SECTION 6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 449 Skill Building In Problems 7–16, if necessary, refer to the graphs of the functions to answer each question. 7. What is the y-intercept of = y x tan ? 8. What is the y-intercept of = y x cot ? 9. What is the y-intercept of = y x sec ? 10. What is the y-intercept of = y x csc ? 11. For what numbers π π − ≤ ≤ x x , 2 2 , does = x sec 1? For what numbers x does = − x sec 1? 12. For what numbers π π − ≤ ≤ x x , 2 2 , does = x csc 1? For what numbers x does = − x csc 1? 13. For what numbers π π − ≤ ≤ x x , 2 2 , does the graph of = y x sec have vertical asymptotes? 14. For what numbers π π − ≤ ≤ x x , 2 2 , does the graph of = y x csc have vertical asymptotes? 15. For what numbers π π − ≤ ≤ x x , 2 2 , does the graph of = y x tan have vertical asymptotes? 16. For what numbers π π − ≤ ≤ x x , 2 2 , does the graph of = y x cot have vertical asymptotes? In Problems 17–40, graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. 17. = y x 3tan 18. = − y x 2tan 19. = y x 4cot 20. = − y x 3cot 21. π( ) = y x tan 2 22. ( ) = y x tan 1 2 23. ( ) = y x cot 1 4 24. π( ) = y x cot 4 25. = y x 2 sec 26. = y x 1 2 csc 27. = − y x 3csc 28. = − y x 4 sec 29. ( ) = y x 4 sec 1 2 30. ( ) = y x 1 2 csc 2 31. π( ) = − y x 2csc 32. π( ) = − y x 3sec 2 33. ( ) = + y x tan 1 4 1 34. = − y x 2cot 1 35. π ( ) = + y x sec 2 3 2 36. π ( ) = y x csc 3 2 37. ( ) = − y x 1 2 tan 2 2 38. ( ) = − y x 3cot 1 2 2 39. π( ) = − y x 2csc 2 1 40. ( ) = + y x 3sec 1 4 1 Concepts and Vocabulary 3. Interactive Figure Exercise Exploring the Graph of the Tangent Function Open the “Graph of the Tangent 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,” “Show Key Points,” and “Show Asymptotes” boxes are unchecked. Use your cursor to slowly move the point on the “Drag me!” slider to the right. Notice the point on the unit circle moves in a counterclockwise direction.As you do this, notice the points on the graph of the tangent function that are traced out.The x-coordinates of these points represent the measure of the angle in the unit circle and the y-coordinates represent the ratio y x of the point corresponding to the angle on the unit circle. Now, uncheck the box “Show Unit Circle” and click on the box “Show Graph.” Based on the graph, the range of the tangent function is . (b) Based on the graph of the tangent function, the tangent function is (odd/even) and its graph is symmetric with respect to the (x-axis, y-axis, origin). (c) Click on the box “Show Key Points.” Click on the box “Show Asymptotes.” Based on the results, the tangent function is periodic with period . (d) The graph of = y x tan has vertical asymptotes at (even/odd) multiples of π π ( ) , 2. (e) The following ordered pairs represent points on the graph of = y x tan . Fill in the ordered pairs with the appropriate values assuming x is the angle and y is the value of the tangent function at x. π π π ( )( )( ) − 4 , 4 , 2 , 4. The graph of =y x sec is symmetric with respect to the and has vertical asymptotes at . 5. Multiple Choice It is easiest to graph = y x sec by first sketching the graph of . (a) = y x sin (b) = y x cos (c) = y x tan (d) = y x csc 6. True or False The graphs of = = = y x y x y x tan , cot , sec , and =y x csc each have infinitely many vertical asymptotes.
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