1. Graphs of Trig Functions

2. Graph of y = sin x
y = sin x
Max sin x = 1 when x = 90°
Min sin x = -1 when x = 270°

3. Graph of y = 2sin x Max 2sin x = 2 when x = 90°
Amplitude = 2
y = sin x
Max 2sin x = 2 when x = 90°
Min 2sin x = -2 when x = 270°

4. Graph of y = 5sin x Max 5sin x = 5 when x = 90°
Amplitude = 5
y = sin x
Max 5sin x = 5 when x = 90°
Min 5sin x = -5 when x = 270°

5. The negative multiplier reflects the initial graph in the x-axis
Graph of y = -sin x
y = sin x
Amplitude = 1
y = -sin x
The negative multiplier reflects the initial graph in the x-axis

6. The negative multiplier reflects the initial graph in the x-axis
Graph of y = -1·5sin x
y = sin x
Amplitude = 1·5
y = -1·5sin x
The negative multiplier reflects the initial graph in the x-axis

7. Graph of y = cos x
y = cos x
Max cos x = 1 when x = 0° or 360°
Min cos x = -1 when x = 180°

8. Graph of y = 3cos x and y = -3cos x

9. Write down the equations of the following graphs2. 1. 4. 3.

10. 6. 5. 8. 7.

11. Graph of y = sin x ± constant
Adding or subtracting a constant to or from sin x moves the graph vertically by the amount of the constant

12. Graph of y = cos x ± constant

13. Graphs of Multiple Angles
y = sin x
y = sin 2x
Max and Min values remain unchanged at +1 and -1
Two complete cycles between 0° and 360°
Period = 180°

14. Graphs of Multiple Angles
y = sin x
y = sin 3x
Max and Min values remain unchanged at +1 and -1
Three complete cycles between 0° and 360°
Period = 120°

15. Graphs of Multiple Angles
y = sin x
y = sin 0·5x
Max and Min values remain unchanged at +1 and -1
Half a complete cycle between 0° and 360°
Period = 720°

16. Graphs of Multiple Angles
y = cos x
y = cos 4x
Max and Min values remain unchanged at +1 and -1
Four complete cycles between 0° and 360°
Period = 90°

17. Write down the equations and period of these graphs

18. b is the number of complete cycles between 0° and 360°
Summary of Graphs so far
y = a sin bx ± c or y = a cos bx ± c
Where a, b and c are constants
a is the amplitude, height above centre line of graph, if a is negative the graph is reflected in the x-axis
b is the number of complete cycles between 0° and 360°
c moves the graph vertically up or down from x-axis

19. Mixed examples – combined effects
There are 2 complete cycles from 0° to 360° so
y = 3sin 2x
Amplitude is 3 so y = 3 sin ?x
Curve rises from origin so is a sine graph

20. There is only one complete cycle from 0° to 360° so y = 6cos x + 4
Curve falls from y-axis so is a cosine graph
Amplitude is 6, so y = 6 cos ?x + 4
Centre line half way between -2 and 10 = 4
So y = ? cos ?x + 4
There is only one complete cycle from 0° to 360° so y = 6cos x + 4

21. There are three complete cycles from 0° to 360° so y = 3cos 3x - 2
Curve falls from y-axis so is a cosine graph
Amplitude is 3, so y = 3 cos ?x - 2
Centre line half way between -5 and 1 = -2
So y = ? cos ?x - 2

22. Find the equations of the following graphs