Transcript 4.6 PP
4.6 Graphs of Composite Trigonometric Functions Copyright © 2011 Pearson, Inc. 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 © 2011 Pearson, Inc. Slide 4.6 - 2 Example Combining the Cosine Function with x2 2 Graph y = ( cos x ) and state whether the function appears to be periodic. Copyright © 2011 Pearson, Inc. Slide 4.6 - 3 Example Combining the Cosine Function with x2 2 Graph y = ( cos x ) and state whether the function appears to be periodic. The function appears to be periodic. Copyright © 2011 Pearson, Inc. Slide 4.6 - 4 Example Combining the Cosine Function with x2 ( ) Graph y = cos x 2 and state whether the function appears to be periodic. Copyright © 2011 Pearson, Inc. Slide 4.6 - 5 Example Combining the Cosine Function with x2 ( ) Graph y = cos x 2 and state whether the function appears to be periodic. The function appears not to be periodic. Copyright © 2011 Pearson, Inc. Slide 4.6 - 6 Example Adding a Sinusoid to a Linear Function x Graph f ( x ) = cos x - and state its domain and range. 3 Copyright © 2011 Pearson, Inc. Slide 4.6 - 7 Example Adding a Sinusoid to a Linear Function x Graph f ( x ) = cos x - and state its domain and range. 3 The function f is the sum of the functions g ( x ) = cos x x and h ( x ) = - . 3 Here's the graph of f = g + h. Domain: ( -¥,¥ ) Range: ( -¥,¥ ) Copyright © 2011 Pearson, Inc. Slide 4.6 - 8 Sums That Are Sinusoids Functions If y1 = a1 sin(b(x - h1 )) and y2 = a2 cos(b(x - h2 )), then y1 + y2 = a1 sin(b(x - h1 )) + a2 cos(b(x - h2 )) is a sinusoid with period 2p b . Copyright © 2011 Pearson, Inc. Slide 4.6 - 9 Example Identifying a Sinusoid Determine whether the following function is or is not a sinusoid. f (x) = 3cos x + 5sin x Copyright © 2011 Pearson, Inc. Slide 4.6 - 10 Example Identifying a Sinusoid Determine whether the following function is or is not a sinusoid. f (x) = 3cos x + 5sin x Yes, since both functions in the sum have period 2p . Copyright © 2011 Pearson, Inc. Slide 4.6 - 11 Example Identifying a Sinusoid Determine whether the following function is or is not a sinusoid. f (x) = cos 3x + sin 5x Copyright © 2011 Pearson, Inc. Slide 4.6 - 12 Example Identifying a Sinusoid Determine whether the following function is or is not a sinusoid. f (x) = cos 3x + sin 5x No, since cos 3x has period 2p / 3 and sin 5x has period 2p / 5. Copyright © 2011 Pearson, Inc. Slide 4.6 - 13 Damped Oscillation The graph of y = f (x)cosbx (or y = f (x)sinbx) 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 © 2011 Pearson, Inc. Slide 4.6 - 14 Quick Review Copyright © 2011 Pearson, Inc. Slide 4.6 - 15 Quick Review Solutions Copyright © 2011 Pearson, Inc. Slide 4.6 - 16