Vertical and Horizontal Translations. Graph each pair of functions on the same coordinate plane See margin

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1 - Lesson Preview What You ll Learn BJECTIVE BJECTIVE To analze vertical translations To analze horizontal translations... And Wh To analze a fabric design, as in Eample BJECTIVE Vertical and Horizontal Translations Translating Part Translating Graphs Verticall Graphs Verticall Reading Math Translate is a snonm for transfer. Check Understanding Check Skills You ll Need (For help, go to Lessons - and -.) Graph each pair of functions on the same coordinate plane.. See margin. =, = +. =-, =-- p. 9.. = «, = «-. = - «, = - «+. = «, = - «. = P P, = New Vocabular translation parent function A translation is an operation that shifts a graph horizontall, verticall, or both. It results in a graph of the same shape and size, in a different position. Comparing Graphs Compare the graphs of = «and = «-. Describe how the graph of = «- relates to the graph of = «.»»»» Make a table of values and graph the equations. For each value of, = «- is less than the value of = «. The graph of = «- is the graph of = «shifted units down. Compare the graphs of each pair of functions. Describe how the graph of the second function relates to the graph of the first function. a. = and = + b. f() = - «and f() = - «+ ± is f()» ± is shifted units up. f()» shifted units up. A famil of functions is a group of functions with common characteristics. A parent function is the simplest function with these characteristics. A parent function and one or more translations make up a famil of functions. Let k be a positive real number. To graph the functions = + k and = «+ k, translate the graph of the parent function up k units. To graph the functions = - k and = «- k, translate the graph down k units. ngoing Assessment and Intervention Before the Lesson Diagnose prerequisite skills using: Check Skills You ll Need During the Lesson Monitor progress using: Check Understanding Additional Eamples Standardized Test Prep Lesson - Vertical and Horizontal Translations 9 P P Interactive lesson includes instant self-check, tutorials, and activities. After the Lesson Assess knowledge using: Lesson Quiz Computer Test Generator CD Chapter Checkpoint (p. 98) -. Plan Lesson Preview Check Skills You ll Need Linear Equations Lesson -: Eample Eercises 8 Etra Practice, p. 8 Absolute Value Functions and Graphs Lesson -: Eamples Eercises 7 Etra Practice, p. 8 Lesson Resources Teaching Resources Practice, Reteaching, Enrichment Checkpoint Quiz Reaching All Students Practice Workbook - Spanish Practice Workbook - Reading and Math Literac C Spanish Reading & Literac C Spanish Checkpoint Quiz Presentation Assistant Plus! Transparencies Check Skills You ll Need - Additional Eamples - Student Edition Answers - Lesson Quiz - PH Presentation Pro CD - Checkpoint Quiz Computer Test Generator CD Technolog Resource Pro CD-RM Computer Test Generator CD Prentice Hall Presentation Pro CD Student Site Teacher Web Code: agk-00 Graphing Calculator, Procedures, Self-grading Lesson Quiz Teacher Center Lesson Planner Resources Plus 9

2 P. Teach 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(). BJECTIVE Teaching Notes Tactile Learners 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 = «-. Math Tip You ma want to point out that the graph of = - «can be obtained b reflecting the graph of = «across the -ais. Teaching Tip Students can confirm that the results are correct b graphing the two equations in each part of Eample on a graphing calculator. Additional Eamples Graph = «and = «+ on the same coordinate plane. Describe how the graphs are related. Check Understanding a.», k b., k Check Understanding Graphing a Vertical Translation For each function, identif the parent function and the value of k. Then graph the function b translating the parent function. a. = - b. = - «+ The parent function is =, and k =. Translate the graph of = down units. The parent function is = - «, and k =. Translate the graph of = - «up units. Identif each parent function and the value of k. Then graph each function b translating the parent function. See left. a. = «- b. = + You can write an equation for a translation. = Writing Equations for Vertical Translations BJECTIVE Translating Graphs Horizontall P P P a. =, units down b. =, unit up shifted units down, shifted unit up, The graph of =, The graph of = P, means k =. means k =. An equation is = -. An equation is = + P. a. = «, units down b. =, units up» Horizontal translations share some of the characteristics of vertical translations. Let h be a positive real number. Then = + h«translates the graph of = «h units to the left, and = - h«translates the graph h units to the right. - = h = +h h h = The graph of» ± is the graph of» shifted units up. 9 9 Chapter Functions, Equations, and Graphs Reaching All Students Below Level 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. Advanced Learners Challenge students to find eamples of the artwork of M. C. Escher, and eplain how the artist used translations to create his intricate drawings. Tactile Learners See note on page 9. Diversit See note on page 9.

