Class Activities For Math 109: Support For Functios & Algebraic Methods [2 ed.]

Citation preview

Class Activities for Functions & Algebraic Methods Support Course Second Edition

Written by Jen Nimtz, Ph.D. Edited by Andrew Richardson

Colophon © 2023 Jennifer (Jen) Nimtz Permission is granted to distribute, remix, adapt, and build upon the material in any medium or format for noncommercial purposes only, and only so long as attribution is given to the creator and the Acknowledgements listed below.

This work is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Note: The above text is used with permission from creativecommons.org.

Acknowledgements Western Washington University This work was produced in part with the funding of a Western Washington University Summer Teaching Grant, Summer 2022.

CPM Educational Program The pattern diagrams and images were created using CPM's eTools. Copyright © CPM Educational Program. All rights reserved. Used with permission. CPM Educational Program, a California 501(c)(3) nonprofit corporation ( www.cpm.org ).

Desmos Studio PBC Graph images are designed with the Desmos Graphing Calculator, used with permission from Desmos Studio PBC ( www.desmos.com ).

Maths Bot Tools for Math Teachers Algebra Tile, Fraction, and Dienes Block images are designed with the Maths Bot online Tools for Math Teachers, used with permission ( www.mathsbot.com ).

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Table of Contents COLOPHON..................................................................................................................... I ACKNOWLEDGEMENTS ..................................................................................................... II TO USERS OF THIS BOOK .................................................................................................. 1 TODOS CUENTAN ............................................................................................................ 2 AXIOM 1 .............................................................................................................................. 2 AXIOM 2 .............................................................................................................................. 2 AXIOM 3 .............................................................................................................................. 2 AXIOM 4 .............................................................................................................................. 2 REFERENCE .......................................................................................................................... 2 DEFINITION OF AN AXIOM ........................................................................................................ 2 PROBLEM SOLVING ......................................................................................................... 3 I. ORIENTATION .................................................................................................................... 3 II. GENERATION .................................................................................................................... 3 III. CONCLUSION ................................................................................................................... 3 CLASS NORMS ................................................................................................................ 4 INTRODUCTIONS .................................................................................................................... 4 THINK—GO AROUND—DISCUSS PROTOCOL ................................................................................. 4 PRACTICE THE PROTOCOL ........................................................................................................ 4 USE THE PROTOCOL TO DEVELOP CLASS NORMS ............................................................................ 4 CHAPTER 1: LINEAR FUNCTIONS................................................................................ 5 PATTERNS & ALGEBRAIC REPRESENTATIONS ....................................................................... 5 ACTIVITY 1: FINDING AND PREDICTING PATTERNS ......................................................................... 5 ACTIVITY 2: ALGEBRA OF PATTERNS: TABLES, GRAPHS, & EQUATIONS ................................................ 6 ACTIVITY 3: MORE ALGEBRA OF PATTERNS (OPTIONAL) .................................................................. 7 ACTIVITY 4: REFLECTION ......................................................................................................... 8 1.1 SOLVING LINEAR EQUATIONS IN ONE VARIABLE ............................................................ 9 ACTIVITY 1: ASSESSING MATHEMATICAL COMMUNICATION AND CORRECTNESS ...................................... 9 ACTIVITY 2: SOLVING APPLIED PROBLEMS GIVEN FORMULA AND A VALUE—LINEAR EQUATIONS IN ONE VARIABLE .......................................................................................................................... 10 ACTIVITY 3: MORE APPLIED PROBLEMS ..................................................................................... 11 ACTIVITY 4: REFLECTION ....................................................................................................... 12 1.2 FUNDAMENTALS OF GRAPHING: SLOPE—LINEAR EQUATIONS IN TWO VARIABLES .............. 13 ACTIVITY 1: EXAMINING SLOPE ............................................................................................... 13

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ACTIVITY 2: CALCULATING SLOPE FROM A GRAPH AND A TABLE ....................................................... 15 ACTIVITY 3: REFLECTION ...................................................................................................... 16 1.3 GRAPHING LINEAR EQUATIONS: SLOPE & INTERCEPTS—LINEAR EQUATIONS IN TWO VARIABLES .................................................................................................................. 17 ACTIVITY 1: EXAMINING INTERCEPTS ....................................................................................... 17 ACTIVITY 2: INTERPRETING 𝒚 = 𝒎𝒙 + 𝒃 IN CONTEXTS ................................................................ 18 ACTIVITY 3: VERTICAL AND HORIZONTAL LINES .......................................................................... 20 ACTIVITY 4: MODELING & INTERPRETING 𝑨𝒙 + 𝑩𝒚 = 𝑪 IN CONTEXT .............................................. 21 ACTIVITY 6: CONNECTING FORMS OF LINEAR EQUATIONS .............................................................. 22 ACTIVITY 7: REFLECTION ON SECTIONS 1.2 & 1.3 ...................................................................... 23 1.4 FINDING EQUATIONS OF LINES ................................................................................. 25 ACTIVITY 1: FINDING EQUATIONS OF LINES GIVEN A GRAPH........................................................... 25 ACTIVITY 2: ASSESSING MATHEMATICAL COMMUNICATION............................................................. 26 ACTIVITY 3: FINDING EQUATIONS OF LINES ............................................................................... 28 ACTIVITY 4: APPLIED PROBLEM ............................................................................................... 30 ACTIVITY 5: REFLECTION—CREATE A CONCEPT MAP .................................................................... 31 1.5 FUNCTIONS ........................................................................................................... 32 ACTIVITY 1: EVALUATING AND SOLVING USING FUNCTION NOTATION ............................................... 33 ACTIVITY 2: FUNCTIONS IN CONTEXTS...................................................................................... 34 ACTIVITY 3: MORE FUNCTIONS IN CONTEXTS ............................................................................. 36 ACTIVITY 4: REFLECTION ...................................................................................................... 37 CHAPTER 2: SYSTEMS OF LINEAR EQUATIONS & INEQUALITIES ............................ 38 2.1 SYSTEMS OF LINEAR EQUATIONS: USING TABLES AND GRAPHS ...................................... 38 ACTIVITY 1: COMPARING PROPORTIONAL AND NON-PROPORTIONAL RELATIONSHIPS ............................ 38 ACTIVITY 2: SYSTEM OF EQUATIONS SOLUTION TYPES—TABLES AND GRAPHS .................................... 40 ACTIVITY 3: REFLECTION ...................................................................................................... 43 2.2 SYSTEMS OF LINEAR EQUATIONS: SUBSTITUTION ........................................................ 44 ACTIVITY 1: SYSTEMS OF LINEAR EQUATIONS—SUBSTITUTION ....................................................... 44 2.3 SYSTEMS OF LINEAR EQUATIONS: ELIMINATION .......................................................... 45 ACTIVITY 1: SYSTEMS OF LINEAR EQUATIONS—ELIMINATION ......................................................... 45 ACTIVITY 2: REFLECTION ...................................................................................................... 46 ACTIVITY 3: SYSTEMS OF LINEAR EQUATIONS—USING ALGEBRAIC STRUCTURE TO SELECT A SOLUTION METHOD ........................................................................................................................... 47 ACTIVITY 4: REFLECTION ...................................................................................................... 48 CHAPTER 3: OPERATIONS ON EXPRESSIONS AND FUNCTIONS .............................. 49

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SECTION 3.1 PROPERTIES OF EXPONENTS......................................................................... 49 ACTIVITY 1: REVIEWING REPEATED ADDITION AND REPEATED MULTIPLICATION .................................. 49 ACTIVITY 2: GENERALIZING THE PROPERTIES OF EXPONENTS .......................................................... 50 ACTIVITY 3: SCIENTIFIC NOTATION AND UNIT CONVERSIONS ......................................................... 55 SECTION 3.2: COMBINING FUNCTIONS............................................................................. 56 ACTIVITY 1: ADDING FUNCTIONS ............................................................................................. 56 ACTIVITY 2: SUBTRACTING FUNCTIONS ..................................................................................... 57 ACTIVITY 3: MULTIPLYING FUNCTIONS ...................................................................................... 58 ACTIVITY 4: DIVIDING FUNCTIONS WITH MONOMIAL DIVISOR ........................................................ 61 ACTIVITY 5: MIXED PRACTICE ................................................................................................. 63 ACTIVITY 6: DIVIDING FUNCTIONS WITH BINOMIAL DIVISOR (OPTIONAL).......................................... 64 ACTIVITY 7: SIMPLIFYING EXPRESSIONS .................................................................................... 66 SECTION 3.4: FACTORING POLYNOMIALS ......................................................................... 67 RELATIONSHIP BETWEEN FACTORING AND MULTIPLYING ................................................................ 67 ACTIVITY 1: UNDERSTANDING FACTORS .................................................................................... 67 ACTIVITY 2: MODELING POLYNOMIALS AND THEIR FACTORS ........................................................... 69 ACTIVITY 3: VOCABULARY OF POLYNOMIALS & FACTORING ............................................................ 70 ACTIVITY 4: SCAFFOLDED TRINOMIAL FACTORING & GENERALIZING ................................................. 72 SECTION 3.5: SPECIAL FACTORING TECHNIQUES ............................................................... 73 ACTIVITY 1: RECOGNIZING STRUCTURE OF SPECIAL QUADRATICS .................................................... 73 ACTIVITY 2: USING SPECIAL FACTORING TECHNIQUES .................................................................. 75 ACTIVITY 3: FACTORING MIXED PRACTICE ................................................................................. 76 CHAPTER 4: QUADRATIC EQUATIONS AND FUNCTIONS .......................................... 77 PREVIEW OF QUADRATIC EQUATIONS AND FUNCTIONS........................................................ 77 ACTIVITY 1: HOW DO WE RECOGNIZE A QUADRATIC EQUATION OR FUNCTION? ................................... 77 ACTIVITY 2: CONNECTING CHARACTERISTICS OF A QUADRATIC GRAPH WITH FORMS OF THE FUNCTION ..... 78 ACTIVITY 3: PROJECTILE HEIGHT APPLICATION ........................................................................... 79 SECTION 4.1 INTRODUCTION TO QUADRATIC FUNCTIONS.................................................... 80 ACTIVITY 1: CHARACTERISTICS OF QUADRATIC FUNCTIONS ............................................................ 80 SECTION 4.5 SOLVING QUADRATIC EQUATIONS WITH FACTORING ........................................ 81 ACTIVITY 1: USING THE ZERO PRODUCT PROPERTY TO SOLVE POLYNOMIAL EQUATIONS ........................ 81 ACTIVITY 2: APPLIED PROBLEMS USING FACTORS OF POLYNOMIALS .................................................. 82 SECTION 4.4 COMPLETING THE SQUARE ........................................................................... 84 ACTIVITY 1: SOLVING USING THE SQUARE ROOT PROPERTY ........................................................... 84 ACTIVITY 2: VISUALIZING COMPLETING THE SQUARE .................................................................... 85 ACTIVITY 3: COMPLETING THE SQUARE ..................................................................................... 86 SECTION 4.6 THE QUADRATIC FORMULA .......................................................................... 88

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ACTIVITY 1: DERIVING THE QUADRATIC FORMULA ....................................................................... 88 ACTIVITY 2: SOLVE APPLIED PROBLEMS USING THE QUADRATIC FORMULA ......................................... 89 SECTION 4.7 GRAPHING STANDARD FORM AND MODELLING ............................................... 91 ACTIVITY 1: WRITE A QUADRATIC EQUATION FROM GRAPH ........................................................... 91 ACTIVITY 2: MODELLING WITH QUADRATICS [REVISE WITH GRAPH] ................................................. 93 CHAPTER 7: RATIONAL FUNCTIONS: EXPRESSIONS AND EQUATIONS ................... 94 SECTION 7.1 RATIONAL FUNCTIONS AND VARIATION ......................................................... 95 ACTIVITY 1: COMPARING LINEAR AND RATIONAL FUNCTIONS.......................................................... 95 SECTIONS 7.2, 7.3, & 7.4: SIMPLIFYING RATIONAL EXPRESSIONS ...................................... 97 ACTIVITY 1: REVIEW OF NUMERIC FRACTIONS ............................................................................ 97 ACTIVITY 2: SIMPLIFYING RATIONAL EXPRESSIONS ...................................................................... 99 ACTIVITY 3: MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS ................................................ 101 ACTIVITY 4: RATIONAL EXPRESSIONS & COMMON DENOMINATORS ................................................ 105 ACTIVITY 5: ADDING AND SUBTRACTING RATIONAL EXPRESSIONS ................................................. 106 ACTIVITY 6: SIMPLIFY RATIONAL EXPRESSIONS—MIXED PRACTICE ............................................... 110 SECTION 7.5: SOLVING RATIONAL EQUATIONS ............................................................... 111 ACTIVITY 1: CLEARING NUMERIC FRACTIONS TO SOLVE LINEAR EQUATIONS (REVIEW) ....................... 111 ACTIVITY 2: CLEARING ALGEBRAIC FRACTIONS TO SOLVE RATIONAL EQUATIONS ............................... 112 ACTIVITY 3: CONSIDERING EXTRANEOUS SOLUTIONS ................................................................. 113 ACTIVITY 4: EVALUATE THE SOLUTION METHODS ...................................................................... 114 ACTIVITY 5: APPLIED PROBLEMS ........................................................................................... 115 CHAPTER 8: RADICAL FUNCTIONS: EXPRESSIONS AND EQUATIONS .................... 117 SECTION 8.1 RADICALS AND RADICAL FUNCTIONS .......................................................... 117 ACTIVITY 1: COMPUTING SQUARES AND SQUARE ROOTS ............................................................. 117 ACTIVITY 2: COMPUTING CUBES AND CUBE ROOTS .................................................................... 119 ACTIVITY 3: RADICALS AND FRACTIONAL EXPONENTS ................................................................. 121 ACTIVITY 4: COMPARING QUADRATIC AND RADICAL FUNCTIONS .................................................... 123 SECTIONS 8.3 ADDING AND SUBTRACTING RADICALS ...................................................... 126 EXPANDING THE DEFINITION OF LIKE TERMS ............................................................................ 126 ACTIVITY 1: ADD AND SUBTRACT RADICALS ............................................................................. 127 SECTION 8.4: MULTIPLYING AND DIVIDING RADICALS .................................................... 129 ACTIVITY 1: MULTIPLYING RADICALS [NOT YET WRITTEN] ........................................................... 129 ACTIVITY 2: DIVIDING RADICALS [NOT YET WRITTEN] ................................................................ 129 SECTION 8.5: SOLVING RADICAL EQUATIONS ................................................................. 130 INTRODUCTION TO SOLVING RADICAL EQUATIONS ..................................................................... 130

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ACTIVITY 1: EVALUATE THE SOLUTION METHODS ...................................................................... 131 ACTIVITY 2: SOLVING RADICAL EQUATIONS WITH ONE VARIABLE TERM .......................................... 133 ACTIVITY 3: APPLIED PROBLEMS............................................................................................ 134 ACTIVITY 4: SOLVING RADICAL EQUATIONS WITH TWO VARIABLE TERMS ........................................ 135 FINAL EXAM REVIEW .............................................................................................. 136 ACTIVITY 1: ALGEBRAIC EQUATIONS CARD SORT ............................................................. 136 ACTIVITY 2: SOLVING VARIOUS ALGEBRAIC EQUATIONS ................................................... 136 ACTIVITY 3: FUNCTIONS CARD SORT ............................................................................. 138 APPENDIX: REVIEW ................................................................................................ 139 RATIONAL NUMBERS ................................................................................................... 139 INTEGERS ........................................................................................................................ 139 FRACTIONS ...................................................................................................................... 139 DECIMALS ........................................................................................................................ 139 ORDER OF OPERATIONS ............................................................................................... 140 EXPONENTS ............................................................................................................... 142 INTEGER EXPONENTS .......................................................................................................... 142 RADICALS: FRACTIONAL EXPONENTS ....................................................................................... 144

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To Users of this Book Hello! I have written the activities in this book to accompany a beginning or intermediate algebra course. The activities in this book are designed to promote algebraic thinking in many ways by engaging in learning actions that include: •

generalizing from patterns we observe,

•

organizing information using tables and graphs,

•

connections between words (applied problems), tables, graphs, equations and functions,

•

evaluating the communication of mathematical reasoning with the goal of improving your own communication of mathematical reasoning and use of commonly accepted mathematical conventions,

•

connections between the mathematics you already know (arithmetic) and algebra,

•

connections between different representations of mathematics and operations, such as rectangular models of the base ten number system, fractions, decimals, and algebra terms.

Many of the activities in this book will ask us to “pause and consider” and/or “reflect.” For example, we may be asked to pause and consider how different aspects of equations, tables and/or graphs are similar and different. Then we will discuss our observations in small groups. The answers to these questions will vary, and that is O.K.! Initially, these questions may make us feel uncomfortable, possibly because in our past mathematical experiences we haven’t been asked to make observations and conjectures about mathematics. There are not wrong answers to these types of questions. However, the answers we provide will be graded on the thoughtfulness and coherence of the response. Write with complete sentences that someone else can read and understand. We may add to and refine our responses as we discuss these questions in small groups and as a class. Over the past 75 years or so, there has a growing body of evidence (from science and mathematics education research) that engaging in deeper mathematical thinking*, discussions and writing, facilitates learning and the retention of what we learn. *Deeper mathematical thinking refers to the effort to understand and connect mathematical concepts and skills rather than simply memorize procedures.

1

Todos Cuentan Todos means “all.” Cuentan means “count.”

Axiom 1 Mathematical talent is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.

Axiom 2 Everyone can have joyful, meaningful, and empowering mathematical experiences.

Axiom 3 Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.

Axiom 4 Every student deserves to be treated with dignity and respect.

Reference Ardila-Mantilla, Federico (2016). Todos Cuentan: Cultivating Diversity in Combinatorics. Notices of the AMS. Volume 63. Number 10. Pages 11641170.

Definition of an Axiom A statement which is regarded as generally accepted or self-evidently true.

2

Problem Solving Phases of Problem Solving I. Orientation

Questions to ask yourself when problem solving

A. Understand: Read the problem to understand what the problem is asking.

o o o

What is the problem asking? How might we restate the problem in our own words? Can we sketch a picture or diagram?

B. Analyze: Re-read the problem, often several times, to determine essential aspects of the problem (concepts, procedures, variables, representations, and/or conditions).

o o o o o o o o

What information is given? What are we asked to find? What relationships are implied? What is staying the same and what is changing? Can we write an expression or equation from what is given? Is there a formula that is provided? Is there a formula (e.g. perimeter, area) that is needed? How is this problem similar to and different from past problems we have solved?

II. Generation C. Explore: Look for patterns, relationships, representations, or algebraic structures that help in solving the problem. Consider if solutions to similar problems might be helpful.

D. Plan: Plan the solution steps. E. Execute: Carry out the plan.

o How might it be helpful to use the following strategies? ▪ Make an organized list or table. ▪ Generalize a pattern from the list or table. ▪ Sketch a picture or diagram ▪ Create a graph. ▪ Write an equation based on the table, graph, or patterns. ▪ Use symmetry. ▪ Solve a similar, yet simpler, problem. ▪ Consider special cases. o What strategies are we going to use to solve this problem? o What is our plan for solving this problem?

