Transcript Slide 1
Homework, Page 392 Find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sin x. 1. y = 2 sin x The graph of y = 2 sin x may be obtained from the graph of y = sin x by applying a vertical stretch of 2. y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1 Homework, Page 392 Find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sin x. 5. y = 0.73 sin x The graph of y = 2 sin x may be obtained from the graph of y = sin x by applying a vertical shrink of 0.73. y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 2 Homework, Page 392 Find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = cos x. 9. y cos 7 x 2 y cos 7 x cos 7 x p 7 The graph of y = cos (–7 x) may be obtained from the graph of y = cos x by applying a horizontal shrink of 1/7. y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 3 Homework, Page 392 Find the amplitude, period, and frequency of the function and use this information to sketch a graph of the function in the window [–3π, 3π] by [–4,4]. x 13. y 3sin 2 y a3 2 p 4 1 2 1 f 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley x Slide 4- 4 Homework, Page 392 Graph one period of the function. Show the scale on both axes 17. y 2sin x y 2sin x a2 2 p 2 1 y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5 Homework, Page 392 Graph one period of the function. Show the scale on both axes 21. y 0.5sin x y 0.5sin x a 0.5 2 p 2 1 y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6 Homework, Page 392 Graph three period of the function. Show the scale on both axes. 25. y 0.5cos3x y 0.5cos 3 x a 0.5 2 p 3 y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7 Homework, Page 392 Specify the period and amplitude of each function. Give the viewing window in which the graph is shown. 29. y 1.5sin 2 x y 1.5cos 2 x a 1.5 2 p 2 The viewing window is 2 , 2 by 2, 2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 8 Homework, Page 392 Specify the period and amplitude of each function. Give the viewing window in which the graph is shown. 33. y 4sin x 3 y 4sin 3 x a4 2 p 6 3 The viewing window is 3,3 by 5,5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 Homework, Page 392 Identify the maximum and minimum values and the zeroes of the function in the interval [–2π, 2π]. 37. y cos 2 x y y cos 2 x a 1 x 2 p 2 The function has a maximum y-value of 1 and a minimum y-value of 1. 7 5 3 3 5 7 The zeroes of the function are at x , , , , , , , 4 4 4 4 4 4 4 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 10 Homework, Page 392 41. Write the functon y sin x as a phase shift of y sin x. y sin x sin x . y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 11 Homework, Page 392 Describe the transformations required to obtain the graph of the given function from a basic trigonometric graph. 2 x 45. y cos 3 3 2 x To obtain the graph of y cos from the graph of y cos x, 3 3 2 apply a vertical shrink of , a horizontal stretch of 3, and reflect 3 about the x-axis. y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 12 Homework, Page 392 Describe the transformations required to obtain the graph of y2 from the graph of y1. 5 49. y1 cos 2 x and y2 cos 2 x 3 To obtain the graph of y2 from the graph of y1 , 5 apply a vertical stretch of . 3 y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 13 Homework, Page 392 Select the pair of functions that have identical graphs.. 53. a y cos x b y sin x 2 c y cos x 2 a and b y x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 14 Homework, Page 392 Construct a sinusoid with the given amplitude that goes through the given point. 57. Amplitude 3, period , point 0, 0 2 a 3; p b2 b y 3sin 2 x c 0 3sin 2 0 c y 3sin 2 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 15 Homework, Page 392 State the amplitude and period of the sinusoid and (relative to the basic function) the phase shift and vertical translation. 61.y 2sin x 1 4 The function has an amplitude of 2, a period of 2 π, a phase shift of 3π/4, and a vertical translation of +1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 16 Homework, Page 392 State the amplitude and period of the sinusoid and (relative to the basic function) the phase shift and vertical translation. 65. y 2cos 2 x 1 The function has an amplitude of 2, a period of 1, no phase shift, and a vertical translation of +1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 17 Homework, Page 392 Find values of a, b, h, and k so that the graph of the function y = a sin (b(x – h)) + k. 69. y 2sin 2 x a 2, b 2, h 0, k 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 18 Homework, Page 392 73. A Ferris wheel 50 ft in diameter makes one revolution every 40 sec. If the center of the wheel is 30 ft above the ground, how long after reaching the low point is a rider 50 ft above the ground? 2 50 p 40 b b 20 a 2 25, k 30 x y 25cos 30 20 The rider will be 50-ft above the ground 15.903 sec after passing the low point. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 19 Homework, Page 392 77. A block mounted on a spring is set into motion directly above a motion detector, which registers the distance to the block in 0.1 sec intervals. When the block is released, it is 7.2 cm above the detector. The table shows the data collected by the motion detector during the first two sec, with distance d measured in cm. t d t 0.1 9.2 1.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 13.9 18.8 21.4 20.0 15.6 10.5 7.4 1.2 1.3 1.4 1.5 1.6 1.7 1.8 d 17.3 20.8 20.8 17.2 12.0 8.1 7.5 0.9 8.1 1.9 1.0 12.1 2.0 10.5 15.6 19.9 a. Make a scatterplot of d as a function of t and estimate the maximum value of d visually. Use this number and the stated minimum of 7.2 to compute the amplitude. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 20 Homework, Page 392 77. a. Make a scatterplot of d as a function of t and estimate the maximum value of d visually. Use this number and the stated minimum of 7.2 to compute the amplitude. M m 21.4 7.2 14.2 a 7.1 2 2 2 b. Estimate the period of the motion from the scatter plot. p 1.25 0.4 0.85sec Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 21 Homework, Page 392 77. c. Model the motion of the block as a sinusoidal function d (t). 2 2 a 7.1; p 0.85 b 2.353 b 0.85 M m 21.4 7.2 k 14.3 d t 7.1cos 2.353 t 14.3 2 2 d. Graph the function with the scatterplot to support the model graphically. