chap04 bk

Full text

(1)Notes. Linear Functions. Introduction In this unit, students will explore data to determine whether a linear relationship exists. They will learn to represent a linear relationship as points on a coordinate plane, and as an equation representing a line. Students will analyze how the equation and the graph of a line are related. They will extend their knowledge of linear graphing to inequalities and systems of linear equations.. Many real-world situations such as Olympic race times can be represented using functions. In this unit, you will learn about linear functions and equations.. Assessment Options Unit 2 Test Pages 453–454 of the Chapter 7 Resource Masters may be used as a test or review for Unit 2. This assessment contains both multiple-choice and short answer items.. Chapter 4 Graphing Relations and Functions. Chapter 5 Analyzing Linear Equations. Chapter 6 Solving Linear Inequalities. TestCheck and Worksheet Builder. Chapter 7 Solving Systems of Linear Equations and Inequalities. This CD-ROM can be used to create additional unit tests and review worksheets.. 188 Unit 2. 188. Unit 2 Linear Functions. Linear Functions.

(2) Teaching Suggestions Have students study the USA TODAY Snapshot. • Ask them if it is possible to represent this data on a coordinate plane. If so, how would they graph the data? not possible on a coordinate plane • Which sport seems to foster the most top medalists? swimming • Point out to students that in their WebQuest they will be analyzing other statistics dealing with Olympic swimming.. The Spirit of the Games The first Olympic Games featured only one event— a foot race. The 2004 Games will include thousands of competitors in about 300 events. In this project, you will explore how linear functions can be illustrated by the Olympics.. USA TODAY Snapshots® America’s top medalists Americans with most Summer Games medals: Mark Spitz, Matt Biondi (swimming), Carl Osburn (shooting) 11 Ray Ewry (track and field). Log on to www.algebra1.com/webquest. Begin your WebQuest by reading the Task.. 10 Carl Lewis, Martin Sheridan (track and field). Then continue working on your WebQuest as you study Unit 2.. 9. Lesson Page. 4-6 230. 5-7 304. 6-6 357. 7-1 373. Additional USA TODAY Snapshots appearing in Unit 2: Chapter 4 Farmers growing bumper crop (p. 210) Chapter 5 Dining out (p. 258) Waiting on weddings (p. 284) Chapter 6 Girls gear up for high school sports (p. 318) Environment first (p. 350) Chapter 7 India's exploding population (p. 386). Shirley Babashoff, Charles Daniels (swimming) 8 Source: U.S. Olympic Committee By Scott Boeck and Julie Stacey, USA TODAY. Unit 2. Linear Functions 189. Internet Project A WebQuest is an online project in which students do research on the Internet, gather data, and make presentations using word processing, graphing, page-making, or presentation software. In each chapter, students advance to the next step in their WebQuest. At the end of Chapter 7, the project culminates with a presentation of their findings. Teaching notes and sample answers are available in the WebQuest and Project Resources. Unit 2 Linear Functions 189.

(3) Graphing Relations and Functions Chapter Overview and Pacing PACING (days) Regular Block. LESSON OBJECTIVES. Basic/ Average. Advanced. Basic/ Average. Advanced. The Coordinate Plane (pp. 192–196) • Locate points on the coordinate plane. • Graph points on a coordinate plane.. 1. 1. 0.5. 0.5. Transformations on the Coordinate Plane (pp. 197–203) • Transform figures by using reflections, translations, dilations, and rotations. • Transform figures on a coordinate plane by using reflections, translations, dilations, and rotations.. 2. 2. 1. 1. 2 (with 4-3 Preview). 2. 1. 1. 1. 1. 0.5. 0.5. 2 (with 4-5 Follow-Up). 2. 1. 1. 1. 1. 0.5. 0.5. 2 (with 4-7 Preview). 2. 1. 1. Writing Equations from Patterns (pp. 240–245) • Look for a pattern. • Write an equation given some of the solutions.. 1. 1. 0.5. 0.5. Study Guide and Practice Test (pp. 246–251) Standardized Test Practice (pp. 252–253). 1. 1. 0.5. 0.5. Chapter Assessment. 1. 1. 0.5. 0.5. 14. 14. 7. 7. Relations (pp. 204–211) Preview: Use a graphing calculator to graph relations. • Represent relations as sets of ordered pairs, tables, mappings, and graphs. • Find the inverse of a relation. Equations as Relations (pp. 212–217) • Use an equation to determine the range for a given domain. • Graph the solution set for a given domain. Graphing Linear Equations (pp. 218–225) • Determine whether an equation is linear. • Graph linear equations. Follow-Up: Use a graphing calculator to graph linear equations. Functions (pp. 226–231) • Determine whether a relation is a function. • Find function values. Arithmetic Sequences (pp. 232–238) Preview: Use a spreadsheet to generate number sequences and patterns. • Recognize arithmetic sequences. • Extend and write formulas for arithmetic sequences.. TOTAL. Pacing suggestions for the entire year can be found on pages T20–T21.. 190A Chapter 4 Graphing Relations and Functions.

(4) Timesaving Tools ™. All-In-One Planner and Resource Center. Chapter Resource Manager. 213–214. 215–216. 217. 218. 219–220. 221–222. 223. 224. 225–226. 227–228. 229. 230. 231–232. 233–234. 235. 236. 237–238. 239–240. 241. 242. 243–244. 245–246. 247. 248. 249–250. 251–252. 253. 254. 255–256. 257–258. 259. 260. Ap plic atio ns* Par Stu ent dy a Gu nd St ide u Wo dent rkb 5-M ook Tra inute nsp Che are nci ck es Int e Cha racti lkb ve oar d Alg ePA Plu SS: T s (l ess utoria ons l ). Ass ess me nt Pre req u Wo isite rkb Ski ook lls. Enr ich me nt. S and tudy Int Guid erv e ent ion (Sk Pra c ills and tice. Ave rag e). Rea di Ma ng to the ma Learn tics. CHAPTER 4 RESOURCE MASTERS. See pages T12–T13.. 29. 4-1. 4-1. GCS 29. 30. 4-2. 4-2. SC 7, SM 45–50. 31. 4-3. 4-3. 32. 4-4. 4-4. GCS 30. 33. 4-5. 4-5. 9. SC 8, SM 139–144. 34. 4-6. 4-6. 10. 35. 4-7. 4-7. 276. 36. 4-8. 4-8. 261–274, 278–280. 37. 275. 275, 277. 276. 5–8. Materials. colored pencils, grid paper, graphing calculator. (Follow-Up: graphing calculator. scissors, string. *Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual. ELL Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish. Chapter 4 Graphing Relations and Functions 190B.

(5) Mathematical Connections and Background Continuity of Instruction Prior Knowledge Students located and identified points on the coordinate plane in previous courses. In Chapter 1, students interpreted and drew graphs of functions from tables. They solved equations and formulas for given variables in Chapter 3.. The Coordinate Plane A coordinate plane contains two perpendicular number lines that intersect at their zero points, called the origin. These number lines, called axes, separate the plane into four quadrants, numbered I, II, III, and IV. Quadrant I is the top, right-hand quadrant, and the rest follow in a counterclockwise direction. The location of points on a plane is described using the numbers on the axes that are vertically and horizontally aligned with the point. The numbers make an ordered pair written as (x, y). An ordered pair is named such because the order in which the coordinates are written is important. The corresponding number from the horizontal axis, the x-coordinate, is written first, followed by the y-coordinate. When graphing an ordered pair, start at (0, 0), the origin. The x-coordinate indicates how many units to move right (positive) or left (negative). The y-coordinate indicates the number of units to move up (positive) or down (negative).. This Chapter This chapter connects algebraic equations to their geometric models. Students identify coordinates, locate points, and perform transformations of figures on a coordinate plane. They also identify relations, functions, the domain, the range, and the inverse of a relation. Students explore graphing linear relations using intercepts and tables. They learn to recognize, extend, and write arithmetic sequences, and then write equations from patterns.. Future Connections The concept of functions is used throughout all mathematics, from algebra to beyond calculus. Functions are used to model many real-world situations that relate two variables. Looking for patterns helps find a trend in data. Many analysts use this concept to determine trends in various fields, such as finance, business, and sociology.. 190C. Chapter 4 Graphing Relations and Functions. Transformations on the Coordinate Plane Transformations of figures include reflections, translations, dilations, and rotations. The preimage is the original figure and the image is the transformed figure. A reflection is a flip over a line. A translation is the sliding of a figure in any direction. In a dilation, a figure is enlarged or reduced. When a figure is turned about a point, it is a called a rotation. There are rules to help find the coordinates of an image when you know the coordinates of the preimage. To find the coordinates of a reflection, either the x- or y-coordinate is multiplied by 1. To translate a figure, add the amount of a translation to the coordinates to find the image. Each coordinate is multiplied by a scale factor in a dilation. In rotations, you sometimes switch the coordinates and multiply one or both of them by 1.. Relations Students have learned in the past that a relation is a set of ordered pairs. Relations can also be presented as a table, a graph, or a mapping. Mappings demonstrate how each element of the domain is paired with elements of the range using arrows. The inverse of a relation is found by switching the coordinates in each ordered pair. In other words, the domain becomes the range and the range becomes the domain..