III. Conclusion F. Verify: Determine if the solution is correct and check for extraneous solutions.

o

How can we check this solution? How can we solve this problem in another way? Do symbolic, tabular and graphical approaches result in the same answer? If an algebraic equation or function is used to solve the problem, then how can we use it to check our answer? Are there any extraneous solutions?

o o

Have I answered what the question was asking? Does the solution make sense in the context of the problem?

o

How do the different representations (equations, tables, graphs, and words/contexts) relate to each other? How is this problem similar to or different from problems we have solved recently? How does this problem connect to what we have learned recently? If you were to create a concept map of mathematics, where would this problem fit in that concept map?

o o o o

G. Interpret: Check to see if the solution makes sense in the context of the problem. H. Reflect: Consider similarities and differences between this problem and other mathematical problems. Consider how the ideas of this problem are connected with other mathematical ideas.

o o o

(Nimtz, 2018, p. 49-50)

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Class Norms Introductions •

Before we begin any small group discussion, take turns introducing yourself to your group. Include your preferred name and pronouns.

Think—Go Around—Discuss Protocol •

First, everyone in the group takes one minute or more to think about the question.

•

Next, each person in the group takes a turn to share their thoughts, without interruption.

•

Finally, the group discusses the different viewpoints.

Practice the Protocol •

If you had to delete all but 4 apps from your cell phone, which 4 apps would you keep? Why?

Use the Protocol to Develop Class Norms •

In this class, what do you need from yourself and each other to feel safe exploring ideas, making conjectures, sharing thinking, and building on and connecting with other’s ideas? Adapted from the WWU Mathematics Educators and Senior Instructor Andrew Richardson class notes.

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Chapter 1: Linear Functions Patterns & Algebraic Representations Activity 1: Finding and Predicting Patterns 1. Examine the visual pattern in the figures below. Sketch Figure 5

Figure 1. Sequence of Figures. Copyright © CPM Educational Program 2. Explain how the number sequence, 4, 10, 16, 22, … relates a characteristic of the sequence of figures. Given the table:

Figure Number

1

2

3

4

5

Line Segments

4

10

16

22

28

Write in Words:

3. Explain how the number sequence, 1, 3, 5, 7, … relates a characteristic of the sequence of figures. Complete the table:

Write in Words:

4. Find another number sequence related to the figures and explain the relationship using a table and words.

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Activity 2: Algebra of Patterns: Tables, Graphs, & Equations We can use tables to help us make sense of patterns and use Algebraic Symbols to predict patterns for larger figures and numbers. 1. Plot the points of the table of values from problem 2 on Activity 1 on the coordinate grid below. Note that the horizontal and vertical axis are labeled, and the tick marks are spaced equally along each axis.

2. Assuming this pattern continues, write an equation that expresses the relationship between the number of Line Segments (L) and the Figure Number (N).

Figure number

Number of line segments written in an unsimplified way that reflects how you see the pattern

The number of line segments written in a more simplified way

The number of line segments written in a completely simplified way

1 2 3 4 N

3. Assuming this pattern continues, use your equation to find the number of Line Segments needed to create Figure 47 for Figure Sequence 1 from Activity 1.

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Activity 3: More Algebra of Patterns (Optional) 1. Examine the visual pattern in the figures below. Sketch Figure 5

Figure 2. Sequence of Figures. Copyright © CPM Educational Program 2. Complete the table of values below represent the relationship of the Figure Number (the input) to the Area, or number of Square Tiles (the output).

Figure Number (N) Tiles (T) 3. Plot the points of the table of values on the coordinate grid below. Be sure to label the horizontal and vertical axis and space the tick marks equally along each axis.

4. Is this a linear or non-linear pattern? How do you know?

7

Activity 4: Reflection In this section, we used tables, graphs, and equations to represent the growth of visual patterns in relation to the figure number. 1. TABLE: What are the strengths and limitations of representing the growth of visual patterns using a table?

2. GRAPH: What are the strengths and limitations of representing the growth of visual patterns using a graph? Why did we plot points instead of sketching lines to represent these problems?

3. EQUATION: What are the strengths and limitations of representing the growth of visual patterns using an equation?

4. How do you think these activities will relate to the work you do in your Math 112 class?

8

1.1 Solving Linear Equations In One Variable Activity 1: Assessing Mathematical Communication and Correctness Two students solved this equation: 4.1𝑥 − 3(3.2𝑥 + 5.7) = 4.9𝑥 − 32.7

1. Which student’s work is easier to read and follow? Which student communicated their thinking using accepted mathematical conventions? Explain.

2. Which student is correct? How do you know? Explain.

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Activity 2: Solving Applied Problems Given Formula and a Value—Linear Equations in One Variable The formula for converting between degrees Fahrenheit, F, and degrees Celsius, C, is: 9 𝐹 = 𝐶 + 32 5 1. In northern Washington state, we can tune in to Canadian radio stations, and when they report the weather, it is in degrees Celsius. In July, a Vancouver radio station reported a temperature of 30 degrees Celsius. Use the given formula to calculate the conversion of 30 degrees Celsius to degrees Fahrenheit. Communicate your method of finding the answer using mathematical conventions (clearly show each step). Be sure to include units in your answer.

2. In southern British Columbia, they can tune in to Washington radio stations. In July, a Bellingham, Washington radio station reported a temperature of 90 degrees Fahrenheit. Use the given formula to calculate the conversion of 90 degrees Fahrenheit to degrees Celsius. Communicate your method of finding the answer using mathematical conventions (clearly show each step). Be sure to include units in your answer.

3. Find an equivalent form of the formula with an input of degrees Fahrenheit and an output of degrees Celsius. In other words, solve the formula for C.

10

Definitions Revenue: The amount of money brought into a company through sales. Revenue is often calculated as:

𝑟𝑒𝑣𝑒𝑛𝑢𝑒 = 𝑝𝑟𝑖𝑐𝑒 ∙ 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑠𝑜𝑙𝑑

Cost: The amount of money spent by a business to create and/or sell a product. Cost typically includes both fixed costs and variable costs. Fixed costs are the same each month or year, such as rent or mortgage payments. Variable costs are directly related to the number of items produced and/or sold. Cost is often calculated as: 𝑐𝑜𝑠𝑡 = 𝑓𝑖𝑥𝑒𝑑 𝑐𝑜𝑠𝑡 + 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑐𝑜𝑠𝑡 𝑐𝑜𝑠𝑡 = 𝑓𝑖𝑥𝑒𝑑 𝑐𝑜𝑠𝑡 + 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑖𝑡𝑒𝑚 ∙ 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 Profit: The amount of money brought in (revenue) left after paying all costs. Profit is calculated as:

𝑝𝑟𝑜𝑓𝑖𝑡 = 𝑟𝑒𝑣𝑒𝑛𝑢𝑒 − 𝑐𝑜𝑠𝑡

Break-even point: A company breaks even when their revenue equals their cost, or when profit is zero.

𝑟𝑒𝑣𝑒𝑛𝑢𝑒 = 𝑐𝑜𝑠𝑡 𝑝𝑟𝑜𝑓𝑖𝑡 = 0 (Clark & Afinson, 2019, p. 4)

Activity 3: More Applied Problems An ice cream cone shop has many costs associated with running the business. Each month, they pay $4800 for salaries, $2955 for utilities, and $3000 for rent of the business space. The cost for each waffle ice cream cone is 34 cents and 84 cents for 2 scoops of ice cream. 1. Assuming they only sell 2 scoop waffle cones, write an equation for the monthly cost of running this ice cream cone shop. 2. If they sell each cone for $5.63, write an equation for revenue. 3. Write an equation for profit. 4. How many ice cream cones do they need to sell to break even? Communicate your method of finding the answer using mathematical conventions and include units.

5. If they sell 5000 cones each month, what is their profit? Communicate your method of finding the answer using mathematical conventions and include units.

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Activity 4: Reflection 1. List important elements of effective communication of reasoning and solving methods using mathematical conventions.

2. How can we verify that our solutions are correct?

3. How might we determine if our solution to an applied problem is reasonable?

12

1.2 Fundamentals of Graphing: Slope—Linear Equations in Two Variables Activity 1: Examining Slope Examine the relationship between symbolic and graphical representations of slope in equations of the form, 𝑦 = 𝑚𝑥 + 𝑏. In this exploration, each equation has a slope, 𝑚, of a different value and the y-intercept has the same value, 𝑏 = 0. 2. The equations on the left are graphed on the right. Label each graph of a line with the matching equation. 𝟏

A. 𝒚 = 𝟐 𝒙 𝟑

B. 𝒚 = 𝟒 𝒙 C. 𝒚 = 𝒙 𝟒

D. 𝒚 = 𝟑 𝒙 E. 𝒚 = 𝟐𝒙

Pause and consider what you notice about each graph in relation to its equation and to the other graphs. How are they different? How are they similar? Discuss in your groups and take notes for question 2 on the next page. Continue with the equations and graphs below. F. 𝒚 = −𝟐𝒙 G. 𝒚 =

−𝟒 𝒙 𝟑

H. 𝒚 = −𝒙 I. 𝒚 =

−𝟑 𝒙 𝟒

J. 𝒚 =

−𝟏 𝒙 𝟐

Pause and consider what you notice about each graph in relation to its equation and to the other graphs. Discuss in your groups and take notes for question 2 on the next page.

13

Consider the set of all the linear functions we’ve examined in this activity thus far. 𝟏

A. 𝒚 = 𝒙 𝟐 𝟑

B. 𝒚 = 𝒙 𝟒

C. 𝒚 = 𝒙 𝟒

D. 𝒚 = 𝒙 𝟑

E. 𝒚 = 𝟐𝒙 F. 𝒚 = −𝟐𝒙 G. 𝒚 =

−𝟒 𝟑

𝒙

H. 𝒚 = −𝒙 I. 𝒚 = J. 𝒚 =

−𝟑 𝟒 −𝟏 𝟐

𝒙 𝒙

Summarize what you discussed in your groups. How are the graphs similar? How are they different? What do you notice about the graphs with a positive coefficient of x (positive slope)? With a negative coefficient of x (negative slope)?

14

Activity 2: Calculating Slope from a Graph and a Table 1. Use the graph below to calculate the slope of the line.

3. Given the table: 𝒙 𝒚

-6 4

-4 11

-2 18

0 25

2 32

4 39

6 46

8 53

a. Assume the pattern continues in the table. Determine whether the table represents a linear equation. Explain.

b. If it is a linear equation, use the table to calculate the slope of the line.

15

Activity 3: Reflection 1. Describe the visual, geometric, characteristics of a positive slope. Of a negative slope. What is the difference between positive slope and negative slope?

2. Describe how the processes of determining the slope of a line from a graph and from a table are similar and different in the following two questions. a. Describe how the formula for slope, 𝑚

=

𝑦2 −𝑦1 𝑥2 −𝑥1

, is similar to the

𝑟𝑖𝑠𝑒 𝑟𝑢𝑛

of a linear

graph. Consider if using a graph would be helpful to illustrate your thinking.

b. Describe how the formula for slope, 𝑚

=

𝑦2 −𝑦1 𝑥2 −𝑥1

, is connected to calculating slope

from a table. Consider if using a table would be helpful to illustrate your thinking.

16

1.3 Graphing Linear Equations: Slope & Intercepts—Linear Equations in Two Variables Activity 1: Examining Intercepts Examine the relationship between symbolic and graphical representations of slope in equations of the form, 𝑦 = 𝑚𝑥 + 𝑏. In this exploration, each equation has a slope of the same value, 𝑚 = 2, and the y-intercept, 𝑏, has a different value. 1. The equations on the left are graphed on the right. Label each graph of a line with the matching equation.

A. 𝐲 = 𝟐𝐱 − 𝟓 B. 𝐲 = 𝟐𝐱 −

𝟓 𝟐

C. 𝐲 = 𝟐𝐱 D. 𝐲 = 𝟐𝐱 +

𝟓 𝟐

E. 𝐲 = 𝟐𝐱 + 𝟓

Pause and consider what you notice about each graph in relation to its equation and to the other graphs. How are they similar? How are they different? Discuss in your groups and take notes for question 2. 2. Describe what you observe.

17

Activity 2: Interpreting 𝒚 = 𝒎𝒙 + 𝒃 in Contexts The formula for converting between degrees Fahrenheit, F, and degrees Celsius, C, is: 9 𝐹 = 𝐶 + 32 5 1. Create a table of values and plot points on the coordinate system provided.

℃ ℉

18

-10

-5

0

5

10

15

20

25

30

35

3. What is the vertical intercept of this equation? What does the vertical intercept mean in this context?

4. What is the slope of this equation? What does the slope mean in this context?

19

Activity 3: Vertical and Horizontal Lines 1. Given the equation: 𝑦 = −3 a. Create a table of values: (Think: “y is always -3, and x can be any number.” ) 𝒙 𝒚 b. Sketch a graph:

2. Given the equation: 𝑥 = 2 a. Create a table of values: (Think: “x is always 2, and y can be any number.” ) 𝒙 𝒚 b. Sketch a graph:

20

Activity 4: Modeling & Interpreting 𝑨𝒙 + 𝑩𝒚 = 𝑪 in Context Jake has $6 to spend on lunch. Drinks cost $1.50 each and the salad bar costs 50 cents per ounce. Assume Jake spends all $6, and there is no tax. 1. Complete the table below with values that satisfy the condition of spending $6. Drinks (𝒙)

0

1

2

3

4

5

Ounces of Salad Bar (𝒚) 2. Write an equation in the Standard Form, 𝐴𝑥 + 𝐵𝑦 = 𝐶, to represent this context.

3. Write an equation in the Slope-Intercept Form of a Linear Equation, 𝑦 = 𝑚𝑥 + 𝑏, to represent this context. In other words, solve the equation in part 2 for y. Show your work.

4. Use a straightedge to sketch a graph of this context. a. What does the vertical-intercept mean in this context?

b. What does the horizontalintercept mean in this context?

21

Activity 6: Connecting Forms of Linear Equations 1. Solve the Standard Form of a Linear Equation, 𝐴𝑥 + 𝐵𝑦 = 𝐶, for 𝑦. Communicate your thinking using mathematical conventions.

a. What is the slope of the Standard Form of a Linear Equation in terms of the parameters, A, B, and/or C?

b. What is the y-intercept of the Standard Form of a Linear Equation in terms of the parameters A, B, and/or C?

c. What are the similarities and differences between the Standard Form of a Linear Equation, 𝐴𝑥 + 𝐵𝑦 = 𝐶, and the Slope-Intercept Form of a Linear Equation, 𝑦 = 𝑚𝑥 + 𝑏.

22

Activity 7: Reflection on Sections 1.2 & 1.3 1. Describe how to find the slope of a line: a. From a table.

b. From a graph.

c. From the equation.

2. Describe how to find the vertical intercept (also called the y-intercept and output intercept) of a line: a. From a table.

b. From a graph.

c. From the equation.

23

3. Describe how to find the horizontal intercept (also called the x-intercept and input intercept) of a line: a. From a table.

b. From a graph.

c. From the equation.

24

1.4 Finding Equations of Lines Activity 1: Finding Equations of Lines given a Graph Parallel Lines 1. Write the equation of the lines below each graph. E

C

A D B F Line A:

Line C:

Line E:

Line B:

Line D:

Line F:

2. What does the slope of each set of the Parallel Lines have in common?

Perpendicular Lines 3. Write the equation of the lines below each graph. C A E

F

D

B Line A:

Line C:

Line E:

Line B:

Line D:

Line F:

4. What does the slope of each set of the Perpendicular Lines have in common?

25

Activity 2: Assessing Mathematical Communication Two students solved this problem. 1 Write the equation of the line that is perpendicular to the line −𝑥 + 2𝑦 = − 2 and passes through the point ( –4, –9 ).

Sketch the given information and graph.

Sketch of the given information and graph.

Find slope of existing line by solving for y:

Find slope of the existing line using the intercepts of graph:

−𝑥 + 2𝑦 = −

1 2

1 −𝑥 + 𝑥 + 2𝑦 = − + 𝑥 2 1 2𝑦 = 𝑥 − 2 1 2𝑦 𝑥 − 2 = 2 2 1 1 𝑦= 𝑥− 2 4

−𝑥 + 2(0) = − 𝑥=

1 2

−(0) + 2𝑦 = −

1 2

1 2 1 2𝑦 − 2 = 2 2 1 𝑦=− 4 2𝑦 = −

1

(2 , 0)

1

(0, − 4)

1

so 𝑚𝐴 = 2. The perpendicular slope is: 𝑚 = −2

Find equation using 𝑦 = 𝑚𝑥 + 𝑏 and substitute slope and point (-4, -9).

1 2

𝑚𝐴 =

𝑦2 − 𝑦1 𝑥2 − 𝑥1

𝑦 = 𝑚𝑥 + 𝑏

1 1 −4 −0 −4 1 1 𝑚𝐴 = = = − ∙ −2 = 1 1 4 2 0−2 −2

−9 = (−2)(−4) + 𝑏

The perpendicular slope is:

−9 = 8 + 𝑏 −9 − 8 = 8 − 8 + 𝑏

Find equation using 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) and substitute slope and point (-4, -9).

so 𝑏 = −17.

𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )

We know the slope and y-intercept, so the equation of the perpendicular line is:

𝑦 − (−9) = −2(𝑥 − (−4))

𝑦 = −2𝑥 − 17

𝑦 + 9 = −2𝑥 − 8

𝑚 = −2

𝑦 + 9 = −2(𝑥 + 4) 𝑦 + 9 − 9 = −2𝑥 − 8 − 9 𝑦 = −2𝑥 − 17

26

1. Did the students follow mathematical conventions for solving this problem? Why or why not?

2. a. How are the students’ solution methods similar (Hint: Pay attention to the italicized text.)?

b. How are the students’ solution methods different?

3. Do you prefer one method over the other? Why or why not?

27

Activity 3: Finding Equations of Lines 1. a. Find the equation of the line with slope, 𝑚 =

−3 5

, and passing through the point ( –2, 4 ). Plot

the point on grid, find a second point using the slope, and sketch the graph. b. Next, find the equation of the line symbolically using methods presented in Activity 2. Communicate your reasoning using mathematical conventions. Check that your sketch of the graphical information and your algebraic work support each other.

2. a. Find the equation of the line that passes through the points ( –5, –2 ) and ( –3, –10 ). Plot the points on the grid and sketch the line. b. Next, find the equation of the line symbolically using methods presented in Activity 2. Communicate your reasoning using mathematical conventions. Check that your sketch of the graphical information and your algebraic work support each other.

28

1

3. a. Write the equation of the line that is parallel to the line −𝑥 + 2𝑦 = − 2 and passes through the point ( –4, –9 ). Sketch the graph of the line on the coordinate grid and plot the point. b. Next, find the equation of the line symbolically using methods presented in Activity 2. Communicate your reasoning using mathematical conventions. Check that your sketch of the graphical information and your algebraic work support each other.

