Lesson How can I transform a sine graph? Transformations of y = sinx

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1 Lesson How can I transform a sine graph? Transformations of = sin Lesson Objective: Mathematical Practices: Length of Activit: Core Problems: Materials: Technolog Notes: Suggested Lesson Activit: Students will appl their understanding of transforming parent graphs to the sine and cosine functions. The will generate general equations for the famil of sine and cosine functions of the form = asin( h) + k. construct viable arguments and critique the reasoning of others, look for and make use of structure, look for and epress regularit in repeated reasoning One da (approimatel 50 minutes) Problems and (with problem if using Further Guidance) Computer and projector Dnamic tool: Transforming Functions A dnamic tool is available for use in this lesson to help students stud the effects of changing parameters in sine and cosine functions on the graphs of these functions. Graphing Tool: Transforming Functions This Internet-based tool allows students to change values of a, b, h, and k in the functions = a sin b( h) + k and = a cos b( h) + k and see the effects on the graphs. Note that students will discover period and the parameter b in subsequent lessons, so for toda s lesson, it is best to limit the use of the tool to manipulating a, h, and k. Toda students will make the (sometimes confusing) leap from representing the input of a cclic function as θ to representing it as. The will then appl their understanding of transforming parent graphs to generate a general equation for sine and cosine functions. Note: Students are not epected to develop ideas about changing the period of these functions toda. In Lessons and 7.2.3, students will develop the idea of period and will epand their general equations to allow for it. Begin b directing teams to read the lesson introduction. Have teams start discussing problem using either Reciprocal Teaching (verball) or Pairs Check (in writing). The will be looking at the meaning of and in the unit circle and the graphs of = sin, = cos, and = tan. After teams have talked for a few minutes, bring the class together for a whole-class discussion. If students have Chapter 7:Trigonometric Functions 653

2 decided that the in the cclic functions represents an angle, challenge the class b asking, Does it alwas have to represent an angle? Could there be another input for a cclic function? You can refer back to the cclic graphs that students made from the dripping pendulum in Lesson and ask the class to identif what would represent in this graph. This is a good preview of Lesson 7.2.2, for which students will graph a sinusoidal function for which the input is time. Be sure students recognize that the input for a cclic function is generall called and, for that reason, the tet transitions from using θ to using. Remind students of their etensive knowledge of transforming parent graphs. Eplain that their task toda is to use sine and cosine as new parent graphs, investigate them completel, and investigate their transformations. Then start teams on problem using a Teammates Consult. If ou prefer a more structured approach, direct them to problem If teams are stuck, ou could ask questions such as, How could ou change the equation to shift the graph up? How could ou make the graph taller? How could ou stretch the graph verticall? If some teams finish ahead of others, direct them to problem (from this lesson s homework), and challenge them to come up with several different was of writing the equation. Closure: (10 minutes) Team Strategies: Ask teams to share their general equations. If teams ideas differ, conduct a class discussion to agree upon a general equation. Use this opportunit to name each of the parameters the teams came up with: amplitude, midline (vertical shift), and horizontal shift. Use the dnamic tool Transforming Functions to reinforce the transformations of the sine function. Note: Do not change the parameter b at this point. If students ask, have them make predictions and tell them that this will be the subject of the net lesson. Save a few minutes after the discussion to allow students to add an new ideas or eplanations to their written work. As students use their graphing calculators, it can be eas for them to slip into working individuall rather than sharing what the are seeing and finding. Encourage Facilitators to ask regularl, What do ou see in the calculator? or What are ou tring to figure out? to help keep the team together. As people eplain, Recorder/Reporters can ask, How did ou enter the equation? and Show us what it looks like, to help their teammates eplain. Before the begin work, remind all students of the importance of active discussion to help them investigate completel. Homework: Problems through Notes to Self: 654 Core Connections Algebra 2