(6) Equations as Relations An equation written in two variables is a relation. Ordered pairs that make a statement true when substituted into the equation are solutions of the equation. The independent variables are the x-values of each ordered pair in the solution set. They represent the domain. The dependent variables are the y-values of each ordered pair in the solution set. They represent the range. When equations are not written so that a variable is isolated on one side of the equal sign, solve for one variable. This variable is the dependent variable. Its value depends on the value of the independent variable substituted into the equation.. Graphing Linear Equations The standard form of a linear equation is Ax By C. If the Properties of Equality can be applied to an equation to rewrite it in standard form, then the equation is linear. To graph a linear equation, first find coordinate pairs that make the statement true. Then plot the points and draw a line through the points. Another graphing method is to find the x- and y-intercepts by alternately replacing x and y with 0. Graph these points, then draw the line that contains these two points. All of the ordered pairs that lie on the line are solutions of the equation.. Arithmetic Sequences A sequence is a set of numbers in a specific order. If the terms of a sequence increase or decrease at a constant rate, it is called it is called an arithmetic sequence. The constant rate or value is called the common difference. The formula for finding a specific term in an arithmetic sequence is an a1 (n 1)d. This means to find a specific term, find the sum of the first term and the product of the common difference times one less than the number of the specific term.. Writing Equations from Patterns Inductive reasoning is used when the problemsolving strategy of looking for a pattern is applied. There can be patterns in sequences of figures or of numbers. To write an equation for a list of data with two variables, use a ratio to compare the common difference of the range variables to the common difference of the domain variables. Next, write the equation as: dependent variable constant ratio(independent variable). Functions A function is a relation in which each element of the range is paired with exactly one element of the domain. This means that an x-coordinate cannot be repeated. To check if a graph is a function, make sure the graph does not touch any vertical line more than once. This is called the vertical line test. Equations can be written using function notation. Solve the equation for the dependent variable, then replace that variable with f(independent variable), such as f(x). The form independent variable, is used on some standardized tests.. www.algebra1.com/key_concepts Additional mathematical information and teaching notes are available in Glencoe’s Algebra 1 Key Concepts: Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows. • Integration: Geometry/Midpoint of a Line Segment (Lesson 14) • Linear Relations and Functions (Lesson 5) • Graphing Linear Equations (Lesson 6) • Writing Linear Equations in Slope-Intercept Form (Lesson 10) • Graphing Technology: Parent and Family Graphs (Lesson 11) • Graphing Linear Equations (Lesson 12) Chapter 4 Graphing Relations and Functions 190D.

(7) and Assessment. ASSESSMENT. INTERVENTION. Type. Student Edition. Teacher Resources. Ongoing. Prerequisite Skills, pp. 191, 196, 203, 211, 217, 223, 231, 238 Practice Quiz 1, p. 211 Practice Quiz 2, p. 231. 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 5–8 Quizzes, CRM pp. 275–276 Mid-Chapter Test, CRM p. 277 Study Guide and Intervention, CRM pp. 213–214, 219–220, 225–226, 231–232, 237–238, 243–244, 249–250, 255–256. Mixed Review. pp. 196, 203, 211, 217, 223, 231, 238, 245. Cumulative Review, CRM p. 278. Error Analysis. Find the Error, pp. 214, 236. Find the Error, TWE pp. 215, 236 Tips for New Teachers, TWE p. 220. Standardized Test Practice. pp. 196, 203, 210, 216, 223, 227, 228, 231, 238, 245, 251, 252–253. TWE pp. 252–253 Standardized Test Practice, CRM pp. 279–280. Open-Ended Assessment. Writing in Math, pp. 196, 203, 210, 216, 222, 231, 238, 245 Open Ended, pp. 194, 200, 208, 214, 221, 228, 236, 243 Standardized Test, p. 253. Modeling: TWE pp. 196, 203, 238, 245 Speaking: TWE pp. 217, 231 Writing: TWE pp. 211, 223 Open-Ended Assessment, CRM p. 273. Chapter Assessment. Study Guide, pp. 246–250 Practice Test, p. 251. Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 261–266 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 267–272 Vocabulary Test/Review, CRM p. 274. Technology/Internet AlgePASS: Tutorial Plus www.algebra1.com/self_check_quiz www.algebra1.com/extra_examples. Standardized Test Practice CD-ROM www.algebra1.com/ standardized_test. TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes www.algebra1.com/ vocabulary_review www.algebra1.com/chapter_test. Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters. Additional Intervention Resources The Princeton Review’s Cracking the SAT & PSAT The Princeton Review’s Cracking the ACT ALEKS. TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment flexibility: • Worksheet Builder to make worksheet and tests • Student Module to take tests on screen (optional) • Management System to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.. 190E. Chapter 4 Graphing Relations and Functions.

(8) Reading and Writing in Mathematics Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 1 Lesson 4-5 4-6. AlgePASS Lesson 9 Graphing Linear Equations on the Coordinate Plane 10 Relations, Functions, Domain, and Range. ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at www.k12aleks.com.. Intervention at Home Parent and Student Study Guide Parents and students may work together to reinforce the concepts and skills of this chapter. (Workbook, pp. 29–37 or log on to www.algebra1.com/parent_student ). Log on for student study help. • For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. www.algebra1.com/extra_examples www.algebra1.com/self_check_quiz. • For chapter review, there is vocabulary review, test practice, and standardized test practice. www.algebra1.com/vocabulary_review www.algebra1.com/chapter_test www.algebra1.com/standardized_test. For more information on Intervention and Assessment, see pp. T8–T11.. Glencoe Algebra 1 provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 191 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 194, 200, 208, 214, 221, 228, 236, 243) • Reading Mathematics, p. 239 • Writing in Math questions in every lesson, pp. 196, 203, 210, 216, 222, 231, 238, 245 • Reading Study Tip, pp. 192, 198, 199, 227, 233, 234 • WebQuest, p. 230 Teacher Wraparound Edition • Foldables Study Organizer, pp. 191, 246 • Study Notebook suggestions, pp. 194, 200, 208, 215, 221, 229, 232, 236, 239, 243 • Modeling activities, pp. 196, 203, 238, 245 • Speaking activities, pp. 217, 231 • Writing activities, pp. 211, 223 • Differentiated Instruction, (Verbal/Linguistic), pp. 196, 213 • ELL Resources, pp. 190, 195, 196, 201, 209, 213, 216, 222, 230, 237, 239, 244, 246 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 4 Resource Masters, pp. vii-viii) • Reading to Learn Mathematics master for each lesson (Chapter 4 Resource Masters, pp. 217, 223, 229, 235, 241, 247, 253, 259) • Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. • Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. • Reading and Writing in the Mathematics Classroom • WebQuest and Project Resources • Hot Words/Hot Topics Sections 1.5, 2.3, 6.1–6.4, 6.6, 6.7, 7.3, 9.4 For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 4 Graphing Relations and Functions 190F.

(9) Notes. Graphing Relations and Functions. Have students read over the list of objectives and make a list of any words with which they are not familiar.. • Lessons 4-1, 4-4, and 4-5 Graph ordered pairs, relations, and equations. • Lesson 4-2 Transform figures on a coordinate plane. • Lesson 4-3 Find the inverse of a relation. • Lesson 4-6 Determine whether a relation is a function. • Lessons 4-7 and 4-8 Look for patterns and write formulas for sequences.. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson.. Key Vocabulary • • • • •. coordinate plane (p. 192) transformation (p. 197) inverse (p. 206) function (p. 226) arithmetic sequence (p. 233). The concept of a function is used throughout higher mathematics, from algebra to calculus. A function is a rule or a formula. You can use a function to describe real-world situations like converting between currencies. For example, if you are in Mexico, you can calculate that an item that costs 100 pesos is equivalent to about 11 U.S. dollars. You will learn how to convert different currencies in Lesson 4-4.. Lesson 4-1 4-2 4-3 Preview 4-3 4-4 4-5 4-5 Follow-Up 4-6 4-7 Preview 4-7 4-8. NCTM Standards. Local Objectives. 2, 3, 6, 8, 9, 10 2, 3, 6, 8, 9, 10 3, 8 2, 3, 6, 8, 9, 10 2, 3, 6, 8, 9, 10 2, 3, 6, 8, 9, 10 2, 3, 10 2, 3, 6, 8, 9, 10 1, 2, 6 1, 2, 6, 8, 9, 10 1, 2, 3, 6, 8, 9, 10. Key to NCTM Standards: 1=Number & Operations, 2=Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 10=Representation 190. 190 Chapter 4 Graphing Relations and Functions. Vocabulary Builder. ELL. The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 4 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 4 test.. Chapter 4 Graphing Relations and Functions.

