Graph each pair of functions on the same coordinate plane See margin. Technology Activity: A Family of Functions

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1 - What You ll Learn To analze translations To analze stretches, shrinks, and reflections...and Wh To analze a fabric design, as in Eample Families of Functions Check Skills You ll Need G for Help Lessons - and -5 Graph each pair of functions on the same coordinate plane.. See margin. =, = +. = -, = - - p = «,= «-. =- «, =- «+ 5. = «,= - «. = P P,= P P New Vocabular parent function translation stretch shrink reflection transformation parameter -. Plan bjectives To analze translations To analze stretches, shrinks, and reflections Eamples Vertical Translation Horizontal Translations Real-World Connection Graphing = a «5 Graphing =-a «Translations Vocabular Tip Translate is a snonm for transfer. Technolog Activit: A Famil of Functions The four functions in each group are related. Graph each set of functions in the same viewing window. Eplain how the are related. 8. See back of book.. () 5 u u () 5 u u () 5 u u () 5 u u. () 5 u u. () 5 u u () 5 u u () 5 u u () 5 u u () 5 u u () 5 u u () 5 u u. () 5 u u () 5uu () 5 u u () 5uu Use the graphs from parts to predict the graph of each function below. n graph paper and without plotting points, sketch what ou think the graph of f should be. Then check our sketch on a graphing calculator. 5. f() 5 u u. f() 5uu 7. Sketch what ou think the graph of f() 5u u should be. Then check our sketch on a graphing calculator. 8. Considering all the functions in Eercises 7, which function would ou call the simplest? Eplain. Math Background If ƒ() is an function and h is a real number, then the graph of ƒ( - h) corresponds to a horizontal translation, of h units, of the graph of ƒ(). If k is a real number, then the graph of ƒ() + k corresponds to a vertical translation, of k units, of the graph of ƒ(). If a is a real number, then the graph of af() is a reflection in the -ais for a 0, and a stretch b a factor of a for a or a shrink b a factor of a for 0 a. More Math Background: p. 5D Lesson Planning and Resources See p. 5E for a list of the resources that support this lesson. Special Needs L Have students each create a tetile design that is made using onl horizontal and vertical translations and then write a description of it. Have students echange descriptions and tr to draw each design. A famil of functions is made up of functions with certain common characteristics. A parent function is the simplest function with these characteristics. The equations of the functions in a famil resemble each other. So do the graphs. ffspring of parent functions include translations, stretches, and shrinks. A translation shifts a graph horizontall, verticall, or both. It results in a graph of the same shape and size but possibl in a different position. Lesson - Families of Functions 9 Below Level L As a memor device to help students recognize whether a function is translated horizontall or verticall, the h in f( - h) could signif a horizontal translation. PowerPoint Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Linear Equations Lesson -: Eample Etra Skills and Word Problems Practice, Ch. Absolute Value Functions and Graphs Lesson -5: Eamples Etra Skills and Word Problem Practice, Ch. learning stle: visual learning stle: verbal 9

2 . Teach Guided Instruction The parent absolute value function is 5 u u. For a positive number k, 5 u u k is a vertical translation. To graph the function 5 u u k, translate the graph of the parent function up k units.to graph the function 5 u u k, translate the graph of the parent down k units. Tactile Learners Vertical Translation Students can trace the graph of = «on tracing paper and slide the traced graph straight down units to check that the get the graph of = «-. Teaching Tip Students ma find it difficult to understand that the negative sign in part (a) means to move to the right and the plus sign in part (b) means to move to the left. Encourage students to make a table of values for each given function along with its parent function. Have students find an ordered pair for the given function and an ordered pair for the parent function that have the same -value. Then, have students eamine their -values and discuss how this helps them determine which direction to shift the given function. Diversit Real-World Connection Translations of two lines result in the diamond shapes in this window. Quick Check a. Describe the translation 5 u u and draw its graph. 5 u u is a translation of 5 u u b units downward. Each -value for 5 u u is less than the corresponding -value for 5 u u. b. Write an equation to translate 5 u u up unit. An equation that translates 5 u u up unit is 5 u u. a. Describe the translation 5 u u. Then draw the graphs of 5 u u and 5 u u in the same coordinate plane. See back of book. b. Write an equation for the translation of 5 u u down unit; Up.5 units.»» ;»» ±.5 Horizontal translations of the parent function 5 u u share some of the characteristics of = h = +h vertical translations. For a positive number h, 5 u h u is a horizontal translation. To graph the function 5 u h u, translate the graph of the parent function right h units. To graph the function h 5 u h u, translate the graph left h units. h = Invite students to look for patterned tetile designs that the can describe with translations. Focus on those that might relate to diverse cultural backgrounds. Horizontal Translations a. The blue graph at the right is a translation of 5 u u. Write an equation for the graph. The graph of 5 u u is translated 5 units to the right. An equation for the graph is 5 u 5 u. = b. Describe the translation 5 u u and draw its graph. 5 u u is a translation of 5 u u b units to the left. = + 9 Chapter Functions, Equations, and Graphs 9 Advanced Learners L Challenge students to find eamples of the artwork of M. C. Escher and to eplain how the artist used translations to create his intricate drawings. learning stle: verbal English Language Learners ELL Make sure students understand the difference between translation and transformation. Suggest that students think about what a translator does. Also, the root word form gives a clue to the meaning of transformation. learning stle: verbal