3 7.2.1 How can I transform a sine graph? Transformations of = sin In Chapter 2, ou developed epertise in investigating functions and transforming parent graphs. In this section, ou will investigate families of cclic functions and their transformations. B the end of this section, ou will be able to graph an sine or cosine equation and write the equation of an sine or cosine graph As ou have seen with man functions in this and other courses, is generall used to represent an input and is used to represent the corresponding output. B this convention, sinusoidal functions should be written = sin, = cos, and = tan. But beware! Something funn is happening. With our team, eamine the unit circle and the three graphs below. What do and represent in the unit circle? What do the represent in each of the graphs? Discuss this with our team and be prepared to share our ideas with the class. [ a: the position along the and aes; b: : the input number or angle, : the output or the height at the given angle; c: : the input number or angle, : the output or the base at the given angle; d: : the input number or angle, : the output or the slope at the given angle. ] a. b. c. d With our team, ou will appl our knowledge about transforming graphs of functions to transform the graphs of = sin and = cos and find their general equations. Your Task: As a team, investigate = sin and = cos completel. You should make graphs, find the domain and range, and label an important points or asmptotes. Then make a sketch and write an equation to demonstrate each transformation of the sine or cosine function ou can find. Finall, find a general equation for a sine and a cosine function. Be prepared to share our summar statements with the class. Chapter 7:Trigonometric Functions 655

4 What can we change in a cclic graph? Which points are important to label? How can we appl the transformations we use with other functions? Are there an new transformations that are special to the sine function? Sketch a graph of at least one ccle of = sin. Label the intercepts. Then work with our team to complete parts (a) through (c) below. a. Write an equation for each part below and sketch a graph of a function that has a parent graph of = sin, but is: [ See graphs below. ] i. Shifted 3 units up. ii. Reflected across the -ais. [ = sin() + 3 ] [ = sin() ] iii. Shifted 2 units to the right. iv. Verticall stretched. [ = sin( 2) ] [ = 2sin() ] i. ii. iii. iv. b. Which points are most important to label in a periodic function? Wh? [ Minimums, maimums, and zeros, because the indicate period, amplitude, and midline. ] c. Write a general equation for the famil of functions with a parent graph of = sin. [ At this point, students will probabl find an equation of the form = asin( h) + k. ] Further Guidance section ends here. 656 Core Connections Algebra 2

5 Imagine the graph = sin() shifted up one unit. a. Sketch what it would look like. [ See graph at right. ] b. What do ou have to change in the equation = sin to move the graph up one unit? Write the new equation. [ = 1 + sin ] c. What are the intercepts of our new equation? Label them with their coordinates on the graph. [ : (0, 1), : ( π 2, 0), ( 3π 2, 0), ( 7π 2, 0),... ] d. When ou listed intercepts in part (c), did ou list more than one -intercept? Should ou have? [ Yes, there are infinitel man, at intervals of 2π. ] The graph at right was made b shifting the first ccle of = sin to the left. a. How man units to the left was it shifted? [ π ] b. Figure out how to change the equation of = sin so that the graph of the new equation will look like the one in part (a). If ou do not have a graphing calculator at home, sketch the graph and check our answer when ou get to class. [ = sin( + π ) ] Which of the situations below (if an) is best modeled b a cclic function? Eplain our reasoning. a. The number of students in each ear s graduating class. [ This ma go up and down, but the ccles are probabl of differing length. ] b. Your hunger level throughout the da. [ This ma or ma not be periodic. ] c. The high-tide level at a point along the coast. [ This is probabl approimatel periodic. ] Chapter 7:Trigonometric Functions 657

6 The CPM Amusement Park has decided to imitate The Screamer but wants to make it even better. Their ride will consist of a circular track with a radius of 100 feet, and the center of the circle will be 50 feet under ground. Passengers will board at the highest point, so the will begin with a blood-curdling drop. Write a function that relates the angle traveled from the starting point to the height of the rider above or below the ground. [ = 100 sin( + π 2 ) 50 or = 100cos 50 ] Should = sin and = cos both be parent graphs, or is one the parent of the other? Give reasons for our decision. [ Onl one needs to be a parent, since = sin( + 90 ) is the same as = cos. ] Find the equation of the eponential function of the form = ab that passes through each of the following pairs of points. a. (1, 18) and (4, 3888) [ = 3 6 ] b. ( 2, 8) and (3, 0.25) [ = 2(0.5) ] Solve each of the following equations. Be sure to check our solutions. a = 5 [ = ± 3 5 ] b = [ = 4, 1 ] c = + 3 [ = 4 ] Evaluate each of the following epressions eactl. a. tan 2π 3 [ 3 ] b. tan 7π 6 [ 3 3 ] David Longshot is known for his long golf drives. Toda he hit the ball 250 ards and estimated that the ball reached a maimum height of 15 ards. Find a quadratic equation that would model the path of the golf ball. [ a = = , possible equation: = ( 125) ] 658 Core Connections Algebra 2