(10) Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 4. For Lesson 4-1. This section provides a review of the basic concepts needed before beginning Chapter 4. Page references are included for additional student help.. Graph Real Numbers. Graph each set of numbers. (For review, see Lesson 2-1.) 1–4. See margin. 1. {1, 3, 5, 7}. 2. 1 1 4. , 1, 1, 2. 3. {8, 5, 2, 1}. 2. {3, 0, 1, 4}. For Lesson 4-2. 2. . Additional review is provided in the Prerequisite Skills Workbook, pp. 5–8.. Distributive Property. Rewrite each expression using the Distributive Property. (For review, see Lesson 1-5.) 5. 3(7 t) 21 3t. 6. 4(w 2). 4w 8. 2. 15b 10. For Lessons 4-4 and 4-5. Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson.. 1 8. (2z 4) z 2. 7. 5(3b 2). Solve Equations for a Specific Variable. Solve each equation for y. (For review, see Lesson 3-8.) 9. 2x y 1 y 1 2x 12. 2x 3y 9 2 y x 3 3 For Lesson 4-6. 10. x 8 y y 8 x. 11. 6x 3y 12 y 2x 4. 1 13. 9 y 4x 2. 14. x 2. y5 3. y 3x 1. y 18 8x. For Lesson. Prerequisite Skill. 4-2. Using the Distributive Property (p. 196) Writing Ordered Pairs (p. 203) Finding Solution Sets (p. 211) Solving Equations (p. 217) Evaluating Expressions (p. 223) Subtracting Integers (p. 231) Writing Ordered Pairs (p. 238). Evaluate Expressions. Evaluate each expression if a 1, b 4, and c 3. (For review, see Lesson 2-3.) 15. a b c 6. 16. 2c b 10. 18. 3a 6b 2c 21. 19. 8a b 3c 3. 4-3 4-4 4-5 4-6 4-7 4-8. 17. c 3a 0. 1 2. 2 3. 20. 6a 8b c 24. Make this Foldable to help you organize your notes about graphing relations and functions. Begin with four sheets of grid paper. Fold. Cut and Staple. Fold each sheet of grid paper in half from top to bottom.. Answers. Cut along fold. Staple the eight halfsheets together to form a booklet.. Cut Tabs into Margin The top tab is 4 lines wide, the next tab is 8 lines wide, and so on.. 1. 0. 4–1. 4–2. 2. 3. 4. 5. 6. 5 4 3 2 1. 0. 1. 7. 8. 9. 2.. Label The Coordinate Plane. 1. 2. 3. 4. 8 7 6 5 4 3 2 1. 0. 1. 3. Label each of the tabs with a lesson number.. 4. 2 11 1 1 0 2. 2. 1 2. 1 11 2 21 2. 2. As you read and study the chapter, use each page to write notes and to graph examples.. Reading and Writing. Chapter 4. Graphing Relations and Functions. 191. TM. For more information about Foldables, see Teaching Mathematics with Foldables.. Summarizing Use this Foldable for student writing about graphing relations and functions. After students make their Foldable, have them label a tab for each lesson in this chapter. Students use their Foldable to take notes, record concepts, and define terms. At the end of each lesson, ask students to write a summary of the lesson in their own words. Summaries are useful for condensing data. Ask students what summaries and algebraic equations have in common. Chapter 4 Graphing Relations and Functions 191.

(11) Lesson Notes. 1 Focus 5-Minute Check Transparency 4-1 Use as a quiz or a review of Chapter 3. Mathematical Background notes are available for this lesson on p. 190C.. Building on Prior Knowledge. The Coordinate Plane • Locate points on the coordinate plane. • Graph points on a coordinate plane.. Vocabulary • • • • • • • • •. axes origin coordinate plane y-axis x-axis x-coordinate y-coordinate quadrant graph. Underwater archaeologists use a grid system to map excavation sites of sunken ships. The grid is used as a point of reference on the ocean floor. The coordinate system is also used to record the location of objects they find. Knowing the position of each object helps archaeologists reconstruct how the ship sank and where to find other artifacts.. IDENTIFY POINTS In mathematics, points are located in reference to two perpendicular number lines called axes.. In Chapter 2, students learned to graph on a number line. In this lesson, they should recognize that graphing on a coordinate plane is comparable to graphing on two perpendicular number lines. do archaeologists use coordinate systems? Ask students: • When archaeologists use a grid system to excavate an archaeological site, what does the grid simulate? a map of the site • What are some reasons underwater archaeologists use a coordinate system at an archaeological site? as a point of reference and to record the location of objects they find • Geography How do you think the grid on a map is similar to the grid system that archaeologists use? The grid on a map is used to locate places on the map, just as the archaeologists’ grid system is used to locate where objects were found.. do archaeologists use coordinate systems?. Study Tip Reading Math The x-coordinate is called the abscissa. The y-coordinate is called the ordinate.. The axes intersect at their zero points, called the origin.. 4 3 2 1. The plane containing the x- and y-axes is called the coordinate plane.. 4 3 2 1 O 1 2 3 4. y. The vertical number line is called the y-axis. 1 2 3 4x. The horizontal number line is called the x-axis.. Points in the coordinate plane are named by ordered pairs of the form (x, y). The first number, or x-coordinate , corresponds to the numbers on the x-axis. The second number, or y-coordinate , corresponds to the numbers on the y-axis. The origin, labeled O, has coordinates (0, 0).. Example 1 Name an Ordered Pair Write the ordered pair for point G. • Follow along a vertical line through the point to find the x-coordinate on the x-axis. The x-coordinate is 4. • Follow along a horizontal line through the point to find the y-coordinate on the y-axis. The y-coordinate is 3. • So, the ordered pair for point G is (4, 3). This can also be written as G(4, 3).. y. G. 4. O. x. Unless marked otherwise, you can assume that each division on the axes represents 1 unit. 192. Chapter 4 Graphing Relations and Functions. Resource Manager Workbook and Reproducible Masters Chapter 4 Resource Masters • Study Guide and Intervention, pp. 213–214 • Skills Practice, p. 215 • Practice, p. 216 • Reading to Learn Mathematics, p. 217 • Enrichment, p. 218. 3. Parent and Student Study Guide Workbook, p. 29. Transparencies 5-Minute Check Transparency 4-1 Answer Key Transparencies. Technology Interactive Chalkboard.

(12) The x-axis and y-axis separate the coordinate plane into four regions, called quadrants. Notice which quadrants contain positive and negative x-coordinates and which quadrants contain positive and negative y-coordinates. The axes are not located in any of the quadrants.. y Quadrant II (, ). 2 Teach. Quadrant I (, ). O. x. Quadrant III (, ). IDENTIFY POINTS. In-Class Examples. Quadrant IV (, ). Power Point®. 1 Write the ordered pair for point B. (3, 2) y. Example 2 Identify Quadrants Write ordered pairs for points A, B, C, and D. Name the quadrant in which each point is located.. y. A. Use a table to help find the coordinates of each point. x-Coordinate. A. 4. Ordered Pair. y-Coordinate 3. B x. C. Quadrant. (4, 3). I none. B. 2. 0. (2, 0). C. 3. 2. (3, 2). III. D. 1. 4. (1, 4). IV. 2 Write ordered pairs for points A, B, C, and D. Name the quadrant in which each point is located.. D. y. A. B O. GRAPH POINTS To graph an ordered pair means to draw a dot at the point on the coordinate plane that corresponds to the ordered pair. This is sometimes called plotting a point. When graphing an ordered pair, start at the origin. The x-coordinate indicates how many units to move right (positive) or left (negative). The y-coordinate indicates how many units to move up (positive) or down (negative).. Plot each point on a coordinate plane. a. R(4, 1) y. Start at the origin. Move left 4 units since the x-coordinate is 4. Move up 1 unit since the y-coordinate is 1. Draw a dot and label it R.. R. D. A. (2, 2). II. B. (0, 2). none. C. (4, 2). IV. D. (5, 4). III. T. • Start at the origin. • Since the x-coordinate is 0, the point will be located on the y-axis. • Move down 5 units. • Draw a dot and label it S.. GRAPH POINTS. S. In-Class Example. Power Point®. 3 Plot each point on a coordinate plane. a. A(3, 1). c. T(3, 2). b. B(2, 0). • Start at the origin. • Move right 3 units and down 2 units. • Draw a dot and label it T.. c. C(2, 5) y Lesson 4-1 The Coordinate Plane. 193. Teacher to Teacher Larry Romary. C. x. O. b. S(0, 5). www.algebra1.com/extra_examples. x. Point Ordered Pair Quadrant. Example 3 Graph Points. • • • •. x. B O. Point. O. A. B O. x. C. Heritage H.S., Monroeville, IN. "I have students trace a picture of an animal from a coloring book (as it will be well outlined) onto a piece of graph paper. Students select about 40-50 points on the figure that would define the outline of the animal. They write the ordered pairs so that when graphed and connected the animal’s picture would result. I give their ordered pairs to my other classes to plot and discover what the animal is." Lesson 4-1 The Coordinate Plane 193.

