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Transformation of sine and cosine functions Sections 8.2 and 8.3 Revisit: Page 142; chapter 4 Copyright © 2009 Pearson Education, Inc.

Section 8.2 and 8.3 Graphs of Transformed Sine and Cosine Functions Graph transformations of y = sin x and y = cos x in the form y = A sin B (x – h) + k and y = A cos B (x – h) + k and determine the amplitude, the period, and the phase shift. Graph sums of functions. Graph functions (damped oscillations) found by multiplying trigonometric functions by other functions. Copyright © 2009 Pearson Education, Inc.

Variations of the Basic Graphs We are interested in the graphs of functions in the form y = A sin B (x – h) + k and y = A cos B (x – h) + k where A, B, h, and k are all constants. These constants have the effect of translating, reflecting, stretching, and shrinking the basic graphs. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant k Let’s observe the effect of the constant k. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant k Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant k The constant D in y = A sin B (x – h) + k and y = A cos B (x – h) + k translates the graphs up k units if k > 0 or down |k| units if k < 0. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant A Let’s observe the effect of the constant A. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant A Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant A If |A| > 1, then there will be a vertical stretching. If |A| < 1, then there will be a vertical shrinking. If A < 0, the graph is also reflected across the x-axis. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Amplitude The amplitude of the graphs of y = A sin B (x – h) + k and y = A cos B (x – h) + k is |A|. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant B Let’s observe the effect of the constant B. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant B Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant B Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant B Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant B If |B| < 1, then there will be a horizontal stretching. If |B| > 1, then there will be a horizontal shrinking. If B < 0, the graph is also reflected across the y-axis. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Period The period of the graphs of y = A sin B (x – h) + k and y = A cos B (x – h) + k is Copyright © 2009 Pearson Education, Inc.

Period: the horizontal distance between two consecutive max/min values The period of the graphs of y = A csc B(x – h) + k and y = A sec B(x – h) + k is Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Period The period of the graphs of y = A tan B(x – h) + k and y = A cot B(x – C) + k is Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant h Let’s observe the effect of the constant C. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant h Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant h Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant h Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. The Constant h If B = 1, then if |h| < 0, then there will be a horizontal translation of |h| units to the right, and if |h| > 0, then there will be a horizontal translation of |h| units to the left. Copyright © 2009 Pearson Education, Inc.

Combined Transformations B careful! y = A sin (Bx – h) + k and y = A cos (Bx – h) + k as and Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Phase Shift The phase shift of the graphs and is the quantity Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Phase Shift If h/B > 0, the graph is translated to the right |h/B| units. If h/B < 0, the graph is translated to the right |h/B| units. Copyright © 2009 Pearson Education, Inc.

Transformations of Sine and Cosine Functions To graph and follow the steps listed below in the order in which they are listed. Copyright © 2009 Pearson Education, Inc.

Transformations of Sine and Cosine Functions 1. Stretch or shrink the graph horizontally according to B. |B| < 1 Stretch horizontally |B| > 1 Shrink horizontally B < 0 Reflect across the y-axis The period is Copyright © 2009 Pearson Education, Inc.

Transformations of Sine and Cosine Functions 2. Stretch or shrink the graph vertically according to A. |A| < 1 Shrink vertically |A| > 1 Stretch vertically A < 0 Reflect across the x-axis The amplitude is A. Copyright © 2009 Pearson Education, Inc.

Transformations of Sine and Cosine Functions 3. Translate the graph horizontally according to C/B. The phase shift is Copyright © 2009 Pearson Education, Inc.

Transformations of Sine and Cosine Functions 4. Translate the graph vertically according to k. k < 0 |k| units down k > 0 k units up Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Homework 1. Transformation of Sine Cosine functions. Sec 8.2 Written exercises #1-10 all. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Sketch the graph of Find the amplitude, the period, and the phase shift. Solution: Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solution continued To create the final graph, we begin with the basic sine curve, y = sin x. Then we sketch graphs of each of the following equations in sequence. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solution continued Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solution continued Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solution continued Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solution continued Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solution continued Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Graph: y = 2 sin x + sin 2x Solution: Graph: y = 2 sin x and y = sin 2x on the same axes. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solution continued Graphically add some y-coordinates, or ordinates, to obtain points on the graph that we seek. At x = π/4, transfer h up to add it to 2 sin x, yielding P1. At x = – π/4, transfer m down to add it to 2 sin x, yielding P2. At x = – 5π/4, add the negative ordinate of sin 2x to the positive ordinate of 2 sin x, yielding P3. This method is called addition of ordinates, because we add the y-values (ordinates) of y = sin 2x to the y-values (ordinates) of y = 2 sin x. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solution continued The period of the sum 2 sin x + sin 2x is 2π, the least common multiple of 2π and π. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Sketch a graph of Solution f is the product of two functions g and h, where To find the function values, we can multiply ordinates. Start with The graph crosses the x-axis at values of x for which sin x = 0, kπ for integer values of k. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solution continued f is constrained between the graphs of y = –e–x/2 and y = e–x/2. Start by graphing these functions using dashed lines. Since f(x) = 0 when x = kπ, k an integer, we mark those points on the graph. Use a calculator to compute other function values. The graph is on the next slide. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solution continued Copyright © 2009 Pearson Education, Inc.