7 Lesson What is missing? One More Parameter for a Cclic Function Lesson Objective: Mathematical Practices: Length of Activit: Students will determine the placement of the parameter b in the general equation for sine and cosine. The will also identif the period of cclic situations. make sense of problems and persevere in solving them, use appropriate tools strategicall, look for and make use of structure, look for and epress regularit in repeated reasoning One da (approimatel 50 minutes) Core Problems: Problems through Materials: Suggested Lesson Activit: Closure: (10 minutes) None Consider using a Participation Quiz toda. Start with a Think-Pair-Share for the question in problem 7-125: Does the general equation students found in Lesson allow for all possible transformations of the sine function? and follow-up with a whole-class discussion. Remind students of the graphs the made from swinging pendulums in Lesson and ask if the have accounted for all tpes of transformations. Students should recognize that the general equation = asin( h) + k does not allow for changing the length of each ccle. Then move teams on to problem Here the will be making a sine graph, but the input will be time instead of angle measurement. The will see an eample of a sine graph in which the period is clearl not 2π. As teams complete their graphs and have addressed all of the discussion points, hand out graphing calculators and direct them to problem In this eploration, the should find the place for the new parameter that controls the period. There should be questions about a general equation, such as whether to write sinb( h) or sin(b h). Teams should test the different forms the generate and discuss which is most useful, but should not necessaril resolve this issue et. The will consider it further in Lesson If time permits, direct teams to problem 7-128, which asks them to determine which situations have a period of 2π and which do not. This can be done verball with Reciprocal Teaching. Lead a brief discussion allowing teams to share their conclusions and their ideas about period and its place in the general equation. Tell students that the will develop these ideas further in the net lesson. If ou used a Participation Quiz toda, reserve a few minutes at the end of class for students to read the written feedback and to reflect on their team s process during the activit. Chapter 7:Trigonometric Functions 659

8 Team Strategies: Consider supporting our students teamwork b conducting a Participation Quiz toda. In a Participation Quiz, the qualit of the teamwork, rather than the mathematical content, is documented and assessed directl b the teacher. Since the problems toda require students to collaborate to bring together all the ideas the have learned so far in this chapter, it is well suited for a Participation Quiz. Homework: Problems through Note: Problem is Checkpoint 7B for completing the square to find the verte of a parabola. Notes to Self: 660 Core Connections Algebra 2

9 7.2.2 What is missing? One More Parameter for a Cclic Function In this lesson, ou will stud one more transformation that is unique to cclic functions. You will also etend our understanding of these functions to include those with input values that do not correspond to angles Does the general equation = asin( h) + k allow for ever possible transformation of the graph of = sin? Are there an transformations possible other than the ones produced b varing values of a, h, and k? Look back at the graphs ou made for the swinging bag of blood in the first lesson of this chapter. Discuss this with our team and be prepared to share our conjectures with the class THE RADAR SCREEN Brianna is an air traffic controller. Ever da she watches the radar line (like a radius of a circle) go around her screen time after time. On one particularl slow travel da, Brianna noticed that it takes 2 seconds for the radar line to travel through an angle of 6 π radians. She decided to make a graph in which the input is time and the output is the distance from the outward end of the radar line to the horizontal ais. Your Task: Following the input and output specifications above, make a table and graph for Brianna s radar. How can we calculate the outputs? How is this graph different from other similar graphs we have made? How long does it take to complete one full ccle on the radar screen? How can we see that on the graph? Chapter 7:Trigonometric Functions 661

10 Now that ou have seen that it is possible to have a sine graph with a ccle length other than 2π, work with our team to make conjectures about how ou could change our general equation to allow for this new transformation. a. In the general equation = asin( h) + k, the quantities a, h, and k are called parameters. Where could a new parameter fit into the equation? [ Some ma come up with a coefficient for as in the possible answers to part (c). ] b. Use our graphing calculator to test the result of putting this new parameter into our general equation. Once ou have found the place for the new parameter, investigate how it works. What happens when it gets larger? What happens when it gets smaller? [ When larger, the waves are shorter. When smaller, the waves are longer. ] c. Write a general equation for a sine function that includes the new parameter ou discovered. [ = a sinb( h) + k or some ma sa = asin(b h) + k ] Another word for ccle length is period. Which of the following have a period of 2π? Which do not? How can ou tell? If the period is not 2π, what is it? a. [ p = π ] b. A pendulum takes 3 seconds to complete one ccle. [ p = 3 seconds ] c. = sinθ [ p = 2π ] d. A radar line takes 1 second to travel through 1 radian. [ p = 2π seconds ] 662 Core Connections Algebra 2