(13) In-Class Example. Example 4 Use a Coordinate System. Power Point®. GEOGRAPHY Latitude and longitude lines form a system of coordinates to designate locations on Earth. Latitude lines run east and west and are the first coordinate of the ordered pairs. Longitude lines run north and south and are the second coordinate of the ordered pairs.. 4 GEOGRAPHY Use the map in Example 4 to answer the following questions.. 125°. a. Name the city at about (33°, 80°). Charleston. 120°. 115°. 105°. 100°. 95°. 90°. 85°. 80°. 75°. 70°. 65°. 45°. Portland. b. Estimate the latitude and longitude of Las Vegas. (37, 115). 110°. Helena. Boise Casper. Geography. San Francisco. The prime meridian, 0° longitude, passes through London’s Greenwich Observatory. A metal marker indicates its exact location.. 3 Practice/Apply. Detroit. 40°. Washington, D.C. Columbus Richmond Louisville. Omaha Denver. Las Vegas Oklahoma City Los Angeles. 35°. Nashville. Raleigh Charleston Albuquerque Little Rock Atlanta Montgomery Dallas. 30°. 70°. Source: www.encarta.msn.com. Austin. New Orleans 25°. Miami. TEACHING TIP. Study Notebook. Point out to students that on maps, the vertical measurement is named first, followed by the horizontal measurement. In the traditional x, y coordinate plane, the horizontal distance will always be named first.. Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 4. • draw a coordinate plane labeling the parts with the vocabulary terms from this lesson. • include any other item(s) that they find helpful in mastering the skills in this lesson.. a. Name the city at (40°, 105°). Locate the latitude line at 40°. Follow the line until it intersects with the longitude line at 105°. The city is Denver. b. Estimate the latitude and longitude of Washington, D.C. Locate Washington, D.C., on the map. It is close to 40° latitude and 75° longitude. There are 5° between each line, so a good estimate is 39° for the latitude and 77° for the longitude.. Concept Check. 3. OPEN ENDED Give the coordinates of a point for each quadrant in the coordinate plane. Sample answer: I(3, 3), II(3, 3), III(3, 3), IV(3, 3).. Guided Practice GUIDED PRACTICE KEY. About the Exercises…. Exercises. Examples. 4–7 8–11 12. 1, 2 3 4. Organization by Objective • Identify Points: 13–24, 37–43 • Graph Points: 25–30. Basic: 13–21 odd, 25–43 odd, 44–47, 51–71 Average: 13–43 odd, 44–47, 51–71 (optional: 48–50) Advanced: 14–42 even, 44–65 (optional: 66–71). 194. Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located. 4. E (2, 5); III. 5. F (1, 1); II. 6. G (4, 4); I. 7. H (4, 2); III. y J. 8. J(2, 5). F. M L E. 11. M(2, 2). 12. ARCHITECTURE Chun Wei has sketched the southern view of a building. If A is located on a coordinate system at (40, 10), locate the coordinates of the other vertices.. E. 30 ft. B(0, 10), C(0, 20), D(20, 20), E(40, 40). Chapter 4 Graphing Relations and Functions. Answers 1.. Chapter 4 Graphing Relations and Functions. y. II. I x. O. III. IV. 2. To graph (1, 4) move 1 unit left from the origin and 4 units up. This point is in quadrant II. To graph (4, 1) move 4 units right from the origin and 1 unit down. This point is in quadrant IV.. 20 ft. D A. 194. x. O. H. 9. K(1, 4). 10. L(0, 3). G. K. Plot each point on a coordinate plane. 8–11. See right.. Application. Odd/Even Assignments Exercises 13–36 are structured so that students practice the same concepts whether they are assigned odd or even problems.. Assignment Guide. 1. Draw a coordinate plane. Label the origin, x-axis, y-axis, and the quadrants. 2. Explain why (1, 4) does not name the same point as (4, 1). See margin.. 1. See margin.. 40 ft. C 10 ft. B.

(14) ★ indicates increased difficulty. NAME ______________________________________________ DATE. 13–24, 39 25–36 37, 38, 40–43. 1, 2 3 4. Extra Practice See page 828.. W. Z. S. R. 13. N (4, 5); II. 14. P (5, 3); IV. 15. Q (1, 3); III. 16. R (5, 2); I. 17. S (3, 3); II. 18. T (2, 0); none. 19. U (2, 1); IV. 20. V (4, 2); III. 21. W (0, 4); none. 22. Z (3, 3); I. y Quadrant II. O. Example 1 Write an ordered pair for point R above.. V Q. P. The x-coordinate is 0 and the y-coordinate is 4. Thus the ordered pair for R is (0, 4).. Example 2 Write ordered pairs for points P and Q above. Then name the quadrant in which each point is located. The x-coordinate of P is 3 and the y-coordinate is 2. Thus the ordered pair for P is (3, 2). P is in Quadrant III. The x-coordinate of Q is 4 and the y-coordinate is 1. Thus the ordered pair for Q is (4, 1). Q is in Quadrant IV.. ★ 23. Write the ordered pair that describes a point 12 units down from and 7 units to the right of the origin. (7, 12) ★ 24. Write the ordered pair for a point that is 9 units to the left of the origin and lies on the x-axis. (9, 0). Exercises Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located. 1. N (3, 0), none. 2. P (2, 3), III. 3. Q (0, 5), none. 4. R (3, 4), II. 5. S (5, 2), IV. 6. T (5, 0), none. 27. C(4, 2). y. T. 8. V (5, 4), I 10. Z (1, 0), none. 28. D(0, 1). 11. A (3, 3), IV. 12. B (0, 4), none. 26. B(2, 2). 29. E(2, 5). 30. F(3, 4). 31. G(4, 4). 32. H(4, 4). 13. Write the ordered pair that describes a point 4 units down from and 3 units to the right of the origin. (3, 4). 33. I(3, 1). 34. J(1, 3). 35. K(4, 0). 36. L(2, 4). 14. Write the ordered pair that is 8 units to the left of the origin and lies on the x-axis.. (8, 0). p. 215 and Practice, p. 216 The Coordinate Plane (shown). Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located.. Sample answer: Austin and Oklahoma City. C. D. 1. E. Huron St.. y vase. Palmer Field. goblet. 5. 5. E (3, 3); I. 6. F (1, 1); II. L. F O. I. x. A K. C G H. D. 8. H (1, 4); III 10. J (2, 5); I 12. L (4, 2); I. Plot each point on the coordinate plane at the right.. plate. 14. N(3, 2). 16. Q(4, 3). 17. R(0, 5). 19. T(5, 1). 20. V(1, 5). y. R Y. M. 15. P(5, 1). Z T. 18. S(1, 2). W. P. O. x. S. 21. W(2, 0). N Q. 22. X(2, 4). 23. Y(4, 4). V. X. 24. Z(1, 2). x. O. 25. CHESS Letters and numbers are used to show the positions of chess pieces and to describe their moves. For example, in the diagram at the right, a white pawn is located at f5. Name the positions of each of the remaining chess pieces.. white pawns: a7, c6; black pawns: b4, d7; white king: g3; black king: e8. 6 5 4. King Pawn. 3 2 1 a. b. c. d. ARCHAEOLOGY For Exercises 26 and 27, use the grid at. 4. the right that shows the location of arrowheads excavated at a midden—a place where people in the past dumped trash, food remains, and other discarded items.. 3. 26. Write the coordinates of each arrowhead. (1, 1), (3, 2), (1, 4). f. g. h. 2 1. 27. Suppose an archaeologist discovers two other arrowheads located at (1, 2) and (3, 3). Draw an arrowhead at each of these locations on the grid.. NAME ______________________________________________ DATE. e. 0. 0. 1. 2 Meters. 3. 4. ____________ PERIOD _____. 4-1. 42. Which street goes from sector (A, 2) to (D, 2)?. Pre-Activity. How do archaeologists use coordinate systems? Read the introduction to Lesson 4-1 at the top of page 192 in your textbook.. 43. B5, C2, D4, E1. What do the terms grid system, grid, and coordinate system mean to you?. See students’ work.. b. xy 0 II, IV. a. xy 0 I, III. 8 7. Reading to Learn Reading to Learn Mathematics ELL Mathematics, p. 217 The Coordinate Plane. 41. In what sector are most of the science buildings?. 44. CRITICAL THINKING Describe the possible locations, in terms of quadrants or axes, for the graph of (x, y) given each condition.. Reading the Lesson 1. Use the coordinate plane shown at the right.. c. xy 0. c. x-axis, y-axis, origin. www.algebra1.com/self_check_quiz. 4. D (5, 5); IV. 13. M(3, 3). 42. E. Huron St. Shapiro Undergraduate Library. 3. C (5, 2); III. 11. K (5, 2); IV. coins. 43. Name the sectors that have bus stops. Division St.. 4. Natural Resources and Environment. Chemistry. 2. B (5, 5); II. 9. I (3, 0); none. 40. In what sector is the Undergraduate Library? C5 Natural Science. 1. A (0, 1); none. 7. G (2, 3); IV. MAPS For Exercises 40–43, use the map at the left. On many maps, letters and numbers are used to define a region or sector. For example, Palmer Field is located in sector E2. Rogelio is a guide for new students at the University of Michigan. He has selected campus landmarks to show the students. 41. C4. E. y J. B E. 38. Name two cities that have approximately the same longitude.. Catherine St.. ____________ PERIOD _____. Skills Practice, 4-1 Practice (Average). 37. Name two cities that have approximately the same latitude.. plate, (7, 2); goblet, (8, 4); vase, (5, 9). 3. x. N A. B. 25. A(3, 5). 39. ARCHAEOLOGY The diagram at the right shows the positions of artifacts found on the ocean floor. Write the coordinates of the location for each object: coins, plate, goblet, and vase. coins, (3, 5);. 2. O. Z P. NAME ______________________________________________ DATE. B. V. W. S. 9. W (2, 1), II. GEOGRAPHY For Exercises 37 and 38, use the map on page 194.. A. Q. R. U. 7. U (1, 1), I. Plot each point on a coordinate plane. 25–36. See margin.. 37. Sample answer: Louisville and Richmond. Quadrant IV. x. U. O. x. Q P Quadrant III. The axes divide the coordinate plane into Quadrants I, II, III, and IV, as shown. The point where the axes intersect is called the origin. The origin has coordinates (0, 0).. T. Quadrant I. R. Lesson 4-1 The Coordinate Plane. 195. y. a. Label the origin O. b. Label the y-axis y. c. Label the x-axis x.. O. x. 2. Explain why the coordinates of the origin are (0, 0).. Sample answer: The origin is the intersection of two number lines at their common zero point. 3. Use the ordered pair (2, 3).. 25–36.. NAME ______________________________________________ DATE. y. E. A. H. 4-1 Enrichment Enrichment,. G. ____________ PERIOD _____. p. 218. The midpoint of a line segment is the point that lies exactly halfway between the two endpoints of the segment. The coordinates of the midpoint of a line segment whose endpoints are (x1, y1) and (x2, y2). I. . K. x x 2. y y 2. . 1 2 1 2 , . are given by . D. c. Describe the steps you would use to locate the point for (2, 3) on the coordinate plane. Start at the origin. Move left 2 units. Then move up 3 units.. the coordinate plane Find the midpoint of each line segment with the given endpoints.. C. 1. (7, 1) and (3, 1). (2, 1). J F. number, the y-coordinate is the second number. b. Name the x- and y-coordinates. The x-coordinate is 2, the y-coordinate. 4. What does the term quadrant mean? Sample answer: one of the four regions in. x. O. a. Explain how to identify the x- and y-coordinates. The x-coordinate is the first. is 3.. Midpoint. B. L. Lesson 4-1. See Examples. Identify Points In the diagram at the right, points are located in reference to two perpendicular number lines called axes. The horizontal number line is the x-axis, and the vertical number line is the y-axis. The plane containing the x- and y-axes is called the coordinate plane. Points in the coordinate plane are named by ordered pairs of the form (x, y). The first number, or x-coordinate corresponds to a number on the x-axis. The second number, or y-coordinate, corresponds to a number on the y-axis.. y. N. Lesson 4-1. For Exercises. p. 213 (shown) The Coordinate Plane and p. 214. Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located.. Meters. Homework Help. ____________ PERIOD _____. Study Guide andIntervention Intervention, 4-1 Study Guide and. Practice and Apply. 3. (4, 4) and (4, 4). (0, 0). 2. (5, 2) and (9, 8). (7, 5) 4. (3, 6) and (10, 15). (6.5, 10.5). Helping You Remember 5. Explain how the way the axes are labeled on the coordinate plane can help you remember how to plot the point for an ordered pair. Sample answer: The right. side of the horizontal axis is labeled with the letter x. This is the side of the horizontal number line where you find positive numbers. The top end of the vertical number line is labeled with the letter y. This is the part of the vertical number line where you find positive numbers.. Lesson 4-1 The Coordinate Plane 195.

