# Transformations and Sinusoidal Functions

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1 Section 3. Curriculum Outcomes Periodic Behaviour Demonstrate an understanding of realworld relationships by translating between graphs, tables, and written descriptions C8 Identify periodic relations and describe their characteristics C3 Drawing scatter plots Analyzing graphs to obtain specific information Graphing using technology Related Activities Review how real-world relationships can be modeled using graphs Collect experimental data regarding a periodic relationship and answer questions regarding that relationship Analyze situations and/or graphs to determine whether the relationship is periodic Analyze sinusoidal functions (one type of periodic function) and use appropriate terminology Use appropriate terminology to analyze graphs and determine whether a function is periodic and/or sinusoidal using graphing technology Assumed Prior Knowledge Page in Text Section 3. Curriculum Outcomes Analyze and apply the graphs of the sine and cosine functions B5 Model situations with sinusoidal functions C Create and analyze scatter plots of periodic data C Determine the equations of sinusoidal functions C3 Analyze tables and graphs of various sine and cosine functions to find patterns, identify characteristics, and determine equations C9 Describe how various changes in the parameters of sinusoidal equations affect their graphs C Transformations and Sinusoidal Functions Related Activities Complete investigations and 3 and discover that graphs of sinusoidal functions can be expressed as transformations of y = sin x Discover that graphing sinusoidal functions using transformations can be accomplished using two techniques Graph sinusoidal functions as transformations of y = sin x Discover the relationship between the functions y = sin x and y = cos x, and recognize that all sinusoidal phenomena can be modeled using transformations of y = sin x as well as y = cos x Graph a sinusoidal function as a transformation of y = cos x; elaborate on the definition of a sinusoidal function Find the equation of a sinusoidal function given a specific situation or graph Assumed Prior Knowledge Graphing using transformations; reflection in the x-axis, vertical stretch, horizontal translation, vertical translation Familiarity with the terms domain, range, maximum, minimum, zeros of a function, absolute value Familiarity with function notation Graphing using technology Page in Text

2 Name: Investigation Let It Roll Police often use trundle wheels in the process of investigating car accidents. A trundle wheel consists of a circular disk (usually one metre in circumference) that is connected to a handle. A counter keeps track of the number of rotations that the wheel has made from its origin. A point on the wheel marks this number. These devices are used to measure the length of skid marks on the pavement so that police can determine the speed of the car at the time of the accident. Purpose: To determine how the height of a point on the edge of a circular object changes with the distance the object rolled. A. Create your own trundle wheel. Place a small piece of masking tape on the bottom edge of an empty can. Measure the diameter and find the circumference of the can. Stick a measuring tape to the floor and place the can on its side to line up the masking tape with 0 cm mark on the tape. Record the diameter of your can. cm B. Note: The horizontal distance must be measured from the start to bottom of the can in each position. Roll the can a short distance. Measure the height of the tape is above the floor and the distance the tape has moved horizontally. Repeat. Record 0 ordered pairs, 0 for each revolution in the table. 5

3 Start of nd Revolution Measurement Number ( rev s, 0 meas. Each) Distance from start (cm) Height of tape (cm) C.. Enter the two columns of data into L and L on the calculator.. Use Stat Plot to draw a scatter plot with the points connected. 3. Sketch the connected graph from your calculator onto the grid below. 53

4 Investigation Questions 3. How far does the can roll before the graph of the function begins to repeat? Describe the function as completely as possible. 4. What does the distance determined in question refer to? Radius of the can The diameter of the can The circumference of the can The area of the bottom of the can 5. What does the maximum height represent? The radius of the can The diameter of the can The circumference of the can The area of the bottom of the can 6. How far do you have to roll the can to be confident that the pattern can be used to predict the height of the tape for any distance? 7. How would the graph change if you placed the tape cm from the edge rather than on the edge? Sketch the graph below

5 Investigation Teacher Notes In doing Investigation, you have a choice; either spend time with the students creating the water wheel or use a different setup on the calculator. To save time use this setup and you will speed things up greatly. If you explain where the nail is on the unit circle when you start you can complete each of the rotations very quickly and the students can readily see the height of the nail for each of the rotations. 55

6 Name: Investigation The Water Wheel Problem Purpose: Determine the equation of the function that models the height of a nail on a water wheel as the wheel rotates. A. Half of a water wheel is submerged below high spring floodwaters. The wheel has a radius of m. A nail on the circumference of the wheel is positioned at water level initially. As the current flows down the river, the wheel rotates. B. Press the MODE key on your calculator and choose Degree and Pol. In WINDOW set θ min = 0, θ max = 70, θstep = 5, X min = 5,. X max = 5,. Xscl =, Y min =. 5, Y max = 5,. and Yscl =. Press Y= and set r=. Push GRAPH. C. Press TRACE to get on the graph and use the values shown in the table to determine the height of the nail to complete the table of values below. Rotation θ a f Height, h, of nail m) D. Use the table to draw a graph of height versus angle of rotation on the graph paper below and label appropriately

