Trigonometric Graphs

1. Trigonometric Graphs Click to continue.

2. You are already familiar with the basic graph of y = sin xo. There are some important points to remember. y It has a maximum value of 1 at 90o. It crosses the x-axis at The curve has a period of It passes through the origin. 1 180o 270o 360o O x 90o y = sin xo -1 It has a minimum value of –1 at 270o. Click to continue.

3. Let us compare the graph of y =sin xoto the family of graphs of the form y = a sin bxo + c where a, b and c are constants. We will begin by looking at graphs of the form y = a sin xo. For example: y = 2 sin xo, y =3.7 sin xo or y = ½ sin xo. Click to continue.

4. Here is the graph of y = sin xo. Click once to see the graph of y = 2 sin xo. y It has a maximum of 2 (twice that of the normal graph). 3 2 Notice the following points on the curve. y = 2 sin xo 1 360o x O 180o y = sin xo It has a period of 360o. -1 It passes through the origin. -2 -3 It has a minimum of –2. Click to continue.

5. Here is the graph of y = sin xo. Click once to see the graph of y = -3 sin xo. y 3 It has a maximum of 3. 2 Notice the following points on the upside-down curve. y = -3 sin xo 1 360o x O 180o y = sin xo It has a period of 360o. -1 It passes through the origin. It has a minimum of -3 (negative three times that of the normal graph). -2 -3 Click to continue.

6. Here is the graph of y = sin xo. Click once to see the graph of y = 2½ sin xo. y It has a maximum of 2½ (two and a half times that of the normal graph). 3 2 Notice the following points on the curve. y = 2½ sin xo 1 360o x O 180o y = sin xo It has a period of 360o. -1 It passes through the origin. -2 -3 It has a minimum of –2½. Click to continue.

7. Here is the graph of y = sin xo. Click once to see the graph of y = ½ sin xo. y 3 It has a maximum of ½ (half of the normal graph). 2 Notice the following points on the curve. 1 y = ½ sin xo 360o x O 180o It has a period of 360o. -1 y = sin xo It passes through the origin. -2 It has a minimum of –½. -3 Click to continue.

8. Here is the graph of y = sin xo. Click once to see the graph of y = a sin xo. y It has a maximum of a (a times that of the normal graph). a Notice the following points on the curve. It still passes through the origin y = a sin xo It has a period of 360o. 1 The period is unaffected. 360o x O 180o y = sin xo -1 It passes through the origin. The height is now “a”. The height is now “a”. -a It has a minimum of –a. Click to continue.

9. This is the graph of which function? y 15 y = sin 15xo 10 5 y = 15 sin xo 360o x O 180o -5 -10 y = sin xo + 15 -15

10. Which of these diagrams shows the graph of y = 7 sin xo? y y 14 7 7 x x O O 180o 360o 360o 720o -7 -14 -7 y y 7 7 3.5 3.5 x x O O 180o 360o 540o 720o 180o 360o -3.5 -3.5 -7 -7

11. For y = a sin xo only the height is affected. The graph will now have an altitude of 1  a. Here are the graphs of y = cos xo and y = tan xo. This is also true for y = a cos xo and y = a tan xo. y y 1 y = tan xo 1 y = cos xo x O O x 90o 180o 270o 360o 45o 90o 180o 270o 360o -1 -1 Click to continue.

12. Here are some examples of the graphs of y = a cos xo. y Click fory = 2 cos xo 1 y = cos xo Click for y = ¾ cos xo x O 90o 180o 270o 360o Click fory = - cos xo -1 Click to continue.

13. Here are some examples of the graphs of y = a tan xo. Click fory = 2tan xo Click fory = -3tan xo y 4 Notice this point y = tan xo 3 Notice this point 2 1 x O 45o 90o 180o 270o 360o 450o -1 Notice this point -2 -3 -4 Click to continue.

14. We will now look at graphs of the form y = sin bxo. For example: y = sin 2xo, y = sin 3xo or y = sin ½xo. Click to continue.

15. You are already familiar with the basic graph of y = sin xo. There are some important points to remember. y 1 180o 270o 360o O x 90o y = sin xo -1 Click to continue.

16. Here is the graph of y = sin xo. Click once to see the graph of y = sin 2xo. y It has a maximum of 1 (the same as a normal graph). 1 y = sin xo It has a period of 360o ÷ 2 = 180o. 360o Notice the following points on the curve. x O 180o It passes through the origin. y = sin 2xo -1 It has a minimum of –1. Click to continue.

17. Here is the graph of y = sin xo. Click once to see the graph of y = sin 3xo. y It has a maximum of 1 (the same as a normal graph). 1 y = sin xo It has a period of 360o÷ 3 = 120o. 360o Notice the following points on the curve. x O 180o It passes through the origin. y = sin 3xo -1 It has a minimum of –1. Click to continue.

18. Here is the graph of y = sin xo. Click once to see the graph of y = sin ½xo. y It has a maximum of 1 (the same as a normal graph). It has a period of 360o ÷ ½ = 720o. 1 Notice the following points on the curve. x O 180o 360o 540o 720o y = sin xo It passes through the origin. -1 y = sin ½ xo It has a minimum of –1. Click to continue.

19. Here is the graph of y = sin bxo. y It still passes through the origin. 1 y = sin bxo x The altitude (or height) is unaffected. O -1 The period is 360o b. The period is 360o÷ b. Period is (360o ÷ b) Click to continue.

