Transformations of Quadratic Functions

Transcription

1 .1 Transormations o Quadratic Functions Essential Question How do the constants a, h, and k aect the raph o the quadratic unction () = a( h) + k? The parent unction o the quadratic amil is. A transormation o the raph o the parent unction is represented b the unction () = a( h) + k, where a 0. Identiin Graphs o Quadratic Functions Work with a partner. Match each quadratic unction with its raph. Eplain our reasonin. Then use a raphin calculator to veri that our answer is correct. a. () = ( ) b. () = ( ) + c. () = ( + ) d. () = 0.5( ) e. () = ( ). () = ( + ) + A. B. C. D. E. F. LOOKING FOR STRUCTURE To be proicient in math, ou need to look closel to discern a pattern or structure. Communicate Your Answer. How do the constants a, h, and k aect the raph o the quadratic unction () = a( h) + k? 3. Write the equation o the quadratic unction whose raph is shown at the riht. Eplain our reasonin. Then use a raphin calculator to veri that our equation is correct. Section.1 Transormations o Quadratic Functions 1

2 .1 Lesson What You Will Learn Core Vocabular quadratic unction, p. parabola, p. verte o a parabola, p. verte orm, p. Previous transormations Describe transormations o quadratic unctions. Write transormations o quadratic unctions. Describin Transormations o Quadratic Functions A quadratic unction is a unction that can be written in the orm () = a( h) + k, where a 0. The U-shaped raph o a quadratic unction is called a parabola. In Section 1., ou raphed quadratic unctions usin tables o values. You can also raph quadratic unctions b applin transormations to the raph o the parent unction. Core Concept Horizontal Translations ( h) = ( h) = ( h), h < 0 = Vertical Translations () + k = + k = + k, k > 0 = shits let when h < 0 shits riht when h > 0 = ( h), h > 0 = + k, k < 0 shits down when k < 0 shits up when k > 0 Translations o a Quadratic Function Describe the transormation o represented b () = ( + ) 1. Then raph each unction. Notice that the unction is o the orm () = ( h) + k. Rewrite the unction to identi h and k. () = ( ( )) + ( 1) h k Because h = and k = 1, the raph o is a translation units let and 1 unit down o the raph o. Monitorin Proress Describe the transormation o represented b. Then raph each unction. 1. () = ( 3). () = ( ) 3. () = ( + 5) + 1 Chapter Quadratic Functions

3 Core Concept Relections in the -Ais () = ( ) = = Relections in the -Ais ( ) = ( ) = = lips over the -ais = Horizontal Stretches and Shrinks (a) = (a) = (a), a > 1 = = is its own relection in the -ais. Vertical Stretches and Shrinks a () = a = a, a > 1 = = (a), 0 < a < 1 = a, 0 < a < 1 horizontal stretch (awa rom -ais) when 0 < a < 1 horizontal shrink (toward -ais) when a > 1 vertical stretch (awa rom -ais) when a > 1 vertical shrink (toward -ais) when 0 < a < 1 Transormations o Quadratic Functions LOOKING FOR STRUCTURE In Eample b, notice that () = + 1. So, ou can also describe the raph o as a vertical stretch b a actor o ollowed b a translation 1 unit up o the raph o. Describe the transormation o represented b. Then raph each unction. a. () = 1 b. () = () + 1 a. Notice that the unction is o the orm () = a, where a = 1. So, the raph o is a relection in the -ais and a vertical shrink b a actor o 1 o the raph o. b. Notice that the unction is o the orm () = (a) + k, where a = and k = 1. So, the raph o is a horizontal shrink b a actor o 1 ollowed b a translation 1 unit up o the raph o. Section.1 Transormations o Quadratic Functions 3

