1 Linear Functions. SEE the Big Idea. Pizza Shop (p. 34) Prom (p. 23) Café Expenses (p. 16) Dirt Bike (p. 7) Swimming (p. 10)

Transcription

1 1 Linear Functions 1.1 Parent Functions and Transormations 1. Transormations o Linear and Absolute Value Functions 1. Modelin with Linear Functions 1. Solvin Linear Sstems Pizza Shop (p. ) Prom (p. ) SEE the Bi Idea Caé Epenses (p. 1) Dirt Bike (p. 7) Swimmin (p. 10)

2 Maintainin Mathematical Proicienc Evaluatin Epressions Eample 1 Evaluate the epression ( ). ( ) = (9 ) Evaluate the power within parentheses. Evaluate. = 18 Multipl within parentheses. = Divide. = 1 Subtract ( + ) ( + ). 1 ( + 18) Transormations o Fiures Eample Relect the black rectanle in the -ais. Then translate the new rectanle 5 units to the let and 1 unit down. A B Move each verte 5 units let and 1 unit down. D C D C D C Take the opposite o each -coordinate. A B A B Graph the transormation o the iure. 7. Translate the rectanle 1 unit riht and units up. 8. Relect the trianle in the -ais. Then translate units let. 9. Translate the trapezoid units down. Then relect in the -ais ABSTRACT REASONING Give an eample to show wh the order o operations is important when evaluatin a numerical epression. Is the order o transormations o iures important? Justi our answer. Dnamic Solutions available at BiIdeasMath.com 1

3 Mathematical Practices Mathematicall proi cient students use technoloical tools to eplore concepts. Usin a Graphin Calculator Core Concept Standard and Square Viewin Windows A tpical screen on a raphin calculator has a heiht-to-width ratio o to. This means that when ou view a raph usin the standard viewin window o 10 to 10 (on each ais), the raph will not be shown in its true perspective. WINDOW Xmin=-10 Xma=10 Xscl=1 Ymin=-10 Yma=10 Yscl=1 This is the standard viewin window. To view a raph in its true perspective, ou need to chane to a square viewin window, where the tick marks on the -ais are spaced the same as the tick marks on the -ais. WINDOW Xmin=-9 Xma=9 Xscl=1 Ymin=- Yma= Yscl=1 This is a square viewin window. Usin a Graphin Calculator Use a raphin calculator to raph =. 10 In the standard viewin window, notice that the tick marks on the -ais are closer toether than those on the -ais. This implies that the raph is not shown in its true perspective. In a square viewin window, notice that the tick marks on both aes have the same spacin. This implies that the raph is shown in its true perspective. Monitorin Proress Use a raphin calculator to raph the equation usin the standard viewin window and a square viewin window. Describe an dierences in the raphs. 1. =. = +. = + 1. = 1 5. =. = This is the raph in the standard viewin window. This is the raph in a square viewin window. Determine whether the viewin window is square. Eplain , , , 8 10., 11. 5, 1., Chapter 1 Linear Functions

4 1.1 Parent Functions and Transormations Essential Question What are the characteristics o some o the basic parent unctions? Identiin Basic Parent Functions JUSTIFYING CONCLUSIONS To be proicient in math, ou need to justi our conclusions and communicate them clearl to others. Work with a partner. Graphs o eiht basic parent unctions are shown below. Classi each unction as constant, linear, absolute value, quadratic, square root, cubic, reciprocal, or eponential. Justi our reasonin. a. b. c. d. e... h. Communicate Your Answer. What are the characteristics o some o the basic parent unctions?. Write an equation or each unction whose raph is shown in Eploration 1. Then use a raphin calculator to veri that our equations are correct. Section 1.1 Parent Functions and Transormations