4. a. Write the equation of the line that is perpendicular to the line 6𝑥 + 𝑦 = −11 and passes through the point ( –4, –3 ). b. Next, find the equation of the line symbolically using methods presented in Activity 2. Communicate your reasoning using mathematical conventions. Check that your sketch of the graphical information and your algebraic work support each other.

29

Activity 4: Applied Problem Health care researchers have noticed that the number of community hospitals in the United States has declined. They examined data beginning in the year 2000. In 2018, there were 5534 community hospitals; but in 2022 there were 5139 community hospitals. Assuming this is a linear trend, find an equation to predict the number of community hospitals t years after 2000. Communicate your reasoning using mathematical conventions. 1. What are the two quantities in this problem? Which quantity is the independent variable (input) and which is the dependent variable (output)?

2. Write coordinate pairs of the related quantities in this problem. 3. Make a sketch of the graphical representation of the problem.

4. Find the equation of the line and use it to predict the number of community hospitals in 2030.

30

Activity 5: Reflection—Create a Concept Map Use this page to create a concept map about Linear Equations in One and Two Variables. Some suggested words, equations, and formulas to include in your concept map: Linear Equations, Linear Equations in One Variable, Linear Equations in Two Variables, 1-Step Equations, 2-Step Equations, Processes, Solve, Write a Linear Equation, Model, Create a Table, Graph, Slope, Y-Intercept, X-Intercept, Standard Form, Slope-Intercept Form, 𝐴𝑥 + 𝐵𝑦 = 𝐶, 𝑦 −𝑦

𝑦 = 𝑚𝑥 + 𝑏, 𝑚 = 𝑥2 −𝑥1 2

1

31

1.5 Functions Definitions Relation: A set of ordered pairs of inputs and outputs. Example: { (1, 5), (3, 7), (9, 4), (-2, 4), (3, -1) }

Domain: The set of all input(s). The set of the first coordinates of all ordered pairs of a relation. Range: The set of all output(s). The set of the second coordinates of all ordered pairs of a relation. Function (formal): A relation in which each input is related to only one output. For each input value in the domain, there is exactly one output in the range. (Clark & Afinson, 2019, p. 4) Function (informal): A function is a particular type of relationship in which you can always predict or determine exactly one output value based on an input value.

Notes on Function Notation Suppose 𝑦 is a function of 𝑥. We can call this function 𝑓, so 𝑦 = 𝑓(𝑥) and 𝑓(𝑥) = 〈 an expression representing the value of 𝑦 given 𝑥 〉 Some Examples:

𝑓(𝑥) =

−3 𝑥 4

+ 2,

𝑓(𝑥) = √2𝑥 − 1 + 3,

𝑓(𝑥) = 𝑥 2 + 5𝑥 + 6, 𝑓(𝑥) =

𝑓(𝑥) = 2𝑥 ,

(𝑥+2) 𝑥 2 +5𝑥+6

Notes about Function Notation: •

𝑓 is the name of the function. The name of the function can be any letter you choose.

•

𝑓(𝑥), which is read as “ 𝑓 𝑜𝑓 𝑥 “ is mathematical notation for the output of a function

•

Coordinate points for the function 𝑦 = 𝑓(𝑥) are (𝑥, 𝑦) = (𝑥, 𝑓(𝑥)).

Notes about misconceptions about Function Notation: •

𝑓(𝑥) has nothing to do with multiplication. It represents the output of a function, 𝑓, with an input of 𝑥.

•

𝑓(𝑥) is not the name of the function. It represents the output of the function for an input of 𝑥. Above, we stated that 𝑓 is the name of the function. Notes adapted from Carlson, Marilyn (2021). College Algebra. Rational Reasoning Inc.

32

Activity 1: Evaluating and Solving Using Function Notation 1. Evaluate each of the functions below for given the input. Be sure to simplify your answers.

a. Given 𝑓(𝑥) = 𝑥 2 + 5𝑥 + 6, evaluate 𝑓(−3).

b. Given 𝑓(𝑥) = 2𝑥 , evaluate 𝑓(3).

c. Given 𝑓(𝑥) = √2𝑥 − 1 + 3, evaluate 𝑓(8.5).

d. Given 𝑓(𝑥) =

2. Given

(𝑥+2) 𝑥 2 +5𝑥+6

𝑓(𝑥) =

−3 𝑥 4

, evaluate 𝑓(−1).

+ 2.

a. Evaluate 𝑓(2).

b. Find the value of 𝑥 if 𝑓(𝑥) = −10.

33

Activity 2: Functions in Contexts This graph represents a person’s distance, D, from their car (in feet) in a parking lot in relation to the time, t, since they began walking (in seconds).

1. Which of the sentences below is a good match for the graph above? Be prepared to justify your selection. a. A person was 5 feet from their car and walked toward it, reaching the car after 20 seconds. b. A person was 20 feet from their car and walked toward it at a rate of 5 feet per second. c. A person was 20 feet from their car and walked toward it, reaching their car after 5 seconds. d. A person was 5 feet from their car and walked away from it, stopping when they were 20 feet away. 2. How far did this person walk in the time represented on the graph? Explain.

34

3. Write the meaning of the function notation, 𝐷(𝑡), representing coordinate pairs, where 𝐷 is distance from the car (in feet) and 𝑡 is time since they started walking (in seconds). Be sure to include the correct units when interpret the meaning. a. 𝐷(0) = ___________. The coordinate pair is ___________. Write the meaning of the function notation and coordinate pair in the context of this problem situation.

b. If 𝐷(𝑡) = 16 𝑓𝑒𝑒𝑡, then 𝑡 = _________. The coordinate pair is ___________. Write the meaning of the function notation and coordinate pair in the context of this problem situation.

c. 𝐷(5) = ___________. The coordinate pair is ___________. Write the meaning of the function notation and coordinate pair in the context of this problem situation.

4. How many feet did they walk per second? How much did the distance between the person and their car change per second? Is the distance between the person and their car increasing or decreasing over time? Explain.

5. Write the function equation that represents the context of this problem situation. Use function notation.

35

Activity 3: More Functions in Contexts 1. You are organizing a small conference and want each person to have their own water bottle. You find a company that sells aluminum water bottles with a business logo etched on the side. The minimum order is 250 bottles for $1875. After the minimum order, each bottle is sold for $6.50. a. Complete the table below, where N is the number of bottles purchased and C(N) is the cost in dollars. N C(N)

0

…

250

251

252

253

254

255

260

… break

b. Sketch a graph of the situation. To make scaling the axis easier, show a break in the axis. Should the graph be drawn as continuous line or as coordinate pairs? Explain.

c. What is the value of 𝐶(252)? What does the 𝐶(252) mean in terms of the input and output in the context of this applied problem?

36

Activity 4: Reflection 1. Give an example of a table describing a function from everyday life. Include descriptions of input and output variables and units of measure.

2. Give an example of a graph describing a function from everyday life. Include descriptions of input and output variables and units of measure.

3. Vocabulary Review: Label each of the words about function variables as “input” or “output.” Write each of these words along the corresponding axis of the coordinate system provided. Y-Axis: ____________ X-Axis: ____________ Independent-Axis: ____________ Dependent-Axis: ___________ Vertical-Axis: _____________ Horizontal-Axis: ____________ Domain Variable: ____________ Range Variable: _____________

37

Chapter 2: Systems of Linear Equations & Inequalities 2.1 Systems of Linear Equations: Using Tables and Graphs Activity 1: Comparing Proportional and Non-Proportional Relationships Tia is comparing two online music download companies. The cost of downloading music from Company A is $0.89 per song. The cost of downloading music from Company B requires first paying $15 to download their app and then $0.59 per song. 1. To determine which is the better deal, complete the table for each company. COMPANY A Number of Downloads

COMPANY B Total Cost ($)

Cost ($) Per Download

Number of Downloads

10

10

20

20

30

30

40

40

50

50

60

60

N

N

Total Cost ($)

Cost ($) Per Download

2. For Company A, why is the Cost per Download the same for any number of downloads?

3. For Company B, why does the Cost per Download decrease as the number of downloads increase?

38

4. A proportional relationship exists if the ratios of corresponding values are always equal. In this problem, the corresponding values are the Number of Downloads and Total Cost. The ratio of those corresponding values is the Cost per Download. Which Company has a proportional relationship between the Total Cost and the Number of Downloaded Songs? Explain.

5. Represent the comparison by graphing the values of the first two columns of each table.

6. For what number of downloaded songs is Company B the better deal? Explain.

7. For what number of downloaded songs is Company A the better deal? Explain.

8. For what number of downloaded songs is the Total Cost the same for both companies? Explain.

39

Activity 2: System of Equations Solution Types—Tables and Graphs Linear System A 3 4

1 2

1. Find the solution to the linear system: 𝑦 = 𝑥 − 4 and 𝑦 = − 𝑥 + 1 a. First complete the tables below. 𝟑 𝒚= 𝒙−𝟒 𝒙 𝟒

𝒙

𝒚=−

–5

–5

–4

–4

–3

–3

–2

–2

–1

–1

0

0

1

1

2

2

3

3

4

4

5

5

𝟏 𝒙+𝟏 𝟐

b. Next graph the linear system on the cartesian coordinate grid below.

c. What is the solution to the linear system?

40

Linear System B 3 4

2. Find the solution to the linear system: 𝑦 = 𝑥 − 4 and a. First complete the tables below. 3

𝑦 = 4𝑥 − 4

𝒙

𝒚

−3𝑥 + 4𝑦 = −16

−3𝑥 + 4𝑦 = −16

𝒙

𝒚

–5

–5

–4

–4

–3

–3

–2

–2

–1

–1

0

0

1

1

2

2

3

3

4

4

5

5

b. Next graph the linear system on the cartesian coordinate grid below.

c. What is the solution to the linear system?

41

Linear System C 3 4

3. Find the solution to the linear system: 𝑦 = 𝑥 + 4 and a. First complete the tables below. 3

𝑦 = 4𝑥 + 4

𝒙

𝒚

−3𝑥 + 4𝑦 = −16

−3𝑥 + 4𝑦 = −16

𝒙

𝒚

–5

–5

–4

–4

–3

–3

–2

–2

–1

–1

0

0

1

1

2

2

3

3

4

4

5

5

b. Next graph the linear system on the cartesian coordinate grid below.

c. What is the solution to the linear system?

42

Activity 3: Reflection There are three types of solutions to systems of equations: a coordinate pair, all points on the line, and no solution. 1. How do we determine the solution to a linear system is a coordinate pair when using a table or graph to solve a linear system?

2. How do we determine if the solution to a linear system is all points on the line when using a table or graph to solve a linear system?

3. How do we determine there is no solution to a linear system is when using a table or graph to solve a linear system? What are the limitations of using a table or graph to determine there is no solution?

4. What are the limitations of using tables or graphs to solve a system of equations?

43

2.2 Systems of Linear Equations: Substitution Activity 1: Systems of Linear Equations—Substitution Linear System A 3

1

1. Find the solution to the linear system, 𝑦 = 4 𝑥 − 4 and 𝑦 = − 2 𝑥 + 1 , using substitution.

Linear System B 3

2. Find the solution to the linear system, 𝑦 = 4 𝑥 − 4 and −3𝑥 + 4𝑦 = −16 , using substitution.

Linear System C 3

3. Find the solution to the linear system, 𝑦 = 4 𝑥 + 4 and −3𝑥 + 4𝑦 = −16 , using substitution.

44

2.3 Systems of Linear Equations: Elimination Activity 1: Systems of Linear Equations—Elimination Linear System A 1. Find the solution to the linear system, −3𝑥 + 4𝑦 = −16 and 𝑥 + 2𝑦 = 2 , using elimination.

Linear System B 2. Find the solution to the linear system, 6𝑥 − 8𝑦 = 7 and −3𝑥 + 4𝑦 = −16 , using elimination.

Linear System C 3. Find the solution to the linear system, −2𝑥 + 4𝑦 = 6 and 𝑥 − 2𝑦 = −3 , using elimination.

45

Activity 2: Reflection There are three types of solutions to systems of equations: a coordinate pair, all Real numbers, and no solution. 1. How do we determine that the solution to a linear system is a coordinate pair when using symbolic methods (substitution and elimination)? How does this compare to finding this type of solution with graphs or tables?

2. How do we determine that the solution to a linear system is all points on a line when using symbolic methods (substitution and elimination)? How does this compare to finding this type of solution with graphs or tables?

3. How do we determine that there is no solution to a linear system when using symbolic methods (substitution and elimination)? How does this compare to finding this type of solution with graphs or tables?

46

Activity 3: Systems of Linear Equations—Using Algebraic Structure to Select a Solution Method Several systems of linear equations are provided below. First, study the structure of the system and use that structure to decide which symbolic method (substitution or elimination) that you will use to solve the system. Second, explain your thinking. Lastly, solve the system. 1.

2𝑥 + 𝑦 = 4 𝑥 = 5 − 2𝑦

2.

𝑦 = 2𝑥 + 1 𝑦 =7−𝑥

3.

2𝑥 + 3𝑦 = 9 4𝑥 + 5𝑦 = 17

47

Activity 4: Reflection Describe the different methods we can use to check the solution to a system of linear equations.

48

Chapter 3: Operations on Expressions and Functions Section 3.1 Properties of Exponents Activity 1: Reviewing Repeated Addition and Repeated Multiplication 1. Simplify the following addition expression:

𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 = ____________

2. Simplify the following multiplication expression:

𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 = ___________

3. Write the terms using repeated addition: 3𝑎 + 2𝑏 = ______________________________________

4. Write the factors using repeated multiplication: 𝑎3 𝑏 2 = ____________________________________

5. What is a more concise operation for repeated addition? Provide an example different from those in problems 1 through 4 above.

6. What is a more concise operation for repeated multiplication? Provide an example different from those in problems 1 through 4 above.

7. Describe how we can tell the difference between the symbols indicating repeated addition and repeated multiplication.

49

Activity 2: Generalizing the Properties of Exponents In Activity 1, we saw that we can write repeated multiplication as an exponent expression and vice versa. Some terms about exponents follow.

Exponent Expression base

𝑎2

exponent

In this activity, we combine that idea with the knowledge that

𝑎 𝑎

= 1 for 𝑎 ≠ 0 to generalize the

Rules of Exponents.

Why is it important to understand algebraic notation and Properties of Exponents? Because we can use algebraic notation to understand and model many situations and to solve problems. For example, we can model an investment with compound interest. We know that 𝑟 𝑛𝑡

the total amount (A) in the account can be modelled with the formula, 𝐴 = 𝑃 (1 + 𝑛) , in which 𝑃 is the principle amount invested, 𝑟 is the interest rate, 𝑛 is the number of times compounded, and 𝑡 is the number of years of the investment. In addition, the Properties of Exponents are essential for Scientific Notation, which is used in calculations of very large and very small numbers throughout STEM careers. For example, see the table of A Few Common Physical and Chemical Constants. These are examples, we do NOT need to memorize these numbers for this class! However, we DO need to understand if the values are large or small quantities and how the exponents are used in both the Scientific Notation and on the units of measure. A Few Common Physical and Chemical Constants

50

Constant

Value

Avogadro's Number

NA = 6.02214 × 1023 mol-1

Faraday Constant

F = 96 485.33 C mol-1

Atomic Mass Constant

1 amu = 1.660538 × 10-27 kg

Molar Gas Constant

R = 0.08205746 L atm K-1 mol-1

Speed of Light (Vacuum)

c = 299 792 458 m s-1

Charge on a Proton/Electron

e = 1.602176 × 10-19 C

Standard acceleration of gravity

g = 9.80665 m s-2

Rydberg constant

R∞ = 1.0973731568539 × 107 m-1

Planck's Constant

h = 6.62607004 × 10-34 J s

For Problem Sets A, B, C, and D below, write each problem in equivalent Expanded Form and then in Simplified Exponent Form.

Problem Set A: Multiplying Expressions with Exponents Problem

1. (𝟏𝟎𝟐 )(𝟏𝟎𝟑 )

Expanded Form

Simplified Exponent Form

= (10 ∙ 10 ) ∙ (10 ∙ 10 ∙ 10) = 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10

2. 𝟓𝟐 ∙ 𝟓𝟒 ∙ 𝟓𝟑

3. (𝒙𝟑 )(𝒙𝟓 )

4. (𝒒𝟑 )(𝒒𝟓 )(𝒒𝟐 )

5. (𝒂𝟐 )(𝒂𝟑 )(𝒃𝟐 )(𝒃𝟑 )

6. Generalize the pattern you see in the exponents into the Property for Multiplying Expressions with Exponents.

51

Problem Set B: Exponent Expressions Raised to an Exponent (a.k.a. Power to a Power) Problem

Expanded Form

Simplified Exponent Form

= (10 ∙ 10 )3 𝟑

1. (𝟏𝟎𝟐 )

= (10 ∙ 10 ) ∙ (10 ∙ 10) ∙ (10 ∙ 10) = 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10

𝟐

2. (𝟓𝟒 )

𝟓

3. (𝒙𝟐 )

𝟐

4. (𝒙𝟑 𝒚𝟐 )

𝒂𝒃𝟑

𝟐

5. ( 𝒄𝟐 )

6. Generalize the pattern you see in the exponents into the Property for Power of a Power with Exponents.

52

Problem Set C: Dividing Expressions with Exponents Problem

1. 𝟏𝟎𝟓 ÷ 𝟏𝟎𝟐

=

𝟏𝟎𝟓

Expanded Form

=

10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 10 ∙ 10 ∙ 1 ∙ 1 ∙ 1

=

10 10 10 10 10 ∙ ∙ ∙ ∙ 1·1 10 10 1 1 1

𝟏𝟎𝟐

Simplified Exponent Form

= 1 ∙ 1 ∙ 10 ∙ 10 ∙ 10

= 2. 𝑪𝟑 ÷ 𝑪𝟓

=

𝑪𝟐

=

𝐶∙𝐶∙𝐶∙1∙1 𝐶∙𝐶∙𝐶∙𝐶∙𝐶 𝐶 𝐶 𝐶 1 1 1·1·1 ∙ ∙ ∙ ∙ 𝐶 𝐶 𝐶 𝐶 𝐶

𝑪𝟓

=1∙1∙1∙

1 1 ∙ 𝐶 𝐶

3. 𝒓𝟔 ÷ 𝒓𝟑

=

𝒓𝟔 𝒓𝟑

4. (𝒂𝟔 𝒃) ÷ (𝒂𝟑 𝒃𝟑 )

=

5.

𝒂𝟔 𝒃 𝒂𝟑 𝒃𝟑

𝒙𝟑 𝒚𝒛𝟑 𝒚 𝟑 𝒙𝟑 𝒛𝟐

6. Generalize the pattern you see in the exponents into the Property for Dividing Expressions with Exponents.

53

Problem Set D: Exploring Zero Exponents and Negative Exponents 1. Use the rule we generalized for Dividing Expressions with Exponents to make sense of Zero Exponents. a. Write in expanded form and simplify: 𝑥3 𝑥3

b. Simplify using the rule for Dividing Expressions with Exponents: 𝑥3 𝑥3

c. Use the results from parts (a) and (b) to write the Property of Zero Exponents.

