Exploration 1-1a: Paper Cup Analysis

Transcription

1 Eploration Masters The Eplorations provide the heart o the cooperative learnin phase o the course. The provide ou with concrete objectives that lead students to discover various concepts or techniques. For instance, Eploration 1-3a (associated with Section 1-3 in the tet) lets students discover raphicall, b pencil-and-paper plottin, the eects o dilations and translations. Other Eplorations allow ou to ollow up on what students have learned. For instance, Eploration 1-3d (the ourth Eploration keed to Section 1-3) lets students show quickl that the have mastered the topic. The latter kind o Eploration can be used with students workin in roups or individuall. The Instructor s Guide tells was in which the Eplorations ma be used and about how lon each takes. Most require 20 0 minutes, thus leavin class time or ou to ollow up on what students have discovered. Do not minimize the importance o this closure activit. Just because students have discovered somethin does not mean the know it. In the eample above, ou must still have classroom discussion and assin homework to make sure students understand these transormations. You miht liken this approach to crossin a turbulent stream. The Eplorations provide a rope, thrown across the stream and anchored on the other side, with which the students can pull themselves throuh the rouh waters without as much daner o bein swept awa. Once students know where the are headed, the are more willin and read to absorb the riors o ettin there. It is recommended that ou intersperse the Eplorations with other modes o instruction. Althouh it is possible to teach practicall the whole course usin just the Eplorations, this mode o instruction would have some o the same disadvantaes as the lecture-onl mode. Students can become tired o just another worksheet! When time is short, eel ree to use the lecture mode. A well-prepared lecture is probabl the most time-eicient means o convein inormation, particularl i ou pause to let students suppl some o the details. We owe it to our students to prepare them or whatever modes o instruction the are likel to encounter in uture mathematics courses.

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3 Eploration 1-1a: Paper Cup Analsis Objective: Find an equation or calculatin the heiht o a stack o paper cups. 1. Obtain several paper cups o the same kind. Measure the heiht o stacks containin,, 3, 2, and just 1 cup. Record the heihts to the nearest 0.1 cm. State what kind o cup ou used. Kind: Number cm Plot the points in the table on this raph paper. Show the scale ou are usin on the vertical ais.. Let be the number o cups in a stack, and let be the heiht o the stack, measured in centimeters. Write an equation or as a unction o. 6. What is the name o the kind o unction whose equation ou wrote in Problem? 7. Show that our equation in Problem ives a heiht close to the measured heiht or a stack o 3 cups. Heiht (cm) 8. Use our equation to predict the heiht o a stack o 3 cups. Round the answer to 1 decimal place Number o cups 3. On averae, b how much did the stack heiht increase or each cup ou added? Show how ou ot our answer. 9. What are the names o the processes o calculatin a value within the rane o the data, as in Problem 7, and outside the rane o data, as in Problem 8? Within: Outside:. A cup manuacturer wants to packae this kind o cup in boes that are cm lon. What is the maimum number o cups the bo could hold? Show how ou et our answer.. How tall would ou epect a -cup stack to be? Show how ou et our answer. Would this be twice as tall as a -cup stack? 11. What did ou learn as a result o doin this Eploration that ou did not know beore? Precalculus with Trionometr: Instructor s Resource Book, Volume 1 Eploration Masters / Ke Curriculum Press

4 Eploration 1-2a: Names o Functions Objective: Recall the names o certain kinds o unctions. 1. () =2 +3is the equation or a linear unction.. () = 2 is the equation or an inverse variation Plot the raph and sketch the result here. Give a power unction. Plot the raph or >0and sketch reason or the name linear. the result. Wh do the words varies inversel with make sense or this unction? Wh can the unction be called a power unction? 2. () = 2 D 6 +is the equation or a quadratic unction. Plot the raph and sketch the result. Eplain how the word quadratic is related to the word quadranle. 6. () = D 3 D is the equation o this quartic unction. Wh do ou think the name quartic is used or this unction? Use our rapher to ind the larest value o at which the raph crosses the -ais. = () 3. () =3 0.7 is the equation or a power unction. Plot the raph and sketch the result. Wh do ou think it is called a power unction? 7. () = D D 3 is the equation o a rational unction. Plot the raph and sketch the result. Wh do ou think it is called a rational unction? What happens to the raph at = 3?. () =3R0.7 is the equation or an eponential unction. Plot the raph and sketch the result. How does an eponential unction dier rom a power unction alebraicall? raphicall? 8. What did ou learn as a result o doin this Eploration that ou did not know beore? 32 / Eploration Masters Precalculus with Trionometr: Instructor s Resource Book, Volume Ke Curriculum Press

