4-4 Graphing Sine and Cosine Functions

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Describe how the graphs of f (x) and g(x) are related.

f (x) = sin x; g(x) = sin x The graph of g(x) is the graph of f (x) compressed vertically.

Create a table listing the coordinates of the ercepts and extrema for f (x) = sin x for one period, 2 interval [0, 2

1 Describe how the graphs of f (x) and g(x) are related. Then find the amplitude of g(x), and sketch two periods of both functions on the same coordinate axes. 1. f (x) = sin x; g(x) = sin x The graph of g(x) is the graph of f (x) compressed vertically. The amplitude of g(x) is or. Create a table listing the coordinates of the ercepts and extrema for f (x) = sin x for one period, 2 interval [0, 2 ]. Then use the amplitude of g(x) to find corresponding points on its graph., on the Functions f (x) = sin x g(x) = sin x (0, 0) (0, 0) Max Min Sketch the curve through the indicated points for each function. Then repeat the pattern to complete a second period. esolutions Manual - Powered by Cognero Page 1

2 3. f (x) = cos x; g(x) = 6 cos x The graph of g(x) is the graph of f (x) expanded vertically. The amplitude of g(x) is or 6. Create a table listing the coordinates of the ercepts and extrema for f (x) = cos x for one period, 2 interval [0, 2 ]. Then use the amplitude of g(x) to find corresponding points on its graph., on the Functions f (x) = cos x g(x) = 6 cos x Max (0, 1) (0, 6) Min Max Sketch the curve through the indicated points for each function. Then repeat the pattern to complete a second period. esolutions Manual - Powered by Cognero Page 2

3 Describe how the graphs of f (x) and g(x) are related. Then find the period of g(x), and sketch at least one period of both functions on the same coordinate axes. 5. f (x) = sin x; g(x) = sin 4x The graph of g(x) is the graph of f (x) compressed horizontally. The period of g(x) is. To find corresponding points on the graph of g(x), change the x-coordinates of those key points on f (x) so that they range from 0 to, increasing by increments of. Functions f (x) = sin x g(x) = sin 4x (0, 0) (0, 0) Max Min Sketch the curve through the indicated points for each function. Then repeat the pattern to complete a second period. esolutions Manual - Powered by Cognero Page 3

4 7. f (x) = cos x; The graph of g(x) is the graph of f (x) expanded horizontally. The period of g(x) is. To find corresponding points on the graph of g(x), change the x-coordinates of those key points on f (x) so that they range from 0 to, increasing by increments of or. Functions f (x) = cos x Max (0, 1) (0, 1) Min Max Sketch the curve through the indicated points for each function. Then repeat the pattern to complete a second period. esolutions Manual - Powered by Cognero Page 4

5 15. State the amplitude, period, frequency, phase shift, and vertical shift of each function. Then graph two periods of the function. In this function, a = 1, b =, c =, and d = 0. Because d = 0, there is no vertical shift. esolutions Manual - Powered by Cognero Page 5

6 17. y = sin 3x 2 In this function, a = 1, b = 3, c = 0, and d = 2. Graph y = sin 3x shifted 2 units down. esolutions Manual - Powered by Cognero Page 6

7 19. In this function, a = 1, b = 1, c =, and d = 4.. Graph y = sin x shifted units to the left and 4 units up. esolutions Manual - Powered by Cognero Page 7

8 Write a sinusoidal function with the given period and amplitude that passes through the given point. 35. period: ; amplitude: 5; point: Use the period to find b. Sample answer: One sinusoidal function in which a = 5 and b = 2 is y = 5 cos 2x. Evaluate the function for The function passes through. Therefore, a sinusoidal function with period π and amplitude 5 that passes through the point is y = 5 cos 2x. esolutions Manual - Powered by Cognero Page 8

9 37. period: ; amplitude: ; point: Use the period to find b. Sample answer: One sinusoidal function in which a = 1.5 and b = 4 is y = 1.5 cos 4x. Evaluate the function for The function passes through. Therefore, a sinusoidal function with period and amplitude 1.5 that passes through the point is y = 1.5 cos 4x. esolutions Manual - Powered by Cognero Page 9

You analyzed graphs of functions. (Lesson 1-5)

You analyzed graphs of functions. (Lesson 1-5) LEQ: How do we graph transformations of the sine and cosine functions & use sinusoidal functions to solve problems? sinusoid amplitude frequency phase shift