3 Quick Check a. The graph is a translation of 5 u u. Write an equation for the graph. 5 ` ` b. Describe the translation 5 u u. Then draw the graphs of 5 u u and 5 u u in the same coordinate plane. See back of book. PowerPoint Additional Eamples a. Describe the translation = «and draw its graph b translating the parent function. translation of»» down units = The parent absolute value function is 5 u u. For values h and k, 5 u h u k is a combined translation of h units horizontall and k units verticall. Ke Concepts Summar The Famil of Absolute Value Functions = Vertical Translation Parent function: Translation up k units, k > 0: Translation down k units, k > 0: Horizontal Translation Parent function: Translation right h units, h > 0: Translation left h units, h > 0: Combined Translation (right h units, up k units) 5 u u 5 u u k 5 u u k 5 u u 5 u h u 5 u h u 5 u h u k 5 ƒ() 5 ƒ() k 5 ƒ() k 5 ƒ() 5 ƒ( h) 5 ƒ( h) 5 ƒ( h) k b. Write an equation for the translation of = «up 8 units.»» 8 a. Describe the translation = «and draw its graph b translating the parent function. translation of»» left units Real-World Connection Fabric Design Describe possible translations of Figures A and B in the Nigerian tetile design below. = = A C B b. Write an equation for the translation of = «right units.»» For: Translating Activit Use: Interactive Tetbook, - page 9 Quick Check Check Skills You ll Need A translation of Figure A A translation of Figure B 5 units up or 5 units down about 5 units left and about units up Describe a possible translation of Figure C in the tetile design. Answers ma var. Sample:.5 units right and units up Lesson - Families of Functions 95 Describe a possible translation of Figures M and N in the design shown below. N M Figure M: units down; Figure N: units down, or else unit right and units down

4 5 9 Teaching Tip Students can confirm that the equation is correct b graphing it on a graphing calculator. PowerPoint Additional Eamples a. Describe and draw the graph of = «. vertical shrink of»» b a factor of = = ( ) b. Write an equation for a vertical stretch of = «b a factor of.»» 5. Write the equation for the graph.»» Resources Dail Notetaking Guide - L Dail Notetaking Guide - Adapted Instruction L Closure 8 = Ask students to describe the was the graph of the parent function = «can be transformed.»» can be a translation up, down, or a combined translation up and down; a reflection in the -ais; a vertical stretch b a factor of a >, or a vertical shrink b a factor of 0 < a <. Stretches, Shrinks, and Reflections A B C D E A B C D E A B C D E A B C D E 5 A B C D E B C D E Quick Check Test-Taking Tip Learn to recognize the parent function from the shape of the graph. Quick Check 9 Chapter Functions, Equations, and Graphs pages Eercises. 5 A vertical stretch multiplies all -values b the same factor greater than, thereb stretching a graph verticall. A vertical shrink reduces -values b a factor between 0 and, thereb compressing the graph verticall. More formall, for the parent function 5 u u and a number a, a., 5 au u is a vertical stretch. For 0, a,, 5 au u is a vertical shrink. Graphing = a a. Describe and then draw the graph of 5 u u. 5 u u is a vertical stretch of 5 u u b a factor of. Each -value for 5 u u is twice the corresponding -value for 5 u u. Note that (, ) lies on 5 u u, whereas (, ) lies on 5 u u. b. Write an equation for a vertical shrink of 5 u u b a factor of. A vertical shrink of 5 u u b a factor of is = «. a. Describe the stretch or shrink = «.Then draw the graphs of 5 u u and = «in the same coordinate plane. See back of book. b. Write an equation for the vertical stretch of 5 u u b a factor of. = A reflection in the -ais changes -values to their opposites. When ou change the -values of a graph to their opposites, the graph reflects across the -ais. For the parent function 5 u u, indeed for an function of the form 5 au u, multipling b gives the reflection 5auu, whose graph is a reflection of 5 au u across the -ais. Graphing = a Multiple Choice Which equation describes the graph? = «= = «= The parent function is 5 u u. This is a reflection across the -ais of = «.The answer is C. 5 A function is a vertical stretch of 5 u u b a factor of 5. Write an equation for the reflection of the function across the -ais. 5..