11 Find an equation for each graph below. [ a: = sin( π 4 ) + 2 ; b: = 1.5sin( + π 2 ) ; c: = sin( π 6 ) + 2 or = sin( + 5π 6 ) + 2 ; d: = 3sin( 2π 3 ) 1 or = 3sin( + π 3 ) 1 ] a. b. c. d Claudia graphed = cosθ and = cos(θ ) on the same set of aes. She did not see an difference in their graphs at all. Wh not? [ 360 is the period of = cosθ, so shifting it 360 left lines up the ccles perfectl. ] Chapter 7:Trigonometric Functions 663

12 This problem is a checkpoint for completing the square to find the verte of a parabola. It will be referred to as Checkpoint 7B. Complete the square to change the equation = into graphing form. Identif the verte of the parabola and sketch the graph. [ Graphing form: = 2( 1) ; verte (1, 3) ; See graph at right. ] Check our answers b referring to the Checkpoint 7B materials located at the back of our book. If ou needed help solving these problems correctl, then ou need more practice. Review the Checkpoint 7B materials and tr the practice problems. Also, consider getting help outside of class time. From this point on, ou will be epected to do problems like these quickl and easil Find the - and -intercepts of the graphs of each of the following equations. a. = b. + 2 = log 3 ( 1) [ = (0, 0), ( 5±3 3 2, 0) and = (0, 0) ] [ = (10, 0), no -intercept ] The average cost of movie tickets is $9.50. If the cost is increasing 4% per ear, in how man ears will the cost double? [ ears ] Change each equation to graphing form. For each equation, find the domain and range and determine if it is a function. [ a: = 2( + 1 4) , = all real numbers, = < < 25 8, Yes it is a function; b: = 3( + 1) , domain: all real numbers, range: < < 15, Yes it is a function. ] a. = b. = Too Tall Thomas has put Rodne s book bag on the snack-shack roof. Rodne goes to borrow a ladder from the school custodian. The tallest ladder available is 10 feet long and the roof is 9 feet from the ground. Rodne places the ladder s tip at the edge of the roof. The ladder is unsafe if the angle it makes with the ground is more than 60º. Is this a safe situation? Justif our conclusion. [ 64.16, unsafe ] 664 Core Connections Algebra 2

13 Deniz s computer is infected with a virus that will erase information from her hard drive. It will erase information quickl at first, but as time goes on, the rate at which information is erased will decrease. In t minutes after the virus starts erasing information, 5,000,000( 1 2 )t btes of information remain on the hard drive. a. Before the virus starts erasing, how man btes of information are on Deniz s hard drive? [ 5,000,000 btes ] b. After how man minutes will there be 1000 btes of information left on the drive? [ 12.3minutes ] c. When will the hard drive be completel erased? [ According to the equation, technicall never, but for all practical purposes, after 23 minutes. ] Graph f () = 6 4. [ See graph at right. ] a. Eplain how ou can graph this without making an table, but using parent graphs. [ The verte of the graph is at (6, 4) with two ras emanating at slopes of ±1. ] b. Graph g() = 6 4. Eplain how ou can graph g() without making an table b using our earlier graph. [ See graph at right. Flip all parts of the graph that are below the -ais above the -ais. ] Chapter 7:Trigonometric Functions 665

14 Lesson What is the period of a function? Period of a Cclic Function Lesson Objective: Mathematical Practices: Length of Activit: Students will find equations for transformed sine curves and will graph transformed sine functions. use appropriate tools strategicall, look for and make use of structure, look for and epress regularit in repeated reasoning One da (approimatel 50 minutes) Core Problems: Problems through Materials: Curves generated b teams in Lesson Meter sticks or long straightedges Suggested Lesson Activit: Start class b leading a whole-class discussion about period. Draw a unit circle on the board and ask what the period must be of an trigonometric function whose input is the angle θ from standard position in the unit circle. In this contet, the onl period that makes sense is 2π. You can ask, Does it follow, then, that all trigonometric functions must have a period of 2π? Challenge students to think of cases in which the period would be different. You can then use the situations in problem as eamples of different periods. Alternatel, this could be done with a Think-Pair-Share. After this discussion, move teams to problem 7-139, in which the will use graphing calculators to eplore the relationship between the value of b in the general equation, = a sin[b( h)] + k, and the period of its graph. When most teams have finished this problem, ask teams to share their conclusions about this relationship. Students should recognize that the value of b tells them the number of ccles in 2π. The must divide 2π b the period in order to find the period length or think of what to multipl b to get 2π. Have teams refer to the curves from the pendulum eperiment in Lesson Direct them to problem 7-140, which asks them to draw in a set of aes and find an equation to describe their curve. Epect this to be challenging! Teams will need to recognize all of the parameters of their particular curve and figure out how these fit into the equation. Encourage them to test their equations in a graphing calculator. If time permits, move teams on to problem 7-141, which could be done as a Pairs Check. The problem asks them to sketch a series of transformed sine graphs, but be sure to allow time for students to do problem 7-142, in which the will clarif the placement of the parameter b in a cclic equation. In problem students will put all of the parameters together into a general equation for sine. 666 Core Connections Algebra 2

