the y value increases

inequality phrase: phrase representing

as the x value increases

each of the inequalities

448

GLOSSARY OF TERMS inference

inverse function

inference: a conclusion based on the

integer exponent: a positive or negative

given data

whole number or zero that tells the number of times a base is multiplied by

infinite: having an unlimited number of

itself

values intercept: the point where the line on a infinite number: the concept of

graph crosses the x-axis or y-axis

something that is unlimited, endless, without bound

interest: money that is a percentage of an original amount typically owed as part

infinite solutions: in systems of

of a debt

equations, coinciding lines have infinite solutions.

interquartile range (IQR): the difference between the upper quartile

input: the set of values supplied to a

(Q3) and the lower quartile (Q1)

function intersecting lines: lines that cross at a input-output pair: an ordered pair

point

in which the input corresponds to the independent variable in the left column

intersection: the point at which two lines

of a function table and the output

cross

corresponds to the right column of a function table; an ordered pair is

interval: the set of continuous input

determined by evaluating the function

values on which a function’s outputs could

using the input.

be increasing, decreasing, or constant

integer: any one of the positive whole

inverse: the opposite number or

numbers, negative whole numbers, and

operation

zero; any member of the set of all whole numbers and their opposites

inverse function: a function that undoes the action of another function

449

GLOSSARY OF TERMS inverse operation

less than or equal to (≤)

inverse operation: the operation that

laws of exponents – multiplication

reverses the effect of another operation

of same bases: can be rewritten as the base raised to the sum of the powers

inverse property of addition: the mathematical property that states that

laws of exponents – negative

when you add a number to its opposite,

exponents: can be rewritten as the

you will always get zero as the sum

multiplicative inverse of the base raised to the positive opposite of the power

inverse property of multiplication: the mathematical property that states

laws of exponents – zero exponents:

that when you multiply a number by its

the mathematical law which states that

reciprocal, you will always get 1

any number raised to the power of zero equals one

irrational number: a decimal number that cannot be expressed as a fraction,

least common multiple: the smallest

is not imaginary, and does not repeat or

multiple that is the same in a set of two

terminate

or more numbers

isosceles triangle: a triangle with two

leg: either of the two sides in a right

or more congruent sides where angles

triangle that form the right angle and are

opposite of the congruent sides are

opposite of acute angles

congruent angles length: the measure of an object from joint frequency: the ratio of the

end to end; the distance from one end to

frequency in a particular category and the

the other end of an object

total number of data values less than (<): smaller than another laws of exponents – division of same

(e.g., 432 < 501)

bases: can be rewritten as the base raised to the difference of the powers

less than or equal to (≤): smaller than or the same as another

450

GLOSSARY OF TERMS like terms

magnitude

like terms: terms that have the same

linear expression: an expression in

variables, including their exponents

which all terms have an exponent of one

likelihood: the probability that an event

linear function: a relationship that when

will occur; also called chance

graphed is a straight line

line: a straight geometric element with

linear graph: a series of points

no thickness, extending endlessly in both

connected on the coordinate plane,

directions; the shortest distance between

forming a straight line that shows a

two points

relationship or rate of change

line of best fit (trend line): a line that

linear inequality: an inequality that

best represents the data on a scatterplot

involves a linear function

line plot: a graph that displays data as

linear parent function: the simplest

points above a number line, to show the

equation of the linear function, y = x or

frequency of each value

f x) = x f(

line segment: a section of a line with two

linear relationship: having a constant

distinct endpoints

rate of change between two quantities/ variables and making a straight line when

linear association: a proportional

graphed; a relationship that creates a

relationship that creates a straight line on

straight line

a graph long division: an algorithm used to find linear equation: an equation in which no

the quotient of two numbers

variable has a power greater than 1; the general form is y = mx + b, where m =

magnitude: the absolute value or

slope and b = y-intercept.

