1. extended learning for chapter 11 (graphs)
E-learning
extended learning for chapter 11 (graphs)

2. Let’s recall first

3. Graph of y = sin  Note: max value = 1 and min value = -1

4. Graph of y = sin 
The graph will repeats itself for every 360˚. The length of interval which the curve repeats is call the period.
Therefore, sine curve has a period of 360˚.
Graph of y = sin 

5. Graph of y = sin  2
Graph of y = sin  2
- 2

6. In general, graph of y = sin  a

7. Graph of y = cos 
Note: max value = 1 and min value = -1

8. Graph of y = cos  3 3
- 3

9. In general, graph of y = cos  a

10. Graph of y = cos  The graph will repeats itself for every 360˚.
Therefore, cosine curve has a period of 360˚.

11. Graph of y = tan 
Note: The graph is not continuous. There are break at 90˚ and 270˚. The curve approach the line at 90˚ and 270˚. Such lines are called asymptotes.
45˚
225˚
135˚
315˚

12. In general, graph of y = tan  a
Note: The graph does not have max and min value. a
- a

13. Summary
Identify the 3 types of graphs:
y = sin 
y = cos 
y = tan 

14. Points to consider when sketching trigonometrical functions:
Easily determined points: a) maximum and minimum points b) points where the graph cuts the axes
Period of the function
Asymptotes (for tangent function) 14

15. Let’s continue learning..

16. Example 1: Sketch y = 4sin x (given y = sin x ) for 0°  x  360° x
Comparing the 2 graphs, what happens to the max and min point of y = 4 sin x?

17. Example 2:
Sketch y = 4 + sin x for 0°  x  360°
Spot the difference between y = 4 sin x and y = 4 + sin x and write down the answer.
y = sin x
y = 4 + sin x
> x x

18. Example 3: Sketch y = - sin x for 0°  x  360° How do we get
y = - sin x graph from y = sin x? x x
y = - sin x x x x x x x
> x
y = sin x x x
Reflection of y = sin x in x axis

19. Example 4: Sketch y = 4 - sin x for 0°  x  360° x x x x
Reflection of y = sin x in x axis
Translation of y = -sin x by 4 units along y axis

20. Example 5: Sketch y = |sin x| for 0°  x  360° > x y = |sin x|

21. Example 6: Sketch y = -|sin x| for 0°  x  360° y = |sin x| > x
Reflection of y = |sin x| in x axis

22. Example 7 Sketch y = -5cos x for 0°  x  360° x 5 x x -5 x x x
Reflection about x axis 5 -5 x x x x x x
y = cos x y = 5cos x y = -5cos x

23. Example 8 Sketch y = 3 + tan x for 0°  x  360° x x x x x x x x
y = tan x y = 3 + tan x

24. Example 9 Sketch y = 2 – sin x, for values of x between 0°  x  360°
y = sin x y = - sin x y = 2 – sin x

25. Example 10 Sketch y = 1 – 3cos x for values of x between 0°  x  360°
y = 3 cos x y = - 3 cos x y = 1 – 3 cos x

26. Example 11 Sketch y = |3cos x| for values of x between 0°  x  360°
y = 3 cos x y = |3 cos x|

27. Example 12
Sketch y = |3 sin x| - 2 for values of x between 0°  x  360°
y = 3 sin x
y = sin x
y = 3 sin x y = |3 sin x| y = y = |3 sin x| - 2

28. Example 13
Sketch y = 2cos x -1 and y = -2|sin x| for values of x between 0  x  360.
Hence find the no. of solutions 2cos x -1 = -2|sin x| in the interval.
Solution:
Answer:
No of solutions = 2
y = 2cos x -1 x x
y = -2|sin x|

29. Example 14
Sketch y = |tan x| and y = 1 - sin x for values of x between 0  x  360.
Hence find the no. of solutions |tan x| = 1 - sin x| in the interval.
Solution:
Answer:
No of solutions = 4
y = |tan x| x x
y = 1 - sin x x x