5 Eploration 1-2b: Restricted Domains and Boolean Variables Objective: Use Boolean variables to plot raphs o unctions in a restricted domain. Weiht Above and Below Earth s Surace: I there were a hole all the wa throuh Earth and it were possible or ou to o throuh it, ou would be weihtless at the center o Earth. This is because ravit would pull ou with the same orce in ever direction. Between the center and the surace, our weiht would var directl with the distance rom the center. 1. Kevin Vader (Darth s son) weihs 200 pounds on the surace o Earth. Earth s radius is about 000 miles. Write the particular equation or Kevin s weiht as a unction o distance rom the center when he is below the surace. 2. Above the surace, our weiht varies inversel with the square o our distance rom the center because the pull o ravit decreases as ou recede rom Earth. Write the particular equation or Kevin s weiht as a unction o distance rom the center when he is above the surace. The linear raph should stop at =000miles, and the inverse square unction raph should start at = 000 miles, the distance rom the center to the surace. Weiht (lb) ,000 Distance (mi). The Boolean variable ( 000) equals 1 i is reater than or equal to 000 and 0 otherwise. Chane our equation or 2 b dividin it b this Boolean variable, then replot the raph. Eplain wh the rapher plots the same raph or 2 when 000 but plots nothin when < Plot the raphs rom Problems 1 and 2 as 1 and 2. Does our raph aree with this one? Where do the two raphs cross each other? Weiht (lb) ,000 Distance (mi). What Boolean variable could ou divide 1 b so that the rapher plots it onl between 0 and 000? Chane the equation or 1. Does the complete raph now match the one above Problem? 6. What word describes the set o -values or which a particular unction is deined? What word describes the correspondin set o -values? -values: -values: 7. What did ou learn as a result o doin this Eploration that ou did not know beore? Precalculus with Trionometr: Instructor s Resource Book, Volume 1 Eploration Masters / Ke Curriculum Press

6 Eploration 1-3a: Translations and Dilations, Numericall Objective: B calculatin values and plottin points, discover the eect on a unction raph o addin and multiplin b constants. 1. The table shows values o a pre-imae unction = (). The raph o is a set o line sements connectin the points, shown dashed in the iure. Find values o the imae unction () = () +3. For instance, (D2)=2+3=. Plot the raph o this transormed unction. () () D2 2 D D The transormation in Problem 1 is a vertical translation b 3 units. Give the meanin o a vertical translation.. Use the values o () in Problem 1 to make a table o values o a new imae unction, () =2 (). For instance, (D1) = 2 (D1) = 2 3 =6. Plot the imae o this transormed unction. D2 D () H 2() 6. The transormation in Problem is a vertical dilation b a actor o 2. Give the meanin o a vertical dilation, and eplain how it diers rom a vertical translation. 3. Use the values o () in Problem 1 to make a table o values o a new imae unction, () = ( D 3). For instance, (1) = (1 D 3)= (D2) = 2. Plot the imae o this transormed unction. () H ( D 3) Describe the transormation in Problem Use the values o () in Problem 1 to make a table o values o a new imae unction, () = ( 1 2 ). For instance, ( ) 1 (D2) = 2 (D2) = (D1) = 3 Plot the imae o this transormed unction. D D () H ( 1 2 ) 8. The transormation in Problem 7 is a horizontal dilation. B what actor is the raph dilated? How 1 is that actor related to the 2 in ( 1 2 )? 9. What did ou learn as a result o doin this Eploration that ou did not know beore? 3 / Eploration Masters Precalculus with Trionometr: Instructor s Resource Book, Volume Ke Curriculum Press

7 Eploration 1-3b: Translations and Dilations, Alebraicall Objective: Find the eect on a unction raph o addin and multiplin b constants. 1. The raph below shows the pre-imae unction () = Plot this raph as 1 on our rapher. Use the window shown, usin GRID ON ormat. 6. Deactivate 3 rom Problem. Then plot Sketch the result here. =3 (). 2. Plot the raph o 2 = () +3. Sketch the result on the raph in Problem The transormation in Problem 2 is a vertical translation o 3. Give the meanin o a vertical translation. 7. The transormation in Problem 6 is a vertical dilation b a actor o 3. Give the meanin o a vertical dilation, and eplain how it diers rom a vertical translation. 8. Deactivate rom Problem 7. Then plot = (3). Sketch the result here.. Deactivate 2 rom Problem 2. Then plot 3 = ( D 3). Sketch the result here. 9. The transormation in Problem 8 is a horizontal dilation. B what actor is the raph dilated?. What words describe the transormation in Problem?. What did ou learn as a result o doin this Eploration that ou did not know beore? Precalculus with Trionometr: Instructor s Resource Book, Volume 1 Eploration Masters / Ke Curriculum Press

8 Eploration 1-3c: Transormations rom Graphs Objective: Given the parent and transormed raphs, identi the transormation. Identi the transormation o (dotted) to et (solid). 1. Verball: Equation: () =. Verball: Equation: () =. Verball: 2. Verball: Equation: () = Equation: () = Graphs coincide. Graphs coincide. 6. Verball: 3. Verball: Equation: () = Equation: () = 7. What did ou learn as a result o doin this Eploration that ou did not know beore? 36 / Eploration Masters Precalculus with Trionometr: Instructor s Resource Book, Volume Ke Curriculum Press