(15) 45. WRITING IN MATH. 4 Assess. How do archaeologists use coordinate systems? Include the following in your answer: • an explanation of how dividing an excavation site into sectors can be helpful in excavating a site, and • a reason why recording the exact location of an artifact is important.. Open-Ended Assessment Modeling Have students draw a map of their dream bedroom on a coordinate plane. The map should include some of the objects and furnishings in the room, but the objects should not be labeled. Students should identify the objects on the map with the ordered pair that makes up the coordinates of each object.. Standardized Test Practice. For Exercises 46 and 47, refer to the figure at the right. 46. ABCD is a rectangle with its center at the origin. If the coordinates of vertex B are (3, 2), what are the coordinates of vertex A? C A C. A C. Extending the Lesson. PREREQUISITE SKILL Students. B D. 6 units 5 units. B (3, 2). A. (3, 2) (3, 2). B. C. 4 units 3 units. The midpoint of a line segment is the point that lies exactly halfway between the two endpoints. The midpoint of a line segment whose endpoints are at (a, b) and ac bd. (c, d) is at , . Find the midpoint of each line segment whose endpoints 2 2 are given. 49. (5, 2) and (9, 8). (7, 5). (2, 1). 50. (4, 4) and (4, 4). (0, 0). Maintain Your Skills Mixed Review. 51. AIRPLANES At 1:30 P.M., an airplane leaves Tucson for Baltimore, a distance of 2240 miles. The plane flies at 280 miles per hour. A second airplane leaves Tucson at 2:15 P.M. and is scheduled to land in Baltimore 15 minutes before the first airplane. At what rate must the second airplane travel to arrive on schedule? (Lesson 3-9) 320 mph Solve each equation or formula for the variable specified. 52. 3x b 2x 5 for x x 5 b 6w b 54. 6w 3h b for h h 3. (Lesson 3-8). 53. 10c 2(2d 3c) for d d c 3a 3(a t) 55. 2t for t t 4 11. Find each square root. Round to the nearest hundredth if necessary. 56. 81 9. Getting Ready for the Next Lesson. 196. 57. 63 7.94. 58. 180 13.42. (Lesson 2-1). 60. 52 18 7 63. 61. 81 47 17 51. 62. 42 60 74 28. 63. 36 15 21 30. 64. 10 16 27 21. 65. 38 65 21 48. PREREQUISITE SKILL Rewrite each expression using the Distributive Property. Then simplify. (To review the Distributive Property, see Lesson 1-5.) 66. 4(x y) 4x 4y. 67. 1(x 3) x 3 2 1 69. 3(2x 5) 6x 15 70. (2x 6y) x 2y 3 3. 68. 3(1 6y) 3 18y 5 1 1 71. (5x 2y) x y 4 4 2. Chapter 4 Graphing Relations and Functions. ELL. Verbal/Linguistic If students have difficulty identifying or plotting points, have them work in pairs. Have students first describe the locations of points on the coordinate plane. Only after both students have agreed on the location of the point should the location be graphed or recorded.. Chapter 4 Graphing Relations and Functions. (Lesson 2-7). 59. 256 16. Evaluate each expression.. Differentiated Instruction. 196. x. O. D. D. 48. (7, 1) and (3, 1). Answer 45. Archaeologists used coordinate systems as a mapping guide and as a system to record locations of artifacts. Answers should include the following. • The grid gives archaeologists a point of reference so they can identify and explain to others the location of artifacts in a site they are excavating. You can divide the space so more people can work at the same time in different areas. • Knowing the exact location of artifacts helps archaeologists reconstruct historical events.. (3, 2) (3, 2). y. 47. What is the length of AD ? B. Getting Ready for Lesson 4-2 will learn about transformations on the coordinate plane in Lesson 4-2. Dilations are one type of transformation. In dilations, the scale factor is distributed by multiplication over each coordinate in an ordered pair. Use Exercises 66–71 to determine your students’ familiarity with the Distributive Property.. Answer the question that was posed at the beginning of the lesson. See margin..

(16) Transformations on the Coordinate Plane. Lesson Notes. • Transform figures by using reflections, translations, dilations, and rotations.. 1 Focus. • Transform figures on a coordinate plane by using reflections, translations, dilations, and rotations.. Vocabulary • • • • • • •. transformation preimage image reflection translation dilation rotation. are transformations used in computer graphics?. 5-Minute Check Transparency 4-2 Use as a quiz or review of Lesson 4-1.. Computer programs can create movement that mimic real-life situations. A new CD-ROM-based flight simulator replicates an actual flight experience so closely that the U.S. Navy is using it for all of their student aviators. The movements of the on-screen graphics are accomplished by using mathematical transformations.. Mathematical Background notes are available for this lesson on p. 190C.. Building on Prior Knowledge. TRANSFORM FIGURES Transformations are movements of geometric. Students may have experienced these geometric transformations under the names flip, slide, turn, and enlarge or shrink. Check students’ familiarity with these terms before presenting the more formal terminology.. figures. The preimage is the position of the figure before the transformation, and the image is the position of the figure after the transformation. reflection a figure is flipped over a line. translation a figure is slid in any direction. dilation a figure is enlarged or reduced. are transformations used in computer graphics? Ask students: • How can actual movement, such as airplane flight, be simulated on a computer screen? Sample answer: Objects on the screen can be made to move to simulate the way objects on the ground appear to move as the plane flies by. • Aviation Why might computer flight simulators be a good way to train beginning student pilots? Sample answer: Student pilots can use the simulator to become familiar with the operation and controls of an airplane without actually flying one.. rotation a figure is turned around a point. Example 1 Identify Transformations Identify each transformation as a reflection, translation, dilation, or rotation. a.. b.. c.. d.. a. The figure has been turned around a point. This is a rotation. b. The figure has been flipped over a line. This is a reflection. c. The figure has been increased in size. This is a dilation. d. The figure has been shifted horizontally to the right. This is a translation. Lesson 4-2 Transformations on the Coordinate Plane. 197. Resource Manager Workbook and Reproducible Masters Chapter 4 Resource Masters • Study Guide and Intervention, pp. 219–220 • Skills Practice, p. 221 • Practice, p. 222 • Reading to Learn Mathematics, p. 223 • Enrichment, p. 224 • Assessment, p. 275. Graphing Calculator and Spreadsheet Masters, p. 29 Parent and Student Study Guide Workbook, p. 30. Transparencies 5-Minute Check Transparency 4-2 Answer Key Transparencies. Technology Interactive Chalkboard. Lesson x-x Lesson Title 197.

