Essential Question How can you transform the graph of a polynomial function? Work with a partner. The graph of the cubic function

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1 3.7 Transormations o Polnomial Functions Essential Question How can ou transorm the raph o a polnomial unction? Transormin the Graph o a Cubic Function Work with a partner. The raph o the cubic unction () = 3 is shown. The raph o each cubic unction represents a transormation o the raph o. Write a rule or. Use a raphin calculator to veri our answers. a. b. c. d. Transormin the Graph o a Quartic Function Work with a partner. The raph o the quartic unction () = is shown. The raph o each quartic unction represents a transormation o the raph o. Write a rule or. Use a raphin calculator to veri our answers. LOOKING FOR STRUCTURE To be proicient in math, ou need to see complicated thins, such as some alebraic epressions, as bein sinle objects or as bein composed o several objects. a. Communicate Your Answer 3. How can ou transorm the raph o a polnomial unction?. Describe the transormation o () = represented b () = ( + ) + 3. Then raph. b. Section 3.7 Transormations o Polnomial Functions

2 3.7 Lesson What You Will Learn Describe transormations o polnomial unctions. Core Vocabular Previous polnomial unction transormations Write transormations o polnomial unctions. Describin Transormations o Polnomial Functions You can transorm raphs o polnomial unctions in the same wa ou transormed raphs o linear unctions, absolute value unctions, and quadratic unctions. Eamples o transormations o the raph o () = are shown below. Core Concept Transormation () Notation Eamples Horizontal Translation Graph shits let or riht. ( h) () = ( 5) () = ( + ) 5 units riht units let Vertical Translation Graph shits up or down. () + k () = + () = unit up units down Relection ( ) () = ( ) = over -ais Graph lips over - or -ais. () () = over -ais Horizontal Stretch or Shrink Graph stretches awa rom or shrinks toward -ais. Vertical Stretch or Shrink Graph stretches awa rom or shrinks toward -ais. (a) a () () = () shrink b a actor o () = ( ) stretch b a actor o () = 8 stretch b a actor o 8 () = shrink b a actor o Translatin a Polnomial Function Describe the transormation o () = 3 represented b () = ( + 5) 3 +. Then raph each unction. Notice that the unction is o the orm () = ( h) 3 + k. Rewrite the unction to identi h and k. () = ( ( 5) ) 3 + h k Because h = 5 and k =, the raph o is a translation 5 units let and units up o the raph o. Monitorin Proress Help in Enlish and Spanish at BiIdeasMath.com. Describe the transormation o () = represented b () = ( 3). Then raph each unction. Chapter 3 Polnomial Functions

3 Transormin Polnomial Functions Describe the transormation o represented b. Then raph each unction. a. () =, () = b. () = 5, () = () 5 3 a. Notice that the unction is o b. Notice that the unction is o the orm () = a, where the orm () = (a) 5 + k, where a =. a = and k = 3. So, the raph o is a So, the raph o is a relection in the -ais and a horizontal shrink b a actor o vertical shrink b a actor o and a translation 3 units down o the raph o. o the raph o. Monitorin Proress Help in Enlish and Spanish at BiIdeasMath.com. Describe the transormation o () = 3 represented b () = ( + ) 3. Then raph each unction. Writin Transormations o Polnomial Functions Writin Transormed Polnomial Functions Let () = Write a rule or and then raph each unction. Describe the raph o as a transormation o the raph o. a. () = ( ) b. () = 3() a. () = ( ) b. () = 3() = ( ) 3 + ( ) + = 3( ) = = REMEMBER Vertical stretches and shrinks do not chane the -intercept(s) o a raph. You can observe this usin the raph in Eample 3(b). 8 The raph o is a relection The raph o is a vertical stretch in the -ais o the raph o. b a actor o 3 o the raph o. Section 3.7 Transormations o Polnomial Functions 3

4 Writin a Transormed Polnomial Function Let the raph o be a vertical stretch b a actor o, ollowed b a translation 3 units up o the raph o () =. Write a rule or. Check 5 3 h Step First write a unction h that represents the vertical stretch o. h() = () Multipl the output b. = ( ) Substitute or (). = Distributive Propert Step Then write a unction that represents the translation o h. () = h() + 3 Add 3 to the output. = + 3 Substitute or h(). The transormed unction is () = + 3. Modelin with Mathematics ( 3) t t t CONNECTIONS TO GEOMETRY You can veri the unction iven in Eample 5 usin the ormula or the volume o a pramid ou learned in a previous course. V = 3 Bh = 3 ( )( 3) = 3 (3 3 ) = 3 3 The unction V() = 3 3 represents the volume (in cubic eet) o the square pramid shown. The unction W() = V(3) represents the volume (in cubic eet) when is measured in ards. Write a rule or W. Find and interpret W(0).. Understand the Problem You are iven a unction V whose inputs are in eet and whose outputs are in cubic eet. You are iven another unction W whose inputs are in ards and whose outputs are in cubic eet. The horizontal shrink shown b W() = V(3) makes sense because there are 3 eet in ard. You are asked to write a rule or W and interpret the output or a iven input.. Make a Plan Write the transormed unction W() and then ind W(0). 3. Solve the Problem W() = V(3) Net, ind W(0). = 3 (3)3 (3) Replace with 3 in V(). = Simpli. W(0) = 9(0) 3 9(0) = = 800 When is 0 ards, the volume o the pramid is 800 cubic eet.. Look Back Because W(0) = V(30), ou can check that our solution is correct b veriin that V(30) = 800. V(30) = 3 (30)3 (30) = = 800 Chapter 3 Polnomial Functions Monitorin Proress Help in Enlish and Spanish at BiIdeasMath.com 3. Let () = 5 + and () = (). Write a rule or and then raph each unction. Describe the raph o as a transormation o the raph o.. Let the raph o be a horizontal stretch b a actor o, ollowed b a translation 3 units to the riht o the raph o () = Write a rule or. 5. WHAT IF? In Eample 5, the heiht o the pramid is, and the volume (in cubic eet) is represented b V() = 3. Write a rule or W. Find and interpret W(7).

