Work with a partner. Compare the graph of the function. to the graph of the parent function. the graph of the function

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1 USING TOOLS STRATEGICALLY To be proicient in math, ou need to use technoloical tools to visualize results and eplore consequences. 1. Transormations o Linear and Absolute Value Functions Essential Question How do the raphs o = () + k, = ( h), and = () compare to the raph o the parent unction? Work with a partner. Compare the raph o the unction = + k to the raph o the parent unction () =. Transormations o the Parent Absolute Value Function Transormation Parent unction = = + = Transormations o the Parent Absolute Value Function Work with a partner. Compare = the raph o the unction = h to the raph o the parent unction () =. Transormation Parent unction = + = Transormation o the Parent Absolute Value Function Work with a partner. Compare the raph o the unction = = to the raph o the parent unction () =. Transormation Parent unction = Communicate Your Answer. How do the raphs o = () + k, = ( h), and = () compare to the raph o the parent unction? 5. Compare the raph o each unction to the raph o its parent unction. Use a raphin calculator to veri our answers are correct. a. = b. = + c. = d. = + 1 e. = ( 1). = Section 1. Transormations o Linear and Absolute Value Functions 17

2 1. Lesson What You Will Learn Write unctions representin translations and relections. Write unctions representin stretches and shrinks. Write unctions representin combinations o transormations. Translations and Relections You can use unction notation to represent transormations o raphs o unctions. Core Concept Horizontal Translations The raph o = ( h) is a horizontal translation o the raph o = (), where h 0. Vertical Translations The raph o = () + k is a vertical translation o the raph o = (), where k 0. = ( h), h < 0 = () = () + k, k > 0 = () = ( h), h > 0 Subtractin h rom the inputs beore evaluatin the unction shits the raph let when h < 0 and riht when h > 0. = () + k, k < 0 Addin k to the outputs shits the raph down when k < 0 and up when k > 0. Writin Translations o Functions Let () = + 1. a. Write a unction whose raph is a translation units down o the raph o. b. Write a unction h whose raph is a translation units to the let o the raph o. a. A translation units down is a vertical translation that adds to each output value. () = () + ( ) Add to the output. = ( ) Substitute + 1 or (). = Simpli. The translated unction is () =. Check h 5 b. A translation units to the let is a horizontal translation that subtracts rom each input value. h() = ( ( )) Subtract rom the input. 5 5 = ( + ) Add the opposite. = ( + ) + 1 Replace with + in (). 5 = + 5 Simpli. The translated unction is h() = Chapter 1 Linear Functions, Linear Sstems, and Matrices

3 STUDY TIP When ou relect a unction in a line, the raphs are smmetric about that line. Core Concept Relections in the -ais The raph o = () is a relection in the -ais o the raph o = (). = () = () Relections in the -ais The raph o = ( ) is a relection in the -ais o the raph o = (). = ( ) = () Multiplin the outputs b 1 chanes their sins. Multiplin the inputs b 1 chanes their sins. Writin Relections o Functions Let () = a. Write a unction whose raph is a relection in the -ais o the raph o. b. Write a unction h whose raph is a relection in the -ais o the raph o. a. A relection in the -ais chanes the sin o each output value. () = () Multipl the output b 1. = ( ) Substitute or (). = + 1 Distributive Propert The relected unction is () = + 1. Check 10 b. A relection in the -ais chanes the sin o each input value. h() = ( ) Multipl the input b h 10 = Replace with in (). = ( ) + 1 Factor out 1. = Product Propert o Absolute Value 10 = + 1 Simpli. The relected unction is h() = + 1. Monitorin Proress Write a unction whose raph represents the indicated transormation o the raph o. Use a raphin calculator to check our answer. 1. () = ; translation 5 units up. () = ; translation units to the riht. () = + 1; relection in the -ais. () = 1 + 1; relection in the -ais Section 1. Transormations o Linear and Absolute Value Functions 19