9 Eploration 1-3d: Transormation Review Objective: Given the parent and transormed raphs, identi the transormation and conirm b rapher. 1. The iure shows the raph o () =D in the domain D1. Duplicate this raph on our rapher. Restrict the domain b dividin b the Boolean variable ( D1 and ). Use GRID ON ormat to et the dots. For Problems 2 6, identi the transormation o (dotted) to et (solid), and conirm b rapher. 2. Verball: Equation: () = Check: 3. Verball: Equation: () = Check:. Verball: Equation: () = Check:. Verball: Equation: () = Check: 6. Verball: and Equation: () = Check: 7. What did ou learn as a result o doin this Eploration that ou did not know beore? Precalculus with Trionometr: Instructor s Resource Book, Volume 1 Eploration Masters / Ke Curriculum Press

10 Eploration 1-a: Composition o Functions Objective: Find the composition o one unction with another. 1. The iure shows two linear unctions, and. Write the domain and rane o each unction. : Domain: Rane: : Domain: Rane: Read values o () rom the raph and write them in this table. I the value o is out o the domain, write none = () () 1 = () 3. The smbol (()) is read o o. It means ind the value o () irst, and then ind o the answer. For instance, () = 1.. So (()) = (1.) =.. Put another column into the table or values o (()). Write none where appropriate.. Show in the table an instance where () is deined but (()) is not deined.. Plot the values o (()) on the iure in Problem 1. I the points do not lie in a straiht line, o back and check our work. 6. The unction in Problem is called the composition o with, which can be written. What are the domain and rane o? Domain: Rane: 7. Find equations or unctions and. 8. Enter in our rapher the and equations as 1 and 2, respectivel. Use Boolean variables to make the unctions have the proper domains. Then plot the raphs. Does the result aree with the iven iure? 9. Enter in 3 b enterin 1 ( 2 ()). Plot this raph. Does it aree with the raph ou drew in Problem?. B suitable alebraic operations on the equations in Problem 7, ind an equation or (()). Simpli the equation as much as possible. 11. What did ou learn as a result o doin this Eploration that ou did not know beore? 38 / Eploration Masters Precalculus with Trionometr: Instructor s Resource Book, Volume Ke Curriculum Press

11 Eploration 1-a: Inverses o Functions Objective: Find the inverse o a unction raphicall, numericall, or alebraicall, and state whether or not the inverse is a unction. Problems 1 6 reer to the linear unction =2 D. 1. Write the equation or the inverse relation b interchanin the variables. Then solve the resultin equation or in terms o. Problems 7 and 8 reer to the quadratic unction = 2 D +7, raphed here The raph shows =2 D. Plot the raph o the inverse relation here The inverse relation in Problems 1 and 2 is a unction. How can ou tell? 7. Plot the line =. Then plot the inverse o the unction b relectin the raph across this line. 8. Eplain wh the inverse o this unction is not a unction. Problems 9 and reer to the eponential unction () =2, raphed here. 1. I the equation or the unction is written as () =2 D, how could ou write the equation or the inverse unction usin the () terminolo? 1 9. Find (0), (1), (2), and (3).. Show that (3) = 1 and D1 (1) = 3. Eplain wh this is true, based on the deinition o the inverse o a unction.. Find D1 (1), D1 (2), D1 (), and D1 (8). Use these points to plot the raph o D1. 6. Plot the line =. How are the raphs o and related to this line? D1 11. What did ou learn as a result o doin this Eploration that ou did not know beore? Precalculus with Trionometr: Instructor s Resource Book, Volume 1 Eploration Masters / Ke Curriculum Press

12 Eploration 1-6a: Translation, Dilation, and Relection Objective: Show that ou know the eects o various constants on the raph o a unction. Name the transormation and sketch the raph. 1. = 1 2 () 3. = () D Transormation: Transormation:. = ( D 7) 2. ( ) 1 = 2 Transormation: Transormation: 0 / Eploration Masters Precalculus with Trionometr: Instructor s Resource Book, Volume Ke Curriculum Press

13 Eploration 1-6a: Translation, Dilation, and Relection continued. = (D) = (D1) 9. = () is an absolute value transormation. Is it a Transormation (as a dilation): transormation in the -direction or a transormation in the -direction?. Sketch the raph o = (). 6. From the results o Problem, ive another name or the transormation = (D). 7. = D () =D1 () Transormation (as a dilation): 11. Sketch the raph o = ( ). 12. What did ou learn as a result o doin this Eploration that ou did not know beore? 8. From the results o Problem 7, ive another name or the transormation = D (). Precalculus with Trionometr: Instructor s Resource Book, Volume 1 Eploration Masters / Ke Curriculum Press