3 a.», h Check Understanding Need Help? Graph the parent function before ou graph the translation. Check Understanding page 9 Graphing Horizontal Translations For each function, identif the parent function and the value of h. Then graph the function b translating the parent function. a. = +«b. =- -«The parent function is = «, and h=. The plus sign means translate to the left. Translate the graph of = «left units. The parent function is =- «, and h=. The minus sign means translate to the right. Translate the graph of =- «right units. Identif each parent function and the value of h. Then graph each function b translating the parent function.», h a. = -«b. = P P You can write a horizontal translation from a graph of a function. Writing Equations for Horizontal Translations The blue graph at the right is a translation of = «. Write an equation for the graph. This is the graph of = «translated units to the right. A shift to the right calls for an equation of the form = -h«. An equation for the graph is = -«. Each graph is a translation of = «. Write an equation for each graph. a. b. Check Skills You ll Need = + Lesson - Vertical and Horizontal Translations 9»±» = Additional Eamples Identif the parent function for =- and the value of k. Then, graph the function b translating the parent function. parent function: ; k Write an equation to translate the graph of = «, down units.» BJECTIVE Teaching Notes Teaching Tip Students ma find it difficult to understand that the plus sign in part (a) means to move to the left and the minus sign in part (b) means to move to the right. 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 Invite students to look for patterned tetile designs that the can described with translations. Focus on those that might relate to diverse cultural backgrounds. 7 Math Tip In each part of the Eample, the student can translate the parent graph horizontall and then translate the resulting graph verticall

4 Additional Eamples Identif the parent function and the value of h for = - «. Graph both functions. parent function:» ; h 8 8 You can combine vertical and horizontal translations to produce diagonal translations. Real-World Connection Fabric Design Describe a possible translation of Figures A and B in the Nigerian tetile design below. The graph is a translation of = - «. Write an equation for the graph. =» ± A C B Describe a possible translation of Figures M and N in the design shown below. N 9 M Figure M: units down; Figure N: units down, or else unit right and units down 7 Graph = + «+. 8 Write an equation for each translation of = «. a. units up, 7 units right» 7 ± b. units down, unit left» ± Closure Suppose h and k are positive numbers. How is the graph of = - h«+ k related to the graph of the parent function = «? The graph of» h ± k is a translation of the parent graph h units right and k units up. Check Understanding 7. Check Understanding 7 9 Chapter Functions, Equations, and Graphs pages 9 98., k Eercises 7 A translation of Figure A: A translation of Figure B: units up or units down about units left and about units up Describe a possible translation of Figure C in the tetile design. Answers ma var. Sample:. units right and units up You can use a parent function to graph a diagonal translation. Graphing Diagonal Translations Graph each function. a. = - «+ b. f() = - + «- The parent function is = «, The parent function is f() = - «, so h =, and k =. so h =, and k =. The minus sign means move h units to the right. The plus sign means move k units up. Place the verte at (, ), and draw the graph opening upward. = - + Graph the function f() = + «+.., k The plus sign means move h units to the left. The minus sign means move k units down. Place the verte at (-,-), and draw the graph opening downward. = + See left. 7.», k

5 8 Check Understanding 8 You can write an equation to describe a diagonal translation. Writing Diagonal Translations a. = «, units down, units left b. f() =- «, unit up, unit right units leftsh=; plus sign unit rightsh= ; minus sign units downsk=; minus sign unit upsk=; plus sign An equation is = + «-. An equation is f() = P P +. a. g() = «, unit down, 7 units right b. =- «, units up, units right g()» 7» ±. Practice Assignment Guide bjective A B C Core Etension bjective A B Core C Etension 7 EXERCISES Practice and Problem Solving For more practice, see Etra Practice. Standardized Test Prep Mied Review 7 A Practice b Eample Eample (page 9) Eample (page 9) Eample (page 9) Eample (page 9) Compare the graphs of each pair of functions. Describe how the graph of the second function relates to the graph of the first function. unit down. =- «, =- «+ units up. f()= «, f()= «-. g()= «, g()= «+. = «, = «- units up units down For each function, identif the parent function and the value of k. Then graph the function b translating the parent function. 7. See margin p. 9.. =-. =-+ 7. = «+ 8. =- «- See back Write an equation for each vertical translation. of book. 9. =, units down 0. = «, units up. =- «, units up» ±» ± For each function, identif the parent function and the value of h. Then graph the function b translating the parent function.. See margin.. = -«. = +«. =- -«. =- +«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. Enrichment - 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()= «-. f()= +«-. f()= -«+. 7. f() u u 8. f() ` 9. ` 0. = «, unit up, units left. = «, units right. =- «, units up, unit right. u u, units down, unit right. = «, units down, units left. u u, unit up f() ` ` f() ` ` Eample (page 9) Eample (page 9) 9. Answers ma var. Sample:. units to the left. 0. Answers ma var. 7 Sample: units to the right..», h Write an equation for each horizontal translation of u u or»» »±»» Describe a possible translation for each figure », h Lesson - Vertical and Horizontal Translations 9.», h.», h Write the equation of each translation of or» Each graph shows a translation of». State the values of h and k Graph each equation.. = -«+. `. =- +«- `. = --«. =- -«+ 7. = +«- Lesson - Practice Algebra Chapter Pearson Education, Inc. All rights reserved. 9