5 1.1 Lesson What You Will Learn Core Vocabular parent unction, p. transormation, p. 5 translation, p. 5 relection, p. 5 vertical stretch, p. vertical shrink, p. Previous unction domain rane slope scatter plot Identi amilies o unctions. Describe transormations o parent unctions. Describe combinations o transormations. Identiin Function Families Functions that belon to the same amil share ke characteristics. The parent unction is the most basic unction in a amil. Functions in the same amil are transormations o their parent unction. Core Concept Parent Functions Famil Constant Linear Absolute Value Quadratic Rule () = 1 () = () = () = Graph Domain All real numbers All real numbers All real numbers All real numbers Rane = 1 All real numbers 0 0 LOOKING FOR STRUCTURE You can also use unction rules to identi unctions. The onl variable term in is an -term, so it is an absolute value unction. Identiin a Function Famil Identi the unction amil to which belons. Compare the raph o to the raph o its parent unction. The raph o is V-shaped, so is an absolute value unction. The raph is shited up and is narrower than the raph o the parent absolute value unction. The domain o each unction is all real numbers, but the rane o is 1 and the rane o the parent absolute value unction is 0. () = + 1 Monitorin Proress Help in Enlish and Spanish at BiIdeasMath.com 1. Identi the unction amil to which belons. Compare the raph o to the raph o its parent unction. 1 () = ( ) Chapter 1 Linear Functions

6 Describin Transormations A transormation chanes the size, shape, position, or orientation o a raph. A translation is a transormation that shits a raph horizontall and/or verticall but does not chane its size, shape, or orientation. REMEMBER The slope-intercept orm o a linear equation is = m + b, where m is the slope and b is the -intercept. Graphin and Describin Translations Graph () = and its parent unction. Then describe the transormation. The unction is a linear unction with a slope o 1 and a -intercept o. So, draw a line throuh the point (0, ) with a slope o 1. The raph o is units below the raph o the parent linear unction. () = So, the raph o () = is a vertical translation units down o the raph o the parent linear unction. (0, ) () = A relection is a transormation that lips a raph over a line called the line o rel ection. A relected point is the same distance rom the line o relection as the oriinal point but on the opposite side o the line. REMEMBER The unction p() = is written in unction notation, where p() is another name or. Graphin and Describin Relections Graph p() = and its parent unction. Then describe the transormation. The unction p is a quadratic unction. Use a table o values to raph each unction. = = () = p() = The raph o p is the raph o the parent unction lipped over the -ais. So, p() = is a relection in the -ais o the parent quadratic unction. Monitorin Proress Help in Enlish and Spanish at BiIdeasMath.com Graph the unction and its parent unction. Then describe the transormation.. () = +. h() = ( ). n() = Section 1.1 Parent Functions and Transormations 5

7 Another wa to transorm the raph o a unction is to multipl all o the -coordinates b the same positive actor (other than 1). When the actor is reater than 1, the transormation is a vertical stretch. When the actor is reater than 0 and less than 1, it is a vertical shrink. Graphin and Describin Stretches and Shrinks Graph each unction and its parent unction. Then describe the transormation. a. () = b. h() = 1 REASONING ABSTRACTLY To visualize a vertical stretch, imaine pullin the points awa rom the -ais. To visualize a vertical shrink, imaine pushin the points toward the -ais. a. The unction is an absolute value unction. Use a table o values to raph the unctions. () = = = The -coordinate o each point on is two times the -coordinate o the correspondin point on the parent unction. () = So, the raph o () = is a vertical stretch o the raph o the parent absolute value unction. b. The unction h is a quadratic unction. Use a table o values to raph the unctions. = = 1 () = h() = Chapter 1 Linear Functions The -coordinate o each point on h is one-hal o the -coordinate o the correspondin point on the parent unction. So, the raph o h() = 1 is a vertical shrink o the raph o the parent quadratic unction. Monitorin Proress Help in Enlish and Spanish at BiIdeasMath.com Graph the unction and its parent unction. Then describe the transormation. 5. () =. h() = 7. c() = 0.