2. Use the rule we generalized for Dividing Expressions with Exponents to make sense of Negative Exponents. a. Write in expanded form and simplify: 𝑥2 𝑥4

b. Simplify using the rule for Dividing Expressions with Exponents: 𝑥2 𝑥4

c. Use the results from parts (a) and (b) to write the Property of Negative Exponents.

54

Activity 3: Scientific Notation and Unit Conversions On a clear night, we can see thousands of stars emitting light in the night sky. We know that light travels at 299,792,458 meters per second. 1. Round the speed of light to the nearest hundred million meters per second.

2. Convert your answer in part 1 to kilometers per second.

3. Estimate how many kilometers light travels in one minute.

4. Our solar system star, the sun, is approximately 147.5 million kilometers from the earth. Use your answer from Problem 3 (above) to calculate how long it takes for the light from the sun to reach the earth.

5. Estimate how many kilometers light travels in one hour.

6. Estimate how far light can travel in one day. For this last problem, your calculator output will likely be in the calculator’s form of scientific notation. Use what you know about exponents and place value to convert this scientific notation to decimal notation.

7. In Imperial units, light travels approximately 5,880,000,000,000 miles in one year, and this distance is called a light year. Use what you know about exponents and place value to convert this decimal notation of one light year to scientific notation.

55

Section 3.2: Combining Functions Activity 1: Adding Functions 1. Add and then compare and contrast the following numeric problems and algebraic problems. Adding Numbers

Adding Functions

Model with Dienes blocks using the app, www.mathsbot.com/manipulatives/blocks

Model with Algebra Tiles using the app, www.mathsbot.com/manipulatives/tiles

a. Given 𝒂 = 𝟓𝟑𝟒 and 𝒃 = 𝟐𝟎𝟑, find 𝒂 + 𝒃.

Given 𝑓(𝑥) = 5𝑥 2 + 3𝑥 + 4 and 𝑔(𝑥) = 2𝑥 2 + 3, find 𝑓(𝑥) + 𝑔(𝑥).

b. Given 𝒂 = 𝟓𝟐𝟑 and 𝒃 = 𝟑𝟗𝟐, find 𝒂 + 𝒃.

Given 𝑓(𝑥) = 5𝑥 2 + 2𝑥 + 3 and 𝑔(𝑥) = 3𝑥 2 + 9𝑥 + 2, find 𝑓(𝑥) + 𝑔(𝑥).

Reflection: Compare and contrast adding numbers and adding polynomial functions. How are they similar? How are they different?

56

Activity 2: Subtracting Functions 1. Subtract and then compare and contrast the numeric problems and algebraic problems. Subtracting Numbers

Subtracting Functions

Model with Dienes blocks using the app, www.mathsbot.com/manipulatives/blocks

Model with Algebra Tiles using the app, www.mathsbot.com/manipulatives/tiles

a. Given 𝒂 = 𝟒𝟑𝟖 and 𝒃 = 𝟏𝟎𝟒, find 𝒂 − 𝒃.

Given 𝑓(𝑥) = 4𝑥 2 + 3𝑥 + 8 and 𝑔(𝑥) = 𝑥 2 + 4, find 𝑓(𝑥) − 𝑔(𝑥).

b. Given 𝒂 = 𝟒𝟎𝟓 and 𝒃 = 𝟐𝟗𝟏, find 𝒂 − 𝒃.

Given 𝑓(𝑥) = 4𝑥 2 + 5 and 𝑔(𝑥) = 2𝑥 2 + 9𝑥 + 1, find 𝑓(𝑥) − 𝑔(𝑥).

Reflection: Compare and contrast subtracting numbers and subtracting polynomial functions. How are they similar? How are they different?

57

Activity 3: Multiplying Functions 1. Multiply and then compare and contrast the numeric problems and algebraic problems. Multiplying Numbers

Multiplying Functions

Model with Dienes blocks using the app, www.mathsbot.com/manipulatives/blocks

Model with Algebra Tiles using the app, www.mathsbot.com/manipulatives/tiles

a. Given 𝒂 = 𝟑𝟏 and 𝒃 = 𝟐𝟓, find 𝒂𝒃.

Given 𝑓(𝑥) = 3𝑥 + 1 and 𝑔(𝑥) = 2𝑥 + 5, find 𝑓(𝑥) ∙ 𝑔(𝑥).

b. Given 𝒂 = 𝟏𝟑 and 𝒃 = 𝟑𝟓, find 𝒂𝒃.

Given 𝑓(𝑥) = 𝑥 + 3 and 𝑔(𝑥) = 3𝑥 + 5, find 𝑓(𝑥) ∙ 𝑔(𝑥).

Reflection: Compare and contrast multiplying numbers and multiplying polynomial functions. How are they similar? How are they different?

58

2. Study the problem solution methods below. In problem 3, you will select one of these methods to solve similar numeric and algebraic problems. Then you will compare and contrast the numeric multiplication and algebraic multiplication problems. Last, you will compare the three different solution methods and consider how the distributive property is evident in each method. Multiplying Numbers Given 𝒂 = 𝟑𝟓𝟏 and 𝒃 = 𝟐𝟑, find 𝒂 ∙ 𝒃.

Multiplying Functions Given 𝑓(𝑥) = 3𝑥 2 + 5𝑥 + 1 and 𝑔(𝑥) = 2𝑥 + 3,

Vertical Multiplication Method 3 × 1 +7 8

0 0 0

5 2 5 2 7

1 3 3 0 3

Box Multiplication Method •

300

+50

20

6000

1000 20

+3

900

1500

find 𝑓(𝑥) ∙ 𝑔(𝑥).

Vertical Multiplication Method 3x2 × 9x2 6x +10x2 6x3 +19x2 3

+5x 2x +15x + 2x +17x

+1 +3 +3 +3

+1

3

8073

Horizontal Multiplication Method (23)(351) = (20+3)(300+50+1) = 20(300+50+1)+3(300+50+1)

Box Multiplication Method •

3x2

+5x

+1

2x

6x3

+10x2

2x

+3

9x2

+15x

3

6x3 + 18x2 + 17x + 3

Horizontal Multiplication Method (2𝑥 + 3)(3𝑥 2 + 5𝑥 + 1)

= 6000+1000+20+900+150+3

= 2𝑥(3𝑥 2 + 5𝑥 + 1) + 3(3𝑥 2 + 5𝑥 + 1)

= 8073

= 6𝑥 3 + 10𝑥 2 + 2𝑥 + 9𝑥 2 + 15𝑥 + 3 = 6𝑥 3 + 19𝑥 2 + 17𝑥 + 3

59

3. Use one of the multiplication methods presented above to multiply. Then compare and contrast the following numeric problems and algebraic problems. Multiplying Numbers a. Given 𝒂 = 𝟔𝟎𝟓 and 𝒃 = 𝟑𝟗𝟐, find 𝒂 ∙ 𝒃.

Multiplying Functions Given 𝑓(𝑥) = 6𝑥 2 + 5 and 𝑔(𝑥) = 3𝑥 2 + 9𝑥 + 2, find 𝑓(𝑥) ∙ 𝑔(𝑥).

Reflection A: Compare and contrast multiplying numbers and multiplying functions. How are they similar? How are they different?

Reflection B: Compare and contrast each multiplication method. How is the distributive property evident in each method? Which method do you prefer? Why?

60

Activity 4: Dividing Functions with Monomial Divisor Examples of Dividing Numbers Given 𝒂 = 𝟔𝟑𝟕 and 𝒃 = 𝟑𝟎, find 𝒂 ÷ 𝒃.

Long Division Method

3

0

6 –6

2

1

3 0 3 3

7 0 7 0 7

𝟕 𝟑𝟎

Notes:

←

–(30·20) = –600

←

–(30·1) = –30

𝟕

so, 𝟔𝟑𝟕 ÷ 𝟑𝟎 = 𝟐𝟏 𝟑𝟎

Expanded Fraction Division Method

Given 𝒂 = 𝟔𝟑𝟕 and 𝒃 = 𝟑𝟎, find 𝒂 ÷ 𝒃.

Given 𝒂 = 𝟗𝟕𝟕 and 𝒃 = 𝟑𝟎, find 𝒂 ÷ 𝒃.

𝟔𝟑𝟕 𝟔𝟎𝟎 + 𝟑𝟎 + 𝟕 = 𝟑𝟎 𝟑𝟎 𝟏 = (𝟔𝟎𝟎 + 𝟑𝟎 + 𝟕) 𝟑𝟎 𝟔𝟎𝟎 𝟑𝟎 𝟕 = + + 𝟑𝟎 𝟑𝟎 𝟑𝟎 𝟕 = 𝟐𝟎 + 𝟏 + 𝟑𝟎 𝟕 = 𝟐𝟏 𝟑𝟎

𝟗𝟕𝟕 𝟗𝟎𝟎 + 𝟕𝟎 + 𝟕 = 𝟑𝟎 𝟑𝟎 𝟏 (𝟗𝟎𝟎 + 𝟕𝟎 + 𝟕) = 𝟑𝟎 𝟗𝟎𝟎 𝟕𝟎 𝟕 = + + 𝟑𝟎 𝟑𝟎 𝟑𝟎 𝟏𝟎 𝟕 = 𝟑𝟎 + 𝟐 + 𝟑𝟎 𝟑𝟎 𝟏𝟕 = 𝟑𝟐 𝟑𝟎

1. Use the Expanded Fraction Division Method for the division problems. a.

2750 𝟓𝟎

b.

842 20

61

Examples: Dividing Functions Given 𝒇(𝒙) = 𝟔𝒙𝟐 + 𝟑𝒙𝟐 + 𝟕 and 𝒈(𝒙) = 𝟑𝒙, find 𝒇(𝒙) ÷ 𝒈(𝒙).

Long Division Method 2x 3x

6x2 –6x2

+3x 3x –3x

+1 +7

+

𝟕 𝟑𝒙

Notes:

←

–(3x)(2x) = –(6x2) = –6x2

←

–(3x)(1) = –(3x) = –3x

+7 7 𝟕

so, (𝟔𝒙𝟐 + 𝟑𝒙 + 𝟕) ÷ (𝟑𝒙) = 𝟐𝒙 + 𝟏 + 𝟑𝒙

Expanded Fraction Division Method 𝟔𝒙𝟐 + 𝟑𝒙 + 𝟕 𝟑𝒙 =

𝟏 (𝟔𝒙𝟐 + 𝟑𝒙 + 𝟕) 𝟑𝒙

=

𝟔𝒙𝟐 𝟑𝒙 𝟕 + + 𝟑𝒙 𝟑𝒙 𝟑𝒙

= 𝟐𝒙 + 𝟏 +

𝟕 𝟑𝒙

2. Use the Expanded Fraction Division Method for the division problems. a.

2𝑥 3 +7𝑥 2 −5𝑥 𝟓𝒙

b.

8𝑥 2 +4𝑥+2 2𝑥

3. Reflection: Compare and contrast dividing numbers and dividing functions. How are they similar? How are they different?

62

Activity 5: Mixed Practice 1. Given 𝑓(𝑥) = 7𝑥 + 5 and 𝑔(𝑥) = 3𝑥 + 2, find: a. 𝑓(𝑥) + 𝑔(𝑥)

b. 𝑓(𝑥) − 𝑔(𝑥)

c. 𝑓(𝑥) ∙ 𝑔(𝑥)

2. Given 𝑓(𝑥) = 6𝑥 2 + 1 and 𝑔(𝑥) = 3𝑥 2 + 9𝑥 + 2, find: a. 𝑓(𝑥) + 𝑔(𝑥)

b. 𝑓(𝑥) − 𝑔(𝑥)

c. 𝑓(𝑥) ∙ 𝑔(𝑥)

3. Given 𝑓(𝑥) = 9𝑥 3 − 3𝑥 2 + 12𝑥 and 𝑔(𝑥) = 3𝑥, find: a. 𝑓(𝑥) + 𝑔(𝑥)

b. 𝑓(𝑥) − 𝑔(𝑥)

c. 𝑓(𝑥) ∙ 𝑔(𝑥)

d. 𝑓(𝑥) ÷ 𝑔(𝑥)

63

Activity 6: Dividing Functions with Binomial Divisor (Optional) 1. Study the examples for dividing numbers and functions below. Then compare and contrast the following numeric problems and algebraic problems. Dividing Numbers Given 𝒂 = 𝟖𝟎𝟕𝟑 and 𝒃 = 𝟐𝟑, find 𝒂 ÷ 𝒃.

Long Division Method 2

3

3 0 9 1 1

8 –6 1 –1

5 7 0 7 5 2 –2

1 3 0 3 0 3 3 0

Notes:

←

–(23·300) = –6900

←

–(23·50) = –1150

←

–(23·1) = –23

so 𝟖𝟎𝟕𝟑 ÷ 𝟐𝟑 = 𝟑𝟓𝟏 Dividing Functions Given 𝒇(𝒙) = 𝟔𝒙𝟑 + 𝟏𝟗𝒙𝟐 + 𝟏𝟕𝒙 + 𝟑 and 𝒈(𝒙) = 𝟐𝒙 + 𝟑, find 𝒇(𝒙) ÷ 𝒈(𝒙).

Long Division Method 2x

+3

3

6x –6x3

3x2 +19x2 –9x2 10x2 –10x2

+5x +17x +17x –15x 2x –2x

+1 +3

Notes:

←

–(2x+3)(3x2) = –(6x3+9x2) = –6x3–9x2

←

–(2x+3)(5x) = –(10x2+15x) = –10x2–15x

←

–(2x+3)(1) = –2x–3

+3 +3 –3 0

so (𝟔𝒙𝟑 + 𝟏𝟗𝒙𝟐 + 𝟏𝟕𝒙 + 𝟑) ÷ (𝟐𝒙 + 𝟑) = 𝟑𝒙𝟐 + 𝟓𝒙 + 𝟏 Reflection: Compare and contrast dividing numbers and dividing functions. How are they similar? How are they different?

64

2. Given 𝑓(𝑥) and 𝑔(𝑥), find 𝑓(𝑥) ÷ 𝑔(𝑥) using long division. Show each step. a. 𝑓(𝑥) = 𝑥 2 + 6𝑥 + 8 𝑔(𝑥) = 𝑥 + 2

b. 𝑓(𝑥) = 𝑥 2 + 12𝑥 − 28 𝑔(𝑥) = 𝑥 − 2

c. 𝒇(𝒙) = 𝒙𝟐 − 𝟔𝒙 + 𝟓 𝒈(𝒙) = 𝒙 − 𝟓

d. 𝑓(𝑥) = 2𝑥 2 + 3𝑥 − 9 𝑔(𝑥) = 2𝑥 − 3

65

Activity 7: Simplifying Expressions In the problems that follow, each expression uses the same symbols, but arranged differently, sometimes with parenthesis. Use the Order of Operations to simplify each expression. 1. 2𝑎−3

2. (−2𝑎)3

3. 23 (−𝑎)

4. (2𝑎)−3

5. (−2𝑎) ∙ 3

6. 2−3 ∙ 𝑎

7. 2 − 𝑎3

8. 23 − 𝑎

9. (2 − 𝑎)3

66

Section 3.4: Factoring Polynomials Relationship Between Factoring and Multiplying Multiply

3𝑥(2𝑥 − 3)(𝑥 + 3)

=

6𝑥 3 + 9𝑥 2 − 27𝑥

Factor

Figure 1. Inverse polynomial operations: multiplying and factoring

Activity 1: Understanding Factors 1. Use the grid below to represent the area of the numbers as rectangles, as many as possible (using only positive integer side lengths). What do the sides of the rectangle represent in relation to the area and the number? a. 15 square units

b. 16 square units

c. 17 square units

67

2. Examine each of the algebra tile diagrams below and answer the related prompts. a.

i. Write an expression for the total area by summing the areas of the six smaller rectangles. Show both the non-simplified and simplified polynomial expression. ii. Write an expression for this area diagram by multiplying the side lengths of the entire figure.

b.

i. Write an expression for the total area by summing the areas of the six smaller rectangles. Show both the non-simplified and simplified polynomial expression. ii. Write an expression for this area diagram by multiplying the side lengths of the entire figure.

c.

i. Write an expression for this area diagram by summing the (signed) areas of all the smaller rectangles. Show both the nonsimplified and simplified polynomial expression. ii. Write an expression for the total area by multiplying the side lengths of the entire figure.

d.

i. Write an expression for this area diagram by summing the (signed) areas of all the smaller rectangles. Show both the nonsimplified and simplified polynomial expression. ii. Write an expression for the total area by multiplying the side lengths of the entire figure.

68

e. Reflection: Use the above Algebra Tile representations of a quadratic polynomial to answer the following questions. i. What do the upper left and lower right portions of the area represent?

ii. What do the lower left and upper right portions of the area represent?

iii. What do you notice about the signs of the terms of the quadratics and related factors?

iii. How might Algebra Tiles help us (or not) to reason about factors of Quadratics?

Activity 2: Modeling Polynomials and their Factors 1. Factor, if possible. Record the factors using accepted mathematical conventions or write PRIME if the expression is not factorable using integers. (Optional: Model factoring with Algebra Tiles using the app, www.mathsbot.com/manipulatives/tiles.) a. 2𝑥 − 6

b. 3𝑥 + 10

c. 3𝑥 − 9

d. 𝑥 2 + 5𝑥 + 3

e. 𝑥 2 + 4𝑥 + 3

f. 2𝑥 2 − 6𝑥

69

Activity 3: Vocabulary of Polynomials & Factoring To communicate about factoring and to use the Factoring Flow Chart below, we need to know the language of polynomials.

Figure 2. Factoring Flowchart

70

Use the polynomial, 6𝑥 3 + 9𝑥 2 − 27𝑥, to answer the following questions. 1. How many terms are in 6𝑥 3 + 9𝑥 2 − 27𝑥 ? Name each term and separate them with commas.

2. The polynomial, 6𝑥 3 + 9𝑥 2 − 27𝑥, is written in standard form of a polynomial. What do you notice about the exponents of each term?

3. The leading term of 6𝑥 3 + 9𝑥 2 − 27𝑥 is _______, and the leading coefficient is _______.

4. GCF is an acronym for Greatest Common Factor. What is the GCF of 6𝑥 3 + 9𝑥 2 − 27𝑥? Complete the blanks with a number or variable to write this first step in factoring. 6𝑥 3 + 9𝑥 2 − 27𝑥 =

𝑥 (2𝑥2 +

−

)

5. The GCF is ________ and is a ________________ factor (fill in the second blank based on the number of terms in the factor). 6. The other factor is _________________________ and is a __________________ quadratic factor (fill in the second blank based on the number of terms in the factor). We need continue factoring and find factors of the quadratic factor, if they exist. 7. Identify Leading Term: The leading term of the quadratic factor above is _______. The leading coefficient of the quadratic factor above is _______. 8. Constant Term: The constant term of the quadratic factor above is: _________. 9. Middle Term: The middle term of the quadratic factor above is _______. The coefficient of the middle term is _______. 10. If (2𝑥 − 3) is a factor of the quadratic factor, what is the remaining factor? Fill in the blanks with a number or a variable. 6𝑥 3 + 9𝑥 2 − 27𝑥 =

𝑥 ( 2𝑥 − 3 )(

+

) 71

11. The polynomial before factoring is a trinomial and is _____________________________. a. The GCF is a monomial factor and is ______________. b. The first binomial factor is ______________________. c. The other binomial factor is _____________________. 12. Describe how we can check that the factored form is correct, and then conduct the check.