5 3.7 Eercises Dnamic Solutions available at BiIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE The raph o () = ( + ) 3 is a translation o the raph o () = 3.. VOCABULARY Describe how the verte orm o quadratic unctions is similar to the orm () = a( h) 3 + k or cubic unctions. Monitorin Proress and Modelin with Mathematics In Eercises 3, describe the transormation o represented b. Then raph each unction. (See Eample.) 3. () =, () = + 3. () =, () = ( 5) 5. () = 5, () = ( ) 5. () =, () = ( + ) ANALYZING RELATIONSHIPS In Eercises 7 0, match the unction with the correct transormation o the raph o. Eplain our reasonin. In Eercises, describe the transormation o represented b. Then raph each unction. (See Eample.). () =, () =. () =, () = 3 3. () = 3, () = () =, () = + 5. () = 5, () = 3 ( + )5. () =, () = () 3 In Eercises 7 0, write a rule or and then raph each unction. Describe the raph o as a transormation o the raph o. (See Eample 3.) 7. () = +, () = ( + ) 8. () = 5 + 3, () = 3() 7. = ( ) 8. = ( + ) + 9. = ( ) + 0. = () 9. () = 3 +, () = () 0. () = + 3, () = ( ) 5 A. B.. ERROR ANALYSIS Describe and correct the error in raphin the unction () = ( + ). C. D. Section 3.7 Transormations o Polnomial Functions 5

6 . ERROR ANALYSIS Describe and correct the error in describin the transormation o the raph o () = 5 represented b the raph o () = (3) 5. The raph o is a horizontal shrink b a actor o 3, ollowed b a translation units down o the raph o. In Eercises 3, write a rule or that represents the indicated transormations o the raph o. (See Eample.) 3. () = 3 ; translation 3 units let, ollowed b a relection in the -ais. () = + + ; vertical stretch b a actor o, ollowed b a translation units riht 5. () = 3 + 9; horizontal shrink b a actor o 3 and a translation units up, ollowed b a relection in the -ais 30. THOUGHT PROVOKING Write and raph a transormation o the raph o () = that results in a raph with a -intercept o. 3. PROBLEM SOLVING A portion o the path that a humminbird lies while eedin can be modeled b the unction () = 5 ( ) ( 7), 0 7 where is the horizontal distance (in meters) and () is the heiht (in meters). The humminbird eeds each time it is at round level. a. At what distances does the humminbird eed? b. A second humminbird eeds meters arther awa than the irst humminbird and lies twice as hih. Write a unction to model the path o the second humminbird.. () = ; relection in the -ais and a vertical stretch b a actor o 3, ollowed b a translation unit down 7. MODELING WITH MATHEMATICS The volume V (in cubic eet) o the pramid is iven b V() = 3. The unction W() = V(3) ives the volume (in cubic eet) o the pramid when is measured in ards. Write a rule or W. Find and interpret W(5). (See Eample 5.) t ( ) t (3 + ) t 8. MAKING AN ARGUMENT The volume o a cube with side lenth is iven b V() = 3. Your riend claims that when ou divide the volume in hal, the volume decreases b a reater amount than when ou divide each side lenth in hal. Is our riend correct? Justi our answer. 9. OPEN-ENDED Describe two transormations o the raph o () = 5 where the order in which the transormations are perormed is important. Then describe two transormations where the order is not important. Eplain our reasonin. Maintainin Mathematical Proicienc 3. HOW DO YOU SEE IT? Determine the real zeros o each unction. Then describe the transormation o the raph o that results in the raph o. 33. MATHEMATICAL CONNECTIONS Write a unction V or the volume (in cubic ards) o the riht circular cone shown. Then write a unction W that ives the volume (in cubic ards) o the cone when is measured in eet. Find and interpret W(3). Reviewin what ou learned in previous rades and lessons Find the minimum value or maimum value o the unction. Describe the domain and rane o the unction, and where the unction is increasin and decreasin. (Section.5) 3. h() = ( + 5) () = 3. () = 3( 0)( + ) 37. () = ( + )( + 8) 38. h() = ( ) () = ( + 3) d 3 d Chapter 3 Polnomial Functions