1. Graphs of Trigonometric Functions
Sections 4.5 – 4.6
Graphs of Trigonometric Functions

2. Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the set of y values such that
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range over an x-interval of
6. The cycle repeats itself indefinitely in both directions of the x-axis.
Properties of Sine and Cosine Functions

3. Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. –1 1
sin x x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x
y = sin x
Sine Function

4. Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1 –1
cos x x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x
y = cos x
Cosine Function

5. Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.
max
x-int
min 3 –3
y = 3 cos x 2  x y x
(0, 3)
( , 3)
( , 0)
( , 0)
( , –3)
Example: y = 3 cos x

6. If |A| > 1, the amplitude stretches the graph vertically.
The amplitude of y = A sin x (or y = A cos x) is half the distance between the maximum and minimum values of the function.
amplitude = |A|
If |A| > 1, the amplitude stretches the graph vertically.
If 0 < |A| < 1, the amplitude shrinks the graph vertically.
If A < 0, the graph is reflected in the x-axis. y x
y = 2sin x
y = sin x
y = sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x
Amplitude

7. If B > 1, the graph of the function is shrunk horizontally.
The period of a function is the x interval needed for the function to complete one cycle.
For B  0, the period of y = A sin Bx is ,
and, the period of y = A cos Bx is also
If B > 1, the graph of the function is shrunk horizontally. y x
period: 2
period:
If 0 < B < 1, the graph of the function is stretched horizontally. y x
period: 4
period: 2
Period of a Function

8. Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. y x
y = sin (–x)
Use the identity sin (–x) = – sin x
y = sin x
Example 2: Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x. y x
Use the identity cos (–x) = cos x
y = cos (–x)
y = cos (–x)
Graph y = f(-x)

9. Use the identity sin (– x) = – sin x:
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = A sin Bx with B > 0
Use the identity sin (– x) = – sin x:
y = 2 sin (–3x) = –2 sin 3x
period: 2 3 =
amplitude: |A| = |–2| = 2
Calculate the five key points. 2 –2
y = –2 sin 3x x y x
( , 2)
(0, 0)
( , 0)
( , 0)
( , -2)
Example: y = 2 sin(-3x)

10. Graph variation functions y = A sin (Bx–C)+D and y = A cos (Bx–C)+D
Exercise: Graph *** Modify the function first using even/odd properties as

11. Function:
Domain:
Range:
Period:

12. Function: 4. Phase shift: 5. Vertical shift: 6
Function: 4. Phase shift: 5. Vertical shift: 6. Parent function: and key-points table of values (one period):
1 unit up x y 1
– 1

13. Function : Adjust the table of values by
dividing every x value by 2 and subtracting ;
multiplying every y value by (–3) and adding 1
1) or ) or
(Notice: B = 2, and phase shift is to the left)

14. Function: =
0 ∙ (–3) + 1 = 1
1 ∙ (–3) + 1 = – 2
–1 ∙ (–3) + 1 = 4

15. *select x-axis unit interval as
Graph new key-points on the coordinate plane and connect with the smooth curve
*select x-axis unit interval as y 3 x -3
The other method – take notes...

16. Graph of the Tangent Function
To graph y = tan x, use the identity
At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes. y x
Properties of y = tan x
1. domain : all real x
2. range: (–, +)
3. period: 
4. vertical asymptotes:
period:
Tangent Function

17. Example: Tangent Function
Example: Find the period and asymptotes and sketch the graph of y x
1. Period of y = tan x is
2. Find consecutive vertical
asymptotes by solving for x:
Vertical asymptotes:
3. Plot several points in
4. Sketch one branch and repeat.
Example: Tangent Function

18. Graph variation functions y = A tan(Bx–C)+D and y = A cot(Bx–C)+D
Exercise: Graph
*** Modify the function first using even/odd properties as 18

19. Function:
Domain:
Range:
Period:
Amplitude: none 19

20. and key-points table of values (one period):
Function:
5. Phase shift:
6. Vertical shift:
7. Parent function:
and key-points table of values (one period):
1 unit up x y
Undefined
– 1 1 20

21. Function: Undefined –1 ∙ (–3) + 1 = 4 0 ∙ (–3) + 1 = 1
1 ∙ (–3) + 1 = – 2 21

22. Graph of the Cotangent Function To graph y = cot x, use the identity .
At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes. y x
Properties of y = cot x
vertical asymptotes
1. domain : all real x
2. range: (–, +)
3. period: 
4. vertical asymptotes:
Cotangent Function

23. Graph function:
Domain:
Range:
Period:
Amplitude: none 23 23

24. and key-points table of values (one period):
Function:
5. Phase shift:
6. Vertical shift:
7. Parent function:
and key-points table of values (one period):
1 unit up x y
Undefined 1
– 1 24 24

25. Function: Undefined 1 ∙ (–3) + 1 = – 2 0 ∙ (–3) + 1 = 1 =
–1 ∙ (–3) + 1 = 4 25 25

26. Graph of the Secant Function The graph y = sec x, use the identity .
At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes. y x
Properties of y = sec x
1. domain : all real x
2. range: (–,–1]  [1, +)
3. period: 2
4. vertical asymptotes:
where cosine is zero.
Secant Function

27. Graph of the Cosecant Function To graph y = csc x, use the identity .
At values of x for which sin x = 0, the cosecant function is undefined and its graph has vertical asymptotes. x y
Properties of y = csc x
1. domain : all real x
2. range: (–,–1]  [1, +)
3. period: 2
4. vertical asymptotes:
where sine is zero.
Cosecant Function