4 STUDY TIP The raphs o = ( a) and = a () represent a stretch or shrink and a relection in the - or -ais o the raph o = (). Stretches and Shrinks In the previous section, ou learned that vertical stretches and shrinks transorm raphs. You can also use horizontal stretches and shrinks to transorm raphs. Core Concept Horizontal Stretches and Shrinks The raph o = (a) is a horizontal stretch or shrink b a actor o 1 o the raph o a = (), where a > 0 and a 1. Multiplin the inputs b a beore evaluatin the unction stretches the raph horizontall (awa rom the -ais) when 0 < a < 1, and shrinks the raph horizontall (toward the -ais) when a > 1. Vertical Stretches and Shrinks The raph o = a () is a vertical stretch or shrink b a actor o a o the raph o = (), where a > 0 and a 1. Multiplin the outputs b a stretches the raph verticall (awa rom the -ais) when a > 1, and shrinks the raph verticall (toward the -ais) when 0 < a < 1. = (a), a > 1 = () = (a), 0 < a < 1 The -intercept stas the same. = a (), a > 1 = () = a (), 0 < a < 1 The -intercept stas the same. Writin Stretches and Shrinks o Functions Let () = 5. Write (a) a unction whose raph is a horizontal shrink o the raph o b a actor o 1, and (b) a unction h whose raph is a vertical stretch o the raph o b a actor o. a. A horizontal shrink b a actor o 1 multiplies each input value b. () = () Multipl the input b. Check = 5 Replace with in (). 10 h 1 The transormed unction is () = 5. b. A vertical stretch b a actor o multiplies each output value b. h() = () Multipl the output b. = ( 5 ) Substitute 5 or (). 1 = 10 Distributive Propert The transormed unction is h() = 10. Monitorin Proress Write a unction whose raph represents the indicated transormation o the raph o. Use a raphin calculator to check our answer. 5. () = + ; horizontal stretch b a actor o. () = ; vertical shrink b a actor o 1 0 Chapter 1 Linear Functions, Linear Sstems, and Matrices

5 Combinations o Transormations You can write a unction that represents a series o transormations on the raph o another unction b applin the transormations one at a time in the stated order. Combinin Transormations Let the raph o be a vertical shrink b a actor o 0.5 ollowed b a translation units up o the raph o () =. Write a rule or. Check 1 Step 1 First write a unction h that represents the vertical shrink o. h() = 0.5 () Multipl the output b 0.5. = 0.5 Substitute or (). Step Then write a unction that represents the translation o h. 8 1 () = h() + Add to the output. 8 = Substitute 0.5 or h(). The transormed unction is () = Modelin with Mathematics You desin a computer ame. Your revenue or downloads is iven b () =. Your proit is $50 less than 90% o the revenue or downloads. Describe how to transorm the raph o to model the proit. What is our proit or 100 downloads? 1. Understand the Problem You are iven a unction that represents our revenue and a verbal statement that represents our proit. You are asked to ind the proit or 100 downloads.. Make a Plan Write a unction p that represents our proit. Then use this unction to ind the proit or 100 downloads.. Solve the Problem proit = 90% revenue 50 Vertical shrink b a actor o 0.9 p() = 0.9 () 50 Translation 50 units down = Substitute or (). = Simpli. 00 p To ind the proit or 100 downloads, evaluate p when = 100. p(100) = 1.8(100) 50 = 10 = Your proit is $10 or 100 downloads. 0 X=100 Y= Look Back The vertical shrink decreases the slope, and the translation shits the raph 50 units down. So, the raph o p is below and not as steep as the raph o. Monitorin Proress 7. Let the raph o be a translation units down ollowed b a relection in the -ais o the raph o () =. Write a rule or. Use a raphin calculator to check our answer. 8. WHAT IF? In Eample 5, our revenue unction is () =. How does this aect our proit or 100 downloads? Section 1. Transormations o Linear and Absolute Value Functions 1

6 1. Eercises Dnamic Solutions available at BiIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The unction () = 5 is a horizontal o the unction () =.. WHICH ONE DOESN'T BELONG? Which transormation does not belon with the other three? Eplain our reasonin. Translate the raph o () = + up units. Stretch the raph o () = + verticall b a actor o. Shrink the raph o () = + 5 horizontall b a actor o 1. Translate the raph o () = + let 1 unit. Monitorin Proress and Modelin with Mathematics In Eercises 8, write a unction whose raph represents the indicated transormation o the raph o. Use a raphin calculator to check our answer. (See Eample 1.). () = 5; translation units to the let. () = + ; translation units to the riht 10. PROBLEM SOLVING You open a caé. The unction () = 000 represents our epected net income (in dollars) ater bein open weeks. Beore ou open, ou incur an etra epense o $1,000. What transormation o is necessar to model this situation? How man weeks will it take to pa o the etra epense? 5. () = + + ; translation units down. () = 9; translation units up 7. () = () = WRITING Describe two dierent translations o the raph o that result in the raph o. 1 1 In Eercises 11 1, write a unction whose raph represents the indicated transormation o the raph o. Use a raphin calculator to check our answer. (See Eample.) 11. () = 5 + ; relection in the -ais 1. () = 1 ; relection in the -ais () = 5 () = 1. () = ; relection in the -ais 1. () = 1 + ; relection in the -ais 15. () = + 11 ; relection in the -ais 1. () = + 1; relection in the -ais Chapter 1 Linear Functions, Linear Sstems, and Matrices

