Practice and Problem-Solving Exercises

A Practice

Determine the value of h in each translation. Describe each phase shift (use a phrase like 3 units to the left). See Problem 1.

6. g ( x ) = f ( x + 1 ) g open x close equals f open x plus 1 close
7. g ( t ) = f ( t + 2 ) g open t close equals f open t plus 2 close
8. f ( z ) = g ( z − 1 . 6 ) f open z close equals g open z minus 1 . 6 close
9. f ( x ) = g ( x − 3 ) f open x close equals g open x minus 3 close
10. y = sin ( x + π ) y equals sine open x plus pi close
11. y = cos ( x − 5 π 7 ) y equals cosine . open . x minus , fraction 5 pi , over 7 end fraction . close

Use the function f ( x ) f open x close  below. Graph each translation. See Problem 2.

12. g ( x ) = f ( x ) + 1 g open x close equals f open x close plus 1
13. g ( x ) = f ( x ) − 3 g open x close equals f open x close minus 3
14. g ( x ) = f ( x + 2 ) g open x close equals f open x plus 2 close
15. g ( x ) = f ( x − 1 ) g open x close equals f open x minus 1 close

Graph each translation of y = cos x y equals cosine x  in the interval from 0 to 2π.

16. y = cos ( x + 3 ) y equals cosine open x plus 3 close
17. y = cos x + 3 y equals cosine x plus 3
18. y = cos x − 4 y equals cosine x minus 4
19. y = cos ( x − 4 ) y equals cosine open x minus 4 close
20. y = cos x + π y equals cosine x plus pi
21. y = cos ( x − π ) y equals cosine open x minus pi close

Describe any phase shift and vertical shift in the graph. See Problem 3.

22. y = 3 sin x + 1 y equals 3 sine x plus 1
23. y = 4 cos ( x + 1 ) − 2 y equals 4 cosine open x plus 1 close minus 2
24. y = sin ( x + π 2 ) + 2 y equals sine . open . x plus , pi over 2 . close . plus 2
25. y = sin ( x − 3 ) + 2 y equals sine open x minus 3 close plus 2

Graph each function in the interval from 0 to 2π.

26. y = 2 sin ( x + π 4 ) − 1 y equals 2 sine . open . x plus , pi over 4 . close . minus 1
27. y = sin ( x + π 3 ) + 1 y equals sine . open . x plus , pi over 3 . close . plus 1
28. y = cos ( x − π ) − 3 y equals cosine open x minus pi close minus 3
29. y = 2 sin ( x − π 6 ) + 2 y equals 2 sine . open . x minus , pi over 6 . close . plus 2

Graph each function in the interval from 0 to 2π. See Problem 4.

30. y = 3 sin 1 2 x y equals 3 sine , 1 half , x
31. y = cos 2 ( x + π 2 ) − 2 y equals cosine 2 . open . x plus , pi over 2 . close . minus 2
32. y = 1 2 sin 2 x − 1 y equals , 1 half sine 2 x minus 1
33. y = sin 3 ( x + π 3 ) y equals sine 3 . open . x plus , pi over 3 . close
34. y = sin 2 ( x + 3 ) − 2 y equals sine 2 open x plus 3 close minus 2
35. y = 3 sin π 2 ( x − 2 ) y equals 3 sine , pi over 2 . open , x minus 2 , close

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