15 Closure: (10 minutes) Lead a brief discussion asking students to clarif what information is needed to determine a unique sine function. This could be done as a Walk and Talk. Have teams share strategies for finding equations for their pendulum graphs and for graphing the functions in problem Homework: Problems through Notes to Self: Chapter 7:Trigonometric Functions 667

16 7.2.3 What is the period of a function? Period of a Cclic Function In Lesson 7.2.2, ou found a place for a new parameter in the general form of a trigonometric equation and discovered that it must have something to do with the period. B the end of this lesson, ou will have the tools ou need to find the equation for an sine or cosine graph and will be able to graph an sine or cosine equation. In other words, ou will learn the graph equation connection. The following questions can help our team sta focused on the purpose of this lesson. Graph Table Unit Circle or Situation Equation How can we write the equation for an sine or cosine graph? How can we graph an sine or cosine function? Find the period for each of the following situations: a. The input is the angle θ in the unit circle and the output is the cosine of θ. [ 2π ] b. The input is time and the output is the average dail temperature in New York. [ One ear. ] c. The input is the distance Nurse Nina has traveled along the hallwa and the output is the distance of blood drips from the midline of the hallwa. [ This depends on Nurse Nina s speed and the speed of the blood pendulum. ] 668 Core Connections Algebra 2

17 Make sure our graphing calculator is in radian mode. a. Set the domain and range of the viewing window so that ou would see just one complete ccle of = sin. What is the domain for one ccle? What is the range? [ domain: 0 2π, range: 1 1 ] b. Graph = sin, = sin(0.5), = sin(2), = sin(3), and = sin(5). Make a sketch and answer the following questions for each equation. [ In the graph below left, = sin is solid and bold, = sin(0.5) is dashed, and = sin(3) is not bold; the graph below middle is = sin(3) and below right is = sin(5). ] i. How man ccles of each graph appear on the screen? [ 1, 1 2, 2, 3, 5 ] ii. iii. The midline is the horizontal ais that goes through the center of the graph. What is the equation for the midline of these graphs? [ All = 0. ] What is the amplitude (height above the midline) of each graph? [ All 1. ] iv. What is the period (ccle length) of each graph? [ 2π, 4π, π, 2π 3, 2π 5 ] v. Is each equation a function? [ Yes. ] c. Make a conjecture about the graph of = sin(b) with respect to each of the questions (i) through (v) above. If ou cannot make a conjecture et, tr more eamples. [ Possible conjectures include: The graph is horizontall stretched if b < 1 or compressed if b > 1. b is the number of ccles between 0 and 2π, so the length of each ccle (i.e., the period) is 2π b. ] d. Create at least three of our own eamples to check our conjectures. Be sure to include sketches of our graphs. [ Answers var. ] e. What is the relationship between the period of a sine graph and the value of b in its equation? [ b = period 2π, b(period) = 2π, and period = 2π b ] Chapter 7:Trigonometric Functions 669

18 Refer to the graph ou made b swinging a pendulum in Lesson Decide where to draw - and -aes and find the equation of our graph. Is there more than one possible equation? Be prepared to share our strategies with the class. [ Equations will var. ] Without using a graphing calculator, describe each of the following functions b stating the amplitude, period, horizontal shift, and midline (vertical shift). Using this information, sketch the graph of each function. After ou have completed each graph, check our sketch with a graphing calculator and correct and eplain an errors. [ See graphs below. ] a. = sin 2( π 6 ) [ a = 1, p = π, h = π 6, midline = 0 ] b. = 3+ sin( 1 3 ) [ a = 1, p = 6π, h = 0, midline = 3 ] c. = 3sin(4) [ a = 3, p = π 2, h = 0, midline = 0 ] d. = sin 1 2 ( +1) [ a = 1, p = 4π, h = 1, midline = 0 ] e. = sin 3( π 3 ) [ a = 1, p = 2π 3, h = π 3, midline = 0 ] f. = 1+ sin(2 π 2 ) [ a = 1, p = π, h = π 4, midline = 1 ] a. b. c. d. e. f. 670 Core Connections Algebra 2