6 Eercise You ma want to point out to students that = +«- is an equivalent equation to = --«- since the absolute values of opposites are equal. Eample 7 (page 9) Graph each function.. See margin.. = +«-. = -«+. = +«+. = -«-. =- +«-. = --«- Eercises, Students ma see that these translations ma be viewed either as horizontal or vertical translations. B Eample 8 (page 9) Appl Your Skills»»± ± 7. = «, unit down, units right 8. =- «, units up, unit left 9. =- «, unit up, units right 0. = «, units up, 7 units right»» 7 ± Describe each translation of f() or f()» as vertical, horizontal, or diagonal. Then graph each translation.. See margin. pages Eercises. f()=-. f()= -«+. f()= +«. f()= «+. f()=+. f()= «- 7. f()= -«+ 8. f()= +«- 9. f()= -« See back of book. Write the equation of each translation of or». Each interval is unit » ±» Each graph shows a translation of». State the values of h and k vertical 7. Answers ma var. Sample: Since a vert. translation affects onl -values and a horiz. translation affects onl -values, order is irrelevant. 8a. h, k 0 h 0, k h, k. 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. 7. 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. 8. a. Graph the equation = on a coordinate plane. b. Translate the graph 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. =+ B. =+ C. =- D. =(-)+ E. =(-)+ e. Critical Thinking Write the equation of the translation of =m that has a graph passing through point (h, k). m( h)±k 9. pen-ended Draw a figure in Quadrant I. Use a translation to move our figure into Quadrant III. Describe our translation. Check students work.. diagonal 9 9 Chapter Functions, Equations, and Graphs. horizontal»±. vertical. vertical. vertical

7 0a. a. Height (ft) Need Help? Use the form u hu k for diagonal translations of absolute value functions. C h(t) Time Challenge b. Answers ma var. Sample: It is a function because each t is paired with a unique height. Multiple Choice t Each graph shows an absolute value function after a translation units up and units left. Write the equation of the original function. 0..» 7. =- «; verte (-, 0). =; through (-, ) ±7»±. = «; verte (a, b). =; through (p, q)» a ±b ±(q p) Graph each pair of functions on the same coordinate plane. Describe the translation that takes the first function to the second function. Standardized Test Prep. = +«, = -«7. = «+, = -«8. = -«, = «+ 9. = +«-, = -«+ 9. See back of book. 0. 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 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 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+) B. g(t)=h(t-) C. g(t)=h(t)+ D. g(t)=h(t)-. Use the graph of the function =f() at the right. Sketch the graph of each function. a. f(+) a c. See margin. b. f()- c. f(+)+. 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. The graph of which equation will NT have a -intercept of? H F. = «+ G. = -«H. = -«+ I. = +«b. Lesson - Vertical and Horizontal Translations 97» c.. Assess Lesson Quiz -. Graph = «and = +«+ on the same coordinate plane. Describe the translation that takes the graph of the parent function to the graph of the other function. Translate the parent graph units left and units up.. Write equations for the graphs obtained b translating = «and =- «as described. a. 0 units right» 0 ;» 0 b. units down» ;» c. 7 units left, units up»±7 ±;»±7 ± 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. 97

8 P Standardized Test Prep Resources For additional practice with a variet of test item formats: Standardized Test Prep, p. Test-Taking Strategies, p. 0 Test-Taking Strategies with Transparencies Eercise ne wa to approach this question is first to identif the verte of each graph. Then, describe the translation that takes the first verte to the second verte. Chapter Checkpoint To check understanding of Lessons - to -: Checkpoint Quiz (p. 98) Teaching Resources Checkpoint Quiz (also in Prentice Hall Assessment Sstem) Reaching All Students Reading and Math Literac C Spanish versions available pages Eercises. [] The graph of»± is» first translated units to the left for»± and then units down for»±. [] includes equations and graph with no eplanation [] includes either the translation equations R graph 9a. Take It to the NET nline lesson quiz at Web Code: aga-00 Etended Response Mied Review Lesson - Lesson - Lesson Chapter Functions, Equations, and Graphs. The graph of = -«is translated units left and units down. What is the equation of the new graph? A A.= +«- B. = -«- C. = +«+ D.= -«+. 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. Evaluate each function for five values of. Then graph each function.. f()= -«+ 7. f()= + «8. f()= P -. See margin See back of book. Solve each inequalit. Graph each solution on a number line #- 70. a b, 7. Geometr Keiko, an orca whale who starred in a number of movies, moved into an outdoor pool with a volume of 8,0 cubic feet. The pool s surface is 0 ft b 7 ft. Write and solve an equation to find the depth of the pool. 0? 7? 8,0; ft Checkpoint Quiz Lessons - through - Instant self-check quiz online and on CD-RM [] attempts to interpret the translation for»± and graphs it but makes an error and does not translate it units left and units down Graph each function.. See back of book.. =-. = +«+. f()= -. f()= «-. = -«-. =+ Write an equation for each graph »± 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 units and right units. What is the equation of the new graph? D A. = +«+ B. = +«- C. = -«+ D. = -«-. Answers ma var. Sample: f(0) ; f() ; f() ; f() ; f( )