distance to zero

451

GLOSSARY OF TERMS mapping

measures of variability

mapping: a function represented by

measure: a number of units that shows

two sets of objects with arrows drawn

the amount or size of something

between them to show relationships between the objects or data

measure of center: a single value used to represent/summarize a collection

marginal frequency: the ratio of the

of data; three commonly used types

sum of the joint relative frequency in a

are mode, median, and mean; also

row or column and the total number of

called measures of central tendency or

data values

measures of average

markdown: a decrease in the cost of an

measurement: a number that shows the

item; a discount

size or amount of something

markup: an increase in the cost of an

measure of variation: a measure of

item to make a profit

how data is spread out, usually including range, interquartile range, variance, and

maximum: the greatest or highest

standard deviation

amount possible or attained measurement system: one of two main maximum value: the place where a

systems of measurement—the metric

function reaches its highest point, or

system and the standard or customary

vertex, on a graph

system, each of which uses different units to measure distance, mass, and volume

mean: the average of a set of numbers calculated by finding the sum of all data

measures of variability: measures of

and dividing by the number of data values

how data is spread out, usually including range, interquartile range, variance, and

mean absolute deviation: the average

standard deviation

difference between the mean and each data point

452

GLOSSARY OF TERMS median

multiplication property of equality

median: the middle number of a set of

mode: the number or value that appears

numbers when the numbers are arranged

the most frequently in a data set

from least to greatest, or the mean of the two middle numbers when the set has two

monomial: an expression containing only

middle numbers

one term

midpoint formula: the formula used to

multi-digit: a number that has more

calculate the point on a line segment that

than one digit

is equidistant from the endpoints (x1, y1) and (x2, y2) on the coordinate plane;

multiple: a product of two integers; one of the numbers that result from multiplying a whole number by the set of whole numbers

minimum: the least or smallest amount or

multiple representations: different

quantity possible, attainable, or required

mathematical ways to represent a relation or a function

minimum value: the place where a function reaches its lowest point, or

multiplicand: the number that is

vertex, on a graph

multiplied by another number; a quantity that is to be multiplied by another quantity

minuend: a number or quantity from which another number is to be

multiplication: a mathematical operation

subtracted; for example, in the equation

consisting of obtaining a product or

7 – 4 = 3, the number 7 is the minuend,

result by joining equal groups, repeated

the number you subtract from.

addition, or forming arrays

mixed number: a whole number and a

multiplication property of equality:

fraction combined; a number made up of

the mathematical property that states

a whole number and a fraction

that multiplying the same number by each side of an equation gives us an equivalent equation

453

GLOSSARY OF TERMS multiplicative comparison

non-proportional relationship

multiplicative comparison: shows the

negative exponent law: the

relationship between two amounts, where

mathematical law which states that any

one quantity is a certain number of times

nonzero number raised to a negative

as large as another quantity; a number is

exponent is equivalent to the reciprocal

multiplied by another number to result in

of the base raised to the opposite of the

a greater or lesser quantity.

negative exponent

multiplicative identity property: the

negative number: a number that is less

mathematical property which states that

than zero

the resulting product of any number and 1 is equal to the original number

negative reciprocal: the result of multiplying the reciprocal by −1

multiplicative inverse: one of two numbers whose product is 1; also called

negative slope: the measure of the

the reciprocal

steepness of a line that shows the slant downward from left to right

multiplier: the number you multiply by; the quantity that the multiplicand

net: a two-dimensional shape that when

is multiplied by; the number being

folded represents a three-dimensional

multiplied

figure

multistep problem: a mathematical

nonlinear association: a relationship

problem involving more than one

that does not create a straight line

operation nonlinear function: a relationship that negative association: a relationship

when graphed does not make a straight

between two variables that move in

line; a relationship that does not create a

opposite directions

straight line; nonlinear association non-proportional relationship: two quantities that do not have equal ratios

454

GLOSSARY OF TERMS nonvertical line

ordered pair

nonvertical line: a line that is horizontal

obtuse angle: an angle that measures

or diagonal

greater than 90°

no solution: in systems of equations,

obtuse triangle: a triangle that contains

parallel lines have no solution.