8 Combinations o Transormations You can use more than one transormation to chane the raph o a unction. Describin Combinations o Transormations Use a raphin calculator to raph () = + 5 and its parent unction. Then describe the transormations. 8 The unction is an absolute value unction. The raph shows that () = + 5 is a relection in the -ais ollowed b a translation 5 units let and units down o the raph o the parent absolute value unction Modelin with Mathematics Time (seconds), Heiht (eet), The table shows the heiht o a dirt bike seconds ater jumpin o a ramp. What tpe o unction can ou use to model the data? Estimate the heiht ater 1.75 seconds. 1. Understand the Problem You are asked to identi the tpe o unction that can model the table o values and then to ind the heiht at a speciic time.. Make a Plan Create a scatter plot o the data. Then use the relationship shown in the scatter plot to estimate the heiht ater 1.75 seconds.. Solve the Problem Create a scatter plot. 0 The data appear to lie on a curve that resembles a quadratic unction. Sketch the curve. So, ou can model the data with a quadratic unction. The raph shows that the heiht is about 15 eet ater 1.75 seconds Look Back To check that our solution is reasonable, analze the values in the table. Notice that the heihts decrease ater 1 second. Because 1.75 is between 1.5 and, the heiht must be between 0 eet and 8 eet. 8 < 15 < Monitorin Proress Help in Enlish and Spanish at BiIdeasMath.com Use a raphin calculator to raph the unction and its parent unction. Then describe the transormations. 8. h() = d() = ( 5) The table shows the amount o uel in a chainsaw over time. What tpe o unction can ou use to model the data? When will the tank be empt? Time (minutes), Fuel remainin (luid ounces), Section 1.1 Parent Functions and Transormations 7

9 1.1 Eercises Dnamic Solutions available at BiIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The unction () = is the o () =.. DIFFERENT WORDS, SAME QUESTION Which is dierent? Find both answers. What are the vertices o the iure ater a relection in the -ais, ollowed b a translation units riht? What are the vertices o the iure ater a translation units up and units riht? What are the vertices o the iure ater a translation units riht, ollowed b a relection in the -ais? What are the vertices o the iure ater a translation units up, ollowed b a relection in the -ais? Monitorin Proress and Modelin with Mathematics In Eercises, identi the unction amil to which belons. Compare the raph o to the raph o its parent unction. (See Eample 1.).. () = () = 5 () = + () = 7. MODELING WITH MATHEMATICS At 8:00 a.m., the temperature is F. The temperature increases F each hour or the net 7 hours. Graph the temperatures over time t (t = 0 represents 8:00 a.m.). What tpe o unction can ou use to model the data? Eplain. 8. MODELING WITH MATHEMATICS You purchase a car rom a dealership or $10,000. The trade-in value o the car each ear ater the purchase is iven b the unction () = 10, What tpe o unction models the trade-in value? In Eercises 9 18, raph the unction and its parent unction. Then describe the transormation. (See Eamples and.) 9. () = () = 11. () = 1 1. h() = ( + ) 1. () = 5 1. () = h() = 1. () = 17. () = 18. () = 8 Chapter 1 Linear Functions

10 In Eercises 19, raph the unction and its parent unction. Then describe the transormation. (See Eample.) 19. () = 1 0. () = 1. () =. h() = 1. h() =. () = 5. h() =. () = 1 In Eercises 7, use a raphin calculator to raph the unction and its parent unction. Then describe the transormations. (See Eample 5.) 7. () = + 8. h() = h() = 1 0. () = () = 1. () =. () = ( + ) + 1. () = 1 1 ERROR ANALYSIS In Eercises 5 and, identi and correct the error in describin the transormation o the parent unction The raph is a relection in the -ais and a vertical shrink o the parent quadratic unction. The raph is a translation units riht o the parent absolute value unction, so the unction is () = +. MATHEMATICAL CONNECTIONS In Eercises 7 and 8, ind the coordinates o the iure ater the transormation. 7. Translate units 8. Relect in the -ais. down. B A C D USING TOOLS In Eercises 9, identi the unction amil and describe the domain and rane. Use a raphin calculator to veri our answer. 9. () = h() = + A C 1. () = +. () = () = 5. () = + B 5. MODELING WITH MATHEMATICS The table shows the speeds o a car as it travels throuh an intersection with a stop sin. What tpe o unction can ou use to model the data? Estimate the speed o the car when it is 0 ards past the intersection. (See Eample.) Displacement rom sin (ards), Speed (miles per hour), THOUGHT PROVOKING In the same coordinate plane, sketch the raph o the parent quadratic unction and the raph o a quadratic unction that has no -intercepts. Describe the transormation(s) o the parent unction. 7. USING STRUCTURE Graph the unctions () = and () =. Are the equivalent? Eplain. Section 1.1 Parent Functions and Transormations 9