(17) TRANSFORM FIGURES ON THE COORDINATE PLANE You can perform transformations on a coordinate plane by changing the coordinates of the points on a figure. The points on the translated figure are indicated by the prime symbol to distinguish them from the original points.. 2 Teach TRANSFORM FIGURES. In-Class Example. Transformations on the Coordinate Plane. Power Point®. Name. 1 Identify each transformation. Reflection. as a reflection, translation, dilation, or rotation. a.. dilation. b.. translation. c.. rotation. d.. reflection. Translation. Dilation. Words. Symbols. To reflect a point over the x-axis, multiply the y-coordinate by 1.. reflection over x-axis: (x, y) → (x, y). To reflect a point over the y-axis, multiply the x-coordinate by 1.. reflection over y-axis: (x, y) → (x, y). To translate a point by an ordered pair (a, b), add a to the x-coordinate and b to the y-coordinate.. (x, y) → (x a, y b). To dilate a figure by a scale factor k, multiply both coordinates by k.. (x, y) → (kx, ky). Model y. (x, y ). (x, y ). x. O (x, y ). y (x, y ). (x a, y b). x. O. y (kx , ky ) (x, y ). If k 1, the figure is enlarged.. x. O. If 0 k 1, the figure is reduced.. Rotation. TRANSFORM FIGURES ON THE COORDINATE PLANE. In-Class Example. Power Point®. 2 A parallelogram has vertices W(1, 4), X(4, 4), Y(4, 1), and Z(3, 1). a. Parallelogram WXYZ is reflected over the y-axis. Find the coordinates of the vertices of the image. W(1, 4), X(4, 4), Y(4, 1), Z(3, 1) b. Graph parallelogram WXYZ and its image WXYZ.. Study Tip. Y. W. W. Z O. Z. 90° rotation: (x, y) → (y, x). To rotate a figure 180° about the origin, multiply both coordinates of each point by 1.. 180° rotation: (x, y) → (x, y). y. The vertices of a polygon are the endpoints of the angles.. O. x. (x, y ). A parallelogram has vertices A(4, 3), B(1, 3), C(0, 1), and D(5, 1). a. Parallelogram ABCD is reflected over the x-axis. Find the coordinates of the vertices of the image. To reflect the figure over the x-axis, multiply each y-coordinate by 1.. X. (x, y) → (x, y) A(4, 3) → A(4, 3) B(1, 3) → B(1, 3). Y. The coordinates of the vertices of the image are A(4, 3), B(1, 3), C(0, 1), and D(5, 1).. x 198. (x, y) → (x, y) C(0, 1) → C(0, 1) D(5, 1) → D(5, 1). Chapter 4 Graphing Relations and Functions. Unlocking Misconceptions Rotations Rotations are traditionally expressed in a counterclockwise direction. This may give students the false impression that figures can only be rotated counterclockwise. To rotate figures 90° clockwise about the origin, switch the two coordinates of each point, and then multiply the new second coordinate by 1. The direction of a 180° rotation about the origin does not matter because whether the figure is rotated clockwise or counterclockwise, the final figure will be in the same position. 198. (x, y ). (y , x ). Example 2 Reflection. Reading Math. y. X. To rotate a figure 90° counterclockwise about the origin, switch the coordinates of each point and then multiply the new first coordinate by 1.. Chapter 4 Graphing Relations and Functions.

(18) Study Tip Reading Math Parallelogram ABCD and its image ABCD are said to be symmetric. The x-axis is called the line of symmetry.. b. Graph parallelogram ABCD and its image ABCD. Graph each vertex of the parallelogram ABCD. Connect the points.. In-Class Examples. y. A. Graph each vertex of the reflected image ABCD. Connect the points.. B. 3 Triangle ABC has vertices A(2, 1), B(2, 4), and C(1, 1).. C. D D'. x. O C'. a. Find the coordinates of the vertices of the image if it is translated 3 units to the right and 5 units down. A(1, 4), B(5, 1), C(4, 4). B'. A'. Power Point®. Example 3 Translation. b. Graph triangle ABC and its image.. Triangle ABC has vertices A(2, 3), B(4, 0), and C(2, 5).. y. a. Find the coordinates of the vertices of the image if it is translated 3 units to the left and 2 units down. To translate the triangle 3 units to the left, add 3 to the x-coordinate of each vertex. To translate the triangle 2 units down, add 2 to the y-coordinate of each vertex.. B. A C B x. O. (x, y) → (x 3, y 2) A(2, 3) → A(2 3, 3 2) → A(5, 1). A. B(4, 0) → B(4 3, 0 2) → B(1, 2). C. C(2, 5) → C(2 3, 5 2) → C(1, 7) The coordinates of the vertices of the image are A(5, 1), B(1, 2), and C(1, 7).. 4 A trapezoid has vertices. y. A A'. B. b. Graph triangle ABC and its image.. x. O. The preimage is ABC.. E(1, 2), F(2, 1), G(2, 1), and H(1, 2).. a. Find the coordinates of the dilated trapezoid E’F’G’H’ if the scale factor is 2. E(2, 4), F(4, 2), G(4, 2), H(2, 4). B'. The translated image is ABC. C C'. b. Graph the preimage and its image.. Example 4 Dilation. y. E. A trapezoid has vertices L(4, 1), M(1, 4), N(7, 0), and P(3, 6).. F. E. 3 4. a. Find the coordinates of the dilated trapezoid LMNP if the scale factor is . 3 4. To dilate the figure multiply the coordinates of each vertex by . (x, y) → x, y 3 4. 3 4. O. 3 3 3 4 4 4 3 3 3 M(1, 4) → M 1, 4 → M , 3 4 4 4 3 3 1 N(7, 0) → N 7, 0 → N 5, 0 4 4 4 3 3 1 1 P(3, 6) → P (3), (6) → P 2, 4 4 4 4 2. . . . . . H. . . . . . 3 4. N5, 0, and P2, 4.. Interactive. 1 2. 3 4. Chalkboard. (continued on the next page). www.algebra1.com/extra_examples. G. H. . The coordinates of the vertices of the image are L3, , M, 3, 1 4. x. G. L(4, 1) → L (4), 1 → L–3, . 1 4. F. Lesson 4-2 Transformations on the Coordinate Plane. PowerPoint® Presentations. 199. Differentiated Instruction Visual/Spatial Some students need to manipulate figures physically to perceive the concepts in this lesson. Have students model each transformation on a large grid with cutouts of the figure. Have them color one side blue and one side red so they can flip, slide, and rotate figures about the plane.. This CD-ROM is a customizable Microsoft® PowerPoint® presentation that includes: • Step-by-step, dynamic solutions of each In-Class Example from the Teacher Wraparound Edition • Additional, Your Turn exercises for each example • The 5-Minute Check Transparencies • Hot links to Glencoe Online Study Tools. Lesson 4-2 Transformations on the Coordinate Plane 199.