one obtuse angle and two acute angles

number line/number line diagram:

odd function: when x is replaced with

a line on which numbers are marked at

−x x in a function and the function is

intervals

simplified, the terms in the resulting function have the opposite signs of those

numerator: the top number within a

in the original function.

fraction, which represents the part of the whole

one solution: in systems of equations, intersecting lines have one solution (x, y).

numeric expression: a mathematical sentence that uses numbers and one or

opposites: numbers the same distance

more operation symbols

away from zero, located on different sides of zero

numerical data: data comprised of numbers, measurements, or quantities

order of operations: a set of rules that dictate which mathematical operation to

numerical radical expression: any

perform first, second, and so on when

numerical expression that contains a

evaluating a mathematical expression

radical ordered pair: the location of a single numerical reasoning: a process

point on a coordinate plane where

using numbers and quantities to draw

the first and second values represent

conclusions

the position relative to the x-axis and y-axis, respectively (x, y); also known as

observation: the value of what is being

coordinate pair

455

GLOSSARY OF TERMS organized data list

percent decrease

organized data list: elements listed in a

part-to-part ratio (comparison): a

particular sequence or order

relationship between one part of a whole and another part of a whole

origin: the center point of a coordinate plane, where the x-axis and y-axis

part-to-whole ratio (comparison): a

intersect, located at (0, 0)

relationship between one part of a whole and the total number of parts in the

outcome: the result of an event

whole

outlier: a number in a set of data that

partial product: the product of the

is much larger or smaller than other

multiplicand and one digit of the multiplier

numbers in the set pattern: a repeating arrangement of output: the result of the input placed in

numbers or shapes

the function pattern of association: a relationship parabola: the shape that a quadratic

between data sets

equation takes when graphed peak: the highest value(s) in a set of data parallel: existing in the same plane and equidistant and not intersecting

per (unit rate): a ratio for an amount for one unit of the other quantity

parallel lines: lines in the same plane that are equidistant and do not intersect

percent: a special ratio that compares a number to 100 using the percent symbol,

parallelogram: a quadrilateral with two

%; a rate per 100

sets of parallel sides percent decrease: the amount by which parameter: a quantity that influences

the cost decreased from the initial value,

the output or behavior of a mathematical

expressed as a percent

object but is viewed as being held constant 456

GLOSSARY OF TERMS percent error

plot

percent error: the measure of how far

periodicity: the tendency of a function

off an estimated value is from the true

to repeat itself in a regular pattern at

value, expressed as a percent

established intervals

percent increase: the amount by which

perpendicular: having the position of

the cost increased from the initial value,

two lines that intersect at a right angle;

expressed as a percent

intersecting at a 90° angle

percent rate of change: the percentage

perpendicular lines: two lines that

increase or decrease of an amount over a

intersect at a 90° angle

unit of time, denoted by r pi: a constant which is found by dividing percentage: a special ratio that

the circumference of a circle by its

compares a number to 100 using the

diameter; approximately 3.142

percent symbol, %; a rate per 100 piecewise function: a function that is perfect cube: an integer that is the result

defined by different formulas at different

of another integer times itself three times

inputs

perfect square: an integer that is the

place value: the numerical value that a

result of another integer times itself

digit has, based on its position within a number

perfect square trinomial: a trinomial whose factored form is the square of a

plane: a flat, two-dimensional surface

binomial; takes the form ax² + bx + c

that continues indefinitely

and satisfies the condition b² = 4ac plot: to indicate the position a number perimeter: the distance around the

is relative to zero on a number line or

outside of a figure or shape

relative to the origin on a coordinate plane

457

GLOSSARY OF TERMS point

power of a power law

point: a dot that represents a specific

positive association: a relationship in

spot on a number line or coordinate

which the values of one variable tend

plane; a geometric object with no

to increase as the values of the other

dimension used to indicate a location

variable increase

point of intersection: the point where

positive number: a number that is

two or more lines cross each other

greater than zero

point-slope form: an equation written

positive rational number: a number to

in the form of y – y1 = m(x x – x1), where

the right of (or greater than) zero that

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

can be expressed as a fraction of two

contained in the line

integers

polygon: a closed figure that has three

positive slope: the measure of the

or more sides, no curved lines, and no

steepness of a line that shows the slant

intersections; a closed figure formed by

upward from left to right

line segments that meet at their endpoints power: a mathematical notation that polynomial: a mathematical expression