11 8. HOW DO YOU SEE IT? Consider the raphs o,, and h. h a. Does the raph o represent a vertical stretch or a vertical shrink o the raph o? Eplain our reasonin. b. Describe how to transorm the raph o to obtain the raph o h. 9. MAKING AN ARGUMENT Your riend sas two dierent translations o the raph o the parent linear unction can result in the raph o () =. Is our riend correct? Eplain. 50. DRAWING CONCLUSIONS A person swims at a constant speed o 1 meter per second. What tpe o unction can be used to model the distance the swimmer travels? I the person has a 10-meter head start, what tpe o transormation does this represent? Eplain. 51. PROBLEM SOLVING You are plain basketball with our riends. The heiht (in eet) o the ball above the round t seconds ater a shot is released rom our hand is modeled b the unction (t) = 1t + t a. Without raphin, identi the tpe o unction that models the heiht o the basketball. b. What is the value o t when the ball is released rom our hand? Eplain our reasonin. c. How man eet above the round is the ball when it is released rom our hand? Eplain. 5. MODELING WITH MATHEMATICS The table shows the batter lives o a computer over time. What tpe o unction can ou use to model the data? Interpret the meanin o the -intercept in this situation. Time (hours), Batter lie remainin, 1 80% 0% 5 0% 0% 8 0% 5. REASONING Compare each unction with its parent unction. State whether it contains a horizontal translation, vertical translation, both, or neither. Eplain our reasonin. a. () = b. () = ( 8) c. () = + + d. () = 5. CRITICAL THINKING Use the values 1, 0, 1, and in the correct bo so the raph o each unction intersects the -ais. Eplain our reasonin. a. () = + 1 b. () = c. () = + 1 d. () = Maintainin Mathematical Proicienc Reviewin what ou learned in previous rades and lessons Determine whether the ordered pair is a solution o the equation. (Skills Review Handbook) 55. () = + ; (1, ) 5. () = ; (, 5) 57. () = ; (5, ) 58. () = ; (1, 8) Find the -intercept and the -intercept o the raph o the equation. (Skills Review Handbook) 59. = 0. = = 1. = 8 10 Chapter 1 Linear Functions

12 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 11

13 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

14 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 Help in Enlish and Spanish at BiIdeasMath.com 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 1

15 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 Help in Enlish and Spanish at BiIdeasMath.com 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 1 Chapter 1 Linear Functions

16 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 Help in Enlish and Spanish at BiIdeasMath.com 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 15

17 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 1 Chapter 1 Linear Functions

18 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 17

19 . 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. 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. Maintainin Mathematical Proicienc Evaluate the unction or the iven value o. (Skills Review Handbook). () = + ; = 7. () = 1; = 1 8. () = + ; = 5 9. () = ; = 1 Create a scatter plot o the data. (Skills Review Handbook). 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

20 What Did You Learn? Core Vocabular parent unction, p. transormation, p. 5 translation, p. 5 relection, p. 5 vertical stretch, p. vertical shrink, p. Core Concepts Section 1.1 Parent Functions, p. Describin Transormations, p. 5 Section 1. Horizontal Translations, p. 1 Vertical Translations, p. 1 Relections in the -ais, p. 1 Relections in the -ais, p. 1 Horizontal Stretches and Shrinks, p. 1 Vertical Stretches and Shrinks, p. 1 Mathematical Practices 1. How can ou analze the values iven in the table in Eercise 5 on pae 9 to help ou determine what tpe o unction models the data?. Eplain how ou would round our answer in Eercise 10 on pae 1 i the etra epense is $1,500. Stud Skills Takin Control o Your Class Time 1. Sit where ou can easil see and hear the teacher, and the teacher can see ou.. Pa attention to what the teacher sas about math, not just what is written on the board.. Ask a question i the teacher is movin throuh the material too ast.. Tr to memorize new inormation while learnin it. 5. Ask or clariication i ou do not understand somethin.. Think as intensel as i ou were oin to take a quiz on the material at the end o class. 7. Volunteer when the teacher asks or someone to o up to the board. 8. At the end o class, identi concepts or problems or which ou still need clariication. 9. Use the tutorials at BiIdeasMath.com or additional help. 19