(19) In-Class Example. b. Graph the preimage and its image.. Power Point®. The image is trapezoid LMNP.. In this age of digital timepieces, some students may not be familiar with clockwise and counterclockwise. Have them use their fingers to draw an imaginary circle in a counterclockwise direction.. M'. L. Notice that the image has sides that are three-fourths the length of the sides of the original figure.. L'. N. N'. O. x. P'. P. 5 Triangle ABC has vertices. Example 5 Rotation. A(1, 3), B(3, 1), and C(5, 2).. Triangle XYZ has vertices X(1, 5), Y(5, 2), and Z(1, 2).. a. Find the coordinates of the image of ABC after it is rotated 180° about the origin. A(1, 3), B(3, 1), C(5, 2). a. Find the coordinates of the image of 䉭XYZ after it is rotated 90° counterclockwise about the origin. To find the coordinates of the vertices after a 90° rotation, switch the coordinates of each point and then multiply the new first coordinate by 1.. b. Graph the preimage and its image. C. M. The preimage is trapezoid LMNP.. Teaching Tip. A. y. (x, y) → (y, x). y. y X. Y'. X(1, 5) → X(5, 1) Y(5, 2) → Y(2, 5) Z(1, 2) → Z(2, 1). B O. X'. b. Graph the preimage and its image.. x. C. x. The rotated image is XYZ.. A. 3 Practice/Apply. Concept Check. 1. Compare and contrast the size, shape, and orientation of a preimage and an image for each type of transformation. See margin. 2. OPEN ENDED Draw a figure on the coordinate plane. Then show a dilation of the object that is an enlargement and a dilation of the object that is a reduction.. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 4. • make drawings to illustrate each transformation. • include any other item(s) that they find helpful in mastering the skills in this lesson.. Y. O. Z'. The image is XYZ.. B. Z. Guided Practice GUIDED PRACTICE KEY Exercises. Examples. 3, 4 5–10. 1 2–5. See margin. Identify each transformation as a reflection, translation, dilation, or rotation.. translation. 3.. rotation. 4.. Find the coordinates of the vertices of each figure after the given transformation is performed. Then graph the preimage and its image.. 5–8. See pp.253A– 253H for graphs. 5. P(1, 2), Q(4, 4), R(2, 3). 5. triangle PQR with P(1, 2), Q(4, 4), and R(2, 3) reflected over the x-axis 6. quadrilateral ABCD with A(4, 2), B(4, 2), C(1, 3), and D(3, 2) translated 3 units up A(4, 5), B(4, 1), C(1, 0), D(3, 5) 7. parallelogram EFGH with E(1, 4), F(5, 1), G(2, 4), and H(4, 1) dilated by a scale factor of 2 E(2, 8), F(10, 2), G(4, 8), H(8, 2) 8. triangle JKL with J(0, 0), K(2, 5), and L(4, 5) rotated 90° counterclockwise about the origin J(0, 0), K(5, 2), L(5, 4). 200. Chapter 4 Graphing Relations and Functions. Answers 1. Transformation Size Reflection same Rotation same Translation same Dilation changes. 200. Chapter 4 Graphing Relations and Functions. 2. Sample answer: y. Shape same same same same. Orientation changes changes same same. x.

(20) NAVIGATION For Exercises 9 and 10, use the following information. A ship was heading on a chartered route when it was blown off course by a storm. The ship is now ten miles west and seven miles south of its original destination.. NAME ______________________________________________ DATE. p. 219 (shown) and p. 220 Transformations on the Coordinate Plane. Transform Figures. 9. Using a coordinate grid, make a drawing to show the original destination A and the current position B of the ship. See margin. 10. Using coordinates (x, y) to represent the original destination of the ship, write an expression to show its current location. (x 10, y 7). ★ indicates increased difficulty. Practice and Apply Homework Help For Exercises. See Examples. 11–16, 37, 38 17–36. 1. Identify each transformation as a reflection, translation, dilation, or rotation.. translation. 11.. rotation. 12.. ____________ PERIOD _____. Study Guide andIntervention Intervention, 4-2 Study Guide and. Transformations are movements of geometric figures. The preimage is the position of the figure before the transformation, and the image is the position of the figure after the transformation. Reflection. A figure is flipped over a line.. Translation. A figure is slid horizontally, vertically, or both.. Dilation. A figure is enlarged or reduced.. Rotation. A figure is turned around a point.. Example Determine whether each transformation is a reflection, translation, dilation, or rotation. a.. The figure has been flipped over a line, so this is a reflection.. b.. The figure has been turned around a point, so this is a rotation.. c.. The figure has been reduced in size, so this is a dilation.. d.. The figure has been shifted horizontally to the right, so this is a translation.. Lesson 4-2. Application. Exercises Determine whether each transformation is a reflection, translation, dilation, or rotation.. 2–5. Extra Practice. reflection. 13.. dilation. 14.. 1.. reflection. 2.. 3.. rotation. 4.. dilation. translation. See page 828.. dilation. 5.. rotation. 6.. NAME ______________________________________________ DATE. reflection. 15.. ____________ PERIOD _____. Skills Practice, 4-2 Practice (Average). p. 221 and Practice, p. 222 Transformations on the(shown) Coordinate Plane. translation. 16.. Identify each transformation as a reflection, translation, dilation, or rotation. 1.. 2.. reflection. For Exercises 17–26, complete parts a and b. 17–26. See pp. 253A-253H for graphs. a. Find the coordinates of the vertices of each figure after the given transformation is performed.. translation. 5. trapezoid EFGH with E(3, 2), F(3, 3), G(1, 2), and H(1, 1) reflected over the y-axis. D(2, 0), E(0, 2), F(3, 4). 17. triangle RST with R(2, 0), S(2, 3), and T(2, 3) reflected over the y-axis 18. trapezoid ABCD with A(2, 3), B(5, 3), C(6, 1), and D(2, 1) reflected over the x-axis A(2, 3), B(5, 3), C(6, 1), D(2, 1). X (1, 3), Y(2, 4), Z(3, 1) Y. y. y. E D. 6. triangle XYZ with X(3, 1), Y(4, 2), and Z(1, 3) rotated 90° counterclockwise about the origin. E(3, 2), F(3, 3), G(1, 2), H(1, 1). y D. X. E. H H. E O. x. E. O. Z. x. G G. F. x. Z. F. Y. y. GRAPHICS For Exercises 7–9, use the diagram at the right and the following information. 1. A designer wants to dilate the rocket by a scale factor of , and 2 1 then translate it 5 units up. 2. 20. parallelogram MNOP with M(6, 0), N(4, 3), O(1, 3), and P(3, 0) translated 3 units right and 2 units down M(3, 2), N(1, 1), O(2, 1), P(0, 2) 21. trapezoid JKLM with J(4, 2), K(2, 4), L(4, 4), and M(4, 4) dilated by a scale. J(2, 1), K(1, 2), L(2, 2), M(2, 2). . . . . Lesson 4-2 Transformations on the Coordinate Plane. B. . . D. . . A y B. F D. C. E O. x. ____________ PERIOD _____. Reading to Learn Reading to Learn Mathematics ELL Mathematics, p. 223 Transformations on the Coordinate Plane. Pre-Activity. How are transformations used in computer graphics? Read the introduction to Lesson 4-2 at the top of page 197 in your textbook. In the sentence, “Computer graphic designers can create movement that mimics real-life situations,” what phrase indicates the use of transformations? create movement. 1. Suppose you look at a diagram that shows two figures ABCDE and ABCDE. If one figure was obtained from the other by using a transformation, how do you tell which was the original figure? The letters that have no prime symbols are used for. vertices of the original figure.. 201. 2. Write the letter of the term and the Roman numeral of the figure that best matches each statement.. ____________ PERIOD _____. C, I. b. A figure is turned around a point.. D, III. A. dilation. B. translation. I.. II.. p. 224 A, II. The Legendary City of Ur. c. A figure is enlarged or reduced.. The city of Ur was founded more than five thousand years ago in Mesopotamia (modern-day Iraq). It was one of the world’s first cities. Between 1922 and 1934, archeologists discovered many treasures from this ancient city. A large cemetery from the 26th century B.C. was found to contain large quantities of gold, silver, bronze, and jewels. The many cultural artifacts that were found, such as musical instruments, weapons, mosaics, and statues, have provided historians with valuable clues about the civilization that existed in early Mesopotamia.. d. A figure is slid horizontally, vertically, or both. B, IV. 1. Suppose that the ordered pairs below represent the volume (cm3) and mass (grams) of ten artifacts from the city of Ur. Plot each point on the graph. A(10, 150) B(150, 1350) C(200, 1760) D(50, 525) E(100, 1500) F(10, 88) G(200 2100). 2200. G. 2000 1800 1600 1400. C. reflection. III.. D. rotation. IV.. Helping You Remember. 2400. Mass (grams). 7 mi. A. . 9. Graph the image on the coordinate plane.. a. A figure is flipped over a line.. NAME ______________________________________________ DATE. C. Reading the Lesson. pentagon PQRST with P(0, 5), Q(3, 4), R(2, 1), S(2, 1), and T(3, 4) reflected over the x-axis, then translated 2 units left and 1 unit up. 10 mi. . NAME ______________________________________________ DATE. parallelogram WXYZ with W(1, 2), X(3, 2), Y(0, 4), and Z(4, 4) reflected over the y-axis, then rotated 180° about the origin. 4-2 Enrichment Enrichment,. . F. 1 1 1 1 1 1 1 A 0, 4 , B , 4 , C , 3 , D 1, 2 , E 1, 2 , F , 3 , G , 4 2 2 2 2 2 2 2. 4-2. quadrilateral TUVW with T(4, 2), U(2, 4), V(0, 2), and W(2, 4) rotated 90° counterclockwise about the origin T(2, 4), U(4, 2), V(2, 0), W(4, 2). 9.. . B. reflection over the x-axis, 90 counterclockwise rotation, and then reflection over the y-axis; three 90 counterclockwise rotations. triangle FGH with F(3, 2), G(2, 5), and H(6, 3) rotated 180° about the origin. www.algebra1.com/self_check_quiz. . G. E. 8. Find the coordinates of the final position of the rocket.. . x. A. A(0, 2), B(1, 3), C(1, 5), D(2, 6), E(2, 6), F(1, 5), G(1, 3). 10. DESIGN Ramona transformed figure ABCDEF to design a pattern for a quilt. Name two different sets of transformations she could have used to design the pattern. Sample answer:. 22. square ABCD with A(2, 1), B(2, 2), C(3, 2), and D(1, 3) dilated by a scale factor of 3 A(6, 3), B(6, 6), C(9, 6), D(3, 9). 23. F (3, 2), G(2, 5), 23. H(6, 3) 24. 25. W(1, 2), X(3, 2), Y(0, 4), Z( 4, 4) ★ 25. 26. P(2, 4), Q(1, 3), R(0, 0), ★ 26. S(4, 0), T (5, 3). O. 7. Write the coordinates for the vertices of the rocket.. C. H E B. 3. Give examples of things in everyday life that can help you remember what reflections, dilations, and rotations are. Sample answer: For a reflection, think of looking. at yourself in a mirror. For a dilation, think of how your hand looks if you hold it far from your face and then move it straight in, very close to your face. For a rotation, think of twisting the top of a jar to open the jar.. 1200 1000 800. I. 600. Lesson 4-2 Transformations on the Coordinate Plane 201. Lesson 4-2. 1 factor of 2. X. O. F. F. 19. quadrilateral RSTU with R(6, 3), S(4, 2), T(1, 5), and U(3, 7) translated 8 units right R(2, 3), S(4, 2), T(7, 5), U(5, 7). rotation. For Exercises 4–6, complete parts a and b. a. Find the coordinates of the vertices of each figure after the given transformation is performed. b. Graph the preimage and its image. 4. triangle DEF with D(2, 3), E(4, 1), and F(1, 1) translated 4 units left and 3 units down. b. Graph the preimage and its image.. 17. R(2, 0), S(2, 3), T (2, 3). 3..

