Exploring Trigonometric Graphs Exploring Trigonometric Graphs Index Syllabus Extract: Relationship to syllabus. . . . . . . . . . . . . . . . . . . . . 3 Leaving Certificate. . . . . . . . . . . . 3 Unit 1 . . . . . . . . . . . . . . . . . . . . . . 4 Unit 2 . . . . . . . . . . . . . . . . . . . . . . 10 Unit 3 . . . . . . . . . . . . . . . . . . . . . . 12 Questions Sheets - Trigonometric Graphs in Context . . . . . . . . . . . . 13 Solutions . . . . . . . . . . . . . . . . . . . 16 Student Activity Sheet 1 Graphs of sinx and cosx . . . . . . . . . . . . . . 34 © Project Maths Development Team 2012 www.projectmaths.ie 2 Exploring Trigonometric Graphs Syllabus Extract: Relationship to syllabus. Leaving Certificate. © Project Maths Development Team 2012 www.projectmaths.ie 3 Exploring Trigonometric Graphs Unit 1 Upon completion of this Unit: Students will: Understand terms such as Maximum Minimum Period Range Amplitude Horizontal midway line Horizontal shape (stretch/shrink) Vertical shape (stretch/shrink) Transformations of the graphs of and In this unit students will explore functions of the type a, b, c ∈ R and examine how the values of “a”, “b” and “c” affect the curves. Prior Knowledge Students should be familiar with graphs of linear and quadratic functions. Students should be familiar with the graphs of from Teaching and Learning Plan 10 and the Student’s CD. At the outset the students should be reminded of the features of the graphs of the functions and by completing Student Activity Sheet 1 [Page 37]. © Project Maths Development Team 2012 www.projectmaths.ie 4 Exploring Trigonometric Graphs Materials required For each student you will need: Mini white board Student Activity Sheet 1 and Student Activity Sheet2a (Mathematical Language / Properties of Trigonometric Graphs) Student Activity Sheet 2b (Mathematical Language / Properties of Trigonometric Graphs) Student Activity Sheet 2c (Mathematical Language / Properties of Trigonometric Graphs) Student Activity Sheet 3 (Mathematical Language / Properties of Trigonometric Graph) Student Activity Sheet 5 (Mathematical Language / Properties of Trigonometric Graphs) Student Activity Sheet 6 (Transformations of Trigonometric Graphs) Student Activity Sheet 7 (Mathematical Language / Properties of Trigonometric Graphs). For each group of students you will need Student’s CD Card Set A1 (Trigonometric Functions ) Card Set A2 (Trigonometric Functions ) Card Set B (Trigonometric Functions Important Features) Card Set C Student Activity 4 (Mathematical Language / Properties of Trigonometric Graphs). Student Activities/Teacher’s support and actions Use the interactive GeoGebra files at: www.projectmaths.ie Working in groups [1] Ask students to work in pairs, distribute one or both of Card Set A1 (Trigonometric Functions g(x) = bsin x) [Page 38] Card Set A2 (Trigonometric Functions g(x) = bcos x) [Page 39] to each pair of student. Ask each group to sort the cards into three sets, selection criteria should be at the group’s discretion. For example, they may group the functions where the coefficient of sin x or cos x is 2. Ask students to write a description of their selection criteria. Ask students to write an equation of their own for each set (if working with cards from one set only). © Project Maths Development Team 2012 www.projectmaths.ie 5 Exploring Trigonometric Graphs Whole group activity using GeoGebra [1] Share all the criteria students have come up with. Hand out one of: Student Activity Sheet 2a (Mathematical Language /Properties of Trigonometric Graphs) [Page 40] and Student Activity Sheet 2b (Mathematical Language /Properties of Trigonometric Graphs) [Page 41] or Student Activity Sheet 2c (Mathematical Language /Properties of Trigonometric Graphs) (getting the students to use a mixture of cards from Card Sets a1 and A2) [Page 42] Now ask the students to use the interactive files f(x)=a+bsin cx and f(x)=a+bcos cx at www.projectmaths.ie a to justify their descriptions of the functions contained on Card Sets A1/A2 and introduce students to the correct use of mathematical language using the previous student activity. Instructions: In the interactive files f(x)=a+bsin cx and f(x)=a+bcos cx Set Slider a to 0. Set Slider c to 1. Set Slider b to desired value. Example: A “b” value of 3 causes a stretch in the direction of the Y axis with scale factor of 3. If student need more practice with transformations and trigonometric graphs use mini whiteboards to explore questions of the type: Give an equation to represent the function which results from: stretching f(x)=sinx in the direction of the Y axis with a scale factor of 4. reflecting f(x)=cosx in the X axis. Describe the transformation required to transform f(x)=sinx to chlaochlú go dti f(x)=-2sinx Define “amplitude”, “range”, “period” and link these to the idea of stretch/shrink Check