Copyright 2009 Pearson Education, Inc. Slide Section 8.2 and 8.3-1

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Rosamond Summers
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1 8.3-1

2 Transformation of sine and cosine functions Sections 8.2 and 8.3 Revisit: Page 142; chapter 4

3 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.

4 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

5 The Constant k Let s observe the effect of the constant k

6 The Constant k 8.3-6

7 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 <

8 The Constant A Let s observe the effect of the constant A

9 The Constant A 8.3-9

10 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

11 Amplitude The amplitude of the graphs of y = A sin B (x h) + k and y = A cos B (x h) + k is A

12 The Constant B Let s observe the effect of the constant B

13 The Constant B

14 The Constant B

15 The Constant B

16 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

17 Period The period of the graphs of y = A sin B (x h) + k and y = A cos B (x h) + k is 2 B

18 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 2 B

19 Period The period of the graphs of y = A tan B(x h) + k and y = A cot B(x C) + k is B

20 The Constant h Let s observe the effect of the constant C

21 The Constant h

22 The Constant h

23 The Constant h

24 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

25 Combined Transformations B careful! y = A sin (Bx h) + k and y = A cos (Bx h) + k as and y Asin B x C B D y Acos B x C B D

26 Phase Shift The phase shift of the graphs and y AsinBx C y AcosBx C D Asin B x C B D Acos B x C B D D is the quantity C B

27 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

28 Transformations of Sine and Cosine Functions To graph and y AsinBx C y AcosBx C follow the steps listed below in the order in which they are listed. D Asin B x C B D Acos B x C B D D

29 Transformations of Sine and Cosine Functions 1. Stretch or shrink the graph horizontally according to B. B < 1 B > 1 B < 0 Stretch horizontally Shrink horizontally Reflect across the y-axis The period is 2 B

30 Transformations of Sine and Cosine Functions 2. Stretch or shrink the graph vertically according to A. A < 1 A > 1 A < 0 Shrink vertically Stretch vertically Reflect across the x-axis The amplitude is A

31 Transformations of Sine and Cosine Functions 3. Translate the graph horizontally according to C/B. C B 0 C B units to the left C B 0 C B units to the right The phase shift is C B

32 Transformations of Sine and Cosine Functions 4. Translate the graph vertically according to k. k < 0 k units down k > 0 k units up

33 Homework 1. Transformation of Sine Cosine functions. 2. Sec 8.2 Written exercises #1-10 all

34 Example Sketch the graph of y 3sin2x / 21. Find the amplitude, the period, and the phase shift. Solution: y 3sin 2x 2 1 3sin 2 x 4 Amplitude A Period 2 B 2 2 Phase shift C B

35 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. 1. y sin2x 2. y 3sin2x 3. y 3sin 2 x 4 4. y 3sin 2 x

36 Example Solution continued y sin x

37 Example Solution continued 1. y sin2x

38 Example Solution continued 2. y 3sin2x

39 Example Solution continued 3. y 3sin 2 x

40 Example Solution continued 4. y 3sin 2 x

41 Example Graph: y = 2 sin x + sin 2x Solution: Graph: y = 2 sin x and y = sin 2x on the same axes

42 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 P 1. At x = π/4, transfer m down to add it to 2 sin x, yielding P 2. At x = 5π/4, add the negative ordinate of sin 2x to the positive ordinate of 2 sin x, yielding P 3. 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

43 Example Solution continued The period of the sum 2 sin x + sin 2x is 2π, the least common multiple of 2π and π

44 Example Sketch a graph of f x e x 2 sin x. Solution f is the product of two functions g and h, where gx e x 2 and hx sin x To find the function values, we can multiply ordinates. Start with 1 sin x 1 e x 2 e x 2 sin x e x 2 The graph crosses the x-axis at values of x for which sin x = 0, kπ for integer values of k

45 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

46 Example Solution continued

Section 5.2 Graphs of the Sine and Cosine Functions

A Periodic Function and Its Period Section 5.2 Graphs of the Sine and Cosine Functions A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in