4 Monitorin Proress Describe the transormation o represented b. Then raph each unction.. () = ( 1 3 ) 5. () = 3( 1). () = ( + 3) + Writin Transormations o Quadratic Functions The lowest point on a parabola that opens up or the hihest point on a parabola that opens down is the verte. The verte orm o a quadratic unction is () = a( h) + k, where a 0 and the verte is (h, k). a indicates a relection in the -ais and/or a vertical stretch or shrink. () = a( h) + k h indicates a horizontal translation. k indicates a vertical translation. Writin a Transormed Quadratic Function Let the raph o be a vertical stretch b a actor o and a relection in the -ais, ollowed b a translation 3 units down o the raph o. Write a rule or and identi the verte. Method 1 Identi how the transormations aect the constants in verte orm. relection in -ais vertical stretch b a = translation 3 units down} k = 3 Write the transormed unction. () = a( h) + k Verte orm o a quadratic unction = ( 0) + ( 3) Substitute or a, 0 or h, and 3 or k. = 3 Simpli. The transormed unction is () = 3. The verte is (0, 3). Method Bein with the parent unction and appl the transormations one at a time in the stated order. Check 0 First write a unction h that represents the relection and vertical stretch o. h() = () Multipl the output b. = Substitute or (). 5 5 Then write a unction that represents the translation o h. 0 () = h() 3 Subtract 3 rom the output. = 3 Substitute or h(). The transormed unction is () = 3. The verte is (0, 3). Chapter Quadratic Functions

5 Writin a Transormed Quadratic Function REMEMBER To multipl two binomials, use the FOIL Method. First Inner ( + 1)( + ) = Outer Last Let the raph o be a translation 3 units riht and units up, ollowed b a relection in the -ais o the raph o 5. Write a rule or. Step 1 First write a unction h that represents the translation o. h() = ( 3) + Subtract 3 rom the input. Add to the output. = ( 3) 5( 3) + Replace with 3 in (). = 11 + Simpli. Step Then write a unction that represents the relection o h. () = h( ) Multipl the input b 1. = ( ) 11( ) + Replace with in h(). = Simpli. 0 = Modelin with Mathematics The heiht h (in eet) o water sprain rom a ire hose can be modeled b h() = , where is the horizontal distance (in eet) rom the ire truck. The crew raises the ladder so that the water hits the round 10 eet arther rom the ire truck. Write a unction that models the new path o the water. 1. Understand the Problem You are iven a unction that represents the path o water sprain rom a ire hose. You are asked to write a unction that represents the path o the water ater the crew raises the ladder.. Make a Plan Analze the raph o the unction to determine the translation o the ladder that causes water to travel 10 eet arther. Then write the unction. 3. Solve the Problem Use a raphin calculator to raph the oriinal unction. Because h(50) = 0, the water oriinall hits the round 50 eet rom the ire truck. The rane o the unction in this contet does not include neative values. However, b observin that h(0) = 3, ou can determine that a translation 3 units (eet) up causes the water to travel 10 eet arther rom the ire truck. () = h() + 3 Add 3 to the output. = Substitute or h() and simpli. 0 X=50 Y= The new path o the water can be modeled b () = Look Back To check that our solution is correct, veri that (0) = 0. (0) = 0.03(0) = = 0 Monitorin Proress 7. Let the raph o be a vertical shrink b a actor o 1 ollowed b a translation units up o the raph o. Write a rule or and identi the verte. 8. Let the raph o be a translation units let ollowed b a horizontal shrink b a actor o 1 3 o the raph o () = +. Write a rule or. 9. WHAT IF? In Eample 5, the water hits the round 10 eet closer to the ire truck ater lowerin the ladder. Write a unction that models the new path o the water. Section.1 Transormations o Quadratic Functions 5

6 .1 Eercises Dnamic Solutions available at BiIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The raph o a quadratic unction is called a(n).. VOCABULARY Identi the verte o the parabola iven b () = ( + ). Monitorin Proress and Modelin with Mathematics In Eercises 3 1, describe the transormation o represented b. Then raph each unction. (See Eample 1.) 3. () = 3. () = () = ( + ). () = ( ) 7. () = ( 1) 8. () = ( + 3) 9. () = ( + ) 10. () = ( 9) () = ( 7) () = ( + 10) 3 ANALYZING RELATIONSHIPS In Eercises 13 1, match the unction with the correct transormation o the raph o. Eplain our reasonin. 13. = ( 1) 1. = () = ( 1) = ( + 1) 1 A. B. In Eercises 17, describe the transormation o represented b. Then raph each unction. (See Eample.) 17. () = 18. () = ( ) 19. () = 3 0. () = () = (). () = () 3. () = 1 5. () = 1 ( 1) ERROR ANALYSIS In Eercises 5 and, describe and correct the error in analzin the raph o The raph is a relection in the -ais and a vertical stretch b a actor o, ollowed b a translation units up o the raph o the parent quadratic unction. The raph is a translation units down, ollowed b a vertical stretch b a actor o and a relection in the -ais o the raph o the parent quadratic unction. C. D. USING STRUCTURE In Eercises 7 30, describe the transormation o the raph o the parent quadratic unction. Then identi the verte. 7. () = 3( + ) () = ( + 1) 5 9. () = () = 1 ( 1) Chapter Quadratic Functions