indicates the number of times the base

consisting of several terms

number is multiplied by itself; also called an exponent

population: a discrete group for the purposes of data collection and analysis

power law: the distribution of an exponent through multiplication to all

positive/negative interval: positive

parts of the base

intervals are those above the x-axis; negative intervals are those below the

power of a power law: the

x-axis.

mathematical law that states that when raising a base with an exponent to another exponent, the exponents are multiplied and the base stays the same

458

GLOSSARY OF TERMS power of one law

protractor

power of one law: the mathematical

probability model: a mathematical

law which states that any number to the

description of an experiment that lists

power of one is equal to that number

all of the possible outcomes and their probabilities

power of zero law: the mathematical law that states that any number to the

product: the solution when multiplying

power of 0 is equal to 1

two or more numbers; the answer to a multiplication problem

prediction: a reasonable guess as to what will happen

proof: evidence or argument establishing a fact or the truth of a statement

preimage: the original figure in a transformation

product of powers law: the mathematical law which states that when

prime number: a number with exactly

multiplying two exponents with the same

two factors—one and itself

base, the exponents are added together

prime factorization: a given set of prime

and the base stays the same

numbers that when multiplied together

proportion: two fractions or ratios that

equals the original number

are equal in value; a type of equation that

prism: a three-dimensional figure that

shows that two ratios are equal

has at least one set of congruent, parallel

proportional corresponding sides:

faces (bases) that are polygons with

sides in the same position in two similar

parallelograms as the remaining faces

polygons that are proportional

probability: the likelihood that something

proportional relationship: when two

will happen

quantities have the same ratio protractor: a mathematical tool for measuring and drawing angles

459

GLOSSARY OF TERMS pyramid

radius

pyramid: a three-dimensional figure in

quantitative data: numerical or

which the base is any polygon and the

measured data that is analyzed for

other faces are triangles that share a

statistical purposes

common vertex quantitative relationship: the Pythagorean theorem: a theorem that

relationship between magnitudes

states that the square of the hypotenuse is equal to the sum of the squares of the

quantity: a number or amount; an

other two sides of a right triangle; a² +

amount that tells how much

b² = c² quotient: the solution when dividing quadrant: one of four sections of

two numbers; the answer to a division

the coordinate plane, formed by the

problem; the result of the division of one

intersection of the x-axis and y-axis

quantity by another quantity

quadratic formula: the formula

quotient of powers law: the

, which gives the

mathematical law that states that when

solutions of equations in the form of ax²

dividing two exponents with the same

+ bx + c = 0, where a ≠ 0

base, one subtracts the exponents and keeps the base the same

quadratic function: a function that can be written in the form f( f x) = ax2 + bx +

radical: a symbol that indicates the root

c, where a, b, and c are real numbers and

of a quantity

a≠0 radicand: the value inside the radical quadratic parent function: the simplest

symbol

equation of the quadratic function, y = x² or f( f x) = x²

radius: the distance from the center of a circle or a sphere to any point that lies on

quadrilateral: a polygon with four sides

the circle or the sphere

and four angles 460

GLOSSARY OF TERMS random sampling/random sample

real-world problem

random sampling/random sample: a

ratio table: a list of pairs of equivalent

selection chosen by chance and which has

ratios used to determine the relationship

no predictability

between the ratios

range: (1) the difference between the

rational exponent: an exponent that can

maximum and minimum values within a

be expressed as

data set; (2) the set of all possible output,

a radical expression where m and n are

or y values, of a relation or function

integers and m represents the power

as a way to rewrite

of the base and n represents the root; rate: a type of ratio where the quantities have two different units rational number: a number that can rate of change: the rate that shows