7 In Eercises 17, write a unction whose raph represents the indicated transormation o the raph o. Use a raphin calculator to check our answer. (See Eample.) 17. () = + ; vertical stretch b a actor o () = + ; vertical shrink b a actor o () = + ; horizontal shrink b a actor o 1 0. () = + ; horizontal stretch b a actor o 1. () = + (, 1). () = (0, ) (, ) In Eercises 7, write a unction whose raph represents the indicated transormations o the raph o. (See Eample.) 7. () = ; vertical stretch b a actor o ollowed b a translation 1 unit up 8. () = ; translation units down ollowed b a vertical shrink b a actor o 1 9. () = ; translation units to the riht ollowed b a horizontal stretch b a actor o 0. () = ; relection in the -ais ollowed b a translation units to the riht 1. () =. () = ERROR ANALYSIS In Eercises and, identi and correct the error in writin the unction whose raph represents the indicated transormations o the raph o. ANALYZING RELATIONSHIPS In Eercises, match the raph o the transormation o with the correct equation shown. Eplain our reasonin.. () = ; translation units to the riht ollowed b a translation units up () = () = ; translation units down ollowed b a vertical stretch b a actor o 5 () = MAKING AN ARGUMENT Your riend claims that when writin a unction whose raph represents a combination o transormations, the order is not important. Is our riend correct? Justi our answer. A. = () B. = () C. = ( + ) D. = () + Section 1. Transormations o Linear and Absolute Value Functions

8 . MODELING WITH MATHEMATICS Durin a recent period o time, bookstore sales have been declinin. The sales (in billions o dollars) can be modeled b the unction (t) = 7 5 t + 17., where t is the number o ears since 00. Suppose sales decreased at twice the rate. How can ou transorm the raph o to model the sales? Eplain how the sales in 010 are aected b this chane. (See Eample 5.) MATHEMATICAL CONNECTIONS For Eercises 7 0, describe the transormation o the raph o to the raph o. Then ind the area o the shaded trianle. 7. () = 8. () =. HOW DO YOU SEE IT? Consider the raph o () = m + b. Describe the eect each transormation has on the slope o the line and the intercepts o the raph. a. Relect the raph o in the -ais. b. Shrink the raph o verticall b a actor o 1. c. Stretch the raph o horizontall b a actor o. 9. () = + 0. () = 5. REASONING The raph o () = + is a relection in the -ais, vertical stretch b a actor o, and a translation units down o the raph o its parent unction. Choose the correct order or the transormations o the raph o the parent unction to obtain the raph o. Eplain our reasonin. Maintainin Mathematical Proicienc Evaluate the unction or the iven value o.. () = + ; = 7. () = 1; = 1 8. () = + ; = 5 9. () = ; = 1 Create a scatter plot o the data. 1. ABSTRACT REASONING The unctions () = m + b and () = m + c represent two parallel lines. a. Write an epression or the vertical translation o the raph o to the raph o. b. Use the deinition o slope to write an epression or the horizontal translation o the raph o to the raph o.. THOUGHT PROVOKING You are plannin a cross-countr biccle trip o 0 miles. Your distance d (in miles) rom the halwa point can be modeled b d = 7 0, where is the time (in das) and = 0 represents June 1. Your plans are altered so that the model is now a riht shit o the oriinal model. Give an eample o how this can happen. Sketch both the oriinal model and the shited model. 5. CRITICAL THINKING Use the correct value 0,, or 1 with a, b, and c so the raph o () = a b + c is a relection in the -ais ollowed b a translation one unit to the let and one unit up o the raph o () = + 1. Eplain our reasonin. Reviewin what ou learned in previous rades and lessons () () Chapter 1 Linear Functions, Linear Sstems, and Matrices