Activity 4: Scaffolded Trinomial Factoring & Generalizing 1. Factor the trinomial quadratic by filling in the blanks with the numbers that make the equality relationship true. Check your work. a.

b.

c.

2𝑥 2 − 𝑥2 +

𝑥 + 20 = (

𝑥−

)(𝑥 − 4)

𝑥 − 24 = (3𝑥 −

)(𝑥 + 6)

𝑥 2 + 𝑥 − 2 = (2𝑥 −

)(3𝑥 +

)

2. Generalize relationships between 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 = (𝑎𝑥 + 𝑐)(𝑏𝑥 + 𝑑). Consider the leading term and last term of the quadratic, and then the middle term, in relation to the factors. a. The leading term of 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 is _______ and, in relation to the factors, (𝑎𝑥 + 𝑐)(𝑏𝑥 + 𝑑), is equal to __________________. b. The constant term of 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 is _______ and, in relation to the factors, (𝑎𝑥 + 𝑐)(𝑏𝑥 + 𝑑), is equal to __________________. c. The middle term of 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 is ________ and, in relation to the factors, (𝑎𝑥 + 𝑐)(𝑏𝑥 + 𝑑), is equal to __________________.

72

Section 3.5: Special Factoring Techniques Activity 1: Recognizing Structure of Special Quadratics 1. Examine each of the algebra tile diagrams below and answer the related prompts. a.

i. Write an expression for the total area by summing the areas of the four smaller rectangles. Show both the non-simplified and simplified polynomial expression.

ii. Write an expression for the total area by multiplying the side lengths of the entire figure. i. Write an expression for the total area by summing the areas of all the smaller rectangles. Show both the non-simplified and simplified polynomial expression.

b.

ii. Write an expression for the total area by multiplying the side lengths of the entire figure.

i. Write an expression for the total area by summing the areas of all the smaller rectangles. Show both the non-simplified and simplified polynomial expression.

c.

ii. Write an expression for the total area by multiplying the side lengths of the entire figure.

d. The Algebra Tiles in the problems above represent a subset of a special set of quadratics called perfect square trinomials. How does the Algebra Tile representation and the simplified polynomial help describe the name of this special quadratic?

73

2. Examine each of the algebra tile diagrams below and answer the related prompts.

a.

i. Write an expression for the total area by summing the (signed) areas of all the smaller rectangles. Show both the non-simplified and simplified polynomial expression. ii. Write an expression for the total area by multiplying the side lengths of the entire figure.

b.

i. Write an expression for the total area by summing the (signed) areas of all the smaller rectangles. Show both the non-simplified and simplified polynomial expression. ii. Write an expression for the total area by multiplying the side lengths of the entire figure.

c.

i. Write an expression for the total area by summing the (signed) areas of all the smaller rectangles. Show both the non-simplified and simplified polynomial expression. ii. Write an expression for the total area by multiplying the side lengths of the entire figure.

d. The Algebra Tiles in the problems above represent a subset of a special set of quadratics called difference of squares. How does the Algebra Tile representation and the simplified polynomial help describe the name of this special quadratic?

3. Compare the diagrams, polynomials, and factors of Problem 1 and of Problem 2? What do you notice? How are they similar? How are they different?

74

Activity 2: Using Special Factoring Techniques 1. Factor, if possible. Record the factors using accepted mathematical conventions or write PRIME if there are no factors. (Optional: Model factoring with Algebra Tiles using the app, www.mathsbot.com/manipulatives/tiles.) a. 𝑥 2 + 6𝑥 + 9

b. 𝑥 2 + 10𝑥 + 16

c. 𝑥 2 + 9

d. 2𝑏 2 − 32

e. 9𝑦 2 + 30𝑦 + 25

f. 𝑥 2 − 36

75

Activity 3: Factoring Mixed Practice 1. Factor, if possible. Record the factors using accepted mathematical conventions or write PRIME if there are no factors. a. 𝑥 2 + 7𝑥 + 12

b. 3𝑥 2 + 9𝑥 − 8𝑥 − 24

c. 𝑥 2 + 4

d. 3𝑏 2 − 12

e. 4𝑦 2 + 14𝑦 + 25

f. 𝑥 2 − 144

h. 𝑦 2 + 5𝑦 − 24

i. 𝑏 2 − 121

g. 𝑥 2 𝑦 + 2𝑥𝑦 − 8𝑥𝑦 − 24𝑥𝑦 2

76

Chapter 4: Quadratic Equations and Functions Preview of Quadratic Equations and Functions We will be studying quadratic equations and quadratic functions. Quadratic equations are formed from quadratic functions by considering one particular output of the quadratic function. In the examples below, the quadratic equation is formed by considering the instance where the output of the quadratic function is zero ( 𝑓(𝑥) = 0), but it could be any number. We will study three forms of quadratic equations and functions and the information each form provides.

Quadratics in Standard Form Equation: 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎

Function: 𝒇(𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄

Quadratics in Factored Form Equation: 𝒂(𝒙 − 𝒑)(𝒙 − 𝒒) = 𝟎

Function: 𝒇(𝒙) = 𝒂(𝒙 − 𝒑)(𝒙 − 𝒒)

Quadratics in Vertex Form Equation: 𝒂(𝒙 − 𝒉)𝟐 + 𝒌 = 𝟎

Function: 𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌

Activity 1: How do we recognize a Quadratic Equation or Function? 1. What indicates that the above three forms are all quadratics? In other words, what is similar about each form of the quadratic equations and functions?

2. How can we distinguish a quadratic equation or function from a linear equation or function?

77

Activity 2: Connecting Characteristics of a Quadratic Graph with Forms of the Function Quadratic in Standard Form Equation: 𝟑𝒙𝟐 − 𝟔𝒙 − 𝟐𝟒 = 𝟎

Function: 𝒇(𝒙) = 𝟑𝒙𝟐 − 𝟔𝒙 − 𝟐𝟒

Quadratic in Factored Form Equation: 𝟑(𝒙 + 𝟐)(𝒙 − 𝟒) = 𝟎

Function: 𝒇(𝒙) = 𝟑(𝒙 + 𝟐)(𝒙 − 𝟒)

Quadratic in Vertex Form Equation: 𝟑(𝒙 − 𝟏)𝟐 − 𝟐𝟕 = 𝟎

Function: 𝒇(𝒙) = 𝟑(𝒙 − 𝟏)𝟐 − 𝟐𝟕

1. Prove that the Quadratic Forms provided above are equivalent equations and functions.

2. Examine the Quadratic Standard Form and its graph on the right. What point(s) can you easily determine from the Standard Form? Explain.

3. Examine Quadratic Factored Form and its graph on the right. What point(s) can you easily determine from the Factored Form? Explain.

4. Examine Quadratic Vertex Form and its graph on the right. What point(s) can you easily determine from the Vertex Form? Explain.

5. What is the axis of symmetry of a Quadratic Graph? What Form(s) of the Quadratic indicate the axis of symmetry?

78

Activity 3: Projectile Height Application 1. The projectile-height equation is ℎ(𝑡) =

−1 𝑔𝑡 2 2

+ 𝑣0 𝑡 + ℎ0 , where g is the acceleration due to

gravity, v0 is the initial vertical velocity (that is, the vertical velocity at time t = 0, the time of launch), and h0 is the initial height of the object (that is, the height at of the object at t = 0). An object is launched from a 58.8-meter-tall building with an initial vertical velocity of 19.6 meters per second (m/s). The acceleration due to gravity is 9.8 m/s2. a. Write the equation for the object's height h at time t seconds after launch.

b. Initial Height i. What is the initial height of the object when launched? ii. What form of a quadratic will show this value? iii. Circle the corresponding point on the graph. Describe the meaning of that point using the quantities for both variables. c. Maximum Height i. When does the object reach its maximum height? Circle this point on the graph. ii. What form of a quadratic will show this value? iii. Describe the meaning of that point using the quantities for both variables.

d. Zero Height i. When does the object strike the ground? Circle this point on the graph. ii. What Form of a Quadratic will show this value? iii. Describe the meaning of that point using the quantities for both variables.

79

Section 4.1 Introduction to Quadratic Functions Activity 1: Characteristics of Quadratic Functions 1. Given the graph, answer the questions. a. Does the parabola open up or open down? b. Estimate the vertex. Is the vertex a maximum or a minimum? c. For what intervals of x is the graph decreasing? d. For what intervals of x is the graph increasing? e. Estimate the horizontal intercept(s). f. Estimate the vertical intercept.

2. Given the graph, answer the questions. a. Does the parabola open up or open down? b. Estimate the vertex. Is the vertex a maximum or a minimum? c. For what intervals of x is the graph decreasing? d. For what intervals of x is the graph increasing? e. Estimate the horizontal intercept(s). f. Estimate the vertical intercept.

80

Section 4.5 Solving Quadratic Equations with Factoring Activity 1: Using the Zero Product Property to Solve Polynomial Equations Zero-Product Property: We know that 0 × 𝑏 = 0 and 𝑎 × 0 = 0. In other words, zero times any number equals zero and any number times zero equals zero. Most importantly, this is the only way for a product to equal zero. If 𝑎𝑏 = 0, then 𝑎 = 0 OR 𝑏 = 0 (or both). To use the zero-product property to solve equations involving polynomials, the equation needs to be set equal to zero. Then we factor and set each factor equal to zero and solve for the variable. Note: If there are more than two factors, this idea still applies and at least one of the factors must be zero. 1. (𝑥 − 2)(𝑥 + 3) = 0

2. 𝑥(2𝑥 − 3)(3𝑥 + 5) = 0

3. 𝑥 2 − 11𝑥 + 18 = 0

4. 𝑥 3 + 3𝑥 2 = 10𝑥

81

Activity 2: Applied Problems using Factors of Polynomials 1. An open box can be made from a rectangular piece of cardboard that is 48 inches by 40 inches by cutting a square from each corner and folding up the sides. a. Sketch and label 2 pictures of this problem. First, draw the flat piece of cardboard. Second, draw the box.

b. Write a function to model the volume, 𝑉(𝑥), of the open box in cubic inches when squares of length x inches are cut from each corner.

c. Evaluate 𝑉(8).

d. Explain the meaning of 𝑉(8). Include both of the variables and associated quantities.

e. For what three values of x will the volume of the box be zero cubic inches? Explain.

82

Recall that Revenue is the amount of money brought in by selling items and can be represented by the equation 𝑅 = 𝑁 ∙ 𝑃, where R is revenue, N is the number of items sold, and P is the price each item sells for. 2. A small business selling camping conversion kits for minivans sells an average of 150 kits each month when the price of each kit is $1000. From researching other campervan kits online, the owner estimates that for each $100 increase in price, the number of kits sold will decrease by 10. a. What is the current average monthly revenue?

b. Find the monthly revenue if the kit price increases to $1200? (Hint: This is two $100 increases.)

c. Write a function, 𝑅(𝑥), to model the monthly revenue if the kit price increases x times by $100.

d. How many $100 increases in price result in a cost of $1500? Use the model, 𝑅(𝑥), to determine the monthly revenue if the price is $1500.

e. Write a sentence explaining the meaning of your result in part d. Include both of the variables and associated quantities.

83

Section 4.4 Completing the Square Activity 1: Solving using the Square Root Property Square Root Property If 𝒄 ≥ 𝟎, the solutions to 𝒙𝟐 = 𝒄 are 𝒙 = ±√𝒄. This is because of the definition of even roots. 𝑛 √𝑎𝑛 = |𝑎| for 𝑛 is even 2 So, to solve 𝑥 = 𝑐, we perform the following steps. 𝑥2 = 𝑐 √𝑥 2 = √𝑐 |𝑥| = √𝑐 𝑥 = ±√𝑐 It is a lot easier to use the Square Root Property, but it is important to know where it comes from!

Solve the Quadratics using the Square Root Property. Find exact values. 1. 3𝑥 2 = 27

`

2. 5𝑥 2 = 405

3. 4(𝑥 − 3)2 = 144

4. 3(𝑥 + 7)2 = 147

5. 4(𝑥 + 3)2 + 12 = 156

6. 3(𝑥 + 7)2 + 15 = 162

84

Visualizing Completing the Square the Area Model Activity 2: Visualizing Completing thewith Square Jen Nimtz, Ph.D.

1. Given the polynomial 𝑥 2 − 6𝑥 = 0, solve by completing the square. Note the coefficient of the leading term is 𝑎 = 1.

Complete the Square for x 2 + 6x.

Visualizing Completing Algebra Tiles Representation

the Square with the Area Model Symbolic Representation Jen Nimtz, Ph.D. x + 3 leting the Square with the Area Model x 2 Complete the Square for x + 6x. Jen Nimtz, Ph.D. 2 𝑥 ++6𝑥 = 0

x

or x 2 + 6x. x

+

x

3

x

+

3

+ x

x2

(𝑥32 + 6𝑥 + _______) _______ Complete− the Square:= x 20+ 6x

3x

x 2 + 6x = (x 2 + 6x + ⇤) − ⇤ = (x 2 + 6x + 9) − 9 = (x + 3) 2 − 9

x x

+ 3x

2

3

9

−9

Complete the Square: x 2 + 6x

3x

Vertex Formx 2 + 6x

Complete the Square: x 2 + 6x

+ 3 −9

3

+ 3

x

3

+

x

3

+

= (x 2 + 6x + ⇤) − ⇤ =2 − (x 2_______ + 6x +=9)0 − 9 (𝑥 + ______) = (x + 3) 2 − 9

2

x + 6x = (x 2 + 6x + ⇤) − ⇤ 3x = (x 2 + 6x9+ 9) − 9 − 9 = (x + 3) 2 − 9

Solve for 𝑥. (𝑥 + ______)2 = _______ 𝑥 + ______ = ±√ 𝑥 = ________ ± _________ 𝑥 = ________ 𝑜𝑟 _________ Complete! Check the answers. 3

85 3

Activity 3: Completing the Square There are many procedures for completing the square of an equation. In this activity, we present a method that more clearly demonstrates the fact that completing the square of a quadratic equation and function are similar procedures. Problem 1. Solve and find Vertex Form by Completing the Square Solve by Completing the Square. First, find Vertex Form, 𝒂(𝒙 − 𝒉)𝟐 + 𝒌 = 𝟎.

Find Vertex Form, 𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌, by Completing the Square.

𝑥 2 + 14𝑥 − 51 = 0

𝑓(𝑥) = 𝑥 2 + 14𝑥 − 51

Group the first two terms.

Group the first two terms.

(𝑥 2 + 14𝑥) − 51 = 0

𝑓(𝑥) = (𝑥 2 + 14𝑥) − 51

Add and subtract half the middle term squared times 𝑎. In this case, 𝑎 = 1.

Add and subtract half the middle term squared times 𝑎. In this case, 𝑎 = 1.

(𝑥 2 + 14𝑥 + ________) − 51 − _________ = 0

𝑓(𝑥) = (𝑥 2 + 14𝑥 + ______) − 51 − _______

Write Quadratic part as a Squared Binomial.

Write Quadratic part as a Squared Binomial.

(𝑥 + __________)2 − __________ = 0

𝑓(𝑥) = (𝑥 + ________)2 − __________

Then, continue to solve for 𝒙 using the Square Root Property. (𝑥 + __________)2 = __________

Rewrite with signs like 𝑓(𝑥) = 𝑎(𝑥 − ℎ )2 + ℎ.

𝑓(𝑥) = (𝑥 − _________)2 + __________ Complete!

𝑥 + ________ = ±√

Vertex ( ________ , _______)

𝑥 = ________ ± √

Now check that

𝑥 = __________ ± ___________ 𝑥 = ________ + ________ 𝑜𝑟 ________ − ________ 𝑥 = __________ 𝑜𝑟 __________ Complete! Now check solutions. (________)2 + 14(________) − 51 = 0 ___________ + _____________ − 51 = 0 Checks! (________)2 + 14(________) − 51 = 0 ___________ + _____________ − 51 = 0 Checks!

86

𝒇(

) = ____________

Problem 2. Solve and find Vertex Form by Completing the Square (Optional) Solve by Completing the Square. First, find Vertex Form, 𝒂(𝒙 − 𝒉)𝟐 + 𝒌 = 𝟎.

Find Vertex Form, 𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌, by Completing the Square.

3𝑥 2 − 8𝑥 + 4 = 0

𝑓(𝑥) = 3𝑥 2 − 8𝑥 + 4

Group the first two terms.

Group the first two terms.

(3𝑥 2 − 8𝑥) + 4 = 0

𝑓(𝑥) = (3𝑥 2 − 8𝑥) + 4

Factor out the coefficient of the Squared term.

Factor out the coefficient of the Squared term.

3(𝑥 2 − ________𝑥) + 4 = 0

𝑓(𝑥) = 3(𝑥 2 − ________𝑥) + 4

Add and subtract half the middle term squared times 𝑎. In this case, 𝑎 = 3.

Add and subtract half the middle term squared times 𝑎. In this case, 𝑎 = 3.

𝟑(𝑥 2 − _______𝑥 + _______) + 4 − 𝟑(_______) = 0

𝑓(𝑥) = 𝟑(𝑥2 − ______𝑥 + ______) + 4 − 𝟑(______)

Write Quadratic part as a Squared Binomial.

Write Quadratic part as a Squared Binomial.

3(𝑥 − _________)2 + _________ = 0

𝑓(𝑥) = 3(𝑥 − _________)2 + _________

Then, continue to solve for 𝒙 using the Square Root Property. )2

3(𝑥 − ________

)2

(𝑥 − ________

= ________

Rewrite with signs like 𝑓(𝑥) = 𝑎(𝑥 − ℎ )2 + ℎ.

𝑓(𝑥) = 3(𝑥 − _________)2 + _________

Complete!

= ________

Vertex ( ________ , _______)

𝑥 − ________ = ±√

Now check that

𝑥 = __________ ± ________

𝒇(

) = ____________

𝑥 = __________ 𝑜𝑟 __________ Complete! Now check solutions. 3(________)2 − 8(__________) + 4 = 0 __________ − __________ + 4 = 0

Checks!

3(__________)2 − 8(__________) + 4 = 0 3(__________) − (__________) + 4 = 0 __________ − __________ + 4 = 0

Checks!