19 Farah and Thu were working on writing the equation of a sine function for the graph at right. The figured out that the amplitude is 3, the horizontal shift is π 4 and the midline is = 2. The can see that the period is π, but the disagree on the equation. Farah has written f () = 3sin2( π ) 2 and Thu has 4 written f () = 3sin(2 π 4 ) 2. a. Whose equation is correct? How can ou be sure? [ Farah s equation is correct, because it matches the horizontal shift on the original graph. ] f() b. Graph the incorrect equation and eplain how it is different from the original graph. [ See graph at right. The horizontal shift is π 8. ] Look back at the general equation ou wrote for the famil of sine functions in problem Now that ou have learned how the period affects the equation, work with our team to add a new parameter (call it b) that allows our general equation to account for an transformation of the sine function, including changes in the length of each ccle. Be prepared to share our general equation with the class. [ = a sin[b( h)] + k ] Use what ou learned in class to complete parts (a) through (c) below. a. Describe what the graph of = 3sin( 1 2 ) will look like compared to the graph of = sin. [ Amplitude 3, period 4π ] b. Sketch both graphs on the same set of aes. [ See graph at right. ] c. Eplain the similarities and differences between the two graphs. [ The differences are the period and amplitude, and therefore some of the -intercepts. The have the same basic shape. ] What is the period of = sin(2π )? How do ou know? [ 1, 2π 2π = 1 or 2π(1) = 2π ] Chapter 7:Trigonometric Functions 671

20 Colleen and Jolleen both used their calculators to find sin 30. Colleen got sin 30º = , but Jolleen got sin 30º = 0.5. Is one of their calculators broken, or is something else going on? Wh did the get different answers? [ Colleen s calculator was in radian mode, while Jolleen s calculator was in degree mode. Colleen s calculation is wrong. ] Ceirin s teacher promised a quiz for the net da, so Ceirin called Adel to review what the had done in class. Suppose I have = sin 2, said Ceirin, what will its graph look like? It will be horizontall compressed b a factor of 2, replied Adel, so the period must be π. Oka, now let s sa I want to shift it one unit to the right. Do I just subtract 1 from, like alwas? I think so, said Adel, but let s check on the graphing calculator. The proceeded to check on their calculators. After a few moments the both spoke at the same time. Rats, said Ceirin, it isn t right. Cool, said Adel, it works. When the arrived at school the net morning, the compared the equations the had put in their graphing calculators while the talked on the phone. One had = sin 2 1, while the other had = sin 2( 1). Which equation was correct? Did the both subtract 1 from? Eplain. Describe the rule for shifting a graph one unit to the right in a wa that avoids this confusion. [ = sin2( 1) is correct. To shift the graph one unit to the right, subtract 1 from before multipling b anthing. ] George was solving the equation (2 1)( + 3) = 4 and he got the solutions = 1 and = 3. Jeffre came along and said, You made 2 a big mistake! You set each factor equal to zero, but it s not equal to zero, it s equal to 4. So ou have to set each factor equal to 4 and then solve. Who is correct? Show George and Jeffre how to solve this equation. To be sure that ou are correct, check our solutions. [ The are both wrong. The equation needs to be set equal to zero before the Zero Product Propert can be applied = 4 is equivalent to (2 + 7)( 1) = 0. = 1 or = 7 2 ] 672 Core Connections Algebra 2

21 Compute the value of each epression without using a calculator. a. log(8) + log(125) [ 3 ] b. log 25 ( 125) [ 1.5 ] c. 1 2 log(25) + log(20) [ 2 ] d. 7log 7 (12) [ 12 ] An eponential function = km + b passes through (3, 7.5) and (4, 6.25). It also has an asmptote at = 5. a. Find the equation of the function. [ = 20( 1 2) + 5 ] b. If the equation also passes through (8, w), what is the value of w? [ w = ] Consider the equation f () = 3( + 4) 2 8. a. Find an equation of a function g() such that f () and g() intersect in onl one point. [ Answers var, if g() is linear, tangent lines onl. ] b. Find an equation of a function h() such that f () and h() intersect in no points. [ An line = b such that b < 8. ] Chapter 7:Trigonometric Functions 673

