680 CHAPTER 6 The Circular Functions and Their Graphs Graph each function over a two-period interval. Identify asymptotes when applicable. 23. y = sin12x + p2 24. y = 2 + cos x 25. y = -1 + 2 sin1x + p2 26. y = tan ax - p 2b 27. y = -2 - cot ax - p 2b 28. y = -csc 2x (Modeling) Solve each problem. 29. Average Monthly Temperature The average monthly temperature (in °F) in San Antonio, Texas, can be modeled by the function ƒ 1x2 = 16.5 sin c p 6 1x - 42d + 68.5, where x is the month, with x = 1 corresponding to January, x = 2 corresponding to February, and so on. (Data from National Climatic Data Center.) (a) Graph ƒ in the window 30, 254 by 340, 904. (b) Determine the amplitude, period, phase shift, and vertical translation of ƒ. (c) What is the average monthly temperature for the month of December? (d) Determine the minimum and maximum average monthly temperatures and the months when they occur. (e) What would be an approximation for the average annual temperature in San Antonio? How is this related to the vertical translation of the sine function in the formula for ƒ? 30. Spring Motion The position of a weight attached to a spring is s1t2 = -4 cos 8pt inches after t seconds. (a) Find the maximum height that the weight rises above the equilibrium position of s1t2 = 0. (b) When does the weight first reach its maximum height if t Ú 0? (c) What are the frequency and period?
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