the students’ understanding of how f(x)=a+bsin cx and f(x)=a+bcos cx Check the students’ understanding of how give information about amplitude, range and period by asking them to find cards from Card Set A1 and/or Card Set A2 that fit certain criteria. For example, the students might be asked to find equation(s) which describe: A graph having an amplitude of 2 Two graphs that have the same period A graph that has been compressed along the X axis A graph that has been stretched vertically Two graphs that are images of each other by reflection. © Project Maths Development Team 2012 www.projectmaths.ie 6 Exploring Trigonometric Graphs Working in groups [2] Distribute to each pair of students Card Set B (Trigonometric Functions-Important Features) [Page 43] Ask each pair to sort the cards into two sets. Selection criteria should be at their own discretion. Ask students to write a description of the selection criteria. Ask student to write an equation which would fit their selection criteria. Whole group activity using GeoGebra [2] Share all the criteria students have come up with. Distribute to each pair of students. Student Activity Sheet 3 (Mathematical Language/Properties of Trigonometric Graphs) [Page 44] Now use the interactive files f(x)=a+bsin cx and f(x)=a+bcos cx at www.projectmaths.ie to justify the descriptions and introduce students to mathematical language. Instructions: In the interactive files f(x)=a+bsin cx and f(x)=a+bcos cx Set Slider a to 0. Set Slider b to 1. Set Slider c to desired value. Example When “c” has a value of 3 this is a horizontal shrink with scale factor 3. If students need more practice with transformations and trigonometric graphs use mini whiteboards to explore questions of the type: Give an equation to represent the function which results from: Stretch f(x)=sin x horizontally by a scale factor of 4 Shrink f(x)=cos x horizontally by a scale factor of 0.5 Decreasing the period of f(x)=sin x by a factor of 2 Increasing the period of f(x)=sin x by a factor of… What transformation converts: f(x)=sin x to f(x)=sin2/5x, f(x)=cos x to f(x)=cos5x? Check student understanding of how the functions f(x)=a+bsin cx and f(x)=a+bcos cx give information about amplitude, range and period by asking them to find cards from Card Set B that fit certain criteria. © Project Maths Development Team 2012 www.projectmaths.ie 7 Exploring Trigonometric Graphs For example, the students might be asked to find equation(s) which describe: A graph having amplitude of 2. Two graphs having the same period. A graph that has been shrunk along the X axis. A graph that has been vertically stretched. A graph that has been horizontally stretched. Two graphs that are images of each other by reflection. Two graphs that have the same amplitude. Working in groups [3] Distribute to each pair of students: Card Set C [Pages 45 – 47] Student Activity Sheet 4 (Mathematical Language/Properties of Trigonometric Graphs) [Page 48] Student Activity Sheet 5 (Mathematical Language /Properties of Trigonometric Graphs) [Page 49] Ask each pair of students to find cards from Card Set C to match the properties described on Student Activity Sheet 4. When the students have found a suitable card they should place it in the correct box on Student Activity Sheet 4 matching the property described therein. They should aim to match as many boxes and cards as possible. Whole-group activity [3] Choose one box from Student Activity Sheet 4 and write down all the cards that were matched with it. Discuss as a group if these cards are appropriate and agree reasons for any decisions reached. Repeat the above for the remaining boxes. Check students understanding of how the equations f(x)=a+bsin cx and f(x)=a+b coscx? give information about amplitude, range and period by asking them to fill in the second section of Student Activity Sheet 5. Working in groups [4] Distribute to each pair of students Student Activity Sheet 6 (Transformation of Trigonometric Graphs)[Page 50] Student Activity Sheet 7 (Mathematical Language / Properties of Trigonometric Graphs) [Page 51] Discuss in groups how the graphs of f(x)=a+bsin cx and f(x)=a+b coscx are affected by changing the value of “a” If required reinforce the students’ prior learning by referring to the properties of the graphs of linear and quadratic functions. . © Project Maths Development Team 2012 www.projectmaths.ie 8 Exploring Trigonometric Graphs Whole group activity using GeoGebra [4] Share all the criteria student have come up with in the previous activity Working in groups [4] and discuss the reasons for these criteria. Now use the interactive files f(x)=a+bsin cx and f(x)=a+b coscx at www.projectmaths.ie to justify the descriptions and introduce student to the appropriate use of mathematical language. Instructions: In the interactive files f(x)=a+bsin cx and f(x)=a+b coscx