87

Section 4.6 The Quadratic Formula Activity 1: Deriving the Quadratic Formula Solve and find Vertex Form by Completing the Square Solve by Completing the Square. First, find Vertex Form, 𝒂(𝒙 − 𝒉)𝟐 + 𝒌 = 𝟎.

Find Vertex Form, 𝒇(𝒙) = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌, by Completing the Square.

𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0

𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐

Group the first two terms.

Group the first two terms.

(𝑎𝑥 2

𝑓(𝑥) = (𝑎𝑥 2 + 𝑏𝑥) + 𝑐

+ 𝑏𝑥) + 𝑐 = 0

Factor out the coefficient of the Squared term.

Factor out the coefficient of the Squared term.

𝑎 (𝑥 2 +

𝑓(𝑥) = 𝑎 (𝑥 2 + 𝑎 𝑥) + 𝑐 = 0

𝑏

𝑥) + 𝑐 = 0

Add and subtract half the middle term squared times 𝑎.

𝒂 (𝑥 2 + (

)𝑥 + (

2

) )+ 𝑐 −𝒂(

2

) =0

Write Quadratic part as a Squared Binomial.

Add and subtract half the middle term squared times 𝑎. 𝑓(𝑥) = 𝒂 (𝑥 2 + (

)𝑥 + (

Write Quadratic part as a Squared Binomial.

2

𝑎 (𝑥 + (

2

)) + (

)=0

Then, continue to solve for 𝒙 using the Square Root Property.

)) = − (

)

Divide both sides of equation by 𝑎.

)) = (

−𝒃

)

)

Subtract constant term from both sides and simplify.

𝑥 = −(

)±

√

Simplify

𝒙=

88

−𝒃±√𝒃𝟐 −𝟒𝒂𝒄 𝟐𝒂

→ Quadratic Formula

)) + (

Complete!

Vertex ( ) = ±√(

)

2

Take the square root of both sides.

𝑥+(

)) + (

Rewrite with signs like 𝑓(𝑥) = 𝑎(𝑥 − ℎ )2 + ℎ.

Vertex (

2

(𝑥 + (

𝑓(𝑥) = 𝑎 (𝑥 + (

𝑓(𝑥) = 𝑎 (𝑥 − (

2

𝑎 (𝑥 + (

2

) )+ 𝑐 −𝒂(

𝟐𝒂

−𝒃 𝟐𝒂

,

𝟒𝒂𝒄−𝒃𝟐 𝟒𝒂

−𝒃

)

, 𝒇 ( )) 𝟐𝒂

)

)

2

Activity 2: Solve Applied Problems using the Quadratic Formula 1. Jackie has a pool that is 40 feet by 30 feet. They plan on spreading a border of crushed stone around the pool and have enough stone to cover 550 square feet. How wide can they make the crushed stone border? a. Sketch a diagram that corresponds to this problem.

b. Write an algebraic equation for the area of the crushed stone border. Hint: If the pool was removed from the outermost rectangle in the diagram, only the area of the stone border would remain.

OR

Add up the areas of the eight rectangles that make up the border (which are shown in blue above).

c. Find the width of the crushed stone border. Round to the nearest hundredth.

89

2. There is a right triangle with two legs of length 𝑥 inches and 𝑥 + 8 inches and a hypotenuse of 20 inches. Find the length of the legs of the triangle. a. Sketch a diagram that corresponds to this problem.

b. Write an algebraic equation to find the length of the legs of the right triangle. (Hint: Use the Pythagorean Theorem.)

c. Find the length of the legs of the right triangle. Round to the nearest hundredth.

d. Find the area of the right triangle.

90

Section 4.7 Graphing Standard Form and Modelling Activity 1: Write a Quadratic Equation from Graph Given the quadratic graph and points on the graph, determine the exact values for: horizontal intercept(s), vertical intercept, vertex, domain, range, and the equation of the function. Assume the coordinate pairs provided near the vertex represent the vertex.

Graph A

Graph C

Graph B

Graph A contains the points (-10, 10), (-8, 0), and (0, 0). Determine exact values for: i. horizontal intercept(s) ii. vertical intercept iii. equation of the function in standard form iv. vertex

v. domain vi. range

91

Graph B contains the points (0, -16) and (8, 0). Determine exact values for: i. horizontal intercept(s) ii. vertical intercept iii. equation of the function in standard form iv. vertex

v. domain vi. range

Graph C contains the points (0, 14) and (4, 2). Determine exact values for: i. horizontal intercept(s) ii. vertical intercept iii. equation of the function in standard form iv. vertex

v. domain vi. range

92

Activity 2: Modeling with Quadratics A baseball is hit so that its height (in feet) t seconds after it is hit can be modeled by ℎ(𝑡) = −16𝑡 2 + 60𝑡 + 4.2. 1. Use the process discussed in class to graph this function by hand. Then, use that information and/or additional algebra to answer the questions that follow. Vertical intercept:

Horizontal intercept(s):

Line of Symmetry:

Vertex:

93

2. What point on the graph gives information about the height of the ball when it is hit? Write the answer as a coordinate pair, then write the coordinate pair using function notation. Lastly, describe the meaning of the point in the context of the situation. Coordinate Pair:

Function Notation:

Meaning: 3. What point on the graph gives information about the ball’s maximum height? Write the answer as a coordinate pair, then write the coordinate pair using function notation. Lastly, describe the meaning of the point in the context of the situation. Coordinate Pair:

Function Notation:

Meaning: 4. If it is not caught, what point(s) on the graph give information about when the ball hits the ground? Write the answer as a coordinate pair, then write the coordinate pair using function notation. Lastly, describe the meaning of the point(s) in the context of the situation. Coordinate Pair:

Function Notation:

Meaning: 5. What point(s) on the graph give information about when the ball reaches a height of 40 feet? Write the answer as a coordinate pair, then write the coordinate pair using function notation. Lastly, describe the meaning of the point(s) in the context of the situation. Additional algebra:

Coordinate Pair(s): Meaning:

94

Function Notation:

Chapter 7: Rational Functions: Expressions and Equations Section 7.1 Rational Functions and Variation Activity 1: Comparing Linear and Rational Functions 1. Miriam is driving from Ferndale, Washington to Portland, Oregon to visit a friend. She is driving at an average speed of 70 miles per hour. Make a table and graph describing how far Miriam has traveled every half an hour. In this situation, the average rate (speed) remains constant at 70 miles per hour, and distance and time are changing. Distance traveled depends upon time. The distance of the trip is about 280 miles. How long will the trip take?

𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 = 𝑹𝒂𝒕𝒆 ∙ 𝑻𝒊𝒎𝒆

→

𝒅 = 𝟕𝟎𝒕

a. Complete the table, plot the points, and sketch the graph.

Time, t (hours)

Distance, d (miles)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

95

2. Miriam is driving from Ferndale, Washington to Portland, Oregon to visit a friend. The trip is a distance of 280 miles. How long the trip will take (time) depends on how fast she drives (average rate). In this situation, the quantity of distance remains constant at 280 miles, and time and average rate (average speed) are changing.

𝑻𝒊𝒎𝒆 =

𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝑹𝒂𝒕𝒆

→

𝑻=

𝟐𝟖𝟎 𝒓

a. Complete the table, plot the points, and sketch the graph.

Rate, r (mph)

Time, T (hours)

2.5 5 10 20 40 60 80 100

b. What happens to the time the trip takes as the average speed gets slower?

c. What happens to the time the trip takes as the average speed gets faster?

d. What is a reasonable domain and range for the mathematical model of this situation?

96

Sections 7.2, 7.3, & 7.4: Simplifying Rational Expressions Activity 1: Review of Numeric Fractions 1. Simplify the fractions using factors and

8

Example 1:

= =

Example 2:

= = =

Example 3:

12 2∙4

a.

3∙4 2

= =

b.

121

12

c.

36

𝑏

𝑏

=1∙𝑐 = 𝑐.

16 72

3

4

2. Multiply fractions and simplify.

7

∙

3 12 (4) (7)

∙

a.

7 4

∙

b.

4 5

7 33

∙

9 42

c.

3

∙

5

10 2

(3) (3∙4) (4∙7) (4∙3∙3) 7 9

4

÷

7

3 12 4 12

= ∙ =

33

𝑎∙𝑏 𝑎∙𝑐

3 7 (4) (3∙4) (3)

∙

3. Divide fractions and simplify. a.

7 4

÷

3 4

b.

7 9

÷

35 27

c.

3 10

÷

9 20

(7)

3∙4∙4 3∙7 16 7

97

Example 4:

5 4

=

5∙ 5∙ 7

5 8

= = = = =

98

a.

+

7

24 22

2∙12 11 12

+

+

24 2∙11

5 9

=

72

c.

5 11

=

121

5. Add and subtract fractions and simplify.

24

8∙

15

45

b.

28

+

8∙ 3

15

=

𝑎

28

5∙ 5∙ 3

7

𝑏∙𝑎

= 𝑐∙𝑎 because 𝑎 = 1 .

35

=

4∙ 7

=

28

=

4∙

Example 5:

𝑏 𝑐

4. Build equivalent fractions using

7 24 7

24

7 24

a.

7 3

+

3 5

b.

7 9

+

31 54

c.

5 11

−

1 2

Activity 2: Simplifying Rational Expressions 1. Simplify the fractions and rational expressions and compare the procedures. If needed, refer back to Activity 1 Example 1 and Problem 1. Simplifying Fractions Simplify the fraction using factors and 𝒂∙𝒃 𝒃 𝒃 =𝟏∙𝒄 = 𝒄. 𝒂∙𝒄 a.

𝟑𝟑

Simplify the fraction using factors and 𝒂∙𝒃 𝒃 𝒃 =𝟏∙ = . c.

Simplify the rational expressions using 𝑎∙𝑏 𝑏 𝑏 factors and =1∙ = . 𝑎∙𝑐

b.

𝟏𝟐𝟏

𝒂∙𝒄

Simplifying Rational Expressions

𝒄

𝒄

𝟐𝟖 𝟒𝟗

𝑐

𝑐

2𝑥 2 +4𝑥 5𝑥 2 +10𝑥

Simplify the fraction using factors and 𝑎∙𝑏 𝑏 𝑏 =1∙ = . 𝑎∙𝑐

d.

𝑐

𝑐

𝑥 2 +3𝑥+2 2𝑥 2 −𝑥−10

e. Reflection: Compare and contrast simplifying fractions and simplifying rational expressions. How are they similar? How are they different?

99

2. Determine which solution method is correct and which is incorrect. Point out the error in the incorrect solution. Student A

Student B

𝒙𝟐 + 𝟔𝒙 + 𝟖 𝒙+𝟒

𝒙𝟐 + 𝟔𝒙 + 𝟖 𝒙+𝟒

=

𝒙 + 𝟔𝒙 + 𝟐 𝟏+𝟏

=

=

𝟕𝒙 + 𝟐 𝟐

= (𝒙 + 𝟐)

= 𝟕𝒙

100

(𝒙 + 𝟐)(𝒙 + 𝟒) (𝒙 + 𝟒)

Activity 3: Multiplying and Dividing Rational Expressions 1. To multiply fractions and rational expressions, first find factors of the numerator and denominator, simplify using factors and

𝑎∙𝑏 𝑎∙𝑐

𝑏 𝑐

𝑏 𝑐

= 1 ∙ = , then multiply the numerators and

multiply the denominators. If needed, refer to Activity 1, Examples 2 & 3, Problems 2 & 3. Multiplying Fractions a.

b.

Multiplying Rational Expressions

𝟕 𝟑

𝑥

𝟑 𝟓

2𝑥−5

∙

∙

𝟗 𝟑𝟎

𝑥

𝟓 𝟓𝟒

𝑥 2 −2𝑥

∙

2𝑥 2 −𝑥−10 𝑥2

∙

𝑥−2 𝑥−4

c. Reflection: Compare and contrast multiplying fractions and multiplying rational expressions. How are they similar? How are they different?

101

2. Determine which solution method is correct. Point out the error(s) in the incorrect solution. Student A

Student B

𝟐𝒙 + 𝟑 𝟕 − 𝒙 ∙ 𝒙−𝟕 𝒙+𝟐

2𝑥 + 3 7 − 𝑥 ∙ 𝑥−7 𝑥+2

=

(𝟐𝒙 + 𝟑) (𝟕 − 𝒙) ∙ (𝒙 − 𝟕) (𝒙 + 𝟐)

=

(2𝑥 + 3) −(𝑥 − 7) ∙ (𝑥 − 7) (𝑥 + 2)

=

(𝟐𝒙 + 𝟑) (𝒙 + 𝟐)

=

−(2𝑥 + 3) (𝑥 + 2)

=

−2𝑥 − 3 𝑥+2

102

3. To divide fractions and rational expressions, first rewrite division as multiplication of the reciprocal of the divisor (the second expression), then multiply as we did above and simplify. Dividing Fractions a.

b.

𝟕 𝟗

𝟕 𝟑

÷

÷

𝟐𝟏

Dividing Rational Expressions

𝑥

𝟓𝟒

2𝑥−5

𝟒𝟗

𝑥

𝟗

𝑥+5

÷

÷

𝑥−3 2𝑥 2 −𝑥−10

𝑥 2 −3𝑥 𝑥 2 +7𝑥+10

c. Reflection: Compare and contrast dividing fractions and dividing rational expressions. How are they similar? How are they different?

103

4. Determine which solution method is correct. Point out the error(s) in the incorrect solution. Student A

Student B

𝒙+𝟓 𝒙−𝟕 ÷ 𝒙−𝟕 𝒙+𝟐

𝒙+𝟓 𝒙−𝟕 ÷ 𝒙−𝟕 𝒙+𝟐

=

(𝒙 + 𝟓) (𝒙 + 𝟐) ∙ (𝒙 − 𝟕) (𝒙 − 𝟕)

=

(𝑥 + 5) (𝑥 − 7) ÷ (𝑥 − 7) (𝑥 + 2)

=

(𝒙 + 𝟓)(𝒙 + 𝟐) (𝒙 − 𝟕)𝟐

=

𝑥+5 1 ÷ 1 𝑥+2

= (𝑥 + 5)(𝑥 + 2)

104

Activity 4: Rational Expressions & Common Denominators 1. Build equivalent fractions and equivalent rational expressions using

𝑏 𝑐

𝑎∙𝑏

= 𝑎∙𝑐 because

𝑎 𝑎

=1.

If needed, refer to Activity 1 Example 4 and Problem 4. Fractions & Common Denominators Build equivalent fractions using 𝒃 𝒃∙𝒂 𝒂 = because = 𝟏 . 𝒄

𝒄∙𝒂

Rational Expressions & Common Denominators Build equivalent rational expressions 𝑏 𝑏∙𝑎 𝑎 using 𝑐 = 𝑐∙𝑎 because 𝑎 = 1 .

𝒂

𝑥−3 a.

𝟓 𝟏𝟐

=

2𝑥−5

b.

𝒄∙𝒂

𝟐 𝟑

Build equivalent rational expressions 𝑏 𝑏∙𝑎 𝑎 using = because = 1 . 𝑐

𝑐∙𝑎

𝑎

𝒂

𝑥−1

=

2𝑥 2 −𝑥−10

𝟏𝟒𝟒

Build equivalent fractions using 𝒃 𝒃∙𝒂 𝒂 = because = 𝟏 . 𝒄

=

𝟖𝟏

𝑥−2

=

3𝑥 3 −24𝑥 2 +36𝑥

c. Reflection: Compare and contrast building equivalent fractions and building equivalent rational expressions. How are they similar? How are they different?

105

Activity 5: Adding and Subtracting Rational Expressions 1. To add the fractions and rational expressions, find common denominators by building equivalent fractions, add the numerators, and then simplify if possible. If needed, refer to Activity 1 Example 5 and Problem 5. Adding Fractions a.

b.

𝟕 𝟗

𝟕 𝟑

+

+

Adding Rational Expressions

𝟑𝟏

−8

𝟓𝟒

𝑥 2 −2𝑥

𝟑

𝑥+2

𝟓

𝑥 2 −4

+1

+

3𝑥 𝑥 2 −2𝑥

c. Reflection: Compare and contrast adding fractions and adding rational expressions. How are they similar? How are they different?

106

2. Determine which solution method is correct and which is incorrect. Point out the error(s) in the incorrect solution. Student A

Student B

(𝒙 + 𝟓) (𝒙 + 𝟑) + (𝒙 − 𝟕) (𝒙 + 𝟐) =

𝟐𝒙 + 𝟖 𝟐𝒙 − 𝟓

(𝑥 + 5) (𝑥 + 3) + (𝑥 − 7) (𝑥 + 2) =

(𝑥 + 5)(𝑥 + 2) (𝑥 + 3)(𝑥 − 7) + (𝑥 − 7)(𝑥 + 2) (𝑥 + 2)(𝑥 − 7)

(𝑥 2 + 7𝑥 + 10) (𝑥 2 − 4𝑥 − 21) = + (𝑥 − 7)(𝑥 + 2) (𝑥 + 2)(𝑥 − 7) =

(2𝑥 2 + 3𝑥 − 11) (𝑥 − 7)(𝑥 + 2)

107

3. To subtract the fractions and rational expressions, first rewrite subtraction as subtraction of the opposite of the subtrahend (the second term), find common denominators by building equivalent fractions, add the numerators as we did above, and then simplify if possible. Subtracting Fractions a.

b.

𝟕 𝟗

𝟓 𝟑

−

−

𝟐𝟑

Subtracting Rational Expressions

𝑥−3

𝟑𝟔

𝑥+1

𝟑

𝑥+2

𝟓

𝑥+5

−

−

𝑥−3 𝑥 2 +3𝑥+2

𝑥−3 𝑥−4

c. Reflection: Compare and contrast subtracting fractions and subtracting rational expressions. How are they similar? How are they different?

108

4. Determine which solution method is correct and which is incorrect. Point out the error(s) in the incorrect solution. Student A

Student B

𝒙+𝟓 𝒙+𝟑 − 𝒙−𝟕 𝒙+𝟐

𝑥+5 𝑥+3 − 𝑥−7 𝑥+2

=

(𝒙 + 𝟓)(𝒙 + 𝟐) −(𝒙 + 𝟑)(𝒙 − 𝟕) + (𝒙 − 𝟕)(𝒙 + 𝟐) (𝒙 + 𝟐)(𝒙 − 𝟕)

=

(𝑥 + 5)(𝑥 + 2) −(𝑥 + 3)(𝑥 − 7) + (𝑥 − 7)(𝑥 + 2) (𝑥 + 2)(𝑥 − 7)

=

𝒙𝟐 + 𝟕𝒙 + 𝟏𝟎 −(𝒙𝟐 − 𝟒𝒙 − 𝟐𝟏) + (𝒙 − 𝟕)(𝒙 + 𝟐) (𝒙 + 𝟐)(𝒙 − 𝟕)

=

𝑥 2 + 7𝑥 + 10 −(𝑥 2 − 4𝑥 − 21) + (𝑥 − 7)(𝑥 + 2) (𝑥 + 2)(𝑥 − 7)

=

𝟑𝒙 − 𝟏𝟏 (𝒙 − 𝟕)(𝒙 + 𝟐)

=

𝑥 2 + 7𝑥 + 10 −𝑥 2 + 4𝑥 + 21 + (𝑥 − 7)(𝑥 + 2) (𝑥 + 2)(𝑥 − 7)

=

11𝑥 + 31 (𝑥 − 7)(𝑥 + 2)

109

Activity 6: Simplify Rational Expressions — Mixed Practice Simplify the rational expressions. Use the Order of Operations. 1.