20. This is the graph of which function? y 1 y = 4 sin xo y = sin 2xo x O 45o 90o 135o 180o y = sin 4xo -1

21. Which of these diagrams shows the graph of y = sin 6xo? y y 1 1 x x O O 180o 360o 90o 180o -1 -1 y y 1 1 0.5 0.5 x x O O 30o 60o 90o 120o 45o 90o -0.5 -0.5 -1 -1

22. For y = sin bxo only the period is affected. The graph will now have a period of 360o b. Here are the graphs of y = cos xo and y = tan xo. This is also true for y = cos bxo and y = tan bxo. y y 1 y = tan xo 1 y = cos xo x O O x 90o 180o 270o 360o 45o 90o 180o 270o 360o -1 -1 Click to continue.

23. Here are some examples of the graphs of y = cos bxo. Click fory = cos 2xoperiod = 360o÷ 2 = 180o Click fory = cos 2/3 xo period = 360o÷ 2/3 = 540o Click fory = cos ½xo period = 360o÷ ½ = 720o y 1 y = cos xo y O 90o 180o 270o 360o 450o 540o 630o 720o -1 Click to continue.

24. Here are some examples of the graphs of y = tan bxo. Click to seey = tan 2xo period = 180o÷ 2 = 90o and 45o÷ 2 = 22.5o y y = tan xo y = tan 2xo y = tan ½xo 4 Notice this point 3 Notice this point 2 1 x O -90o -45o 45o 90o 180o Click to seey = tan ½xo period = 180o÷ ½ = 360o and 45o÷ ½ = 90o -1 Notice this point -2 -3 -4 Click to continue.

25. We will now look at graphs of the form y = sin xo + c. For example: y = sin xo+ 2, y =sin xo+ 3 or y = sin xo– 1. Click to continue.

26. You are already familiar with the basic graph of y = sin xo. There are some important points to remember. y 1 180o 270o 360o O x 90o y = sin xo -1 Click to continue.

27. Here is the graph of y = sin xo. Click once to see the graph of y = sin xo+ 1. It has a maximum of 1 + 1 = 2. y 3 It has a period of 360o. 2 1 Notice the following points on the curve. y = sin xo + 1 The whole graph has been moved up one unit. The whole graph has been moved up one unit. y = sin xo 360o x O 180o -1 It passes through the origin + 1 = (0, 1). It has a minimum of –1 + 1 = 0. -2 -3 Click to continue.

28. Here is the graph of y = cos xo. Click once to see the graph ofy = cos xo– 1. y 1 y = cos xo x O 90o 180o 270o 360o y = cos xo– 1 -1 The whole graph has been moved down one unit. The whole graph has been moved down one unit. Click to continue.

29. Here is the graph of y = tan xo. y = tan xo+ 2 y Notice this point 4 y = tan xo 3 Notice this point Click once to see the graph of y = tan xo+ 2. 2 1 x O 45o 90o 180o 270o 360o 450o -1 -2 The whole graph has been moved up two units. The whole graph has been moved up two units. Click to continue.

30. This is the graph of which function? y 3 y = -3 sin xo 2 1 y = sin xo – 2 x O 180o 360o 540o 720o -1 -2 y = sin xo + 2 -3

31. Which of these diagrams shows the graph of y = cos xo+ 2? y y 4 2 2 x x O O 180o 360o 180o 360o 540o -2 -4 -2 y y 3 6 2 4 1 2 O x O x 180o 360o 180o 360o 540o -1 -2

32. We will now look at graphs of the form y = a sin bxo + c, y = a cos bxo + c and y = a tan bxo + c. For example: y = 2 sin 3xo– 1, y =½ cos 4xo+ 3 or y = ¾ tan ¼xo– 12. Click to continue.

33. Let us look at the graph of y = 2 sin 3xo – 1. Begin by considering the simple curve of y = sin xo. Now, think on the graph of y = 2 sin xo: the 2 will double the height. The graph of y = 2 sin 3xo: the 3 makes the period  as long (360o÷ 3 = 120o) Finally, y = 2 sin 3xo – 1, where the –1 moves the whole graph down one unit. y 2 1 x O 120o 180o 360o 540o -1 y = 2 sin 3xo – 1 -2 -3 Click to continue.

34. a = 1.5 b = 2 c = - 1 Look at this graph. What function does it show? y The first complete wave finishes here. 2 1. First, decide on the type. Maximum of 0.5 1 x It must be a COSINE graph because the first bump is on the y-axis. O 90o 180o 270o 360o -1 2. Next, look at the height. -2 Therefore, the height is 3 units. This is the middle of the wave and it has been moved 1 unit down from the x-axis. -3 Normally, a COSINE graph has a height of 2. Therefore the height has been multiplied by 3 ÷ 2 = 1.5 Minimum of –2.5 4. Finally, find out how much it has been moved down (or up). Click to continue. Therefore, we get –1. 3. Now, consider the period. This means the period is 180o so 360o÷ 180o = 2. y = 1.5 cos 2xo - 1 y = 1.5 cos 2xo - 1

35. Which of these graphs shows the function y = 2 sin 3xo + 1? y y 2 2 x x O 1 180o 360o 540o O 180o 360o -1 -2 y y 3 6 2 4 1 2 x O O x 120o 240o 360o 720o 1080o -1 -2 -2 -3