Set Slider b to 1. Set Slider c to 1. Set Slider a to desired value. For example When “a” has a value of 3 this causes a vertical translation upwards of length 3 units. If student need more practice with transformations and trigonometric graphs use mini whiteboards to answer questions of the type: Give an equation to represent the function which results from translating: f(x)=sin x vertically upwards by 2 units. Give the amplitude of the function which results from translating f(x)=cos x vertically downwards by 3 units. Give the range of the function which results from translating f(x)=sin x vertically upwards by 4 units. Give the period of the function which results from translating f(x)=cos x vertically downwards by 2.5 units. Give the transformation which has been used in transforming f(x)=cos x to f(x)=7 + cos x Check student understanding of how the equation gives information about transformations amplitude, range and period by getting them to complete Student Activity Sheet 7. © Project Maths Development Team 2012 www.projectmaths.ie 9 Exploring Trigonometric Graphs Unit 2 Having completed this Unit: Students will: Understand how to identify the: Period Range Amplitude Horizontal midway line Horizontal shape in the direction of the X axis Vertical shape in the direction of the Y axis Transformations of trigonometric functions from their graphs. Furthermore, the students will be able to sketch trigonometric graphs given their equations. Prior Knowledge Student should be familiar with the graphs f(x)=a+bsin cx and f(x)=a+b cos cx from Unit 1. Materials required For each student you will need: Mini white board Student Activity Sheet 8 (Trigonometric Graphs) For each group of student you will need: Card Set D Card Set E Card Set F Working in groups [5] Distribute to each pair of students Card Set D Trigonometric Graphs [Pages 52 – 53] Student Activity Sheet 8 (Trigonometric Graphs) [Page 54] Ask each group of students to fill in Student Activity Sheet 8 using details from the graphs on Card Set D to do so. Distribute to each pair of student Card Set E [Page 55] Ask each group to match the graphs from Card Set D with the trigonometric functions from Card Set E using the information derived from the earlier group activity. Check student understanding by asking them to explain why they matched the cards as they did. © Project Maths Development Team 2012 www.projectmaths.ie 10 Exploring Trigonometric Graphs Working in groups [6] Distribute Card Set D Trigonometric Graphs [Pages 52 – 53] to each pair of student if not already handed out. Ask each group of students to name and identify a number of points which would allow them to draw a sketch of trigonometric graphs of the form: f(x)=a+bsin cx and f(x)=a+b coscx Student can use mini whiteboards to draw sketches as part of their discussions. Whole group activity [6] Choose one group’s set of points. Discuss as a group the merits or otherwise of this set of points. Ask questions such as: Will this set of points give a unique graph? Will this set of points give the graph for all possible trigonometric functions of the form f(x)=a+bsin cx and f(x)=a+b coscx is this the smallest set of points that will allow a sketch to be drawn? Taking feedback from all the groups identify the five points required: Maximum Minimum Three points of intersection with the horizontal midway line. Working in groups [7] Distribute to each pair of students Card Set F [Page 56] Check students understanding of how the equation gives the information required to sketch a trigonometric function of the form f(x)=a+bsin cx and f(x)=a+b cos cx by asking them to use mini whiteboards to sketch the graphs from Card Set F. Each group of students can check their sketches by using the interactive files f(x)=a+bsin cx and f(x)=a+b cos cx at www.projectmaths.ie Note 1: if students do not have Card Set D or Student Activity Sheet 8 in their possession they could use Card Set E for this activity. Note 2: Students could use the functions from Student Activity Sheet 7 instead of using Card Set F, if this is considered desirable. © Project Maths Development Team 2012 www.projectmaths.ie 11 Exploring Trigonometric Graphs Unit 3 Having completed this Unit: Student will: be able to formulate conjectures form patterns understand the need to explain their findings and justify their conclusions have developed the skills to communicate mathematics verbally and in written form using appropriate mathematical language be able to apply their knowledge and skills to solve problems in familiar and unfamiliar contexts be able to analyse information presented verbally and translate it into mathematical form devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions. Prior Knowledge Students should be familiar with the graphs of f(x)=a+bsin cx and f(x)=a+b cos cx from Units 1 and 2 of Exploring