be written as a fraction of integers a/b,

how one quantity changes in relation to

where b ≠ 0; a number that can be

another quantity

written as a ratio using two integers

ratio: a comparison of two quantities

ray: part of a line with a fixed starting

that shows their sizes in relation to one

point and no endpoint

another real number: any one of the set of all ratio language: language used to

rational and irrational numbers

mathematically describe the relationship between any two units that are being

real solution: a value that satisfies the

compared in a ratio using the phrase for

equation; called roots, x-intercepts, or

every… there are… or the word to

zeros

ratio relationship: equivalent ratios

real-world problem: a contextual-

form a ratio relationship between the two

based problem that can be interpreted,

quantities being compared

represented, and analyzed through the application of mathematics

461

GLOSSARY OF TERMS reciprocal

right angle

reciprocal: one of two numbers whose

relative frequency: how often a number

product is 1; also called the multiplicative

occurs in a data set divided by the total

inverse

number of outcomes

rectangle: a parallelogram with opposite

relative maximum: a point that is higher

equal sides and four right angles

than the points directly beside it on both sides

recursive formula: a formula that defines each term of a sequence using

relative minimum: a point that is lower

preceding term(s)

than the points directly beside it on both sides

recursive process: the calculation of the next number in a sequence by repeated

remainder: a leftover quantity resulting

application of a rule

from the quotient of 2 integers

reduction: the creation of a similar image

repeating decimal: a decimal number

that is now smaller than the original image

in which a digit or group of digits is repeated indefinitely, as in 0.333… or

reflect: to transform a point so that it is

1.851851851…

equidistant on opposite sides of the x- or y-axis

representative sample: a sample that matches or reflects a population

reflection: the mirror image of a figure; the flipping of a figure

residual: the difference between the observed y value (from the scatterplot)

regression: the process of drawing a line

and the predicted y value (from the

through data in a scatterplot

regression equation line)

relationship: the rule in a pattern

right angle: an angle that measures 90°

462

GLOSSARY OF TERMS right polygon

scientific notation

right polygon: a polygon with at least

scale: the representation of the

one right angle

relationship between a measurement on a model and the corresponding

right prism: a solid composed of a

measurement on the actual object

polygon as its base and vertical sides perpendicular to the base

scale drawing: a smaller or larger representation of an object that is

right rectangular prism: a prism with six

proportional to the original object

rectangular faces where the lateral edge is perpendicular to the plane of the base

scale factor: the ratio of corresponding side lengths in a scale drawing to those of

right triangle: a triangle with one 90º

the original figure

angle scaled interval: a measurement scale rotation: the turning of a figure around a

used on a graph with the distance

fixed point

between marks being equal and the marks counting by a constant value

rounding: the process of raising or lowering a number to a specific

scalene triangle: a triangle with no

place value position; representing an

congruent sides

approximate worth scatterplot: a series of plotted points ruler: a tool used to measure length and

that show the relationship between two

to draw straight lines

sets of data

sample: one part of the given population

scientific notation: a method of expression used to write very small and

sample space: all possible outcomes of

very large numbers by representing them

an experiment

with decimal numbers between 1 and 10, with each decimal being multiplied to a power of 10

463

GLOSSARY OF TERMS sequence

solution

sequence: an ordered arrangement of

simplest form: (1) the smallest possible

numbers or objects

way to write an equivalent fraction for the fraction given; (2) the smallest way to

set: (1) a collection of objects or things;

write an equivalent expression

(2) a group of unique numbers or objects called members or elements

simplify: to replace a numerical expression with the simplest name for its

shape: a description of the type of graph

value by using the substitution principle

seen, as symmetrical, peaks, skewed, or uniform

simulation: a model of random events

side: the line segment that connects two

skewed data: when data on a graph is

vertices in a figure

not symmetrical; when the graphed data shows a tail on one side or the other

signed number: a positive or negative number; a number that has the sign + for

slope: how steep a line is; represented as

positive or − for negative

m in the slope-intercept equation

similar figures: two or more figures that

slope formula: the formula used to find

are the same shape but different sizes

the slope between two points (x1, y1) and

similar triangles: two or more triangles

(x2, y2) ,

that have congruent angles and

slope-intercept form: a way to write

proportional sides

the equation of a line so that it is easy to view the slope and y-intercept of the line;