(21) ANIMATION For Exercises 27–29, use the diagram at the right. An animator places an arrow representing an airplane on a coordinate grid. She wants to move the arrow 2 units right and then reflect it across the x-axis.. About the Exercises… Organization by Objective • Transform Figures: 11–16, 34, 37–38 • Transform Figures on the Coordinate Plane: 17–33, 35–36 Odd/Even Assignments Exercises 11–26 are structured so that students practice the same concepts whether they are assigned odd or even problems.. 27. A(5, 1), B(3, 3), C(5, 5), D(5, 4), E(8, 4), F(8, 2), G(5, 2) 28. A(3, 1), B(1, 3), C(3, 5), D(3, 4), E(6, 4), F(6, 2), G(3, 2). 27. Write the coordinates for the vertices of the arrow. 28. Find the coordinates of the final position of the arrow.. y. C' E' B'. D' G' F' A' A. O. x. F B. G D E. 29. Graph the image. See right.. C. 30. Trapezoid JKLM with J(6, 0), K(1, 5), L(1, 1), and M(3, 1) is translated to JKLM with J(3, 2), K(2, 3), L(2, 1), M(0, 3). Describe this translation. 3 units right, 2 units down 31. Triangle QRS with vertices Q(2, 6), R(8, 0), and S(6, 4) is dilated. If the image QRS has vertices Q(1, 3), R(4, 0), and S(3, 2), what is the scale factor? 1 2. Assignment Guide Basic: 11–23 odd, 27–29, 31, 34–36, 39–43, 47–59 Average: 11–25 odd, 27–29, 31, 33–36, 39–43, 47–59 (optional: 44–46) Advanced: 12–26 even, 30, 32, 37–57 (optional: 58, 59). ★ 32. Describe the transformation of parallelogram WXYZ with W(5, 3), X(2, 5),. Y(0, 3), and Z(3, 1) if the coordinates of its image are W(5, 3), X(2, 5), Y(0, 3), and Z(3, 1). reflection over the y-axis. ★ 33. Describe the transformation of triangle XYZ with X(2, 1), Y(5, 3), and. Z(4, 0) if the coordinates of its image are X(1, 2), Y(3, 5 ), and Z(0, 4).. 90° counterclockwise rotation. DIGITAL PHOTOGRAPHY For Exercises 34–36, use the following information. Soto wants to enlarge a digital photograph that is 1800 pixels wide and 1600 pixels 1 high (1800 1600) by a scale factor of 2. 2 34. What will be the dimensions of the new digital photograph? 4500 4000. Answer 37.. 35. Use a coordinate grid to draw a picture representing the 1800 1600 digital photograph. Place one corner of the photograph at the origin and write the coordinates of the other three vertices. See pp. 253A–253H. 36. Draw the enlarged photograph and write its coordinates. See pp. 253A–253H. ART For Exercises 37 and 38, use the following information. On grid paper, draw a regular octagon like the one shown.. Digital Photography Digital photographs contain hundreds of thousands or millions of tiny squares called pixels. Source: www.shortcourses.com. 37. See margin for drawing; the pattern resembles a snowflake. 202. 202. Chapter 4 Graphing Relations and Functions. ★ 37. Reflect the octagon over each of its sides. Describe the pattern that results. 38. Could this same pattern be drawn using any of the other transformations? If so, which kind? yes; translation 39. CRITICAL THINKING Make a conjecture about the coordinates of a point (x, y) that has been rotated 90° clockwise about the origin. (y, x) 40. CRITICAL THINKING Determine whether the following statement is sometimes, always, or never true. A reflection over the x-axis followed by a reflection over the y-axis gives the same result as a rotation of 180°. always. Chapter 4 Graphing Relations and Functions.

(22) 41. WRITING IN MATH. Answer the question that was posed at the beginning of the lesson. See margin.. 4 Assess. How are transformations used in computer graphics? Include the following in your answer: • examples of movements that could be simulated by transformations, and • types of other industries that might use transformations in computer graphics to simulate movement.. Standardized Test Practice. 42. The coordinates of the vertices of quadrilateral QRST are Q(2, 4), R(3, 7), S(4, 2), and T(5, 3). If the quadrilateral is moved up 3 units and right 1 unit, which point below has the correct coordinates? C Q(1, 5). A. B. R(4, 4). C. S(5, 1). D. T(6, 0). 84.. D. 96.. 2 1 43. x is of y and y is of z. If x 14, then z C 3 4. 48.. A. Extending the Lesson. 44. A(3, 4), B(2, 2), C(3, 2), D(4, 0). 72.. C. Graph the image of each figure after a reflection over the graph of the given equation. Find the coordinates of the vertices. 44. x 0. 45. y 3 y. 45. J(3, 5), K(2, 8), L(1, 8), M(3, 5) 46. R(3, 3), S(0, 4), T (4, 1). B. y. y L. S'. A' B. B'. R. M. D'. D. T'. O. S x. x. O. M'. J' C. R'. x. O. J. Modeling Draw a coordinate plane on the chalkboard or overhead projector. Give a student volunteer a cardboard cutout figure to place on the coordinate plane. Once the student volunteer has placed the figure on the coordinate plane, ask the student to model one of the transformations described in this lesson.. Getting Ready for Lesson 4-3. 46. y x. K. A. Open-Ended Assessment. C' T. PREREQUISITE SKILL Students will learn about relations in Lesson 4-3. Students will learn that a relation is a set of ordered pairs. Use Exercises 58–59 to determine your students’ familiarity with writing ordered pairs.. L'. K'. Assessment Options Quiz (Lessons 4-1 and 4-2) is available on p. 275 of the Chapter 4 Resource Masters.. Maintain Your Skills Mixed Review. Plot each point on a coordinate plane. (Lesson 4-1) 47–52. See margin. 47. A(2, 1). 48. B(4, 0). 49. C(1, 5). 50. D(1, 1). 51. E(2, 3). 52. F(4, 3). Answers. 53. CHEMISTRY Jamaal needs a 25% solution of nitric acid. He has 20 milliliters of a 30% solution. How many milliliters of a 15% solution should he add to obtain the required 25% solution? (Lesson 3-9) 10 mL. 58. {(1, 9.95), Two dice are rolled and their sum is recorded. Find each probability. (Lesson 2-6) (2, 11.45), (3, 12.95), 1 13 72% 55. P(sum is greater than 10) 8% (4, 14.45), (5, 15.95), 54. P(sum is less than 9) 1 1 85 5 2 (6, 17.45)} 56. P(sum is less than 7) 42% 57. P(sum is greater than 4) 83%. Getting Ready for the Next Lesson. 12 6 PREREQUISITE SKILL Write a set of ordered pairs that represents the data in the table. (To review ordered pairs, see Lesson 1-8.). 58.. Number of toppings Cost of large pizza ($). 59. {(0, 100), (5, 90), (10, 81), (15, 73), (20, 66), (25, 60), (30, 55)}. 59.. Time (minutes) Temperature of boiled water as it cools (°C). 1. 2. 3. 4. 5. 6. 9.95 11.45 12.95 14.45 15.95 17.45 0. 5. 10. 15. 20. 25. 30. 100. 90. 81. 73. 66. 60. 55. Lesson 4-2 Transformations on the Coordinate Plane. 203. 41. Artists use computer graphics to simulate movement, change the size of objects, and create designs. Answers should include the following. • Objects can appear to move by using a series of translations. Moving forward can be simulated by enlarging objects using dilations so they appear to be getting closer. • Computer graphics are used in special effects in movies, animated cartoons, and web design. 47–52. C E. B D. A F. Lesson 4-2 Transformations on the Coordinate Plane 203.

No results found