21 Quiz Identi the unction amil to which belons. Compare the raph o the unction to the raph o its parent unction. (Section 1.1) 1. 1 () = () = + 1 () = ( + 1) Graph the unction and its parent unction. Then describe the transormation. (Section 1.1). () = 5. () =. () = ( 1) 7. () = () = () = 1 Write a unction whose raph represents the indicated transormation o the raph o. (Section 1.) 10. () = + 1; translation units up 11. () = ; vertical shrink b a actor o 1 1. () = + 5 ; relection in the -ais 1. () = 1 ; translation units let Write a unction whose raph represents the indicated transormations o the raph o. (Section 1.) 1. Let be a translation units down and a horizontal shrink b a actor o o the raph o () =. 15. Let be a translation 9 units down ollowed b a relection in the -ais o the raph o () =. 1. Let be a relection in the -ais and a vertical stretch b a actor o ollowed b a translation 7 units down and 1 unit riht o the raph o () =. 17. Let be a translation 1 unit down and units let ollowed b a vertical shrink b a actor o 1 o the raph o () =. 18. The table shows the total distance a new car travels each month ater it is purchased. What tpe o unction can ou use to model the data? Estimate the mileae ater 1 ear. (Section 1.1) Time (months), Distance (miles), , The total cost o an annual pass plus campin or das in a National Park can be modeled b the unction () = Senior citizens pa hal o this price and receive an additional $0 discount. Describe how to transorm the raph o to model the total cost or a senior citizen. What is the total cost or a senior citizen to o campin or three das? (Section 1.) 0 Chapter 1 Linear Functions

22 1. Modelin with Linear Functions Essential Question How can ou use a linear unction to model and analze a real-lie situation? Modelin with a Linear Function MODELING WITH MATHEMATICS To be proicient in math, ou need to routinel interpret our results in the contet o the situation. Work with a partner. A compan purchases a copier or $1,000. The spreadsheet shows how the copier depreciates over an 8-ear period. a. Write a linear unction to represent the value V o the copier as a unction o the number t o ears. b. Sketch a raph o the unction. Eplain wh this tpe o depreciation is called straiht line depreciation. c. Interpret the slope o the raph in the contet o the problem. Modelin with Linear Functions Work with a partner. Match each description o the situation with its correspondin raph. Eplain our reasonin A Year, t B Value, V $1,000 $10,750 $9,500 $8,50 $7,000 $5,750 $,500 $,50 $,000 a. A person ives $0 per week to a riend to repa a $00 loan. b. An emploee receives $1.50 per hour plus $ or each unit produced per hour. c. A sales representative receives $0 per da or ood plus $0.55 or each mile driven. d. A computer that was purchased or $750 depreciates $100 per ear. A. B C. D Communicate Your Answer. How can ou use a linear unction to model and analze a real-lie situation?. Use the Internet or some other reerence to ind a real-lie eample o straiht line depreciation. a. Use a spreadsheet to show the depreciation. b. Write a unction that models the depreciation. c. Sketch a raph o the unction. Section 1. Modelin with Linear Functions 1

23 1. Lesson Core Vocabular line o it, p. line o best it, p. 5 correlation coeicient, p. 5 Previous slope slope-intercept orm point-slope orm scatter plot What You Will Learn Write equations o linear unctions usin points and slopes. Find lines o it and lines o best it. Writin Linear Equations Core Concept Writin an Equation o a Line Given slope m and -intercept b Given slope m and a point ( 1, 1 ) Use slope-intercept orm: = m + b Use point-slope orm: 1 = m( 1 ) Given points ( 1, 1 ) and (, ) First use the slope ormula to ind m. Then use point-slope orm with either iven point. Distance (miles) Asteroid 01 DA1 1 8 (5, ) 0 0 Time (seconds) REMEMBER An equation o the orm = m indicates that and are in a proportional relationship. Writin a Linear Equation rom a Graph The raph shows the distance Asteroid 01 DA1 travels in seconds. Write an equation o the line and interpret the slope. The asteroid came within 17,00 miles o Earth in Februar, 01. About how lon does it take the asteroid to travel that distance? From the raph, ou can see the slope is m = =.8 and the -intercept is b = 0. 5 Use slope-intercept orm to write an equation o the line. = m + b Slope-intercept orm = Substitute.8 or m and 0 or b. The equation is =.8. The slope indicates that the asteroid travels.8 miles per second. Use the equation to ind how lon it takes the asteroid to travel 17,00 miles. 17,00 =.8 Substitute 17,00 or. 58 Divide each side b.8. Because there are 00 seconds in 1 hour, it takes the asteroid about 1 hour to travel 17,00 miles. Monitorin Proress 1. The raph shows the remainin balance on a car loan ater makin monthl paments. Write an equation o the line and interpret the slope and -intercept. What is the remainin balance ater paments? Help in Enlish and Spanish at BiIdeasMath.com Balance (thousands o dollars) 18 1 (10, 15) (0, 18) Car Loan Number o paments Chapter 1 Linear Functions