5 Ke Concepts Summar Families of Functions: Absolute Value Functions EXERCISES Eample (page 9) Eample (page 9) Vertical Stretch or Shrink, and Reflection in -ais Parent function: Reflection in -ais: Stretch Shrink Practice and Problem Solving A G Practice b Eample for Help What ou have learned about the absolute value function etends to functions in general. Each member of a famil of functions is a transformation, or change, of a parent function. Algebraicall, the transformations take the same form using parameters, like h, k, and a. Graphicall, the results are similar shifts, stretches, shrinks, and reflections of the parent function. (a. ) (0, a, ) Reflection in -ais: Combined Transformation For each function, graph the function b translating the parent function.. See margin p u u. 5 u u. 5 u u. 5 u u Write an equation for each vertical translation of 5 u u. 5. unit down. units up 7. units up 5 u u» ±» ± For each function, identif the translation of the parent function. Then graph the function. 8. See margin. 8. = - «9. = + 5«0. = - «. = + «Write an equation for each horizontal translation of 5 u u.... f b factor a: 5 u u 5uu 5 au u 5auu 5 au h u k 5 ƒ() 5ƒ() 5 aƒ() 5aƒ() 5 aƒ( h) k For more eercises, see Etra Skill and Word Problem Practice.. Practice Assignment Guide A B -, 9-, 8, 9,,,, 8-5 A B 7-8, 7, 0, 5, 7 C Challenge 5-5 Test Prep 5-57 Mied Review 58- Homework Quick Check To check students understanding of ke skills and concepts, go over Eercises, 7, 8, 9, 0. Error Prevention! Eercises 8 Students often forget that when h 0, replacing with - h in the parent function translates the graph to the right. Replacing with + h when h 0 translates the graph to the left. GPS Enrichment Guided Problem Solving Reteaching Practice Name Class Date Practice - Describe each translation of f()» as vertical, horizontal, or diagonal. Then graph each translation. Vertical and Horizontal Translations. f() = + «. f() = + «. f() = «-5. f() = + «- 5. f() = - «+. ` ` f() f() 5 ` ` f() 5 ` ` L L L L Eample (page 95) 5. Answers ma var. Sample:.5 units to the left.. Answers ma var. 7 Sample: units to the right.» ±»» Describe a possible translation for each figure. 5.. Write an equation for each translation. 0. = «, unit up, units left. = «, units right. =- «, units up, unit right. 5uu, units down, unit right. = «, units down, units left 5. 5uu, unit up 5 Write the equation of each translation of or» Each graph shows a translation of». State the values of h and k Graph each equation.. = - «+. 5`. =- + «- ` 5. = - - «. =- - «+ 7. = + «- Pearson Education, Inc. All rights reserved. Lesson - Families of Functions right units 5 9. left 5 units 0. right unit. left units 97

6 Eample (page 9) Graph each function. 7. See back of book u u 8. = «9. 5 u u 0. = u u. 5.5u u u u Write the equation for each graph. Each interval is unit. 5 u u u u. 5 u u Eample 5 (page 9). See left B Appl Your Skills P P 5 u u 5 u u 5 5 u u Describe each translation of f()» as vertical, horizontal, or combined. Then graph each translation. 9. See back of book. 9. f() = f() = - 5«+. f() = + «. f() = -. f() = «-. f() = - «- Write the equation for each graph. Each interval is unit Answers ma var. Sample: Since a vert. translation affects onl -values and a horiz. translation affects onl -values, order is irrelevant. 0a. G nline Homework Help Visit: PHSchool.com Web Code: age-00 5 u u 5 u u 5 u u GPS 8. Data Analsis Suppose ou plot data with ears as the independent variable. What tpe of translation are ou making when ou start with = 0 rather than a ear such as 998? Eplain. Horizontal; the graph is shifted left. 9. Writing Eplain wh appling a vertical translation and then a horizontal translation produces the same result as appling a horizontal translation and then a vertical translation. See left. 0. a. Graph the parent function ƒ() 5 and the function ƒ() = on a coordinate plane. ± 5 b. Translate the second graph 5 units up. Write an equation for the new line. c. Translate the graph from part (b) units right. Write an equation for the new line. d. Which equation describes the line ou graphed in part (c)? E A. = + 5 B. = + C. = - D. = ( - 5) + E. = ( - ) + 5 e. Critical Thinking Write the equation of the translation of = m that has a graph passing through point (h, k). m( h) ± k. pen-ended Draw a figure in Quadrant I. Use a translation to move our figure into Quadrant III. Describe our translation. Check students work. 98 Chapter Functions, Equations, and Graphs 98

