CV

1. 1
MATH 130
Lecture on
The Graphs of
Trigonometric Functions

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

3. 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.
0
-1
0
1
0
sin x
0
x
2

2
3

2

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
2
3


 2



2
2
3

2

2
5
1

1
x
y = sin x

4. 4
y
1
1

2

3

2
x
 
3

2

 
4
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
x-int
max
3
0
-3
0
3
y = 3 cos x
2

0
x
2

2
3
(0, 3)
2
3
( , 0)
( , 0)
2


2
( , 3)

( , –3)

5. 5
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
0
-1
0
1
cos x
0
x 2

2
3

2

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
2
3


 2



2
2
3

2

2
5
1

1
x
y = cos x

6. 6
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.
2
3
2

4
y
x
4


2

y = –4 sin x
reflection of y = 4 sin x y = 4 sin x
y = 2sin x
2
1
y = sin x
y = sin x

7. 7
y
x

 
2

sin x
y 

period: 2
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 .
b

2
For b  0, the period of y = a cos bx is also .
b

2
If 0 < b < 1, the graph of the function is stretched horizontally.
If b > 1, the graph of the function is shrunk horizontally.
y
x

 
2
 
3 
4
cos x
y 

period: 2
2
1
cos x
y 

period: 4
sin 2
y x

:
period 

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

9. 9
2
y
2

6

x
2


6
5
3

3
2
6

6

3

2

3
2
0
2
0
–2
0
y = –2 sin 3x
0
x
Example: Sketch the graph of y = 2 sin(–3x).
Rewrite the function in the form y = a sin bx with b > 0
amplitude: |a| = |–2| = 2
Calculate the five key points.
(0, 0) ( , 0)
3

( , 2)
2

( ,-2)
6

( , 0)
3
2
Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x
period:
b

2 
2
3
=

10. 10
The graph of y = A sin (Bx – C) is obtained by horizontally shifting the graph
of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to
x = C
/B. The number C
/B is called the phase shift.
amplitude = | A|
period = 2 /B.
x
y
Amplitude: | A|
Period: 2/B
y = A sin Bx
Starting point: x = C/B
The Graph of y = Asin(Bx - C)

11. 11
Example
Determine the amplitude, period, and phase
shift of y = 2sin(3x-)
Solution:
Amplitude = |A| = 2
period = 2/B = 2/3
phase shift = C/B = /3

12. 12
Example cont.
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-3
-2
-1
1
2
3
• y = 2sin(3x- )

13. 13
A common mistake…
a is not amplitude; is amplitude.
a may be positive or negative; amplitude is always
positive.
a
The standard forms for sine and cosine functions are:
where a,b,c and d are constants.
( ) sin( )
f t a bt c d
  
( ) cos( )
g t a bt c d
  

14. 14
In the standard form:
•a controls amplitude
•b controls period
•c controls phase shift
•d controls vertical shift
( ) sin( )
f t a bt c d
  
( ) cos( )
g t a bt c d
  

15. 15
d
c
bx
a 
 )
sin(
Amplitude
Period:
2π/b Phase Shift:
c/b
Vertical
Shift