22 Lesson What are the connections? Graph Equation Lesson Objective: Mathematical Practices: Length of Activit: Students will consolidate their understanding of the connections between cclic graphs and their equations. The will also practice graphing equations and writing equations from graphs. Finall, the will determine that sine and cosine functions are just horizontall shifted versions of each other. make sense of problems and persevere in solving them, model with mathematics, use appropriate tools strategicall, look for and make use of structure, look for and epress regularit in repeated reasoning One da (approimatel 50 minutes) Core Problems: Problems through Materials: Suggested Lesson Activit: None Start b pointing out the objective of this lesson. Then ask teams to read the lesson introduction and make a note of the focus questions. Point out that b the end of the lesson, the should be able to answer them. Direct teams to work on problems and In problem 7-152, teams will generate a list of the attributes the need to know in order to generate a sine or cosine equation or graph. In problem 7-153, teams will split into pairs. Each pair will decide on a value for each attribute and will create an equation and sketch its graph. This is also where vocabular is formalized, if that has not alread happened through class discussion. In problem 7-154, pairs will trade equations with another pair, graph the equation the have received, and check the other team s work. Have teams move on to problem Teams should recognize that the graph could be a sine or a cosine function and the could shift it either direction b eactl one period and get an identical graph. If time permits, teams should start problem 7-156, in which the will write an equation for a cclic situation. To assist teams to go from situation equation, ask them, Is there an other connection in the web that could help ou figure out the equation? Man teams ma decide that the need to sketch a graph first in order to determine the information the need to write the equation. To sketch a graph, teams will need to use the highest and lowest points to find the amplitude, and the will have to determine the period and how to place the aes. There are several possible solutions. As ou circulate look at their sketches and ask for eplanations of their choices. 674 Core Connections Algebra 2

23 Closure: (10 minutes) Team Strategies: Ask teams to share their strategies and conclusions from problem using a Swapmeet. Then direct students to complete the Learning Log entr in problem 7-157, in which the revisit the target questions for the lesson and write their ideas. As ou open toda s lesson, ask Facilitators to read the target questions in the lesson introduction aloud and to continue to revisit them as their stud teams work. Homework: Problems through Notes to Self: Chapter 7:Trigonometric Functions 675

24 7.2.4 What are the connections? Graph Equation In the past few lessons, ou have been developing the understanding necessar to graph a cclic equation without making a table and to write an equation from a cclic graph. In toda s lesson, ou will strengthen our understanding of the connections between a cclic equation and its graph. B the end of this lesson, ou will be able to answer the following questions: Graph Table Unit Circle or Situation Equation Does it matter if we use sine or cosine? What do we need to know to make a complete graph or write an equation? What do ou need to know about the sine or cosine functions to graph them or write their equations? Talk with our team and write a list of all of the attributes of a sine or cosine function that ou need to know to write an equation and graph it. [ Teams should generate descriptions of the following attributes, although the ma not et have the formal vocabular: amplitude, period, horizontal shift, midline, and orientation. ] CREATE-A-CURVE Split our team into pairs. With our partner, ou will create our own sine or cosine function, write its equation, and draw its graph. Be sure to keep our equation and graph a secret! Start b choosing whether ou will work with a sine or a cosine function. a. Half the distance from the highest point to the lowest point is called the amplitude. You can also think of amplitude as the vertical stretch. What is the amplitude of our function? b. How far to the left or right of the -ais will our graph begin? In other words, what will be the horizontal shift of our function? c. How much above or below the -ais will the center of our graph be? In other words, what will be the midline of our function? d. What will the period of our function be? e. What will the orientation of our graph in relation to = sin or = cos be? Is it the same or is it flipped? f. Now that ou have decided on all of the attributes for our function, write its equation. 676 Core Connections Algebra 2

