SECTION 6.4 Graphs of the Sine and Cosine Functions 437 points in red that are traced out. The x-coordinates of the points in red represent the measure of the angle in the unit circle and the y-coordinates of the points in red represent the y-coordinates of the point corresponding to the angle on the unit circle. As the angle increases from 0 to π ( ) ≈ 2 1.57 , to π( ) ≈3.14 , the value of = y x cos decreases from 1 to 0 to . (b) As the angle increases from π( ) ≈3.14 to π ( ) ≈ 3 2 4.71 to π( ) ≈ 2 6.28 , the value of = y x cos increases from −1 to to . (c) Check the box “Show Graph.” Notice that the graph of the cosine function continues indefinitely in both the negative and positive directions, suggesting the domain of the cosine function is the set of all real numbers. Now, uncheck the box “Show Unit Circle” and check the box “Show Key Points.” Based on the graph, what is the range of = y x cos ? Use interval notation to write your answer. (d) The graph of the function = y x cos is symmetric with respect to the (y-axis, x-axis, origin). The cosine function is a(n) (even/odd) function. (e) The following ordered pairs represent points on the graph of = y x cos . Fill in the ordered pairs with the appropriate values assuming x is the angle and y is the value of the cosine function at x. π π π ( ) ( ) ( ) 6 , 4 , 3 2 , 5. The maximum value of π = ≤ ≤ y x x sin ,0 2 , is and occurs at = x . 6. If the function ω( ) = > y A x A sin , 0, has amplitude 3 and period 2, then = A and ω = . 7. The function ( ) = − y x 3cos 6 has amplitude and period . 8. True or False The graphs of = y x sin and = y x cos are identical except for a horizontal shift. 9. True or False For π( ) = y x 2sin , the amplitude is 2 and the period is π 2 . 10. True or False The graph of the sine function has infinitely many x-intercepts. 11. Multiple Choice One period of the graph of ω( ) = y x sin or ω( ) = y x cos is called a(n) . (a) amplitude (b) phase shift (c) transformation (d) cycle 12. Multiple Choice To graph ( ) = − y x 3 sin 2 using key points, the equivalent form could be graphed instead. (a) ( ) = − − y x 3sin 2 (b) ( ) = − y x 2 sin 3 (c) ( ) = y x 3sin 2 (d) ( ) = − y x 3sin 2 Concepts and Vocabulary 3. Interactive Figure Exercise Exploring the Graph of the Sine Function Open the “Graph of the Sine Curve” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Check the box “Show Unit Circle.” Be sure the “Show Graph” and “Show Key Points” boxes are unchecked. Use your cursor to slowly move the point on the “Drag me!” slider. Notice the point in blue on the unit circle moves in a counterclockwise direction. Also, notice the points in red that are traced out. The x-coordinates of the points in red represent the measure of the angle in the unit circle and the y-coordinates of the points in red represent the y-coordinates of the point corresponding to the angle on the unit circle. As the angle increases from 0 to π ( ) ≈ 2 1.57 , the value of = y x sin increases from 0 to . (b) As the angle increases from π ( ) ≈ 2 1.57 to π( ) ≈3.14 , the value of = y x sin decreases from 1 to . (c) As the angle increases from π to π ( ) ≈ 3 2 4.71 , the value of = y x sin decreases from 0 to . (d) As the angle increases from π ( ) ≈ 3 2 4.71 to π( ) ≈ 2 6.28 , the value of = y x sin increases from −1 to . (e) Check the box “Show Graph.” Notice that the graph of the sine function continues indefinitely in both the negative and positive directions, suggesting the domain of the sine function is the set of all real numbers. Uncheck the box “Show Unit Circle” and check the box “Show Key Points.” Based on the graph, what is the range of = y x sin ? Use interval notation to write your answer. (f) The graph of the function = y x sin is symmetric with respect to the (y-axis, x-axis, origin). The sine function is a(n) (even/odd) function. (g) The following ordered pairs represent points on the graph of = y x sin . Fill in the ordered pairs with the appropriate values assuming x is the angle and y is the value of the sine function at x. π π π ( ) ( ) ( ) 6 , 4 , 3 2 , 4. Interactive Figure Exercise Exploring the Graph of the Cosine Function Open the “Graph of the Cosine Curve” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Check the box “Show Unit Circle.” Be sure the “Show Graph” and “Show Key Points” boxes are unchecked. Use your cursor to slowly move the point on the “Drag me!” slider. Notice the point in blue on the unit circle moves in a counterclockwise direction. Also, notice the Skill Building 13. ( ) = f x x sin (a) What is the y-intercept of the graph of f ? (b) For what numbers x, π π − ≤ ≤ x , is the graph of f increasing? (c) What is the absolute maximum of f ? (d) For what numbers x, π ≤ ≤ x 0 2 , does ( ) = f x 0? (e) For what numbers x, π π − ≤ ≤ x 2 2 , does ( ) = f x 1? Where does ( ) = − f x 1? (f) For what numbers x, π π − ≤ ≤ x 2 2 , does ( ) = − f x 1 2 ? (g) What are the x-intercepts of f ?
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