The document discusses how to draw trigonometric graphs by modifying basic trigonometric functions. It explains that: 1) Adding a constant K in front of sin(x) or cos(x) changes the amplitude to K. 2) Adding a constant K in the argument (e.g. sin(Kx)) changes the period to 360/K degrees. 3) Adding a constant to the function (e.g. sin(x)+K) translates the graph K units parallel to the y-axis. 4) Adding a negative sign reflects the graph across the x-axis. More complex graphs can be drawn by combining these effects.

1. Drawing Trigonometric Graphs.

2. The Basic Graphs.
You should already be familiar with the following graphs:
Y = SIN X

3. Y= COS X

4. Y = TAN X

5. Changing Trigonometric Graphs.
You should know how the following graphs differ from the basic
trigonometric graphs:
Y= 2 SIN X
The 2 in front of the sin x changes the “amplitude” of the graph.

6. Y = 5 COS X
As expected the amplitude of the graph is now 5. Hence the
graph has a maximum value of 5 and a minimum value of –5.

7. Y = SIN 2X
By introducing the 2 in front of the X , the “period” of the
graph now becomes 360o ÷ 2 = 180o.

8. Y= COS 6X
The period of the graph has now become 360o ÷ 6 = 60o as
expected.

9. Y= SIN X + 1
The plus 1 has the effect of “translating” the graph one
square parallel to the y axis.

10. Y = COS X - 5
The cosine graph is translated 5 squares downwards parallel to
the y axis.

11. Y= – SIN X
The minus sign in front of the function “reflects” the whole graph
in the X axis.

12. Y = – COS X
As expected the cosine graph is reflected in the X axis.

13. Summary Of Effects.
(1) Y= K COS X & Y = K SINX
+k
-k
The amplitude of the function is “K” .
(2) Y = COS KX & Y = SIN KX
360 ÷ K
The period of the function is “360 ÷ k” .

14. (3) Y= COS X + K & Y = SIN X + K
+k
-k
Translates the graph + K or – K
parallel to the y axis.
(4) Y = - COS X & Y = - SIN X.
Y = - COS X Reflects the graph in the x axis.

15. Combining The Effects.
We are now going to draw more complex trigonometric graphs
like the one shown above, by considering what each part of the
equation does to the graph of the equation.

16. y = 4 sin x
Example 1.
Draw the graph of : 4
y = 4sin2x + 3
Solution.
y = 4 sin 2x
Draw the graph of :
y = sin x
180o
y = 4sin 2x + 3
+3

17. Example 2 y = - 6cos5x
Draw the graph y = 2 – 6 cos 5x
Solution.
Draw the graph of :
y = cos x
y = 2 – 6 cos 5x
y= 6cos5x
72o
6

18. Creating A Phase Shift.
Shown below is the graph of y = sin xo
Now compare it with the graph of y= sin( x - 60o)
+ 60o
The graph is translated 60o to the right parallel to the x axis.

19. Shown below is the graph of y = cos x.
Now compare it to the graph of y = cos ( x + 45o)
-45o
The graph is translated 45o to the left parallel to the x axis.