25 Cop the equation for our curve from problem on a clean sheet of paper. Trade papers with another pair of students. a. Sketch a graph of the equation ou received from the other pair of students. b. When ou are finished with our graph, give it back to the other pair so the can check the accurac of our graph When ou look at a graph and prepare to write an equation for it, do ou think it matters if ou choose sine or cosine? Which do ou think will work best? With our team, find at least four different equations for the graph at right. Be prepared to share our equations with the class. [ Possible equations include = 2cos, = 2sin( π 2 ), = 2cos( π ), and = 2sin( + π 2 ) ] a. Did it matter if ou choose sine or cosine? [ No. ] b. Which of our equations do ou prefer? Wh? [ Answers var. ] Brenna s mom, Mrs. Herstone, is watching Brenna plaing at the park. Some children are pushing Brenna around the merr-go-round. Mrs. Herstone decides to take some data, so she started her stopwatch. At 0.5 seconds Brenna is farthest from Mrs. Herstone, 26 feet awa. When the stopwatch reads 4.2 seconds, Brenna is closest at 12 feet awa. Find a cclic equation that models the distance Brenna is from Mrs. Herstone over time if the merr-go-round is spinning at a constant rate. [ Some possibilities include: d(t) = 7sin 7.4 2π (t 2.35) + 19, d(t) = 7cos 7.4 2π (t 0.5) + 19, d(t) = 7cos 7.4 2π (t + 3.2) + 19, d(t) = 7sin 7.4 2π (t ) + 19 ] Chapter 7:Trigonometric Functions 677

26 LEARNING LOG In our Learning Log, write our ideas about the target questions for this lesson: Does it matter if I use sine or cosine? What do I need to know to make a complete graph or write an equation? Title this entr Cclic Equations and Graphs and label it with toda s date. MATH NOTES ETHODS AND MEANINGS General Equation for Sine Functions The general equation for the sine function is = asin[b( h)] + k. p a h k a The amplitude (half of the distance between the highest and the lowest points) is a. The period is the length of one ccle. It is labeled p on the graph. The number of ccles in 2π is b. The horizontal shift is h. The vertical shift is k. The midline is = k. 678 Core Connections Algebra 2

27 Susan knew how to shift = sin to get the graph at right, but she wondered if it would be possible to get the same graph b shifting = cos. a. Is it possible to write a cosine function for this graph? [ Yes. ] b. If ou think it is possible, find an equation that does it. If ou think it is impossible, eplain wh. [ = cos( + π 2 ) ] c. Adlai said, I can get that graph without shifting to the right or left. What equation did he write? [ = sin ] In the function = 4sin(6), how man ccles of sine are there from 0 to 2π? How long is each ccle (i.e., what is the period)? [ 6 ccles, period: π 3 ] Write the equation of a cclic function that has an amplitude of 7 and a period of 8π. Sketch its graph. [ Answers ma var, but = 7sin( 4) works. ] Recall the strategies ou developed for converting degrees to radians. How could ou reverse that? Convert each of the following angle measures. Be sure to show all of our work. a. π radians to degrees [ 180 ] b. 3π radians to degrees [ 540 ] c. 30 degrees to radians [ π π 6 radians ] d. 4 radians to degrees [ 45 ] e. 225 degrees to radians [ 5π 3π 4 radians ] f. 2 radians to degrees [ 270 ] Find the eact value for each of the following trig epressions. For parts (g) and (h), assume that 0 θ 2π. a. cos( 3π 4 )= [ 2 2 ] b. tan( 4π 3 )= [ 3 ] c. sin( 11π 6 )= [ 1 2 ] d. sin 3π 2 ( 4 )= [ 2 ] e. tan( 5π 4 )= [ 1 ] f. tan( 17π 6 )= [ 1 or ] g. tan(θ) = 1 [ π 4 or 5π 4 ] h. tan(θ) = 1 [ 3π 4 or 7π 4 ] Chapter 7:Trigonometric Functions 679

28 Solve this sstem of equations: 5 4 6z = z = z = 16 [ ( 1, 1 2, 2) ] Use the Zero Product Propert to solve each equation in parts (a) and (b) below. a. (2 + 1)(3 5) = 0 b. ( 3)( 2) = 12 [ = 0, = 1 2, or = 5 3 ] [ = 6 or = 1 ] c. Write an equation and show how ou can use the Zero Product Propert to solve it Find a quadratic equation whose graph has each of the following characteristics: a. No -intercepts and a negative -intercept. [ Answers var, sample answer: = 2 2 ] b. One -intercept and a positive -intercept. [ Answers var, sample answer: = ( 3) 2 ] c. Two -intercepts and a negative -intercept. [ Answers var, sample answer: = ( + 1)( + 3) ] A two-bedroom house in Seattle was worth $400,000 in If it appreciates at a rate of 3.5% each ear: a. How much will it be worth in 2015? [ About $564,240 ] b. When will it be worth $800,000? [ In 2025 ] c. In Jacksonville, houses are depreciating at 2% per ear. If a house is worth $200,000 now, how much value will it have lost in 10 ears? [ About $36,585 ] 680 Core Connections Algebra 2