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 22 Homework, Page 392 81. The graph of y = sin 2x has half the period of the graph of y = sin 4x. Justify your answer. False, the graph of y = sin 2x has twice the period of the graph of 2 2 y = sin 4x because p 2 p Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 2 2 2 Slide 4- 23 Homework, Page 392 85. The period of the function f (x) = 210 sin (420x +840) is a. 840 b. c. d. e. 420 210 2 2 p b 420 210 210 420 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 24 Homework, Page 392 89. A piano tuner strikes a tuning fork for the note middle C and creates a sound wave modeled by y = 1.5 sin 524 πt, where t is the time in seconds. 2 2 1 (a) What is the period of the function? p b 524 262 (b) What is the frequency f = 1/p of this note? 1 1 f 262 1 p (c) Graph the function. 262 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 25 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What you’ll learn about The Tangent Function The Cotangent Function The Secant Function The Cosecant Function … and why This will give us functions for the remaining trigonometric ratios. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 27 Asymptotes of the Tangent Function Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 28 Zeros of the Tangent Function Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 29 Asymptotes of the Cotangent Function Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 30 Zeros of the Cotangent Function Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 31 The Secant Function Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 32 The Cosecant Function Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 33 Basic Trigonometry Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 34 Example Analyzing Trigonometric Functions Analyze the function for domain, range, continuity, increasing or decreasing, symmetry, boundedness, extrema, asymptotes, and end behavior f x sec x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 35 Example Transformations of Trigonometric Functions Describe the transformations required to obtain the graph of the given function from a basic trigonometric function. 1 f x 2sec x 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 36 Example Solving Trigonometric Equations Solve the equation for x in the given interval. sec x 2, x 3 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 37 Example Solving Trigonometric Equations With a Calculator Solve the equation for x in the given interval. csc x 1.5, x 3 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 38 Example Solving Trigonometric Word Problems A hot air balloon is being blow due east from point P and traveling at a constant height of 800 ft. The angle y is formed by the ground and the line of vision from point P to the balloon. The angle changes as the balloon travels. a. Express the horizontal distance x as a function of the angle y. b. When the angle is , what is the horizontal distance from 20 P? c. An angle of 20 is equivalent to how many degrees? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 39 Homework Homework Assignment #30 Read Section 4.6 Page 401, Exercises: 1 – 65 (EOO) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 40 4.6 Graphs of Composite Trigonometric Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review State the domain and range of the function. 1. f ( x) -3sin 2 x 2. f ( x) | x | 2 3. f ( x) 2 cos 3 x 4. Describe the behavior of y e as x . -3 x 5. Find f g and g f , given f ( x) x 3 and g ( x) x 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 42 Quick Review Solutions State the domain and range of the function. 1. f ( x) -3sin 2 x Domain: , Range: 3,3 2. f ( x) | x | 2 Domain: , Range: 2, 3. f ( x) 2 cos 3 x Domain: , Range: 2, 2 4. Describe the behavior of y e as x . -3 x lim e x 3 x 0 5. Find f g and g f , given f ( x) x 3 and g ( x ) x 2 f g x 3; g f x 3 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 43 What you’ll learn about Combining Trigonometric and Algebraic Functions Sums and Differences of Sinusoids Damped Oscillation … and why Function composition extends our ability to model periodic phenomena like heartbeats and sound waves. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 44 Example Combining the Cosine Function with x2 Graph y cos x and state whether the function 2 appears to be periodic. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 45 Example Combining the Cosine Function with x2 Graph y cos x 2 and state whether the function appears to be periodic. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 46 Sums That Are Sinusoidal Functions If y a sin(b( x h )) and y a cos(b( x h )), then 1 1 1 2 2 2 y y a sin(b( x h )) a cos(b( x h )) is a 1 2 1 1 2 2 sinusoid with period 2 / | b|. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 47 Sums That Are Not Sinusoidal Functions If y a sin(b( x h )) and y f ( x) where f ( x) is not 1 1 1 2 a sin(b( x h )) or a cos(b( x h )), but another 2 2 2 2 trigonometric function, then y y is a periodic 1 2 function, but not a sinusoid. If y f ( x) is not a trigonometric function, then y y 2 1 2 is neither periodic nor sinusoidal. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 48 Example Identifying a Sinusoid Determine whether the following function is or is not a sinusoid: f ( x) 3cos x 5sin x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 49 Example Identifying a Sinusoid Determine whether the following function is or is not a sinusoid: f ( x) cos3x sin 5x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 50 Example Identifying a Non-Sinusoid Determine whether the following function is or is not a sinusoid: f ( x) 3x sin 5x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 51 Damped Oscillation The graph of y f ( x)cos bx (or y f ( x)sin bx) oscillates between the graphs of y f ( x) and y - f ( x). When this reduces the amplitude of the wave, it is called damped oscillation. The factor f ( x) is called the damping factor. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 52 Example Working with Damped Oscillation The oscillations of a spring subject to friction are modeled by the equation y 0.43e cos1.8t. 0.55 t a Graph y and its two damping curves in the same viewing window for 0 t 12. b Approximately how long does it take for the spring to be damped so that 0.2 y 0.2? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 53