Trigonometric Graphs. Materials required For each students you will need: Mini white board Question Sheets (Trigonometric Graphs in Context) Working in groups [7] Ask student to work in pairs. Distribute to each student Question Sheets (Trigonometric Graphs in Context) [Page 15 - 17] Ask each group to attempt Question 1. Take feedback from groups using appropriate questioning to establish and develop the students’ understanding. Repeat the above for two additional questions. Now ask each student to work independently on the remainder of the questions on the Question Sheets(Trigonometric Graphs in Context). Provide assistance to individuals as required and ensure that the students’ learning is of a high quality and that their understanding of the material is being developed and deepened. © Project Maths Development Team 2012 www.projectmaths.ie 12 Exploring Trigonometric Graphs Questions Sheets Trigonometric Graphs in Context Question 1 A wrecking ball attached to a crane swings back and forth. The distance that the ball moves to the left and to the right of its resting position with respect to time is represented by the following graph. a. What is the period of the crane’s motion. Explain your answer? b. What is the equation of the horizontal midway line of the curve and what does it represent? c. What is the amplitude of the crane’s motion? Draw a diagram to represent what the amplitude represents in terms of the motion of the ball? d. How many complete swings will the wrecking ball make in four minutes? e. What does point A represent? Question 2 As a dolphin swims along by the side of a cruise ship he jumps to a height of four metres above the water surface and dives to a depth of four meters below the surface. He does this in a regular motion (simple harmonic motion). A passenger using a stopwatch and starting it just as the dolphin is at the surface determines that the dolphin completes one cycle every 8 seconds. a. Describe the displacement of the dolphin relative to the water surface using a sinusoid curve and sketch the graph of the dolphin’s displacement relative to the water surface b After three seconds how high above the surface is the dolphin? c Is the dolphin above or below the surface of the water after 37 seconds? © Project Maths Development Team 2012 www.projectmaths.ie 13 Exploring Trigonometric Graphs Question 3 The inside rim of a bicycle wheel whose diameter is 25 inches, is 3 inches off the ground. An ant is sitting on the inside rim of the wheel at the point 3 inches off the ground. Sean starts riding the bicycle at a steady rate. The wheel makes one revolution every 1.6 seconds. a. Find the equation of a sinusoid curve which describes the motion of the ant and draw a graph of the function. b What height in centimetres from the ground will the ant be 25 seconds into the trip given that 1 inch = 2.54 cm.? c Within the first 10 seconds how many times will the ant be at its starting height? Question 4 The number of people in thousands employed in a resort town is represented by the function Take t = 0 as last day of January Take t = 1 as last day of February Take t = 2 as last day of March a. Draw a rough sketch showing the variation in the number of people employed in the town for one complete period. b When is the maximum number of people employed and what is this maximum number? c During which months of the year will the number of people employed be 4,650 or greater? d Does this model have any drawbacks and if yes identify one? Question 5 A tsunami (tidal wave) is a fast moving wave caused by an underwater earthquake. The water oscillates about its normal level, with equal amplitudes above and below this level The period is fifteen minutes. Suppose that a tsunami with an amplitude of ten metres approaches the pier at Honolulu, where the normal depth of water is nine metres. Assuming that the depth of water varies sinusoidally with time as the tsunami passes, predict the depth of the water at the following times after the tsunami first reaches the pier. a Two minutes, four minutes and twelve minutes. b According to your model what will be the minimum depth of the water? c How do you interpret this answer in terms of what will happen in the real world? © Project Maths Development Team 2012 www.projectmaths.ie 14 Exploring Trigonometric Graphs Question 6 A buoy in the ocean is bobbing up and down in harmonic motion. At t = 0 seconds, the buoy is at its high point and returns to that high point every 8 seconds. The buoy moves a distance of 1.44 meters from its highest point to its lowest point. a. Using a sinusoidal function create a mathematical model to represent the depth of water under the buoy and sketch the curve. b How high is the buoy at time = 29 seconds? c Is it rising or falling at that time? d How many times in 2 minutes will the buoy be at sea