24 Modelin with Mathematics Lakeside Inn Number o students, Total cost, 100 $ $ $ $00 00 $700 Two prom venues chare a rental ee plus a ee per student. The table shows the total costs or dierent numbers o students at Lakeside Inn. The total cost (in dollars) or students at Sunview Resort is represented b the equation = Which venue chares less per student? How man students must attend or the total costs to be the same? 1. Understand the Problem You are iven an equation that represents the total cost at one venue and a table o values showin total costs at another venue. You need to compare the costs.. Make a Plan Write an equation that models the total cost at Lakeside Inn. Then compare the slopes to determine which venue chares less per student. Finall, equate the cost epressions and solve to determine the number o students or which the total costs are equal.. Solve the Problem First ind the slope usin an two points rom the table. Use ( 1, 1 ) = (100, 1500) and (, ) = (15, 1800). m = = = 00 5 = 1 Write an equation that represents the total cost at Lakeside Inn usin the slope o 1 and a point rom the table. Use ( 1, 1 ) = (100, 1500). 1 = m( 1 ) Point-slope orm 1500 = 1( 100) Substitute or m, 1, and = Distributive Propert = Add 1500 to each side. Equate the cost epressions and solve = Set cost epressions equal. 00 = Combine like terms. 150 = Divide each side b. Comparin the slopes o the equations, Sunview Resort chares $10 per student, which is less than the $1 per student that Lakeside Inn chares. The total costs are the same or 150 students.. Look Back Notice that the table shows the total cost or 150 students at Lakeside Inn is $100. To check that our solution is correct, veri that the total cost at Sunview Resort is also $100 or 150 students. = 10(150) + 00 Substitute 150 or. = 100 Simpli. Monitorin Proress Help in Enlish and Spanish at BiIdeasMath.com. WHAT IF? Maple Ride chares a rental ee plus a $10 ee per student. The total cost is $1900 or 10 students. Describe the number o students that must attend or the total cost at Maple Ride to be less than the total costs at the other two venues. Use a raph to justi our answer. Section 1. Modelin with Linear Functions

25 Findin Lines o Fit and Lines o Best Fit Data do not alwas show an eact linear relationship. When the data in a scatter plot show an approimatel linear relationship, ou can model the data with a line o it. Core Concept Findin a Line o Fit Step 1 Create a scatter plot o the data. Step Sketch the line that most closel appears to ollow the trend iven b the data points. There should be about as man points above the line as below it. Step Choose two points on the line and estimate the coordinates o each point. These points do not have to be oriinal data points. Step Write an equation o the line that passes throuh the two points rom Step. This equation is a model or the data. Findin a Line o Fit Femur lenth, Heiht, The table shows the emur lenths (in centimeters) and heihts (in centimeters) o several people. Do the data show a linear relationship? I so, write an equation o a line o it and use it to estimate the heiht o a person whose emur is 5 centimeters lon. Step 1 Create a scatter plot o the data. The data show a linear relationship. Step Sketch the line that most closel appears to it the data. One possibilit is shown. Step Choose two points on the line. For the line shown, ou miht choose (0, 170) and (50, 195). Step Write an equation o the line. First, ind the slope. m = = = 5 10 =.5 Use point-slope orm to write an equation. Use ( 1, 1 ) = (0, 170). 1 = m( 1 ) Point-slope orm 170 =.5( 0) Substitute or m, 1, and = = Use the equation to estimate the heiht o the person. Heiht (centimeters) Distributive Propert Add 170 to each side. Human Skeleton (0, 170) (50, 195) 50 Femur lenth (centimeters) =.5(5) + 70 Substitute 5 or. = Simpli. The approimate heiht o a person with a 5-centimeter emur is centimeters. Chapter 1 Linear Functions

