Exploring Graphs of Periodic Functions
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1 8.2 Eploring Graphs of Periodic Functions GOAL Investigate the characteristics of the graphs of sine and cosine functions. EXPLORE the Math Carissa and Benjamin created a spinner. The glued graph paper to a board and centred the pointer at the origin. The turned the pointer counterclockwise through two complete turns. YOU WILL NEED ruler templates graphing technolog Carissa graphed the distance from the tip of the pointer to the horizontal ais as a function of the angle,, it had rotated. Benjamin graphed the distance from the tip of the pointer to the vertical ais as a function of the angle,, it had rotated. Benjamin s distances Carissa s distances The students compared their graphs and eamined: the number of -intercepts the domain the -intercept the range? What would Carissa s and Benjamin s graphs look like for two complete spins, and what are the characteristics of these graphs? NEL 8.2 Eploring Graphs of Periodic Functions 52
2 ic function A function whose graph repeats in regular intervals or ccles. The horizontal line halfwa between the maimum and minimum values of a ic function. The distance from the to either the maimum or minimum value of a ic function; the is alwas epressed as a positive number. maimum The length of the interval of the domain to complete one ccle. maimum ccle minimum minimum Reflecting A. Based on our observations, which characteristics of the graphs of ic functions are similar to characteristics of the graphs of polnomial, eponential, and logarithmic functions ou have studied? B. Based on our observations, which characteristics of the graphs of ic functions differ from characteristics of the graphs of polnomial, eponential, and logarithmic functions ou have studied? C. Benjamin claims that if he continued drawing his graph for several more revolutions, he would notice a repeating pattern. This repeating pattern would result in a ic function with several more -intercepts. Do ou agree or disagree? Eplain. D. Carissa claims that if she continued drawing her graph for several more complete revolutions, the range would remain the same. Do ou agree or disagree? Eplain. E. Describe the relationships among the range, the, and the of each graph. F. What is the of each graph? Eplain how the helps to describe a characteristic of the graph of a ic function. G. Ensure that our graphing technolog is in degree mode, then graph the ic function 5 sin for the domain 5 # # 72, [ R6. How does this graph compare with Carissa s graph? Eplain the reasons for an similarities or differences. H. Graph the ic function 5 cos for the domain 5 # # 72, [ R6 ou used in part G. How does this graph compare with Benjamin s graph? Eplain the reasons for an similarities or differences. I. Ensure that our graphing technolog is in radian mode, then graph 5 sin and 5 cos for the domain 5 # # 4p, [ R6. Compare these graphs with the graphs ou drew in parts G and H. 522 Chapter 8 Sinusoidal Functions NEL
3 In Summar Ke Ideas The function 5 sin is a ic function. - - sin The function 5 cos is a ic function. cos sin cos The graphs of 5 sin and 5 cos have the following common characteristics: - multiple -intercepts - one -intercept - a domain of 5 [ R6 - a range of 5 2 # #, [ R6 - an of - a of 36 or 2p - a defined b the equation 5 Need to Know The graphs of 5 sin and 5 cos are congruent curves. - 9 cos() sin() sin() - 2 cos() sin() The of the curves, 5, is the horizontal line halfwa between the maimum and minimum values. The two graphs oscillate about this line. The of a graph is the length of one complete ccle. NEL 8.2 Eploring Graphs of Periodic Functions 523
4 Tip Communication Graphing technolog can graph ic functions in both degree and radian mode. Make sure our technolog is in the desired mode before ou enter the function and graph it. further Your Understanding. a) Using graphing technolog and degree measure, graph 5 sin and 5 cos on the same aes, for the domain 5 # # 72, [ R6. b) What is the value of cos when the value of sin is a maimum? When is the value of cos a minimum? c) What is the value of sin when the value of cos is a maimum? When is the value of sin a minimum? d) Repeat part a) in radian measure for the domain 5 # # 4p, [ R6. 2. The of the function is 36 or 2p. Eplain wh. 5 sin 3. Compare the graph of the function 5 sin from to 8 and from 8 to 36. Is the graph smmetrical? If so, in what wa? 4. Compare the graph of the function 5 cos from 9 to 27 and from 27 to 45. Is the graph smmetrical? If so, in what wa? 5. What is the -intercept of the graph of each function? a) 5 sin b) 5 cos 6. Determine the -intercepts of each graph over the interval from to 72. a) 5 sin b) 5 cos 524 Chapter 8 Sinusoidal Functions NEL
5 7. The graph of 5 sin is sometimes called a sine wave. Discuss with a partner wh this description is appropriate. 8. Adriana sas that a sine graph is alwas a cosine graph translated left b 9. Do ou agree? Eplain. 9. Suppose that Carissa put the pointer at the top of the circle and spun it counterclockwise. What would the graph representing the distance from the tip of the pointer to the horizontal ais look like? Eplain NEL 8.2 Eploring Graphs of Periodic Functions 525
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