level? Question 7 If the depth of water in a canal continually varies between a minimum 2m. below a specified buoy mark and a maximum of 2m above this mark over a 24-hour period. a Construct a formula involving a trigonometric function to describe this situation. The road to an island close to the shore is sometimes covered with water. When the water rises to the level of the road, the road is closed. On a particular day, the water at high tide is 5 m above the mean sea level. b Construct a formula involving a trigonometric function to model this situation if high tide occurs every 12 hours and the tide behaves in simple harmonic motion. Draw a sketch of the model. c Find the height of the road above sea level if the road is closed for 3 hours on the day in question. d If the road were raised so that it is impassable for only 2 hours 20 minutes, by how much was it raised? Question 8 At a certain latitude the number (d) of hours of daylight in each day is given by, d = A + B sin kt° where A and B are positive constants and t is the time in days after the spring equinox. Assuming the number of hours of daylight follows an annual cycle of 365 days; a Find the value of k correct to three decimal places. b If the shortest and longest days have 6 and 18 hours of daylight respectively state the values of A and B. c Find in hours and minutes the amount of daylight on New Year’s day which is 80 days before the spring equinox. d A town at this latitude holds a fair twice a year on days that have exactly 10 hours of daylight. Find, in relation to the spring equinox, which two days these are. © Project Maths Development Team 2012 www.projectmaths.ie 15 Exploring Trigonometric Graphs Solutions Question 1 A wrecking ball attached to a crane swings back and forth. The distance that the ball moves to the left and to the right of its resting position with respect to time is represented by the following graph. a. What is the period of the crane’s motion. Explain your answer? b. What is the equation of the horizontal midway line of the curve and what does it represent? c. What is the amplitude of the crane’s motion? Draw a diagram to represent what the amplitude represents in terms of the motion of the ball? d. How many complete swings will the wrecking ball make in four minutes? e. What does point A represent? a. The period is 8 seconds. It represents the time taken for the wrecking ball to complete a full swing (i.e. from the extreme left to the extreme right and back to the extreme left again). © Project Maths Development Team 2012 www.projectmaths.ie 16 Exploring Trigonometric Graphs b. The equation of the horizontal midway line is d = 0. This represents the resting position of the ball. c. The amplitude is 4 metres. This distance represents the maximum distance the wrecking ball swings to the left and right of its resting position. © Project Maths Development Team 2012 www.projectmaths.ie 17 Exploring Trigonometric Graphs d. 4 minutes = 240 seconds period = 8 seconds number of swings = e. Point A represents the position of the wrecking ball when collecting and recording of the data began Question 2 As a dolphin swims along by the side of a cruise ship he jumps to a height of four metres above the water surface and dives to a depth of four meters below the surface. He does this in a regular motion (simple harmonic motion). A passenger using a stopwatch and starting it just as the dolphin is at the surface determines that the dolphin completes one cycle every 8 seconds. a Describe the displacement of the dolphin relative to the water surface using a sinusoid curve and sketch the graph of the dolphin’s displacement relative to the water surface b After three seconds how high above the surface is the dolphin? c Is the dolphin above or below the surface of the water after 37 seconds? © Project Maths Development Team 2012 www.projectmaths.ie 18 Exploring Trigonometric Graphs a. a ± b sin cx a ± b cos cx a = horizontal midway line a=0 b = aimplitude b=4 period = shape = function normal © Project Maths Development Team 2012 www.projectmaths.ie 19 Exploring Trigonometric Graphs b) After 3 seconds h = 2.828 After 3 seconds the dolphin is 2.828m above the water c) After 37 seconds h = -2.828 The dolphin is below the surface of the water Question 3 The inside rim of a bicycle wheel whose diameter is 25 inches, is 3 inches off the ground. An ant is sitting on the inside rim of the wheel at the point 3 inches off the ground. Sean starts riding the bicycle at a steady rate. The wheel makes one revolution every 1.6 seconds. a Find the equation of a sinusoid curve which describes the motion of the ant and draw a graph of the function. b What height in centimetres from the ground will the ant be 25 seconds into the trip given that 1 inch = 2.54 cm.? c Within the first 10 seconds how many times will the ant be at its