3.

5.

𝑥−𝑦 2𝑥+𝑦

1

∙

−

2𝑥−𝑦

2.

𝑥+𝑦

𝑥 2 −𝑦 2

4.

𝑥+𝑦 𝑥 2 +2𝑥𝑦+𝑦 2

1

1

( 𝑥+𝑦 − 𝑥−𝑦 ) ÷

110

4𝑦 (𝑥 2 +2𝑥𝑦+𝑦 2 )

6.

𝑥 𝑥 2 −4

𝑥 𝑥 2𝑦

−

𝑥+3 2𝑥−5

+(

𝑥 𝑦2 𝑧

÷

1 𝑥−2

+

𝑦 𝑥 2𝑧2

2𝑥 2 +𝑥−15 𝑥−3

∙

3−𝑥 𝑥+2

)

Section 7.5: Solving Rational Equations Activity 1: Clearing Numeric Fractions to Solve Linear Equations (Review) Recall solving linear equations by clearing fractions using common denominators. 1.

3.

5 12

3 4

𝑥=

7 12

𝑥+

1

3

2

8

𝑥= 𝑥+

1 12

2.

4.

1 8

1 6

1

19

2

24

1

−1

2

4

𝑥+ =

𝑥− =

𝑥+

𝑥+

7 12

3 4

111

Activity 2: Clearing Algebraic Fractions to Solve Rational Equations 2. Solve rational equations by clearing fractions using common denominators. a.

1 𝑥

c.

112

5

= + 2, 𝑥

1 𝑥+2

+1=

b.

𝑥≠0

6 𝑥+2

,

𝑥 ≠ −2

d.

3 𝑥−3

3 𝑥

+

−

2 𝑥−1

2 𝑥+3

=

=

1 2𝑥

12 𝑥 2 −9

,

,

𝑥 ≠ 0 AND 𝑥 ≠ 1

𝑥 ≠ −3 AND 𝑥 ≠ 3

Activity 3: Considering Extraneous Solutions 1. Solve rational equations by clearing fractions using common denominators. Check for

extraneous solutions. a.

b.

c.

3 𝑥−2

+

2−

2𝑥 3𝑥+3

2 𝑥+2

1 𝑥−1

−

=

=

12 𝑥 2 −4

𝑥−2 𝑥−1

𝑥+2 6𝑥+6

=

𝑥−6 8𝑥+8

+

5 12

113

Activity 4: Evaluate the Solution Methods 1. Determine which solution method is correct. Point out the error(s) in the incorrect solution. Student A 𝟑

a.

(𝒙−𝟐)

− 𝟕=

Student B 𝒙 (𝒙−𝟐)

The LCD is (𝒙 − 𝟐).

3 𝑥 − 7= (𝑥 − 2) (𝑥 − 2) The LCD is (𝑥 − 2).

𝟑 𝒙 3 𝑥 ) ∙ (𝒙 − 𝟐) (𝑥 − 2) ∙ (𝒙 − 𝟐) ∙ ( − 𝟕) = ( − 7 = ∙ (𝑥 − 2) (𝒙 − 𝟐) (𝒙 − 𝟐) (𝑥 − 2) (𝑥 − 2)

𝟑(𝒙 − 𝟐) 𝒙(𝒙 − 𝟐) − 𝟕(𝒙 − 𝟐) = (𝒙 − 𝟐) (𝒙 − 𝟐)

3−7=𝑥 −4 = 𝑥

𝟑 − 𝟕(𝒙 − 𝟐) = 𝒙 𝟑 − 𝟕𝒙 + 𝟏𝟒 = 𝒙 𝟏𝟕 = 𝟖𝒙 𝟏𝟕 𝟖

𝒙

b.

𝟑𝒙

−

𝟒 𝟑

=

=𝒙

𝟐𝒙 𝟑𝒙

𝑥 4 2𝑥 − = 3𝑥 3 3𝑥 The LCD is 3𝑥.

The LCD is 𝟑𝒙. 𝒙 𝟒 𝟐𝒙 𝟑𝒙 ∙ ( − ) = ( ) ∙ 𝟑𝒙 𝟑𝒙 𝟑 𝟑𝒙

𝑥 4 2𝑥 3𝑥 ∙ ( − ) = ( ) ∙ 3𝑥 3𝑥 3 3𝑥

𝟑𝒙 ∙ 𝒙 𝟑𝒙 ∙ 𝟒 𝟐𝒙 ∙ 𝟑𝒙 − = 𝟑𝒙 𝟑 𝟑𝒙

3𝑥 ∙ 𝑥 3𝑥 ∙ 4 2𝑥 ∙ 3𝑥 − = 3𝑥 3 3𝑥

𝒙 − 𝟒𝒙 = 𝟐𝒙

𝑥 − 4𝑥 = 2𝑥

− 𝟑𝒙 = 𝟐𝒙

− 3𝑥 = 2𝑥

𝟎 = 𝟓𝒙

0 = 5𝑥

𝟎=𝒙

0=𝑥

No Solution because 𝒙 ≠ 𝟎 .

114

Activity 5: Applied Problems Review Section 7.1 Activity 2 before completing this problem. 1. A kayaker can paddle 14 miles downstream in the same amount of time that they can paddle 2 miles upstream. If the river current averages 3 miles per hour, how fast can the kayaker paddle in still water? a. Define the variable. b. Make a sketch of this problem and label quantities.

c. Read the problem and complete the table below. Write Rate in terms of the speed of paddling (the variable) and the speed of the current. Write an expression for Time based on Distance and Rate using 𝐷 = 𝑅𝑇 Distance (miles)

Rate (mph)

Time (hours)

Downstream (with current) Upstream (against current) d. Set up the rational equation using the information that the time is the same for the downstream and upstream scenarios, then solve.

e. Check your answer by substitution. You can also use your graphing calculator to graph the equation of time downstream and graph the equation of time upstream, and then find the intersection.

115

2. It takes Tom 36 minutes to mow the lawn, while it takes Jake 45 minutes to mow the same lawn. If Tom and Jake work together, using two lawn mowers, how long would it take them to mow the lawn? a. What are we asked to find? Use this to define the variable.

b. How much of the lawn does Tom mow in 1 minute?

c. How much of the lawn does Jake mow in 1 minute?

d. Use the verbal model below to write a rational equation. Part of lawn Tom mows in one minute.

e. Solve the equation.

116

+

Part of lawn Jake mows in one minute.

=

Part of lawn both mow in one minute (use variable from part a)

Chapter 8: Radical Functions: Expressions and Equations Section 8.1 Radicals and Radical Functions Activity 1: Computing Squares and Square Roots 1. Calculate the squared numbers and the related square roots. Memorizing these facts will be helpful.

Squared Number

Result

Square Root

𝟏𝟐

√𝟏

𝟐𝟐

√𝟒

𝟑𝟐

√𝟗

𝟒𝟐

√𝟏𝟔

𝟓𝟐

√𝟐𝟓

𝟔𝟐

√𝟑𝟔

𝟕𝟐

√𝟒𝟗

𝟖𝟐

√𝟔𝟒

𝟗𝟐

√𝟖𝟏

𝟏𝟎𝟐

√𝟏𝟎𝟎

𝟏𝟏𝟐

√𝟏𝟐𝟏

𝟏𝟐𝟐

√𝟏𝟒𝟒

Result

REFLECT: What relationship do you notice between squared numbers and square roots?

117

2. Consider the sign (positive or negative) and format of the squared numbers and the sign of the result.

𝟏𝟐

(−𝟏)𝟐

Opposite of Squared Number −𝟏𝟐

𝟐𝟐 𝟑𝟐

(−𝟐)𝟐 (−𝟑)𝟐

−𝟐𝟐 −𝟑𝟐

Squared Number

Result

Squared Number

Result

Result

REFLECT: What do you notice about the signs and syntax of the squared numbers in relation to the results? Why is this the case?

3. Consider the sign (positive or negative) and format of the square roots of numbers and the sign of the real number result, if it exists.

Square Root

Result

Square Root

Result

Square Root

𝟐

√ 𝟏𝟐

𝟐

√(−𝟏)𝟐

𝟐

𝟐

√ 𝟐𝟐

𝟐

√(−𝟐)𝟐

𝟐

𝟐

𝟐

𝟐

√ 𝟑𝟐

√(−𝟑)𝟐

Result

√−𝟏𝟐 √−𝟐𝟐 √−𝟑𝟐

2

NOTE: In the first two tables, the numeric patterns suggest that √𝑥 2 = |𝑥|

REFLECT: What else do you notice about the signs and syntax of the square roots in relation to the results? Why is this the case?

118

Activity 2: Computing Cubes and Cube Roots Radical Expression index radical

𝑛

√𝑎

radicand

1. Calculate the cubed numbers and the related cube roots. Memorizing these facts will be helpful.

Number Cubed

Result

Cube Root

𝟏𝟑

𝟑

𝟐𝟑

𝟑

Result

√𝟏 √𝟖

𝟑𝟑

𝟑

𝟒𝟑

𝟑

√𝟐𝟕 √𝟔𝟒

𝟓𝟑

𝟑

𝟔𝟑

𝟑

𝟕𝟑

𝟑

𝟖𝟑

𝟑

𝟗𝟑

𝟑

√𝟏𝟐𝟓 √𝟐𝟏𝟔 √𝟑𝟒𝟑 √𝟓𝟏𝟐 √𝟕𝟐𝟗

𝟏𝟎𝟑

𝟑

𝟏𝟏𝟑

𝟑

𝟏𝟐𝟑

𝟑

√𝟏𝟎𝟎𝟎 √𝟏𝟑𝟑𝟏 √𝟏𝟕𝟐𝟖

REFLECT: What relationship do you notice between cubed numbers and cube roots?

119

2. Consider the sign (positive or negative) and format of the cubed numbers and the sign of the result.

Cubed Number

Cubed Number

Result

Opposite of Cubed Number

Result

𝟏𝟑

(−𝟏)𝟑

−𝟏𝟑

𝟐𝟑

(−𝟐)𝟑

−𝟐𝟑

𝟑𝟑

(−𝟑)𝟑

−𝟑𝟑

Result

REFLECT: What do you notice about the signs and syntax of the cubed numbers in relation to the results? Why is this the case?

3. Consider the sign (positive or negative) and format of the cube roots of numbers and the sign of the result.

Cube Root

Cube Root

Result

Cube Root

Result

𝟑

√𝟏𝟑

𝟑

√(−𝟏)𝟑

𝟑

𝟑

√𝟐𝟑

𝟑

√(−𝟐)𝟑

𝟑

𝟑

𝟑

𝟑

√𝟑𝟑

√(−𝟑)𝟑

Result

√−𝟏𝟑 √−𝟐𝟑 √−𝟑𝟑

REFLECT: What do you notice about the signs and syntax of the cube roots in relation to the results? Why is this the case?

120

Activity 3: Radicals and Fractional Exponents Radicals as Rational Exponents

1 𝑎𝑛

𝑛

√𝑎 =

Review the Properties of Exponents and the Property of Radicals in the Appendix. We can use either radicals or rational exponents to write equivalent expressions. For example: Using Rational Exponents 𝟑

√𝟔𝟒𝒙𝟏𝟎 𝒚𝟒

Using Radicals

𝟑

√𝒙𝒚−𝟐

𝟑

√𝟔𝟒𝒙𝟏𝟎 𝒚𝟒

1

𝟑

√𝒙𝒚−𝟐

=

(𝟔𝟒𝒙𝟏𝟎 𝒚𝟒 )3

=√ =

1

(𝒙𝒚−𝟐 )3

𝟑 𝟔𝟒𝒙𝟏𝟎 𝒚𝟒

1 𝟏𝟎 𝟒 3 𝟔𝟒𝒙 𝒚

𝒙𝒚−𝟐

=(

𝟑

√𝟔𝟒𝒙𝟗 𝒚𝟔

𝒙𝒚−𝟐

)

1

= 𝟒𝒙𝟑 𝒚𝟐

= (𝟔𝟒𝒙𝟗 𝒚𝟔 )3 = 𝟒𝒙𝟑 𝒚𝟐

At times, it is necessary to change radicals to rational exponents. If we need to multiply radicals with different indices, then we need to use rational exponents. See the example below. 𝟏

𝟏

√𝒙𝟑 𝟒√𝒙 = (𝒙𝟑 )𝟐 𝒙𝟒 𝟑

𝟏

𝟔

𝟏

= 𝒙𝟐 𝒙𝟒 = 𝒙𝟒 𝒙𝟒 𝟕

= 𝒙𝟒 𝟒

𝟑

= 𝒙𝟒 𝒙𝟒 𝟒

= 𝒙 √𝒙𝟑

121

1. Rewrite the radical expressions using rational exponents and simplify if possible. a.

c.

e.

3

√𝑦 6

b.

3

√𝑥 2 𝑦 6

d.

3

√𝑥 2

√𝑥 2 𝑦 6

3

√5𝑥 3

f. ( √𝑥𝑦 2 )

2

2. Rewrite the rational exponent expressions with radical(s) and simplify if possible. 3

2

a.

𝑥5

c.

𝑥 3𝑦3

e.

(27𝑥 3 )3

b.

𝑦3

d.

(𝑥 3 𝑦)3

f.

(27𝑥 3 )2

2

2

1

1

We can rewrite the radicand as factors, some of which are perfect squares. √27𝑥 3 = √3 ∙ 9 ∙ 𝑥 ∙ 𝑥 2

122

Activity 4: Comparing Quadratic and Radical Functions In this activity, we examine the relationship between the speed a car is traveling and its braking distance. The distance needed to stop a car depends on its speed at the moment of braking, and two times a constant, 𝜇 , based on the road conditions (see the table below). Road Surface Asphalt or concrete road surface Gravel surface Unsurfaced road Road covered with packed snow Icy road

𝝁, Dry Surface

𝝁, Wet Surface

22.4 19.2 16.0 6.4 3.2

12.8 9.6 6.4 6.4 3.2

1. Researchers conducted thousands of tests to determine the quadratic relationship used to predict the distance (in feet), D, needed to stop a car based on the speed (in mph), or velocity, 𝑉, of the car and the road conditions (𝜇):

𝑽𝟐 𝑫(𝑽) = 𝟐𝝁

Note: This equation does not take into consideration the road incline.

a. Write the equation for the stopping distance on a wet asphalt road.

b. Use the equation to complete the table (round to the nearest tenth). c. Plot the points and sketch the graph. Car Speed (Velocity), V (mph)

Stopping Distance, D (feet)

0 10 20 30 40 50 60 70 80 90 100

123

2. If we solve the relationship for the velocity (𝑉), we can predict the speed a car was being driven, by the distance (𝐷) needed to stop the car and the road conditions (𝜇):

𝑉(𝐷) = √2𝜇𝐷

Note: This equation does not take into consideration the road incline.

a. Show and describe the steps used to solve the equation 𝐷(𝑉) =

𝑉2 2𝜇

for the velocity (𝑉).

b. Write the equation for the velocity (based on the stopping distance) on a wet asphalt road.

c. Use the equation to complete the table (round to the nearest tenth). d. Plot the points and sketch the graph. Stopping Distance (D) Feet 0 10 20 30 40 50 60 70 80 90 100

124

Car Speed Velocity (V) MPH

3. The person that caused an accident claimed they were driving their car at the speed limit, which was 30 miles per hour. The road surface was a wet asphalt road. The skid marks were measured at 34 feet long.

a. Use the equation to determine if the evidence supports the driver’s claims. Explain.

b. Use the table to determine if the evidence supports the driver’s claims. Explain.

c. Use the graph to determine if the evidence supports the driver’s claims. Explain.

125

Sections 8.3 Adding and Subtracting Radicals Expanding the Definition of Like Terms Definition of Like Terms Recall the prior definition of Like Terms that we used when adding and subtracting functions. A polynomial is a mathematical expression that contains the sum of powers of variables multiplied by coefficients. A term of a polynomial is a constant or the product of a constant and power of variable(s). Terms are separated by the operation of addition, and subtraction is defined as addition of the opposite. Like Terms have the same variable(s) raised to the same exponent.

polynomial with four terms Example 1:

4𝑥 2 + 6𝑥𝑦 − 10𝑥𝑦 − 15𝑦 2 like terms

Expanding Definition of Like Terms To add and subtract radicals, we need to expand the definition of like terms. Like terms are expanded to include radical expressions with the same index and radicand inside. These radical expressions may also be multiplied outside by variables with the same exponents. Recall that radicals can be written with fractional exponents, so this “new” definition is similar to before! Example 2:

2𝑥𝑦√𝑥𝑦 + √𝑥 3 𝑦 3 = 2𝑥𝑦√𝑥𝑦 + √𝑥 2 𝑥𝑦 2 𝑦 = 2𝑥𝑦√𝑥𝑦 + 𝑥𝑦√𝑥𝑦

like terms = 3𝑥𝑦√𝑥𝑦

126

Activity 1: Add and Subtract Radicals 1. For each problem, simplify the radicals in each term of the expression, and then add like terms if possible. a.

7√𝑞 − 3√𝑞

b. 2𝑥√5 + 7𝑥√5 − 4𝑥√5

c.

2√4𝑥 2 𝑦 − 3𝑥√𝑦

d. √10𝑎 − 3𝑏√7 + 15𝑏√7 + 2√10𝑎

e.