7 In Eercises 31 3, write a rule or described b the transormations o the raph o. Then identi the verte. (See Eamples 3 and.) 31. ; vertical stretch b a actor o and a relection in the -ais, ollowed b a translation units up 3. ; vertical shrink b a actor o 1 3 and a relection in the -ais, ollowed b a translation 3 units riht 33. () = 8 ; horizontal stretch b a actor o and a translation units up, ollowed b a relection in the -ais 3. () = ( + ) + 3; horizontal shrink b a actor o 1 and a translation 1 unit down, ollowed b a relection in the -ais USING TOOLS In Eercises 35 0, match the unction with its raph. Eplain our reasonin. 35. () = ( 1) 3. () = 1 ( + 1) 37. () = ( 1) () = ( + 1) () = ( + 1) 0. () = ( 1) + A. B. JUSTIFYING STEPS In Eercises 1 and, justi each step in writin a unction based on the transormations o () = translation units down ollowed b a relection in the -ais h() = () = + () = h() = ( + ) = +. relection in the -ais ollowed b a translation units riht h() = ( ) = ( ) + ( ) = () = h( ) = ( ) ( ) = MODELING WITH MATHEMATICS The unction h() = 0.03( 1) + models the jump o a red kanaroo, where is the horizontal distance traveled (in eet) and h() is the heiht (in eet). When the kanaroo jumps rom a hiher location, it lands 5 eet arther awa. Write a unction that models the second jump. (See Eample 5.) C. D. E. F.. MODELING WITH MATHEMATICS The unction (t) = 1t + 10 models the heiht (in eet) o an object t seconds ater it is dropped rom a heiht o 10 eet on Earth. The same object dropped rom the same heiht on the moon is modeled b (t) = 8 3 t Describe the transormation o the raph o to obtain. From what heiht must the object be dropped on the moon so it hits the round at the same time as on Earth? Section.1 Transormations o Quadratic Functions 7

8 5. MODELING WITH MATHEMATICS Flin ish use their pectoral ins like airplane wins to lide throuh the air. a. Write an equation o the orm = a( h) + k with verte (33, 5) that models the liht path, assumin the ish leaves the water at (0, 0). b. What are the domain and rane o the unction? What do the represent in this situation? c. Does the value o a chane when the liht path has verte (30, )? Justi our answer. 7. COMPARING METHODS Let the raph o be a translation 3 units up and 1 unit riht ollowed b a vertical stretch b a actor o o the raph o. a. Identi the values o a, h, and k and use verte orm to write the transormed unction. b. Use unction notation to write the transormed unction. Compare this unction with our unction in part (a). c. Suppose the vertical stretch was perormed irst, ollowed b the translations. Repeat parts (a) and (b). d. Which method do ou preer when writin a transormed unction? Eplain.. HOW DO YOU SEE IT? Describe the raph o as a transormation o the raph o. 8. THOUGHT PROVOKING A jump on a poo stick with a conventional sprin can be modeled b () = 0.5( ) + 18, where is the horizontal distance (in inches) and () is the vertical distance (in inches). Write at least one transormation o the unction and provide a possible reason or our transormation. 9. MATHEMATICAL CONNECTIONS The area o a circle depends on the radius, as shown in the raph. A circular earrin with a radius o r millimeters has a circular hole with a radius o 3r millimeters. Describe a transormation o the raph below that models the area o the blue portion o the earrin. Area (square units) A Circle A = πr r Radius (units) Maintainin Mathematical Proicienc Reviewin what ou learned in previous rades and lessons A line o smmetr or the iure is shown in red. Find the coordinates o point A. 50. (, 3) = (0, ) A 5. A = A = (, ) 8 Chapter Quadratic Functions