starting height? a. Horizontal Midway Line Horizontal Midway Line © Project Maths Development Team 2012 www.projectmaths.ie 20 Exploring Trigonometric Graphs a ± b sin cx a ± b cos cx a = horizontal midway line a = 12.5 b = aimplitude b = 9.5 period c = 1.25π shape = cos function reflected h = height off the ground h = 12.5-9.5cxos1.25πt b. 25 seconds into the trip t = 25 h = 12.5-9.5 cos 1.2π (25) h = 19.218 inches = (19.218) (2.54) = 48.81372cm Tar éis 25 soicind tá an seangán 48.81372cm ón talamh. c. period = 1.6 seconds in 10 seconds cycle 6.25 cycle at starting point: 7 times © Project Maths Development Team 2012 www.projectmaths.ie 21 Exploring Trigonometric Graphs Question 4 The number of people in thousands employed in a resort town is represented by the function Take t = 0 as last day of January Take t = 1 as last day of February Take t = 2 as last day of March a Draw a rough sketch showing the variation in the number of people employed in the town for one complete period. b When is the maximum number of people employed and what is this maximum number? c During which months of the year will the number of people employed be 4650 or greater? d Does this model have any drawbacks and if yes identify one? a. a - b cos cx 3.8-1.7 cos a = horizontal midway line = 3.8 b = aimplitude = 1.7 period shape = cos function reflected © Project Maths Development Team 2012 www.projectmaths.ie 22 Exploring Trigonometric Graphs a. b. Range = [3.8 - 1.7, 3.8 + 1.7] Range = [2.1, 5.5] Maximum number employed = 5,500 people on the last day of July c. © Project Maths Development Team 2012 www.projectmaths.ie 23 Exploring Trigonometric Graphs d. A drawback of this model is that it assumes all months of the year have the same number of days. Question 5 A tsunami (tidal wave) is a fast moving wave caused by an underwater earthquake. The water oscillates about its normal level, with equal amplitudes above and below this level The period is fifteen minutes. Suppose that a tsunami with an amplitude of ten metres approaches the pier at Honolulu, where the normal depth of water is nine metres. Assuming that the depth of water varies sinusoidally with time as the tsunami passes, predict the depth of the water at the following times after the tsunami first reaches the pier. a Two minutes, four minutes and twelve minutes. b According to your model what will be the minimum depth of the water? c How do you interpret this answer in terms of what will happen in the real world? a. © Project Maths Development Team 2012 www.projectmaths.ie 24 Exploring Trigonometric Graphs © Project Maths Development Team 2012 www.projectmaths.ie 25 Exploring Trigonometric Graphs b The minimum level of the water appears to be -1m. c The water level can never be less than 0m. The water would drain away from the pier for a short period of time. Question 6 A buoy in the ocean is bobbing up and down in harmonic motion. At t = 0 seconds, the buoy is at its high point and returns to that high point every 8 seconds. The buoy moves a distance of 1.44 meters from its highest point to its lowest point. a Using a sinusoidal function create a mathematical model to represent the depth of water under the buoy and sketch the curve. b How high is the buoy at time = 29 seconds? c Is it rising or falling at that time? d How many times in 2 minutes will the buoy be at sea level? a. © Project Maths Development Team 2012 www.projectmaths.ie 26 Exploring Trigonometric Graphs normal © Project Maths Development Team 2012 www.projectmaths.ie 27 Exploring Trigonometric Graphs b. c. d. Question 7 If the depth of water in a canal continually varies between a minimum 2m. below a specified buoy mark and a maximum of 2m above this mark over a 24-hour period. a. Construct a formula involving a trigonometric function to describe this situation. The road to an island close to the shore is sometimes covered with water. When the water rises to the level of the road, the road is closed. On a particular day, the water at high tide is 5 m above the mean sea level. b Construct a formula involving a trigonometric function to model this situation if high tide occurs every 12 hours and the tide behaves in simple harmonic motion. Draw a sketch of the model. c Find the height of the road above sea level if the road is closed for 3 hours on the day in question. d If the road were raised so that it is impassable for only 2 hours 20 minutes, by how much was it raised? © Project Maths Development Team 2012 www.projectmaths.ie 28 Exploring Trigonometric Graphs a. b. © Project Maths Development Team 2012 www.projectmaths.ie 29 Exploring Trigonometric Graphs © Project Maths Development Team 2012 www.projectmaths.ie 30 Exploring Trigonometric Graphs c. d. © Project Maths Development Team 2012 www.projectmaths.ie 31 Exploring Trigonometric Graphs Question 8 At a certain latitude