5√2𝑏 + 4√2𝑏 − 3√2𝑏

3

f. √125 + √27 − √20 + √12

127

2. Determine which method is correct and which is incorrect. Point out the error in the incorrect solution. Student A √𝟐𝟓𝒂𝟑 𝒃𝒄𝟓 + 𝟐𝒂√𝒂𝒃𝒄𝟓 − √𝟒𝒂𝟒 𝒃𝟐 𝒄𝟔

Student B √25𝑎3 𝑏𝑐 5 + 2𝑎√𝑎𝑏𝑐 5 − √4𝑎4 𝑏 2 𝑐 6

= 𝟓𝒂𝟐 𝒄𝟒 √𝒂𝒃𝒄 + 𝟐𝒂𝒄𝟒 √𝒂𝒃𝒄 − 𝟐𝒂𝟐 𝒃𝒄𝟑

= √52 𝑎2 𝑎𝑏𝑐 4 𝑐 + 2𝑎√𝑎𝑏𝑐 4 𝑐 − √22 𝑎4 𝑏2 𝑐 6

= 𝟓𝒂𝒄𝟐 √𝒂𝒃𝒄 + 𝟐𝒂𝒄𝟐 √𝒂𝒃𝒄 − 𝟐𝒂𝟐 𝒃𝒄𝟑

= 5𝑎𝑐 2 √𝑎𝑏𝑐 + 2𝑎𝑐 2 √𝑎𝑏𝑐 − 2𝑎2 𝑏𝑐 3

= 𝟕𝒂𝒄𝟐 √𝒂𝒃𝒄 − 𝟐𝒂𝟐 𝒃𝒄𝟑

= 7𝑎𝑐 2 √𝑎𝑏𝑐 − 2𝑎2 𝑏𝑐 3

128

Section 8.4: Multiplying and Dividing Radicals Activity 1: Multiplying Radicals [Not yet written]

Activity 2: Dividing Radicals [Not yet written]

129

Section 8.5: Solving Radical Equations Introduction to Solving Radical Equations Recall that when we learned about adding and subtracting radicals, we expanded our definition of like terms to include terms with radical expressions that have the same index and radicand. In Example 1 below, we show combining like terms as equivalent expressions. Example 1:

3

3

2 √ 𝑥 + 2 − √𝑥 + 2 + √ 𝑥 + 2 = 3 √𝑥 + 2 − √𝑥 + 2 like terms

We also know that the radical symbol can serve two purposes: a fractional exponent, and a grouping symbol. In Example 2 below, we show this as equivalent expressions. Example 2:

3

1 2

3√𝑥 + 2 − √𝑥 + 2 = 3(𝑥 + 2) − (𝑥 + 2)

1 3

DISCUSS: We can use these ideas to solve Radical Equations. This is similar to, yet different from, how we have solved Quadratic Equations presented in Vertex Form. Compare the two solutions below, step-by-step, and discuss how they are similar and how they are different. Solving a Quadratic Equation presented in Vertex Form

Solving a Radical Equation

𝟒(𝒙 + 𝟑)𝟐 − 𝟏𝟒𝟒 = 𝟎

2√𝑦 − 2 − 8 = 2

𝟒(𝒙 + 𝟑)𝟐 − 𝟏𝟒𝟒 + 𝟏𝟒𝟒 = 𝟎 + 𝟏𝟒𝟒

2√𝑦 − 2 − 8 + 8 = 2 + 8

𝟐

𝟒(𝒙 + 𝟑) = 𝟏𝟒𝟒 𝟒(𝒙+𝟑)𝟐 𝟒

=

𝟏𝟒𝟒 𝟒

(𝒙 + 𝟑)𝟐 = 𝟑𝟔 √(𝒙 + 𝟑)𝟐 = ±√𝟑𝟔 𝒙 + 𝟑 = ±𝟔 𝒙 + 𝟑 − 𝟑 = −𝟑 ± 𝟔 𝒙 = 𝟑 𝒐𝒓 𝒙 = −𝟗

2√𝑦 − 2 = 10 2√𝑦−2 2

=

10 2

√𝑦 − 2 = 5 2

(√𝑦 − 2) = 52 𝑦 − 2 = 25 𝑦 − 2 + 2 = 25 + 2 𝑦 = 27

130

Activity 1: Evaluate the Solution Methods 1. Based on what we have learned so far about radicals, determine which solution is correct. Point out the error(s) and/or misconception in the incorrect solution. Student A

Student B

√𝒗 − 𝟐 = 𝟑

√𝑣 − 2 = 3

𝟐

√𝑣 − 2 + 2 = 3 + 2

(√𝒗 − 𝟐) = 𝟑𝟐 𝒗−𝟐=𝟗 𝒗−𝟐+𝟐=𝟗+𝟐 𝒗 = 𝟏𝟏

√𝑣 = 5 2

(√𝑣) = 52 𝑣 = 25

2. Based on what we have learned about the far about radicals, determine which solution is correct. Point out the error(s) and/or misconception in the incorrect solution. Student A

𝟓√𝒙 − 𝟐 = 𝟏𝟓 𝟓√𝒙−𝟐 𝟓

=

𝟏𝟓 𝟓

√𝒙 − 𝟐 = 𝟑 𝟐

(√𝒙 − 𝟐) = 𝟑𝟐

Student B

5√𝑥 − 2 = 15 √5𝑥 − 10 = 15 2

(√5𝑥 − 10) = 152 5𝑥 − 10 = 225 5𝑥 − 10 + 10 = 225 + 10

𝒙−𝟐=𝟗 𝒙−𝟐+𝟐=𝟗+𝟐 𝒙 = 𝟏𝟏

5𝑥 = 235 5𝑥 5

=

235 5

𝑥 = 47

131

3. Based on what we have learned so far about radicals, determine which solution is correct. Point out the error(s) and/or misconception in the incorrect solution. Student A

Student B

𝟕 + 𝟓√𝒗 − 𝟑 = −𝟑 −𝟕 + 𝟕 + 𝟓√𝒗 − 𝟑 = −𝟑 − 𝟕

7 + 5√𝑣 − 3 = −3 −7 + 7 + 5√𝑣 − 3 = −3 − 7 5√𝑣 − 3 = −10

𝟓√𝒗 − 𝟑 = −𝟏𝟎 𝟓√𝒗−𝟑 𝟓

=

−𝟏𝟎 𝟓

5√𝑣−3 5

=

−10 5

√𝑣 − 3 = −2 2

√𝒗 − 𝟑 = −𝟐

(√𝑣 − 3) = (−2)2

No Solution

𝑣−3 =4 𝑣=7

4. Based on what we have learned about the far about radicals, determine which solution is correct. Point out the error(s) and/or misconception in the incorrect solution. Student A 𝟑

𝟓 + 𝟐 √𝒙 − 𝟏 = −𝟑

Student B 3

5 + 2√𝑥 − 1 = −3 3

𝟑

𝟓 − 𝟓 + 𝟐 √𝒙 − 𝟏 = −𝟑 − 𝟓

5 − 5 + 2 √𝑥 − 1 = −3 − 5 3

2√𝑥 − 1 = −8

𝟑

𝟐 √𝒙 − 𝟏 = −𝟖 𝟑

𝟐 √𝒙−𝟏 𝟐

=

3

2 √𝑥−1 2

−𝟖

√𝒙 − 𝟏 = −𝟒 No Solution

−8 2

3

𝟐

𝟑

=

√𝑥 − 1 = −4

3

3

( √𝑥 − 1) = (−4)3 𝑥 − 1 = −64 𝑥 = −63

132

Activity 2: Solving Radical Equations with One Variable Term 1. To solve a radical equation with one variable term, we begin by isolating the radical. We also need to check for extraneous solutions. a.

5 = √𝑥 + 3

b.

c.

8 + 2√𝑣 − 2 = −2

d.

e.

3 √2𝑥 − 1 = −15

3

f.

−10√𝑦 − 10 = −60

3

√𝑏 + 2 = 4

3

4 − 2 √𝑎 − 1 = 12

133

Activity 3: Applied Problems 1. When an object is dropped from a height of ℎ feet, the time it takes to hit the ground is given by the formula,

𝑡=√

2ℎ 32

, in which 𝑡 is the time in seconds for the object to fall ℎ

feet. Note: This formula is for objects near the Earth’s surface and ignores air resistance. Round your answers to two decimal places. a. Find how long it will take an object to fall 100 feet.

b. From what height would an object be dropped if it takes 10 seconds to hit the ground?

2. The distance a person can see to the horizon depends upon the height of their eyes above ground. This assumes that there are no objects obstructing their view of the horizon, such as buildings, hills, mountains, and islands. The distance to the horizon can be estimated by the formula, 𝑑 = √1.5ℎ , in which 𝑑 is the distance in miles and ℎ is the height in feet of point of vision above sea level. Round your answers to two decimal places. a. Find the distance someone can see if their eyes are 54 feet above sea level.

b. If a person in an airplane can see 150 miles to the horizon, what is the approximate altitude of the airplane?

134

Activity 4: Solving Radical Equations with Two Variable Terms 1. To solve each equation, get a single radical variable term on one side of the equation and solve. Check for extraneous solutions. a.

√3𝑥 − √4𝑥 − 1 = 0

b.

c.

𝑛 + 3 = √4𝑛 + 8

d.

−𝑥 + √6𝑥 + 19 = 2

e.

√𝑥 + √𝑥 + 3 = 3

f.

√2𝑥 + 6 = √𝑥 + 4 + 1

√30 − 𝑥 − 𝑥 = 0

135

Final Exam Review Activity 1: Algebraic Equations Card Sort

Activity 2: Solving Various Algebraic Equations Solve the following equations algebraically without a calculator. Show your work. 3

1

1. 4(3𝑥 + 1) − 2(3𝑥 − 2) = 15

2.

3. √3𝑡 + 4 + 𝑡 = 8

4. √𝑞 2 + 2𝑞 − 2 = 0

136

4

3

1

(𝑟 + 3) − (3𝑟 + 2) = 𝑟 8 2

5.

7.

6𝑝−12 𝑝+3

5

6. (𝑥 − 1)2 − 5 = 4

+ 𝑝−2 = 6

1 𝑦 2 +3𝑦+2

1

2

+ 𝑦−1 = 𝑦 2−1

9. 2𝑧 2 = 2𝑧 + 1

8. 3(2𝑐 − 7)2 = 15

10. 2𝑧 2 = 2𝑧 − 1

137

𝑏

11. 𝐴 = 𝑏−𝑐 , solve for 𝑏

12. 2𝑝 = 3𝑝𝑐 + 𝑓, solve for 𝑝

13. 𝑔𝑓 + 3𝑔 = 4𝑓 − 6𝑔𝑓, solve for 𝑓.

Activity 3: Functions Card Sort

138

Appendix: Review Rational Numbers Integers See Clark & Anfinson, Intermediate Algebra Text, Appendix A: Basic Algebra Review

Fractions See Clark & Anfinson, Intermediate Algebra Text, Appendix A: Basic Algebra Review

Decimals See Clark & Anfinson, Intermediate Algebra Text, Appendix A: Basic Algebra Review

139

Order of Operations When we encounter an expression with many operations, such as 3 + (21 ÷

7+2 ) × 22+3 , 3

it makes a difference which operations to perform first. You may recall learning the mnemonic, “Please Excuse My Dear Aunt Sally,” or PEMDAS, to help you remember the Order of Operations, but be careful! There can be misconceptions associated with PEMDAS that we need to be aware of and discuss briefly in this review. We also will find that the Properties of Real Numbers and Properties of Exponents are also important in the simplification of expressions. P for PARENTHESIS: “Parenthesis” in the mnemonic actually means any type of grouping symbol. There are many different types of grouping symbols, ( ), [ ], { }, | |, √ Also, the fraction bar,

2+3 2+18

, and √

.

𝑛

, is considered a grouping symbol, so the numerator is grouped, the

denominator is grouped, and each needs to be simplified before dividing.

2+3 2+18

5

1

= 20 = 4.

Furthermore, if there is an expression in an exponent, 𝑥 1+3 = 𝑥 4 , that is considered grouped and should be simplified first. Lastly, if there are nested groupings, we work from the innermost grouping first and work outwards. E for EXPONENTS: 3

1

We will learn that Radicals can be written using Exponent form, such as √27 = 273 = 3. In addition to exponents, special functions such as log

, sin

, cos

, etcetera, are calculated in

this step. M for MULTIPLICATION and D for DIVISION (Left to Right): The left to right order does not matter if only multiplication is involved, but it matters for division. A for ADDITION and S for SUBTRACTION (Left to Right): The left to right order does not matter if only addition is involved, but it matters for subtraction.

140

Order of Operations Example: 3 + (21 ÷

7+2 ) × 22+3 3

9 = 3 + (21 ÷ ) × 22+3 3 3 + (21 ÷ 3) × 22+3 = 3 + (7) × 25 = 3 + 7 × 32 = 3 + 224 = 227

141

Exponents Integer Exponents Exponent Expression

𝑎2

base

exponent

Table of Common Integer Exponents Base (B)

𝑩−𝟐

𝑩−𝟏

𝑩𝟎

𝑩𝟏

𝑩𝟐

𝑩𝟑

2

2−2 =

1 4

2−1 =

1 2

20 = 1

21 = 2

22 = 4

23 = 8

3

3−2 =

1 9

3−1 =

1 3

30 = 1

31 = 3

32 = 9

33 = 27

4

4−2 =

1 16

4−1 =

1 4

40 = 1

41 = 4

42 = 16

43 = 64

5

5−2 =

1 25

5−1 =

1 5

50 = 1

51 = 5

52 = 25

53 = 125

6

6−2 =

1 36

6−1 =

1 6

60 = 1

61 = 6

62 = 36

63 = 216

7

7−2 =

1 49

7−1 =

1 7

70 = 1

71 = 7

72 = 49

73 = 343

8

8−2 =

1 64

8−1 =

1 8

80 = 1

81 = 8

82 = 64

83 = 512

9

9−2 =

1 81

9−1 =

1 9

90 = 1

91 = 9

92 = 81

93 = 729

10

10−2 =

1 100

10−1 =

1 10

100 = 1

101 = 10

102 = 100

103 = 1000

𝑎

𝑎−2 =

𝑎−1 =

1 𝑎

𝑎0 = 1

𝑎1 = 𝑎

𝑎2

𝑎3

142

1 𝑎2

Properties of Exponents Property

Example 𝑥2𝑥3

𝑎𝑚 𝑎𝑛 = 𝑎𝑚+𝑛

= (𝑥 ∙ 𝑥)(𝑥 ∙ 𝑥 ∙ 𝑥) = 𝑥5

𝑎𝑚 𝑎𝑛

= 𝑎𝑚−𝑛 for 𝑎 ≠ 0

𝑥3 𝑥2 𝑥∙𝑥∙𝑥 = 𝑥∙𝑥 = 𝑥 3−2 = 𝑥 1 = 𝑥 for 𝑥 ≠ 0 (𝑥 3 )2

(𝑎𝑚 )𝑛 = 𝑎𝑚𝑛

= (𝑥 ∙ 𝑥 ∙ 𝑥)2 = (𝑥 ∙ 𝑥 ∙ 𝑥)(𝑥 ∙ 𝑥 ∙ 𝑥) = 𝑥6 (𝑥𝑦)3

(𝑎𝑏)𝑚 = 𝑎𝑚 𝑏 𝑚

= 𝑥𝑦 ∙ 𝑥𝑦 ∙ 𝑥𝑦 = 𝑥3𝑦3

𝑎 𝑚

𝑎𝑚

(𝑏) = 𝑏𝑚 for 𝑏 ≠ 0

𝑥 2 ( ) 𝑦 𝑥 𝑥 = ∙ 𝑦 𝑦 𝑥2

= 𝑦2 for 𝑦 ≠ 0 𝑥2 𝑥4 1

𝑎−𝑚 = 𝑎𝑚 for 𝑎 ≠ 0

=

𝑥∙𝑥 𝑥∙𝑥∙𝑥∙𝑥

= 𝑥 2−4 = 𝑥 −2 =

1 𝑥2

for 𝑥 ≠ 0

𝑥3 𝑥3

𝑎0 = 1 for 𝑎 ≠ 0

=

𝑥∙𝑥∙𝑥 𝑥∙𝑥∙𝑥

= 𝑥 3−3 = 𝑥 0 = 1 for 𝑥 ≠ 0

143

Radicals: Rational Exponents Radical Expression index

3

√𝑎

radical

radicand

Radicals as Rational Exponents 3

√𝑎 =

1 𝑎3

Table of Common Radicals and Equivalent Rational Exponents 𝟏

𝟐

√𝑩 = √𝑩 = 𝑩𝟐 = 𝑺𝒒𝒖𝒂𝒓𝒆 𝑹𝒐𝒐𝒕 2

𝟏

𝟑

√𝑩 = 𝑩𝟑 = 𝑪𝒖𝒃𝒆 𝑹𝒐𝒐𝒕

1

3

√4 = √4 = 42 = 2 2

1

3

√9 = √9 = 92 = 3 2

1

3

1

3

1

3

1

3

1

3

2

1

1

144

1

√512 = 5123 = 8

3

√81 = √81 = 812 = 10 √100 = √100 = 1002 = 10

1

√343 = 3433 = 7

√64 = √64 = 642 = 8 2

1

√216 = 2163 = 6

√49 = √49 = 492 = 7 2

1

√125 = 1253 = 5

√36 = √36 = 362 = 6 2

1

√64 = 643 = 4

√25 = √25 = 252 = 5 2

1

√27 = 273 = 3

√16 = √16 = 162 = 4 2

1

√8 = 83 = 2

1

√729 = 7293 = 9

3

1

√1000 = 10003 = 10

Properties of Radicals Property

Examples 𝟑 𝟑

1 𝑛

√𝑎 = 𝑎 for 𝑛 ≠ 0

𝑛

if √𝑎 exists. 𝑛

√𝟔𝟒

=

=𝟒

√𝑎𝑛 = |𝑎| for 𝑛 is even

𝑛

if

𝑛

√𝑎 𝑛

exists.

𝟏 (𝟔𝟒)𝟑

√𝑎𝑛 = 𝑎 for 𝑛 is odd

𝑛

𝑛

𝑛

√𝑎𝑏 = √𝑎 ∙ √𝑏 𝑛

if √𝑎 and √𝑏 exist. 𝑛

Similar to Properties of Exponents.

𝑎 √𝑏

𝑛

𝑛

=

√𝑎 𝑛 √𝑏

for 𝑏 ≠ 0

𝑛

Similar to Properties of Exponents.

= |𝟑|

if √𝑎 and 𝑚

𝟐

√𝒙𝟐

= 𝒙, 𝒇𝒐𝒓 𝒙 ≥ 𝟎

√−𝟔𝟒

𝟑

= −𝟒

=𝒙

√𝟕𝟐

√𝒙𝟒 𝒚𝟓

√𝒙𝟑

= √𝟐 ∙ 𝟑𝟔 = √𝟐 ∙ √𝟑𝟔

= √𝒙𝟒 ∙ √𝒚𝟓

= √𝟐 ∙ 𝟔

= √𝒙𝟒 ∙ √𝒚𝟒 ∙ 𝒚

= 𝟔√𝟐

= 𝒙𝟐 ∙ 𝒚𝟐 √𝒚

𝟑

√

𝟔 𝟖

𝟑

√

=

√𝟔

𝟑

√𝟖

𝒙𝟕 𝒚𝟔 𝟑

𝟑

√𝟔 = 𝟐 √ √𝑎 =

=𝒙

𝟑

𝟑

𝑛 𝑚

= 𝒙𝟏

√𝟖𝟏

𝑛

if √𝑎 and √𝑏 exist.

𝟏

= (𝒙𝟑 )𝟑

𝟒

=𝟑 𝑛

√𝒙𝟑

=

√𝒙𝟔 ∙ 𝒙 𝟑

√𝒚𝟔

𝒙𝟐 ∙ 𝟑√𝒙 = 𝒚𝟐

𝑛𝑚

√𝑎

√𝑎 exist.

𝟑 𝟐

√ √𝟔𝟒

𝑛𝑚

Similar to Properties of Exponents.

𝟑 𝟐

√ √𝒙

𝟔

= √𝟔𝟒 =𝟐

= 𝟔√𝒙

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