7 Investigation Questions. a. What do the local maxima on this graph represent? b. What do the local minima on this graph represent? c. What do the zeros on this graph represent? a f Rotation θ Sinθ. Complete this table using your calculator using rotations that are multiples of How does this compare with the one in Step C of this Investigation? 4. Which one of these functions would best model this situation? i.) θ = sinh ii.) h = sinθ iii.) h = cos θ iv.) h = 90θ v.) θ = h vi.) h = θ Confirm your answer by using the TRACE or TABLE features on the calculator. Note the WINDOW settings and write the domain and the range of this function. 5. Determine the height of the nail if the wheel rotates 60 from its initial position. 6. Determine the amplitude, period, and equation of the sinusoidal axis for this function. 7. Determine and describe five key points that would allow you to draw a sine curve quickly. 57

8 Name: Investigation 3 Purpose: Transformations through the water wheel problem Determine how the equation of the sinusoidal function for the water wheel situation would be affected by changes to some aspects of the situation. A. From Investigation, review the information and complete the table below. Notice that the table does not include as many ordered pairs as before. af Rotation θ Height, h, of nail (m) B. In each of the cases, the water wheel situation has been changed slightly so that the table of values, the equation of the function, and the resulting graph are different from those in investigation. The mapping rule must be obtained by comparing the ordered pairs in the original situation to the ordered pairs in the new situation. Look at each of the cases on pages 06 and 07 and complete each of the tables below. Case Table θ h (m) 00 Equation Function Form: Graph (include a scale) Transformational Form: 0 Mapping Rule ( θ, h) à (, ) Transformation: Local Maximum: (, ) Y-Intercept: Local Minimum: (, ) Sinusoidal Axis: Amplitude: Period: 58

9 Case Table θ h (m) 00 Equation Function Form: 70 0 Graph (include a scale) Transformational Form: 3600 Mapping Rule ( θ, h) à (, ) Transformation: Local Maximum: (, ) Y-Intercept: Local Minimum: (, ) Sinusoidal Axis: Amplitude: Period: Case 3 Table θ h (m) -900 Graph (include a scale) Equation Transformational Form: 0 70 Function Form: 0 Mapping Rule ( θ, h) à (, ) Transformation: Local Maximum: (, ) Y-Intercept: Local Minimum: (, ) Sinusoidal Axis: Amplitude: Period: 59

10 Case 4 Table θ h (m) 00 Equation Function Form: 35 0 Graph (include a scale) Transformational Form: 800 Mapping Rule ( θ, h) à (, ) Transformation: Local Maximum: (, ) Y-Intercept: Local Minimum: (, ) Sinusoidal Axis: Amplitude: Period: 60

11 Name: Investigation 4 y = cos x Purpose: Compare the functions y = sin x and y = cos x in all respects. E. Fill in the following tables below with the appropriate values. θ sinθ θ cosθ F. Using a different color for each graph, plot the points on the grid below

12 Investigation Questions. Determine and describe the five key points that would allow you to draw the cosine curve quickly?. Compare the functions y = cos x and y = sin x. How are they the same? How are they different? Include appropriate terms in your comparison, such as periodic, sinusoidal, domain, range, local maxima, amplitude, and so on. 3. What transformation would map the function y = cos x onto y = sin x? 4. Can all sinusoidal phenomena be modeled using transformations of y = sin x? Can all sinusoidal phenomena be modeled using transformations of y = cos x? Explain. 6

13 Worksheet Math 04/05 Graphs of Trig Functions. Graph each function using transformations b g b g b g a f b g b g a.) y = cos x + 60 b.) y = cos x c.) by + g = sin 3b x 45 g d.) y 3 = cos x e.) y = sinbx + 45 g f.) y = cos ( x ) Determine the amplitude, period, and equation of the sinusoidal axis for each function. State the mapping rule. a.) by 6g = sin bx + 30 g b.) 3y = cos4bx 45 g c.) by + g = cos x d.) by + = x g cos b 60 g e.) 4 y 5 = sin 0.5 x 50 f.) y = sin5 x g.) y b g b g b g = cos3bx 40 g + 5 h.) y 3sin bx g = Determine the equation of each sinusoidal function, first as a transformation ofy = sin x, and then as a transformation of y = cos x. a.) b.)

14 c.) d.) e.) f.)

15 g.) h.) i.) j.)

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