the number (d) of hours of daylight in each day is given by, d = A + B sin kt° where A and B are positive constants and t is the time in days after the spring equinox. Assuming the number of hours of daylight follows an annual cycle of 365 days; a Find the value of k correct to three decimal places. b If the shortest and longest days have 6 and 18 hours of daylight respectively state the values of A and B. c Find in hours and minutes the amount of daylight on New Year’s day which is 80 days before the spring equinox. d A town at this latitude holds a fair twice a year on days that have exactly 10 hours of daylight. Find, in relation to the spring equinox, which two days these are. a. © Project Maths Development Team 2012 www.projectmaths.ie 32 Exploring Trigonometric Graphs b. c. d. © Project Maths Development Team 2012 www.projectmaths.ie 33 Exploring Trigonometric Graphs Student Activity Sheet 1 Graphs of sinx and cosx Complete the table below and then draw the graph of sinx and cosx x y = sinx y = cosx 0° 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° 360° 0° x y = sin x y = cos x 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° 360° What is the horizontal midway line of each graph? sin x____________________________________cos x_____________________________________ What is the maximum of each graph? sin x____________________________________cos x_____________________________________ What is the minimum of each graph? sin x____________________________________cos x_____________________________________ What is the range of each graph? sin x____________________________________cos x_____________________________________ The amplitude of a graph is the greatest height the graph is above the horizontal midway line. What is the amplitude of each graph? sin x____________________________________cos x_____________________________________ When a graph repeats itself after a particular interval the interval is known as the period. What is the period of each graph? sin x____________________________________cos x_____________________________________ What difference if any is there between the curves of sin x and cos x? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ © Project Maths Development Team 2012 www.projectmaths.ie 34 Exploring Trigonometric Graphs Card Set A1 Trigonometric Functions g (x) = bsin x A1 A2 g (x) = -3 sin x A3 g (x) = 2 sin x A4 g (x) = ½ sin x A5 g (x) = 5 sin x A6 g (x) = -4 sin x g (x) = ²⁄3 sin x A7 A8 g (x) = -¼ sin x A9 g (x) = sin x A10 g (x) = -sin x © Project Maths Development Team 2012 www.projectmaths.ie g (x) = -¹⁄3 sin x 35 Exploring Trigonometric Graphs Card Set A2 Trigonometric Functions g (x) = bcos x A11 A12 g (x) = -2 cos x A13 g (x) = cos x A14 g (x) = ¼ cos x A15 g (x) = cos x A16 g (x) = -5 cos x g (x) = ½ cos x A17 A18 g (x) = -³⁄2 cos x g (x) = 6cos x A19 A20 g (x) = -cos x g (x) = -¹⁄5 cos x © Project Maths Development Team 2012 www.projectmaths.ie 36 Exploring Trigonometric Graphs Student Activity Sheet 2a: Mathematical Language / Properties of Trigonometric Graphs Card A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 Curve Amplitude Range g (x) = -3 sin x g (x) = 2 sin x g (x) = ½ sin x g (x) = 5 sin x g (x) = -4 sin x g (x) = ²⁄3 sin x g (x) = -¼ sin x g (x) = sin x g (x) = -sin x g (x) = -¹⁄3 sin x Period Vertical Transformation (3) Vertical Shape (3) Horizontal Shape (3) Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink From your observations above what is the effect of b on the curve? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ © Project Maths Development Team 2012 www.projectmaths.ie 37 Exploring Trigonometric Graphs Student Activity Sheet 2b: Mathematical Language / Properties of Trigonometric Graphs Card A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 Curve Amplitude Range g (x) = -2 cos x g (x) = cos x g (x) = ¼ cos x g (x) = cos x g (x) = -5 cos x g (x) = ½ cos x g (x) = -³⁄2 cos x g (x) = 6cos x g (x) = -cos x g (x) = -¹⁄5 cos x Period Vertical Transformation (3) Vertical Shape (3) Horizontal Shape (3) Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink From your observations above what is the effect of b on the curve? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ © Project Maths Development Team 2012 www.projectmaths.ie 38 Exploring Trigonometric Graphs Student Activity Sheet 2c: Mathematical Language / Properties of Trigonometric Graphs Card Curve Amplitude Range Period Vertical Transformation (3) Vertical Shape (3) Horizontal Shape (3) Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink From your observations above what is the effect of b on the curve? