Trigonometric graphs

Trig Graphs

1. Block 3
Trigonometric Graphs

2. What is to be learned?
•A reminder of how to draw and identify trig
graphs.
•Take it a bit further.

3. 90 180 270 360
1
0
-1
Y = sinx
Maximum Value = 1
Minimum Value = -1

4. 90 180 270 360
1
0
-1
Y = cosx
Maximum Value = 1
Minimum Value = -1

5. 90 180 270 360
7
0
-7
Y = 7sinx
Maximum Value = 7
Minimum Value = -7
Range = Max - Min
Range = 7 – (-7)
= 14
→range = 14
Range

6. 90 180 270 360
4
0
-4
Y = 4cosx
Maximum Value = 4
Minimum Value = -4
→range = 8

7. 90 180 270 360
8
0
-8
Y = - 8sinx
Maximum Value = 8
Minimum Value = -8
“Opposite” to Sin x

8. 90 180 270 360
6
0
-6
Y = - 6cosx
Maximum Value = 6
Minimum Value = -6
“Opposite” to Cos x

9. 90
0
180
0
270
0
360
0
90
0
180
0
270
0
360
0
3
-3
6
-6
Write the Equations
1.
2.
y = -3sinx y = -6cosx
y = 9sinx y = cosx
3.
4.
9
-9
1
-1
90
0
180
0
270
0
360
0
90
0
180
0
270
0
360
0

10. 90 180 270 360
1
0
-1
Y = sin x
540450
Period of graph is 360
0
Cycle starts again
Also applies to Y = cos x
Between 0
0
and 360
0
there is 1 cycle
Taking it Further

11. 90 180 270 360
1
0
-1
Y = sin 2x
Period of graph is 180
0
There are 2 cycles between 0
0
and 360
0

12. Combining these rules
Draw y = 6sin2x
Max 6
Min -6
2 cycles
Period = 360 ÷ 2 = 180
0
90 180 270 360
6
0
-6
Y = 6sin 2x

13. Recognising Graph
Max 8
Min -8
4 cycles
90 180 270 360
8
0
-8
Y = 8cos4x
Cosine

14. 90
0
180
0
270
0
360
0 90
0
180
0
270
0
360
0
90
0
180
0
270
0
360
0 90
0
180
0
270
0
360
0
7
-7
5
-5
3
- 3
2
-2
Write the Equations
1.
2.
3. 4.
y = 7sin2x y = 5cos2x
y = 3cos4x y = 2sin3x

15. Changing the Scale
Nice for Drawing Graphs 
y = 4 Sin 6x
Cycles?
Period
6
360 ÷ 6 = 60
0
15 30 45 60
4
0
-4

16. 30
0
60
0
90
0
120
0
7
0
-7
Not so nice for recognising graphs 
Period = 120
0
No of Cycles in 360?360 ÷ 120 = 3
y = 7 cos 3x
240
0
360
0

17. Find equation of graph below.
Cycles
Max 7
Negative sin
360 ÷ 60 = 6
15 30 45 60
7
0
-7
y = -7sin6x

18. Remember rules for y = (x – 3 )
2
+ 5
Same rules for trig graphs!
3 units to right Up 5
Extra Trig Graph Rules

19. 90 180 270 360
4
0
-4
Y = 4cos (x – 45
0
)
450
Y = 4cosx
45
0
to right
Sketch Normal Graph
Move each point
right/left
y =4cos(x – 45
0
)

20. 90 180 270 360
11
0
-11
Recognising
Sin Graph
30
0
to right
y = 11 sin(x – 30
0
)
30
0

21. 90 180 270 360
13
0
-13
Recognising
Cos Graph
20
0
to left
y = 13 cos(x + 20
0
)
-20
0

22. 90 180 270 360
11
0
-11
A Bit of Confusion
Sin Graph
30
0
to left
y = 11 sin(x + 30
0
)
-30
0
60
0
Cos Graph
60
0
to right
y = 11 cos(x – 60
0
)
Both correct

23. 6
-6
y = 6cos(x + 30
0
)
-30
0
Identify this graph
90
0
180
0
270
0
360
0

24. 90 180 270 360
1
0
-1
Y = sinx + 2
Y = sinx
2
3

25. 90 180 270 360
4
0
-4
Y = 4cosx + 6
8
12
range = 8
Graph Type
y = 4cosx
2
6
10
-2
Equation?

26. 90 180 270 360
0
No Maximum (or minimum)
What about y = Tanx ???
Goes to infinity
Cycle complete
Period is 180
0

27. 90 180 270 360
0
Changing the period
Cycle complete
Normal Period is 180
0
2 cycles
y = tan2x

28. 90 180 270 360
0
y = -Tanx

29. Also
Can now use radians!

30. 90 180 270 360
1
0
-1
Y = sinx
π
/
2
π
3π
/
2 2π

31. Trigonometric Graphs
Follow all the same rules as other function
graphs.
Range is handy for identifying (max – min)
e.g. for y = 7sinx →range = 14

32. π
/
2
π 2π
2
0
-2
y = 2cos(x –
π
/
4
)
4
6
y = 2cosx
Sketch y = 2cos(x –
π
/
4
) + 1
y = 2cos(x –
π
/
4
)

+ 1
3π
/
2

33. 0
-2
-4
Sketch y = 3sin(x +
π
/
4) – 1
Y = 3sinx
2
4
Y = 3sin(x +
π
/
4
)
Y = 3sin(x +
π
/
4
) – 1
Key Question
2π
3π
/
2π
π
/
2