simple event: one event at a time with

y = mx + b

one single outcome solution: any number that makes an simple interest: a way to calculate

equation true

interest accrued using the formula I = Prt

464

GLOSSARY OF TERMS solution of a system of inequalities

stem-and-leaf plot

solution of a system of inequalities:

square unit: a unit of area, specifically

the overlapping region that makes both

square centimeters, inches, feet, and

inequalities true

meters

solution set: a set of numbers that

standard deviation: a measure of how

makes an inequality statement true

spread out numbers are; calculated by finding the square root of the variance

sphere: a three-dimensional round figure where every surface point is equidistant

standard form: a way to write numbers

from the center of the figure

by using the digits 0–9, with each digit having a place value

spread: a measure of how far the numbers in a data set are from the mean

standard form (linear): Ax + By = C,

or median; including the commonly used

where A, B, and C are constants and A

types range and quartiles; also known as

and B are not both 0

measures of variation or dispersion standard form (quadratic): y = ax² + square: any number or variable times itself

bx + c or ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0

square number: a number to the power of 2, i.e., 3² represents the square

statistical question: a question that

number 9 and can be read as “three

anticipates differences in data

squared” or “three to the power of two.” statistics: the study of data and square root: a number that, when

collecting, organizing, representing, and

multiplied by itself, produces the given

interpreting data

number stem-and-leaf plot: a plot where each square root function: a function of the

data value is split into a “leaf” (usually

form f( f x) =

the last digit) and a “stem” (the other

, where x is greater than

digits) 465

GLOSSARY OF TERMS step function

system of equations

step function: a piecewise-defined

sum: the solution when adding two or

function where each piece’s formula is a

more numbers; the answer to an addition

constant

problem

straight angle: an angle that measures

supplementary angles: two adjacent

exactly 180°

angles that, when added, make 180°; two angles whose sum is 180°

strict inequality: an inequality that has no equality conditions; the strict inequality

surface area: the total area of each of the

is either greater than or less than.

faces and curved surfaces of a solid figure

subcategory: a category within a category;

survey: a data collection tool or list of

a collection of objects with even more

questions used to gather information

specific characteristics than a category

about individuals or groups of people

substitution: replacing letters in an

symbol: a mark or character used as a

algebraic expression with known values

representation of an object, function, or process

substitution method: a method of solving systems by substituting equations

symmetrical: the relationship between

within one another

objects that are the same size and shape after a flip, slide, or turn

subtraction property of equality: the mathematical property that states that

symmetrical distribution: data that is

subtracting the same number from each

in the shape of a bell; it can be equally

side of an equation gives us an equivalent

divided in half.

equation system of equations: two or more subtrahend: a quantity or number to

equations with two or more variables

be subtracted from another; the number being subtracted 466

GLOSSARY OF TERMS system of inequalities

triangle

system of inequalities: two or more

thousandths: the third digit to the right

inequalities with two or more variables

of the decimal point; a thousandth is one out of 1,000 equal parts of a whole.

table: a chart that uses rows and columns to organize information

three-dimensional figure: a solid having three measurable dimensions

tape diagram: a rectangular visual model that represents equal parts, used to

transformation: changing a shape

model word problems involving part-part-

through movement on a coordinate plane

whole relationships translation: moving a figure along a line tax: a fee added to a good or service,

for a specific distance

usually a percentage of the total transversal: a line that cuts through two tenths: the first digit to the right of the

or more lines in the same plane

decimal point; a tenth is one out of 10 equal parts of a whole.