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ © Project Maths Development Team 2012 www.projectmaths.ie 39 Exploring Trigonometric Graphs Card Set B: Trigonometric Functions Important features B1 B2 g (x) = cos ½x B3 g (x) = sin 2x B4 g (x) = sin 3x B5 g (x) = cos 4x B6 g (x) = cos 3x B7 g (x) = sin 4⁄5x B8 g (x) = sin 4x B9 g (x) = cos 2⁄3x B10 g (x) = cos 2x © Project Maths Development Team 2012 www.projectmaths.ie g (x) = sin ¹⁄4x 40 Exploring Trigonometric Graphs Student Activity Sheet 3: Mathematical Language / Properties of Trigonometric Graphs Card B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 Curve Amplitude Range g (x) = cos ½x g (x) = sin 2x g (x) = sin 3x g (x) = cos 4x g (x) = cos 3x g (x) = sin 4⁄5x g (x) = sin 4x g (x) = cos 2⁄3x g (x) = cos 2x g (x) = sin ¹⁄4x Period Vertical Transformation (3) Vertical Shape (3) Horizontal Shape (3) Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink Normal Stretch Stretch Reflected Shrink Shrink From your observations above what is the effect of c on the curve? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ © Project Maths Development Team 2012 www.projectmaths.ie 41 Exploring Trigonometric Graphs Card Set C g (x) = 3sin ½x g (x) = 3sin ½x g (x) = 3sin ½x g (x) = -5cos 3x g (x) = -5cos 3x g (x) = -5cos 3x g (x) = ½ sin 4x g (x) = ½ sin 4x g (x) = ½ sin 4x g (x) = -2⁄3 cos ¹⁄4x g (x) = -2⁄3 cos ¹⁄4x g (x) = -2⁄3 cos ¹⁄4x © Project Maths Development Team 2012 www.projectmaths.ie 42 Exploring Trigonometric Graphs Card Set C (continued) g (x) = 5cos 2x g (x) = 5cos 2x g (x) = 5cos 2x g (x) = 1⁄2 sin 2⁄3 x g (x) = 1⁄2 sin 2⁄3 x g (x) = 1⁄2 sin 2⁄3 x g (x) = -4 cos 1⁄3x g (x) = -4 cos 1⁄3x g (x) = -4 cos 1⁄3x g (x) = -2⁄5 cos 6x g (x) = -2⁄5 cos 6x g (x) = -2⁄5 cos 6x © Project Maths Development Team 2012 www.projectmaths.ie 43 Exploring Trigonometric Graphs Card Set C (continued) g (x) = 5cos 2x g (x) = 1⁄2 sin 2⁄3 x g (x) = 1⁄2 sin 2⁄3 x g (x) = 1⁄2 sin 2⁄3 x g (x) = -4 cos 1⁄3x g (x) = -4 cos 1⁄3x © Project Maths Development Team 2012 www.projectmaths.ie 44 Exploring Trigonometric Graphs Student Activity Sheet 4 Mathematical Language / Properties of Trigonometric Graphs Vertical Transformation Normal Reflected Vertical Shape Stretch Shrink Horizontal Shape Shrink © Project Maths Development Team 2012 www.projectmaths.ie Stretch 45 Exploring Trigonometric Graphs Student Activity Sheet 5 Mathematical Language / Properties of Trigonometric Graphs Curve g (x) = g (x) = g (x) = g (x) = g (x) = g (x) = g (x) = g (x) = g (x) = © Project Maths Development Team 2012 Amplitude Range Period 6 ∕3 2π [-4, 4] 2 ∕2 π [-1∕2, 1∕2] 6π [-2, 2] 2π π ¹⁄5 3 3π [-3, 3] 4π ¼ www.projectmaths.ie 46 Exploring Trigonometric Graphs Student Activity Sheet 6: Transformation of Trigonometric Graphs Curve Effect of the value a in Horizontal Midway Line Amplitude Range h(x) =1 + sin x h(x) =2 + sin x h(x) =3 + sin x h(x) =-1 + sin x h(x) =-2 + sin x h(x) =-3 + sin x h(x) =1 + cos x h(x) =2 + cos x h(x) =3 + cos x h(x) =-1 + cos x h(x) =-2 + cos x h(x) =-3 + cos x From your observations above what is the effect of a on the curve? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ © Project Maths Development Team 2012 www.projectmaths.ie 47 Exploring Trigonometric Graphs Student Activity Sheet 7: Mathematical Language / Properties of Trigonometric Graphs Curve a b c d e f g h i j k l m n o Vertical Translation Vertical Shape Amplitude Range Period g (x) = 2 + 3 sin 4 g (x) = 3 - 4 cos 2x g (x) = -4 - 2 sin 3 g (x) = -1 + 5 sin 2 g (x) = 5 - cos x⁄2 g (x) = 3 sin 3x g (x) = 1 + cos 4x g (x) = ½ + 5 cos x g (x) = 3 - 3 sin3x g (x) = ¼ - ½ sin ¹∕3x g (x) = 2 + 3 sin 4x g (x) = 1 + 3 cosx g (x) = 4 + 3 cos ½x g (x) = 5 sinx +1 g (x) = -1 -sinx 2∕3x © Project Maths Development Team 2012 www.projectmaths.ie 48 Exploring Trigonometric Graphs Card Set D: Trigonometric Graphs D1 D2 D3 D4 © Project Maths Development Team 2012 www.projectmaths.ie 49 Exploring Trigonometric Graphs Card Set D: Trigonometric Graphs (continued) D5 D6 D7 D8 © Project Maths Development Team 2012 www.projectmaths.ie 50 Exploring Trigonometric Graphs Student Activity Sheet 8: Trigonometric Graphs Graph Equation Horizontal Midway line Function Type And Vertical Shape Amplitude Range Number of cycles in 2π Period D1 D2 D3 D4 D5 D6 D7 D8 © Project Maths Development Team 2012 www.projectmaths.ie 51 Exploring Trigonometric Graphs Card Set E: Trigonometric Graphs E1 E2 f (x) = ½ -2 sin 3x f (x) = 1 + sin ³⁄2 x E3 E4 f (x) = -2 + 3 cos ¹⁄3x f (x) = 1 + 2 sin 3x E5 E6 f (x) = 2 - ½ sin 5x f (x) = -1 + ½ cos 3x E7 E8 f (x) = 3 -2 cos ½x f (x) = -2 - cos 4x © Project Maths Development Team 2012 www.projectmaths.ie 52 Exploring Trigonometric Graphs Card Set F F1 F2 k (x) = 2 + 3 sin 4x k (x) = 5 - 4 cos 2x F3 F4 k (x) = 1 + 3 cos ½ x k (x) = ½ - sin 3x F5 F6 k (x) = 3 + 2 cos ¹⁄4 x k (x) = 1 - ½ sin x F7 F8 k (x) = 2 + ¹⁄4 cos 3x F9 k (x) = 4 sin 2x F10 k (x) = -1 -2 sin 3x k (x) = -2 + 3 cos 4x © Project Maths Development Team 2012 www.projectmaths.ie 53

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