26 The line o best it is the line that lies as close as possible to all o the data points. Man technolo tools have a linear reression eature that ou can use to ind the line o best it or a set o data. The correlation coeicient, denoted b r, is a number rom 1 to 1 that measures how well a line its a set o data pairs (, ). When r is near 1, the points lie close to a line with a positive slope. When r is near 1, the points lie close to a line with a neative slope. When r is near 0, the points do not lie close to an line. Usin a Graphin Calculator humerus emur Use the linear reression eature on a raphin calculator to ind an equation o the line o best it or the data in Eample. Estimate the heiht o a person whose emur is 5 centimeters lon. Compare this heiht to our estimate in Eample. Step 1 Enter the data into two lists. L1 L L1(1)=0 L Step Use the linear reression eature. The line o best it is = The value o r is close to 1. LinRe =a+b a= b= r = r= Step Graph the reression equation with the scatter plot. 10 Step Use the trace eature to ind the value o when = =. + 5 ATTENDING TO PRECISION Be sure to analze the data values to help ou select an appropriate viewin window or our raph X=5 Y=15 10 The approimate heiht o a person with a 5-centimeter emur is 15 centimeters. This is less than the estimate ound in Eample. Monitorin Proress Help in Enlish and Spanish at BiIdeasMath.com. The table shows the humerus lenths (in centimeters) and heihts (in centimeters) o several emales. 55 Humerus lenth, Heiht, a. Do the data show a linear relationship? I so, write an equation o a line o it and use it to estimate the heiht o a emale whose humerus is 0 centimeters lon. b. Use the linear reression eature on a raphin calculator to ind an equation o the line o best it or the data. Estimate the heiht o a emale whose humerus is 0 centimeters lon. Compare this heiht to our estimate in part (a). Section 1. Modelin with Linear Functions 5

27 1. Eercises Dnamic Solutions available at BiIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The linear equation = 1 + is written in orm.. VOCABULARY A line o best it has a correlation coeicient o What can ou conclude about the slope o the line? Monitorin Proress and Modelin with Mathematics In Eercises 8, use the raph to write an equation o the line and interpret the slope. (See Eample 1.). Tippin. Gasoline Tank Tip (dollars) (10, ) Cost o meal (dollars) Fuel (allons) 8 90 (90, 9) Distance (miles) 9. MODELING WITH MATHEMATICS Two newspapers chare a ee or placin an advertisement in their paper plus a ee based on the number o lines in the advertisement. The table shows the total costs or dierent lenth advertisements at the Dail Times. The total cost (in dollars) or an advertisement that is lines lon at the Greenville Journal is represented b the equation = + 0. Which newspaper chares less per line? How man lines must be in an advertisement or the total costs to be the same? (See Eample.) Dail Times 5. Savins Account Balance (dollars) (, 00) Time (weeks) 7. Tpin Speed Words tped (1, 55) (, 15) 0 0 Time (minutes). Tree Growth 8. Tree heiht (eet) Volume (cubic eet) 0 0 Ae (ears) Swimmin Pool (, 00) (5, 180) 0 0 Time (hours) Number o lines, Total cost, PROBLEM SOLVING While on vacation in Canada, ou notice that temperatures are reported in derees Celsius. You know there is a linear relationship between Fahrenheit and Celsius, but ou oret the ormula. From science class, ou remember the reezin point o water is 0 C or F, and its boilin point is 100 C or 1 F. a. Write an equation that represents derees Fahrenheit in terms o derees Celsius. b. The temperature outside is C. What is this temperature in derees Fahrenheit? c. Rewrite our equation in part (a) to represent derees Celsius in terms o derees Fahrenheit. d. The temperature o the hotel pool water is 8 F. What is this temperature in derees Celsius? Chapter 1 Linear Functions