trapezoid: a quadrilateral with one set of parallel sides

term: (1) a number, a variable, or a product of numbers and variables in

trend: the general direction that data

an expression separated by addition,

points seem to follow

subtraction, or sometimes division; (2) in an algebraic expression, a number

tree diagram: a diagram with connecting

or variable, or a product or quotient of

lines to calculate the number of possible

numbers and variables

outcomes of an event

terminating decimal: a decimal number

triangle: a polygon with exactly three

that has a finite number of digits

straight sides and three angles

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GLOSSARY OF TERMS triangle angle sum property

triangle angle sum property: the

variation

union: a combination of two or more things

mathematical property of a triangle which states that the angles of a triangle always

unit: a type of measurement such as an

add up to 180°

inch, a pound, or a second

triangle inequality theorem: the

unit cube: a cube in which all sides have

theorem that states that the sum of any

a length of one unit

2 sides of a triangle must be greater than the measure of the third side

unit of measurement: a standard amount that is used to measure

trinomial: a polynomial expression containing three terms

unit price: the price of goods per one unit of measure

truncated decimal: a decimal number where some digits are left off and the

unit rate: a rate with a denominator of 1

number is approximated at a certain point

that shows how many units of the first type

without rounding

correspond to one unit of the second type

two-dimensional figure: a flat figure

upward: the direction a parabola opens

with two measurable dimensions

when the value of a > 0

two-way relative frequency table: a

variability: how spread out data is

two-way table that displays percentages or ratios, called relative frequencies

variable: a letter or symbol that takes

two-way table: a chart used to show

a letter that can stand for an unknown

the relationship between two categorical

number or a set of numbers

variables

the place of a number that can change;

variation: how spread out data is

undefined slope: the slope of a vertical line

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GLOSSARY OF TERMS vertex/vertices

x-axis

vertex/vertices: the common point of

vertical number line: a number line that

two rays that form an angle; the common

travels up and down, perpendicular to a

point to any two sides of a polygon

corresponding horizontal line; from top to bottom; perpendicular to the horizon

vertex: the minimum or maximum point in a quadratic function; identified as (h, k)

vertical reflection: a reflection over a horizontal line such as the x-axis

vertex form: y = a(x – h)² + k, where a, h, and k are constants and a ≠ 0

vertical shift: a change in a function that moves the function up or down

vertical: describes the direction of a line that travels up and down, perpendicular

vertical translation: a shift in the base

to a corresponding horizontal line; from

of the graph up or down

top to bottom; perpendicular to the horizon

volume: the amount of space an object occupies; the measured amount of cubic

vertical angles: angles opposite from

units that fit inside a solid figure

one another when two lines cross; opposite congruent angles that are

whole number: a number zero or above

formed on either side of intersecting lines

that contains no fractional or decimal part; a positive number without a

vertical dilation: expansion or

fractional piece

contraction in the vertical direction width: how many units wide something is vertical line test: a visual way to tell whether a line is a function; if any vertical

withdrawal: a sum of money that is

line intersects the graph more than once,

taken out of a bank account

then the graph is not a function. x-axis: a horizontal number line on a coordinate plane

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GLOSSARY OF TERMS x-coordinate

x-coordinate: the first term in an

zero slope

zero slope: the slope of a horizontal line

ordered pair; provides the location along the x-axis within the coordinate plane x-intercept: the x-coordinate or coordinates where a graph intersects the x-axis, identified as (x, 0) y-axis: a vertical number line on a coordinate plane y-coordinate: the second term in an ordered pair; provides the location along the y-axis within the coordinate plane y-intercept: the point on a graph of an equation where the line crosses the y-axis zero: (1) the only integer that is neither negative nor positive and is its own opposite; (2) the value of x where an expression is equal to zero; this is the x-coordinate of the x-intercept of the expression’s graph. zero product property: the mathematical property which states that when multiplying two numbers together results in zero, then either a, b, or both a and b are zero; if ab = 0, then either a = 0 or b = 0 or both 470

Workspace

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A Part of STEMscopes Math Developed by Accelerate Learning Inc. 800-531-0864

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