7 5a. 5a. Height (ft) C h(t) Time Challenge b. Yes; sample eplanation: Each t is paired with a unique height. t Test Prep Multiple Choice lesson quiz, PHSchool.com, Web Code: aga-00 Each graph shows an absolute value function after a translation units up and units left. Write the equation of the original function...» 7» Write an equation for the translation so the graph has the given verte.. =- «; verte (-5, 0) 5. = «; verte (-, )» ± 5» ±. = «; verte (a, b) 7. =- «; verte (p, q)» a ±b» p ±q Graph each pair of functions on the same coordinate plane. Describe the translation that takes the first function to the second function. 8. = + «, = - 5«9. = «+, = - «50. = - «, = «+ 5. = + «-, = - « See back of book. 5. Suppose ou are plaing with a o-o, as shown at the right. a. Sketch a graph of h(t) to show the height h of the o-o above the floor over time t. At t = 0, the o-o leaves our hand. ft b. Critical Thinking Should our graph be that of a function? Eplain. c. Suppose ou demonstrate our o-o abilit on an auditorium stage that is 5 ft above the floor. Describe the translation of the graph of h(t) that represents the height of the o-o above the auditorium floor. vertical translation 5 ft up d. Choose a function g(t) that represents the height of the o-o above the auditorium floor when ou are on stage. C A. g(t) = h(t + 5) B. g(t) = h(t - 5) C. g(t) = h(t) + 5 D. g(t) = h(t) Use the graph of the function = f() at the right. Sketch the graph of each function. a. f( + ) a d. See margin. b. f() - c. f( + ) + d. - f () 5. Which translation takes = + «- to = «+? C A. units right, units down B. units left, units up C. units right, units up D. units left, units down 55. The graph of which equation will NT have a -intercept of 5? H F. = «+5 G. = - 5«H. = - 5«+5 J. = + 5«b. c. d. Lesson - Families of Functions 99. Assess & Reteach PowerPoint Lesson Quiz. Write equations for the graphs obtained b translating = «. a. 0 units right» 0 b. units down» c. 7 units left, units up» ± 7 ±;» ± 7 ± d. reflection in the -ais» e. vertical shrink b a factor of» f. vertical stretch b a factor of 5»5. Graph =- «and = «in the same coordinate plane. = = Alternative Assessment Have students work in groups of three. Students create si cards, three with = (sign) h«(sign) k and three with =- (sign) h«(sign) k, respectivel. The place the cards in a bo. Students then create si different positive values of h on one card each and si different positive values of k on one card each and place the values of h in one bo and the values of k in another bo. In a fourth bo students place twelve cards, si with a minus sign written and si with a plus sign. A student draws once from the first three boes and twice from the fourth bo. Then, the student writes and graphs the function represented, using the signs in the order drawn. Students repeat until all cards have been drawn and check each other s work. 99

8 P P Test Prep Resources For additional practice with a variet of test item formats: Standardized Test Prep, p. Test-Taking Strategies, p. 08 Test-Taking Strategies with Transparencies Etended Response Mied Review 5. The graph of = - «is translated units left and units down. What is the equation of the new graph? A A. = + «- B. = - «- C. = + «+ D. = - «+ 57. Start with the parent function of = + «-. Eplain how to describe the graph of = + «- as two consecutive translations of the parent function. Include graphs of the parent function and each translation. See margin. Use this Checkpoint Quiz to check students understanding of the skills and concepts of Lessons - through - G for Help Lesson -5 Lesson - Evaluate each function for five values of. Then graph each function. 58. f() = - «+ 59. f() = + «0. f() = See margin. 59. See back of book. Solve each inequalit. Graph each solution on a number line. Resources Grab & Go Checkpoint Quiz Lesson # -. a b,. Geometr Keiko, an orca whale who starred in a number of movies, moved into an outdoor pool with a volume of 8,50 cubic feet.the pool s surface is a 50 ft-b-75 ft rectangle.write and solve an equation to find the depth of the pool. 50? 75? 8,50; 5 ft Checkpoint Quiz Lessons - through - Graph each function.. See back of book.. = - 5. = + «+. f() = -. f() = «- 5. = - «-. = + Write an equation for each graph pages Eercises 57. [] The graph of» ± is» first translated units to the left for» ± and then units down for» ±. 9a. 5 7» ± 5 9. a. Use the table below. Model the relation with a scatter plot and a trend line. b. Predict the value of when = 0. about The graph of = «is translated down 5 units and right units. What is the equation of the new graph? D A. = + «+5 B. = + «-5 C. = - «+5 D. = - «-5 00 Chapter Functions, Equations, and Graphs [] includes equations and graph with no eplanation [] includes either the translation equations R graph 00 [] attempts to interpret the translation for» ± and graphs it but makes an error and does not translate it units left and units down 58. Answers ma var. Sample: f(0) 5; f() ; f() ; f() ; f( )